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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>The measure transfer for subshifts induced by a morphism of free monoids</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="subshift, invariant measure, recognizable monoid morphism" lang="en" name="keywords"/> <base href="/html/2211.11234v4/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S1" title="In The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2" title="In The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Notation and conventions</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS1" title="In 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Standard terminology and well known facts</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS2" title="In 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>“Not so standard” basic facts and terminology</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS3" title="In 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>About injectivity</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS3.SSS1" title="In 2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3.1 </span>Typical injectivity problems</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS3.SSS2" title="In 2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3.2 </span>Shift-period preservation</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3" title="In The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>The measure transfer</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.SS1" title="In 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Subdivision morphisms</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.SS2" title="In 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Letter-to-letter morphisms</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.SS3" title="In 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>The induced measure morphisms</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.SS4" title="In 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>Basic properties of the measure transfer map</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4" title="In The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Evaluation of the transferred measure <math alttext="\sigma M(\mu)" class="ltx_Math" display="inline"><semantics><mrow><mi>σ</mi><mo>⁢</mo><mi>M</mi><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content"><apply><times></times><ci>𝜎</ci><ci>𝑀</ci><ci>𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex">\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun">italic_σ italic_M ( italic_μ )</annotation></semantics></math></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.SS1" title="In 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>A first example for the measure transfer</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.SS2" title="In 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>An alternative evaluation method</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5" title="In The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Shift-orbit injectivity and related notions</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6" title="In The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>The injectivity of the measure transfer for letter-to-letter morphisms</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line ltx_leqno"> <h1 class="ltx_title ltx_title_document">The measure transfer for subshifts induced by a morphism of free monoids</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Nicolas Bédaride </span></span> <span class="ltx_author_before">, </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Arnaud Hilion </span></span> <span class="ltx_author_before"> and </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Martin Lustig </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_address"><span class="ltx_text ltx_font_typewriter" id="id18.1.id1">Aix Marseille Université, CNRS, I2M UMR 7373, 13453 Marseille, France</span> </span> <span class="ltx_contact ltx_role_email"><a href="mailto:"><span class="ltx_text ltx_font_typewriter" id="id19.2.id1">nicolas.bedaride@univ-amu.fr</span></a> </span> <span class="ltx_contact ltx_role_address"><span class="ltx_text ltx_font_typewriter" id="id20.3.id1">Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France</span> </span> <span class="ltx_contact ltx_role_email"><a href="mailto:"><span class="ltx_text ltx_font_typewriter" id="id21.4.id1">arnaud.hilion@math.univ-toulouse.fr</span></a> </span> <span class="ltx_contact ltx_role_address"><span class="ltx_text ltx_font_typewriter" id="id22.5.id1">Aix Marseille Université, CNRS, I2M UMR 7373, 13453 Marseille, France</span> </span> <span class="ltx_contact ltx_role_email"><a href="mailto:"><span class="ltx_text ltx_font_typewriter" id="id23.6.id1">martinlustig@gmx.de</span></a> </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract.</h6> <p class="ltx_p" id="id9.9">Every non-erasing monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mrow id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml"><mi id="id1.1.m1.1.1.2" xref="id1.1.m1.1.1.2.cmml">σ</mi><mo id="id1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="id1.1.m1.1.1.1.cmml">:</mo><mrow id="id1.1.m1.1.1.3" xref="id1.1.m1.1.1.3.cmml"><msup id="id1.1.m1.1.1.3.2" xref="id1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="id1.1.m1.1.1.3.2.2" xref="id1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="id1.1.m1.1.1.3.2.3" xref="id1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="id1.1.m1.1.1.3.1" stretchy="false" xref="id1.1.m1.1.1.3.1.cmml">→</mo><msup id="id1.1.m1.1.1.3.3" xref="id1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id1.1.m1.1.1.3.3.2" xref="id1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="id1.1.m1.1.1.3.3.3" xref="id1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><apply id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1"><ci id="id1.1.m1.1.1.1.cmml" xref="id1.1.m1.1.1.1">:</ci><ci id="id1.1.m1.1.1.2.cmml" xref="id1.1.m1.1.1.2">𝜎</ci><apply id="id1.1.m1.1.1.3.cmml" xref="id1.1.m1.1.1.3"><ci id="id1.1.m1.1.1.3.1.cmml" xref="id1.1.m1.1.1.3.1">→</ci><apply id="id1.1.m1.1.1.3.2.cmml" xref="id1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="id1.1.m1.1.1.3.2.1.cmml" xref="id1.1.m1.1.1.3.2">superscript</csymbol><ci id="id1.1.m1.1.1.3.2.2.cmml" xref="id1.1.m1.1.1.3.2.2">𝒜</ci><times id="id1.1.m1.1.1.3.2.3.cmml" xref="id1.1.m1.1.1.3.2.3"></times></apply><apply id="id1.1.m1.1.1.3.3.cmml" xref="id1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="id1.1.m1.1.1.3.3.1.cmml" xref="id1.1.m1.1.1.3.3">superscript</csymbol><ci id="id1.1.m1.1.1.3.3.2.cmml" xref="id1.1.m1.1.1.3.3.2">ℬ</ci><times id="id1.1.m1.1.1.3.3.3.cmml" xref="id1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> induces a <span class="ltx_text ltx_font_italic" id="id9.9.1">measure transfer map</span> <math alttext="\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))" class="ltx_Math" display="inline" id="id2.2.m2.3"><semantics id="id2.2.m2.3a"><mrow id="id2.2.m2.3.3" xref="id2.2.m2.3.3.cmml"><mrow id="id2.2.m2.3.3.3" xref="id2.2.m2.3.3.3.cmml"><msub id="id2.2.m2.3.3.3.2" xref="id2.2.m2.3.3.3.2.cmml"><mi id="id2.2.m2.3.3.3.2.2" xref="id2.2.m2.3.3.3.2.2.cmml">σ</mi><mi id="id2.2.m2.3.3.3.2.3" xref="id2.2.m2.3.3.3.2.3.cmml">X</mi></msub><mo id="id2.2.m2.3.3.3.1" xref="id2.2.m2.3.3.3.1.cmml">⁢</mo><mi id="id2.2.m2.3.3.3.3" xref="id2.2.m2.3.3.3.3.cmml">M</mi></mrow><mo id="id2.2.m2.3.3.2" lspace="0.278em" rspace="0.278em" xref="id2.2.m2.3.3.2.cmml">:</mo><mrow id="id2.2.m2.3.3.1" xref="id2.2.m2.3.3.1.cmml"><mrow id="id2.2.m2.3.3.1.3" xref="id2.2.m2.3.3.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id2.2.m2.3.3.1.3.2" xref="id2.2.m2.3.3.1.3.2.cmml">ℳ</mi><mo id="id2.2.m2.3.3.1.3.1" xref="id2.2.m2.3.3.1.3.1.cmml">⁢</mo><mrow id="id2.2.m2.3.3.1.3.3.2" xref="id2.2.m2.3.3.1.3.cmml"><mo id="id2.2.m2.3.3.1.3.3.2.1" stretchy="false" xref="id2.2.m2.3.3.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml">𝒳</mi><mo id="id2.2.m2.3.3.1.3.3.2.2" stretchy="false" xref="id2.2.m2.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="id2.2.m2.3.3.1.2" stretchy="false" xref="id2.2.m2.3.3.1.2.cmml">→</mo><mrow id="id2.2.m2.3.3.1.1" xref="id2.2.m2.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="id2.2.m2.3.3.1.1.3" xref="id2.2.m2.3.3.1.1.3.cmml">ℳ</mi><mo id="id2.2.m2.3.3.1.1.2" xref="id2.2.m2.3.3.1.1.2.cmml">⁢</mo><mrow id="id2.2.m2.3.3.1.1.1.1" xref="id2.2.m2.3.3.1.1.1.1.1.cmml"><mo id="id2.2.m2.3.3.1.1.1.1.2" stretchy="false" xref="id2.2.m2.3.3.1.1.1.1.1.cmml">(</mo><mrow id="id2.2.m2.3.3.1.1.1.1.1" xref="id2.2.m2.3.3.1.1.1.1.1.cmml"><mi id="id2.2.m2.3.3.1.1.1.1.1.2" xref="id2.2.m2.3.3.1.1.1.1.1.2.cmml">σ</mi><mo id="id2.2.m2.3.3.1.1.1.1.1.1" xref="id2.2.m2.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="id2.2.m2.3.3.1.1.1.1.1.3.2" xref="id2.2.m2.3.3.1.1.1.1.1.cmml"><mo id="id2.2.m2.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="id2.2.m2.3.3.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="id2.2.m2.2.2" xref="id2.2.m2.2.2.cmml">𝒳</mi><mo id="id2.2.m2.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="id2.2.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="id2.2.m2.3.3.1.1.1.1.3" stretchy="false" xref="id2.2.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="id2.2.m2.3b"><apply id="id2.2.m2.3.3.cmml" xref="id2.2.m2.3.3"><ci id="id2.2.m2.3.3.2.cmml" xref="id2.2.m2.3.3.2">:</ci><apply id="id2.2.m2.3.3.3.cmml" xref="id2.2.m2.3.3.3"><times id="id2.2.m2.3.3.3.1.cmml" xref="id2.2.m2.3.3.3.1"></times><apply id="id2.2.m2.3.3.3.2.cmml" xref="id2.2.m2.3.3.3.2"><csymbol cd="ambiguous" id="id2.2.m2.3.3.3.2.1.cmml" xref="id2.2.m2.3.3.3.2">subscript</csymbol><ci id="id2.2.m2.3.3.3.2.2.cmml" xref="id2.2.m2.3.3.3.2.2">𝜎</ci><ci id="id2.2.m2.3.3.3.2.3.cmml" xref="id2.2.m2.3.3.3.2.3">𝑋</ci></apply><ci id="id2.2.m2.3.3.3.3.cmml" xref="id2.2.m2.3.3.3.3">𝑀</ci></apply><apply id="id2.2.m2.3.3.1.cmml" xref="id2.2.m2.3.3.1"><ci id="id2.2.m2.3.3.1.2.cmml" xref="id2.2.m2.3.3.1.2">→</ci><apply id="id2.2.m2.3.3.1.3.cmml" xref="id2.2.m2.3.3.1.3"><times id="id2.2.m2.3.3.1.3.1.cmml" xref="id2.2.m2.3.3.1.3.1"></times><ci id="id2.2.m2.3.3.1.3.2.cmml" xref="id2.2.m2.3.3.1.3.2">ℳ</ci><ci id="id2.2.m2.1.1.cmml" xref="id2.2.m2.1.1">𝒳</ci></apply><apply id="id2.2.m2.3.3.1.1.cmml" xref="id2.2.m2.3.3.1.1"><times id="id2.2.m2.3.3.1.1.2.cmml" xref="id2.2.m2.3.3.1.1.2"></times><ci id="id2.2.m2.3.3.1.1.3.cmml" xref="id2.2.m2.3.3.1.1.3">ℳ</ci><apply id="id2.2.m2.3.3.1.1.1.1.1.cmml" xref="id2.2.m2.3.3.1.1.1.1"><times id="id2.2.m2.3.3.1.1.1.1.1.1.cmml" xref="id2.2.m2.3.3.1.1.1.1.1.1"></times><ci id="id2.2.m2.3.3.1.1.1.1.1.2.cmml" xref="id2.2.m2.3.3.1.1.1.1.1.2">𝜎</ci><ci id="id2.2.m2.2.2.cmml" xref="id2.2.m2.2.2">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id2.2.m2.3c">\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="id2.2.m2.3d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) )</annotation></semantics></math> between the measure cones <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="id3.3.m3.1"><semantics id="id3.3.m3.1a"><mrow id="id3.3.m3.1.2" xref="id3.3.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="id3.3.m3.1.2.2" xref="id3.3.m3.1.2.2.cmml">ℳ</mi><mo id="id3.3.m3.1.2.1" xref="id3.3.m3.1.2.1.cmml">⁢</mo><mrow id="id3.3.m3.1.2.3.2" xref="id3.3.m3.1.2.cmml"><mo id="id3.3.m3.1.2.3.2.1" stretchy="false" xref="id3.3.m3.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="id3.3.m3.1.1" xref="id3.3.m3.1.1.cmml">𝒳</mi><mo id="id3.3.m3.1.2.3.2.2" stretchy="false" xref="id3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id3.3.m3.1b"><apply id="id3.3.m3.1.2.cmml" xref="id3.3.m3.1.2"><times id="id3.3.m3.1.2.1.cmml" xref="id3.3.m3.1.2.1"></times><ci id="id3.3.m3.1.2.2.cmml" xref="id3.3.m3.1.2.2">ℳ</ci><ci id="id3.3.m3.1.1.cmml" xref="id3.3.m3.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id3.3.m3.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="id3.3.m3.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> and <math alttext="\cal M(\sigma(X))" class="ltx_Math" display="inline" id="id4.4.m4.2"><semantics id="id4.4.m4.2a"><mrow id="id4.4.m4.2.2" xref="id4.4.m4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="id4.4.m4.2.2.3" xref="id4.4.m4.2.2.3.cmml">ℳ</mi><mo id="id4.4.m4.2.2.2" xref="id4.4.m4.2.2.2.cmml">⁢</mo><mrow id="id4.4.m4.2.2.1.1" xref="id4.4.m4.2.2.1.1.1.cmml"><mo id="id4.4.m4.2.2.1.1.2" stretchy="false" xref="id4.4.m4.2.2.1.1.1.cmml">(</mo><mrow id="id4.4.m4.2.2.1.1.1" xref="id4.4.m4.2.2.1.1.1.cmml"><mi id="id4.4.m4.2.2.1.1.1.2" xref="id4.4.m4.2.2.1.1.1.2.cmml">σ</mi><mo id="id4.4.m4.2.2.1.1.1.1" xref="id4.4.m4.2.2.1.1.1.1.cmml">⁢</mo><mrow id="id4.4.m4.2.2.1.1.1.3.2" xref="id4.4.m4.2.2.1.1.1.cmml"><mo id="id4.4.m4.2.2.1.1.1.3.2.1" stretchy="false" xref="id4.4.m4.2.2.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="id4.4.m4.1.1" xref="id4.4.m4.1.1.cmml">𝒳</mi><mo id="id4.4.m4.2.2.1.1.1.3.2.2" stretchy="false" xref="id4.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="id4.4.m4.2.2.1.1.3" stretchy="false" xref="id4.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id4.4.m4.2b"><apply id="id4.4.m4.2.2.cmml" xref="id4.4.m4.2.2"><times id="id4.4.m4.2.2.2.cmml" xref="id4.4.m4.2.2.2"></times><ci id="id4.4.m4.2.2.3.cmml" xref="id4.4.m4.2.2.3">ℳ</ci><apply id="id4.4.m4.2.2.1.1.1.cmml" xref="id4.4.m4.2.2.1.1"><times id="id4.4.m4.2.2.1.1.1.1.cmml" xref="id4.4.m4.2.2.1.1.1.1"></times><ci id="id4.4.m4.2.2.1.1.1.2.cmml" xref="id4.4.m4.2.2.1.1.1.2">𝜎</ci><ci id="id4.4.m4.1.1.cmml" xref="id4.4.m4.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id4.4.m4.2c">\cal M(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="id4.4.m4.2d">caligraphic_M ( italic_σ ( caligraphic_X ) )</annotation></semantics></math>, associated to any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="id5.5.m5.1"><semantics id="id5.5.m5.1a"><mrow id="id5.5.m5.1.1" xref="id5.5.m5.1.1.cmml"><mi id="id5.5.m5.1.1.2" xref="id5.5.m5.1.1.2.cmml">X</mi><mo id="id5.5.m5.1.1.1" xref="id5.5.m5.1.1.1.cmml">⊆</mo><msup id="id5.5.m5.1.1.3" xref="id5.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id5.5.m5.1.1.3.2" xref="id5.5.m5.1.1.3.2.cmml">𝒜</mi><mi id="id5.5.m5.1.1.3.3" xref="id5.5.m5.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="id5.5.m5.1b"><apply id="id5.5.m5.1.1.cmml" xref="id5.5.m5.1.1"><subset id="id5.5.m5.1.1.1.cmml" xref="id5.5.m5.1.1.1"></subset><ci id="id5.5.m5.1.1.2.cmml" xref="id5.5.m5.1.1.2">𝑋</ci><apply id="id5.5.m5.1.1.3.cmml" xref="id5.5.m5.1.1.3"><csymbol cd="ambiguous" id="id5.5.m5.1.1.3.1.cmml" xref="id5.5.m5.1.1.3">superscript</csymbol><ci id="id5.5.m5.1.1.3.2.cmml" xref="id5.5.m5.1.1.3.2">𝒜</ci><ci id="id5.5.m5.1.1.3.3.cmml" xref="id5.5.m5.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id5.5.m5.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="id5.5.m5.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and its image subshift <math alttext="\sigma(X)\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="id6.6.m6.1"><semantics id="id6.6.m6.1a"><mrow id="id6.6.m6.1.2" xref="id6.6.m6.1.2.cmml"><mrow id="id6.6.m6.1.2.2" xref="id6.6.m6.1.2.2.cmml"><mi id="id6.6.m6.1.2.2.2" xref="id6.6.m6.1.2.2.2.cmml">σ</mi><mo id="id6.6.m6.1.2.2.1" xref="id6.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="id6.6.m6.1.2.2.3.2" xref="id6.6.m6.1.2.2.cmml"><mo id="id6.6.m6.1.2.2.3.2.1" stretchy="false" xref="id6.6.m6.1.2.2.cmml">(</mo><mi id="id6.6.m6.1.1" xref="id6.6.m6.1.1.cmml">X</mi><mo id="id6.6.m6.1.2.2.3.2.2" stretchy="false" xref="id6.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="id6.6.m6.1.2.1" xref="id6.6.m6.1.2.1.cmml">⊆</mo><msup id="id6.6.m6.1.2.3" xref="id6.6.m6.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id6.6.m6.1.2.3.2" xref="id6.6.m6.1.2.3.2.cmml">ℬ</mi><mi id="id6.6.m6.1.2.3.3" xref="id6.6.m6.1.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="id6.6.m6.1b"><apply id="id6.6.m6.1.2.cmml" xref="id6.6.m6.1.2"><subset id="id6.6.m6.1.2.1.cmml" xref="id6.6.m6.1.2.1"></subset><apply id="id6.6.m6.1.2.2.cmml" xref="id6.6.m6.1.2.2"><times id="id6.6.m6.1.2.2.1.cmml" xref="id6.6.m6.1.2.2.1"></times><ci id="id6.6.m6.1.2.2.2.cmml" xref="id6.6.m6.1.2.2.2">𝜎</ci><ci id="id6.6.m6.1.1.cmml" xref="id6.6.m6.1.1">𝑋</ci></apply><apply id="id6.6.m6.1.2.3.cmml" xref="id6.6.m6.1.2.3"><csymbol cd="ambiguous" id="id6.6.m6.1.2.3.1.cmml" xref="id6.6.m6.1.2.3">superscript</csymbol><ci id="id6.6.m6.1.2.3.2.cmml" xref="id6.6.m6.1.2.3.2">ℬ</ci><ci id="id6.6.m6.1.2.3.3.cmml" xref="id6.6.m6.1.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id6.6.m6.1c">\sigma(X)\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="id6.6.m6.1d">italic_σ ( italic_X ) ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>. Furthermore, an efficient technique to compute the value of the transferred measure <math alttext="\sigma_{X}M(\mu)" class="ltx_Math" display="inline" id="id7.7.m7.1"><semantics id="id7.7.m7.1a"><mrow id="id7.7.m7.1.2" xref="id7.7.m7.1.2.cmml"><msub id="id7.7.m7.1.2.2" xref="id7.7.m7.1.2.2.cmml"><mi id="id7.7.m7.1.2.2.2" xref="id7.7.m7.1.2.2.2.cmml">σ</mi><mi id="id7.7.m7.1.2.2.3" xref="id7.7.m7.1.2.2.3.cmml">X</mi></msub><mo id="id7.7.m7.1.2.1" xref="id7.7.m7.1.2.1.cmml">⁢</mo><mi id="id7.7.m7.1.2.3" xref="id7.7.m7.1.2.3.cmml">M</mi><mo id="id7.7.m7.1.2.1a" xref="id7.7.m7.1.2.1.cmml">⁢</mo><mrow id="id7.7.m7.1.2.4.2" xref="id7.7.m7.1.2.cmml"><mo id="id7.7.m7.1.2.4.2.1" stretchy="false" xref="id7.7.m7.1.2.cmml">(</mo><mi id="id7.7.m7.1.1" xref="id7.7.m7.1.1.cmml">μ</mi><mo id="id7.7.m7.1.2.4.2.2" stretchy="false" xref="id7.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="id7.7.m7.1b"><apply id="id7.7.m7.1.2.cmml" xref="id7.7.m7.1.2"><times id="id7.7.m7.1.2.1.cmml" xref="id7.7.m7.1.2.1"></times><apply id="id7.7.m7.1.2.2.cmml" xref="id7.7.m7.1.2.2"><csymbol cd="ambiguous" id="id7.7.m7.1.2.2.1.cmml" xref="id7.7.m7.1.2.2">subscript</csymbol><ci id="id7.7.m7.1.2.2.2.cmml" xref="id7.7.m7.1.2.2.2">𝜎</ci><ci id="id7.7.m7.1.2.2.3.cmml" xref="id7.7.m7.1.2.2.3">𝑋</ci></apply><ci id="id7.7.m7.1.2.3.cmml" xref="id7.7.m7.1.2.3">𝑀</ci><ci id="id7.7.m7.1.1.cmml" xref="id7.7.m7.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id7.7.m7.1c">\sigma_{X}M(\mu)</annotation><annotation encoding="application/x-llamapun" id="id7.7.m7.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M ( italic_μ )</annotation></semantics></math> on any cylinder <math alttext="[w]" class="ltx_Math" display="inline" id="id8.8.m8.1"><semantics id="id8.8.m8.1a"><mrow id="id8.8.m8.1.2.2" xref="id8.8.m8.1.2.1.cmml"><mo id="id8.8.m8.1.2.2.1" stretchy="false" xref="id8.8.m8.1.2.1.1.cmml">[</mo><mi id="id8.8.m8.1.1" xref="id8.8.m8.1.1.cmml">w</mi><mo id="id8.8.m8.1.2.2.2" stretchy="false" xref="id8.8.m8.1.2.1.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="id8.8.m8.1b"><apply id="id8.8.m8.1.2.1.cmml" xref="id8.8.m8.1.2.2"><csymbol cd="latexml" id="id8.8.m8.1.2.1.1.cmml" xref="id8.8.m8.1.2.2.1">delimited-[]</csymbol><ci id="id8.8.m8.1.1.cmml" xref="id8.8.m8.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id8.8.m8.1c">[w]</annotation><annotation encoding="application/x-llamapun" id="id8.8.m8.1d">[ italic_w ]</annotation></semantics></math> (for <math alttext="w\in\cal B^{*}" class="ltx_Math" display="inline" id="id9.9.m9.1"><semantics id="id9.9.m9.1a"><mrow id="id9.9.m9.1.1" xref="id9.9.m9.1.1.cmml"><mi id="id9.9.m9.1.1.2" xref="id9.9.m9.1.1.2.cmml">w</mi><mo id="id9.9.m9.1.1.1" xref="id9.9.m9.1.1.1.cmml">∈</mo><msup id="id9.9.m9.1.1.3" xref="id9.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id9.9.m9.1.1.3.2" xref="id9.9.m9.1.1.3.2.cmml">ℬ</mi><mo id="id9.9.m9.1.1.3.3" xref="id9.9.m9.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="id9.9.m9.1b"><apply id="id9.9.m9.1.1.cmml" xref="id9.9.m9.1.1"><in id="id9.9.m9.1.1.1.cmml" xref="id9.9.m9.1.1.1"></in><ci id="id9.9.m9.1.1.2.cmml" xref="id9.9.m9.1.1.2">𝑤</ci><apply id="id9.9.m9.1.1.3.cmml" xref="id9.9.m9.1.1.3"><csymbol cd="ambiguous" id="id9.9.m9.1.1.3.1.cmml" xref="id9.9.m9.1.1.3">superscript</csymbol><ci id="id9.9.m9.1.1.3.2.cmml" xref="id9.9.m9.1.1.3.2">ℬ</ci><times id="id9.9.m9.1.1.3.3.cmml" xref="id9.9.m9.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id9.9.m9.1c">w\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="id9.9.m9.1d">italic_w ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>) is presented.</p> <p class="ltx_p" id="id12.12"><span class="ltx_text ltx_font_bold" id="id12.12.1">Theorem:</span> If a non-erasing morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="id10.10.m1.1"><semantics id="id10.10.m1.1a"><mrow id="id10.10.m1.1.1" xref="id10.10.m1.1.1.cmml"><mi id="id10.10.m1.1.1.2" xref="id10.10.m1.1.1.2.cmml">σ</mi><mo id="id10.10.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="id10.10.m1.1.1.1.cmml">:</mo><mrow id="id10.10.m1.1.1.3" xref="id10.10.m1.1.1.3.cmml"><msup id="id10.10.m1.1.1.3.2" xref="id10.10.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="id10.10.m1.1.1.3.2.2" xref="id10.10.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="id10.10.m1.1.1.3.2.3" xref="id10.10.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="id10.10.m1.1.1.3.1" stretchy="false" xref="id10.10.m1.1.1.3.1.cmml">→</mo><msup id="id10.10.m1.1.1.3.3" xref="id10.10.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id10.10.m1.1.1.3.3.2" xref="id10.10.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="id10.10.m1.1.1.3.3.3" xref="id10.10.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="id10.10.m1.1b"><apply id="id10.10.m1.1.1.cmml" xref="id10.10.m1.1.1"><ci id="id10.10.m1.1.1.1.cmml" xref="id10.10.m1.1.1.1">:</ci><ci id="id10.10.m1.1.1.2.cmml" xref="id10.10.m1.1.1.2">𝜎</ci><apply id="id10.10.m1.1.1.3.cmml" xref="id10.10.m1.1.1.3"><ci id="id10.10.m1.1.1.3.1.cmml" xref="id10.10.m1.1.1.3.1">→</ci><apply id="id10.10.m1.1.1.3.2.cmml" xref="id10.10.m1.1.1.3.2"><csymbol cd="ambiguous" id="id10.10.m1.1.1.3.2.1.cmml" xref="id10.10.m1.1.1.3.2">superscript</csymbol><ci id="id10.10.m1.1.1.3.2.2.cmml" xref="id10.10.m1.1.1.3.2.2">𝒜</ci><times id="id10.10.m1.1.1.3.2.3.cmml" xref="id10.10.m1.1.1.3.2.3"></times></apply><apply id="id10.10.m1.1.1.3.3.cmml" xref="id10.10.m1.1.1.3.3"><csymbol cd="ambiguous" id="id10.10.m1.1.1.3.3.1.cmml" xref="id10.10.m1.1.1.3.3">superscript</csymbol><ci id="id10.10.m1.1.1.3.3.2.cmml" xref="id10.10.m1.1.1.3.3.2">ℬ</ci><times id="id10.10.m1.1.1.3.3.3.cmml" xref="id10.10.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id10.10.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="id10.10.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is injective on the shift-orbits of some subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="id11.11.m2.1"><semantics id="id11.11.m2.1a"><mrow id="id11.11.m2.1.1" xref="id11.11.m2.1.1.cmml"><mi id="id11.11.m2.1.1.2" xref="id11.11.m2.1.1.2.cmml">X</mi><mo id="id11.11.m2.1.1.1" xref="id11.11.m2.1.1.1.cmml">⊆</mo><msup id="id11.11.m2.1.1.3" xref="id11.11.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="id11.11.m2.1.1.3.2" xref="id11.11.m2.1.1.3.2.cmml">𝒜</mi><mi id="id11.11.m2.1.1.3.3" xref="id11.11.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="id11.11.m2.1b"><apply id="id11.11.m2.1.1.cmml" xref="id11.11.m2.1.1"><subset id="id11.11.m2.1.1.1.cmml" xref="id11.11.m2.1.1.1"></subset><ci id="id11.11.m2.1.1.2.cmml" xref="id11.11.m2.1.1.2">𝑋</ci><apply id="id11.11.m2.1.1.3.cmml" xref="id11.11.m2.1.1.3"><csymbol cd="ambiguous" id="id11.11.m2.1.1.3.1.cmml" xref="id11.11.m2.1.1.3">superscript</csymbol><ci id="id11.11.m2.1.1.3.2.cmml" xref="id11.11.m2.1.1.3.2">𝒜</ci><ci id="id11.11.m2.1.1.3.3.cmml" xref="id11.11.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id11.11.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="id11.11.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="\sigma M_{X}" class="ltx_Math" display="inline" id="id12.12.m3.1"><semantics id="id12.12.m3.1a"><mrow id="id12.12.m3.1.1" xref="id12.12.m3.1.1.cmml"><mi id="id12.12.m3.1.1.2" xref="id12.12.m3.1.1.2.cmml">σ</mi><mo id="id12.12.m3.1.1.1" xref="id12.12.m3.1.1.1.cmml">⁢</mo><msub id="id12.12.m3.1.1.3" xref="id12.12.m3.1.1.3.cmml"><mi id="id12.12.m3.1.1.3.2" xref="id12.12.m3.1.1.3.2.cmml">M</mi><mi id="id12.12.m3.1.1.3.3" xref="id12.12.m3.1.1.3.3.cmml">X</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="id12.12.m3.1b"><apply id="id12.12.m3.1.1.cmml" xref="id12.12.m3.1.1"><times id="id12.12.m3.1.1.1.cmml" xref="id12.12.m3.1.1.1"></times><ci id="id12.12.m3.1.1.2.cmml" xref="id12.12.m3.1.1.2">𝜎</ci><apply id="id12.12.m3.1.1.3.cmml" xref="id12.12.m3.1.1.3"><csymbol cd="ambiguous" id="id12.12.m3.1.1.3.1.cmml" xref="id12.12.m3.1.1.3">subscript</csymbol><ci id="id12.12.m3.1.1.3.2.cmml" xref="id12.12.m3.1.1.3.2">𝑀</ci><ci id="id12.12.m3.1.1.3.3.cmml" xref="id12.12.m3.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="id12.12.m3.1c">\sigma M_{X}</annotation><annotation encoding="application/x-llamapun" id="id12.12.m3.1d">italic_σ italic_M start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> is injective.</p> <p class="ltx_p" id="id17.17">The assumption on <math alttext="\sigma" class="ltx_Math" display="inline" id="id13.13.m1.1"><semantics id="id13.13.m1.1a"><mi id="id13.13.m1.1.1" xref="id13.13.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="id13.13.m1.1b"><ci id="id13.13.m1.1.1.cmml" xref="id13.13.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="id13.13.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="id13.13.m1.1d">italic_σ</annotation></semantics></math> that it is “injective on the shift-orbits of <math alttext="X" class="ltx_Math" display="inline" id="id14.14.m2.1"><semantics id="id14.14.m2.1a"><mi id="id14.14.m2.1.1" xref="id14.14.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="id14.14.m2.1b"><ci id="id14.14.m2.1.1.cmml" xref="id14.14.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="id14.14.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="id14.14.m2.1d">italic_X</annotation></semantics></math>” is strictly weaker than “recognizable in <math alttext="X" class="ltx_Math" display="inline" id="id15.15.m3.1"><semantics id="id15.15.m3.1a"><mi id="id15.15.m3.1.1" xref="id15.15.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="id15.15.m3.1b"><ci id="id15.15.m3.1.1.cmml" xref="id15.15.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="id15.15.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="id15.15.m3.1d">italic_X</annotation></semantics></math>”, and strictly stronger than “recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="id16.16.m4.1"><semantics id="id16.16.m4.1a"><mi id="id16.16.m4.1.1" xref="id16.16.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="id16.16.m4.1b"><ci id="id16.16.m4.1.1.cmml" xref="id16.16.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="id16.16.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="id16.16.m4.1d">italic_X</annotation></semantics></math>”. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="id17.17.m5.1"><semantics id="id17.17.m5.1a"><mrow id="id17.17.m5.1.1" xref="id17.17.m5.1.1.cmml"><msub id="id17.17.m5.1.1.2" xref="id17.17.m5.1.1.2.cmml"><mi id="id17.17.m5.1.1.2.2" xref="id17.17.m5.1.1.2.2.cmml">σ</mi><mi id="id17.17.m5.1.1.2.3" xref="id17.17.m5.1.1.2.3.cmml">X</mi></msub><mo id="id17.17.m5.1.1.1" xref="id17.17.m5.1.1.1.cmml">⁢</mo><mi id="id17.17.m5.1.1.3" xref="id17.17.m5.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="id17.17.m5.1b"><apply id="id17.17.m5.1.1.cmml" xref="id17.17.m5.1.1"><times id="id17.17.m5.1.1.1.cmml" xref="id17.17.m5.1.1.1"></times><apply id="id17.17.m5.1.1.2.cmml" xref="id17.17.m5.1.1.2"><csymbol cd="ambiguous" id="id17.17.m5.1.1.2.1.cmml" xref="id17.17.m5.1.1.2">subscript</csymbol><ci id="id17.17.m5.1.1.2.2.cmml" xref="id17.17.m5.1.1.2.2">𝜎</ci><ci id="id17.17.m5.1.1.2.3.cmml" xref="id17.17.m5.1.1.2.3">𝑋</ci></apply><ci id="id17.17.m5.1.1.3.cmml" xref="id17.17.m5.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id17.17.m5.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="id17.17.m5.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math>.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Key words and phrases: </h6>subshift, invariant measure, recognizable monoid morphism </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">2010 Mathematics Subject Classification: </h6>Primary 37B10, Secondary 37A25, 37E25 </div> <div class="ltx_acknowledgements">The first author was partially supported by ANR Project IZES ANR-22-CE40-0011 </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1. </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.19">The prime objects of this paper are morphisms <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mrow id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml"><mi id="S1.p1.1.m1.1.1.2" xref="S1.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.p1.1.m1.1.1.3" xref="S1.p1.1.m1.1.1.3.cmml"><msup id="S1.p1.1.m1.1.1.3.2" xref="S1.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.1.m1.1.1.3.2.2" xref="S1.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.p1.1.m1.1.1.3.2.3" xref="S1.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.p1.1.m1.1.1.3.1" stretchy="false" xref="S1.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S1.p1.1.m1.1.1.3.3" xref="S1.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.1.m1.1.1.3.3.2" xref="S1.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S1.p1.1.m1.1.1.3.3.3" xref="S1.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><apply id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1"><ci id="S1.p1.1.m1.1.1.1.cmml" xref="S1.p1.1.m1.1.1.1">:</ci><ci id="S1.p1.1.m1.1.1.2.cmml" xref="S1.p1.1.m1.1.1.2">𝜎</ci><apply id="S1.p1.1.m1.1.1.3.cmml" xref="S1.p1.1.m1.1.1.3"><ci id="S1.p1.1.m1.1.1.3.1.cmml" xref="S1.p1.1.m1.1.1.3.1">→</ci><apply id="S1.p1.1.m1.1.1.3.2.cmml" xref="S1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.p1.1.m1.1.1.3.2.1.cmml" xref="S1.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.p1.1.m1.1.1.3.2.2.cmml" xref="S1.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S1.p1.1.m1.1.1.3.2.3.cmml" xref="S1.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S1.p1.1.m1.1.1.3.3.cmml" xref="S1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.p1.1.m1.1.1.3.3.1.cmml" xref="S1.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.p1.1.m1.1.1.3.3.2.cmml" xref="S1.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S1.p1.1.m1.1.1.3.3.3.cmml" xref="S1.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of free monoids <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><msup id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.2.m2.1.1.2" xref="S1.p1.2.m2.1.1.2.cmml">𝒜</mi><mo id="S1.p1.2.m2.1.1.3" xref="S1.p1.2.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><apply id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p1.2.m2.1.1.1.cmml" xref="S1.p1.2.m2.1.1">superscript</csymbol><ci id="S1.p1.2.m2.1.1.2.cmml" xref="S1.p1.2.m2.1.1.2">𝒜</ci><times id="S1.p1.2.m2.1.1.3.cmml" xref="S1.p1.2.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\cal B^{*}" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><msup id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.3.m3.1.1.2" xref="S1.p1.3.m3.1.1.2.cmml">ℬ</mi><mo id="S1.p1.3.m3.1.1.3" xref="S1.p1.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><apply id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p1.3.m3.1.1.1.cmml" xref="S1.p1.3.m3.1.1">superscript</csymbol><ci id="S1.p1.3.m3.1.1.2.cmml" xref="S1.p1.3.m3.1.1.2">ℬ</ci><times id="S1.p1.3.m3.1.1.3.cmml" xref="S1.p1.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> over finite sets <math alttext="\cal A" class="ltx_Math" display="inline" id="S1.p1.4.m4.1"><semantics id="S1.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.1b"><ci id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">caligraphic_A</annotation></semantics></math> and <math alttext="\cal B" class="ltx_Math" display="inline" id="S1.p1.5.m5.1"><semantics id="S1.p1.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p1.5.m5.1.1" xref="S1.p1.5.m5.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.5.m5.1b"><ci id="S1.p1.5.m5.1.1.cmml" xref="S1.p1.5.m5.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m5.1c">\cal B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.1d">caligraphic_B</annotation></semantics></math> respectively. These sets are called <span class="ltx_text ltx_font_italic" id="S1.p1.19.1">alphabets</span>, and their elements are <span class="ltx_text ltx_font_italic" id="S1.p1.19.2">letters</span>, denoted here by <math alttext="a_{k}\in\cal A" class="ltx_Math" display="inline" id="S1.p1.6.m6.1"><semantics id="S1.p1.6.m6.1a"><mrow id="S1.p1.6.m6.1.1" xref="S1.p1.6.m6.1.1.cmml"><msub id="S1.p1.6.m6.1.1.2" xref="S1.p1.6.m6.1.1.2.cmml"><mi id="S1.p1.6.m6.1.1.2.2" xref="S1.p1.6.m6.1.1.2.2.cmml">a</mi><mi id="S1.p1.6.m6.1.1.2.3" xref="S1.p1.6.m6.1.1.2.3.cmml">k</mi></msub><mo id="S1.p1.6.m6.1.1.1" xref="S1.p1.6.m6.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.6.m6.1.1.3" xref="S1.p1.6.m6.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.6.m6.1b"><apply id="S1.p1.6.m6.1.1.cmml" xref="S1.p1.6.m6.1.1"><in id="S1.p1.6.m6.1.1.1.cmml" xref="S1.p1.6.m6.1.1.1"></in><apply id="S1.p1.6.m6.1.1.2.cmml" xref="S1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.p1.6.m6.1.1.2.1.cmml" xref="S1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S1.p1.6.m6.1.1.2.2.cmml" xref="S1.p1.6.m6.1.1.2.2">𝑎</ci><ci id="S1.p1.6.m6.1.1.2.3.cmml" xref="S1.p1.6.m6.1.1.2.3">𝑘</ci></apply><ci id="S1.p1.6.m6.1.1.3.cmml" xref="S1.p1.6.m6.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.6.m6.1c">a_{k}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.6.m6.1d">italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> or <math alttext="b_{j}\in\cal B" class="ltx_Math" display="inline" id="S1.p1.7.m7.1"><semantics id="S1.p1.7.m7.1a"><mrow id="S1.p1.7.m7.1.1" xref="S1.p1.7.m7.1.1.cmml"><msub id="S1.p1.7.m7.1.1.2" xref="S1.p1.7.m7.1.1.2.cmml"><mi id="S1.p1.7.m7.1.1.2.2" xref="S1.p1.7.m7.1.1.2.2.cmml">b</mi><mi id="S1.p1.7.m7.1.1.2.3" xref="S1.p1.7.m7.1.1.2.3.cmml">j</mi></msub><mo id="S1.p1.7.m7.1.1.1" xref="S1.p1.7.m7.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.7.m7.1.1.3" xref="S1.p1.7.m7.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.7.m7.1b"><apply id="S1.p1.7.m7.1.1.cmml" xref="S1.p1.7.m7.1.1"><in id="S1.p1.7.m7.1.1.1.cmml" xref="S1.p1.7.m7.1.1.1"></in><apply id="S1.p1.7.m7.1.1.2.cmml" xref="S1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S1.p1.7.m7.1.1.2.1.cmml" xref="S1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S1.p1.7.m7.1.1.2.2.cmml" xref="S1.p1.7.m7.1.1.2.2">𝑏</ci><ci id="S1.p1.7.m7.1.1.2.3.cmml" xref="S1.p1.7.m7.1.1.2.3">𝑗</ci></apply><ci id="S1.p1.7.m7.1.1.3.cmml" xref="S1.p1.7.m7.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.7.m7.1c">b_{j}\in\cal B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.7.m7.1d">italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ caligraphic_B</annotation></semantics></math>. To <math alttext="\cal A" class="ltx_Math" display="inline" id="S1.p1.8.m8.1"><semantics id="S1.p1.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p1.8.m8.1.1" xref="S1.p1.8.m8.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S1.p1.8.m8.1b"><ci id="S1.p1.8.m8.1.1.cmml" xref="S1.p1.8.m8.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.8.m8.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.8.m8.1d">caligraphic_A</annotation></semantics></math> and <math alttext="\cal B" class="ltx_Math" display="inline" id="S1.p1.9.m9.1"><semantics id="S1.p1.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p1.9.m9.1.1" xref="S1.p1.9.m9.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.9.m9.1b"><ci id="S1.p1.9.m9.1.1.cmml" xref="S1.p1.9.m9.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.9.m9.1c">\cal B</annotation><annotation encoding="application/x-llamapun" id="S1.p1.9.m9.1d">caligraphic_B</annotation></semantics></math> there are canonically associated the spaces <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p1.10.m10.1"><semantics id="S1.p1.10.m10.1a"><msup id="S1.p1.10.m10.1.1" xref="S1.p1.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.10.m10.1.1.2" xref="S1.p1.10.m10.1.1.2.cmml">𝒜</mi><mi id="S1.p1.10.m10.1.1.3" xref="S1.p1.10.m10.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p1.10.m10.1b"><apply id="S1.p1.10.m10.1.1.cmml" xref="S1.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S1.p1.10.m10.1.1.1.cmml" xref="S1.p1.10.m10.1.1">superscript</csymbol><ci id="S1.p1.10.m10.1.1.2.cmml" xref="S1.p1.10.m10.1.1.2">𝒜</ci><ci id="S1.p1.10.m10.1.1.3.cmml" xref="S1.p1.10.m10.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.10.m10.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.10.m10.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p1.11.m11.1"><semantics id="S1.p1.11.m11.1a"><msup id="S1.p1.11.m11.1.1" xref="S1.p1.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.11.m11.1.1.2" xref="S1.p1.11.m11.1.1.2.cmml">ℬ</mi><mi id="S1.p1.11.m11.1.1.3" xref="S1.p1.11.m11.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p1.11.m11.1b"><apply id="S1.p1.11.m11.1.1.cmml" xref="S1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S1.p1.11.m11.1.1.1.cmml" xref="S1.p1.11.m11.1.1">superscript</csymbol><ci id="S1.p1.11.m11.1.1.2.cmml" xref="S1.p1.11.m11.1.1.2">ℬ</ci><ci id="S1.p1.11.m11.1.1.3.cmml" xref="S1.p1.11.m11.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.11.m11.1c">\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.11.m11.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, equipped both with the product topology and a <span class="ltx_text ltx_font_italic" id="S1.p1.19.3">shift operator</span>, here always denoted by <math alttext="T" class="ltx_Math" display="inline" id="S1.p1.12.m12.1"><semantics id="S1.p1.12.m12.1a"><mi id="S1.p1.12.m12.1.1" xref="S1.p1.12.m12.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S1.p1.12.m12.1b"><ci id="S1.p1.12.m12.1.1.cmml" xref="S1.p1.12.m12.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.12.m12.1c">T</annotation><annotation encoding="application/x-llamapun" id="S1.p1.12.m12.1d">italic_T</annotation></semantics></math>. All morphisms <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p1.13.m13.1"><semantics id="S1.p1.13.m13.1a"><mi id="S1.p1.13.m13.1.1" xref="S1.p1.13.m13.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.13.m13.1b"><ci id="S1.p1.13.m13.1.1.cmml" xref="S1.p1.13.m13.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.13.m13.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p1.13.m13.1d">italic_σ</annotation></semantics></math> in this paper are <span class="ltx_text ltx_font_italic" id="S1.p1.19.4">non-erasing</span> (i.e. none of the <math alttext="a_{k}\in\cal A" class="ltx_Math" display="inline" id="S1.p1.14.m14.1"><semantics id="S1.p1.14.m14.1a"><mrow id="S1.p1.14.m14.1.1" xref="S1.p1.14.m14.1.1.cmml"><msub id="S1.p1.14.m14.1.1.2" xref="S1.p1.14.m14.1.1.2.cmml"><mi id="S1.p1.14.m14.1.1.2.2" xref="S1.p1.14.m14.1.1.2.2.cmml">a</mi><mi id="S1.p1.14.m14.1.1.2.3" xref="S1.p1.14.m14.1.1.2.3.cmml">k</mi></msub><mo id="S1.p1.14.m14.1.1.1" xref="S1.p1.14.m14.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.14.m14.1.1.3" xref="S1.p1.14.m14.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.14.m14.1b"><apply id="S1.p1.14.m14.1.1.cmml" xref="S1.p1.14.m14.1.1"><in id="S1.p1.14.m14.1.1.1.cmml" xref="S1.p1.14.m14.1.1.1"></in><apply id="S1.p1.14.m14.1.1.2.cmml" xref="S1.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S1.p1.14.m14.1.1.2.1.cmml" xref="S1.p1.14.m14.1.1.2">subscript</csymbol><ci id="S1.p1.14.m14.1.1.2.2.cmml" xref="S1.p1.14.m14.1.1.2.2">𝑎</ci><ci id="S1.p1.14.m14.1.1.2.3.cmml" xref="S1.p1.14.m14.1.1.2.3">𝑘</ci></apply><ci id="S1.p1.14.m14.1.1.3.cmml" xref="S1.p1.14.m14.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.14.m14.1c">a_{k}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S1.p1.14.m14.1d">italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> are mapped to the empty word), and such <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p1.15.m15.1"><semantics id="S1.p1.15.m15.1a"><mi id="S1.p1.15.m15.1.1" xref="S1.p1.15.m15.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.15.m15.1b"><ci id="S1.p1.15.m15.1.1.cmml" xref="S1.p1.15.m15.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.15.m15.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p1.15.m15.1d">italic_σ</annotation></semantics></math> induces canonically a map <math alttext="\sigma^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p1.16.m16.1"><semantics id="S1.p1.16.m16.1a"><mrow id="S1.p1.16.m16.1.1" xref="S1.p1.16.m16.1.1.cmml"><msup id="S1.p1.16.m16.1.1.2" xref="S1.p1.16.m16.1.1.2.cmml"><mi id="S1.p1.16.m16.1.1.2.2" xref="S1.p1.16.m16.1.1.2.2.cmml">σ</mi><mi id="S1.p1.16.m16.1.1.2.3" xref="S1.p1.16.m16.1.1.2.3.cmml">ℤ</mi></msup><mo id="S1.p1.16.m16.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p1.16.m16.1.1.1.cmml">:</mo><mrow id="S1.p1.16.m16.1.1.3" xref="S1.p1.16.m16.1.1.3.cmml"><msup id="S1.p1.16.m16.1.1.3.2" xref="S1.p1.16.m16.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.16.m16.1.1.3.2.2" xref="S1.p1.16.m16.1.1.3.2.2.cmml">𝒜</mi><mi id="S1.p1.16.m16.1.1.3.2.3" xref="S1.p1.16.m16.1.1.3.2.3.cmml">ℤ</mi></msup><mo id="S1.p1.16.m16.1.1.3.1" stretchy="false" xref="S1.p1.16.m16.1.1.3.1.cmml">→</mo><msup id="S1.p1.16.m16.1.1.3.3" xref="S1.p1.16.m16.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.16.m16.1.1.3.3.2" xref="S1.p1.16.m16.1.1.3.3.2.cmml">ℬ</mi><mi id="S1.p1.16.m16.1.1.3.3.3" xref="S1.p1.16.m16.1.1.3.3.3.cmml">ℤ</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.16.m16.1b"><apply id="S1.p1.16.m16.1.1.cmml" xref="S1.p1.16.m16.1.1"><ci id="S1.p1.16.m16.1.1.1.cmml" xref="S1.p1.16.m16.1.1.1">:</ci><apply id="S1.p1.16.m16.1.1.2.cmml" xref="S1.p1.16.m16.1.1.2"><csymbol cd="ambiguous" id="S1.p1.16.m16.1.1.2.1.cmml" xref="S1.p1.16.m16.1.1.2">superscript</csymbol><ci id="S1.p1.16.m16.1.1.2.2.cmml" xref="S1.p1.16.m16.1.1.2.2">𝜎</ci><ci id="S1.p1.16.m16.1.1.2.3.cmml" xref="S1.p1.16.m16.1.1.2.3">ℤ</ci></apply><apply id="S1.p1.16.m16.1.1.3.cmml" xref="S1.p1.16.m16.1.1.3"><ci id="S1.p1.16.m16.1.1.3.1.cmml" xref="S1.p1.16.m16.1.1.3.1">→</ci><apply id="S1.p1.16.m16.1.1.3.2.cmml" xref="S1.p1.16.m16.1.1.3.2"><csymbol cd="ambiguous" id="S1.p1.16.m16.1.1.3.2.1.cmml" xref="S1.p1.16.m16.1.1.3.2">superscript</csymbol><ci id="S1.p1.16.m16.1.1.3.2.2.cmml" xref="S1.p1.16.m16.1.1.3.2.2">𝒜</ci><ci id="S1.p1.16.m16.1.1.3.2.3.cmml" xref="S1.p1.16.m16.1.1.3.2.3">ℤ</ci></apply><apply id="S1.p1.16.m16.1.1.3.3.cmml" xref="S1.p1.16.m16.1.1.3.3"><csymbol cd="ambiguous" id="S1.p1.16.m16.1.1.3.3.1.cmml" xref="S1.p1.16.m16.1.1.3.3">superscript</csymbol><ci id="S1.p1.16.m16.1.1.3.3.2.cmml" xref="S1.p1.16.m16.1.1.3.3.2">ℬ</ci><ci id="S1.p1.16.m16.1.1.3.3.3.cmml" xref="S1.p1.16.m16.1.1.3.3.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.16.m16.1c">\sigma^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.16.m16.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT : caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. But while <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p1.17.m17.1"><semantics id="S1.p1.17.m17.1a"><msup id="S1.p1.17.m17.1.1" xref="S1.p1.17.m17.1.1.cmml"><mi id="S1.p1.17.m17.1.1.2" xref="S1.p1.17.m17.1.1.2.cmml">σ</mi><mi id="S1.p1.17.m17.1.1.3" xref="S1.p1.17.m17.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p1.17.m17.1b"><apply id="S1.p1.17.m17.1.1.cmml" xref="S1.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S1.p1.17.m17.1.1.1.cmml" xref="S1.p1.17.m17.1.1">superscript</csymbol><ci id="S1.p1.17.m17.1.1.2.cmml" xref="S1.p1.17.m17.1.1.2">𝜎</ci><ci id="S1.p1.17.m17.1.1.3.cmml" xref="S1.p1.17.m17.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.17.m17.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.17.m17.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is continuous, in general it fails to be a “morphism of dynamical systems” from <math alttext="(\cal A^{\mathbb{Z}},T)" class="ltx_Math" display="inline" id="S1.p1.18.m18.2"><semantics id="S1.p1.18.m18.2a"><mrow id="S1.p1.18.m18.2.2.1" xref="S1.p1.18.m18.2.2.2.cmml"><mo id="S1.p1.18.m18.2.2.1.2" stretchy="false" xref="S1.p1.18.m18.2.2.2.cmml">(</mo><msup id="S1.p1.18.m18.2.2.1.1" xref="S1.p1.18.m18.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.18.m18.2.2.1.1.2" xref="S1.p1.18.m18.2.2.1.1.2.cmml">𝒜</mi><mi id="S1.p1.18.m18.2.2.1.1.3" xref="S1.p1.18.m18.2.2.1.1.3.cmml">ℤ</mi></msup><mo id="S1.p1.18.m18.2.2.1.3" xref="S1.p1.18.m18.2.2.2.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.18.m18.1.1" xref="S1.p1.18.m18.1.1.cmml">𝒯</mi><mo id="S1.p1.18.m18.2.2.1.4" stretchy="false" xref="S1.p1.18.m18.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.18.m18.2b"><interval closure="open" id="S1.p1.18.m18.2.2.2.cmml" xref="S1.p1.18.m18.2.2.1"><apply id="S1.p1.18.m18.2.2.1.1.cmml" 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xref="S1.p1.19.m19.1.1">𝒯</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.19.m19.2c">(\cal B^{\mathbb{Z}},T)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.19.m19.2d">( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT , caligraphic_T )</annotation></semantics></math> in the classical topological dynamics meaning, in that most of the time we have</p> <table class="ltx_equation ltx_eqn_table" id="S1.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(1.1)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="T\circ\sigma^{\mathbb{Z}}\,\,\neq\,\,\sigma^{\mathbb{Z}}\circ T\,." class="ltx_Math" display="block" id="S1.E1.m1.1"><semantics id="S1.E1.m1.1a"><mrow id="S1.E1.m1.1.1.1" xref="S1.E1.m1.1.1.1.1.cmml"><mrow id="S1.E1.m1.1.1.1.1" xref="S1.E1.m1.1.1.1.1.cmml"><mrow id="S1.E1.m1.1.1.1.1.2" xref="S1.E1.m1.1.1.1.1.2.cmml"><mi id="S1.E1.m1.1.1.1.1.2.2" xref="S1.E1.m1.1.1.1.1.2.2.cmml">T</mi><mo id="S1.E1.m1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S1.E1.m1.1.1.1.1.2.1.cmml">∘</mo><msup id="S1.E1.m1.1.1.1.1.2.3" xref="S1.E1.m1.1.1.1.1.2.3.cmml"><mi id="S1.E1.m1.1.1.1.1.2.3.2" xref="S1.E1.m1.1.1.1.1.2.3.2.cmml">σ</mi><mi id="S1.E1.m1.1.1.1.1.2.3.3" xref="S1.E1.m1.1.1.1.1.2.3.3.cmml">ℤ</mi></msup></mrow><mo id="S1.E1.m1.1.1.1.1.1" rspace="0.608em" xref="S1.E1.m1.1.1.1.1.1.cmml">≠</mo><mrow id="S1.E1.m1.1.1.1.1.3" xref="S1.E1.m1.1.1.1.1.3.cmml"><msup id="S1.E1.m1.1.1.1.1.3.2" xref="S1.E1.m1.1.1.1.1.3.2.cmml"><mi id="S1.E1.m1.1.1.1.1.3.2.2" xref="S1.E1.m1.1.1.1.1.3.2.2.cmml">σ</mi><mi id="S1.E1.m1.1.1.1.1.3.2.3" xref="S1.E1.m1.1.1.1.1.3.2.3.cmml">ℤ</mi></msup><mo id="S1.E1.m1.1.1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S1.E1.m1.1.1.1.1.3.1.cmml">∘</mo><mi id="S1.E1.m1.1.1.1.1.3.3" xref="S1.E1.m1.1.1.1.1.3.3.cmml">T</mi></mrow></mrow><mo id="S1.E1.m1.1.1.1.2" lspace="0.170em" xref="S1.E1.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.E1.m1.1b"><apply id="S1.E1.m1.1.1.1.1.cmml" xref="S1.E1.m1.1.1.1"><neq id="S1.E1.m1.1.1.1.1.1.cmml" xref="S1.E1.m1.1.1.1.1.1"></neq><apply id="S1.E1.m1.1.1.1.1.2.cmml" xref="S1.E1.m1.1.1.1.1.2"><compose id="S1.E1.m1.1.1.1.1.2.1.cmml" xref="S1.E1.m1.1.1.1.1.2.1"></compose><ci id="S1.E1.m1.1.1.1.1.2.2.cmml" xref="S1.E1.m1.1.1.1.1.2.2">𝑇</ci><apply id="S1.E1.m1.1.1.1.1.2.3.cmml" xref="S1.E1.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S1.E1.m1.1.1.1.1.2.3.1.cmml" xref="S1.E1.m1.1.1.1.1.2.3">superscript</csymbol><ci id="S1.E1.m1.1.1.1.1.2.3.2.cmml" xref="S1.E1.m1.1.1.1.1.2.3.2">𝜎</ci><ci id="S1.E1.m1.1.1.1.1.2.3.3.cmml" xref="S1.E1.m1.1.1.1.1.2.3.3">ℤ</ci></apply></apply><apply id="S1.E1.m1.1.1.1.1.3.cmml" xref="S1.E1.m1.1.1.1.1.3"><compose id="S1.E1.m1.1.1.1.1.3.1.cmml" xref="S1.E1.m1.1.1.1.1.3.1"></compose><apply id="S1.E1.m1.1.1.1.1.3.2.cmml" xref="S1.E1.m1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S1.E1.m1.1.1.1.1.3.2.1.cmml" xref="S1.E1.m1.1.1.1.1.3.2">superscript</csymbol><ci id="S1.E1.m1.1.1.1.1.3.2.2.cmml" xref="S1.E1.m1.1.1.1.1.3.2.2">𝜎</ci><ci id="S1.E1.m1.1.1.1.1.3.2.3.cmml" xref="S1.E1.m1.1.1.1.1.3.2.3">ℤ</ci></apply><ci id="S1.E1.m1.1.1.1.1.3.3.cmml" xref="S1.E1.m1.1.1.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.m1.1c">T\circ\sigma^{\mathbb{Z}}\,\,\neq\,\,\sigma^{\mathbb{Z}}\circ T\,.</annotation><annotation encoding="application/x-llamapun" id="S1.E1.m1.1d">italic_T ∘ italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ≠ italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ∘ italic_T .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p1.22">This creates a number of well known technical problems; for instance any non-empty, closed, shift-invariant subset (called a <span class="ltx_text ltx_font_italic" id="S1.p1.22.1">subshift</span>) <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p1.20.m1.1"><semantics id="S1.p1.20.m1.1a"><mrow id="S1.p1.20.m1.1.1" xref="S1.p1.20.m1.1.1.cmml"><mi id="S1.p1.20.m1.1.1.2" xref="S1.p1.20.m1.1.1.2.cmml">X</mi><mo id="S1.p1.20.m1.1.1.1" xref="S1.p1.20.m1.1.1.1.cmml">⊆</mo><msup id="S1.p1.20.m1.1.1.3" xref="S1.p1.20.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p1.20.m1.1.1.3.2" xref="S1.p1.20.m1.1.1.3.2.cmml">𝒜</mi><mi id="S1.p1.20.m1.1.1.3.3" xref="S1.p1.20.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.20.m1.1b"><apply id="S1.p1.20.m1.1.1.cmml" xref="S1.p1.20.m1.1.1"><subset id="S1.p1.20.m1.1.1.1.cmml" xref="S1.p1.20.m1.1.1.1"></subset><ci id="S1.p1.20.m1.1.1.2.cmml" xref="S1.p1.20.m1.1.1.2">𝑋</ci><apply id="S1.p1.20.m1.1.1.3.cmml" xref="S1.p1.20.m1.1.1.3"><csymbol cd="ambiguous" id="S1.p1.20.m1.1.1.3.1.cmml" xref="S1.p1.20.m1.1.1.3">superscript</csymbol><ci id="S1.p1.20.m1.1.1.3.2.cmml" xref="S1.p1.20.m1.1.1.3.2">𝒜</ci><ci id="S1.p1.20.m1.1.1.3.3.cmml" xref="S1.p1.20.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.20.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.20.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> has <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p1.21.m2.1"><semantics id="S1.p1.21.m2.1a"><msup id="S1.p1.21.m2.1.1" xref="S1.p1.21.m2.1.1.cmml"><mi id="S1.p1.21.m2.1.1.2" xref="S1.p1.21.m2.1.1.2.cmml">σ</mi><mi id="S1.p1.21.m2.1.1.3" xref="S1.p1.21.m2.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p1.21.m2.1b"><apply id="S1.p1.21.m2.1.1.cmml" xref="S1.p1.21.m2.1.1"><csymbol cd="ambiguous" id="S1.p1.21.m2.1.1.1.cmml" xref="S1.p1.21.m2.1.1">superscript</csymbol><ci id="S1.p1.21.m2.1.1.2.cmml" xref="S1.p1.21.m2.1.1.2">𝜎</ci><ci id="S1.p1.21.m2.1.1.3.cmml" xref="S1.p1.21.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.21.m2.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.21.m2.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>-image that is in general not shift-invariant, and thus not a subshift by itself. Still, there is a well defined image subshift (see Definition-Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm2" title="Definition-Remark 2.2. ‣ 2.2. “Not so standard” basic facts and terminology ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.2</span></a> below), denoted here by <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S1.p1.22.m3.1"><semantics id="S1.p1.22.m3.1a"><mrow id="S1.p1.22.m3.1.2" xref="S1.p1.22.m3.1.2.cmml"><mi id="S1.p1.22.m3.1.2.2" xref="S1.p1.22.m3.1.2.2.cmml">σ</mi><mo id="S1.p1.22.m3.1.2.1" xref="S1.p1.22.m3.1.2.1.cmml">⁢</mo><mrow id="S1.p1.22.m3.1.2.3.2" xref="S1.p1.22.m3.1.2.cmml"><mo id="S1.p1.22.m3.1.2.3.2.1" stretchy="false" xref="S1.p1.22.m3.1.2.cmml">(</mo><mi id="S1.p1.22.m3.1.1" xref="S1.p1.22.m3.1.1.cmml">X</mi><mo id="S1.p1.22.m3.1.2.3.2.2" stretchy="false" xref="S1.p1.22.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.22.m3.1b"><apply id="S1.p1.22.m3.1.2.cmml" xref="S1.p1.22.m3.1.2"><times id="S1.p1.22.m3.1.2.1.cmml" xref="S1.p1.22.m3.1.2.1"></times><ci id="S1.p1.22.m3.1.2.2.cmml" xref="S1.p1.22.m3.1.2.2">𝜎</ci><ci id="S1.p1.22.m3.1.1.cmml" xref="S1.p1.22.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.22.m3.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.22.m3.1d">italic_σ ( italic_X )</annotation></semantics></math>, which has been studied previously in many occasions.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.13">The focus of this paper is on the set <math alttext="\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mrow id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.1.m1.1.1.3" xref="S1.p2.1.m1.1.1.3.cmml">ℳ</mi><mo id="S1.p2.1.m1.1.1.2" xref="S1.p2.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.p2.1.m1.1.1.1.1" xref="S1.p2.1.m1.1.1.1.1.1.cmml"><mo id="S1.p2.1.m1.1.1.1.1.2" stretchy="false" xref="S1.p2.1.m1.1.1.1.1.1.cmml">(</mo><msup id="S1.p2.1.m1.1.1.1.1.1" xref="S1.p2.1.m1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.1.m1.1.1.1.1.1.2" xref="S1.p2.1.m1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S1.p2.1.m1.1.1.1.1.1.3" xref="S1.p2.1.m1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S1.p2.1.m1.1.1.1.1.3" stretchy="false" xref="S1.p2.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><apply id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1"><times id="S1.p2.1.m1.1.1.2.cmml" xref="S1.p2.1.m1.1.1.2"></times><ci id="S1.p2.1.m1.1.1.3.cmml" xref="S1.p2.1.m1.1.1.3">ℳ</ci><apply id="S1.p2.1.m1.1.1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p2.1.m1.1.1.1.1.1.1.cmml" xref="S1.p2.1.m1.1.1.1.1">superscript</csymbol><ci id="S1.p2.1.m1.1.1.1.1.1.2.cmml" xref="S1.p2.1.m1.1.1.1.1.1.2">𝒜</ci><ci id="S1.p2.1.m1.1.1.1.1.1.3.cmml" xref="S1.p2.1.m1.1.1.1.1.1.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> of measures <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><msup id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">𝒜</mi><mi id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p2.3.m3.1.1.1.cmml" xref="S1.p2.3.m3.1.1">superscript</csymbol><ci id="S1.p2.3.m3.1.1.2.cmml" xref="S1.p2.3.m3.1.1.2">𝒜</ci><ci id="S1.p2.3.m3.1.1.3.cmml" xref="S1.p2.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> that are <span class="ltx_text ltx_font_italic" id="S1.p2.13.1">invariant</span>, which means that <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">italic_μ</annotation></semantics></math> is a finite Borel measure, and that the shift operator preserves <math alttext="\mu\," class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><ci id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">\mu\,</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_μ</annotation></semantics></math>. The map <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p2.6.m6.1"><semantics id="S1.p2.6.m6.1a"><msup id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml"><mi id="S1.p2.6.m6.1.1.2" xref="S1.p2.6.m6.1.1.2.cmml">σ</mi><mi id="S1.p2.6.m6.1.1.3" xref="S1.p2.6.m6.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.1b"><apply id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S1.p2.6.m6.1.1.1.cmml" xref="S1.p2.6.m6.1.1">superscript</csymbol><ci id="S1.p2.6.m6.1.1.2.cmml" xref="S1.p2.6.m6.1.1.2">𝜎</ci><ci id="S1.p2.6.m6.1.1.3.cmml" xref="S1.p2.6.m6.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> gives us directly the classical “push-forward” measure <math alttext="\mu_{*}" class="ltx_Math" display="inline" id="S1.p2.7.m7.1"><semantics id="S1.p2.7.m7.1a"><msub id="S1.p2.7.m7.1.1" xref="S1.p2.7.m7.1.1.cmml"><mi id="S1.p2.7.m7.1.1.2" xref="S1.p2.7.m7.1.1.2.cmml">μ</mi><mo id="S1.p2.7.m7.1.1.3" xref="S1.p2.7.m7.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S1.p2.7.m7.1b"><apply id="S1.p2.7.m7.1.1.cmml" xref="S1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S1.p2.7.m7.1.1.1.cmml" xref="S1.p2.7.m7.1.1">subscript</csymbol><ci id="S1.p2.7.m7.1.1.2.cmml" xref="S1.p2.7.m7.1.1.2">𝜇</ci><times id="S1.p2.7.m7.1.1.3.cmml" xref="S1.p2.7.m7.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.7.m7.1c">\mu_{*}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.7.m7.1d">italic_μ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p2.8.m8.1"><semantics id="S1.p2.8.m8.1a"><msup id="S1.p2.8.m8.1.1" xref="S1.p2.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.8.m8.1.1.2" xref="S1.p2.8.m8.1.1.2.cmml">ℬ</mi><mi id="S1.p2.8.m8.1.1.3" xref="S1.p2.8.m8.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p2.8.m8.1b"><apply id="S1.p2.8.m8.1.1.cmml" xref="S1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S1.p2.8.m8.1.1.1.cmml" xref="S1.p2.8.m8.1.1">superscript</csymbol><ci id="S1.p2.8.m8.1.1.2.cmml" xref="S1.p2.8.m8.1.1.2">ℬ</ci><ci id="S1.p2.8.m8.1.1.3.cmml" xref="S1.p2.8.m8.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.8.m8.1c">\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.8.m8.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, but because of (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S1.E1" title="In 1. Introduction ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">1.1</span></a>) this measure will almost never be invariant, as it will not be preserved by the shift operator on <math alttext="\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p2.9.m9.1"><semantics id="S1.p2.9.m9.1a"><msup id="S1.p2.9.m9.1.1" xref="S1.p2.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.9.m9.1.1.2" xref="S1.p2.9.m9.1.1.2.cmml">ℬ</mi><mi id="S1.p2.9.m9.1.1.3" xref="S1.p2.9.m9.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p2.9.m9.1b"><apply id="S1.p2.9.m9.1.1.cmml" xref="S1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S1.p2.9.m9.1.1.1.cmml" xref="S1.p2.9.m9.1.1">superscript</csymbol><ci id="S1.p2.9.m9.1.1.2.cmml" xref="S1.p2.9.m9.1.1.2">ℬ</ci><ci id="S1.p2.9.m9.1.1.3.cmml" xref="S1.p2.9.m9.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.9.m9.1c">\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.9.m9.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Nevertheless, any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p2.10.m10.1"><semantics id="S1.p2.10.m10.1a"><mi id="S1.p2.10.m10.1.1" xref="S1.p2.10.m10.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.10.m10.1b"><ci id="S1.p2.10.m10.1.1.cmml" xref="S1.p2.10.m10.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.10.m10.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p2.10.m10.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p2.11.m11.1"><semantics id="S1.p2.11.m11.1a"><msup id="S1.p2.11.m11.1.1" xref="S1.p2.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.11.m11.1.1.2" xref="S1.p2.11.m11.1.1.2.cmml">𝒜</mi><mi id="S1.p2.11.m11.1.1.3" xref="S1.p2.11.m11.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p2.11.m11.1b"><apply id="S1.p2.11.m11.1.1.cmml" xref="S1.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S1.p2.11.m11.1.1.1.cmml" xref="S1.p2.11.m11.1.1">superscript</csymbol><ci id="S1.p2.11.m11.1.1.2.cmml" xref="S1.p2.11.m11.1.1.2">𝒜</ci><ci id="S1.p2.11.m11.1.1.3.cmml" xref="S1.p2.11.m11.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.11.m11.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.11.m11.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> can canonically be “transferred” by <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p2.12.m12.1"><semantics id="S1.p2.12.m12.1a"><mi id="S1.p2.12.m12.1.1" xref="S1.p2.12.m12.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.12.m12.1b"><ci id="S1.p2.12.m12.1.1.cmml" xref="S1.p2.12.m12.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.12.m12.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p2.12.m12.1d">italic_σ</annotation></semantics></math> to define an invariant measure on <math alttext="\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p2.13.m13.1"><semantics id="S1.p2.13.m13.1a"><msup id="S1.p2.13.m13.1.1" xref="S1.p2.13.m13.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.13.m13.1.1.2" xref="S1.p2.13.m13.1.1.2.cmml">ℬ</mi><mi id="S1.p2.13.m13.1.1.3" xref="S1.p2.13.m13.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p2.13.m13.1b"><apply id="S1.p2.13.m13.1.1.cmml" xref="S1.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S1.p2.13.m13.1.1.1.cmml" xref="S1.p2.13.m13.1.1">superscript</csymbol><ci id="S1.p2.13.m13.1.1.2.cmml" xref="S1.p2.13.m13.1.1.2">ℬ</ci><ci id="S1.p2.13.m13.1.1.3.cmml" xref="S1.p2.13.m13.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.13.m13.1c">\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.13.m13.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Setting up properly and studying carefully this <span class="ltx_text ltx_font_italic" id="S1.p2.13.2">measure transfer map</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})" class="ltx_Math" display="block" id="S1.Ex1.m1.2"><semantics id="S1.Ex1.m1.2a"><mrow id="S1.Ex1.m1.2.2" xref="S1.Ex1.m1.2.2.cmml"><mrow id="S1.Ex1.m1.2.2.4" xref="S1.Ex1.m1.2.2.4.cmml"><mi id="S1.Ex1.m1.2.2.4.2" xref="S1.Ex1.m1.2.2.4.2.cmml">σ</mi><mo id="S1.Ex1.m1.2.2.4.1" xref="S1.Ex1.m1.2.2.4.1.cmml">⁢</mo><mi id="S1.Ex1.m1.2.2.4.3" xref="S1.Ex1.m1.2.2.4.3.cmml">M</mi></mrow><mo id="S1.Ex1.m1.2.2.3" lspace="0.278em" rspace="0.278em" xref="S1.Ex1.m1.2.2.3.cmml">:</mo><mrow id="S1.Ex1.m1.2.2.2" xref="S1.Ex1.m1.2.2.2.cmml"><mrow id="S1.Ex1.m1.1.1.1.1" xref="S1.Ex1.m1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex1.m1.1.1.1.1.3" xref="S1.Ex1.m1.1.1.1.1.3.cmml">ℳ</mi><mo id="S1.Ex1.m1.1.1.1.1.2" xref="S1.Ex1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.1.1.1.1.1.1" xref="S1.Ex1.m1.1.1.1.1.1.1.1.cmml"><mo id="S1.Ex1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S1.Ex1.m1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S1.Ex1.m1.1.1.1.1.1.1.1" xref="S1.Ex1.m1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex1.m1.1.1.1.1.1.1.1.2" xref="S1.Ex1.m1.1.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S1.Ex1.m1.1.1.1.1.1.1.1.3" xref="S1.Ex1.m1.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S1.Ex1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S1.Ex1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Ex1.m1.2.2.2.3" stretchy="false" xref="S1.Ex1.m1.2.2.2.3.cmml">→</mo><mrow id="S1.Ex1.m1.2.2.2.2" xref="S1.Ex1.m1.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex1.m1.2.2.2.2.3" xref="S1.Ex1.m1.2.2.2.2.3.cmml">ℳ</mi><mo id="S1.Ex1.m1.2.2.2.2.2" xref="S1.Ex1.m1.2.2.2.2.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.2.2.2.2.1.1" xref="S1.Ex1.m1.2.2.2.2.1.1.1.cmml"><mo id="S1.Ex1.m1.2.2.2.2.1.1.2" stretchy="false" xref="S1.Ex1.m1.2.2.2.2.1.1.1.cmml">(</mo><msup id="S1.Ex1.m1.2.2.2.2.1.1.1" xref="S1.Ex1.m1.2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex1.m1.2.2.2.2.1.1.1.2" xref="S1.Ex1.m1.2.2.2.2.1.1.1.2.cmml">ℬ</mi><mi id="S1.Ex1.m1.2.2.2.2.1.1.1.3" xref="S1.Ex1.m1.2.2.2.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S1.Ex1.m1.2.2.2.2.1.1.3" stretchy="false" xref="S1.Ex1.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex1.m1.2b"><apply id="S1.Ex1.m1.2.2.cmml" xref="S1.Ex1.m1.2.2"><ci id="S1.Ex1.m1.2.2.3.cmml" xref="S1.Ex1.m1.2.2.3">:</ci><apply id="S1.Ex1.m1.2.2.4.cmml" xref="S1.Ex1.m1.2.2.4"><times id="S1.Ex1.m1.2.2.4.1.cmml" xref="S1.Ex1.m1.2.2.4.1"></times><ci id="S1.Ex1.m1.2.2.4.2.cmml" xref="S1.Ex1.m1.2.2.4.2">𝜎</ci><ci id="S1.Ex1.m1.2.2.4.3.cmml" xref="S1.Ex1.m1.2.2.4.3">𝑀</ci></apply><apply id="S1.Ex1.m1.2.2.2.cmml" xref="S1.Ex1.m1.2.2.2"><ci id="S1.Ex1.m1.2.2.2.3.cmml" xref="S1.Ex1.m1.2.2.2.3">→</ci><apply id="S1.Ex1.m1.1.1.1.1.cmml" xref="S1.Ex1.m1.1.1.1.1"><times id="S1.Ex1.m1.1.1.1.1.2.cmml" xref="S1.Ex1.m1.1.1.1.1.2"></times><ci id="S1.Ex1.m1.1.1.1.1.3.cmml" xref="S1.Ex1.m1.1.1.1.1.3">ℳ</ci><apply id="S1.Ex1.m1.1.1.1.1.1.1.1.cmml" xref="S1.Ex1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Ex1.m1.1.1.1.1.1.1.1.1.cmml" xref="S1.Ex1.m1.1.1.1.1.1.1">superscript</csymbol><ci id="S1.Ex1.m1.1.1.1.1.1.1.1.2.cmml" xref="S1.Ex1.m1.1.1.1.1.1.1.1.2">𝒜</ci><ci id="S1.Ex1.m1.1.1.1.1.1.1.1.3.cmml" xref="S1.Ex1.m1.1.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S1.Ex1.m1.2.2.2.2.cmml" xref="S1.Ex1.m1.2.2.2.2"><times id="S1.Ex1.m1.2.2.2.2.2.cmml" xref="S1.Ex1.m1.2.2.2.2.2"></times><ci id="S1.Ex1.m1.2.2.2.2.3.cmml" xref="S1.Ex1.m1.2.2.2.2.3">ℳ</ci><apply id="S1.Ex1.m1.2.2.2.2.1.1.1.cmml" xref="S1.Ex1.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S1.Ex1.m1.2.2.2.2.1.1.1.1.cmml" xref="S1.Ex1.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S1.Ex1.m1.2.2.2.2.1.1.1.2.cmml" xref="S1.Ex1.m1.2.2.2.2.1.1.1.2">ℬ</ci><ci id="S1.Ex1.m1.2.2.2.2.1.1.1.3.cmml" xref="S1.Ex1.m1.2.2.2.2.1.1.1.3">ℤ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex1.m1.2c">\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S1.Ex1.m1.2d">italic_σ italic_M : caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p2.14">induced by any non-erasing free monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S1.p2.14.m1.1"><semantics id="S1.p2.14.m1.1a"><mrow id="S1.p2.14.m1.1.1" xref="S1.p2.14.m1.1.1.cmml"><mi id="S1.p2.14.m1.1.1.2" xref="S1.p2.14.m1.1.1.2.cmml">σ</mi><mo id="S1.p2.14.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p2.14.m1.1.1.1.cmml">:</mo><mrow id="S1.p2.14.m1.1.1.3" xref="S1.p2.14.m1.1.1.3.cmml"><msup id="S1.p2.14.m1.1.1.3.2" xref="S1.p2.14.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.14.m1.1.1.3.2.2" xref="S1.p2.14.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.p2.14.m1.1.1.3.2.3" xref="S1.p2.14.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.p2.14.m1.1.1.3.1" stretchy="false" xref="S1.p2.14.m1.1.1.3.1.cmml">→</mo><msup id="S1.p2.14.m1.1.1.3.3" xref="S1.p2.14.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p2.14.m1.1.1.3.3.2" xref="S1.p2.14.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S1.p2.14.m1.1.1.3.3.3" xref="S1.p2.14.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.14.m1.1b"><apply id="S1.p2.14.m1.1.1.cmml" xref="S1.p2.14.m1.1.1"><ci id="S1.p2.14.m1.1.1.1.cmml" xref="S1.p2.14.m1.1.1.1">:</ci><ci id="S1.p2.14.m1.1.1.2.cmml" xref="S1.p2.14.m1.1.1.2">𝜎</ci><apply id="S1.p2.14.m1.1.1.3.cmml" xref="S1.p2.14.m1.1.1.3"><ci id="S1.p2.14.m1.1.1.3.1.cmml" xref="S1.p2.14.m1.1.1.3.1">→</ci><apply id="S1.p2.14.m1.1.1.3.2.cmml" xref="S1.p2.14.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.p2.14.m1.1.1.3.2.1.cmml" xref="S1.p2.14.m1.1.1.3.2">superscript</csymbol><ci id="S1.p2.14.m1.1.1.3.2.2.cmml" xref="S1.p2.14.m1.1.1.3.2.2">𝒜</ci><times id="S1.p2.14.m1.1.1.3.2.3.cmml" xref="S1.p2.14.m1.1.1.3.2.3"></times></apply><apply id="S1.p2.14.m1.1.1.3.3.cmml" xref="S1.p2.14.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.p2.14.m1.1.1.3.3.1.cmml" xref="S1.p2.14.m1.1.1.3.3">superscript</csymbol><ci id="S1.p2.14.m1.1.1.3.3.2.cmml" xref="S1.p2.14.m1.1.1.3.3.2">ℬ</ci><times id="S1.p2.14.m1.1.1.3.3.3.cmml" xref="S1.p2.14.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.14.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.14.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is the main purpose of this paper. We show (see Subsection <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.SS4" title="3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.4</span></a> and Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm5" title="Lemma 5.5. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.5</span></a>):</p> </div> <div class="ltx_theorem ltx_theorem_prop" id="S1.Thmthm1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmthm1.1.1.1">Proposition 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmthm1.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmthm1.p1"> <p class="ltx_p" id="S1.Thmthm1.p1.5"><span class="ltx_text ltx_font_italic" id="S1.Thmthm1.p1.5.5">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S1.Thmthm1.p1.1.1.m1.1"><semantics id="S1.Thmthm1.p1.1.1.m1.1a"><mrow id="S1.Thmthm1.p1.1.1.m1.1.1" xref="S1.Thmthm1.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmthm1.p1.1.1.m1.1.1.2" xref="S1.Thmthm1.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S1.Thmthm1.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm1.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmthm1.p1.1.1.m1.1.1.3" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.cmml"><msup id="S1.Thmthm1.p1.1.1.m1.1.1.3.2" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm1.p1.1.1.m1.1.1.3.2.2" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.Thmthm1.p1.1.1.m1.1.1.3.2.3" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.Thmthm1.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S1.Thmthm1.p1.1.1.m1.1.1.3.3" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm1.p1.1.1.m1.1.1.3.3.2" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S1.Thmthm1.p1.1.1.m1.1.1.3.3.3" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm1.p1.1.1.m1.1b"><apply id="S1.Thmthm1.p1.1.1.m1.1.1.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1"><ci id="S1.Thmthm1.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.1">:</ci><ci id="S1.Thmthm1.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S1.Thmthm1.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3"><ci id="S1.Thmthm1.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.1">→</ci><apply id="S1.Thmthm1.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmthm1.p1.1.1.m1.1.1.3.2.1.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.Thmthm1.p1.1.1.m1.1.1.3.2.2.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S1.Thmthm1.p1.1.1.m1.1.1.3.2.3.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S1.Thmthm1.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmthm1.p1.1.1.m1.1.1.3.3.1.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.Thmthm1.p1.1.1.m1.1.1.3.3.2.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S1.Thmthm1.p1.1.1.m1.1.1.3.3.3.cmml" xref="S1.Thmthm1.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm1.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm1.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be any non-erasing morphism, and let <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.Thmthm1.p1.2.2.m2.1"><semantics id="S1.Thmthm1.p1.2.2.m2.1a"><mrow id="S1.Thmthm1.p1.2.2.m2.1.1" xref="S1.Thmthm1.p1.2.2.m2.1.1.cmml"><mi id="S1.Thmthm1.p1.2.2.m2.1.1.2" xref="S1.Thmthm1.p1.2.2.m2.1.1.2.cmml">X</mi><mo id="S1.Thmthm1.p1.2.2.m2.1.1.1" xref="S1.Thmthm1.p1.2.2.m2.1.1.1.cmml">⊆</mo><msup id="S1.Thmthm1.p1.2.2.m2.1.1.3" xref="S1.Thmthm1.p1.2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm1.p1.2.2.m2.1.1.3.2" xref="S1.Thmthm1.p1.2.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S1.Thmthm1.p1.2.2.m2.1.1.3.3" xref="S1.Thmthm1.p1.2.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm1.p1.2.2.m2.1b"><apply id="S1.Thmthm1.p1.2.2.m2.1.1.cmml" xref="S1.Thmthm1.p1.2.2.m2.1.1"><subset id="S1.Thmthm1.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmthm1.p1.2.2.m2.1.1.1"></subset><ci id="S1.Thmthm1.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmthm1.p1.2.2.m2.1.1.2">𝑋</ci><apply id="S1.Thmthm1.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmthm1.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.Thmthm1.p1.2.2.m2.1.1.3.1.cmml" xref="S1.Thmthm1.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S1.Thmthm1.p1.2.2.m2.1.1.3.2.cmml" xref="S1.Thmthm1.p1.2.2.m2.1.1.3.2">𝒜</ci><ci id="S1.Thmthm1.p1.2.2.m2.1.1.3.3.cmml" xref="S1.Thmthm1.p1.2.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm1.p1.2.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm1.p1.2.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> be any subshift over <math alttext="\cal A" class="ltx_Math" display="inline" id="S1.Thmthm1.p1.3.3.m3.1"><semantics id="S1.Thmthm1.p1.3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm1.p1.3.3.m3.1.1" xref="S1.Thmthm1.p1.3.3.m3.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm1.p1.3.3.m3.1b"><ci id="S1.Thmthm1.p1.3.3.m3.1.1.cmml" xref="S1.Thmthm1.p1.3.3.m3.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm1.p1.3.3.m3.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm1.p1.3.3.m3.1d">caligraphic_A</annotation></semantics></math>, with image subshift <math alttext="Y:=\sigma(X)\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.Thmthm1.p1.4.4.m4.1"><semantics id="S1.Thmthm1.p1.4.4.m4.1a"><mrow id="S1.Thmthm1.p1.4.4.m4.1.2" xref="S1.Thmthm1.p1.4.4.m4.1.2.cmml"><mi id="S1.Thmthm1.p1.4.4.m4.1.2.2" xref="S1.Thmthm1.p1.4.4.m4.1.2.2.cmml">Y</mi><mo id="S1.Thmthm1.p1.4.4.m4.1.2.3" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm1.p1.4.4.m4.1.2.3.cmml">:=</mo><mrow id="S1.Thmthm1.p1.4.4.m4.1.2.4" xref="S1.Thmthm1.p1.4.4.m4.1.2.4.cmml"><mi id="S1.Thmthm1.p1.4.4.m4.1.2.4.2" xref="S1.Thmthm1.p1.4.4.m4.1.2.4.2.cmml">σ</mi><mo id="S1.Thmthm1.p1.4.4.m4.1.2.4.1" xref="S1.Thmthm1.p1.4.4.m4.1.2.4.1.cmml">⁢</mo><mrow id="S1.Thmthm1.p1.4.4.m4.1.2.4.3.2" xref="S1.Thmthm1.p1.4.4.m4.1.2.4.cmml"><mo id="S1.Thmthm1.p1.4.4.m4.1.2.4.3.2.1" stretchy="false" xref="S1.Thmthm1.p1.4.4.m4.1.2.4.cmml">(</mo><mi id="S1.Thmthm1.p1.4.4.m4.1.1" xref="S1.Thmthm1.p1.4.4.m4.1.1.cmml">X</mi><mo id="S1.Thmthm1.p1.4.4.m4.1.2.4.3.2.2" stretchy="false" xref="S1.Thmthm1.p1.4.4.m4.1.2.4.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm1.p1.4.4.m4.1.2.5" xref="S1.Thmthm1.p1.4.4.m4.1.2.5.cmml">⊆</mo><msup id="S1.Thmthm1.p1.4.4.m4.1.2.6" xref="S1.Thmthm1.p1.4.4.m4.1.2.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm1.p1.4.4.m4.1.2.6.2" xref="S1.Thmthm1.p1.4.4.m4.1.2.6.2.cmml">ℬ</mi><mi id="S1.Thmthm1.p1.4.4.m4.1.2.6.3" xref="S1.Thmthm1.p1.4.4.m4.1.2.6.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm1.p1.4.4.m4.1b"><apply id="S1.Thmthm1.p1.4.4.m4.1.2.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2"><and id="S1.Thmthm1.p1.4.4.m4.1.2a.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2"></and><apply id="S1.Thmthm1.p1.4.4.m4.1.2b.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2"><csymbol cd="latexml" id="S1.Thmthm1.p1.4.4.m4.1.2.3.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.3">assign</csymbol><ci id="S1.Thmthm1.p1.4.4.m4.1.2.2.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.2">𝑌</ci><apply id="S1.Thmthm1.p1.4.4.m4.1.2.4.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.4"><times id="S1.Thmthm1.p1.4.4.m4.1.2.4.1.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.4.1"></times><ci id="S1.Thmthm1.p1.4.4.m4.1.2.4.2.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.4.2">𝜎</ci><ci id="S1.Thmthm1.p1.4.4.m4.1.1.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.1">𝑋</ci></apply></apply><apply id="S1.Thmthm1.p1.4.4.m4.1.2c.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2"><subset id="S1.Thmthm1.p1.4.4.m4.1.2.5.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.5"></subset><share href="https://arxiv.org/html/2211.11234v4#S1.Thmthm1.p1.4.4.m4.1.2.4.cmml" id="S1.Thmthm1.p1.4.4.m4.1.2d.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2"></share><apply id="S1.Thmthm1.p1.4.4.m4.1.2.6.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.6"><csymbol cd="ambiguous" id="S1.Thmthm1.p1.4.4.m4.1.2.6.1.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.6">superscript</csymbol><ci id="S1.Thmthm1.p1.4.4.m4.1.2.6.2.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.6.2">ℬ</ci><ci id="S1.Thmthm1.p1.4.4.m4.1.2.6.3.cmml" xref="S1.Thmthm1.p1.4.4.m4.1.2.6.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm1.p1.4.4.m4.1c">Y:=\sigma(X)\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm1.p1.4.4.m4.1d">italic_Y := italic_σ ( italic_X ) ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Then the induced measure transfer map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S1.Thmthm1.p1.5.5.m5.1"><semantics id="S1.Thmthm1.p1.5.5.m5.1a"><mrow id="S1.Thmthm1.p1.5.5.m5.1.1" xref="S1.Thmthm1.p1.5.5.m5.1.1.cmml"><mi id="S1.Thmthm1.p1.5.5.m5.1.1.2" xref="S1.Thmthm1.p1.5.5.m5.1.1.2.cmml">σ</mi><mo id="S1.Thmthm1.p1.5.5.m5.1.1.1" xref="S1.Thmthm1.p1.5.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.Thmthm1.p1.5.5.m5.1.1.3" xref="S1.Thmthm1.p1.5.5.m5.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm1.p1.5.5.m5.1b"><apply id="S1.Thmthm1.p1.5.5.m5.1.1.cmml" xref="S1.Thmthm1.p1.5.5.m5.1.1"><times id="S1.Thmthm1.p1.5.5.m5.1.1.1.cmml" xref="S1.Thmthm1.p1.5.5.m5.1.1.1"></times><ci id="S1.Thmthm1.p1.5.5.m5.1.1.2.cmml" xref="S1.Thmthm1.p1.5.5.m5.1.1.2">𝜎</ci><ci id="S1.Thmthm1.p1.5.5.m5.1.1.3.cmml" xref="S1.Thmthm1.p1.5.5.m5.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm1.p1.5.5.m5.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm1.p1.5.5.m5.1d">italic_σ italic_M</annotation></semantics></math> restricts/co-restricts to a well defined map</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma_{X}M:\cal M(X)\to\cal M(Y)\,,\,\,\mu\mapsto\mu^{\sigma}" class="ltx_Math" display="block" id="S1.Ex2.m1.4"><semantics id="S1.Ex2.m1.4a"><mrow id="S1.Ex2.m1.4.4" xref="S1.Ex2.m1.4.4.cmml"><mrow id="S1.Ex2.m1.4.4.4" xref="S1.Ex2.m1.4.4.4.cmml"><msub id="S1.Ex2.m1.4.4.4.2" xref="S1.Ex2.m1.4.4.4.2.cmml"><mi id="S1.Ex2.m1.4.4.4.2.2" xref="S1.Ex2.m1.4.4.4.2.2.cmml">σ</mi><mi id="S1.Ex2.m1.4.4.4.2.3" xref="S1.Ex2.m1.4.4.4.2.3.cmml">X</mi></msub><mo id="S1.Ex2.m1.4.4.4.1" xref="S1.Ex2.m1.4.4.4.1.cmml">⁢</mo><mi id="S1.Ex2.m1.4.4.4.3" xref="S1.Ex2.m1.4.4.4.3.cmml">M</mi></mrow><mo id="S1.Ex2.m1.4.4.3" lspace="0.278em" rspace="0.278em" xref="S1.Ex2.m1.4.4.3.cmml">:</mo><mrow id="S1.Ex2.m1.4.4.2.2" xref="S1.Ex2.m1.4.4.2.3.cmml"><mrow id="S1.Ex2.m1.3.3.1.1.1" xref="S1.Ex2.m1.3.3.1.1.1.cmml"><mrow id="S1.Ex2.m1.3.3.1.1.1.2" xref="S1.Ex2.m1.3.3.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.3.3.1.1.1.2.2" xref="S1.Ex2.m1.3.3.1.1.1.2.2.cmml">ℳ</mi><mo id="S1.Ex2.m1.3.3.1.1.1.2.1" xref="S1.Ex2.m1.3.3.1.1.1.2.1.cmml">⁢</mo><mrow id="S1.Ex2.m1.3.3.1.1.1.2.3.2" xref="S1.Ex2.m1.3.3.1.1.1.2.cmml"><mo id="S1.Ex2.m1.3.3.1.1.1.2.3.2.1" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.1.1" xref="S1.Ex2.m1.1.1.cmml">𝒳</mi><mo id="S1.Ex2.m1.3.3.1.1.1.2.3.2.2" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.Ex2.m1.3.3.1.1.1.1" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.1.1.cmml">→</mo><mrow id="S1.Ex2.m1.3.3.1.1.1.3" xref="S1.Ex2.m1.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.3.3.1.1.1.3.2" xref="S1.Ex2.m1.3.3.1.1.1.3.2.cmml">ℳ</mi><mo id="S1.Ex2.m1.3.3.1.1.1.3.1" xref="S1.Ex2.m1.3.3.1.1.1.3.1.cmml">⁢</mo><mrow id="S1.Ex2.m1.3.3.1.1.1.3.3.2" xref="S1.Ex2.m1.3.3.1.1.1.3.cmml"><mo id="S1.Ex2.m1.3.3.1.1.1.3.3.2.1" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex2.m1.2.2" xref="S1.Ex2.m1.2.2.cmml">𝒴</mi><mo id="S1.Ex2.m1.3.3.1.1.1.3.3.2.2" rspace="0.170em" stretchy="false" xref="S1.Ex2.m1.3.3.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S1.Ex2.m1.4.4.2.2.3" rspace="0.497em" xref="S1.Ex2.m1.4.4.2.3a.cmml">,</mo><mrow id="S1.Ex2.m1.4.4.2.2.2" xref="S1.Ex2.m1.4.4.2.2.2.cmml"><mi id="S1.Ex2.m1.4.4.2.2.2.2" xref="S1.Ex2.m1.4.4.2.2.2.2.cmml">μ</mi><mo id="S1.Ex2.m1.4.4.2.2.2.1" stretchy="false" xref="S1.Ex2.m1.4.4.2.2.2.1.cmml">↦</mo><msup id="S1.Ex2.m1.4.4.2.2.2.3" xref="S1.Ex2.m1.4.4.2.2.2.3.cmml"><mi id="S1.Ex2.m1.4.4.2.2.2.3.2" xref="S1.Ex2.m1.4.4.2.2.2.3.2.cmml">μ</mi><mi id="S1.Ex2.m1.4.4.2.2.2.3.3" xref="S1.Ex2.m1.4.4.2.2.2.3.3.cmml">σ</mi></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex2.m1.4b"><apply id="S1.Ex2.m1.4.4.cmml" xref="S1.Ex2.m1.4.4"><ci id="S1.Ex2.m1.4.4.3.cmml" xref="S1.Ex2.m1.4.4.3">:</ci><apply id="S1.Ex2.m1.4.4.4.cmml" xref="S1.Ex2.m1.4.4.4"><times id="S1.Ex2.m1.4.4.4.1.cmml" xref="S1.Ex2.m1.4.4.4.1"></times><apply id="S1.Ex2.m1.4.4.4.2.cmml" xref="S1.Ex2.m1.4.4.4.2"><csymbol cd="ambiguous" id="S1.Ex2.m1.4.4.4.2.1.cmml" xref="S1.Ex2.m1.4.4.4.2">subscript</csymbol><ci id="S1.Ex2.m1.4.4.4.2.2.cmml" xref="S1.Ex2.m1.4.4.4.2.2">𝜎</ci><ci id="S1.Ex2.m1.4.4.4.2.3.cmml" xref="S1.Ex2.m1.4.4.4.2.3">𝑋</ci></apply><ci id="S1.Ex2.m1.4.4.4.3.cmml" xref="S1.Ex2.m1.4.4.4.3">𝑀</ci></apply><apply id="S1.Ex2.m1.4.4.2.3.cmml" xref="S1.Ex2.m1.4.4.2.2"><csymbol cd="ambiguous" id="S1.Ex2.m1.4.4.2.3a.cmml" xref="S1.Ex2.m1.4.4.2.2.3">formulae-sequence</csymbol><apply id="S1.Ex2.m1.3.3.1.1.1.cmml" xref="S1.Ex2.m1.3.3.1.1.1"><ci id="S1.Ex2.m1.3.3.1.1.1.1.cmml" xref="S1.Ex2.m1.3.3.1.1.1.1">→</ci><apply id="S1.Ex2.m1.3.3.1.1.1.2.cmml" xref="S1.Ex2.m1.3.3.1.1.1.2"><times id="S1.Ex2.m1.3.3.1.1.1.2.1.cmml" xref="S1.Ex2.m1.3.3.1.1.1.2.1"></times><ci id="S1.Ex2.m1.3.3.1.1.1.2.2.cmml" xref="S1.Ex2.m1.3.3.1.1.1.2.2">ℳ</ci><ci id="S1.Ex2.m1.1.1.cmml" xref="S1.Ex2.m1.1.1">𝒳</ci></apply><apply id="S1.Ex2.m1.3.3.1.1.1.3.cmml" xref="S1.Ex2.m1.3.3.1.1.1.3"><times id="S1.Ex2.m1.3.3.1.1.1.3.1.cmml" xref="S1.Ex2.m1.3.3.1.1.1.3.1"></times><ci id="S1.Ex2.m1.3.3.1.1.1.3.2.cmml" xref="S1.Ex2.m1.3.3.1.1.1.3.2">ℳ</ci><ci id="S1.Ex2.m1.2.2.cmml" xref="S1.Ex2.m1.2.2">𝒴</ci></apply></apply><apply id="S1.Ex2.m1.4.4.2.2.2.cmml" xref="S1.Ex2.m1.4.4.2.2.2"><csymbol cd="latexml" id="S1.Ex2.m1.4.4.2.2.2.1.cmml" xref="S1.Ex2.m1.4.4.2.2.2.1">maps-to</csymbol><ci id="S1.Ex2.m1.4.4.2.2.2.2.cmml" xref="S1.Ex2.m1.4.4.2.2.2.2">𝜇</ci><apply id="S1.Ex2.m1.4.4.2.2.2.3.cmml" xref="S1.Ex2.m1.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S1.Ex2.m1.4.4.2.2.2.3.1.cmml" xref="S1.Ex2.m1.4.4.2.2.2.3">superscript</csymbol><ci id="S1.Ex2.m1.4.4.2.2.2.3.2.cmml" xref="S1.Ex2.m1.4.4.2.2.2.3.2">𝜇</ci><ci id="S1.Ex2.m1.4.4.2.2.2.3.3.cmml" xref="S1.Ex2.m1.4.4.2.2.2.3.3">𝜎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex2.m1.4c">\sigma_{X}M:\cal M(X)\to\cal M(Y)\,,\,\,\mu\mapsto\mu^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S1.Ex2.m1.4d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( caligraphic_Y ) , italic_μ ↦ italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.Thmthm1.p1.7"><span class="ltx_text ltx_font_italic" id="S1.Thmthm1.p1.7.1">which has the following properties:</span></p> <ol class="ltx_enumerate" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.2"><math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S1.I1.i1.p1.1.m1.1"><semantics id="S1.I1.i1.p1.1.m1.1a"><mrow id="S1.I1.i1.p1.1.m1.1.1" xref="S1.I1.i1.p1.1.m1.1.1.cmml"><msub id="S1.I1.i1.p1.1.m1.1.1.2" xref="S1.I1.i1.p1.1.m1.1.1.2.cmml"><mi id="S1.I1.i1.p1.1.m1.1.1.2.2" xref="S1.I1.i1.p1.1.m1.1.1.2.2.cmml">σ</mi><mi id="S1.I1.i1.p1.1.m1.1.1.2.3" xref="S1.I1.i1.p1.1.m1.1.1.2.3.cmml">X</mi></msub><mo id="S1.I1.i1.p1.1.m1.1.1.1" xref="S1.I1.i1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.I1.i1.p1.1.m1.1.1.3" xref="S1.I1.i1.p1.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.1.m1.1b"><apply id="S1.I1.i1.p1.1.m1.1.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1"><times id="S1.I1.i1.p1.1.m1.1.1.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.1"></times><apply id="S1.I1.i1.p1.1.m1.1.1.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S1.I1.i1.p1.1.m1.1.1.2.2.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2.2">𝜎</ci><ci id="S1.I1.i1.p1.1.m1.1.1.2.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.2.3">𝑋</ci></apply><ci id="S1.I1.i1.p1.1.m1.1.1.3.cmml" xref="S1.I1.i1.p1.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.1.m1.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.2.1"> is an </span><math alttext="\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S1.I1.i1.p1.2.m2.1"><semantics id="S1.I1.i1.p1.2.m2.1a"><msub id="S1.I1.i1.p1.2.m2.1.1" xref="S1.I1.i1.p1.2.m2.1.1.cmml"><mi id="S1.I1.i1.p1.2.m2.1.1.2" xref="S1.I1.i1.p1.2.m2.1.1.2.cmml">ℝ</mi><mrow id="S1.I1.i1.p1.2.m2.1.1.3" xref="S1.I1.i1.p1.2.m2.1.1.3.cmml"><mi id="S1.I1.i1.p1.2.m2.1.1.3.2" xref="S1.I1.i1.p1.2.m2.1.1.3.2.cmml"></mi><mo id="S1.I1.i1.p1.2.m2.1.1.3.1" xref="S1.I1.i1.p1.2.m2.1.1.3.1.cmml">≥</mo><mn id="S1.I1.i1.p1.2.m2.1.1.3.3" xref="S1.I1.i1.p1.2.m2.1.1.3.3.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.I1.i1.p1.2.m2.1b"><apply id="S1.I1.i1.p1.2.m2.1.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.I1.i1.p1.2.m2.1.1.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1">subscript</csymbol><ci id="S1.I1.i1.p1.2.m2.1.1.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.2">ℝ</ci><apply id="S1.I1.i1.p1.2.m2.1.1.3.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3"><geq id="S1.I1.i1.p1.2.m2.1.1.3.1.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.1"></geq><csymbol cd="latexml" id="S1.I1.i1.p1.2.m2.1.1.3.2.cmml" xref="S1.I1.i1.p1.2.m2.1.1.3.2">absent</csymbol><cn id="S1.I1.i1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S1.I1.i1.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i1.p1.2.m2.1c">\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i1.p1.2.m2.1d">blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i1.p1.2.2">-linear map of cones.</span></p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1"><math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S1.I1.i2.p1.1.m1.1"><semantics id="S1.I1.i2.p1.1.m1.1a"><mrow id="S1.I1.i2.p1.1.m1.1.1" xref="S1.I1.i2.p1.1.m1.1.1.cmml"><msub id="S1.I1.i2.p1.1.m1.1.1.2" xref="S1.I1.i2.p1.1.m1.1.1.2.cmml"><mi id="S1.I1.i2.p1.1.m1.1.1.2.2" xref="S1.I1.i2.p1.1.m1.1.1.2.2.cmml">σ</mi><mi id="S1.I1.i2.p1.1.m1.1.1.2.3" xref="S1.I1.i2.p1.1.m1.1.1.2.3.cmml">X</mi></msub><mo id="S1.I1.i2.p1.1.m1.1.1.1" xref="S1.I1.i2.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.I1.i2.p1.1.m1.1.1.3" xref="S1.I1.i2.p1.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.1.m1.1b"><apply id="S1.I1.i2.p1.1.m1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1"><times id="S1.I1.i2.p1.1.m1.1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1.1"></times><apply id="S1.I1.i2.p1.1.m1.1.1.2.cmml" xref="S1.I1.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.1.m1.1.1.2.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S1.I1.i2.p1.1.m1.1.1.2.2.cmml" xref="S1.I1.i2.p1.1.m1.1.1.2.2">𝜎</ci><ci id="S1.I1.i2.p1.1.m1.1.1.2.3.cmml" xref="S1.I1.i2.p1.1.m1.1.1.2.3">𝑋</ci></apply><ci id="S1.I1.i2.p1.1.m1.1.1.3.cmml" xref="S1.I1.i2.p1.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.1.m1.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i2.p1.1.1"> is continuous.</span></p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.2"><math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S1.I1.i3.p1.1.m1.1"><semantics id="S1.I1.i3.p1.1.m1.1a"><mrow id="S1.I1.i3.p1.1.m1.1.1" xref="S1.I1.i3.p1.1.m1.1.1.cmml"><msub id="S1.I1.i3.p1.1.m1.1.1.2" xref="S1.I1.i3.p1.1.m1.1.1.2.cmml"><mi id="S1.I1.i3.p1.1.m1.1.1.2.2" xref="S1.I1.i3.p1.1.m1.1.1.2.2.cmml">σ</mi><mi id="S1.I1.i3.p1.1.m1.1.1.2.3" xref="S1.I1.i3.p1.1.m1.1.1.2.3.cmml">X</mi></msub><mo id="S1.I1.i3.p1.1.m1.1.1.1" xref="S1.I1.i3.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.I1.i3.p1.1.m1.1.1.3" xref="S1.I1.i3.p1.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.1.m1.1b"><apply id="S1.I1.i3.p1.1.m1.1.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1"><times id="S1.I1.i3.p1.1.m1.1.1.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1.1"></times><apply id="S1.I1.i3.p1.1.m1.1.1.2.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i3.p1.1.m1.1.1.2.1.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2">subscript</csymbol><ci id="S1.I1.i3.p1.1.m1.1.1.2.2.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2.2">𝜎</ci><ci id="S1.I1.i3.p1.1.m1.1.1.2.3.cmml" xref="S1.I1.i3.p1.1.m1.1.1.2.3">𝑋</ci></apply><ci id="S1.I1.i3.p1.1.m1.1.1.3.cmml" xref="S1.I1.i3.p1.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.1.m1.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i3.p1.2.1"> is functorial: </span><math alttext="(\sigma^{\prime}\circ\sigma)_{X}M={\sigma^{\prime}}_{\sigma(X)}M\circ\sigma_{X}M" class="ltx_Math" display="inline" id="S1.I1.i3.p1.2.m2.2"><semantics id="S1.I1.i3.p1.2.m2.2a"><mrow id="S1.I1.i3.p1.2.m2.2.2" xref="S1.I1.i3.p1.2.m2.2.2.cmml"><mrow id="S1.I1.i3.p1.2.m2.2.2.1" xref="S1.I1.i3.p1.2.m2.2.2.1.cmml"><msub id="S1.I1.i3.p1.2.m2.2.2.1.1" xref="S1.I1.i3.p1.2.m2.2.2.1.1.cmml"><mrow id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.cmml"><mo id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.2" stretchy="false" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.cmml"><msup id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.cmml"><mi id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.2" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.2.cmml">σ</mi><mo id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.3" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.1.cmml">∘</mo><mi id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.3" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.3.cmml">σ</mi></mrow><mo id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.3" stretchy="false" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mi id="S1.I1.i3.p1.2.m2.2.2.1.1.3" xref="S1.I1.i3.p1.2.m2.2.2.1.1.3.cmml">X</mi></msub><mo id="S1.I1.i3.p1.2.m2.2.2.1.2" xref="S1.I1.i3.p1.2.m2.2.2.1.2.cmml">⁢</mo><mi id="S1.I1.i3.p1.2.m2.2.2.1.3" xref="S1.I1.i3.p1.2.m2.2.2.1.3.cmml">M</mi></mrow><mo id="S1.I1.i3.p1.2.m2.2.2.2" xref="S1.I1.i3.p1.2.m2.2.2.2.cmml">=</mo><mrow id="S1.I1.i3.p1.2.m2.2.2.3" xref="S1.I1.i3.p1.2.m2.2.2.3.cmml"><mrow id="S1.I1.i3.p1.2.m2.2.2.3.2" xref="S1.I1.i3.p1.2.m2.2.2.3.2.cmml"><mrow id="S1.I1.i3.p1.2.m2.2.2.3.2.2" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.cmml"><mmultiscripts id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.cmml"><mi id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.2" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.2.cmml">σ</mi><mrow id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2a" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.cmml"></mrow><mo id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.3" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.3.cmml">′</mo><mrow id="S1.I1.i3.p1.2.m2.1.1.1" xref="S1.I1.i3.p1.2.m2.1.1.1.cmml"><mi id="S1.I1.i3.p1.2.m2.1.1.1.3" xref="S1.I1.i3.p1.2.m2.1.1.1.3.cmml">σ</mi><mo id="S1.I1.i3.p1.2.m2.1.1.1.2" xref="S1.I1.i3.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S1.I1.i3.p1.2.m2.1.1.1.4.2" xref="S1.I1.i3.p1.2.m2.1.1.1.cmml"><mo id="S1.I1.i3.p1.2.m2.1.1.1.4.2.1" stretchy="false" xref="S1.I1.i3.p1.2.m2.1.1.1.cmml">(</mo><mi id="S1.I1.i3.p1.2.m2.1.1.1.1" xref="S1.I1.i3.p1.2.m2.1.1.1.1.cmml">X</mi><mo id="S1.I1.i3.p1.2.m2.1.1.1.4.2.2" stretchy="false" xref="S1.I1.i3.p1.2.m2.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2b" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.cmml"></mrow></mmultiscripts><mo id="S1.I1.i3.p1.2.m2.2.2.3.2.2.1" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.1.cmml">⁢</mo><mi id="S1.I1.i3.p1.2.m2.2.2.3.2.2.3" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.3.cmml">M</mi></mrow><mo id="S1.I1.i3.p1.2.m2.2.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S1.I1.i3.p1.2.m2.2.2.3.2.1.cmml">∘</mo><msub id="S1.I1.i3.p1.2.m2.2.2.3.2.3" xref="S1.I1.i3.p1.2.m2.2.2.3.2.3.cmml"><mi id="S1.I1.i3.p1.2.m2.2.2.3.2.3.2" xref="S1.I1.i3.p1.2.m2.2.2.3.2.3.2.cmml">σ</mi><mi id="S1.I1.i3.p1.2.m2.2.2.3.2.3.3" xref="S1.I1.i3.p1.2.m2.2.2.3.2.3.3.cmml">X</mi></msub></mrow><mo id="S1.I1.i3.p1.2.m2.2.2.3.1" xref="S1.I1.i3.p1.2.m2.2.2.3.1.cmml">⁢</mo><mi id="S1.I1.i3.p1.2.m2.2.2.3.3" xref="S1.I1.i3.p1.2.m2.2.2.3.3.cmml">M</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.2.m2.2b"><apply id="S1.I1.i3.p1.2.m2.2.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2"><eq id="S1.I1.i3.p1.2.m2.2.2.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.2"></eq><apply id="S1.I1.i3.p1.2.m2.2.2.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1"><times id="S1.I1.i3.p1.2.m2.2.2.1.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.2"></times><apply id="S1.I1.i3.p1.2.m2.2.2.1.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.2.2.1.1.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1">subscript</csymbol><apply id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1"><compose id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.1"></compose><apply id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2">superscript</csymbol><ci id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.2">𝜎</ci><ci id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.3">′</ci></apply><ci id="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.1.1.1.3">𝜎</ci></apply><ci id="S1.I1.i3.p1.2.m2.2.2.1.1.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.1.3">𝑋</ci></apply><ci id="S1.I1.i3.p1.2.m2.2.2.1.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.1.3">𝑀</ci></apply><apply id="S1.I1.i3.p1.2.m2.2.2.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3"><times id="S1.I1.i3.p1.2.m2.2.2.3.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.1"></times><apply id="S1.I1.i3.p1.2.m2.2.2.3.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2"><compose id="S1.I1.i3.p1.2.m2.2.2.3.2.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.1"></compose><apply id="S1.I1.i3.p1.2.m2.2.2.3.2.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2"><times id="S1.I1.i3.p1.2.m2.2.2.3.2.2.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.1"></times><apply id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2">subscript</csymbol><apply id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2">superscript</csymbol><ci id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.2">𝜎</ci><ci id="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.2.2.3">′</ci></apply><apply id="S1.I1.i3.p1.2.m2.1.1.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1.1"><times id="S1.I1.i3.p1.2.m2.1.1.1.2.cmml" xref="S1.I1.i3.p1.2.m2.1.1.1.2"></times><ci id="S1.I1.i3.p1.2.m2.1.1.1.3.cmml" xref="S1.I1.i3.p1.2.m2.1.1.1.3">𝜎</ci><ci id="S1.I1.i3.p1.2.m2.1.1.1.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1.1.1">𝑋</ci></apply></apply><ci id="S1.I1.i3.p1.2.m2.2.2.3.2.2.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.2.3">𝑀</ci></apply><apply id="S1.I1.i3.p1.2.m2.2.2.3.2.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.3"><csymbol cd="ambiguous" id="S1.I1.i3.p1.2.m2.2.2.3.2.3.1.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.3">subscript</csymbol><ci id="S1.I1.i3.p1.2.m2.2.2.3.2.3.2.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.3.2">𝜎</ci><ci id="S1.I1.i3.p1.2.m2.2.2.3.2.3.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.2.3.3">𝑋</ci></apply></apply><ci id="S1.I1.i3.p1.2.m2.2.2.3.3.cmml" xref="S1.I1.i3.p1.2.m2.2.2.3.3">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.2.m2.2c">(\sigma^{\prime}\circ\sigma)_{X}M={\sigma^{\prime}}_{\sigma(X)}M\circ\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.2.m2.2d">( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∘ italic_σ ) start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M = italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ ( italic_X ) end_POSTSUBSCRIPT italic_M ∘ italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i3.p1.2.2"></span></p> </div> </li> <li class="ltx_item" id="S1.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(4)</span> <div class="ltx_para" id="S1.I1.i4.p1"> <p class="ltx_p" id="S1.I1.i4.p1.4"><span class="ltx_text ltx_font_italic" id="S1.I1.i4.p1.4.1">If the subshift </span><math alttext="X^{\prime}\subseteq X" class="ltx_Math" display="inline" id="S1.I1.i4.p1.1.m1.1"><semantics id="S1.I1.i4.p1.1.m1.1a"><mrow id="S1.I1.i4.p1.1.m1.1.1" xref="S1.I1.i4.p1.1.m1.1.1.cmml"><msup id="S1.I1.i4.p1.1.m1.1.1.2" xref="S1.I1.i4.p1.1.m1.1.1.2.cmml"><mi id="S1.I1.i4.p1.1.m1.1.1.2.2" xref="S1.I1.i4.p1.1.m1.1.1.2.2.cmml">X</mi><mo id="S1.I1.i4.p1.1.m1.1.1.2.3" xref="S1.I1.i4.p1.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S1.I1.i4.p1.1.m1.1.1.1" xref="S1.I1.i4.p1.1.m1.1.1.1.cmml">⊆</mo><mi id="S1.I1.i4.p1.1.m1.1.1.3" xref="S1.I1.i4.p1.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i4.p1.1.m1.1b"><apply id="S1.I1.i4.p1.1.m1.1.1.cmml" xref="S1.I1.i4.p1.1.m1.1.1"><subset id="S1.I1.i4.p1.1.m1.1.1.1.cmml" xref="S1.I1.i4.p1.1.m1.1.1.1"></subset><apply id="S1.I1.i4.p1.1.m1.1.1.2.cmml" xref="S1.I1.i4.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.I1.i4.p1.1.m1.1.1.2.1.cmml" xref="S1.I1.i4.p1.1.m1.1.1.2">superscript</csymbol><ci id="S1.I1.i4.p1.1.m1.1.1.2.2.cmml" xref="S1.I1.i4.p1.1.m1.1.1.2.2">𝑋</ci><ci id="S1.I1.i4.p1.1.m1.1.1.2.3.cmml" xref="S1.I1.i4.p1.1.m1.1.1.2.3">′</ci></apply><ci id="S1.I1.i4.p1.1.m1.1.1.3.cmml" xref="S1.I1.i4.p1.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i4.p1.1.m1.1c">X^{\prime}\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i4.p1.1.m1.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i4.p1.4.2"> is the support of a measure </span><math alttext="\mu\in\cal M(X)" class="ltx_Math" display="inline" id="S1.I1.i4.p1.2.m2.1"><semantics id="S1.I1.i4.p1.2.m2.1a"><mrow id="S1.I1.i4.p1.2.m2.1.2" xref="S1.I1.i4.p1.2.m2.1.2.cmml"><mi id="S1.I1.i4.p1.2.m2.1.2.2" xref="S1.I1.i4.p1.2.m2.1.2.2.cmml">μ</mi><mo id="S1.I1.i4.p1.2.m2.1.2.1" xref="S1.I1.i4.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="S1.I1.i4.p1.2.m2.1.2.3" xref="S1.I1.i4.p1.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i4.p1.2.m2.1.2.3.2" xref="S1.I1.i4.p1.2.m2.1.2.3.2.cmml">ℳ</mi><mo id="S1.I1.i4.p1.2.m2.1.2.3.1" xref="S1.I1.i4.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S1.I1.i4.p1.2.m2.1.2.3.3.2" xref="S1.I1.i4.p1.2.m2.1.2.3.cmml"><mo id="S1.I1.i4.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S1.I1.i4.p1.2.m2.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.I1.i4.p1.2.m2.1.1" xref="S1.I1.i4.p1.2.m2.1.1.cmml">𝒳</mi><mo id="S1.I1.i4.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S1.I1.i4.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i4.p1.2.m2.1b"><apply id="S1.I1.i4.p1.2.m2.1.2.cmml" xref="S1.I1.i4.p1.2.m2.1.2"><in id="S1.I1.i4.p1.2.m2.1.2.1.cmml" xref="S1.I1.i4.p1.2.m2.1.2.1"></in><ci id="S1.I1.i4.p1.2.m2.1.2.2.cmml" xref="S1.I1.i4.p1.2.m2.1.2.2">𝜇</ci><apply id="S1.I1.i4.p1.2.m2.1.2.3.cmml" xref="S1.I1.i4.p1.2.m2.1.2.3"><times id="S1.I1.i4.p1.2.m2.1.2.3.1.cmml" xref="S1.I1.i4.p1.2.m2.1.2.3.1"></times><ci id="S1.I1.i4.p1.2.m2.1.2.3.2.cmml" xref="S1.I1.i4.p1.2.m2.1.2.3.2">ℳ</ci><ci id="S1.I1.i4.p1.2.m2.1.1.cmml" xref="S1.I1.i4.p1.2.m2.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i4.p1.2.m2.1c">\mu\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i4.p1.2.m2.1d">italic_μ ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i4.p1.4.3">, then </span><math alttext="\sigma(X^{\prime})" class="ltx_Math" display="inline" id="S1.I1.i4.p1.3.m3.1"><semantics id="S1.I1.i4.p1.3.m3.1a"><mrow id="S1.I1.i4.p1.3.m3.1.1" xref="S1.I1.i4.p1.3.m3.1.1.cmml"><mi id="S1.I1.i4.p1.3.m3.1.1.3" xref="S1.I1.i4.p1.3.m3.1.1.3.cmml">σ</mi><mo id="S1.I1.i4.p1.3.m3.1.1.2" xref="S1.I1.i4.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S1.I1.i4.p1.3.m3.1.1.1.1" xref="S1.I1.i4.p1.3.m3.1.1.1.1.1.cmml"><mo id="S1.I1.i4.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S1.I1.i4.p1.3.m3.1.1.1.1.1.cmml">(</mo><msup id="S1.I1.i4.p1.3.m3.1.1.1.1.1" xref="S1.I1.i4.p1.3.m3.1.1.1.1.1.cmml"><mi id="S1.I1.i4.p1.3.m3.1.1.1.1.1.2" xref="S1.I1.i4.p1.3.m3.1.1.1.1.1.2.cmml">X</mi><mo id="S1.I1.i4.p1.3.m3.1.1.1.1.1.3" xref="S1.I1.i4.p1.3.m3.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.I1.i4.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S1.I1.i4.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i4.p1.3.m3.1b"><apply id="S1.I1.i4.p1.3.m3.1.1.cmml" xref="S1.I1.i4.p1.3.m3.1.1"><times id="S1.I1.i4.p1.3.m3.1.1.2.cmml" xref="S1.I1.i4.p1.3.m3.1.1.2"></times><ci id="S1.I1.i4.p1.3.m3.1.1.3.cmml" xref="S1.I1.i4.p1.3.m3.1.1.3">𝜎</ci><apply id="S1.I1.i4.p1.3.m3.1.1.1.1.1.cmml" xref="S1.I1.i4.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S1.I1.i4.p1.3.m3.1.1.1.1.1.1.cmml" xref="S1.I1.i4.p1.3.m3.1.1.1.1">superscript</csymbol><ci id="S1.I1.i4.p1.3.m3.1.1.1.1.1.2.cmml" xref="S1.I1.i4.p1.3.m3.1.1.1.1.1.2">𝑋</ci><ci id="S1.I1.i4.p1.3.m3.1.1.1.1.1.3.cmml" xref="S1.I1.i4.p1.3.m3.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i4.p1.3.m3.1c">\sigma(X^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i4.p1.3.m3.1d">italic_σ ( italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i4.p1.4.4"> is the support of </span><math alttext="\sigma_{X}M(\mu)" class="ltx_Math" display="inline" id="S1.I1.i4.p1.4.m4.1"><semantics id="S1.I1.i4.p1.4.m4.1a"><mrow id="S1.I1.i4.p1.4.m4.1.2" xref="S1.I1.i4.p1.4.m4.1.2.cmml"><msub id="S1.I1.i4.p1.4.m4.1.2.2" xref="S1.I1.i4.p1.4.m4.1.2.2.cmml"><mi id="S1.I1.i4.p1.4.m4.1.2.2.2" xref="S1.I1.i4.p1.4.m4.1.2.2.2.cmml">σ</mi><mi id="S1.I1.i4.p1.4.m4.1.2.2.3" xref="S1.I1.i4.p1.4.m4.1.2.2.3.cmml">X</mi></msub><mo id="S1.I1.i4.p1.4.m4.1.2.1" xref="S1.I1.i4.p1.4.m4.1.2.1.cmml">⁢</mo><mi id="S1.I1.i4.p1.4.m4.1.2.3" xref="S1.I1.i4.p1.4.m4.1.2.3.cmml">M</mi><mo id="S1.I1.i4.p1.4.m4.1.2.1a" xref="S1.I1.i4.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S1.I1.i4.p1.4.m4.1.2.4.2" xref="S1.I1.i4.p1.4.m4.1.2.cmml"><mo id="S1.I1.i4.p1.4.m4.1.2.4.2.1" stretchy="false" xref="S1.I1.i4.p1.4.m4.1.2.cmml">(</mo><mi id="S1.I1.i4.p1.4.m4.1.1" xref="S1.I1.i4.p1.4.m4.1.1.cmml">μ</mi><mo id="S1.I1.i4.p1.4.m4.1.2.4.2.2" stretchy="false" xref="S1.I1.i4.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i4.p1.4.m4.1b"><apply id="S1.I1.i4.p1.4.m4.1.2.cmml" xref="S1.I1.i4.p1.4.m4.1.2"><times id="S1.I1.i4.p1.4.m4.1.2.1.cmml" xref="S1.I1.i4.p1.4.m4.1.2.1"></times><apply id="S1.I1.i4.p1.4.m4.1.2.2.cmml" xref="S1.I1.i4.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S1.I1.i4.p1.4.m4.1.2.2.1.cmml" xref="S1.I1.i4.p1.4.m4.1.2.2">subscript</csymbol><ci id="S1.I1.i4.p1.4.m4.1.2.2.2.cmml" xref="S1.I1.i4.p1.4.m4.1.2.2.2">𝜎</ci><ci id="S1.I1.i4.p1.4.m4.1.2.2.3.cmml" xref="S1.I1.i4.p1.4.m4.1.2.2.3">𝑋</ci></apply><ci id="S1.I1.i4.p1.4.m4.1.2.3.cmml" xref="S1.I1.i4.p1.4.m4.1.2.3">𝑀</ci><ci id="S1.I1.i4.p1.4.m4.1.1.cmml" xref="S1.I1.i4.p1.4.m4.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i4.p1.4.m4.1c">\sigma_{X}M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i4.p1.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M ( italic_μ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i4.p1.4.5">.</span></p> </div> </li> <li class="ltx_item" id="S1.I1.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(5)</span> <div class="ltx_para" id="S1.I1.i5.p1"> <p class="ltx_p" id="S1.I1.i5.p1.2"><span class="ltx_text ltx_font_italic" id="S1.I1.i5.p1.2.1">If a measure </span><math alttext="\mu\in\cal M(X)" class="ltx_Math" display="inline" id="S1.I1.i5.p1.1.m1.1"><semantics id="S1.I1.i5.p1.1.m1.1a"><mrow id="S1.I1.i5.p1.1.m1.1.2" xref="S1.I1.i5.p1.1.m1.1.2.cmml"><mi id="S1.I1.i5.p1.1.m1.1.2.2" xref="S1.I1.i5.p1.1.m1.1.2.2.cmml">μ</mi><mo id="S1.I1.i5.p1.1.m1.1.2.1" xref="S1.I1.i5.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S1.I1.i5.p1.1.m1.1.2.3" xref="S1.I1.i5.p1.1.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i5.p1.1.m1.1.2.3.2" xref="S1.I1.i5.p1.1.m1.1.2.3.2.cmml">ℳ</mi><mo id="S1.I1.i5.p1.1.m1.1.2.3.1" xref="S1.I1.i5.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S1.I1.i5.p1.1.m1.1.2.3.3.2" xref="S1.I1.i5.p1.1.m1.1.2.3.cmml"><mo id="S1.I1.i5.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S1.I1.i5.p1.1.m1.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.I1.i5.p1.1.m1.1.1" xref="S1.I1.i5.p1.1.m1.1.1.cmml">𝒳</mi><mo id="S1.I1.i5.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S1.I1.i5.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i5.p1.1.m1.1b"><apply id="S1.I1.i5.p1.1.m1.1.2.cmml" xref="S1.I1.i5.p1.1.m1.1.2"><in id="S1.I1.i5.p1.1.m1.1.2.1.cmml" xref="S1.I1.i5.p1.1.m1.1.2.1"></in><ci id="S1.I1.i5.p1.1.m1.1.2.2.cmml" xref="S1.I1.i5.p1.1.m1.1.2.2">𝜇</ci><apply id="S1.I1.i5.p1.1.m1.1.2.3.cmml" xref="S1.I1.i5.p1.1.m1.1.2.3"><times id="S1.I1.i5.p1.1.m1.1.2.3.1.cmml" xref="S1.I1.i5.p1.1.m1.1.2.3.1"></times><ci id="S1.I1.i5.p1.1.m1.1.2.3.2.cmml" xref="S1.I1.i5.p1.1.m1.1.2.3.2">ℳ</ci><ci id="S1.I1.i5.p1.1.m1.1.1.cmml" xref="S1.I1.i5.p1.1.m1.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i5.p1.1.m1.1c">\mu\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i5.p1.1.m1.1d">italic_μ ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i5.p1.2.2"> is ergodic, then so is </span><math alttext="\mu^{\sigma}\in\cal M(Y)" class="ltx_Math" display="inline" id="S1.I1.i5.p1.2.m2.1"><semantics id="S1.I1.i5.p1.2.m2.1a"><mrow id="S1.I1.i5.p1.2.m2.1.2" xref="S1.I1.i5.p1.2.m2.1.2.cmml"><msup id="S1.I1.i5.p1.2.m2.1.2.2" xref="S1.I1.i5.p1.2.m2.1.2.2.cmml"><mi id="S1.I1.i5.p1.2.m2.1.2.2.2" xref="S1.I1.i5.p1.2.m2.1.2.2.2.cmml">μ</mi><mi id="S1.I1.i5.p1.2.m2.1.2.2.3" xref="S1.I1.i5.p1.2.m2.1.2.2.3.cmml">σ</mi></msup><mo id="S1.I1.i5.p1.2.m2.1.2.1" xref="S1.I1.i5.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="S1.I1.i5.p1.2.m2.1.2.3" xref="S1.I1.i5.p1.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.I1.i5.p1.2.m2.1.2.3.2" xref="S1.I1.i5.p1.2.m2.1.2.3.2.cmml">ℳ</mi><mo id="S1.I1.i5.p1.2.m2.1.2.3.1" xref="S1.I1.i5.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S1.I1.i5.p1.2.m2.1.2.3.3.2" xref="S1.I1.i5.p1.2.m2.1.2.3.cmml"><mo id="S1.I1.i5.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S1.I1.i5.p1.2.m2.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.I1.i5.p1.2.m2.1.1" xref="S1.I1.i5.p1.2.m2.1.1.cmml">𝒴</mi><mo id="S1.I1.i5.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S1.I1.i5.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i5.p1.2.m2.1b"><apply id="S1.I1.i5.p1.2.m2.1.2.cmml" xref="S1.I1.i5.p1.2.m2.1.2"><in id="S1.I1.i5.p1.2.m2.1.2.1.cmml" xref="S1.I1.i5.p1.2.m2.1.2.1"></in><apply id="S1.I1.i5.p1.2.m2.1.2.2.cmml" xref="S1.I1.i5.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S1.I1.i5.p1.2.m2.1.2.2.1.cmml" xref="S1.I1.i5.p1.2.m2.1.2.2">superscript</csymbol><ci id="S1.I1.i5.p1.2.m2.1.2.2.2.cmml" xref="S1.I1.i5.p1.2.m2.1.2.2.2">𝜇</ci><ci id="S1.I1.i5.p1.2.m2.1.2.2.3.cmml" xref="S1.I1.i5.p1.2.m2.1.2.2.3">𝜎</ci></apply><apply id="S1.I1.i5.p1.2.m2.1.2.3.cmml" xref="S1.I1.i5.p1.2.m2.1.2.3"><times id="S1.I1.i5.p1.2.m2.1.2.3.1.cmml" xref="S1.I1.i5.p1.2.m2.1.2.3.1"></times><ci id="S1.I1.i5.p1.2.m2.1.2.3.2.cmml" xref="S1.I1.i5.p1.2.m2.1.2.3.2">ℳ</ci><ci id="S1.I1.i5.p1.2.m2.1.1.cmml" xref="S1.I1.i5.p1.2.m2.1.1">𝒴</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i5.p1.2.m2.1c">\mu^{\sigma}\in\cal M(Y)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i5.p1.2.m2.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ∈ caligraphic_M ( caligraphic_Y )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I1.i5.p1.2.3">.</span></p> </div> </li> </ol> <p class="ltx_p" id="S1.Thmthm1.p1.6"><span class="ltx_text ltx_font_italic" id="S1.Thmthm1.p1.6.1">(In addition, the map <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S1.Thmthm1.p1.6.1.m1.1"><semantics id="S1.Thmthm1.p1.6.1.m1.1a"><mrow id="S1.Thmthm1.p1.6.1.m1.1.1" xref="S1.Thmthm1.p1.6.1.m1.1.1.cmml"><msub id="S1.Thmthm1.p1.6.1.m1.1.1.2" xref="S1.Thmthm1.p1.6.1.m1.1.1.2.cmml"><mi id="S1.Thmthm1.p1.6.1.m1.1.1.2.2" xref="S1.Thmthm1.p1.6.1.m1.1.1.2.2.cmml">σ</mi><mi id="S1.Thmthm1.p1.6.1.m1.1.1.2.3" xref="S1.Thmthm1.p1.6.1.m1.1.1.2.3.cmml">X</mi></msub><mo id="S1.Thmthm1.p1.6.1.m1.1.1.1" xref="S1.Thmthm1.p1.6.1.m1.1.1.1.cmml">⁢</mo><mi id="S1.Thmthm1.p1.6.1.m1.1.1.3" xref="S1.Thmthm1.p1.6.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm1.p1.6.1.m1.1b"><apply id="S1.Thmthm1.p1.6.1.m1.1.1.cmml" xref="S1.Thmthm1.p1.6.1.m1.1.1"><times id="S1.Thmthm1.p1.6.1.m1.1.1.1.cmml" xref="S1.Thmthm1.p1.6.1.m1.1.1.1"></times><apply id="S1.Thmthm1.p1.6.1.m1.1.1.2.cmml" xref="S1.Thmthm1.p1.6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.Thmthm1.p1.6.1.m1.1.1.2.1.cmml" xref="S1.Thmthm1.p1.6.1.m1.1.1.2">subscript</csymbol><ci id="S1.Thmthm1.p1.6.1.m1.1.1.2.2.cmml" xref="S1.Thmthm1.p1.6.1.m1.1.1.2.2">𝜎</ci><ci id="S1.Thmthm1.p1.6.1.m1.1.1.2.3.cmml" xref="S1.Thmthm1.p1.6.1.m1.1.1.2.3">𝑋</ci></apply><ci id="S1.Thmthm1.p1.6.1.m1.1.1.3.cmml" xref="S1.Thmthm1.p1.6.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm1.p1.6.1.m1.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm1.p1.6.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> is surjective, see Proposition 4.4 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>.)</span></p> </div> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.6">As indicated already before, the above introduced measure <math alttext="\mu^{\sigma}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><msup id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mi id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml">μ</mi><mi id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml">σ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1">superscript</csymbol><ci id="S1.p3.1.m1.1.1.2.cmml" xref="S1.p3.1.m1.1.1.2">𝜇</ci><ci id="S1.p3.1.m1.1.1.3.cmml" xref="S1.p3.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">\mu^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math>, obtained from a given measure <math alttext="\mu\in\cal M(X)" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.2" xref="S1.p3.2.m2.1.2.cmml"><mi id="S1.p3.2.m2.1.2.2" xref="S1.p3.2.m2.1.2.2.cmml">μ</mi><mo id="S1.p3.2.m2.1.2.1" xref="S1.p3.2.m2.1.2.1.cmml">∈</mo><mrow id="S1.p3.2.m2.1.2.3" xref="S1.p3.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.2.m2.1.2.3.2" xref="S1.p3.2.m2.1.2.3.2.cmml">ℳ</mi><mo id="S1.p3.2.m2.1.2.3.1" xref="S1.p3.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S1.p3.2.m2.1.2.3.3.2" xref="S1.p3.2.m2.1.2.3.cmml"><mo id="S1.p3.2.m2.1.2.3.3.2.1" stretchy="false" xref="S1.p3.2.m2.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">𝒳</mi><mo id="S1.p3.2.m2.1.2.3.3.2.2" stretchy="false" xref="S1.p3.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.2.cmml" xref="S1.p3.2.m2.1.2"><in id="S1.p3.2.m2.1.2.1.cmml" xref="S1.p3.2.m2.1.2.1"></in><ci id="S1.p3.2.m2.1.2.2.cmml" xref="S1.p3.2.m2.1.2.2">𝜇</ci><apply id="S1.p3.2.m2.1.2.3.cmml" xref="S1.p3.2.m2.1.2.3"><times id="S1.p3.2.m2.1.2.3.1.cmml" xref="S1.p3.2.m2.1.2.3.1"></times><ci id="S1.p3.2.m2.1.2.3.2.cmml" xref="S1.p3.2.m2.1.2.3.2">ℳ</ci><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\mu\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">italic_μ ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math> via the measure transfer induced by a morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mi id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><ci id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_σ</annotation></semantics></math>, is almost always quite different from the push-forward measure <math alttext="\mu_{*}" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><msub id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml"><mi id="S1.p3.4.m4.1.1.2" xref="S1.p3.4.m4.1.1.2.cmml">μ</mi><mo id="S1.p3.4.m4.1.1.3" xref="S1.p3.4.m4.1.1.3.cmml">∗</mo></msub><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><apply id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S1.p3.4.m4.1.1.1.cmml" xref="S1.p3.4.m4.1.1">subscript</csymbol><ci id="S1.p3.4.m4.1.1.2.cmml" xref="S1.p3.4.m4.1.1.2">𝜇</ci><times id="S1.p3.4.m4.1.1.3.cmml" xref="S1.p3.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">\mu_{*}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">italic_μ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math> as defined by <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><mi id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><ci id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">italic_σ</annotation></semantics></math>. It turns out that the popular sources for basics in symbolic dynamics do only consider this very particular case <math alttext="\mu^{\sigma}=\mu_{*}\," class="ltx_Math" display="inline" id="S1.p3.6.m6.1"><semantics id="S1.p3.6.m6.1a"><mrow id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml"><msup id="S1.p3.6.m6.1.1.2" xref="S1.p3.6.m6.1.1.2.cmml"><mi id="S1.p3.6.m6.1.1.2.2" xref="S1.p3.6.m6.1.1.2.2.cmml">μ</mi><mi id="S1.p3.6.m6.1.1.2.3" xref="S1.p3.6.m6.1.1.2.3.cmml">σ</mi></msup><mo id="S1.p3.6.m6.1.1.1" xref="S1.p3.6.m6.1.1.1.cmml">=</mo><msub id="S1.p3.6.m6.1.1.3" xref="S1.p3.6.m6.1.1.3.cmml"><mi id="S1.p3.6.m6.1.1.3.2" xref="S1.p3.6.m6.1.1.3.2.cmml">μ</mi><mo id="S1.p3.6.m6.1.1.3.3" xref="S1.p3.6.m6.1.1.3.3.cmml">∗</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.6.m6.1b"><apply id="S1.p3.6.m6.1.1.cmml" xref="S1.p3.6.m6.1.1"><eq id="S1.p3.6.m6.1.1.1.cmml" xref="S1.p3.6.m6.1.1.1"></eq><apply id="S1.p3.6.m6.1.1.2.cmml" xref="S1.p3.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.p3.6.m6.1.1.2.1.cmml" xref="S1.p3.6.m6.1.1.2">superscript</csymbol><ci id="S1.p3.6.m6.1.1.2.2.cmml" xref="S1.p3.6.m6.1.1.2.2">𝜇</ci><ci id="S1.p3.6.m6.1.1.2.3.cmml" xref="S1.p3.6.m6.1.1.2.3">𝜎</ci></apply><apply id="S1.p3.6.m6.1.1.3.cmml" xref="S1.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.p3.6.m6.1.1.3.1.cmml" xref="S1.p3.6.m6.1.1.3">subscript</csymbol><ci id="S1.p3.6.m6.1.1.3.2.cmml" xref="S1.p3.6.m6.1.1.3.2">𝜇</ci><times id="S1.p3.6.m6.1.1.3.3.cmml" xref="S1.p3.6.m6.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m6.1c">\mu^{\sigma}=\mu_{*}\,</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m6.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = italic_μ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>, see for instance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib15" title="">15</a>]</cite>, Proposition 5.22 . The general set-up for the measure transfer, as studied here, doesn’t seem to be available in the existing literature. The basic properties of the transferred measure proved below are in particular used in our cousin paper <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite> for the purpose of an explicit treatment of the transferred measure by means of S-adic expansions of a given subshift.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.5">In order to underline that, despite its divergence from the usual push-forward concept, the measure transfer is a very natural tool under the given circumstances, we’d like to mention that there are several “ad hoc” occasions where the measure transfer has already appeared beforehand, see Example <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S1.Thmthm6" title="Example 1.6. ‣ 1. Introduction ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">1.6</span></a> below. A special but rather relevant such case takes place if the given morphism is an endomorphisms <math alttext="\sigma:\cal A^{*}\to\cal A^{*}" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mi id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml">σ</mi><mo id="S1.p4.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.p4.1.m1.1.1.1.cmml">:</mo><mrow id="S1.p4.1.m1.1.1.3" xref="S1.p4.1.m1.1.1.3.cmml"><msup id="S1.p4.1.m1.1.1.3.2" xref="S1.p4.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p4.1.m1.1.1.3.2.2" xref="S1.p4.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.p4.1.m1.1.1.3.2.3" xref="S1.p4.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.p4.1.m1.1.1.3.1" stretchy="false" xref="S1.p4.1.m1.1.1.3.1.cmml">→</mo><msup id="S1.p4.1.m1.1.1.3.3" xref="S1.p4.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p4.1.m1.1.1.3.3.2" xref="S1.p4.1.m1.1.1.3.3.2.cmml">𝒜</mi><mo id="S1.p4.1.m1.1.1.3.3.3" xref="S1.p4.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"><ci id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1.1">:</ci><ci id="S1.p4.1.m1.1.1.2.cmml" xref="S1.p4.1.m1.1.1.2">𝜎</ci><apply id="S1.p4.1.m1.1.1.3.cmml" xref="S1.p4.1.m1.1.1.3"><ci id="S1.p4.1.m1.1.1.3.1.cmml" xref="S1.p4.1.m1.1.1.3.1">→</ci><apply id="S1.p4.1.m1.1.1.3.2.cmml" xref="S1.p4.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.p4.1.m1.1.1.3.2.1.cmml" xref="S1.p4.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.p4.1.m1.1.1.3.2.2.cmml" xref="S1.p4.1.m1.1.1.3.2.2">𝒜</ci><times id="S1.p4.1.m1.1.1.3.2.3.cmml" xref="S1.p4.1.m1.1.1.3.2.3"></times></apply><apply id="S1.p4.1.m1.1.1.3.3.cmml" xref="S1.p4.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.p4.1.m1.1.1.3.3.1.cmml" xref="S1.p4.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.p4.1.m1.1.1.3.3.2.cmml" xref="S1.p4.1.m1.1.1.3.3.2">𝒜</ci><times id="S1.p4.1.m1.1.1.3.3.3.cmml" xref="S1.p4.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">\sigma:\cal A^{*}\to\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, thus called a <span class="ltx_text ltx_font_italic" id="S1.p4.5.1">substitution</span>. We then have the symbolic dynamics version of the well known and important situation where a system of zero entropy coexists with a system of positive entropy, giving rise to a wealth of subtle and profound interactions. For geometers, a geometric counterpart is obtained, for example, by considering the attractive lamination of a pseudo-Anosov homeomorphism <math alttext="f" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">italic_f</annotation></semantics></math> on a hyperbolic surface: this is a geodesic lamination of zero entropy (for the geodesic flow), but the action of the pseudo-Anosov is of positive entropy (given by the dilation coefficient <math alttext="\lambda" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">\lambda</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">italic_λ</annotation></semantics></math> of this homeomorphism). In such contexts, it is natural to consider the invariant measures of the zero-entropy system that are <span class="ltx_text ltx_font_italic" id="S1.p4.5.2">projectively</span> invariant under the positive entropy transformation operator (the substitution <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p4.4.m4.1"><semantics id="S1.p4.4.m4.1a"><mi id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.1b"><ci id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.1d">italic_σ</annotation></semantics></math> or the pseudo-Anosov <math alttext="f" class="ltx_Math" display="inline" id="S1.p4.5.m5.1"><semantics id="S1.p4.5.m5.1a"><mi id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.1b"><ci id="S1.p4.5.m5.1.1.cmml" xref="S1.p4.5.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.1d">italic_f</annotation></semantics></math> in the above examples). Relating back to our main topic, the measure transfer, we observe:</p> </div> <div class="ltx_theorem ltx_theorem_rem" id="S1.Thmthm2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmthm2.1.1.1">Remark 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmthm2.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmthm2.p1"> <p class="ltx_p" id="S1.Thmthm2.p1.14">Any non-erasing substitution <math alttext="\sigma:\cal A^{*}\to\cal A^{*}" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.1.m1.1"><semantics id="S1.Thmthm2.p1.1.m1.1a"><mrow id="S1.Thmthm2.p1.1.m1.1.1" xref="S1.Thmthm2.p1.1.m1.1.1.cmml"><mi id="S1.Thmthm2.p1.1.m1.1.1.2" xref="S1.Thmthm2.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S1.Thmthm2.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm2.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmthm2.p1.1.m1.1.1.3" xref="S1.Thmthm2.p1.1.m1.1.1.3.cmml"><msup id="S1.Thmthm2.p1.1.m1.1.1.3.2" xref="S1.Thmthm2.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm2.p1.1.m1.1.1.3.2.2" xref="S1.Thmthm2.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.Thmthm2.p1.1.m1.1.1.3.2.3" xref="S1.Thmthm2.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.Thmthm2.p1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmthm2.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S1.Thmthm2.p1.1.m1.1.1.3.3" xref="S1.Thmthm2.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm2.p1.1.m1.1.1.3.3.2" xref="S1.Thmthm2.p1.1.m1.1.1.3.3.2.cmml">𝒜</mi><mo id="S1.Thmthm2.p1.1.m1.1.1.3.3.3" xref="S1.Thmthm2.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.1.m1.1b"><apply id="S1.Thmthm2.p1.1.m1.1.1.cmml" xref="S1.Thmthm2.p1.1.m1.1.1"><ci id="S1.Thmthm2.p1.1.m1.1.1.1.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.1">:</ci><ci id="S1.Thmthm2.p1.1.m1.1.1.2.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.2">𝜎</ci><apply id="S1.Thmthm2.p1.1.m1.1.1.3.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3"><ci id="S1.Thmthm2.p1.1.m1.1.1.3.1.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.1">→</ci><apply id="S1.Thmthm2.p1.1.m1.1.1.3.2.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.1.m1.1.1.3.2.1.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.Thmthm2.p1.1.m1.1.1.3.2.2.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S1.Thmthm2.p1.1.m1.1.1.3.2.3.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S1.Thmthm2.p1.1.m1.1.1.3.3.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.1.m1.1.1.3.3.1.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.Thmthm2.p1.1.m1.1.1.3.3.2.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.3.2">𝒜</ci><times id="S1.Thmthm2.p1.1.m1.1.1.3.3.3.cmml" xref="S1.Thmthm2.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\sigma^{n}(\cal A)\cap\cal A=\emptyset" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.2.m2.1"><semantics id="S1.Thmthm2.p1.2.m2.1a"><mrow id="S1.Thmthm2.p1.2.m2.1.2" xref="S1.Thmthm2.p1.2.m2.1.2.cmml"><mrow id="S1.Thmthm2.p1.2.m2.1.2.2" xref="S1.Thmthm2.p1.2.m2.1.2.2.cmml"><mrow id="S1.Thmthm2.p1.2.m2.1.2.2.2" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.cmml"><msup id="S1.Thmthm2.p1.2.m2.1.2.2.2.2" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.2.cmml"><mi id="S1.Thmthm2.p1.2.m2.1.2.2.2.2.2" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.2.2.cmml">σ</mi><mi id="S1.Thmthm2.p1.2.m2.1.2.2.2.2.3" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.2.3.cmml">n</mi></msup><mo id="S1.Thmthm2.p1.2.m2.1.2.2.2.1" 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id="S1.Thmthm2.p1.2.m2.1.2.1.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.1"></eq><apply id="S1.Thmthm2.p1.2.m2.1.2.2.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2"><intersect id="S1.Thmthm2.p1.2.m2.1.2.2.1.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2.1"></intersect><apply id="S1.Thmthm2.p1.2.m2.1.2.2.2.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2.2"><times id="S1.Thmthm2.p1.2.m2.1.2.2.2.1.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.1"></times><apply id="S1.Thmthm2.p1.2.m2.1.2.2.2.2.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.2"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.2.m2.1.2.2.2.2.1.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.2">superscript</csymbol><ci id="S1.Thmthm2.p1.2.m2.1.2.2.2.2.2.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.2.2">𝜎</ci><ci id="S1.Thmthm2.p1.2.m2.1.2.2.2.2.3.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2.2.2.3">𝑛</ci></apply><ci id="S1.Thmthm2.p1.2.m2.1.1.cmml" xref="S1.Thmthm2.p1.2.m2.1.1">𝒜</ci></apply><ci id="S1.Thmthm2.p1.2.m2.1.2.2.3.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.2.3">𝒜</ci></apply><emptyset id="S1.Thmthm2.p1.2.m2.1.2.3.cmml" xref="S1.Thmthm2.p1.2.m2.1.2.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.2.m2.1c">\sigma^{n}(\cal A)\cap\cal A=\emptyset</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( caligraphic_A ) ∩ caligraphic_A = ∅</annotation></semantics></math> for some sufficiently large <math alttext="n\geq 1" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.3.m3.1"><semantics id="S1.Thmthm2.p1.3.m3.1a"><mrow id="S1.Thmthm2.p1.3.m3.1.1" xref="S1.Thmthm2.p1.3.m3.1.1.cmml"><mi id="S1.Thmthm2.p1.3.m3.1.1.2" xref="S1.Thmthm2.p1.3.m3.1.1.2.cmml">n</mi><mo id="S1.Thmthm2.p1.3.m3.1.1.1" xref="S1.Thmthm2.p1.3.m3.1.1.1.cmml">≥</mo><mn id="S1.Thmthm2.p1.3.m3.1.1.3" xref="S1.Thmthm2.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.3.m3.1b"><apply id="S1.Thmthm2.p1.3.m3.1.1.cmml" xref="S1.Thmthm2.p1.3.m3.1.1"><geq id="S1.Thmthm2.p1.3.m3.1.1.1.cmml" xref="S1.Thmthm2.p1.3.m3.1.1.1"></geq><ci id="S1.Thmthm2.p1.3.m3.1.1.2.cmml" xref="S1.Thmthm2.p1.3.m3.1.1.2">𝑛</ci><cn id="S1.Thmthm2.p1.3.m3.1.1.3.cmml" type="integer" xref="S1.Thmthm2.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.3.m3.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.3.m3.1d">italic_n ≥ 1</annotation></semantics></math> defines a <span class="ltx_text ltx_font_italic" id="S1.Thmthm2.p1.14.1">substitutive subshift</span> <math alttext="X_{\sigma}" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.4.m4.1"><semantics id="S1.Thmthm2.p1.4.m4.1a"><msub id="S1.Thmthm2.p1.4.m4.1.1" xref="S1.Thmthm2.p1.4.m4.1.1.cmml"><mi id="S1.Thmthm2.p1.4.m4.1.1.2" xref="S1.Thmthm2.p1.4.m4.1.1.2.cmml">X</mi><mi id="S1.Thmthm2.p1.4.m4.1.1.3" xref="S1.Thmthm2.p1.4.m4.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.4.m4.1b"><apply id="S1.Thmthm2.p1.4.m4.1.1.cmml" xref="S1.Thmthm2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.4.m4.1.1.1.cmml" xref="S1.Thmthm2.p1.4.m4.1.1">subscript</csymbol><ci id="S1.Thmthm2.p1.4.m4.1.1.2.cmml" xref="S1.Thmthm2.p1.4.m4.1.1.2">𝑋</ci><ci id="S1.Thmthm2.p1.4.m4.1.1.3.cmml" xref="S1.Thmthm2.p1.4.m4.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.4.m4.1c">X_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.4.m4.1d">italic_X start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> which has the property that its image subshift <math alttext="\sigma(X_{\sigma})" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.5.m5.1"><semantics id="S1.Thmthm2.p1.5.m5.1a"><mrow id="S1.Thmthm2.p1.5.m5.1.1" xref="S1.Thmthm2.p1.5.m5.1.1.cmml"><mi id="S1.Thmthm2.p1.5.m5.1.1.3" xref="S1.Thmthm2.p1.5.m5.1.1.3.cmml">σ</mi><mo id="S1.Thmthm2.p1.5.m5.1.1.2" xref="S1.Thmthm2.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S1.Thmthm2.p1.5.m5.1.1.1.1" xref="S1.Thmthm2.p1.5.m5.1.1.1.1.1.cmml"><mo id="S1.Thmthm2.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S1.Thmthm2.p1.5.m5.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmthm2.p1.5.m5.1.1.1.1.1" xref="S1.Thmthm2.p1.5.m5.1.1.1.1.1.cmml"><mi id="S1.Thmthm2.p1.5.m5.1.1.1.1.1.2" xref="S1.Thmthm2.p1.5.m5.1.1.1.1.1.2.cmml">X</mi><mi id="S1.Thmthm2.p1.5.m5.1.1.1.1.1.3" xref="S1.Thmthm2.p1.5.m5.1.1.1.1.1.3.cmml">σ</mi></msub><mo id="S1.Thmthm2.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S1.Thmthm2.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.5.m5.1b"><apply id="S1.Thmthm2.p1.5.m5.1.1.cmml" xref="S1.Thmthm2.p1.5.m5.1.1"><times id="S1.Thmthm2.p1.5.m5.1.1.2.cmml" xref="S1.Thmthm2.p1.5.m5.1.1.2"></times><ci id="S1.Thmthm2.p1.5.m5.1.1.3.cmml" xref="S1.Thmthm2.p1.5.m5.1.1.3">𝜎</ci><apply id="S1.Thmthm2.p1.5.m5.1.1.1.1.1.cmml" xref="S1.Thmthm2.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.5.m5.1.1.1.1.1.1.cmml" xref="S1.Thmthm2.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="S1.Thmthm2.p1.5.m5.1.1.1.1.1.2.cmml" xref="S1.Thmthm2.p1.5.m5.1.1.1.1.1.2">𝑋</ci><ci id="S1.Thmthm2.p1.5.m5.1.1.1.1.1.3.cmml" xref="S1.Thmthm2.p1.5.m5.1.1.1.1.1.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.5.m5.1c">\sigma(X_{\sigma})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.5.m5.1d">italic_σ ( italic_X start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT )</annotation></semantics></math> is equal to <math alttext="X_{\sigma}\," class="ltx_Math" display="inline" id="S1.Thmthm2.p1.6.m6.1"><semantics id="S1.Thmthm2.p1.6.m6.1a"><msub id="S1.Thmthm2.p1.6.m6.1.1" xref="S1.Thmthm2.p1.6.m6.1.1.cmml"><mi id="S1.Thmthm2.p1.6.m6.1.1.2" xref="S1.Thmthm2.p1.6.m6.1.1.2.cmml">X</mi><mi id="S1.Thmthm2.p1.6.m6.1.1.3" xref="S1.Thmthm2.p1.6.m6.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.6.m6.1b"><apply id="S1.Thmthm2.p1.6.m6.1.1.cmml" xref="S1.Thmthm2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.6.m6.1.1.1.cmml" xref="S1.Thmthm2.p1.6.m6.1.1">subscript</csymbol><ci id="S1.Thmthm2.p1.6.m6.1.1.2.cmml" xref="S1.Thmthm2.p1.6.m6.1.1.2">𝑋</ci><ci id="S1.Thmthm2.p1.6.m6.1.1.3.cmml" xref="S1.Thmthm2.p1.6.m6.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.6.m6.1c">X_{\sigma}\,</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.6.m6.1d">italic_X start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. To every non-negative eigenvector <math alttext="\vec{v}" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.7.m7.1"><semantics id="S1.Thmthm2.p1.7.m7.1a"><mover accent="true" id="S1.Thmthm2.p1.7.m7.1.1" xref="S1.Thmthm2.p1.7.m7.1.1.cmml"><mi id="S1.Thmthm2.p1.7.m7.1.1.2" xref="S1.Thmthm2.p1.7.m7.1.1.2.cmml">v</mi><mo id="S1.Thmthm2.p1.7.m7.1.1.1" stretchy="false" xref="S1.Thmthm2.p1.7.m7.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.7.m7.1b"><apply id="S1.Thmthm2.p1.7.m7.1.1.cmml" xref="S1.Thmthm2.p1.7.m7.1.1"><ci id="S1.Thmthm2.p1.7.m7.1.1.1.cmml" xref="S1.Thmthm2.p1.7.m7.1.1.1">→</ci><ci id="S1.Thmthm2.p1.7.m7.1.1.2.cmml" xref="S1.Thmthm2.p1.7.m7.1.1.2">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.7.m7.1c">\vec{v}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.7.m7.1d">over→ start_ARG italic_v end_ARG</annotation></semantics></math> of the transition matrix <math alttext="M(\sigma)" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.8.m8.1"><semantics id="S1.Thmthm2.p1.8.m8.1a"><mrow id="S1.Thmthm2.p1.8.m8.1.2" xref="S1.Thmthm2.p1.8.m8.1.2.cmml"><mi id="S1.Thmthm2.p1.8.m8.1.2.2" xref="S1.Thmthm2.p1.8.m8.1.2.2.cmml">M</mi><mo id="S1.Thmthm2.p1.8.m8.1.2.1" xref="S1.Thmthm2.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm2.p1.8.m8.1.2.3.2" xref="S1.Thmthm2.p1.8.m8.1.2.cmml"><mo id="S1.Thmthm2.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S1.Thmthm2.p1.8.m8.1.2.cmml">(</mo><mi id="S1.Thmthm2.p1.8.m8.1.1" xref="S1.Thmthm2.p1.8.m8.1.1.cmml">σ</mi><mo id="S1.Thmthm2.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S1.Thmthm2.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.8.m8.1b"><apply id="S1.Thmthm2.p1.8.m8.1.2.cmml" xref="S1.Thmthm2.p1.8.m8.1.2"><times id="S1.Thmthm2.p1.8.m8.1.2.1.cmml" xref="S1.Thmthm2.p1.8.m8.1.2.1"></times><ci id="S1.Thmthm2.p1.8.m8.1.2.2.cmml" xref="S1.Thmthm2.p1.8.m8.1.2.2">𝑀</ci><ci id="S1.Thmthm2.p1.8.m8.1.1.cmml" xref="S1.Thmthm2.p1.8.m8.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.8.m8.1c">M(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.8.m8.1d">italic_M ( italic_σ )</annotation></semantics></math>, with eigenvalue <math alttext="\lambda>1" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.9.m9.1"><semantics id="S1.Thmthm2.p1.9.m9.1a"><mrow id="S1.Thmthm2.p1.9.m9.1.1" xref="S1.Thmthm2.p1.9.m9.1.1.cmml"><mi id="S1.Thmthm2.p1.9.m9.1.1.2" xref="S1.Thmthm2.p1.9.m9.1.1.2.cmml">λ</mi><mo id="S1.Thmthm2.p1.9.m9.1.1.1" xref="S1.Thmthm2.p1.9.m9.1.1.1.cmml">></mo><mn id="S1.Thmthm2.p1.9.m9.1.1.3" xref="S1.Thmthm2.p1.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.9.m9.1b"><apply id="S1.Thmthm2.p1.9.m9.1.1.cmml" xref="S1.Thmthm2.p1.9.m9.1.1"><gt id="S1.Thmthm2.p1.9.m9.1.1.1.cmml" xref="S1.Thmthm2.p1.9.m9.1.1.1"></gt><ci id="S1.Thmthm2.p1.9.m9.1.1.2.cmml" xref="S1.Thmthm2.p1.9.m9.1.1.2">𝜆</ci><cn id="S1.Thmthm2.p1.9.m9.1.1.3.cmml" type="integer" xref="S1.Thmthm2.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.9.m9.1c">\lambda>1</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.9.m9.1d">italic_λ > 1</annotation></semantics></math>, there is canonically associated an invariant measure <math alttext="\mu_{\vec{v}}" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.10.m10.1"><semantics id="S1.Thmthm2.p1.10.m10.1a"><msub id="S1.Thmthm2.p1.10.m10.1.1" xref="S1.Thmthm2.p1.10.m10.1.1.cmml"><mi id="S1.Thmthm2.p1.10.m10.1.1.2" xref="S1.Thmthm2.p1.10.m10.1.1.2.cmml">μ</mi><mover accent="true" id="S1.Thmthm2.p1.10.m10.1.1.3" xref="S1.Thmthm2.p1.10.m10.1.1.3.cmml"><mi id="S1.Thmthm2.p1.10.m10.1.1.3.2" xref="S1.Thmthm2.p1.10.m10.1.1.3.2.cmml">v</mi><mo id="S1.Thmthm2.p1.10.m10.1.1.3.1" stretchy="false" xref="S1.Thmthm2.p1.10.m10.1.1.3.1.cmml">→</mo></mover></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.10.m10.1b"><apply id="S1.Thmthm2.p1.10.m10.1.1.cmml" xref="S1.Thmthm2.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.10.m10.1.1.1.cmml" xref="S1.Thmthm2.p1.10.m10.1.1">subscript</csymbol><ci id="S1.Thmthm2.p1.10.m10.1.1.2.cmml" xref="S1.Thmthm2.p1.10.m10.1.1.2">𝜇</ci><apply id="S1.Thmthm2.p1.10.m10.1.1.3.cmml" xref="S1.Thmthm2.p1.10.m10.1.1.3"><ci id="S1.Thmthm2.p1.10.m10.1.1.3.1.cmml" xref="S1.Thmthm2.p1.10.m10.1.1.3.1">→</ci><ci id="S1.Thmthm2.p1.10.m10.1.1.3.2.cmml" xref="S1.Thmthm2.p1.10.m10.1.1.3.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.10.m10.1c">\mu_{\vec{v}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.10.m10.1d">italic_μ start_POSTSUBSCRIPT over→ start_ARG italic_v end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> on the subshift <math alttext="X_{\sigma}" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.11.m11.1"><semantics id="S1.Thmthm2.p1.11.m11.1a"><msub id="S1.Thmthm2.p1.11.m11.1.1" xref="S1.Thmthm2.p1.11.m11.1.1.cmml"><mi id="S1.Thmthm2.p1.11.m11.1.1.2" xref="S1.Thmthm2.p1.11.m11.1.1.2.cmml">X</mi><mi id="S1.Thmthm2.p1.11.m11.1.1.3" xref="S1.Thmthm2.p1.11.m11.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.11.m11.1b"><apply id="S1.Thmthm2.p1.11.m11.1.1.cmml" xref="S1.Thmthm2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.11.m11.1.1.1.cmml" xref="S1.Thmthm2.p1.11.m11.1.1">subscript</csymbol><ci id="S1.Thmthm2.p1.11.m11.1.1.2.cmml" xref="S1.Thmthm2.p1.11.m11.1.1.2">𝑋</ci><ci id="S1.Thmthm2.p1.11.m11.1.1.3.cmml" xref="S1.Thmthm2.p1.11.m11.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.11.m11.1c">X_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.11.m11.1d">italic_X start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib2" title="">2</a>]</cite>). It can be shown (using Proposition 4.3 and Remark 4.2 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>) that the measure transfer map induced by <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.12.m12.1"><semantics id="S1.Thmthm2.p1.12.m12.1a"><mi id="S1.Thmthm2.p1.12.m12.1.1" xref="S1.Thmthm2.p1.12.m12.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.12.m12.1b"><ci id="S1.Thmthm2.p1.12.m12.1.1.cmml" xref="S1.Thmthm2.p1.12.m12.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.12.m12.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.12.m12.1d">italic_σ</annotation></semantics></math> acts on <math alttext="\mu_{\vec{v}}" class="ltx_Math" display="inline" id="S1.Thmthm2.p1.13.m13.1"><semantics id="S1.Thmthm2.p1.13.m13.1a"><msub id="S1.Thmthm2.p1.13.m13.1.1" xref="S1.Thmthm2.p1.13.m13.1.1.cmml"><mi id="S1.Thmthm2.p1.13.m13.1.1.2" xref="S1.Thmthm2.p1.13.m13.1.1.2.cmml">μ</mi><mover accent="true" id="S1.Thmthm2.p1.13.m13.1.1.3" xref="S1.Thmthm2.p1.13.m13.1.1.3.cmml"><mi id="S1.Thmthm2.p1.13.m13.1.1.3.2" xref="S1.Thmthm2.p1.13.m13.1.1.3.2.cmml">v</mi><mo id="S1.Thmthm2.p1.13.m13.1.1.3.1" stretchy="false" xref="S1.Thmthm2.p1.13.m13.1.1.3.1.cmml">→</mo></mover></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.13.m13.1b"><apply id="S1.Thmthm2.p1.13.m13.1.1.cmml" xref="S1.Thmthm2.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S1.Thmthm2.p1.13.m13.1.1.1.cmml" xref="S1.Thmthm2.p1.13.m13.1.1">subscript</csymbol><ci id="S1.Thmthm2.p1.13.m13.1.1.2.cmml" xref="S1.Thmthm2.p1.13.m13.1.1.2">𝜇</ci><apply id="S1.Thmthm2.p1.13.m13.1.1.3.cmml" xref="S1.Thmthm2.p1.13.m13.1.1.3"><ci id="S1.Thmthm2.p1.13.m13.1.1.3.1.cmml" xref="S1.Thmthm2.p1.13.m13.1.1.3.1">→</ci><ci id="S1.Thmthm2.p1.13.m13.1.1.3.2.cmml" xref="S1.Thmthm2.p1.13.m13.1.1.3.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.13.m13.1c">\mu_{\vec{v}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.13.m13.1d">italic_μ start_POSTSUBSCRIPT over→ start_ARG italic_v end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> as homothety with stretching factor <math alttext="\lambda\," class="ltx_Math" display="inline" id="S1.Thmthm2.p1.14.m14.1"><semantics id="S1.Thmthm2.p1.14.m14.1a"><mi id="S1.Thmthm2.p1.14.m14.1.1" xref="S1.Thmthm2.p1.14.m14.1.1.cmml">λ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm2.p1.14.m14.1b"><ci id="S1.Thmthm2.p1.14.m14.1.1.cmml" xref="S1.Thmthm2.p1.14.m14.1.1">𝜆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm2.p1.14.m14.1c">\lambda\,</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm2.p1.14.m14.1d">italic_λ</annotation></semantics></math>, giving</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu_{\vec{v}}^{\sigma}=\lambda\cdot\mu_{\vec{v}}\,." class="ltx_Math" display="block" id="S1.Ex3.m1.1"><semantics id="S1.Ex3.m1.1a"><mrow id="S1.Ex3.m1.1.1.1" xref="S1.Ex3.m1.1.1.1.1.cmml"><mrow id="S1.Ex3.m1.1.1.1.1" xref="S1.Ex3.m1.1.1.1.1.cmml"><msubsup id="S1.Ex3.m1.1.1.1.1.2" xref="S1.Ex3.m1.1.1.1.1.2.cmml"><mi id="S1.Ex3.m1.1.1.1.1.2.2.2" xref="S1.Ex3.m1.1.1.1.1.2.2.2.cmml">μ</mi><mover accent="true" id="S1.Ex3.m1.1.1.1.1.2.2.3" xref="S1.Ex3.m1.1.1.1.1.2.2.3.cmml"><mi id="S1.Ex3.m1.1.1.1.1.2.2.3.2" xref="S1.Ex3.m1.1.1.1.1.2.2.3.2.cmml">v</mi><mo id="S1.Ex3.m1.1.1.1.1.2.2.3.1" stretchy="false" xref="S1.Ex3.m1.1.1.1.1.2.2.3.1.cmml">→</mo></mover><mi id="S1.Ex3.m1.1.1.1.1.2.3" xref="S1.Ex3.m1.1.1.1.1.2.3.cmml">σ</mi></msubsup><mo id="S1.Ex3.m1.1.1.1.1.1" xref="S1.Ex3.m1.1.1.1.1.1.cmml">=</mo><mrow id="S1.Ex3.m1.1.1.1.1.3" xref="S1.Ex3.m1.1.1.1.1.3.cmml"><mi id="S1.Ex3.m1.1.1.1.1.3.2" xref="S1.Ex3.m1.1.1.1.1.3.2.cmml">λ</mi><mo id="S1.Ex3.m1.1.1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S1.Ex3.m1.1.1.1.1.3.1.cmml">⋅</mo><msub id="S1.Ex3.m1.1.1.1.1.3.3" xref="S1.Ex3.m1.1.1.1.1.3.3.cmml"><mi id="S1.Ex3.m1.1.1.1.1.3.3.2" xref="S1.Ex3.m1.1.1.1.1.3.3.2.cmml">μ</mi><mover accent="true" id="S1.Ex3.m1.1.1.1.1.3.3.3" xref="S1.Ex3.m1.1.1.1.1.3.3.3.cmml"><mi id="S1.Ex3.m1.1.1.1.1.3.3.3.2" xref="S1.Ex3.m1.1.1.1.1.3.3.3.2.cmml">v</mi><mo id="S1.Ex3.m1.1.1.1.1.3.3.3.1" stretchy="false" xref="S1.Ex3.m1.1.1.1.1.3.3.3.1.cmml">→</mo></mover></msub></mrow></mrow><mo id="S1.Ex3.m1.1.1.1.2" lspace="0em" xref="S1.Ex3.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex3.m1.1b"><apply id="S1.Ex3.m1.1.1.1.1.cmml" xref="S1.Ex3.m1.1.1.1"><eq id="S1.Ex3.m1.1.1.1.1.1.cmml" xref="S1.Ex3.m1.1.1.1.1.1"></eq><apply id="S1.Ex3.m1.1.1.1.1.2.cmml" xref="S1.Ex3.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.Ex3.m1.1.1.1.1.2.1.cmml" xref="S1.Ex3.m1.1.1.1.1.2">superscript</csymbol><apply id="S1.Ex3.m1.1.1.1.1.2.2.cmml" xref="S1.Ex3.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.Ex3.m1.1.1.1.1.2.2.1.cmml" xref="S1.Ex3.m1.1.1.1.1.2">subscript</csymbol><ci id="S1.Ex3.m1.1.1.1.1.2.2.2.cmml" xref="S1.Ex3.m1.1.1.1.1.2.2.2">𝜇</ci><apply id="S1.Ex3.m1.1.1.1.1.2.2.3.cmml" xref="S1.Ex3.m1.1.1.1.1.2.2.3"><ci id="S1.Ex3.m1.1.1.1.1.2.2.3.1.cmml" xref="S1.Ex3.m1.1.1.1.1.2.2.3.1">→</ci><ci id="S1.Ex3.m1.1.1.1.1.2.2.3.2.cmml" xref="S1.Ex3.m1.1.1.1.1.2.2.3.2">𝑣</ci></apply></apply><ci id="S1.Ex3.m1.1.1.1.1.2.3.cmml" xref="S1.Ex3.m1.1.1.1.1.2.3">𝜎</ci></apply><apply id="S1.Ex3.m1.1.1.1.1.3.cmml" xref="S1.Ex3.m1.1.1.1.1.3"><ci id="S1.Ex3.m1.1.1.1.1.3.1.cmml" xref="S1.Ex3.m1.1.1.1.1.3.1">⋅</ci><ci id="S1.Ex3.m1.1.1.1.1.3.2.cmml" xref="S1.Ex3.m1.1.1.1.1.3.2">𝜆</ci><apply id="S1.Ex3.m1.1.1.1.1.3.3.cmml" xref="S1.Ex3.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S1.Ex3.m1.1.1.1.1.3.3.1.cmml" xref="S1.Ex3.m1.1.1.1.1.3.3">subscript</csymbol><ci id="S1.Ex3.m1.1.1.1.1.3.3.2.cmml" xref="S1.Ex3.m1.1.1.1.1.3.3.2">𝜇</ci><apply id="S1.Ex3.m1.1.1.1.1.3.3.3.cmml" xref="S1.Ex3.m1.1.1.1.1.3.3.3"><ci id="S1.Ex3.m1.1.1.1.1.3.3.3.1.cmml" xref="S1.Ex3.m1.1.1.1.1.3.3.3.1">→</ci><ci id="S1.Ex3.m1.1.1.1.1.3.3.3.2.cmml" xref="S1.Ex3.m1.1.1.1.1.3.3.3.2">𝑣</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex3.m1.1c">\mu_{\vec{v}}^{\sigma}=\lambda\cdot\mu_{\vec{v}}\,.</annotation><annotation encoding="application/x-llamapun" id="S1.Ex3.m1.1d">italic_μ start_POSTSUBSCRIPT over→ start_ARG italic_v end_ARG end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = italic_λ ⋅ italic_μ start_POSTSUBSCRIPT over→ start_ARG italic_v end_ARG end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">The last equality indicates already that the transferred measure of a probability measure will in general not be probability. Indeed, we have (see Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.Thmthm4" title="Remark 4.4. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.4</span></a> and Proposition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.Thmthm5" title="Proposition 4.5. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.5</span></a> below):</p> </div> <div class="ltx_theorem ltx_theorem_prop" id="S1.Thmthm3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmthm3.1.1.1">Proposition 1.3</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmthm3.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmthm3.p1"> <p class="ltx_p" id="S1.Thmthm3.p1.4"><span class="ltx_text ltx_font_italic" id="S1.Thmthm3.p1.4.4">For any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S1.Thmthm3.p1.1.1.m1.1"><semantics id="S1.Thmthm3.p1.1.1.m1.1a"><mi id="S1.Thmthm3.p1.1.1.m1.1.1" xref="S1.Thmthm3.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm3.p1.1.1.m1.1b"><ci id="S1.Thmthm3.p1.1.1.m1.1.1.cmml" xref="S1.Thmthm3.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm3.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm3.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> on the full shift <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.Thmthm3.p1.2.2.m2.1"><semantics id="S1.Thmthm3.p1.2.2.m2.1a"><msup id="S1.Thmthm3.p1.2.2.m2.1.1" xref="S1.Thmthm3.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm3.p1.2.2.m2.1.1.2" xref="S1.Thmthm3.p1.2.2.m2.1.1.2.cmml">𝒜</mi><mi id="S1.Thmthm3.p1.2.2.m2.1.1.3" xref="S1.Thmthm3.p1.2.2.m2.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.Thmthm3.p1.2.2.m2.1b"><apply id="S1.Thmthm3.p1.2.2.m2.1.1.cmml" xref="S1.Thmthm3.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.Thmthm3.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmthm3.p1.2.2.m2.1.1">superscript</csymbol><ci id="S1.Thmthm3.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmthm3.p1.2.2.m2.1.1.2">𝒜</ci><ci id="S1.Thmthm3.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmthm3.p1.2.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm3.p1.2.2.m2.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm3.p1.2.2.m2.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and any non-erasing morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S1.Thmthm3.p1.3.3.m3.1"><semantics id="S1.Thmthm3.p1.3.3.m3.1a"><mrow id="S1.Thmthm3.p1.3.3.m3.1.1" xref="S1.Thmthm3.p1.3.3.m3.1.1.cmml"><mi id="S1.Thmthm3.p1.3.3.m3.1.1.2" xref="S1.Thmthm3.p1.3.3.m3.1.1.2.cmml">σ</mi><mo id="S1.Thmthm3.p1.3.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm3.p1.3.3.m3.1.1.1.cmml">:</mo><mrow id="S1.Thmthm3.p1.3.3.m3.1.1.3" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.cmml"><msup id="S1.Thmthm3.p1.3.3.m3.1.1.3.2" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm3.p1.3.3.m3.1.1.3.2.2" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.Thmthm3.p1.3.3.m3.1.1.3.2.3" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.Thmthm3.p1.3.3.m3.1.1.3.1" stretchy="false" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.1.cmml">→</mo><msup id="S1.Thmthm3.p1.3.3.m3.1.1.3.3" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm3.p1.3.3.m3.1.1.3.3.2" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.3.2.cmml">ℬ</mi><mo id="S1.Thmthm3.p1.3.3.m3.1.1.3.3.3" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm3.p1.3.3.m3.1b"><apply id="S1.Thmthm3.p1.3.3.m3.1.1.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1"><ci id="S1.Thmthm3.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.1">:</ci><ci id="S1.Thmthm3.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.2">𝜎</ci><apply id="S1.Thmthm3.p1.3.3.m3.1.1.3.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3"><ci id="S1.Thmthm3.p1.3.3.m3.1.1.3.1.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.1">→</ci><apply id="S1.Thmthm3.p1.3.3.m3.1.1.3.2.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmthm3.p1.3.3.m3.1.1.3.2.1.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.2">superscript</csymbol><ci id="S1.Thmthm3.p1.3.3.m3.1.1.3.2.2.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.2.2">𝒜</ci><times id="S1.Thmthm3.p1.3.3.m3.1.1.3.2.3.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.2.3"></times></apply><apply id="S1.Thmthm3.p1.3.3.m3.1.1.3.3.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmthm3.p1.3.3.m3.1.1.3.3.1.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.3">superscript</csymbol><ci id="S1.Thmthm3.p1.3.3.m3.1.1.3.3.2.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.3.2">ℬ</ci><times id="S1.Thmthm3.p1.3.3.m3.1.1.3.3.3.cmml" xref="S1.Thmthm3.p1.3.3.m3.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm3.p1.3.3.m3.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm3.p1.3.3.m3.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the transferred measure <math alttext="\mu^{\sigma}=\sigma M(\mu)" class="ltx_Math" display="inline" id="S1.Thmthm3.p1.4.4.m4.1"><semantics id="S1.Thmthm3.p1.4.4.m4.1a"><mrow id="S1.Thmthm3.p1.4.4.m4.1.2" xref="S1.Thmthm3.p1.4.4.m4.1.2.cmml"><msup id="S1.Thmthm3.p1.4.4.m4.1.2.2" xref="S1.Thmthm3.p1.4.4.m4.1.2.2.cmml"><mi id="S1.Thmthm3.p1.4.4.m4.1.2.2.2" xref="S1.Thmthm3.p1.4.4.m4.1.2.2.2.cmml">μ</mi><mi id="S1.Thmthm3.p1.4.4.m4.1.2.2.3" xref="S1.Thmthm3.p1.4.4.m4.1.2.2.3.cmml">σ</mi></msup><mo id="S1.Thmthm3.p1.4.4.m4.1.2.1" xref="S1.Thmthm3.p1.4.4.m4.1.2.1.cmml">=</mo><mrow id="S1.Thmthm3.p1.4.4.m4.1.2.3" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.cmml"><mi id="S1.Thmthm3.p1.4.4.m4.1.2.3.2" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.2.cmml">σ</mi><mo id="S1.Thmthm3.p1.4.4.m4.1.2.3.1" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.1.cmml">⁢</mo><mi id="S1.Thmthm3.p1.4.4.m4.1.2.3.3" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.3.cmml">M</mi><mo id="S1.Thmthm3.p1.4.4.m4.1.2.3.1a" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S1.Thmthm3.p1.4.4.m4.1.2.3.4.2" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.cmml"><mo id="S1.Thmthm3.p1.4.4.m4.1.2.3.4.2.1" stretchy="false" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.cmml">(</mo><mi id="S1.Thmthm3.p1.4.4.m4.1.1" xref="S1.Thmthm3.p1.4.4.m4.1.1.cmml">μ</mi><mo id="S1.Thmthm3.p1.4.4.m4.1.2.3.4.2.2" stretchy="false" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm3.p1.4.4.m4.1b"><apply id="S1.Thmthm3.p1.4.4.m4.1.2.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2"><eq id="S1.Thmthm3.p1.4.4.m4.1.2.1.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.1"></eq><apply id="S1.Thmthm3.p1.4.4.m4.1.2.2.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S1.Thmthm3.p1.4.4.m4.1.2.2.1.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.2">superscript</csymbol><ci id="S1.Thmthm3.p1.4.4.m4.1.2.2.2.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.2.2">𝜇</ci><ci id="S1.Thmthm3.p1.4.4.m4.1.2.2.3.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.2.3">𝜎</ci></apply><apply id="S1.Thmthm3.p1.4.4.m4.1.2.3.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.3"><times id="S1.Thmthm3.p1.4.4.m4.1.2.3.1.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.1"></times><ci id="S1.Thmthm3.p1.4.4.m4.1.2.3.2.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.2">𝜎</ci><ci id="S1.Thmthm3.p1.4.4.m4.1.2.3.3.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.2.3.3">𝑀</ci><ci id="S1.Thmthm3.p1.4.4.m4.1.1.cmml" xref="S1.Thmthm3.p1.4.4.m4.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm3.p1.4.4.m4.1c">\mu^{\sigma}=\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm3.p1.4.4.m4.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = italic_σ italic_M ( italic_μ )</annotation></semantics></math> satisfies</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu^{\sigma}(\cal B^{\mathbb{Z}})=\sum_{a_{k}\in\cal A}\sum_{b_{j}\in\cal B}|% \sigma(a_{k})|_{b_{j}}\cdot\mu([a_{k}])=\sum_{a_{k}\in\cal A}|\sigma(a_{k})|% \cdot\mu([a_{k}])\,." class="ltx_Math" display="block" id="S1.Ex4.m1.1"><semantics id="S1.Ex4.m1.1a"><mrow id="S1.Ex4.m1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.cmml"><mrow id="S1.Ex4.m1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.cmml"><mrow id="S1.Ex4.m1.1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.1.cmml"><msup id="S1.Ex4.m1.1.1.1.1.1.3" xref="S1.Ex4.m1.1.1.1.1.1.3.cmml"><mi id="S1.Ex4.m1.1.1.1.1.1.3.2" xref="S1.Ex4.m1.1.1.1.1.1.3.2.cmml">μ</mi><mi id="S1.Ex4.m1.1.1.1.1.1.3.3" xref="S1.Ex4.m1.1.1.1.1.1.3.3.cmml">σ</mi></msup><mo id="S1.Ex4.m1.1.1.1.1.1.2" xref="S1.Ex4.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex4.m1.1.1.1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S1.Ex4.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.Ex4.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S1.Ex4.m1.1.1.1.1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.1.1.1.1.2" xref="S1.Ex4.m1.1.1.1.1.1.1.1.1.2.cmml">ℬ</mi><mi id="S1.Ex4.m1.1.1.1.1.1.1.1.1.3" xref="S1.Ex4.m1.1.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S1.Ex4.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S1.Ex4.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Ex4.m1.1.1.1.1.7" rspace="0.111em" xref="S1.Ex4.m1.1.1.1.1.7.cmml">=</mo><mrow id="S1.Ex4.m1.1.1.1.1.3" xref="S1.Ex4.m1.1.1.1.1.3.cmml"><munder id="S1.Ex4.m1.1.1.1.1.3.3" xref="S1.Ex4.m1.1.1.1.1.3.3.cmml"><mo id="S1.Ex4.m1.1.1.1.1.3.3.2" movablelimits="false" rspace="0em" xref="S1.Ex4.m1.1.1.1.1.3.3.2.cmml">∑</mo><mrow id="S1.Ex4.m1.1.1.1.1.3.3.3" xref="S1.Ex4.m1.1.1.1.1.3.3.3.cmml"><msub id="S1.Ex4.m1.1.1.1.1.3.3.3.2" xref="S1.Ex4.m1.1.1.1.1.3.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.3.3.3.2.2" xref="S1.Ex4.m1.1.1.1.1.3.3.3.2.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.3.3.3.2.3" xref="S1.Ex4.m1.1.1.1.1.3.3.3.2.3.cmml">𝓀</mi></msub><mo id="S1.Ex4.m1.1.1.1.1.3.3.3.1" xref="S1.Ex4.m1.1.1.1.1.3.3.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.3.3.3.3" xref="S1.Ex4.m1.1.1.1.1.3.3.3.3.cmml">𝒜</mi></mrow></munder><mrow id="S1.Ex4.m1.1.1.1.1.3.2" xref="S1.Ex4.m1.1.1.1.1.3.2.cmml"><munder id="S1.Ex4.m1.1.1.1.1.3.2.3" xref="S1.Ex4.m1.1.1.1.1.3.2.3.cmml"><mo id="S1.Ex4.m1.1.1.1.1.3.2.3.2" movablelimits="false" rspace="0em" xref="S1.Ex4.m1.1.1.1.1.3.2.3.2.cmml">∑</mo><mrow id="S1.Ex4.m1.1.1.1.1.3.2.3.3" xref="S1.Ex4.m1.1.1.1.1.3.2.3.3.cmml"><msub id="S1.Ex4.m1.1.1.1.1.3.2.3.3.2" xref="S1.Ex4.m1.1.1.1.1.3.2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.3.2.3.3.2.2" xref="S1.Ex4.m1.1.1.1.1.3.2.3.3.2.2.cmml">𝒷</mi><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.3.2.3.3.2.3" xref="S1.Ex4.m1.1.1.1.1.3.2.3.3.2.3.cmml">𝒿</mi></msub><mo id="S1.Ex4.m1.1.1.1.1.3.2.3.3.1" xref="S1.Ex4.m1.1.1.1.1.3.2.3.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.3.2.3.3.3" xref="S1.Ex4.m1.1.1.1.1.3.2.3.3.3.cmml">ℬ</mi></mrow></munder><mrow id="S1.Ex4.m1.1.1.1.1.3.2.2" xref="S1.Ex4.m1.1.1.1.1.3.2.2.cmml"><mrow id="S1.Ex4.m1.1.1.1.1.2.1.1.1" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.cmml"><msub id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.cmml"><mrow id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.2.cmml"><mo id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.2" stretchy="false" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.2.1.cmml">|</mo><mrow id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.cmml"><mi id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.3" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.3.cmml">σ</mi><mo id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.2" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.2" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.3" xref="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.3.cmml">𝓀</mi></msub><mo id="S1.Ex4.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3" stretchy="false" 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caligraphic_a start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT caligraphic_b start_POSTSUBSCRIPT caligraphic_j end_POSTSUBSCRIPT ∈ caligraphic_B end_POSTSUBSCRIPT | italic_σ ( caligraphic_a start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT caligraphic_b start_POSTSUBSCRIPT caligraphic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋅ italic_μ ( [ caligraphic_a start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ] ) = ∑ start_POSTSUBSCRIPT caligraphic_a start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT | italic_σ ( caligraphic_a start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ) | ⋅ italic_μ ( [ caligraphic_a start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ] ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.Thmthm3.p1.5"><span class="ltx_text ltx_font_italic" 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italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ] ) start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.11">Here <math alttext="|w|_{u}" class="ltx_Math" display="inline" id="S1.p6.1.m1.1"><semantics id="S1.p6.1.m1.1a"><msub id="S1.p6.1.m1.1.2" xref="S1.p6.1.m1.1.2.cmml"><mrow id="S1.p6.1.m1.1.2.2.2" xref="S1.p6.1.m1.1.2.2.1.cmml"><mo id="S1.p6.1.m1.1.2.2.2.1" stretchy="false" xref="S1.p6.1.m1.1.2.2.1.1.cmml">|</mo><mi id="S1.p6.1.m1.1.1" xref="S1.p6.1.m1.1.1.cmml">w</mi><mo id="S1.p6.1.m1.1.2.2.2.2" stretchy="false" xref="S1.p6.1.m1.1.2.2.1.1.cmml">|</mo></mrow><mi id="S1.p6.1.m1.1.2.3" xref="S1.p6.1.m1.1.2.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p6.1.m1.1b"><apply id="S1.p6.1.m1.1.2.cmml" xref="S1.p6.1.m1.1.2"><csymbol cd="ambiguous" id="S1.p6.1.m1.1.2.1.cmml" xref="S1.p6.1.m1.1.2">subscript</csymbol><apply id="S1.p6.1.m1.1.2.2.1.cmml" xref="S1.p6.1.m1.1.2.2.2"><abs id="S1.p6.1.m1.1.2.2.1.1.cmml" xref="S1.p6.1.m1.1.2.2.2.1"></abs><ci id="S1.p6.1.m1.1.1.cmml" xref="S1.p6.1.m1.1.1">𝑤</ci></apply><ci id="S1.p6.1.m1.1.2.3.cmml" xref="S1.p6.1.m1.1.2.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.1.m1.1c">|w|_{u}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.1.m1.1d">| italic_w | start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> denotes the number of (possibly overlapping) occurrences of the word <math alttext="u" class="ltx_Math" display="inline" id="S1.p6.2.m2.1"><semantics id="S1.p6.2.m2.1a"><mi id="S1.p6.2.m2.1.1" xref="S1.p6.2.m2.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S1.p6.2.m2.1b"><ci id="S1.p6.2.m2.1.1.cmml" xref="S1.p6.2.m2.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.2.m2.1c">u</annotation><annotation encoding="application/x-llamapun" id="S1.p6.2.m2.1d">italic_u</annotation></semantics></math> as factor in the word <math alttext="w" class="ltx_Math" display="inline" id="S1.p6.3.m3.1"><semantics id="S1.p6.3.m3.1a"><mi id="S1.p6.3.m3.1.1" xref="S1.p6.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S1.p6.3.m3.1b"><ci id="S1.p6.3.m3.1.1.cmml" xref="S1.p6.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S1.p6.3.m3.1d">italic_w</annotation></semantics></math>, and <math alttext="[w]" class="ltx_Math" display="inline" id="S1.p6.4.m4.1"><semantics id="S1.p6.4.m4.1a"><mrow id="S1.p6.4.m4.1.2.2" xref="S1.p6.4.m4.1.2.1.cmml"><mo id="S1.p6.4.m4.1.2.2.1" stretchy="false" xref="S1.p6.4.m4.1.2.1.1.cmml">[</mo><mi id="S1.p6.4.m4.1.1" xref="S1.p6.4.m4.1.1.cmml">w</mi><mo id="S1.p6.4.m4.1.2.2.2" stretchy="false" xref="S1.p6.4.m4.1.2.1.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.4.m4.1b"><apply id="S1.p6.4.m4.1.2.1.cmml" xref="S1.p6.4.m4.1.2.2"><csymbol cd="latexml" id="S1.p6.4.m4.1.2.1.1.cmml" xref="S1.p6.4.m4.1.2.2.1">delimited-[]</csymbol><ci id="S1.p6.4.m4.1.1.cmml" xref="S1.p6.4.m4.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.4.m4.1c">[w]</annotation><annotation encoding="application/x-llamapun" id="S1.p6.4.m4.1d">[ italic_w ]</annotation></semantics></math> is the “cylinder” that consists of all biinfinite words <math alttext="\ldots x_{n}x_{n+1}\ldots" class="ltx_Math" display="inline" id="S1.p6.5.m5.1"><semantics id="S1.p6.5.m5.1a"><mrow id="S1.p6.5.m5.1.1" xref="S1.p6.5.m5.1.1.cmml"><mi id="S1.p6.5.m5.1.1.2" mathvariant="normal" xref="S1.p6.5.m5.1.1.2.cmml">…</mi><mo id="S1.p6.5.m5.1.1.1" xref="S1.p6.5.m5.1.1.1.cmml">⁢</mo><msub id="S1.p6.5.m5.1.1.3" xref="S1.p6.5.m5.1.1.3.cmml"><mi id="S1.p6.5.m5.1.1.3.2" xref="S1.p6.5.m5.1.1.3.2.cmml">x</mi><mi id="S1.p6.5.m5.1.1.3.3" xref="S1.p6.5.m5.1.1.3.3.cmml">n</mi></msub><mo id="S1.p6.5.m5.1.1.1a" xref="S1.p6.5.m5.1.1.1.cmml">⁢</mo><msub id="S1.p6.5.m5.1.1.4" xref="S1.p6.5.m5.1.1.4.cmml"><mi id="S1.p6.5.m5.1.1.4.2" xref="S1.p6.5.m5.1.1.4.2.cmml">x</mi><mrow id="S1.p6.5.m5.1.1.4.3" xref="S1.p6.5.m5.1.1.4.3.cmml"><mi id="S1.p6.5.m5.1.1.4.3.2" xref="S1.p6.5.m5.1.1.4.3.2.cmml">n</mi><mo id="S1.p6.5.m5.1.1.4.3.1" xref="S1.p6.5.m5.1.1.4.3.1.cmml">+</mo><mn id="S1.p6.5.m5.1.1.4.3.3" xref="S1.p6.5.m5.1.1.4.3.3.cmml">1</mn></mrow></msub><mo id="S1.p6.5.m5.1.1.1b" xref="S1.p6.5.m5.1.1.1.cmml">⁢</mo><mi id="S1.p6.5.m5.1.1.5" mathvariant="normal" xref="S1.p6.5.m5.1.1.5.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.5.m5.1b"><apply id="S1.p6.5.m5.1.1.cmml" xref="S1.p6.5.m5.1.1"><times id="S1.p6.5.m5.1.1.1.cmml" xref="S1.p6.5.m5.1.1.1"></times><ci id="S1.p6.5.m5.1.1.2.cmml" xref="S1.p6.5.m5.1.1.2">…</ci><apply id="S1.p6.5.m5.1.1.3.cmml" xref="S1.p6.5.m5.1.1.3"><csymbol cd="ambiguous" id="S1.p6.5.m5.1.1.3.1.cmml" xref="S1.p6.5.m5.1.1.3">subscript</csymbol><ci id="S1.p6.5.m5.1.1.3.2.cmml" xref="S1.p6.5.m5.1.1.3.2">𝑥</ci><ci id="S1.p6.5.m5.1.1.3.3.cmml" xref="S1.p6.5.m5.1.1.3.3">𝑛</ci></apply><apply id="S1.p6.5.m5.1.1.4.cmml" xref="S1.p6.5.m5.1.1.4"><csymbol cd="ambiguous" id="S1.p6.5.m5.1.1.4.1.cmml" xref="S1.p6.5.m5.1.1.4">subscript</csymbol><ci id="S1.p6.5.m5.1.1.4.2.cmml" xref="S1.p6.5.m5.1.1.4.2">𝑥</ci><apply id="S1.p6.5.m5.1.1.4.3.cmml" xref="S1.p6.5.m5.1.1.4.3"><plus id="S1.p6.5.m5.1.1.4.3.1.cmml" xref="S1.p6.5.m5.1.1.4.3.1"></plus><ci id="S1.p6.5.m5.1.1.4.3.2.cmml" xref="S1.p6.5.m5.1.1.4.3.2">𝑛</ci><cn id="S1.p6.5.m5.1.1.4.3.3.cmml" type="integer" xref="S1.p6.5.m5.1.1.4.3.3">1</cn></apply></apply><ci id="S1.p6.5.m5.1.1.5.cmml" xref="S1.p6.5.m5.1.1.5">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.5.m5.1c">\ldots x_{n}x_{n+1}\ldots</annotation><annotation encoding="application/x-llamapun" id="S1.p6.5.m5.1d">… italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT …</annotation></semantics></math> for which the positive half-word <math alttext="x_{1}x_{2}\ldots" class="ltx_Math" display="inline" id="S1.p6.6.m6.1"><semantics id="S1.p6.6.m6.1a"><mrow id="S1.p6.6.m6.1.1" xref="S1.p6.6.m6.1.1.cmml"><msub id="S1.p6.6.m6.1.1.2" xref="S1.p6.6.m6.1.1.2.cmml"><mi id="S1.p6.6.m6.1.1.2.2" xref="S1.p6.6.m6.1.1.2.2.cmml">x</mi><mn id="S1.p6.6.m6.1.1.2.3" xref="S1.p6.6.m6.1.1.2.3.cmml">1</mn></msub><mo id="S1.p6.6.m6.1.1.1" xref="S1.p6.6.m6.1.1.1.cmml">⁢</mo><msub id="S1.p6.6.m6.1.1.3" xref="S1.p6.6.m6.1.1.3.cmml"><mi id="S1.p6.6.m6.1.1.3.2" xref="S1.p6.6.m6.1.1.3.2.cmml">x</mi><mn id="S1.p6.6.m6.1.1.3.3" xref="S1.p6.6.m6.1.1.3.3.cmml">2</mn></msub><mo id="S1.p6.6.m6.1.1.1a" xref="S1.p6.6.m6.1.1.1.cmml">⁢</mo><mi id="S1.p6.6.m6.1.1.4" mathvariant="normal" xref="S1.p6.6.m6.1.1.4.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.6.m6.1b"><apply id="S1.p6.6.m6.1.1.cmml" xref="S1.p6.6.m6.1.1"><times id="S1.p6.6.m6.1.1.1.cmml" xref="S1.p6.6.m6.1.1.1"></times><apply id="S1.p6.6.m6.1.1.2.cmml" xref="S1.p6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.p6.6.m6.1.1.2.1.cmml" xref="S1.p6.6.m6.1.1.2">subscript</csymbol><ci id="S1.p6.6.m6.1.1.2.2.cmml" xref="S1.p6.6.m6.1.1.2.2">𝑥</ci><cn id="S1.p6.6.m6.1.1.2.3.cmml" type="integer" xref="S1.p6.6.m6.1.1.2.3">1</cn></apply><apply id="S1.p6.6.m6.1.1.3.cmml" xref="S1.p6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.p6.6.m6.1.1.3.1.cmml" xref="S1.p6.6.m6.1.1.3">subscript</csymbol><ci id="S1.p6.6.m6.1.1.3.2.cmml" xref="S1.p6.6.m6.1.1.3.2">𝑥</ci><cn id="S1.p6.6.m6.1.1.3.3.cmml" type="integer" xref="S1.p6.6.m6.1.1.3.3">2</cn></apply><ci id="S1.p6.6.m6.1.1.4.cmml" xref="S1.p6.6.m6.1.1.4">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.6.m6.1c">x_{1}x_{2}\ldots</annotation><annotation encoding="application/x-llamapun" id="S1.p6.6.m6.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT …</annotation></semantics></math> starts with <math alttext="w" class="ltx_Math" display="inline" id="S1.p6.7.m7.1"><semantics id="S1.p6.7.m7.1a"><mi id="S1.p6.7.m7.1.1" xref="S1.p6.7.m7.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S1.p6.7.m7.1b"><ci id="S1.p6.7.m7.1.1.cmml" xref="S1.p6.7.m7.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.7.m7.1c">w</annotation><annotation encoding="application/x-llamapun" id="S1.p6.7.m7.1d">italic_w</annotation></semantics></math> as prefix. By <math alttext="M(\sigma)" class="ltx_Math" display="inline" id="S1.p6.8.m8.1"><semantics id="S1.p6.8.m8.1a"><mrow id="S1.p6.8.m8.1.2" xref="S1.p6.8.m8.1.2.cmml"><mi id="S1.p6.8.m8.1.2.2" xref="S1.p6.8.m8.1.2.2.cmml">M</mi><mo id="S1.p6.8.m8.1.2.1" xref="S1.p6.8.m8.1.2.1.cmml">⁢</mo><mrow id="S1.p6.8.m8.1.2.3.2" xref="S1.p6.8.m8.1.2.cmml"><mo id="S1.p6.8.m8.1.2.3.2.1" stretchy="false" xref="S1.p6.8.m8.1.2.cmml">(</mo><mi id="S1.p6.8.m8.1.1" xref="S1.p6.8.m8.1.1.cmml">σ</mi><mo id="S1.p6.8.m8.1.2.3.2.2" stretchy="false" xref="S1.p6.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.8.m8.1b"><apply id="S1.p6.8.m8.1.2.cmml" xref="S1.p6.8.m8.1.2"><times id="S1.p6.8.m8.1.2.1.cmml" xref="S1.p6.8.m8.1.2.1"></times><ci id="S1.p6.8.m8.1.2.2.cmml" xref="S1.p6.8.m8.1.2.2">𝑀</ci><ci id="S1.p6.8.m8.1.1.cmml" xref="S1.p6.8.m8.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.8.m8.1c">M(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.8.m8.1d">italic_M ( italic_σ )</annotation></semantics></math> we denote the <span class="ltx_text ltx_font_italic" id="S1.p6.11.1">incidence matrix</span> of <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p6.9.m9.1"><semantics id="S1.p6.9.m9.1a"><mi id="S1.p6.9.m9.1.1" xref="S1.p6.9.m9.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p6.9.m9.1b"><ci id="S1.p6.9.m9.1.1.cmml" xref="S1.p6.9.m9.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.9.m9.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p6.9.m9.1d">italic_σ</annotation></semantics></math>, which has coefficient <math alttext="|\sigma(a_{k})|_{b_{j}}" class="ltx_Math" display="inline" id="S1.p6.10.m10.1"><semantics id="S1.p6.10.m10.1a"><msub id="S1.p6.10.m10.1.1" xref="S1.p6.10.m10.1.1.cmml"><mrow id="S1.p6.10.m10.1.1.1.1" xref="S1.p6.10.m10.1.1.1.2.cmml"><mo id="S1.p6.10.m10.1.1.1.1.2" stretchy="false" xref="S1.p6.10.m10.1.1.1.2.1.cmml">|</mo><mrow id="S1.p6.10.m10.1.1.1.1.1" xref="S1.p6.10.m10.1.1.1.1.1.cmml"><mi id="S1.p6.10.m10.1.1.1.1.1.3" xref="S1.p6.10.m10.1.1.1.1.1.3.cmml">σ</mi><mo id="S1.p6.10.m10.1.1.1.1.1.2" xref="S1.p6.10.m10.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.p6.10.m10.1.1.1.1.1.1.1" xref="S1.p6.10.m10.1.1.1.1.1.1.1.1.cmml"><mo id="S1.p6.10.m10.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.p6.10.m10.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S1.p6.10.m10.1.1.1.1.1.1.1.1" xref="S1.p6.10.m10.1.1.1.1.1.1.1.1.cmml"><mi id="S1.p6.10.m10.1.1.1.1.1.1.1.1.2" xref="S1.p6.10.m10.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S1.p6.10.m10.1.1.1.1.1.1.1.1.3" xref="S1.p6.10.m10.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S1.p6.10.m10.1.1.1.1.1.1.1.3" stretchy="false" xref="S1.p6.10.m10.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p6.10.m10.1.1.1.1.3" stretchy="false" xref="S1.p6.10.m10.1.1.1.2.1.cmml">|</mo></mrow><msub id="S1.p6.10.m10.1.1.3" xref="S1.p6.10.m10.1.1.3.cmml"><mi id="S1.p6.10.m10.1.1.3.2" xref="S1.p6.10.m10.1.1.3.2.cmml">b</mi><mi id="S1.p6.10.m10.1.1.3.3" xref="S1.p6.10.m10.1.1.3.3.cmml">j</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S1.p6.10.m10.1b"><apply id="S1.p6.10.m10.1.1.cmml" xref="S1.p6.10.m10.1.1"><csymbol cd="ambiguous" id="S1.p6.10.m10.1.1.2.cmml" xref="S1.p6.10.m10.1.1">subscript</csymbol><apply id="S1.p6.10.m10.1.1.1.2.cmml" xref="S1.p6.10.m10.1.1.1.1"><abs id="S1.p6.10.m10.1.1.1.2.1.cmml" xref="S1.p6.10.m10.1.1.1.1.2"></abs><apply id="S1.p6.10.m10.1.1.1.1.1.cmml" xref="S1.p6.10.m10.1.1.1.1.1"><times id="S1.p6.10.m10.1.1.1.1.1.2.cmml" xref="S1.p6.10.m10.1.1.1.1.1.2"></times><ci id="S1.p6.10.m10.1.1.1.1.1.3.cmml" xref="S1.p6.10.m10.1.1.1.1.1.3">𝜎</ci><apply id="S1.p6.10.m10.1.1.1.1.1.1.1.1.cmml" xref="S1.p6.10.m10.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p6.10.m10.1.1.1.1.1.1.1.1.1.cmml" xref="S1.p6.10.m10.1.1.1.1.1.1.1">subscript</csymbol><ci id="S1.p6.10.m10.1.1.1.1.1.1.1.1.2.cmml" xref="S1.p6.10.m10.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S1.p6.10.m10.1.1.1.1.1.1.1.1.3.cmml" xref="S1.p6.10.m10.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply><apply id="S1.p6.10.m10.1.1.3.cmml" xref="S1.p6.10.m10.1.1.3"><csymbol cd="ambiguous" id="S1.p6.10.m10.1.1.3.1.cmml" xref="S1.p6.10.m10.1.1.3">subscript</csymbol><ci id="S1.p6.10.m10.1.1.3.2.cmml" xref="S1.p6.10.m10.1.1.3.2">𝑏</ci><ci id="S1.p6.10.m10.1.1.3.3.cmml" xref="S1.p6.10.m10.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.10.m10.1c">|\sigma(a_{k})|_{b_{j}}</annotation><annotation encoding="application/x-llamapun" id="S1.p6.10.m10.1d">| italic_σ ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> at the position <math alttext="(j,k)" class="ltx_Math" display="inline" id="S1.p6.11.m11.2"><semantics id="S1.p6.11.m11.2a"><mrow id="S1.p6.11.m11.2.3.2" xref="S1.p6.11.m11.2.3.1.cmml"><mo id="S1.p6.11.m11.2.3.2.1" stretchy="false" xref="S1.p6.11.m11.2.3.1.cmml">(</mo><mi id="S1.p6.11.m11.1.1" xref="S1.p6.11.m11.1.1.cmml">j</mi><mo id="S1.p6.11.m11.2.3.2.2" xref="S1.p6.11.m11.2.3.1.cmml">,</mo><mi id="S1.p6.11.m11.2.2" xref="S1.p6.11.m11.2.2.cmml">k</mi><mo id="S1.p6.11.m11.2.3.2.3" stretchy="false" xref="S1.p6.11.m11.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p6.11.m11.2b"><interval closure="open" id="S1.p6.11.m11.2.3.1.cmml" xref="S1.p6.11.m11.2.3.2"><ci id="S1.p6.11.m11.1.1.cmml" xref="S1.p6.11.m11.1.1">𝑗</ci><ci id="S1.p6.11.m11.2.2.cmml" xref="S1.p6.11.m11.2.2">𝑘</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S1.p6.11.m11.2c">(j,k)</annotation><annotation encoding="application/x-llamapun" id="S1.p6.11.m11.2d">( italic_j , italic_k )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.13">In order to compute the transferred measure <math alttext="\mu^{\sigma}([w^{\prime}])" class="ltx_Math" display="inline" id="S1.p7.1.m1.1"><semantics id="S1.p7.1.m1.1a"><mrow id="S1.p7.1.m1.1.1" xref="S1.p7.1.m1.1.1.cmml"><msup id="S1.p7.1.m1.1.1.3" xref="S1.p7.1.m1.1.1.3.cmml"><mi id="S1.p7.1.m1.1.1.3.2" xref="S1.p7.1.m1.1.1.3.2.cmml">μ</mi><mi id="S1.p7.1.m1.1.1.3.3" xref="S1.p7.1.m1.1.1.3.3.cmml">σ</mi></msup><mo id="S1.p7.1.m1.1.1.2" xref="S1.p7.1.m1.1.1.2.cmml">⁢</mo><mrow id="S1.p7.1.m1.1.1.1.1" xref="S1.p7.1.m1.1.1.cmml"><mo id="S1.p7.1.m1.1.1.1.1.2" stretchy="false" xref="S1.p7.1.m1.1.1.cmml">(</mo><mrow id="S1.p7.1.m1.1.1.1.1.1.1" xref="S1.p7.1.m1.1.1.1.1.1.2.cmml"><mo id="S1.p7.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S1.p7.1.m1.1.1.1.1.1.2.1.cmml">[</mo><msup id="S1.p7.1.m1.1.1.1.1.1.1.1" xref="S1.p7.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S1.p7.1.m1.1.1.1.1.1.1.1.2" xref="S1.p7.1.m1.1.1.1.1.1.1.1.2.cmml">w</mi><mo id="S1.p7.1.m1.1.1.1.1.1.1.1.3" xref="S1.p7.1.m1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.p7.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S1.p7.1.m1.1.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S1.p7.1.m1.1.1.1.1.3" stretchy="false" xref="S1.p7.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.1.m1.1b"><apply id="S1.p7.1.m1.1.1.cmml" xref="S1.p7.1.m1.1.1"><times id="S1.p7.1.m1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.2"></times><apply id="S1.p7.1.m1.1.1.3.cmml" xref="S1.p7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.p7.1.m1.1.1.3.1.cmml" xref="S1.p7.1.m1.1.1.3">superscript</csymbol><ci id="S1.p7.1.m1.1.1.3.2.cmml" xref="S1.p7.1.m1.1.1.3.2">𝜇</ci><ci id="S1.p7.1.m1.1.1.3.3.cmml" xref="S1.p7.1.m1.1.1.3.3">𝜎</ci></apply><apply id="S1.p7.1.m1.1.1.1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.1.1.1.1"><csymbol cd="latexml" id="S1.p7.1.m1.1.1.1.1.1.2.1.cmml" xref="S1.p7.1.m1.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S1.p7.1.m1.1.1.1.1.1.1.1.cmml" xref="S1.p7.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p7.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S1.p7.1.m1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S1.p7.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S1.p7.1.m1.1.1.1.1.1.1.1.2">𝑤</ci><ci id="S1.p7.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S1.p7.1.m1.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.1.m1.1c">\mu^{\sigma}([w^{\prime}])</annotation><annotation encoding="application/x-llamapun" id="S1.p7.1.m1.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( [ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] )</annotation></semantics></math> of an arbitrary cylinder <math alttext="[w^{\prime}]\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.p7.2.m2.1"><semantics id="S1.p7.2.m2.1a"><mrow id="S1.p7.2.m2.1.1" xref="S1.p7.2.m2.1.1.cmml"><mrow id="S1.p7.2.m2.1.1.1.1" xref="S1.p7.2.m2.1.1.1.2.cmml"><mo id="S1.p7.2.m2.1.1.1.1.2" stretchy="false" xref="S1.p7.2.m2.1.1.1.2.1.cmml">[</mo><msup id="S1.p7.2.m2.1.1.1.1.1" xref="S1.p7.2.m2.1.1.1.1.1.cmml"><mi id="S1.p7.2.m2.1.1.1.1.1.2" xref="S1.p7.2.m2.1.1.1.1.1.2.cmml">w</mi><mo id="S1.p7.2.m2.1.1.1.1.1.3" xref="S1.p7.2.m2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.p7.2.m2.1.1.1.1.3" stretchy="false" xref="S1.p7.2.m2.1.1.1.2.1.cmml">]</mo></mrow><mo id="S1.p7.2.m2.1.1.2" xref="S1.p7.2.m2.1.1.2.cmml">⊆</mo><msup id="S1.p7.2.m2.1.1.3" xref="S1.p7.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p7.2.m2.1.1.3.2" xref="S1.p7.2.m2.1.1.3.2.cmml">ℬ</mi><mi id="S1.p7.2.m2.1.1.3.3" xref="S1.p7.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.2.m2.1b"><apply id="S1.p7.2.m2.1.1.cmml" xref="S1.p7.2.m2.1.1"><subset id="S1.p7.2.m2.1.1.2.cmml" xref="S1.p7.2.m2.1.1.2"></subset><apply id="S1.p7.2.m2.1.1.1.2.cmml" xref="S1.p7.2.m2.1.1.1.1"><csymbol cd="latexml" id="S1.p7.2.m2.1.1.1.2.1.cmml" xref="S1.p7.2.m2.1.1.1.1.2">delimited-[]</csymbol><apply id="S1.p7.2.m2.1.1.1.1.1.cmml" xref="S1.p7.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p7.2.m2.1.1.1.1.1.1.cmml" xref="S1.p7.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S1.p7.2.m2.1.1.1.1.1.2.cmml" xref="S1.p7.2.m2.1.1.1.1.1.2">𝑤</ci><ci id="S1.p7.2.m2.1.1.1.1.1.3.cmml" xref="S1.p7.2.m2.1.1.1.1.1.3">′</ci></apply></apply><apply id="S1.p7.2.m2.1.1.3.cmml" xref="S1.p7.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.p7.2.m2.1.1.3.1.cmml" xref="S1.p7.2.m2.1.1.3">superscript</csymbol><ci id="S1.p7.2.m2.1.1.3.2.cmml" xref="S1.p7.2.m2.1.1.3.2">ℬ</ci><ci id="S1.p7.2.m2.1.1.3.3.cmml" xref="S1.p7.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.2.m2.1c">[w^{\prime}]\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.p7.2.m2.1d">[ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, as done in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S1.Thmthm3" title="Proposition 1.3. ‣ 1. Introduction ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">1.3</span></a> for the special case <math alttext="w^{\prime}=b_{j}\in\cal B" class="ltx_Math" display="inline" id="S1.p7.3.m3.1"><semantics id="S1.p7.3.m3.1a"><mrow id="S1.p7.3.m3.1.1" xref="S1.p7.3.m3.1.1.cmml"><msup id="S1.p7.3.m3.1.1.2" xref="S1.p7.3.m3.1.1.2.cmml"><mi id="S1.p7.3.m3.1.1.2.2" xref="S1.p7.3.m3.1.1.2.2.cmml">w</mi><mo id="S1.p7.3.m3.1.1.2.3" xref="S1.p7.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S1.p7.3.m3.1.1.3" xref="S1.p7.3.m3.1.1.3.cmml">=</mo><msub id="S1.p7.3.m3.1.1.4" xref="S1.p7.3.m3.1.1.4.cmml"><mi id="S1.p7.3.m3.1.1.4.2" xref="S1.p7.3.m3.1.1.4.2.cmml">b</mi><mi id="S1.p7.3.m3.1.1.4.3" xref="S1.p7.3.m3.1.1.4.3.cmml">j</mi></msub><mo id="S1.p7.3.m3.1.1.5" xref="S1.p7.3.m3.1.1.5.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S1.p7.3.m3.1.1.6" xref="S1.p7.3.m3.1.1.6.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.3.m3.1b"><apply id="S1.p7.3.m3.1.1.cmml" xref="S1.p7.3.m3.1.1"><and id="S1.p7.3.m3.1.1a.cmml" xref="S1.p7.3.m3.1.1"></and><apply id="S1.p7.3.m3.1.1b.cmml" xref="S1.p7.3.m3.1.1"><eq id="S1.p7.3.m3.1.1.3.cmml" xref="S1.p7.3.m3.1.1.3"></eq><apply id="S1.p7.3.m3.1.1.2.cmml" xref="S1.p7.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.p7.3.m3.1.1.2.1.cmml" xref="S1.p7.3.m3.1.1.2">superscript</csymbol><ci id="S1.p7.3.m3.1.1.2.2.cmml" xref="S1.p7.3.m3.1.1.2.2">𝑤</ci><ci id="S1.p7.3.m3.1.1.2.3.cmml" xref="S1.p7.3.m3.1.1.2.3">′</ci></apply><apply id="S1.p7.3.m3.1.1.4.cmml" xref="S1.p7.3.m3.1.1.4"><csymbol cd="ambiguous" id="S1.p7.3.m3.1.1.4.1.cmml" xref="S1.p7.3.m3.1.1.4">subscript</csymbol><ci id="S1.p7.3.m3.1.1.4.2.cmml" xref="S1.p7.3.m3.1.1.4.2">𝑏</ci><ci id="S1.p7.3.m3.1.1.4.3.cmml" xref="S1.p7.3.m3.1.1.4.3">𝑗</ci></apply></apply><apply id="S1.p7.3.m3.1.1c.cmml" xref="S1.p7.3.m3.1.1"><in id="S1.p7.3.m3.1.1.5.cmml" xref="S1.p7.3.m3.1.1.5"></in><share href="https://arxiv.org/html/2211.11234v4#S1.p7.3.m3.1.1.4.cmml" id="S1.p7.3.m3.1.1d.cmml" xref="S1.p7.3.m3.1.1"></share><ci id="S1.p7.3.m3.1.1.6.cmml" xref="S1.p7.3.m3.1.1.6">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.3.m3.1c">w^{\prime}=b_{j}\in\cal B</annotation><annotation encoding="application/x-llamapun" id="S1.p7.3.m3.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ caligraphic_B</annotation></semantics></math>, it turns out that it is useful to introduce the number <math alttext="\lfloor\sigma(w)\rfloor_{u}" class="ltx_Math" display="inline" id="S1.p7.4.m4.2"><semantics id="S1.p7.4.m4.2a"><msub id="S1.p7.4.m4.2.2" xref="S1.p7.4.m4.2.2.cmml"><mrow id="S1.p7.4.m4.2.2.1.1" xref="S1.p7.4.m4.2.2.1.2.cmml"><mo id="S1.p7.4.m4.2.2.1.1.2" stretchy="false" xref="S1.p7.4.m4.2.2.1.2.1.cmml">⌊</mo><mrow id="S1.p7.4.m4.2.2.1.1.1" xref="S1.p7.4.m4.2.2.1.1.1.cmml"><mi id="S1.p7.4.m4.2.2.1.1.1.2" xref="S1.p7.4.m4.2.2.1.1.1.2.cmml">σ</mi><mo id="S1.p7.4.m4.2.2.1.1.1.1" xref="S1.p7.4.m4.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S1.p7.4.m4.2.2.1.1.1.3.2" xref="S1.p7.4.m4.2.2.1.1.1.cmml"><mo id="S1.p7.4.m4.2.2.1.1.1.3.2.1" stretchy="false" xref="S1.p7.4.m4.2.2.1.1.1.cmml">(</mo><mi id="S1.p7.4.m4.1.1" xref="S1.p7.4.m4.1.1.cmml">w</mi><mo id="S1.p7.4.m4.2.2.1.1.1.3.2.2" stretchy="false" xref="S1.p7.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p7.4.m4.2.2.1.1.3" stretchy="false" xref="S1.p7.4.m4.2.2.1.2.1.cmml">⌋</mo></mrow><mi id="S1.p7.4.m4.2.2.3" xref="S1.p7.4.m4.2.2.3.cmml">u</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p7.4.m4.2b"><apply id="S1.p7.4.m4.2.2.cmml" xref="S1.p7.4.m4.2.2"><csymbol cd="ambiguous" id="S1.p7.4.m4.2.2.2.cmml" xref="S1.p7.4.m4.2.2">subscript</csymbol><apply id="S1.p7.4.m4.2.2.1.2.cmml" xref="S1.p7.4.m4.2.2.1.1"><floor id="S1.p7.4.m4.2.2.1.2.1.cmml" xref="S1.p7.4.m4.2.2.1.1.2"></floor><apply id="S1.p7.4.m4.2.2.1.1.1.cmml" xref="S1.p7.4.m4.2.2.1.1.1"><times id="S1.p7.4.m4.2.2.1.1.1.1.cmml" xref="S1.p7.4.m4.2.2.1.1.1.1"></times><ci id="S1.p7.4.m4.2.2.1.1.1.2.cmml" xref="S1.p7.4.m4.2.2.1.1.1.2">𝜎</ci><ci id="S1.p7.4.m4.1.1.cmml" xref="S1.p7.4.m4.1.1">𝑤</ci></apply></apply><ci id="S1.p7.4.m4.2.2.3.cmml" xref="S1.p7.4.m4.2.2.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.4.m4.2c">\lfloor\sigma(w)\rfloor_{u}</annotation><annotation encoding="application/x-llamapun" id="S1.p7.4.m4.2d">⌊ italic_σ ( italic_w ) ⌋ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT</annotation></semantics></math> of <span class="ltx_text ltx_font_italic" id="S1.p7.13.1">essential occurrences</span> of <math alttext="u" class="ltx_Math" display="inline" id="S1.p7.5.m5.1"><semantics id="S1.p7.5.m5.1a"><mi id="S1.p7.5.m5.1.1" xref="S1.p7.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S1.p7.5.m5.1b"><ci id="S1.p7.5.m5.1.1.cmml" xref="S1.p7.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S1.p7.5.m5.1d">italic_u</annotation></semantics></math> as a factor of <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S1.p7.6.m6.1"><semantics id="S1.p7.6.m6.1a"><mrow id="S1.p7.6.m6.1.2" xref="S1.p7.6.m6.1.2.cmml"><mi id="S1.p7.6.m6.1.2.2" xref="S1.p7.6.m6.1.2.2.cmml">σ</mi><mo id="S1.p7.6.m6.1.2.1" xref="S1.p7.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.p7.6.m6.1.2.3.2" xref="S1.p7.6.m6.1.2.cmml"><mo id="S1.p7.6.m6.1.2.3.2.1" stretchy="false" xref="S1.p7.6.m6.1.2.cmml">(</mo><mi id="S1.p7.6.m6.1.1" xref="S1.p7.6.m6.1.1.cmml">w</mi><mo id="S1.p7.6.m6.1.2.3.2.2" stretchy="false" xref="S1.p7.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.6.m6.1b"><apply id="S1.p7.6.m6.1.2.cmml" xref="S1.p7.6.m6.1.2"><times id="S1.p7.6.m6.1.2.1.cmml" xref="S1.p7.6.m6.1.2.1"></times><ci id="S1.p7.6.m6.1.2.2.cmml" xref="S1.p7.6.m6.1.2.2">𝜎</ci><ci id="S1.p7.6.m6.1.1.cmml" xref="S1.p7.6.m6.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.6.m6.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S1.p7.6.m6.1d">italic_σ ( italic_w )</annotation></semantics></math>, by which we mean that the first letter of <math alttext="u" class="ltx_Math" display="inline" id="S1.p7.7.m7.1"><semantics id="S1.p7.7.m7.1a"><mi id="S1.p7.7.m7.1.1" xref="S1.p7.7.m7.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S1.p7.7.m7.1b"><ci id="S1.p7.7.m7.1.1.cmml" xref="S1.p7.7.m7.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.7.m7.1c">u</annotation><annotation encoding="application/x-llamapun" id="S1.p7.7.m7.1d">italic_u</annotation></semantics></math> occurs in the <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p7.8.m8.1"><semantics id="S1.p7.8.m8.1a"><mi id="S1.p7.8.m8.1.1" xref="S1.p7.8.m8.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p7.8.m8.1b"><ci id="S1.p7.8.m8.1.1.cmml" xref="S1.p7.8.m8.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.8.m8.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p7.8.m8.1d">italic_σ</annotation></semantics></math>-image of first letter of <math alttext="w" class="ltx_Math" display="inline" id="S1.p7.9.m9.1"><semantics id="S1.p7.9.m9.1a"><mi id="S1.p7.9.m9.1.1" xref="S1.p7.9.m9.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S1.p7.9.m9.1b"><ci id="S1.p7.9.m9.1.1.cmml" xref="S1.p7.9.m9.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.9.m9.1c">w</annotation><annotation encoding="application/x-llamapun" id="S1.p7.9.m9.1d">italic_w</annotation></semantics></math>, and the last letter of <math alttext="u" class="ltx_Math" display="inline" id="S1.p7.10.m10.1"><semantics id="S1.p7.10.m10.1a"><mi id="S1.p7.10.m10.1.1" xref="S1.p7.10.m10.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S1.p7.10.m10.1b"><ci id="S1.p7.10.m10.1.1.cmml" xref="S1.p7.10.m10.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.10.m10.1c">u</annotation><annotation encoding="application/x-llamapun" id="S1.p7.10.m10.1d">italic_u</annotation></semantics></math> occurs in the <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p7.11.m11.1"><semantics id="S1.p7.11.m11.1a"><mi id="S1.p7.11.m11.1.1" xref="S1.p7.11.m11.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p7.11.m11.1b"><ci id="S1.p7.11.m11.1.1.cmml" xref="S1.p7.11.m11.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.11.m11.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p7.11.m11.1d">italic_σ</annotation></semantics></math>-image of last letter of <math alttext="w" class="ltx_Math" display="inline" id="S1.p7.12.m12.1"><semantics id="S1.p7.12.m12.1a"><mi id="S1.p7.12.m12.1.1" xref="S1.p7.12.m12.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S1.p7.12.m12.1b"><ci id="S1.p7.12.m12.1.1.cmml" xref="S1.p7.12.m12.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.12.m12.1c">w</annotation><annotation encoding="application/x-llamapun" id="S1.p7.12.m12.1d">italic_w</annotation></semantics></math>. The following is proved in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.Thmthm2" title="Proposition 4.2. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.2</span></a> below, and several examples of concrete computations of cylinder values <math alttext="\mu^{\sigma}([w^{\prime}])" class="ltx_Math" display="inline" id="S1.p7.13.m13.1"><semantics id="S1.p7.13.m13.1a"><mrow id="S1.p7.13.m13.1.1" xref="S1.p7.13.m13.1.1.cmml"><msup id="S1.p7.13.m13.1.1.3" xref="S1.p7.13.m13.1.1.3.cmml"><mi id="S1.p7.13.m13.1.1.3.2" xref="S1.p7.13.m13.1.1.3.2.cmml">μ</mi><mi id="S1.p7.13.m13.1.1.3.3" xref="S1.p7.13.m13.1.1.3.3.cmml">σ</mi></msup><mo id="S1.p7.13.m13.1.1.2" xref="S1.p7.13.m13.1.1.2.cmml">⁢</mo><mrow id="S1.p7.13.m13.1.1.1.1" xref="S1.p7.13.m13.1.1.cmml"><mo id="S1.p7.13.m13.1.1.1.1.2" stretchy="false" xref="S1.p7.13.m13.1.1.cmml">(</mo><mrow id="S1.p7.13.m13.1.1.1.1.1.1" xref="S1.p7.13.m13.1.1.1.1.1.2.cmml"><mo id="S1.p7.13.m13.1.1.1.1.1.1.2" stretchy="false" xref="S1.p7.13.m13.1.1.1.1.1.2.1.cmml">[</mo><msup id="S1.p7.13.m13.1.1.1.1.1.1.1" xref="S1.p7.13.m13.1.1.1.1.1.1.1.cmml"><mi id="S1.p7.13.m13.1.1.1.1.1.1.1.2" xref="S1.p7.13.m13.1.1.1.1.1.1.1.2.cmml">w</mi><mo id="S1.p7.13.m13.1.1.1.1.1.1.1.3" xref="S1.p7.13.m13.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.p7.13.m13.1.1.1.1.1.1.3" stretchy="false" xref="S1.p7.13.m13.1.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S1.p7.13.m13.1.1.1.1.3" stretchy="false" xref="S1.p7.13.m13.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p7.13.m13.1b"><apply id="S1.p7.13.m13.1.1.cmml" xref="S1.p7.13.m13.1.1"><times id="S1.p7.13.m13.1.1.2.cmml" xref="S1.p7.13.m13.1.1.2"></times><apply id="S1.p7.13.m13.1.1.3.cmml" xref="S1.p7.13.m13.1.1.3"><csymbol cd="ambiguous" id="S1.p7.13.m13.1.1.3.1.cmml" xref="S1.p7.13.m13.1.1.3">superscript</csymbol><ci id="S1.p7.13.m13.1.1.3.2.cmml" xref="S1.p7.13.m13.1.1.3.2">𝜇</ci><ci id="S1.p7.13.m13.1.1.3.3.cmml" xref="S1.p7.13.m13.1.1.3.3">𝜎</ci></apply><apply id="S1.p7.13.m13.1.1.1.1.1.2.cmml" xref="S1.p7.13.m13.1.1.1.1.1.1"><csymbol cd="latexml" id="S1.p7.13.m13.1.1.1.1.1.2.1.cmml" xref="S1.p7.13.m13.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S1.p7.13.m13.1.1.1.1.1.1.1.cmml" xref="S1.p7.13.m13.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p7.13.m13.1.1.1.1.1.1.1.1.cmml" xref="S1.p7.13.m13.1.1.1.1.1.1.1">superscript</csymbol><ci id="S1.p7.13.m13.1.1.1.1.1.1.1.2.cmml" xref="S1.p7.13.m13.1.1.1.1.1.1.1.2">𝑤</ci><ci id="S1.p7.13.m13.1.1.1.1.1.1.1.3.cmml" xref="S1.p7.13.m13.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p7.13.m13.1c">\mu^{\sigma}([w^{\prime}])</annotation><annotation encoding="application/x-llamapun" id="S1.p7.13.m13.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( [ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] )</annotation></semantics></math> are given in Sections <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3" title="3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4" title="4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_prop" id="S1.Thmthm4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmthm4.1.1.1">Proposition 1.4</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmthm4.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmthm4.p1"> <p class="ltx_p" id="S1.Thmthm4.p1.7"><span class="ltx_text ltx_font_italic" id="S1.Thmthm4.p1.7.7">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S1.Thmthm4.p1.1.1.m1.1"><semantics id="S1.Thmthm4.p1.1.1.m1.1a"><mrow id="S1.Thmthm4.p1.1.1.m1.1.1" xref="S1.Thmthm4.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmthm4.p1.1.1.m1.1.1.2" xref="S1.Thmthm4.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S1.Thmthm4.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm4.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmthm4.p1.1.1.m1.1.1.3" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.cmml"><msup id="S1.Thmthm4.p1.1.1.m1.1.1.3.2" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm4.p1.1.1.m1.1.1.3.2.2" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.Thmthm4.p1.1.1.m1.1.1.3.2.3" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.Thmthm4.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S1.Thmthm4.p1.1.1.m1.1.1.3.3" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm4.p1.1.1.m1.1.1.3.3.2" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S1.Thmthm4.p1.1.1.m1.1.1.3.3.3" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm4.p1.1.1.m1.1b"><apply id="S1.Thmthm4.p1.1.1.m1.1.1.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1"><ci id="S1.Thmthm4.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.1">:</ci><ci id="S1.Thmthm4.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S1.Thmthm4.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3"><ci id="S1.Thmthm4.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.1">→</ci><apply id="S1.Thmthm4.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmthm4.p1.1.1.m1.1.1.3.2.1.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.Thmthm4.p1.1.1.m1.1.1.3.2.2.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S1.Thmthm4.p1.1.1.m1.1.1.3.2.3.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S1.Thmthm4.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmthm4.p1.1.1.m1.1.1.3.3.1.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.Thmthm4.p1.1.1.m1.1.1.3.3.2.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S1.Thmthm4.p1.1.1.m1.1.1.3.3.3.cmml" xref="S1.Thmthm4.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm4.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm4.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be any non-erasing monoid morphism, and let <math alttext="\mu" class="ltx_Math" display="inline" id="S1.Thmthm4.p1.2.2.m2.1"><semantics id="S1.Thmthm4.p1.2.2.m2.1a"><mi id="S1.Thmthm4.p1.2.2.m2.1.1" xref="S1.Thmthm4.p1.2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm4.p1.2.2.m2.1b"><ci id="S1.Thmthm4.p1.2.2.m2.1.1.cmml" xref="S1.Thmthm4.p1.2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm4.p1.2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm4.p1.2.2.m2.1d">italic_μ</annotation></semantics></math> be any invariant measure on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.Thmthm4.p1.3.3.m3.1"><semantics id="S1.Thmthm4.p1.3.3.m3.1a"><msup id="S1.Thmthm4.p1.3.3.m3.1.1" xref="S1.Thmthm4.p1.3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm4.p1.3.3.m3.1.1.2" xref="S1.Thmthm4.p1.3.3.m3.1.1.2.cmml">𝒜</mi><mi id="S1.Thmthm4.p1.3.3.m3.1.1.3" xref="S1.Thmthm4.p1.3.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.Thmthm4.p1.3.3.m3.1b"><apply id="S1.Thmthm4.p1.3.3.m3.1.1.cmml" xref="S1.Thmthm4.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S1.Thmthm4.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmthm4.p1.3.3.m3.1.1">superscript</csymbol><ci id="S1.Thmthm4.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmthm4.p1.3.3.m3.1.1.2">𝒜</ci><ci id="S1.Thmthm4.p1.3.3.m3.1.1.3.cmml" xref="S1.Thmthm4.p1.3.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm4.p1.3.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm4.p1.3.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Then for any <math alttext="w^{\prime}\in\cal B^{*}" class="ltx_Math" display="inline" id="S1.Thmthm4.p1.4.4.m4.1"><semantics id="S1.Thmthm4.p1.4.4.m4.1a"><mrow id="S1.Thmthm4.p1.4.4.m4.1.1" xref="S1.Thmthm4.p1.4.4.m4.1.1.cmml"><msup id="S1.Thmthm4.p1.4.4.m4.1.1.2" xref="S1.Thmthm4.p1.4.4.m4.1.1.2.cmml"><mi id="S1.Thmthm4.p1.4.4.m4.1.1.2.2" xref="S1.Thmthm4.p1.4.4.m4.1.1.2.2.cmml">w</mi><mo id="S1.Thmthm4.p1.4.4.m4.1.1.2.3" xref="S1.Thmthm4.p1.4.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S1.Thmthm4.p1.4.4.m4.1.1.1" xref="S1.Thmthm4.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="S1.Thmthm4.p1.4.4.m4.1.1.3" xref="S1.Thmthm4.p1.4.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm4.p1.4.4.m4.1.1.3.2" xref="S1.Thmthm4.p1.4.4.m4.1.1.3.2.cmml">ℬ</mi><mo id="S1.Thmthm4.p1.4.4.m4.1.1.3.3" xref="S1.Thmthm4.p1.4.4.m4.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm4.p1.4.4.m4.1b"><apply id="S1.Thmthm4.p1.4.4.m4.1.1.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1"><in id="S1.Thmthm4.p1.4.4.m4.1.1.1.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.1"></in><apply id="S1.Thmthm4.p1.4.4.m4.1.1.2.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S1.Thmthm4.p1.4.4.m4.1.1.2.1.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.2">superscript</csymbol><ci id="S1.Thmthm4.p1.4.4.m4.1.1.2.2.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.2.2">𝑤</ci><ci id="S1.Thmthm4.p1.4.4.m4.1.1.2.3.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.2.3">′</ci></apply><apply id="S1.Thmthm4.p1.4.4.m4.1.1.3.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.Thmthm4.p1.4.4.m4.1.1.3.1.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S1.Thmthm4.p1.4.4.m4.1.1.3.2.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.3.2">ℬ</ci><times id="S1.Thmthm4.p1.4.4.m4.1.1.3.3.cmml" xref="S1.Thmthm4.p1.4.4.m4.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm4.p1.4.4.m4.1c">w^{\prime}\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm4.p1.4.4.m4.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="|w^{\prime}|\geq 2" class="ltx_Math" display="inline" id="S1.Thmthm4.p1.5.5.m5.1"><semantics id="S1.Thmthm4.p1.5.5.m5.1a"><mrow id="S1.Thmthm4.p1.5.5.m5.1.1" xref="S1.Thmthm4.p1.5.5.m5.1.1.cmml"><mrow id="S1.Thmthm4.p1.5.5.m5.1.1.1.1" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.2.cmml"><mo id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.2" stretchy="false" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.2.1.cmml">|</mo><msup id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.cmml"><mi id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.2" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.2.cmml">w</mi><mo id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.3" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.3" stretchy="false" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.2.1.cmml">|</mo></mrow><mo id="S1.Thmthm4.p1.5.5.m5.1.1.2" xref="S1.Thmthm4.p1.5.5.m5.1.1.2.cmml">≥</mo><mn id="S1.Thmthm4.p1.5.5.m5.1.1.3" xref="S1.Thmthm4.p1.5.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm4.p1.5.5.m5.1b"><apply id="S1.Thmthm4.p1.5.5.m5.1.1.cmml" xref="S1.Thmthm4.p1.5.5.m5.1.1"><geq id="S1.Thmthm4.p1.5.5.m5.1.1.2.cmml" xref="S1.Thmthm4.p1.5.5.m5.1.1.2"></geq><apply id="S1.Thmthm4.p1.5.5.m5.1.1.1.2.cmml" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1"><abs id="S1.Thmthm4.p1.5.5.m5.1.1.1.2.1.cmml" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1.2"></abs><apply id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.cmml" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.2.cmml" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.2">𝑤</ci><ci id="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.3.cmml" xref="S1.Thmthm4.p1.5.5.m5.1.1.1.1.1.3">′</ci></apply></apply><cn id="S1.Thmthm4.p1.5.5.m5.1.1.3.cmml" type="integer" xref="S1.Thmthm4.p1.5.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm4.p1.5.5.m5.1c">|w^{\prime}|\geq 2</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm4.p1.5.5.m5.1d">| italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≥ 2</annotation></semantics></math> the transferred measure <math alttext="\mu^{\sigma}:=\sigma M(\mu)" class="ltx_Math" display="inline" id="S1.Thmthm4.p1.6.6.m6.1"><semantics id="S1.Thmthm4.p1.6.6.m6.1a"><mrow id="S1.Thmthm4.p1.6.6.m6.1.2" xref="S1.Thmthm4.p1.6.6.m6.1.2.cmml"><msup id="S1.Thmthm4.p1.6.6.m6.1.2.2" xref="S1.Thmthm4.p1.6.6.m6.1.2.2.cmml"><mi id="S1.Thmthm4.p1.6.6.m6.1.2.2.2" xref="S1.Thmthm4.p1.6.6.m6.1.2.2.2.cmml">μ</mi><mi id="S1.Thmthm4.p1.6.6.m6.1.2.2.3" xref="S1.Thmthm4.p1.6.6.m6.1.2.2.3.cmml">σ</mi></msup><mo id="S1.Thmthm4.p1.6.6.m6.1.2.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm4.p1.6.6.m6.1.2.1.cmml">:=</mo><mrow id="S1.Thmthm4.p1.6.6.m6.1.2.3" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.cmml"><mi id="S1.Thmthm4.p1.6.6.m6.1.2.3.2" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.2.cmml">σ</mi><mo id="S1.Thmthm4.p1.6.6.m6.1.2.3.1" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.1.cmml">⁢</mo><mi id="S1.Thmthm4.p1.6.6.m6.1.2.3.3" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.3.cmml">M</mi><mo id="S1.Thmthm4.p1.6.6.m6.1.2.3.1a" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S1.Thmthm4.p1.6.6.m6.1.2.3.4.2" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.cmml"><mo id="S1.Thmthm4.p1.6.6.m6.1.2.3.4.2.1" stretchy="false" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.cmml">(</mo><mi id="S1.Thmthm4.p1.6.6.m6.1.1" xref="S1.Thmthm4.p1.6.6.m6.1.1.cmml">μ</mi><mo id="S1.Thmthm4.p1.6.6.m6.1.2.3.4.2.2" stretchy="false" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm4.p1.6.6.m6.1b"><apply id="S1.Thmthm4.p1.6.6.m6.1.2.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2"><csymbol cd="latexml" id="S1.Thmthm4.p1.6.6.m6.1.2.1.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.1">assign</csymbol><apply id="S1.Thmthm4.p1.6.6.m6.1.2.2.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.2"><csymbol cd="ambiguous" id="S1.Thmthm4.p1.6.6.m6.1.2.2.1.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.2">superscript</csymbol><ci id="S1.Thmthm4.p1.6.6.m6.1.2.2.2.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.2.2">𝜇</ci><ci id="S1.Thmthm4.p1.6.6.m6.1.2.2.3.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.2.3">𝜎</ci></apply><apply id="S1.Thmthm4.p1.6.6.m6.1.2.3.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.3"><times id="S1.Thmthm4.p1.6.6.m6.1.2.3.1.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.1"></times><ci id="S1.Thmthm4.p1.6.6.m6.1.2.3.2.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.2">𝜎</ci><ci id="S1.Thmthm4.p1.6.6.m6.1.2.3.3.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.2.3.3">𝑀</ci><ci id="S1.Thmthm4.p1.6.6.m6.1.1.cmml" xref="S1.Thmthm4.p1.6.6.m6.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm4.p1.6.6.m6.1c">\mu^{\sigma}:=\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm4.p1.6.6.m6.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT := italic_σ italic_M ( italic_μ )</annotation></semantics></math>, evaluated on the cylinder <math alttext="[w^{\prime}]" class="ltx_Math" display="inline" id="S1.Thmthm4.p1.7.7.m7.1"><semantics id="S1.Thmthm4.p1.7.7.m7.1a"><mrow id="S1.Thmthm4.p1.7.7.m7.1.1.1" xref="S1.Thmthm4.p1.7.7.m7.1.1.2.cmml"><mo id="S1.Thmthm4.p1.7.7.m7.1.1.1.2" stretchy="false" xref="S1.Thmthm4.p1.7.7.m7.1.1.2.1.cmml">[</mo><msup id="S1.Thmthm4.p1.7.7.m7.1.1.1.1" xref="S1.Thmthm4.p1.7.7.m7.1.1.1.1.cmml"><mi id="S1.Thmthm4.p1.7.7.m7.1.1.1.1.2" xref="S1.Thmthm4.p1.7.7.m7.1.1.1.1.2.cmml">w</mi><mo id="S1.Thmthm4.p1.7.7.m7.1.1.1.1.3" xref="S1.Thmthm4.p1.7.7.m7.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.Thmthm4.p1.7.7.m7.1.1.1.3" stretchy="false" xref="S1.Thmthm4.p1.7.7.m7.1.1.2.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm4.p1.7.7.m7.1b"><apply id="S1.Thmthm4.p1.7.7.m7.1.1.2.cmml" xref="S1.Thmthm4.p1.7.7.m7.1.1.1"><csymbol cd="latexml" id="S1.Thmthm4.p1.7.7.m7.1.1.2.1.cmml" xref="S1.Thmthm4.p1.7.7.m7.1.1.1.2">delimited-[]</csymbol><apply id="S1.Thmthm4.p1.7.7.m7.1.1.1.1.cmml" xref="S1.Thmthm4.p1.7.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmthm4.p1.7.7.m7.1.1.1.1.1.cmml" xref="S1.Thmthm4.p1.7.7.m7.1.1.1.1">superscript</csymbol><ci id="S1.Thmthm4.p1.7.7.m7.1.1.1.1.2.cmml" xref="S1.Thmthm4.p1.7.7.m7.1.1.1.1.2">𝑤</ci><ci id="S1.Thmthm4.p1.7.7.m7.1.1.1.1.3.cmml" xref="S1.Thmthm4.p1.7.7.m7.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm4.p1.7.7.m7.1c">[w^{\prime}]</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm4.p1.7.7.m7.1d">[ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ]</annotation></semantics></math>, is given by the finite sum</span></p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu^{\sigma}([w^{\prime}])=\sum_{w\,\in\,W(w^{\prime})}{\lfloor\sigma(w)% \rfloor}_{w^{\prime}}\cdot\mu([w])\,," class="ltx_Math" display="block" id="S1.Ex6.m1.4"><semantics id="S1.Ex6.m1.4a"><mrow id="S1.Ex6.m1.4.4.1" xref="S1.Ex6.m1.4.4.1.1.cmml"><mrow id="S1.Ex6.m1.4.4.1.1" xref="S1.Ex6.m1.4.4.1.1.cmml"><mrow id="S1.Ex6.m1.4.4.1.1.1" xref="S1.Ex6.m1.4.4.1.1.1.cmml"><msup id="S1.Ex6.m1.4.4.1.1.1.3" xref="S1.Ex6.m1.4.4.1.1.1.3.cmml"><mi id="S1.Ex6.m1.4.4.1.1.1.3.2" xref="S1.Ex6.m1.4.4.1.1.1.3.2.cmml">μ</mi><mi id="S1.Ex6.m1.4.4.1.1.1.3.3" xref="S1.Ex6.m1.4.4.1.1.1.3.3.cmml">σ</mi></msup><mo id="S1.Ex6.m1.4.4.1.1.1.2" xref="S1.Ex6.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex6.m1.4.4.1.1.1.1.1" 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xref="S1.Ex6.m1.4.4.1.1.2.1.1.1.3"><csymbol cd="ambiguous" id="S1.Ex6.m1.4.4.1.1.2.1.1.1.3.1.cmml" xref="S1.Ex6.m1.4.4.1.1.2.1.1.1.3">superscript</csymbol><ci id="S1.Ex6.m1.4.4.1.1.2.1.1.1.3.2.cmml" xref="S1.Ex6.m1.4.4.1.1.2.1.1.1.3.2">𝑤</ci><ci id="S1.Ex6.m1.4.4.1.1.2.1.1.1.3.3.cmml" xref="S1.Ex6.m1.4.4.1.1.2.1.1.1.3.3">′</ci></apply></apply><ci id="S1.Ex6.m1.4.4.1.1.2.1.1.3.cmml" xref="S1.Ex6.m1.4.4.1.1.2.1.1.3">𝜇</ci></apply><apply id="S1.Ex6.m1.4.4.1.1.3.2.2.1.1.1.cmml" xref="S1.Ex6.m1.4.4.1.1.3.2.2.1.1.2"><csymbol cd="latexml" id="S1.Ex6.m1.4.4.1.1.3.2.2.1.1.1.1.cmml" xref="S1.Ex6.m1.4.4.1.1.3.2.2.1.1.2.1">delimited-[]</csymbol><ci id="S1.Ex6.m1.3.3.cmml" xref="S1.Ex6.m1.3.3">𝑤</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex6.m1.4c">\mu^{\sigma}([w^{\prime}])=\sum_{w\,\in\,W(w^{\prime})}{\lfloor\sigma(w)% \rfloor}_{w^{\prime}}\cdot\mu([w])\,,</annotation><annotation encoding="application/x-llamapun" id="S1.Ex6.m1.4d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( [ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ] ) = ∑ start_POSTSUBSCRIPT italic_w ∈ italic_W ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT ⌊ italic_σ ( italic_w ) ⌋ start_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⋅ italic_μ ( [ italic_w ] ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.Thmthm4.p1.8"><span class="ltx_text ltx_font_italic" id="S1.Thmthm4.p1.8.1">for <math alttext="W(w^{\prime})={\big{\{}}w\in\cal A^{*}\,:\,|w|\leq\frac{|w^{\prime}|-2}{\min\{% |\sigma(a_{k})|\,\,:\,\,a_{k}\in\cal A\}}+2{\big{\}}}" class="ltx_Math" display="inline" id="S1.Thmthm4.p1.8.1.m1.7"><semantics id="S1.Thmthm4.p1.8.1.m1.7a"><mrow id="S1.Thmthm4.p1.8.1.m1.7.7" xref="S1.Thmthm4.p1.8.1.m1.7.7.cmml"><mrow id="S1.Thmthm4.p1.8.1.m1.5.5.1" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.cmml"><mi id="S1.Thmthm4.p1.8.1.m1.5.5.1.3" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.3.cmml">W</mi><mo id="S1.Thmthm4.p1.8.1.m1.5.5.1.2" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.2.cmml">⁢</mo><mrow id="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1.cmml"><mo id="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.2" stretchy="false" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1.cmml">(</mo><msup id="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1.cmml"><mi id="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1.2" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1.2.cmml">w</mi><mo id="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1.3" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.3" stretchy="false" xref="S1.Thmthm4.p1.8.1.m1.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm4.p1.8.1.m1.7.7.4" xref="S1.Thmthm4.p1.8.1.m1.7.7.4.cmml">=</mo><mrow id="S1.Thmthm4.p1.8.1.m1.7.7.3.2" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.3.cmml"><mo id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.3" maxsize="120%" minsize="120%" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.3.1.cmml">{</mo><mrow id="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1" xref="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.cmml"><mi id="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.2" xref="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.2.cmml">w</mi><mo id="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.1" xref="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.1.cmml">∈</mo><msup id="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.3" xref="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.3.2" xref="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.3.2.cmml">𝒜</mi><mo id="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.3.3" xref="S1.Thmthm4.p1.8.1.m1.6.6.2.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.4" lspace="0.278em" rspace="0.448em" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.3.1.cmml">:</mo><mrow id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.cmml"><mrow id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.2.2" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.2.1.cmml"><mo id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.2.2.1" stretchy="false" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm4.p1.8.1.m1.4.4" xref="S1.Thmthm4.p1.8.1.m1.4.4.cmml">𝓌</mi><mo id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.2.2.2" stretchy="false" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.2.1.1.cmml">|</mo></mrow><mo id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.1" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.1.cmml">≤</mo><mrow id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.3" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.3.cmml"><mfrac id="S1.Thmthm4.p1.8.1.m1.3.3" xref="S1.Thmthm4.p1.8.1.m1.3.3.cmml"><mrow id="S1.Thmthm4.p1.8.1.m1.1.1.1" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.cmml"><mrow id="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.1.2.cmml"><mo id="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1.2" stretchy="false" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.1.2.1.cmml">|</mo><msup id="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1.1" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1.1.2" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1.1.2.cmml">𝓌</mi><mo id="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1.1.3" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S1.Thmthm4.p1.8.1.m1.1.1.1.1.1.3" stretchy="false" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S1.Thmthm4.p1.8.1.m1.1.1.1.2" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.2.cmml">−</mo><mn class="ltx_font_mathcaligraphic" id="S1.Thmthm4.p1.8.1.m1.1.1.1.3" mathvariant="script" xref="S1.Thmthm4.p1.8.1.m1.1.1.1.3.cmml">2</mn></mrow><mrow id="S1.Thmthm4.p1.8.1.m1.3.3.3.2" xref="S1.Thmthm4.p1.8.1.m1.3.3.3.3.cmml"><mi id="S1.Thmthm4.p1.8.1.m1.2.2.2.1" xref="S1.Thmthm4.p1.8.1.m1.2.2.2.1.cmml">min</mi><mo id="S1.Thmthm4.p1.8.1.m1.3.3.3.2a" xref="S1.Thmthm4.p1.8.1.m1.3.3.3.3.cmml">⁡</mo><mrow id="S1.Thmthm4.p1.8.1.m1.3.3.3.2.1" xref="S1.Thmthm4.p1.8.1.m1.3.3.3.3.cmml"><mo 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id="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.3.2.cmml" type="integer" xref="S1.Thmthm4.p1.8.1.m1.7.7.3.2.2.3.2">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm4.p1.8.1.m1.7c">W(w^{\prime})={\big{\{}}w\in\cal A^{*}\,:\,|w|\leq\frac{|w^{\prime}|-2}{\min\{% |\sigma(a_{k})|\,\,:\,\,a_{k}\in\cal A\}}+2{\big{\}}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm4.p1.8.1.m1.7d">italic_W ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = { italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : | caligraphic_w | ≤ divide start_ARG | caligraphic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | - caligraphic_2 end_ARG start_ARG roman_min { | italic_σ ( caligraphic_a start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ) | : caligraphic_a start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ∈ caligraphic_A } end_ARG + caligraphic_2 }</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.8">As stated above in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S1.Thmthm1" title="Proposition 1.1. ‣ 1. Introduction ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">1.1</span></a>, any invariant measure <math alttext="\mu^{\prime}" class="ltx_Math" display="inline" id="S1.p8.1.m1.1"><semantics id="S1.p8.1.m1.1a"><msup id="S1.p8.1.m1.1.1" xref="S1.p8.1.m1.1.1.cmml"><mi id="S1.p8.1.m1.1.1.2" xref="S1.p8.1.m1.1.1.2.cmml">μ</mi><mo id="S1.p8.1.m1.1.1.3" xref="S1.p8.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.p8.1.m1.1b"><apply id="S1.p8.1.m1.1.1.cmml" xref="S1.p8.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p8.1.m1.1.1.1.cmml" xref="S1.p8.1.m1.1.1">superscript</csymbol><ci id="S1.p8.1.m1.1.1.2.cmml" xref="S1.p8.1.m1.1.1.2">𝜇</ci><ci id="S1.p8.1.m1.1.1.3.cmml" xref="S1.p8.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.1.m1.1c">\mu^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.1.m1.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> on the image subshift <math alttext="Y=\sigma(X)" class="ltx_Math" display="inline" id="S1.p8.2.m2.1"><semantics id="S1.p8.2.m2.1a"><mrow id="S1.p8.2.m2.1.2" xref="S1.p8.2.m2.1.2.cmml"><mi id="S1.p8.2.m2.1.2.2" xref="S1.p8.2.m2.1.2.2.cmml">Y</mi><mo id="S1.p8.2.m2.1.2.1" xref="S1.p8.2.m2.1.2.1.cmml">=</mo><mrow id="S1.p8.2.m2.1.2.3" xref="S1.p8.2.m2.1.2.3.cmml"><mi id="S1.p8.2.m2.1.2.3.2" xref="S1.p8.2.m2.1.2.3.2.cmml">σ</mi><mo id="S1.p8.2.m2.1.2.3.1" xref="S1.p8.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S1.p8.2.m2.1.2.3.3.2" xref="S1.p8.2.m2.1.2.3.cmml"><mo id="S1.p8.2.m2.1.2.3.3.2.1" stretchy="false" xref="S1.p8.2.m2.1.2.3.cmml">(</mo><mi id="S1.p8.2.m2.1.1" xref="S1.p8.2.m2.1.1.cmml">X</mi><mo id="S1.p8.2.m2.1.2.3.3.2.2" stretchy="false" xref="S1.p8.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.2.m2.1b"><apply id="S1.p8.2.m2.1.2.cmml" xref="S1.p8.2.m2.1.2"><eq id="S1.p8.2.m2.1.2.1.cmml" xref="S1.p8.2.m2.1.2.1"></eq><ci id="S1.p8.2.m2.1.2.2.cmml" xref="S1.p8.2.m2.1.2.2">𝑌</ci><apply id="S1.p8.2.m2.1.2.3.cmml" xref="S1.p8.2.m2.1.2.3"><times id="S1.p8.2.m2.1.2.3.1.cmml" xref="S1.p8.2.m2.1.2.3.1"></times><ci id="S1.p8.2.m2.1.2.3.2.cmml" xref="S1.p8.2.m2.1.2.3.2">𝜎</ci><ci id="S1.p8.2.m2.1.1.cmml" xref="S1.p8.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.2.m2.1c">Y=\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S1.p8.2.m2.1d">italic_Y = italic_σ ( italic_X )</annotation></semantics></math> is equal to the transfer <math alttext="\sigma_{X}M(\mu)" class="ltx_Math" display="inline" id="S1.p8.3.m3.1"><semantics id="S1.p8.3.m3.1a"><mrow id="S1.p8.3.m3.1.2" xref="S1.p8.3.m3.1.2.cmml"><msub id="S1.p8.3.m3.1.2.2" xref="S1.p8.3.m3.1.2.2.cmml"><mi id="S1.p8.3.m3.1.2.2.2" xref="S1.p8.3.m3.1.2.2.2.cmml">σ</mi><mi id="S1.p8.3.m3.1.2.2.3" xref="S1.p8.3.m3.1.2.2.3.cmml">X</mi></msub><mo id="S1.p8.3.m3.1.2.1" xref="S1.p8.3.m3.1.2.1.cmml">⁢</mo><mi id="S1.p8.3.m3.1.2.3" xref="S1.p8.3.m3.1.2.3.cmml">M</mi><mo id="S1.p8.3.m3.1.2.1a" xref="S1.p8.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.p8.3.m3.1.2.4.2" xref="S1.p8.3.m3.1.2.cmml"><mo id="S1.p8.3.m3.1.2.4.2.1" stretchy="false" xref="S1.p8.3.m3.1.2.cmml">(</mo><mi id="S1.p8.3.m3.1.1" xref="S1.p8.3.m3.1.1.cmml">μ</mi><mo id="S1.p8.3.m3.1.2.4.2.2" stretchy="false" xref="S1.p8.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.3.m3.1b"><apply id="S1.p8.3.m3.1.2.cmml" xref="S1.p8.3.m3.1.2"><times id="S1.p8.3.m3.1.2.1.cmml" xref="S1.p8.3.m3.1.2.1"></times><apply id="S1.p8.3.m3.1.2.2.cmml" xref="S1.p8.3.m3.1.2.2"><csymbol cd="ambiguous" id="S1.p8.3.m3.1.2.2.1.cmml" xref="S1.p8.3.m3.1.2.2">subscript</csymbol><ci id="S1.p8.3.m3.1.2.2.2.cmml" xref="S1.p8.3.m3.1.2.2.2">𝜎</ci><ci id="S1.p8.3.m3.1.2.2.3.cmml" xref="S1.p8.3.m3.1.2.2.3">𝑋</ci></apply><ci id="S1.p8.3.m3.1.2.3.cmml" xref="S1.p8.3.m3.1.2.3">𝑀</ci><ci id="S1.p8.3.m3.1.1.cmml" xref="S1.p8.3.m3.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.3.m3.1c">\sigma_{X}M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S1.p8.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M ( italic_μ )</annotation></semantics></math> of some measure <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p8.4.m4.1"><semantics id="S1.p8.4.m4.1a"><mi id="S1.p8.4.m4.1.1" xref="S1.p8.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p8.4.m4.1b"><ci id="S1.p8.4.m4.1.1.cmml" xref="S1.p8.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p8.4.m4.1d">italic_μ</annotation></semantics></math> on the given preimage subshift <math alttext="X" class="ltx_Math" display="inline" id="S1.p8.5.m5.1"><semantics id="S1.p8.5.m5.1a"><mi id="S1.p8.5.m5.1.1" xref="S1.p8.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p8.5.m5.1b"><ci id="S1.p8.5.m5.1.1.cmml" xref="S1.p8.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p8.5.m5.1d">italic_X</annotation></semantics></math>. However, for a general non-erasing morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p8.6.m6.1"><semantics id="S1.p8.6.m6.1a"><mi id="S1.p8.6.m6.1.1" xref="S1.p8.6.m6.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p8.6.m6.1b"><ci id="S1.p8.6.m6.1.1.cmml" xref="S1.p8.6.m6.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.6.m6.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p8.6.m6.1d">italic_σ</annotation></semantics></math>, this measure <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p8.7.m7.1"><semantics id="S1.p8.7.m7.1a"><mi id="S1.p8.7.m7.1.1" xref="S1.p8.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p8.7.m7.1b"><ci id="S1.p8.7.m7.1.1.cmml" xref="S1.p8.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p8.7.m7.1d">italic_μ</annotation></semantics></math> will be far from uniquely determined by <math alttext="\mu^{\prime}" class="ltx_Math" display="inline" id="S1.p8.8.m8.1"><semantics id="S1.p8.8.m8.1a"><msup id="S1.p8.8.m8.1.1" xref="S1.p8.8.m8.1.1.cmml"><mi id="S1.p8.8.m8.1.1.2" xref="S1.p8.8.m8.1.1.2.cmml">μ</mi><mo id="S1.p8.8.m8.1.1.3" xref="S1.p8.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.p8.8.m8.1b"><apply id="S1.p8.8.m8.1.1.cmml" xref="S1.p8.8.m8.1.1"><csymbol cd="ambiguous" id="S1.p8.8.m8.1.1.1.cmml" xref="S1.p8.8.m8.1.1">superscript</csymbol><ci id="S1.p8.8.m8.1.1.2.cmml" xref="S1.p8.8.m8.1.1.2">𝜇</ci><ci id="S1.p8.8.m8.1.1.3.cmml" xref="S1.p8.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.8.m8.1c">\mu^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.8.m8.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.3">On the other hand, the injectivity of the measure transfer map, if given, is a strong and useful tool in many circumstances, see Example <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm9" title="Example 6.9. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.9</span></a>. In particular, it can be used as key ingredient for the construction of subshifts with entropy 0 but infinitely many distinct ergodic probability measures. In §7 of our cousin paper <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite> this has been detailed out for the special case of minimal subshifts, so that the additional assumption that <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.p9.1.m1.1"><semantics id="S1.p9.1.m1.1a"><mi id="S1.p9.1.m1.1.1" xref="S1.p9.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.p9.1.m1.1b"><ci id="S1.p9.1.m1.1.1.cmml" xref="S1.p9.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.p9.1.m1.1d">italic_σ</annotation></semantics></math> is recognizable in <math alttext="X" class="ltx_Math" display="inline" id="S1.p9.2.m2.1"><semantics id="S1.p9.2.m2.1a"><mi id="S1.p9.2.m2.1.1" xref="S1.p9.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p9.2.m2.1b"><ci id="S1.p9.2.m2.1.1.cmml" xref="S1.p9.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p9.2.m2.1d">italic_X</annotation></semantics></math> could be used. This additional assumption implies indeed the injectivity of the measure transfer map <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S1.p9.3.m3.1"><semantics id="S1.p9.3.m3.1a"><mrow id="S1.p9.3.m3.1.1" xref="S1.p9.3.m3.1.1.cmml"><msub id="S1.p9.3.m3.1.1.2" xref="S1.p9.3.m3.1.1.2.cmml"><mi id="S1.p9.3.m3.1.1.2.2" xref="S1.p9.3.m3.1.1.2.2.cmml">σ</mi><mi id="S1.p9.3.m3.1.1.2.3" xref="S1.p9.3.m3.1.1.2.3.cmml">X</mi></msub><mo id="S1.p9.3.m3.1.1.1" xref="S1.p9.3.m3.1.1.1.cmml">⁢</mo><mi id="S1.p9.3.m3.1.1.3" xref="S1.p9.3.m3.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.3.m3.1b"><apply id="S1.p9.3.m3.1.1.cmml" xref="S1.p9.3.m3.1.1"><times id="S1.p9.3.m3.1.1.1.cmml" xref="S1.p9.3.m3.1.1.1"></times><apply id="S1.p9.3.m3.1.1.2.cmml" xref="S1.p9.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.p9.3.m3.1.1.2.1.cmml" xref="S1.p9.3.m3.1.1.2">subscript</csymbol><ci id="S1.p9.3.m3.1.1.2.2.cmml" xref="S1.p9.3.m3.1.1.2.2">𝜎</ci><ci id="S1.p9.3.m3.1.1.2.3.cmml" xref="S1.p9.3.m3.1.1.2.3">𝑋</ci></apply><ci id="S1.p9.3.m3.1.1.3.cmml" xref="S1.p9.3.m3.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.3.m3.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S1.p9.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> (see Corollary 3.9 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>). We show here the following stronger result (see Theorem <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm6" title="Theorem 5.6. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.6</span></a> below):</p> </div> <div class="ltx_theorem ltx_theorem_thm" id="S1.Thmthm5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmthm5.1.1.1">Theorem 1.5</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmthm5.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmthm5.p1"> <p class="ltx_p" id="S1.Thmthm5.p1.2"><span class="ltx_text ltx_font_italic" id="S1.Thmthm5.p1.2.2">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S1.Thmthm5.p1.1.1.m1.1"><semantics id="S1.Thmthm5.p1.1.1.m1.1a"><mrow id="S1.Thmthm5.p1.1.1.m1.1.1" xref="S1.Thmthm5.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmthm5.p1.1.1.m1.1.1.2" xref="S1.Thmthm5.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S1.Thmthm5.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm5.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmthm5.p1.1.1.m1.1.1.3" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.cmml"><msup id="S1.Thmthm5.p1.1.1.m1.1.1.3.2" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm5.p1.1.1.m1.1.1.3.2.2" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.Thmthm5.p1.1.1.m1.1.1.3.2.3" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.Thmthm5.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S1.Thmthm5.p1.1.1.m1.1.1.3.3" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm5.p1.1.1.m1.1.1.3.3.2" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S1.Thmthm5.p1.1.1.m1.1.1.3.3.3" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm5.p1.1.1.m1.1b"><apply id="S1.Thmthm5.p1.1.1.m1.1.1.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1"><ci id="S1.Thmthm5.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.1">:</ci><ci id="S1.Thmthm5.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S1.Thmthm5.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3"><ci id="S1.Thmthm5.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.1">→</ci><apply id="S1.Thmthm5.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmthm5.p1.1.1.m1.1.1.3.2.1.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.Thmthm5.p1.1.1.m1.1.1.3.2.2.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S1.Thmthm5.p1.1.1.m1.1.1.3.2.3.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S1.Thmthm5.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmthm5.p1.1.1.m1.1.1.3.3.1.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.Thmthm5.p1.1.1.m1.1.1.3.3.2.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S1.Thmthm5.p1.1.1.m1.1.1.3.3.3.cmml" xref="S1.Thmthm5.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm5.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm5.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-erasing morphism of free monoids on finite alphabets, and let <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.Thmthm5.p1.2.2.m2.1"><semantics id="S1.Thmthm5.p1.2.2.m2.1a"><mrow id="S1.Thmthm5.p1.2.2.m2.1.1" xref="S1.Thmthm5.p1.2.2.m2.1.1.cmml"><mi id="S1.Thmthm5.p1.2.2.m2.1.1.2" xref="S1.Thmthm5.p1.2.2.m2.1.1.2.cmml">X</mi><mo id="S1.Thmthm5.p1.2.2.m2.1.1.1" xref="S1.Thmthm5.p1.2.2.m2.1.1.1.cmml">⊆</mo><msup id="S1.Thmthm5.p1.2.2.m2.1.1.3" xref="S1.Thmthm5.p1.2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm5.p1.2.2.m2.1.1.3.2" xref="S1.Thmthm5.p1.2.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S1.Thmthm5.p1.2.2.m2.1.1.3.3" xref="S1.Thmthm5.p1.2.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm5.p1.2.2.m2.1b"><apply id="S1.Thmthm5.p1.2.2.m2.1.1.cmml" xref="S1.Thmthm5.p1.2.2.m2.1.1"><subset id="S1.Thmthm5.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmthm5.p1.2.2.m2.1.1.1"></subset><ci id="S1.Thmthm5.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmthm5.p1.2.2.m2.1.1.2">𝑋</ci><apply id="S1.Thmthm5.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmthm5.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.Thmthm5.p1.2.2.m2.1.1.3.1.cmml" xref="S1.Thmthm5.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S1.Thmthm5.p1.2.2.m2.1.1.3.2.cmml" xref="S1.Thmthm5.p1.2.2.m2.1.1.3.2">𝒜</ci><ci id="S1.Thmthm5.p1.2.2.m2.1.1.3.3.cmml" xref="S1.Thmthm5.p1.2.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm5.p1.2.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm5.p1.2.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> be any subshift.</span></p> </div> <div class="ltx_para" id="S1.Thmthm5.p2"> <p class="ltx_p" id="S1.Thmthm5.p2.3"><span class="ltx_text ltx_font_italic" id="S1.Thmthm5.p2.3.3">If the map induced by <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.Thmthm5.p2.1.1.m1.1"><semantics id="S1.Thmthm5.p2.1.1.m1.1a"><mi id="S1.Thmthm5.p2.1.1.m1.1.1" xref="S1.Thmthm5.p2.1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm5.p2.1.1.m1.1b"><ci id="S1.Thmthm5.p2.1.1.m1.1.1.cmml" xref="S1.Thmthm5.p2.1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm5.p2.1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm5.p2.1.1.m1.1d">italic_σ</annotation></semantics></math> on the shift-orbits of <math alttext="X" class="ltx_Math" display="inline" id="S1.Thmthm5.p2.2.2.m2.1"><semantics id="S1.Thmthm5.p2.2.2.m2.1a"><mi id="S1.Thmthm5.p2.2.2.m2.1.1" xref="S1.Thmthm5.p2.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm5.p2.2.2.m2.1b"><ci id="S1.Thmthm5.p2.2.2.m2.1.1.cmml" xref="S1.Thmthm5.p2.2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm5.p2.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm5.p2.2.2.m2.1d">italic_X</annotation></semantics></math> is injective, then the measure transfer map <math alttext="\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))" class="ltx_Math" display="inline" id="S1.Thmthm5.p2.3.3.m3.3"><semantics id="S1.Thmthm5.p2.3.3.m3.3a"><mrow id="S1.Thmthm5.p2.3.3.m3.3.3" xref="S1.Thmthm5.p2.3.3.m3.3.3.cmml"><mrow id="S1.Thmthm5.p2.3.3.m3.3.3.3" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.cmml"><msub id="S1.Thmthm5.p2.3.3.m3.3.3.3.2" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.2.cmml"><mi id="S1.Thmthm5.p2.3.3.m3.3.3.3.2.2" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.2.2.cmml">σ</mi><mi id="S1.Thmthm5.p2.3.3.m3.3.3.3.2.3" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.2.3.cmml">X</mi></msub><mo id="S1.Thmthm5.p2.3.3.m3.3.3.3.1" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.1.cmml">⁢</mo><mi id="S1.Thmthm5.p2.3.3.m3.3.3.3.3" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.3.cmml">M</mi></mrow><mo id="S1.Thmthm5.p2.3.3.m3.3.3.2" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm5.p2.3.3.m3.3.3.2.cmml">:</mo><mrow id="S1.Thmthm5.p2.3.3.m3.3.3.1" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.cmml"><mrow id="S1.Thmthm5.p2.3.3.m3.3.3.1.3" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm5.p2.3.3.m3.3.3.1.3.2" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3.2.cmml">ℳ</mi><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.3.1" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3.1.cmml">⁢</mo><mrow id="S1.Thmthm5.p2.3.3.m3.3.3.1.3.3.2" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3.cmml"><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.3.3.2.1" stretchy="false" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm5.p2.3.3.m3.1.1" xref="S1.Thmthm5.p2.3.3.m3.1.1.cmml">𝒳</mi><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.3.3.2.2" stretchy="false" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.2" stretchy="false" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.2.cmml">→</mo><mrow id="S1.Thmthm5.p2.3.3.m3.3.3.1.1" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.3" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.3.cmml">ℳ</mi><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.2" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.2.cmml">⁢</mo><mrow id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.cmml"><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.2" stretchy="false" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.cmml"><mi id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.2" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.2.cmml">σ</mi><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.1" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.3.2" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.cmml"><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm5.p2.3.3.m3.2.2" xref="S1.Thmthm5.p2.3.3.m3.2.2.cmml">𝒳</mi><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.3" stretchy="false" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm5.p2.3.3.m3.3b"><apply id="S1.Thmthm5.p2.3.3.m3.3.3.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3"><ci id="S1.Thmthm5.p2.3.3.m3.3.3.2.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.2">:</ci><apply id="S1.Thmthm5.p2.3.3.m3.3.3.3.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.3"><times id="S1.Thmthm5.p2.3.3.m3.3.3.3.1.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.1"></times><apply id="S1.Thmthm5.p2.3.3.m3.3.3.3.2.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.2"><csymbol cd="ambiguous" id="S1.Thmthm5.p2.3.3.m3.3.3.3.2.1.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.2">subscript</csymbol><ci id="S1.Thmthm5.p2.3.3.m3.3.3.3.2.2.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.2.2">𝜎</ci><ci id="S1.Thmthm5.p2.3.3.m3.3.3.3.2.3.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.2.3">𝑋</ci></apply><ci id="S1.Thmthm5.p2.3.3.m3.3.3.3.3.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.3.3">𝑀</ci></apply><apply id="S1.Thmthm5.p2.3.3.m3.3.3.1.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1"><ci id="S1.Thmthm5.p2.3.3.m3.3.3.1.2.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.2">→</ci><apply id="S1.Thmthm5.p2.3.3.m3.3.3.1.3.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3"><times id="S1.Thmthm5.p2.3.3.m3.3.3.1.3.1.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3.1"></times><ci id="S1.Thmthm5.p2.3.3.m3.3.3.1.3.2.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.3.2">ℳ</ci><ci id="S1.Thmthm5.p2.3.3.m3.1.1.cmml" xref="S1.Thmthm5.p2.3.3.m3.1.1">𝒳</ci></apply><apply id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1"><times id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.2.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.2"></times><ci id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.3.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.3">ℳ</ci><apply id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1"><times id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.1.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.1"></times><ci id="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.2.cmml" xref="S1.Thmthm5.p2.3.3.m3.3.3.1.1.1.1.1.2">𝜎</ci><ci id="S1.Thmthm5.p2.3.3.m3.2.2.cmml" xref="S1.Thmthm5.p2.3.3.m3.2.2">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm5.p2.3.3.m3.3c">\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm5.p2.3.3.m3.3d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) )</annotation></semantics></math> is injective.</span></p> </div> </div> <div class="ltx_para" id="S1.p10"> <p class="ltx_p" id="S1.p10.4">To be specific, we’d like to note that the hypothesis “injectivity on the set of shift-orbits of <math alttext="X" class="ltx_Math" display="inline" id="S1.p10.1.m1.1"><semantics id="S1.p10.1.m1.1a"><mi id="S1.p10.1.m1.1.1" xref="S1.p10.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p10.1.m1.1b"><ci id="S1.p10.1.m1.1.1.cmml" xref="S1.p10.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p10.1.m1.1d">italic_X</annotation></semantics></math>” is strictly weaker than “recognizable in <math alttext="X" class="ltx_Math" display="inline" id="S1.p10.2.m2.1"><semantics id="S1.p10.2.m2.1a"><mi id="S1.p10.2.m2.1.1" xref="S1.p10.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p10.2.m2.1b"><ci id="S1.p10.2.m2.1.1.cmml" xref="S1.p10.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p10.2.m2.1d">italic_X</annotation></semantics></math>”, and strictly stronger than “recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S1.p10.3.m3.1"><semantics id="S1.p10.3.m3.1a"><mi id="S1.p10.3.m3.1.1" xref="S1.p10.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p10.3.m3.1b"><ci id="S1.p10.3.m3.1.1.cmml" xref="S1.p10.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p10.3.m3.1d">italic_X</annotation></semantics></math>”. Also, the converse to the conclusion of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S1.Thmthm5" title="Theorem 1.5. ‣ 1. Introduction ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">1.5</span></a> does not hold in full generality. Furthermore, the two properties “recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S1.p10.4.m4.1"><semantics id="S1.p10.4.m4.1a"><mi id="S1.p10.4.m4.1.1" xref="S1.p10.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p10.4.m4.1b"><ci id="S1.p10.4.m4.1.1.cmml" xref="S1.p10.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p10.4.m4.1d">italic_X</annotation></semantics></math>” and “injectivity of the induced measure transfer map” are logically independent. The implications among all of these properties and their refusals are conveniently summarized in Fig. 1, and an exhaustive discussion with proofs and counter-examples is given in Section <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5" title="5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5</span></a>.</p> </div> <div class="ltx_para" id="S1.p11"> <p class="ltx_p" id="S1.p11.1">To terminate this introduction, we’d like to list some incidents where our readers may already have encountered the measure transfer map, perhaps “disguised” in a different setting or language:</p> </div> <div class="ltx_theorem ltx_theorem_example" id="S1.Thmthm6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmthm6.1.1.1">Example 1.6</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmthm6.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmthm6.p1"> <p class="ltx_p" id="S1.Thmthm6.p1.16">(1) Let <math alttext="S" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.1.m1.1"><semantics id="S1.Thmthm6.p1.1.m1.1a"><mi id="S1.Thmthm6.p1.1.m1.1.1" xref="S1.Thmthm6.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.1.m1.1b"><ci id="S1.Thmthm6.p1.1.m1.1.1.cmml" xref="S1.Thmthm6.p1.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.1.m1.1d">italic_S</annotation></semantics></math> be a compact surface with non-empty boundary, and <math alttext="\tau\subseteq S" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.2.m2.1"><semantics id="S1.Thmthm6.p1.2.m2.1a"><mrow id="S1.Thmthm6.p1.2.m2.1.1" xref="S1.Thmthm6.p1.2.m2.1.1.cmml"><mi id="S1.Thmthm6.p1.2.m2.1.1.2" xref="S1.Thmthm6.p1.2.m2.1.1.2.cmml">τ</mi><mo id="S1.Thmthm6.p1.2.m2.1.1.1" xref="S1.Thmthm6.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S1.Thmthm6.p1.2.m2.1.1.3" xref="S1.Thmthm6.p1.2.m2.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.2.m2.1b"><apply id="S1.Thmthm6.p1.2.m2.1.1.cmml" xref="S1.Thmthm6.p1.2.m2.1.1"><subset id="S1.Thmthm6.p1.2.m2.1.1.1.cmml" xref="S1.Thmthm6.p1.2.m2.1.1.1"></subset><ci id="S1.Thmthm6.p1.2.m2.1.1.2.cmml" xref="S1.Thmthm6.p1.2.m2.1.1.2">𝜏</ci><ci id="S1.Thmthm6.p1.2.m2.1.1.3.cmml" xref="S1.Thmthm6.p1.2.m2.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.2.m2.1c">\tau\subseteq S</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.2.m2.1d">italic_τ ⊆ italic_S</annotation></semantics></math> a train track that fills <math alttext="S" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.3.m3.1"><semantics id="S1.Thmthm6.p1.3.m3.1a"><mi id="S1.Thmthm6.p1.3.m3.1.1" xref="S1.Thmthm6.p1.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.3.m3.1b"><ci id="S1.Thmthm6.p1.3.m3.1.1.cmml" xref="S1.Thmthm6.p1.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.3.m3.1d">italic_S</annotation></semantics></math> and satisfies the usual conditions on its complementary components (see for instance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib10" title="">10</a>]</cite> for details). Then a proper choice of arcs <math alttext="\alpha_{1},\ldots,\alpha_{d}" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.4.m4.3"><semantics id="S1.Thmthm6.p1.4.m4.3a"><mrow id="S1.Thmthm6.p1.4.m4.3.3.2" xref="S1.Thmthm6.p1.4.m4.3.3.3.cmml"><msub id="S1.Thmthm6.p1.4.m4.2.2.1.1" xref="S1.Thmthm6.p1.4.m4.2.2.1.1.cmml"><mi id="S1.Thmthm6.p1.4.m4.2.2.1.1.2" xref="S1.Thmthm6.p1.4.m4.2.2.1.1.2.cmml">α</mi><mn id="S1.Thmthm6.p1.4.m4.2.2.1.1.3" xref="S1.Thmthm6.p1.4.m4.2.2.1.1.3.cmml">1</mn></msub><mo id="S1.Thmthm6.p1.4.m4.3.3.2.3" xref="S1.Thmthm6.p1.4.m4.3.3.3.cmml">,</mo><mi id="S1.Thmthm6.p1.4.m4.1.1" mathvariant="normal" xref="S1.Thmthm6.p1.4.m4.1.1.cmml">…</mi><mo id="S1.Thmthm6.p1.4.m4.3.3.2.4" xref="S1.Thmthm6.p1.4.m4.3.3.3.cmml">,</mo><msub id="S1.Thmthm6.p1.4.m4.3.3.2.2" xref="S1.Thmthm6.p1.4.m4.3.3.2.2.cmml"><mi id="S1.Thmthm6.p1.4.m4.3.3.2.2.2" xref="S1.Thmthm6.p1.4.m4.3.3.2.2.2.cmml">α</mi><mi id="S1.Thmthm6.p1.4.m4.3.3.2.2.3" xref="S1.Thmthm6.p1.4.m4.3.3.2.2.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.4.m4.3b"><list id="S1.Thmthm6.p1.4.m4.3.3.3.cmml" xref="S1.Thmthm6.p1.4.m4.3.3.2"><apply id="S1.Thmthm6.p1.4.m4.2.2.1.1.cmml" xref="S1.Thmthm6.p1.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.4.m4.2.2.1.1.1.cmml" xref="S1.Thmthm6.p1.4.m4.2.2.1.1">subscript</csymbol><ci id="S1.Thmthm6.p1.4.m4.2.2.1.1.2.cmml" xref="S1.Thmthm6.p1.4.m4.2.2.1.1.2">𝛼</ci><cn id="S1.Thmthm6.p1.4.m4.2.2.1.1.3.cmml" type="integer" xref="S1.Thmthm6.p1.4.m4.2.2.1.1.3">1</cn></apply><ci id="S1.Thmthm6.p1.4.m4.1.1.cmml" xref="S1.Thmthm6.p1.4.m4.1.1">…</ci><apply id="S1.Thmthm6.p1.4.m4.3.3.2.2.cmml" xref="S1.Thmthm6.p1.4.m4.3.3.2.2"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.4.m4.3.3.2.2.1.cmml" xref="S1.Thmthm6.p1.4.m4.3.3.2.2">subscript</csymbol><ci id="S1.Thmthm6.p1.4.m4.3.3.2.2.2.cmml" xref="S1.Thmthm6.p1.4.m4.3.3.2.2.2">𝛼</ci><ci id="S1.Thmthm6.p1.4.m4.3.3.2.2.3.cmml" xref="S1.Thmthm6.p1.4.m4.3.3.2.2.3">𝑑</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.4.m4.3c">\alpha_{1},\ldots,\alpha_{d}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.4.m4.3d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> transverse to <math alttext="\tau" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.5.m5.1"><semantics id="S1.Thmthm6.p1.5.m5.1a"><mi id="S1.Thmthm6.p1.5.m5.1.1" xref="S1.Thmthm6.p1.5.m5.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.5.m5.1b"><ci id="S1.Thmthm6.p1.5.m5.1.1.cmml" xref="S1.Thmthm6.p1.5.m5.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.5.m5.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.5.m5.1d">italic_τ</annotation></semantics></math> gives rise to intervals with the following property: Any oriented geodesic lamination <math alttext="\Lambda\subseteq S" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.6.m6.1"><semantics id="S1.Thmthm6.p1.6.m6.1a"><mrow id="S1.Thmthm6.p1.6.m6.1.1" xref="S1.Thmthm6.p1.6.m6.1.1.cmml"><mi id="S1.Thmthm6.p1.6.m6.1.1.2" mathvariant="normal" xref="S1.Thmthm6.p1.6.m6.1.1.2.cmml">Λ</mi><mo id="S1.Thmthm6.p1.6.m6.1.1.1" xref="S1.Thmthm6.p1.6.m6.1.1.1.cmml">⊆</mo><mi id="S1.Thmthm6.p1.6.m6.1.1.3" xref="S1.Thmthm6.p1.6.m6.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.6.m6.1b"><apply id="S1.Thmthm6.p1.6.m6.1.1.cmml" xref="S1.Thmthm6.p1.6.m6.1.1"><subset id="S1.Thmthm6.p1.6.m6.1.1.1.cmml" xref="S1.Thmthm6.p1.6.m6.1.1.1"></subset><ci id="S1.Thmthm6.p1.6.m6.1.1.2.cmml" xref="S1.Thmthm6.p1.6.m6.1.1.2">Λ</ci><ci id="S1.Thmthm6.p1.6.m6.1.1.3.cmml" xref="S1.Thmthm6.p1.6.m6.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.6.m6.1c">\Lambda\subseteq S</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.6.m6.1d">roman_Λ ⊆ italic_S</annotation></semantics></math>, for which we assume that it can be isotoped into an interval-fibered neighborhood <math alttext="\cal N(\tau)" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.7.m7.1"><semantics id="S1.Thmthm6.p1.7.m7.1a"><mrow id="S1.Thmthm6.p1.7.m7.1.2" xref="S1.Thmthm6.p1.7.m7.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p1.7.m7.1.2.2" xref="S1.Thmthm6.p1.7.m7.1.2.2.cmml">𝒩</mi><mo id="S1.Thmthm6.p1.7.m7.1.2.1" xref="S1.Thmthm6.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p1.7.m7.1.2.3.2" xref="S1.Thmthm6.p1.7.m7.1.2.cmml"><mo id="S1.Thmthm6.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S1.Thmthm6.p1.7.m7.1.2.cmml">(</mo><mi id="S1.Thmthm6.p1.7.m7.1.1" xref="S1.Thmthm6.p1.7.m7.1.1.cmml">τ</mi><mo id="S1.Thmthm6.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S1.Thmthm6.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.7.m7.1b"><apply id="S1.Thmthm6.p1.7.m7.1.2.cmml" xref="S1.Thmthm6.p1.7.m7.1.2"><times id="S1.Thmthm6.p1.7.m7.1.2.1.cmml" xref="S1.Thmthm6.p1.7.m7.1.2.1"></times><ci id="S1.Thmthm6.p1.7.m7.1.2.2.cmml" xref="S1.Thmthm6.p1.7.m7.1.2.2">𝒩</ci><ci id="S1.Thmthm6.p1.7.m7.1.1.cmml" xref="S1.Thmthm6.p1.7.m7.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.7.m7.1c">\cal N(\tau)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.7.m7.1d">caligraphic_N ( italic_τ )</annotation></semantics></math> of <math alttext="\tau" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.8.m8.1"><semantics id="S1.Thmthm6.p1.8.m8.1a"><mi id="S1.Thmthm6.p1.8.m8.1.1" xref="S1.Thmthm6.p1.8.m8.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.8.m8.1b"><ci id="S1.Thmthm6.p1.8.m8.1.1.cmml" xref="S1.Thmthm6.p1.8.m8.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.8.m8.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.8.m8.1d">italic_τ</annotation></semantics></math> in such a way that <math alttext="\Lambda" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.9.m9.1"><semantics id="S1.Thmthm6.p1.9.m9.1a"><mi id="S1.Thmthm6.p1.9.m9.1.1" mathvariant="normal" xref="S1.Thmthm6.p1.9.m9.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.9.m9.1b"><ci id="S1.Thmthm6.p1.9.m9.1.1.cmml" xref="S1.Thmthm6.p1.9.m9.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.9.m9.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.9.m9.1d">roman_Λ</annotation></semantics></math> becomes transverse to all interval fibers, defines an interval exchange transformation system (IET) on the intervals <math alttext="\alpha_{k}" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.10.m10.1"><semantics id="S1.Thmthm6.p1.10.m10.1a"><msub id="S1.Thmthm6.p1.10.m10.1.1" xref="S1.Thmthm6.p1.10.m10.1.1.cmml"><mi id="S1.Thmthm6.p1.10.m10.1.1.2" xref="S1.Thmthm6.p1.10.m10.1.1.2.cmml">α</mi><mi id="S1.Thmthm6.p1.10.m10.1.1.3" xref="S1.Thmthm6.p1.10.m10.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.10.m10.1b"><apply id="S1.Thmthm6.p1.10.m10.1.1.cmml" xref="S1.Thmthm6.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.10.m10.1.1.1.cmml" xref="S1.Thmthm6.p1.10.m10.1.1">subscript</csymbol><ci id="S1.Thmthm6.p1.10.m10.1.1.2.cmml" xref="S1.Thmthm6.p1.10.m10.1.1.2">𝛼</ci><ci id="S1.Thmthm6.p1.10.m10.1.1.3.cmml" xref="S1.Thmthm6.p1.10.m10.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.10.m10.1c">\alpha_{k}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.10.m10.1d">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, and thus a subshift <math alttext="X_{\Lambda}\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.11.m11.1"><semantics id="S1.Thmthm6.p1.11.m11.1a"><mrow id="S1.Thmthm6.p1.11.m11.1.1" xref="S1.Thmthm6.p1.11.m11.1.1.cmml"><msub id="S1.Thmthm6.p1.11.m11.1.1.2" xref="S1.Thmthm6.p1.11.m11.1.1.2.cmml"><mi id="S1.Thmthm6.p1.11.m11.1.1.2.2" xref="S1.Thmthm6.p1.11.m11.1.1.2.2.cmml">X</mi><mi id="S1.Thmthm6.p1.11.m11.1.1.2.3" mathvariant="normal" xref="S1.Thmthm6.p1.11.m11.1.1.2.3.cmml">Λ</mi></msub><mo id="S1.Thmthm6.p1.11.m11.1.1.1" xref="S1.Thmthm6.p1.11.m11.1.1.1.cmml">⊆</mo><msup id="S1.Thmthm6.p1.11.m11.1.1.3" xref="S1.Thmthm6.p1.11.m11.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p1.11.m11.1.1.3.2" xref="S1.Thmthm6.p1.11.m11.1.1.3.2.cmml">𝒜</mi><mi id="S1.Thmthm6.p1.11.m11.1.1.3.3" xref="S1.Thmthm6.p1.11.m11.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.11.m11.1b"><apply id="S1.Thmthm6.p1.11.m11.1.1.cmml" xref="S1.Thmthm6.p1.11.m11.1.1"><subset id="S1.Thmthm6.p1.11.m11.1.1.1.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.1"></subset><apply id="S1.Thmthm6.p1.11.m11.1.1.2.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.11.m11.1.1.2.1.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.2">subscript</csymbol><ci id="S1.Thmthm6.p1.11.m11.1.1.2.2.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.2.2">𝑋</ci><ci id="S1.Thmthm6.p1.11.m11.1.1.2.3.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.2.3">Λ</ci></apply><apply id="S1.Thmthm6.p1.11.m11.1.1.3.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.11.m11.1.1.3.1.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.3">superscript</csymbol><ci id="S1.Thmthm6.p1.11.m11.1.1.3.2.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.3.2">𝒜</ci><ci id="S1.Thmthm6.p1.11.m11.1.1.3.3.cmml" xref="S1.Thmthm6.p1.11.m11.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.11.m11.1c">X_{\Lambda}\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.11.m11.1d">italic_X start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="\cal A=\{\alpha_{1},\ldots,\alpha_{d}\}" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.12.m12.3"><semantics id="S1.Thmthm6.p1.12.m12.3a"><mrow id="S1.Thmthm6.p1.12.m12.3.3" xref="S1.Thmthm6.p1.12.m12.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p1.12.m12.3.3.4" xref="S1.Thmthm6.p1.12.m12.3.3.4.cmml">𝒜</mi><mo id="S1.Thmthm6.p1.12.m12.3.3.3" xref="S1.Thmthm6.p1.12.m12.3.3.3.cmml">=</mo><mrow id="S1.Thmthm6.p1.12.m12.3.3.2.2" xref="S1.Thmthm6.p1.12.m12.3.3.2.3.cmml"><mo id="S1.Thmthm6.p1.12.m12.3.3.2.2.3" stretchy="false" xref="S1.Thmthm6.p1.12.m12.3.3.2.3.cmml">{</mo><msub id="S1.Thmthm6.p1.12.m12.2.2.1.1.1" xref="S1.Thmthm6.p1.12.m12.2.2.1.1.1.cmml"><mi id="S1.Thmthm6.p1.12.m12.2.2.1.1.1.2" xref="S1.Thmthm6.p1.12.m12.2.2.1.1.1.2.cmml">α</mi><mn class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p1.12.m12.2.2.1.1.1.3" mathvariant="script" xref="S1.Thmthm6.p1.12.m12.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S1.Thmthm6.p1.12.m12.3.3.2.2.4" xref="S1.Thmthm6.p1.12.m12.3.3.2.3.cmml">,</mo><mi id="S1.Thmthm6.p1.12.m12.1.1" mathvariant="normal" xref="S1.Thmthm6.p1.12.m12.1.1.cmml">…</mi><mo id="S1.Thmthm6.p1.12.m12.3.3.2.2.5" xref="S1.Thmthm6.p1.12.m12.3.3.2.3.cmml">,</mo><msub id="S1.Thmthm6.p1.12.m12.3.3.2.2.2" xref="S1.Thmthm6.p1.12.m12.3.3.2.2.2.cmml"><mi id="S1.Thmthm6.p1.12.m12.3.3.2.2.2.2" xref="S1.Thmthm6.p1.12.m12.3.3.2.2.2.2.cmml">α</mi><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p1.12.m12.3.3.2.2.2.3" xref="S1.Thmthm6.p1.12.m12.3.3.2.2.2.3.cmml">𝒹</mi></msub><mo id="S1.Thmthm6.p1.12.m12.3.3.2.2.6" stretchy="false" xref="S1.Thmthm6.p1.12.m12.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.12.m12.3b"><apply id="S1.Thmthm6.p1.12.m12.3.3.cmml" xref="S1.Thmthm6.p1.12.m12.3.3"><eq id="S1.Thmthm6.p1.12.m12.3.3.3.cmml" xref="S1.Thmthm6.p1.12.m12.3.3.3"></eq><ci id="S1.Thmthm6.p1.12.m12.3.3.4.cmml" xref="S1.Thmthm6.p1.12.m12.3.3.4">𝒜</ci><set id="S1.Thmthm6.p1.12.m12.3.3.2.3.cmml" xref="S1.Thmthm6.p1.12.m12.3.3.2.2"><apply id="S1.Thmthm6.p1.12.m12.2.2.1.1.1.cmml" xref="S1.Thmthm6.p1.12.m12.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.12.m12.2.2.1.1.1.1.cmml" xref="S1.Thmthm6.p1.12.m12.2.2.1.1.1">subscript</csymbol><ci id="S1.Thmthm6.p1.12.m12.2.2.1.1.1.2.cmml" xref="S1.Thmthm6.p1.12.m12.2.2.1.1.1.2">𝛼</ci><cn id="S1.Thmthm6.p1.12.m12.2.2.1.1.1.3.cmml" type="integer" xref="S1.Thmthm6.p1.12.m12.2.2.1.1.1.3">1</cn></apply><ci id="S1.Thmthm6.p1.12.m12.1.1.cmml" xref="S1.Thmthm6.p1.12.m12.1.1">…</ci><apply id="S1.Thmthm6.p1.12.m12.3.3.2.2.2.cmml" xref="S1.Thmthm6.p1.12.m12.3.3.2.2.2"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.12.m12.3.3.2.2.2.1.cmml" xref="S1.Thmthm6.p1.12.m12.3.3.2.2.2">subscript</csymbol><ci id="S1.Thmthm6.p1.12.m12.3.3.2.2.2.2.cmml" xref="S1.Thmthm6.p1.12.m12.3.3.2.2.2.2">𝛼</ci><ci id="S1.Thmthm6.p1.12.m12.3.3.2.2.2.3.cmml" xref="S1.Thmthm6.p1.12.m12.3.3.2.2.2.3">𝒹</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.12.m12.3c">\cal A=\{\alpha_{1},\ldots,\alpha_{d}\}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.12.m12.3d">caligraphic_A = { italic_α start_POSTSUBSCRIPT caligraphic_1 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT caligraphic_d end_POSTSUBSCRIPT }</annotation></semantics></math>. Any transverse measure <math alttext="\mu_{\Lambda}" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.13.m13.1"><semantics id="S1.Thmthm6.p1.13.m13.1a"><msub id="S1.Thmthm6.p1.13.m13.1.1" xref="S1.Thmthm6.p1.13.m13.1.1.cmml"><mi id="S1.Thmthm6.p1.13.m13.1.1.2" xref="S1.Thmthm6.p1.13.m13.1.1.2.cmml">μ</mi><mi id="S1.Thmthm6.p1.13.m13.1.1.3" mathvariant="normal" xref="S1.Thmthm6.p1.13.m13.1.1.3.cmml">Λ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.13.m13.1b"><apply id="S1.Thmthm6.p1.13.m13.1.1.cmml" xref="S1.Thmthm6.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.13.m13.1.1.1.cmml" xref="S1.Thmthm6.p1.13.m13.1.1">subscript</csymbol><ci id="S1.Thmthm6.p1.13.m13.1.1.2.cmml" xref="S1.Thmthm6.p1.13.m13.1.1.2">𝜇</ci><ci id="S1.Thmthm6.p1.13.m13.1.1.3.cmml" xref="S1.Thmthm6.p1.13.m13.1.1.3">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.13.m13.1c">\mu_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.13.m13.1d">italic_μ start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="\Lambda" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.14.m14.1"><semantics id="S1.Thmthm6.p1.14.m14.1a"><mi id="S1.Thmthm6.p1.14.m14.1.1" mathvariant="normal" xref="S1.Thmthm6.p1.14.m14.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.14.m14.1b"><ci id="S1.Thmthm6.p1.14.m14.1.1.cmml" xref="S1.Thmthm6.p1.14.m14.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.14.m14.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.14.m14.1d">roman_Λ</annotation></semantics></math> defines an invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.15.m15.1"><semantics id="S1.Thmthm6.p1.15.m15.1a"><mi id="S1.Thmthm6.p1.15.m15.1.1" xref="S1.Thmthm6.p1.15.m15.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.15.m15.1b"><ci id="S1.Thmthm6.p1.15.m15.1.1.cmml" xref="S1.Thmthm6.p1.15.m15.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.15.m15.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.15.m15.1d">italic_μ</annotation></semantics></math> on <math alttext="X_{\Lambda}" class="ltx_Math" display="inline" id="S1.Thmthm6.p1.16.m16.1"><semantics id="S1.Thmthm6.p1.16.m16.1a"><msub id="S1.Thmthm6.p1.16.m16.1.1" xref="S1.Thmthm6.p1.16.m16.1.1.cmml"><mi id="S1.Thmthm6.p1.16.m16.1.1.2" xref="S1.Thmthm6.p1.16.m16.1.1.2.cmml">X</mi><mi id="S1.Thmthm6.p1.16.m16.1.1.3" mathvariant="normal" xref="S1.Thmthm6.p1.16.m16.1.1.3.cmml">Λ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p1.16.m16.1b"><apply id="S1.Thmthm6.p1.16.m16.1.1.cmml" xref="S1.Thmthm6.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p1.16.m16.1.1.1.cmml" xref="S1.Thmthm6.p1.16.m16.1.1">subscript</csymbol><ci id="S1.Thmthm6.p1.16.m16.1.1.2.cmml" xref="S1.Thmthm6.p1.16.m16.1.1.2">𝑋</ci><ci id="S1.Thmthm6.p1.16.m16.1.1.3.cmml" xref="S1.Thmthm6.p1.16.m16.1.1.3">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p1.16.m16.1c">X_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p1.16.m16.1d">italic_X start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.Thmthm6.p2"> <p class="ltx_p" id="S1.Thmthm6.p2.12">Assume now that (as shown by Thurston for any “pseudo-Anosov” homeomorphism) that some homeomorphism <math alttext="h:S\to S" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.1.m1.1"><semantics id="S1.Thmthm6.p2.1.m1.1a"><mrow id="S1.Thmthm6.p2.1.m1.1.1" xref="S1.Thmthm6.p2.1.m1.1.1.cmml"><mi id="S1.Thmthm6.p2.1.m1.1.1.2" xref="S1.Thmthm6.p2.1.m1.1.1.2.cmml">h</mi><mo id="S1.Thmthm6.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm6.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmthm6.p2.1.m1.1.1.3" xref="S1.Thmthm6.p2.1.m1.1.1.3.cmml"><mi id="S1.Thmthm6.p2.1.m1.1.1.3.2" xref="S1.Thmthm6.p2.1.m1.1.1.3.2.cmml">S</mi><mo id="S1.Thmthm6.p2.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmthm6.p2.1.m1.1.1.3.1.cmml">→</mo><mi id="S1.Thmthm6.p2.1.m1.1.1.3.3" xref="S1.Thmthm6.p2.1.m1.1.1.3.3.cmml">S</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.1.m1.1b"><apply id="S1.Thmthm6.p2.1.m1.1.1.cmml" xref="S1.Thmthm6.p2.1.m1.1.1"><ci id="S1.Thmthm6.p2.1.m1.1.1.1.cmml" xref="S1.Thmthm6.p2.1.m1.1.1.1">:</ci><ci id="S1.Thmthm6.p2.1.m1.1.1.2.cmml" xref="S1.Thmthm6.p2.1.m1.1.1.2">ℎ</ci><apply id="S1.Thmthm6.p2.1.m1.1.1.3.cmml" xref="S1.Thmthm6.p2.1.m1.1.1.3"><ci id="S1.Thmthm6.p2.1.m1.1.1.3.1.cmml" xref="S1.Thmthm6.p2.1.m1.1.1.3.1">→</ci><ci id="S1.Thmthm6.p2.1.m1.1.1.3.2.cmml" xref="S1.Thmthm6.p2.1.m1.1.1.3.2">𝑆</ci><ci id="S1.Thmthm6.p2.1.m1.1.1.3.3.cmml" xref="S1.Thmthm6.p2.1.m1.1.1.3.3">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.1.m1.1c">h:S\to S</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.1.m1.1d">italic_h : italic_S → italic_S</annotation></semantics></math>, after being properly isotoped, maps <math alttext="\cal N(\tau)" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.2.m2.1"><semantics id="S1.Thmthm6.p2.2.m2.1a"><mrow id="S1.Thmthm6.p2.2.m2.1.2" xref="S1.Thmthm6.p2.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p2.2.m2.1.2.2" xref="S1.Thmthm6.p2.2.m2.1.2.2.cmml">𝒩</mi><mo id="S1.Thmthm6.p2.2.m2.1.2.1" xref="S1.Thmthm6.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p2.2.m2.1.2.3.2" xref="S1.Thmthm6.p2.2.m2.1.2.cmml"><mo id="S1.Thmthm6.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S1.Thmthm6.p2.2.m2.1.2.cmml">(</mo><mi id="S1.Thmthm6.p2.2.m2.1.1" xref="S1.Thmthm6.p2.2.m2.1.1.cmml">τ</mi><mo id="S1.Thmthm6.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S1.Thmthm6.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.2.m2.1b"><apply id="S1.Thmthm6.p2.2.m2.1.2.cmml" xref="S1.Thmthm6.p2.2.m2.1.2"><times id="S1.Thmthm6.p2.2.m2.1.2.1.cmml" xref="S1.Thmthm6.p2.2.m2.1.2.1"></times><ci id="S1.Thmthm6.p2.2.m2.1.2.2.cmml" xref="S1.Thmthm6.p2.2.m2.1.2.2">𝒩</ci><ci id="S1.Thmthm6.p2.2.m2.1.1.cmml" xref="S1.Thmthm6.p2.2.m2.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.2.m2.1c">\cal N(\tau)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.2.m2.1d">caligraphic_N ( italic_τ )</annotation></semantics></math> in an interval-fiber preserving fashion into <math alttext="\cal N(\tau)" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.3.m3.1"><semantics id="S1.Thmthm6.p2.3.m3.1a"><mrow id="S1.Thmthm6.p2.3.m3.1.2" xref="S1.Thmthm6.p2.3.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p2.3.m3.1.2.2" xref="S1.Thmthm6.p2.3.m3.1.2.2.cmml">𝒩</mi><mo id="S1.Thmthm6.p2.3.m3.1.2.1" xref="S1.Thmthm6.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p2.3.m3.1.2.3.2" xref="S1.Thmthm6.p2.3.m3.1.2.cmml"><mo id="S1.Thmthm6.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S1.Thmthm6.p2.3.m3.1.2.cmml">(</mo><mi id="S1.Thmthm6.p2.3.m3.1.1" xref="S1.Thmthm6.p2.3.m3.1.1.cmml">τ</mi><mo id="S1.Thmthm6.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S1.Thmthm6.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.3.m3.1b"><apply id="S1.Thmthm6.p2.3.m3.1.2.cmml" xref="S1.Thmthm6.p2.3.m3.1.2"><times id="S1.Thmthm6.p2.3.m3.1.2.1.cmml" xref="S1.Thmthm6.p2.3.m3.1.2.1"></times><ci id="S1.Thmthm6.p2.3.m3.1.2.2.cmml" xref="S1.Thmthm6.p2.3.m3.1.2.2">𝒩</ci><ci id="S1.Thmthm6.p2.3.m3.1.1.cmml" xref="S1.Thmthm6.p2.3.m3.1.1">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.3.m3.1c">\cal N(\tau)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.3.m3.1d">caligraphic_N ( italic_τ )</annotation></semantics></math>. Then any transverse measure <math alttext="\mu_{\Lambda}" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.4.m4.1"><semantics id="S1.Thmthm6.p2.4.m4.1a"><msub id="S1.Thmthm6.p2.4.m4.1.1" xref="S1.Thmthm6.p2.4.m4.1.1.cmml"><mi id="S1.Thmthm6.p2.4.m4.1.1.2" xref="S1.Thmthm6.p2.4.m4.1.1.2.cmml">μ</mi><mi id="S1.Thmthm6.p2.4.m4.1.1.3" mathvariant="normal" xref="S1.Thmthm6.p2.4.m4.1.1.3.cmml">Λ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.4.m4.1b"><apply id="S1.Thmthm6.p2.4.m4.1.1.cmml" xref="S1.Thmthm6.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p2.4.m4.1.1.1.cmml" xref="S1.Thmthm6.p2.4.m4.1.1">subscript</csymbol><ci id="S1.Thmthm6.p2.4.m4.1.1.2.cmml" xref="S1.Thmthm6.p2.4.m4.1.1.2">𝜇</ci><ci id="S1.Thmthm6.p2.4.m4.1.1.3.cmml" xref="S1.Thmthm6.p2.4.m4.1.1.3">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.4.m4.1c">\mu_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.4.m4.1d">italic_μ start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math> gives rise to an “<math alttext="h" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.5.m5.1"><semantics id="S1.Thmthm6.p2.5.m5.1a"><mi id="S1.Thmthm6.p2.5.m5.1.1" xref="S1.Thmthm6.p2.5.m5.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.5.m5.1b"><ci id="S1.Thmthm6.p2.5.m5.1.1.cmml" xref="S1.Thmthm6.p2.5.m5.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.5.m5.1c">h</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.5.m5.1d">italic_h</annotation></semantics></math>-image transverse measure” <math alttext="\mu^{\prime}_{\Lambda}" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.6.m6.1"><semantics id="S1.Thmthm6.p2.6.m6.1a"><msubsup id="S1.Thmthm6.p2.6.m6.1.1" xref="S1.Thmthm6.p2.6.m6.1.1.cmml"><mi id="S1.Thmthm6.p2.6.m6.1.1.2.2" xref="S1.Thmthm6.p2.6.m6.1.1.2.2.cmml">μ</mi><mi id="S1.Thmthm6.p2.6.m6.1.1.3" mathvariant="normal" xref="S1.Thmthm6.p2.6.m6.1.1.3.cmml">Λ</mi><mo id="S1.Thmthm6.p2.6.m6.1.1.2.3" xref="S1.Thmthm6.p2.6.m6.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.6.m6.1b"><apply id="S1.Thmthm6.p2.6.m6.1.1.cmml" xref="S1.Thmthm6.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p2.6.m6.1.1.1.cmml" xref="S1.Thmthm6.p2.6.m6.1.1">subscript</csymbol><apply id="S1.Thmthm6.p2.6.m6.1.1.2.cmml" xref="S1.Thmthm6.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p2.6.m6.1.1.2.1.cmml" xref="S1.Thmthm6.p2.6.m6.1.1">superscript</csymbol><ci id="S1.Thmthm6.p2.6.m6.1.1.2.2.cmml" xref="S1.Thmthm6.p2.6.m6.1.1.2.2">𝜇</ci><ci id="S1.Thmthm6.p2.6.m6.1.1.2.3.cmml" xref="S1.Thmthm6.p2.6.m6.1.1.2.3">′</ci></apply><ci id="S1.Thmthm6.p2.6.m6.1.1.3.cmml" xref="S1.Thmthm6.p2.6.m6.1.1.3">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.6.m6.1c">\mu^{\prime}_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.6.m6.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math>, which, when translated back to the corresponding invariant measure <math alttext="\mu^{\prime}" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.7.m7.1"><semantics id="S1.Thmthm6.p2.7.m7.1a"><msup id="S1.Thmthm6.p2.7.m7.1.1" xref="S1.Thmthm6.p2.7.m7.1.1.cmml"><mi id="S1.Thmthm6.p2.7.m7.1.1.2" xref="S1.Thmthm6.p2.7.m7.1.1.2.cmml">μ</mi><mo id="S1.Thmthm6.p2.7.m7.1.1.3" xref="S1.Thmthm6.p2.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.7.m7.1b"><apply id="S1.Thmthm6.p2.7.m7.1.1.cmml" xref="S1.Thmthm6.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p2.7.m7.1.1.1.cmml" xref="S1.Thmthm6.p2.7.m7.1.1">superscript</csymbol><ci id="S1.Thmthm6.p2.7.m7.1.1.2.cmml" xref="S1.Thmthm6.p2.7.m7.1.1.2">𝜇</ci><ci id="S1.Thmthm6.p2.7.m7.1.1.3.cmml" xref="S1.Thmthm6.p2.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.7.m7.1c">\mu^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.7.m7.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> on <math alttext="X_{h(\Lambda)}" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.8.m8.1"><semantics id="S1.Thmthm6.p2.8.m8.1a"><msub id="S1.Thmthm6.p2.8.m8.1.2" xref="S1.Thmthm6.p2.8.m8.1.2.cmml"><mi id="S1.Thmthm6.p2.8.m8.1.2.2" xref="S1.Thmthm6.p2.8.m8.1.2.2.cmml">X</mi><mrow id="S1.Thmthm6.p2.8.m8.1.1.1" xref="S1.Thmthm6.p2.8.m8.1.1.1.cmml"><mi id="S1.Thmthm6.p2.8.m8.1.1.1.3" xref="S1.Thmthm6.p2.8.m8.1.1.1.3.cmml">h</mi><mo id="S1.Thmthm6.p2.8.m8.1.1.1.2" xref="S1.Thmthm6.p2.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmthm6.p2.8.m8.1.1.1.4.2" xref="S1.Thmthm6.p2.8.m8.1.1.1.cmml"><mo id="S1.Thmthm6.p2.8.m8.1.1.1.4.2.1" stretchy="false" xref="S1.Thmthm6.p2.8.m8.1.1.1.cmml">(</mo><mi id="S1.Thmthm6.p2.8.m8.1.1.1.1" mathvariant="normal" xref="S1.Thmthm6.p2.8.m8.1.1.1.1.cmml">Λ</mi><mo id="S1.Thmthm6.p2.8.m8.1.1.1.4.2.2" stretchy="false" xref="S1.Thmthm6.p2.8.m8.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.8.m8.1b"><apply id="S1.Thmthm6.p2.8.m8.1.2.cmml" xref="S1.Thmthm6.p2.8.m8.1.2"><csymbol cd="ambiguous" id="S1.Thmthm6.p2.8.m8.1.2.1.cmml" xref="S1.Thmthm6.p2.8.m8.1.2">subscript</csymbol><ci id="S1.Thmthm6.p2.8.m8.1.2.2.cmml" xref="S1.Thmthm6.p2.8.m8.1.2.2">𝑋</ci><apply id="S1.Thmthm6.p2.8.m8.1.1.1.cmml" xref="S1.Thmthm6.p2.8.m8.1.1.1"><times id="S1.Thmthm6.p2.8.m8.1.1.1.2.cmml" xref="S1.Thmthm6.p2.8.m8.1.1.1.2"></times><ci id="S1.Thmthm6.p2.8.m8.1.1.1.3.cmml" xref="S1.Thmthm6.p2.8.m8.1.1.1.3">ℎ</ci><ci id="S1.Thmthm6.p2.8.m8.1.1.1.1.cmml" xref="S1.Thmthm6.p2.8.m8.1.1.1.1">Λ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.8.m8.1c">X_{h(\Lambda)}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.8.m8.1d">italic_X start_POSTSUBSCRIPT italic_h ( roman_Λ ) end_POSTSUBSCRIPT</annotation></semantics></math>, turns out to be precisely the transferred measure <math alttext="\sigma M(\mu)" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.9.m9.1"><semantics id="S1.Thmthm6.p2.9.m9.1a"><mrow id="S1.Thmthm6.p2.9.m9.1.2" xref="S1.Thmthm6.p2.9.m9.1.2.cmml"><mi id="S1.Thmthm6.p2.9.m9.1.2.2" xref="S1.Thmthm6.p2.9.m9.1.2.2.cmml">σ</mi><mo id="S1.Thmthm6.p2.9.m9.1.2.1" xref="S1.Thmthm6.p2.9.m9.1.2.1.cmml">⁢</mo><mi id="S1.Thmthm6.p2.9.m9.1.2.3" xref="S1.Thmthm6.p2.9.m9.1.2.3.cmml">M</mi><mo id="S1.Thmthm6.p2.9.m9.1.2.1a" xref="S1.Thmthm6.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p2.9.m9.1.2.4.2" xref="S1.Thmthm6.p2.9.m9.1.2.cmml"><mo id="S1.Thmthm6.p2.9.m9.1.2.4.2.1" stretchy="false" xref="S1.Thmthm6.p2.9.m9.1.2.cmml">(</mo><mi id="S1.Thmthm6.p2.9.m9.1.1" xref="S1.Thmthm6.p2.9.m9.1.1.cmml">μ</mi><mo id="S1.Thmthm6.p2.9.m9.1.2.4.2.2" stretchy="false" xref="S1.Thmthm6.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.9.m9.1b"><apply id="S1.Thmthm6.p2.9.m9.1.2.cmml" xref="S1.Thmthm6.p2.9.m9.1.2"><times id="S1.Thmthm6.p2.9.m9.1.2.1.cmml" xref="S1.Thmthm6.p2.9.m9.1.2.1"></times><ci id="S1.Thmthm6.p2.9.m9.1.2.2.cmml" xref="S1.Thmthm6.p2.9.m9.1.2.2">𝜎</ci><ci id="S1.Thmthm6.p2.9.m9.1.2.3.cmml" xref="S1.Thmthm6.p2.9.m9.1.2.3">𝑀</ci><ci id="S1.Thmthm6.p2.9.m9.1.1.cmml" xref="S1.Thmthm6.p2.9.m9.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.9.m9.1c">\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.9.m9.1d">italic_σ italic_M ( italic_μ )</annotation></semantics></math>, where <math alttext="\sigma" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.10.m10.1"><semantics id="S1.Thmthm6.p2.10.m10.1a"><mi id="S1.Thmthm6.p2.10.m10.1.1" xref="S1.Thmthm6.p2.10.m10.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.10.m10.1b"><ci id="S1.Thmthm6.p2.10.m10.1.1.cmml" xref="S1.Thmthm6.p2.10.m10.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.10.m10.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.10.m10.1d">italic_σ</annotation></semantics></math> is the morphism on <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.11.m11.1"><semantics id="S1.Thmthm6.p2.11.m11.1a"><msup id="S1.Thmthm6.p2.11.m11.1.1" xref="S1.Thmthm6.p2.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p2.11.m11.1.1.2" xref="S1.Thmthm6.p2.11.m11.1.1.2.cmml">𝒜</mi><mo id="S1.Thmthm6.p2.11.m11.1.1.3" xref="S1.Thmthm6.p2.11.m11.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.11.m11.1b"><apply id="S1.Thmthm6.p2.11.m11.1.1.cmml" xref="S1.Thmthm6.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p2.11.m11.1.1.1.cmml" xref="S1.Thmthm6.p2.11.m11.1.1">superscript</csymbol><ci id="S1.Thmthm6.p2.11.m11.1.1.2.cmml" xref="S1.Thmthm6.p2.11.m11.1.1.2">𝒜</ci><times id="S1.Thmthm6.p2.11.m11.1.1.3.cmml" xref="S1.Thmthm6.p2.11.m11.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.11.m11.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.11.m11.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> induced by <math alttext="h" class="ltx_Math" display="inline" id="S1.Thmthm6.p2.12.m12.1"><semantics id="S1.Thmthm6.p2.12.m12.1a"><mi id="S1.Thmthm6.p2.12.m12.1.1" xref="S1.Thmthm6.p2.12.m12.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p2.12.m12.1b"><ci id="S1.Thmthm6.p2.12.m12.1.1.cmml" xref="S1.Thmthm6.p2.12.m12.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p2.12.m12.1c">h</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p2.12.m12.1d">italic_h</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.Thmthm6.p3"> <p class="ltx_p" id="S1.Thmthm6.p3.7">In the most frequently considered pseudo-Anosov case the <math alttext="h" class="ltx_Math" display="inline" id="S1.Thmthm6.p3.1.m1.1"><semantics id="S1.Thmthm6.p3.1.m1.1a"><mi id="S1.Thmthm6.p3.1.m1.1.1" xref="S1.Thmthm6.p3.1.m1.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p3.1.m1.1b"><ci id="S1.Thmthm6.p3.1.m1.1.1.cmml" xref="S1.Thmthm6.p3.1.m1.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p3.1.m1.1c">h</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p3.1.m1.1d">italic_h</annotation></semantics></math>-invariant lamination <math alttext="\Lambda" class="ltx_Math" display="inline" id="S1.Thmthm6.p3.2.m2.1"><semantics id="S1.Thmthm6.p3.2.m2.1a"><mi id="S1.Thmthm6.p3.2.m2.1.1" mathvariant="normal" xref="S1.Thmthm6.p3.2.m2.1.1.cmml">Λ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p3.2.m2.1b"><ci id="S1.Thmthm6.p3.2.m2.1.1.cmml" xref="S1.Thmthm6.p3.2.m2.1.1">Λ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p3.2.m2.1c">\Lambda</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p3.2.m2.1d">roman_Λ</annotation></semantics></math> as well as the corresponding subshift <math alttext="X_{\Lambda}" class="ltx_Math" display="inline" id="S1.Thmthm6.p3.3.m3.1"><semantics id="S1.Thmthm6.p3.3.m3.1a"><msub id="S1.Thmthm6.p3.3.m3.1.1" xref="S1.Thmthm6.p3.3.m3.1.1.cmml"><mi id="S1.Thmthm6.p3.3.m3.1.1.2" xref="S1.Thmthm6.p3.3.m3.1.1.2.cmml">X</mi><mi id="S1.Thmthm6.p3.3.m3.1.1.3" mathvariant="normal" xref="S1.Thmthm6.p3.3.m3.1.1.3.cmml">Λ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p3.3.m3.1b"><apply id="S1.Thmthm6.p3.3.m3.1.1.cmml" xref="S1.Thmthm6.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p3.3.m3.1.1.1.cmml" xref="S1.Thmthm6.p3.3.m3.1.1">subscript</csymbol><ci id="S1.Thmthm6.p3.3.m3.1.1.2.cmml" xref="S1.Thmthm6.p3.3.m3.1.1.2">𝑋</ci><ci id="S1.Thmthm6.p3.3.m3.1.1.3.cmml" xref="S1.Thmthm6.p3.3.m3.1.1.3">Λ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p3.3.m3.1c">X_{\Lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p3.3.m3.1d">italic_X start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT</annotation></semantics></math> turn out to be both, minimal and uniquely ergodic, so that we have <math alttext="\sigma M(\mu)=\lambda\,\mu\," class="ltx_Math" display="inline" id="S1.Thmthm6.p3.4.m4.1"><semantics id="S1.Thmthm6.p3.4.m4.1a"><mrow id="S1.Thmthm6.p3.4.m4.1.2" xref="S1.Thmthm6.p3.4.m4.1.2.cmml"><mrow id="S1.Thmthm6.p3.4.m4.1.2.2" xref="S1.Thmthm6.p3.4.m4.1.2.2.cmml"><mi id="S1.Thmthm6.p3.4.m4.1.2.2.2" xref="S1.Thmthm6.p3.4.m4.1.2.2.2.cmml">σ</mi><mo id="S1.Thmthm6.p3.4.m4.1.2.2.1" xref="S1.Thmthm6.p3.4.m4.1.2.2.1.cmml">⁢</mo><mi id="S1.Thmthm6.p3.4.m4.1.2.2.3" xref="S1.Thmthm6.p3.4.m4.1.2.2.3.cmml">M</mi><mo id="S1.Thmthm6.p3.4.m4.1.2.2.1a" xref="S1.Thmthm6.p3.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p3.4.m4.1.2.2.4.2" xref="S1.Thmthm6.p3.4.m4.1.2.2.cmml"><mo id="S1.Thmthm6.p3.4.m4.1.2.2.4.2.1" stretchy="false" xref="S1.Thmthm6.p3.4.m4.1.2.2.cmml">(</mo><mi id="S1.Thmthm6.p3.4.m4.1.1" xref="S1.Thmthm6.p3.4.m4.1.1.cmml">μ</mi><mo id="S1.Thmthm6.p3.4.m4.1.2.2.4.2.2" stretchy="false" xref="S1.Thmthm6.p3.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm6.p3.4.m4.1.2.1" xref="S1.Thmthm6.p3.4.m4.1.2.1.cmml">=</mo><mrow id="S1.Thmthm6.p3.4.m4.1.2.3" xref="S1.Thmthm6.p3.4.m4.1.2.3.cmml"><mi id="S1.Thmthm6.p3.4.m4.1.2.3.2" xref="S1.Thmthm6.p3.4.m4.1.2.3.2.cmml">λ</mi><mo id="S1.Thmthm6.p3.4.m4.1.2.3.1" lspace="0.170em" xref="S1.Thmthm6.p3.4.m4.1.2.3.1.cmml">⁢</mo><mi id="S1.Thmthm6.p3.4.m4.1.2.3.3" xref="S1.Thmthm6.p3.4.m4.1.2.3.3.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p3.4.m4.1b"><apply id="S1.Thmthm6.p3.4.m4.1.2.cmml" xref="S1.Thmthm6.p3.4.m4.1.2"><eq id="S1.Thmthm6.p3.4.m4.1.2.1.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.1"></eq><apply id="S1.Thmthm6.p3.4.m4.1.2.2.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.2"><times id="S1.Thmthm6.p3.4.m4.1.2.2.1.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.2.1"></times><ci id="S1.Thmthm6.p3.4.m4.1.2.2.2.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.2.2">𝜎</ci><ci id="S1.Thmthm6.p3.4.m4.1.2.2.3.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.2.3">𝑀</ci><ci id="S1.Thmthm6.p3.4.m4.1.1.cmml" xref="S1.Thmthm6.p3.4.m4.1.1">𝜇</ci></apply><apply id="S1.Thmthm6.p3.4.m4.1.2.3.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.3"><times id="S1.Thmthm6.p3.4.m4.1.2.3.1.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.3.1"></times><ci id="S1.Thmthm6.p3.4.m4.1.2.3.2.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.3.2">𝜆</ci><ci id="S1.Thmthm6.p3.4.m4.1.2.3.3.cmml" xref="S1.Thmthm6.p3.4.m4.1.2.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p3.4.m4.1c">\sigma M(\mu)=\lambda\,\mu\,</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p3.4.m4.1d">italic_σ italic_M ( italic_μ ) = italic_λ italic_μ</annotation></semantics></math>, where <math alttext="\lambda>1" class="ltx_Math" display="inline" id="S1.Thmthm6.p3.5.m5.1"><semantics id="S1.Thmthm6.p3.5.m5.1a"><mrow id="S1.Thmthm6.p3.5.m5.1.1" xref="S1.Thmthm6.p3.5.m5.1.1.cmml"><mi id="S1.Thmthm6.p3.5.m5.1.1.2" xref="S1.Thmthm6.p3.5.m5.1.1.2.cmml">λ</mi><mo id="S1.Thmthm6.p3.5.m5.1.1.1" xref="S1.Thmthm6.p3.5.m5.1.1.1.cmml">></mo><mn id="S1.Thmthm6.p3.5.m5.1.1.3" xref="S1.Thmthm6.p3.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p3.5.m5.1b"><apply id="S1.Thmthm6.p3.5.m5.1.1.cmml" xref="S1.Thmthm6.p3.5.m5.1.1"><gt id="S1.Thmthm6.p3.5.m5.1.1.1.cmml" xref="S1.Thmthm6.p3.5.m5.1.1.1"></gt><ci id="S1.Thmthm6.p3.5.m5.1.1.2.cmml" xref="S1.Thmthm6.p3.5.m5.1.1.2">𝜆</ci><cn id="S1.Thmthm6.p3.5.m5.1.1.3.cmml" type="integer" xref="S1.Thmthm6.p3.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p3.5.m5.1c">\lambda>1</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p3.5.m5.1d">italic_λ > 1</annotation></semantics></math> is the celebrated “stretching factor” (= the Perron-Frobenious eigenvalue of <math alttext="M(\sigma)" class="ltx_Math" display="inline" id="S1.Thmthm6.p3.6.m6.1"><semantics id="S1.Thmthm6.p3.6.m6.1a"><mrow id="S1.Thmthm6.p3.6.m6.1.2" xref="S1.Thmthm6.p3.6.m6.1.2.cmml"><mi id="S1.Thmthm6.p3.6.m6.1.2.2" xref="S1.Thmthm6.p3.6.m6.1.2.2.cmml">M</mi><mo id="S1.Thmthm6.p3.6.m6.1.2.1" xref="S1.Thmthm6.p3.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p3.6.m6.1.2.3.2" xref="S1.Thmthm6.p3.6.m6.1.2.cmml"><mo id="S1.Thmthm6.p3.6.m6.1.2.3.2.1" stretchy="false" xref="S1.Thmthm6.p3.6.m6.1.2.cmml">(</mo><mi id="S1.Thmthm6.p3.6.m6.1.1" xref="S1.Thmthm6.p3.6.m6.1.1.cmml">σ</mi><mo id="S1.Thmthm6.p3.6.m6.1.2.3.2.2" stretchy="false" xref="S1.Thmthm6.p3.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p3.6.m6.1b"><apply id="S1.Thmthm6.p3.6.m6.1.2.cmml" xref="S1.Thmthm6.p3.6.m6.1.2"><times id="S1.Thmthm6.p3.6.m6.1.2.1.cmml" xref="S1.Thmthm6.p3.6.m6.1.2.1"></times><ci id="S1.Thmthm6.p3.6.m6.1.2.2.cmml" xref="S1.Thmthm6.p3.6.m6.1.2.2">𝑀</ci><ci id="S1.Thmthm6.p3.6.m6.1.1.cmml" xref="S1.Thmthm6.p3.6.m6.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p3.6.m6.1c">M(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p3.6.m6.1d">italic_M ( italic_σ )</annotation></semantics></math>) for <math alttext="h" class="ltx_Math" display="inline" id="S1.Thmthm6.p3.7.m7.1"><semantics id="S1.Thmthm6.p3.7.m7.1a"><mi id="S1.Thmthm6.p3.7.m7.1.1" xref="S1.Thmthm6.p3.7.m7.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p3.7.m7.1b"><ci id="S1.Thmthm6.p3.7.m7.1.1.cmml" xref="S1.Thmthm6.p3.7.m7.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p3.7.m7.1c">h</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p3.7.m7.1d">italic_h</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S1.Thmthm6.p4"> <p class="ltx_p" id="S1.Thmthm6.p4.8">(2) Any word <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S1.Thmthm6.p4.1.m1.1"><semantics id="S1.Thmthm6.p4.1.m1.1a"><mrow id="S1.Thmthm6.p4.1.m1.1.1" xref="S1.Thmthm6.p4.1.m1.1.1.cmml"><mi id="S1.Thmthm6.p4.1.m1.1.1.2" xref="S1.Thmthm6.p4.1.m1.1.1.2.cmml">w</mi><mo id="S1.Thmthm6.p4.1.m1.1.1.1" xref="S1.Thmthm6.p4.1.m1.1.1.1.cmml">∈</mo><msup id="S1.Thmthm6.p4.1.m1.1.1.3" xref="S1.Thmthm6.p4.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p4.1.m1.1.1.3.2" xref="S1.Thmthm6.p4.1.m1.1.1.3.2.cmml">𝒜</mi><mo id="S1.Thmthm6.p4.1.m1.1.1.3.3" xref="S1.Thmthm6.p4.1.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p4.1.m1.1b"><apply id="S1.Thmthm6.p4.1.m1.1.1.cmml" xref="S1.Thmthm6.p4.1.m1.1.1"><in id="S1.Thmthm6.p4.1.m1.1.1.1.cmml" xref="S1.Thmthm6.p4.1.m1.1.1.1"></in><ci id="S1.Thmthm6.p4.1.m1.1.1.2.cmml" xref="S1.Thmthm6.p4.1.m1.1.1.2">𝑤</ci><apply id="S1.Thmthm6.p4.1.m1.1.1.3.cmml" xref="S1.Thmthm6.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.Thmthm6.p4.1.m1.1.1.3.1.cmml" xref="S1.Thmthm6.p4.1.m1.1.1.3">superscript</csymbol><ci id="S1.Thmthm6.p4.1.m1.1.1.3.2.cmml" xref="S1.Thmthm6.p4.1.m1.1.1.3.2">𝒜</ci><times id="S1.Thmthm6.p4.1.m1.1.1.3.3.cmml" xref="S1.Thmthm6.p4.1.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p4.1.m1.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p4.1.m1.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> (or rather, its conjugacy class in the free group <math alttext="F(\cal A)" class="ltx_Math" display="inline" id="S1.Thmthm6.p4.2.m2.1"><semantics id="S1.Thmthm6.p4.2.m2.1a"><mrow id="S1.Thmthm6.p4.2.m2.1.2" xref="S1.Thmthm6.p4.2.m2.1.2.cmml"><mi id="S1.Thmthm6.p4.2.m2.1.2.2" xref="S1.Thmthm6.p4.2.m2.1.2.2.cmml">F</mi><mo id="S1.Thmthm6.p4.2.m2.1.2.1" xref="S1.Thmthm6.p4.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p4.2.m2.1.2.3.2" xref="S1.Thmthm6.p4.2.m2.1.2.cmml"><mo id="S1.Thmthm6.p4.2.m2.1.2.3.2.1" stretchy="false" xref="S1.Thmthm6.p4.2.m2.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p4.2.m2.1.1" xref="S1.Thmthm6.p4.2.m2.1.1.cmml">𝒜</mi><mo id="S1.Thmthm6.p4.2.m2.1.2.3.2.2" stretchy="false" xref="S1.Thmthm6.p4.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p4.2.m2.1b"><apply id="S1.Thmthm6.p4.2.m2.1.2.cmml" xref="S1.Thmthm6.p4.2.m2.1.2"><times id="S1.Thmthm6.p4.2.m2.1.2.1.cmml" xref="S1.Thmthm6.p4.2.m2.1.2.1"></times><ci id="S1.Thmthm6.p4.2.m2.1.2.2.cmml" xref="S1.Thmthm6.p4.2.m2.1.2.2">𝐹</ci><ci id="S1.Thmthm6.p4.2.m2.1.1.cmml" xref="S1.Thmthm6.p4.2.m2.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p4.2.m2.1c">F(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p4.2.m2.1d">italic_F ( caligraphic_A )</annotation></semantics></math>) defines a finite subshift <math alttext="X_{w}" class="ltx_Math" display="inline" id="S1.Thmthm6.p4.3.m3.1"><semantics id="S1.Thmthm6.p4.3.m3.1a"><msub id="S1.Thmthm6.p4.3.m3.1.1" xref="S1.Thmthm6.p4.3.m3.1.1.cmml"><mi id="S1.Thmthm6.p4.3.m3.1.1.2" xref="S1.Thmthm6.p4.3.m3.1.1.2.cmml">X</mi><mi id="S1.Thmthm6.p4.3.m3.1.1.3" xref="S1.Thmthm6.p4.3.m3.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p4.3.m3.1b"><apply id="S1.Thmthm6.p4.3.m3.1.1.cmml" xref="S1.Thmthm6.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p4.3.m3.1.1.1.cmml" xref="S1.Thmthm6.p4.3.m3.1.1">subscript</csymbol><ci id="S1.Thmthm6.p4.3.m3.1.1.2.cmml" xref="S1.Thmthm6.p4.3.m3.1.1.2">𝑋</ci><ci id="S1.Thmthm6.p4.3.m3.1.1.3.cmml" xref="S1.Thmthm6.p4.3.m3.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p4.3.m3.1c">X_{w}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p4.3.m3.1d">italic_X start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> that consists of the sequence <math alttext="w^{\pm\infty}=\ldots www\ldots\,\," class="ltx_Math" display="inline" id="S1.Thmthm6.p4.4.m4.1"><semantics id="S1.Thmthm6.p4.4.m4.1a"><mrow id="S1.Thmthm6.p4.4.m4.1.1" xref="S1.Thmthm6.p4.4.m4.1.1.cmml"><msup id="S1.Thmthm6.p4.4.m4.1.1.2" xref="S1.Thmthm6.p4.4.m4.1.1.2.cmml"><mi id="S1.Thmthm6.p4.4.m4.1.1.2.2" xref="S1.Thmthm6.p4.4.m4.1.1.2.2.cmml">w</mi><mrow id="S1.Thmthm6.p4.4.m4.1.1.2.3" xref="S1.Thmthm6.p4.4.m4.1.1.2.3.cmml"><mo id="S1.Thmthm6.p4.4.m4.1.1.2.3a" xref="S1.Thmthm6.p4.4.m4.1.1.2.3.cmml">±</mo><mi id="S1.Thmthm6.p4.4.m4.1.1.2.3.2" mathvariant="normal" xref="S1.Thmthm6.p4.4.m4.1.1.2.3.2.cmml">∞</mi></mrow></msup><mo id="S1.Thmthm6.p4.4.m4.1.1.1" xref="S1.Thmthm6.p4.4.m4.1.1.1.cmml">=</mo><mrow id="S1.Thmthm6.p4.4.m4.1.1.3" xref="S1.Thmthm6.p4.4.m4.1.1.3.cmml"><mi id="S1.Thmthm6.p4.4.m4.1.1.3.2" mathvariant="normal" xref="S1.Thmthm6.p4.4.m4.1.1.3.2.cmml">…</mi><mo id="S1.Thmthm6.p4.4.m4.1.1.3.1" xref="S1.Thmthm6.p4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S1.Thmthm6.p4.4.m4.1.1.3.3" xref="S1.Thmthm6.p4.4.m4.1.1.3.3.cmml">w</mi><mo id="S1.Thmthm6.p4.4.m4.1.1.3.1a" xref="S1.Thmthm6.p4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S1.Thmthm6.p4.4.m4.1.1.3.4" xref="S1.Thmthm6.p4.4.m4.1.1.3.4.cmml">w</mi><mo id="S1.Thmthm6.p4.4.m4.1.1.3.1b" xref="S1.Thmthm6.p4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S1.Thmthm6.p4.4.m4.1.1.3.5" xref="S1.Thmthm6.p4.4.m4.1.1.3.5.cmml">w</mi><mo id="S1.Thmthm6.p4.4.m4.1.1.3.1c" xref="S1.Thmthm6.p4.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S1.Thmthm6.p4.4.m4.1.1.3.6" mathvariant="normal" xref="S1.Thmthm6.p4.4.m4.1.1.3.6.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p4.4.m4.1b"><apply id="S1.Thmthm6.p4.4.m4.1.1.cmml" xref="S1.Thmthm6.p4.4.m4.1.1"><eq id="S1.Thmthm6.p4.4.m4.1.1.1.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.1"></eq><apply id="S1.Thmthm6.p4.4.m4.1.1.2.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S1.Thmthm6.p4.4.m4.1.1.2.1.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.2">superscript</csymbol><ci id="S1.Thmthm6.p4.4.m4.1.1.2.2.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.2.2">𝑤</ci><apply id="S1.Thmthm6.p4.4.m4.1.1.2.3.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.2.3"><csymbol cd="latexml" id="S1.Thmthm6.p4.4.m4.1.1.2.3.1.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.2.3">plus-or-minus</csymbol><infinity id="S1.Thmthm6.p4.4.m4.1.1.2.3.2.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.2.3.2"></infinity></apply></apply><apply id="S1.Thmthm6.p4.4.m4.1.1.3.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.3"><times id="S1.Thmthm6.p4.4.m4.1.1.3.1.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.3.1"></times><ci id="S1.Thmthm6.p4.4.m4.1.1.3.2.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.3.2">…</ci><ci id="S1.Thmthm6.p4.4.m4.1.1.3.3.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.3.3">𝑤</ci><ci id="S1.Thmthm6.p4.4.m4.1.1.3.4.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.3.4">𝑤</ci><ci id="S1.Thmthm6.p4.4.m4.1.1.3.5.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.3.5">𝑤</ci><ci id="S1.Thmthm6.p4.4.m4.1.1.3.6.cmml" xref="S1.Thmthm6.p4.4.m4.1.1.3.6">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p4.4.m4.1c">w^{\pm\infty}=\ldots www\ldots\,\,</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p4.4.m4.1d">italic_w start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT = … italic_w italic_w italic_w …</annotation></semantics></math> and its finitely many shift translates. To <math alttext="w" class="ltx_Math" display="inline" id="S1.Thmthm6.p4.5.m5.1"><semantics id="S1.Thmthm6.p4.5.m5.1a"><mi id="S1.Thmthm6.p4.5.m5.1.1" xref="S1.Thmthm6.p4.5.m5.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p4.5.m5.1b"><ci id="S1.Thmthm6.p4.5.m5.1.1.cmml" xref="S1.Thmthm6.p4.5.m5.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p4.5.m5.1c">w</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p4.5.m5.1d">italic_w</annotation></semantics></math> there is canonically associated a <span class="ltx_text ltx_font_italic" id="S1.Thmthm6.p4.8.1">characteristic measure</span> <math alttext="\mu_{w}\in\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S1.Thmthm6.p4.6.m6.1"><semantics id="S1.Thmthm6.p4.6.m6.1a"><mrow id="S1.Thmthm6.p4.6.m6.1.1" xref="S1.Thmthm6.p4.6.m6.1.1.cmml"><msub id="S1.Thmthm6.p4.6.m6.1.1.3" xref="S1.Thmthm6.p4.6.m6.1.1.3.cmml"><mi id="S1.Thmthm6.p4.6.m6.1.1.3.2" xref="S1.Thmthm6.p4.6.m6.1.1.3.2.cmml">μ</mi><mi id="S1.Thmthm6.p4.6.m6.1.1.3.3" xref="S1.Thmthm6.p4.6.m6.1.1.3.3.cmml">w</mi></msub><mo id="S1.Thmthm6.p4.6.m6.1.1.2" xref="S1.Thmthm6.p4.6.m6.1.1.2.cmml">∈</mo><mrow id="S1.Thmthm6.p4.6.m6.1.1.1" xref="S1.Thmthm6.p4.6.m6.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p4.6.m6.1.1.1.3" xref="S1.Thmthm6.p4.6.m6.1.1.1.3.cmml">ℳ</mi><mo id="S1.Thmthm6.p4.6.m6.1.1.1.2" xref="S1.Thmthm6.p4.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S1.Thmthm6.p4.6.m6.1.1.1.1.1" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.cmml"><mo id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.2" stretchy="false" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.2" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.3" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.3" stretchy="false" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p4.6.m6.1b"><apply id="S1.Thmthm6.p4.6.m6.1.1.cmml" xref="S1.Thmthm6.p4.6.m6.1.1"><in id="S1.Thmthm6.p4.6.m6.1.1.2.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.2"></in><apply id="S1.Thmthm6.p4.6.m6.1.1.3.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.Thmthm6.p4.6.m6.1.1.3.1.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.3">subscript</csymbol><ci id="S1.Thmthm6.p4.6.m6.1.1.3.2.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.3.2">𝜇</ci><ci id="S1.Thmthm6.p4.6.m6.1.1.3.3.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.3.3">𝑤</ci></apply><apply id="S1.Thmthm6.p4.6.m6.1.1.1.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.1"><times id="S1.Thmthm6.p4.6.m6.1.1.1.2.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.1.2"></times><ci id="S1.Thmthm6.p4.6.m6.1.1.1.3.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.1.3">ℳ</ci><apply id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.1.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1">superscript</csymbol><ci id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.2.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.2">𝒜</ci><ci id="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.3.cmml" xref="S1.Thmthm6.p4.6.m6.1.1.1.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p4.6.m6.1c">\mu_{w}\in\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p4.6.m6.1d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ∈ caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> with support equal to <math alttext="X_{w}" class="ltx_Math" display="inline" id="S1.Thmthm6.p4.7.m7.1"><semantics id="S1.Thmthm6.p4.7.m7.1a"><msub id="S1.Thmthm6.p4.7.m7.1.1" xref="S1.Thmthm6.p4.7.m7.1.1.cmml"><mi id="S1.Thmthm6.p4.7.m7.1.1.2" xref="S1.Thmthm6.p4.7.m7.1.1.2.cmml">X</mi><mi id="S1.Thmthm6.p4.7.m7.1.1.3" xref="S1.Thmthm6.p4.7.m7.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p4.7.m7.1b"><apply id="S1.Thmthm6.p4.7.m7.1.1.cmml" xref="S1.Thmthm6.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p4.7.m7.1.1.1.cmml" xref="S1.Thmthm6.p4.7.m7.1.1">subscript</csymbol><ci id="S1.Thmthm6.p4.7.m7.1.1.2.cmml" xref="S1.Thmthm6.p4.7.m7.1.1.2">𝑋</ci><ci id="S1.Thmthm6.p4.7.m7.1.1.3.cmml" xref="S1.Thmthm6.p4.7.m7.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p4.7.m7.1c">X_{w}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p4.7.m7.1d">italic_X start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> and total measure <math alttext="\mu_{w}(X_{w})=\mu_{w}(\cal A^{\mathbb{Z}})=|w|" class="ltx_Math" display="inline" id="S1.Thmthm6.p4.8.m8.3"><semantics id="S1.Thmthm6.p4.8.m8.3a"><mrow id="S1.Thmthm6.p4.8.m8.3.3" xref="S1.Thmthm6.p4.8.m8.3.3.cmml"><mrow id="S1.Thmthm6.p4.8.m8.2.2.1" xref="S1.Thmthm6.p4.8.m8.2.2.1.cmml"><msub id="S1.Thmthm6.p4.8.m8.2.2.1.3" xref="S1.Thmthm6.p4.8.m8.2.2.1.3.cmml"><mi id="S1.Thmthm6.p4.8.m8.2.2.1.3.2" xref="S1.Thmthm6.p4.8.m8.2.2.1.3.2.cmml">μ</mi><mi id="S1.Thmthm6.p4.8.m8.2.2.1.3.3" xref="S1.Thmthm6.p4.8.m8.2.2.1.3.3.cmml">w</mi></msub><mo id="S1.Thmthm6.p4.8.m8.2.2.1.2" xref="S1.Thmthm6.p4.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="S1.Thmthm6.p4.8.m8.2.2.1.1.1" xref="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.cmml"><mo id="S1.Thmthm6.p4.8.m8.2.2.1.1.1.2" stretchy="false" xref="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.cmml">(</mo><msub id="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1" xref="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.cmml"><mi id="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.2" xref="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.2.cmml">X</mi><mi id="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.3" xref="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.3.cmml">w</mi></msub><mo id="S1.Thmthm6.p4.8.m8.2.2.1.1.1.3" stretchy="false" xref="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm6.p4.8.m8.3.3.4" xref="S1.Thmthm6.p4.8.m8.3.3.4.cmml">=</mo><mrow id="S1.Thmthm6.p4.8.m8.3.3.2" xref="S1.Thmthm6.p4.8.m8.3.3.2.cmml"><msub id="S1.Thmthm6.p4.8.m8.3.3.2.3" xref="S1.Thmthm6.p4.8.m8.3.3.2.3.cmml"><mi id="S1.Thmthm6.p4.8.m8.3.3.2.3.2" xref="S1.Thmthm6.p4.8.m8.3.3.2.3.2.cmml">μ</mi><mi id="S1.Thmthm6.p4.8.m8.3.3.2.3.3" xref="S1.Thmthm6.p4.8.m8.3.3.2.3.3.cmml">w</mi></msub><mo id="S1.Thmthm6.p4.8.m8.3.3.2.2" xref="S1.Thmthm6.p4.8.m8.3.3.2.2.cmml">⁢</mo><mrow id="S1.Thmthm6.p4.8.m8.3.3.2.1.1" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.cmml"><mo id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.2" stretchy="false" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.cmml">(</mo><msup id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.2" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.2.cmml">𝒜</mi><mi id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.3" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.3" stretchy="false" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm6.p4.8.m8.3.3.5" xref="S1.Thmthm6.p4.8.m8.3.3.5.cmml">=</mo><mrow id="S1.Thmthm6.p4.8.m8.3.3.6.2" xref="S1.Thmthm6.p4.8.m8.3.3.6.1.cmml"><mo id="S1.Thmthm6.p4.8.m8.3.3.6.2.1" stretchy="false" xref="S1.Thmthm6.p4.8.m8.3.3.6.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p4.8.m8.1.1" xref="S1.Thmthm6.p4.8.m8.1.1.cmml">𝓌</mi><mo id="S1.Thmthm6.p4.8.m8.3.3.6.2.2" stretchy="false" xref="S1.Thmthm6.p4.8.m8.3.3.6.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p4.8.m8.3b"><apply id="S1.Thmthm6.p4.8.m8.3.3.cmml" xref="S1.Thmthm6.p4.8.m8.3.3"><and id="S1.Thmthm6.p4.8.m8.3.3a.cmml" 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xref="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.2">𝑋</ci><ci id="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.3.cmml" xref="S1.Thmthm6.p4.8.m8.2.2.1.1.1.1.3">𝑤</ci></apply></apply><apply id="S1.Thmthm6.p4.8.m8.3.3.2.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2"><times id="S1.Thmthm6.p4.8.m8.3.3.2.2.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.2"></times><apply id="S1.Thmthm6.p4.8.m8.3.3.2.3.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.3"><csymbol cd="ambiguous" id="S1.Thmthm6.p4.8.m8.3.3.2.3.1.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.3">subscript</csymbol><ci id="S1.Thmthm6.p4.8.m8.3.3.2.3.2.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.3.2">𝜇</ci><ci id="S1.Thmthm6.p4.8.m8.3.3.2.3.3.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.3.3">𝑤</ci></apply><apply id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.1.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1">superscript</csymbol><ci id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.2.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.2">𝒜</ci><ci id="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.3.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.2.1.1.1.3">ℤ</ci></apply></apply></apply><apply id="S1.Thmthm6.p4.8.m8.3.3c.cmml" xref="S1.Thmthm6.p4.8.m8.3.3"><eq id="S1.Thmthm6.p4.8.m8.3.3.5.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S1.Thmthm6.p4.8.m8.3.3.2.cmml" id="S1.Thmthm6.p4.8.m8.3.3d.cmml" xref="S1.Thmthm6.p4.8.m8.3.3"></share><apply id="S1.Thmthm6.p4.8.m8.3.3.6.1.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.6.2"><abs id="S1.Thmthm6.p4.8.m8.3.3.6.1.1.cmml" xref="S1.Thmthm6.p4.8.m8.3.3.6.2.1"></abs><ci id="S1.Thmthm6.p4.8.m8.1.1.cmml" xref="S1.Thmthm6.p4.8.m8.1.1">𝓌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p4.8.m8.3c">\mu_{w}(X_{w})=\mu_{w}(\cal A^{\mathbb{Z}})=|w|</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p4.8.m8.3d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) = | caligraphic_w |</annotation></semantics></math> (see (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E5" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.5</span></a>) for more details).</p> </div> <div class="ltx_para" id="S1.Thmthm6.p5"> <p class="ltx_p" id="S1.Thmthm6.p5.1">For any non-erasing morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S1.Thmthm6.p5.1.m1.1"><semantics id="S1.Thmthm6.p5.1.m1.1a"><mrow id="S1.Thmthm6.p5.1.m1.1.1" xref="S1.Thmthm6.p5.1.m1.1.1.cmml"><mi id="S1.Thmthm6.p5.1.m1.1.1.2" xref="S1.Thmthm6.p5.1.m1.1.1.2.cmml">σ</mi><mo id="S1.Thmthm6.p5.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm6.p5.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmthm6.p5.1.m1.1.1.3" xref="S1.Thmthm6.p5.1.m1.1.1.3.cmml"><msup id="S1.Thmthm6.p5.1.m1.1.1.3.2" xref="S1.Thmthm6.p5.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p5.1.m1.1.1.3.2.2" xref="S1.Thmthm6.p5.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.Thmthm6.p5.1.m1.1.1.3.2.3" xref="S1.Thmthm6.p5.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.Thmthm6.p5.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmthm6.p5.1.m1.1.1.3.1.cmml">→</mo><msup id="S1.Thmthm6.p5.1.m1.1.1.3.3" xref="S1.Thmthm6.p5.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p5.1.m1.1.1.3.3.2" xref="S1.Thmthm6.p5.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S1.Thmthm6.p5.1.m1.1.1.3.3.3" xref="S1.Thmthm6.p5.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p5.1.m1.1b"><apply id="S1.Thmthm6.p5.1.m1.1.1.cmml" xref="S1.Thmthm6.p5.1.m1.1.1"><ci id="S1.Thmthm6.p5.1.m1.1.1.1.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.1">:</ci><ci id="S1.Thmthm6.p5.1.m1.1.1.2.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.2">𝜎</ci><apply id="S1.Thmthm6.p5.1.m1.1.1.3.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3"><ci id="S1.Thmthm6.p5.1.m1.1.1.3.1.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.1">→</ci><apply id="S1.Thmthm6.p5.1.m1.1.1.3.2.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmthm6.p5.1.m1.1.1.3.2.1.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.Thmthm6.p5.1.m1.1.1.3.2.2.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.2.2">𝒜</ci><times id="S1.Thmthm6.p5.1.m1.1.1.3.2.3.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.2.3"></times></apply><apply id="S1.Thmthm6.p5.1.m1.1.1.3.3.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmthm6.p5.1.m1.1.1.3.3.1.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.Thmthm6.p5.1.m1.1.1.3.3.2.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.3.2">ℬ</ci><times id="S1.Thmthm6.p5.1.m1.1.1.3.3.3.cmml" xref="S1.Thmthm6.p5.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p5.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p5.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the transferred measure of any characteristic measure is again a characteristic measure, given by</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma M(\mu_{w})=\mu_{\sigma(w)}\,." class="ltx_Math" display="block" id="S1.Ex7.m1.2"><semantics id="S1.Ex7.m1.2a"><mrow id="S1.Ex7.m1.2.2.1" xref="S1.Ex7.m1.2.2.1.1.cmml"><mrow id="S1.Ex7.m1.2.2.1.1" xref="S1.Ex7.m1.2.2.1.1.cmml"><mrow id="S1.Ex7.m1.2.2.1.1.1" xref="S1.Ex7.m1.2.2.1.1.1.cmml"><mi id="S1.Ex7.m1.2.2.1.1.1.3" xref="S1.Ex7.m1.2.2.1.1.1.3.cmml">σ</mi><mo id="S1.Ex7.m1.2.2.1.1.1.2" xref="S1.Ex7.m1.2.2.1.1.1.2.cmml">⁢</mo><mi id="S1.Ex7.m1.2.2.1.1.1.4" xref="S1.Ex7.m1.2.2.1.1.1.4.cmml">M</mi><mo id="S1.Ex7.m1.2.2.1.1.1.2a" xref="S1.Ex7.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex7.m1.2.2.1.1.1.1.1" xref="S1.Ex7.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S1.Ex7.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S1.Ex7.m1.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Ex7.m1.2.2.1.1.1.1.1.1" xref="S1.Ex7.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S1.Ex7.m1.2.2.1.1.1.1.1.1.2" xref="S1.Ex7.m1.2.2.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S1.Ex7.m1.2.2.1.1.1.1.1.1.3" xref="S1.Ex7.m1.2.2.1.1.1.1.1.1.3.cmml">w</mi></msub><mo id="S1.Ex7.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S1.Ex7.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Ex7.m1.2.2.1.1.2" xref="S1.Ex7.m1.2.2.1.1.2.cmml">=</mo><msub id="S1.Ex7.m1.2.2.1.1.3" xref="S1.Ex7.m1.2.2.1.1.3.cmml"><mi id="S1.Ex7.m1.2.2.1.1.3.2" xref="S1.Ex7.m1.2.2.1.1.3.2.cmml">μ</mi><mrow id="S1.Ex7.m1.1.1.1" xref="S1.Ex7.m1.1.1.1.cmml"><mi id="S1.Ex7.m1.1.1.1.3" xref="S1.Ex7.m1.1.1.1.3.cmml">σ</mi><mo id="S1.Ex7.m1.1.1.1.2" xref="S1.Ex7.m1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex7.m1.1.1.1.4.2" xref="S1.Ex7.m1.1.1.1.cmml"><mo id="S1.Ex7.m1.1.1.1.4.2.1" stretchy="false" xref="S1.Ex7.m1.1.1.1.cmml">(</mo><mi id="S1.Ex7.m1.1.1.1.1" xref="S1.Ex7.m1.1.1.1.1.cmml">w</mi><mo id="S1.Ex7.m1.1.1.1.4.2.2" stretchy="false" xref="S1.Ex7.m1.1.1.1.cmml">)</mo></mrow></mrow></msub></mrow><mo id="S1.Ex7.m1.2.2.1.2" lspace="0em" xref="S1.Ex7.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex7.m1.2b"><apply id="S1.Ex7.m1.2.2.1.1.cmml" xref="S1.Ex7.m1.2.2.1"><eq id="S1.Ex7.m1.2.2.1.1.2.cmml" xref="S1.Ex7.m1.2.2.1.1.2"></eq><apply id="S1.Ex7.m1.2.2.1.1.1.cmml" xref="S1.Ex7.m1.2.2.1.1.1"><times id="S1.Ex7.m1.2.2.1.1.1.2.cmml" xref="S1.Ex7.m1.2.2.1.1.1.2"></times><ci id="S1.Ex7.m1.2.2.1.1.1.3.cmml" xref="S1.Ex7.m1.2.2.1.1.1.3">𝜎</ci><ci id="S1.Ex7.m1.2.2.1.1.1.4.cmml" xref="S1.Ex7.m1.2.2.1.1.1.4">𝑀</ci><apply id="S1.Ex7.m1.2.2.1.1.1.1.1.1.cmml" xref="S1.Ex7.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Ex7.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S1.Ex7.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S1.Ex7.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S1.Ex7.m1.2.2.1.1.1.1.1.1.2">𝜇</ci><ci id="S1.Ex7.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S1.Ex7.m1.2.2.1.1.1.1.1.1.3">𝑤</ci></apply></apply><apply id="S1.Ex7.m1.2.2.1.1.3.cmml" xref="S1.Ex7.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="S1.Ex7.m1.2.2.1.1.3.1.cmml" xref="S1.Ex7.m1.2.2.1.1.3">subscript</csymbol><ci id="S1.Ex7.m1.2.2.1.1.3.2.cmml" xref="S1.Ex7.m1.2.2.1.1.3.2">𝜇</ci><apply id="S1.Ex7.m1.1.1.1.cmml" xref="S1.Ex7.m1.1.1.1"><times id="S1.Ex7.m1.1.1.1.2.cmml" xref="S1.Ex7.m1.1.1.1.2"></times><ci id="S1.Ex7.m1.1.1.1.3.cmml" xref="S1.Ex7.m1.1.1.1.3">𝜎</ci><ci id="S1.Ex7.m1.1.1.1.1.cmml" xref="S1.Ex7.m1.1.1.1.1">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex7.m1.2c">\sigma M(\mu_{w})=\mu_{\sigma(w)}\,.</annotation><annotation encoding="application/x-llamapun" id="S1.Ex7.m1.2d">italic_σ italic_M ( italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_σ ( italic_w ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para ltx_noindent" id="S1.Thmthm6.p6"> <p class="ltx_p" id="S1.Thmthm6.p6.9">(3) For the special case that <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.1.m1.1"><semantics id="S1.Thmthm6.p6.1.m1.1a"><mrow id="S1.Thmthm6.p6.1.m1.1.1" xref="S1.Thmthm6.p6.1.m1.1.1.cmml"><mi id="S1.Thmthm6.p6.1.m1.1.1.2" xref="S1.Thmthm6.p6.1.m1.1.1.2.cmml">σ</mi><mo id="S1.Thmthm6.p6.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm6.p6.1.m1.1.1.1.cmml">:</mo><mrow id="S1.Thmthm6.p6.1.m1.1.1.3" xref="S1.Thmthm6.p6.1.m1.1.1.3.cmml"><msup id="S1.Thmthm6.p6.1.m1.1.1.3.2" xref="S1.Thmthm6.p6.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.1.m1.1.1.3.2.2" xref="S1.Thmthm6.p6.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S1.Thmthm6.p6.1.m1.1.1.3.2.3" xref="S1.Thmthm6.p6.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S1.Thmthm6.p6.1.m1.1.1.3.1" stretchy="false" xref="S1.Thmthm6.p6.1.m1.1.1.3.1.cmml">→</mo><msup id="S1.Thmthm6.p6.1.m1.1.1.3.3" xref="S1.Thmthm6.p6.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.1.m1.1.1.3.3.2" xref="S1.Thmthm6.p6.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S1.Thmthm6.p6.1.m1.1.1.3.3.3" xref="S1.Thmthm6.p6.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.1.m1.1b"><apply id="S1.Thmthm6.p6.1.m1.1.1.cmml" xref="S1.Thmthm6.p6.1.m1.1.1"><ci id="S1.Thmthm6.p6.1.m1.1.1.1.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.1">:</ci><ci id="S1.Thmthm6.p6.1.m1.1.1.2.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.2">𝜎</ci><apply id="S1.Thmthm6.p6.1.m1.1.1.3.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3"><ci id="S1.Thmthm6.p6.1.m1.1.1.3.1.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.1">→</ci><apply id="S1.Thmthm6.p6.1.m1.1.1.3.2.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmthm6.p6.1.m1.1.1.3.2.1.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.2">superscript</csymbol><ci id="S1.Thmthm6.p6.1.m1.1.1.3.2.2.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.2.2">𝒜</ci><times id="S1.Thmthm6.p6.1.m1.1.1.3.2.3.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.2.3"></times></apply><apply id="S1.Thmthm6.p6.1.m1.1.1.3.3.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmthm6.p6.1.m1.1.1.3.3.1.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.3">superscript</csymbol><ci id="S1.Thmthm6.p6.1.m1.1.1.3.3.2.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.3.2">ℬ</ci><times id="S1.Thmthm6.p6.1.m1.1.1.3.3.3.cmml" xref="S1.Thmthm6.p6.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> extends to a free group automorphism <math alttext="\varphi_{\sigma}:F(\cal A)\to F(\cal B)" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.2.m2.2"><semantics id="S1.Thmthm6.p6.2.m2.2a"><mrow id="S1.Thmthm6.p6.2.m2.2.3" xref="S1.Thmthm6.p6.2.m2.2.3.cmml"><msub id="S1.Thmthm6.p6.2.m2.2.3.2" xref="S1.Thmthm6.p6.2.m2.2.3.2.cmml"><mi id="S1.Thmthm6.p6.2.m2.2.3.2.2" xref="S1.Thmthm6.p6.2.m2.2.3.2.2.cmml">φ</mi><mi id="S1.Thmthm6.p6.2.m2.2.3.2.3" xref="S1.Thmthm6.p6.2.m2.2.3.2.3.cmml">σ</mi></msub><mo id="S1.Thmthm6.p6.2.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmthm6.p6.2.m2.2.3.1.cmml">:</mo><mrow id="S1.Thmthm6.p6.2.m2.2.3.3" xref="S1.Thmthm6.p6.2.m2.2.3.3.cmml"><mrow id="S1.Thmthm6.p6.2.m2.2.3.3.2" xref="S1.Thmthm6.p6.2.m2.2.3.3.2.cmml"><mi id="S1.Thmthm6.p6.2.m2.2.3.3.2.2" xref="S1.Thmthm6.p6.2.m2.2.3.3.2.2.cmml">F</mi><mo id="S1.Thmthm6.p6.2.m2.2.3.3.2.1" xref="S1.Thmthm6.p6.2.m2.2.3.3.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.2.m2.2.3.3.2.3.2" xref="S1.Thmthm6.p6.2.m2.2.3.3.2.cmml"><mo id="S1.Thmthm6.p6.2.m2.2.3.3.2.3.2.1" stretchy="false" xref="S1.Thmthm6.p6.2.m2.2.3.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.2.m2.1.1" xref="S1.Thmthm6.p6.2.m2.1.1.cmml">𝒜</mi><mo id="S1.Thmthm6.p6.2.m2.2.3.3.2.3.2.2" stretchy="false" xref="S1.Thmthm6.p6.2.m2.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm6.p6.2.m2.2.3.3.1" stretchy="false" xref="S1.Thmthm6.p6.2.m2.2.3.3.1.cmml">→</mo><mrow id="S1.Thmthm6.p6.2.m2.2.3.3.3" xref="S1.Thmthm6.p6.2.m2.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.2.m2.2.3.3.3.2" xref="S1.Thmthm6.p6.2.m2.2.3.3.3.2.cmml">ℱ</mi><mo id="S1.Thmthm6.p6.2.m2.2.3.3.3.1" xref="S1.Thmthm6.p6.2.m2.2.3.3.3.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.2.m2.2.3.3.3.3.2" xref="S1.Thmthm6.p6.2.m2.2.3.3.3.cmml"><mo id="S1.Thmthm6.p6.2.m2.2.3.3.3.3.2.1" stretchy="false" xref="S1.Thmthm6.p6.2.m2.2.3.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.2.m2.2.2" xref="S1.Thmthm6.p6.2.m2.2.2.cmml">ℬ</mi><mo id="S1.Thmthm6.p6.2.m2.2.3.3.3.3.2.2" stretchy="false" xref="S1.Thmthm6.p6.2.m2.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.2.m2.2b"><apply id="S1.Thmthm6.p6.2.m2.2.3.cmml" xref="S1.Thmthm6.p6.2.m2.2.3"><ci id="S1.Thmthm6.p6.2.m2.2.3.1.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.1">:</ci><apply id="S1.Thmthm6.p6.2.m2.2.3.2.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.2"><csymbol cd="ambiguous" id="S1.Thmthm6.p6.2.m2.2.3.2.1.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.2">subscript</csymbol><ci id="S1.Thmthm6.p6.2.m2.2.3.2.2.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.2.2">𝜑</ci><ci id="S1.Thmthm6.p6.2.m2.2.3.2.3.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.2.3">𝜎</ci></apply><apply id="S1.Thmthm6.p6.2.m2.2.3.3.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.3"><ci id="S1.Thmthm6.p6.2.m2.2.3.3.1.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.3.1">→</ci><apply id="S1.Thmthm6.p6.2.m2.2.3.3.2.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.3.2"><times id="S1.Thmthm6.p6.2.m2.2.3.3.2.1.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.3.2.1"></times><ci id="S1.Thmthm6.p6.2.m2.2.3.3.2.2.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.3.2.2">𝐹</ci><ci id="S1.Thmthm6.p6.2.m2.1.1.cmml" xref="S1.Thmthm6.p6.2.m2.1.1">𝒜</ci></apply><apply id="S1.Thmthm6.p6.2.m2.2.3.3.3.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.3.3"><times id="S1.Thmthm6.p6.2.m2.2.3.3.3.1.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.3.3.1"></times><ci id="S1.Thmthm6.p6.2.m2.2.3.3.3.2.cmml" xref="S1.Thmthm6.p6.2.m2.2.3.3.3.2">ℱ</ci><ci id="S1.Thmthm6.p6.2.m2.2.2.cmml" xref="S1.Thmthm6.p6.2.m2.2.2">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.2.m2.2c">\varphi_{\sigma}:F(\cal A)\to F(\cal B)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.2.m2.2d">italic_φ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT : italic_F ( caligraphic_A ) → caligraphic_F ( caligraphic_B )</annotation></semantics></math>, one can use the well established 1-1 relationship between invariant measures on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.3.m3.1"><semantics id="S1.Thmthm6.p6.3.m3.1a"><msup id="S1.Thmthm6.p6.3.m3.1.1" xref="S1.Thmthm6.p6.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.3.m3.1.1.2" xref="S1.Thmthm6.p6.3.m3.1.1.2.cmml">𝒜</mi><mi id="S1.Thmthm6.p6.3.m3.1.1.3" xref="S1.Thmthm6.p6.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.3.m3.1b"><apply id="S1.Thmthm6.p6.3.m3.1.1.cmml" xref="S1.Thmthm6.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p6.3.m3.1.1.1.cmml" xref="S1.Thmthm6.p6.3.m3.1.1">superscript</csymbol><ci id="S1.Thmthm6.p6.3.m3.1.1.2.cmml" xref="S1.Thmthm6.p6.3.m3.1.1.2">𝒜</ci><ci id="S1.Thmthm6.p6.3.m3.1.1.3.cmml" xref="S1.Thmthm6.p6.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> on one hand and currents on <math alttext="F(\cal A)" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.4.m4.1"><semantics id="S1.Thmthm6.p6.4.m4.1a"><mrow id="S1.Thmthm6.p6.4.m4.1.2" xref="S1.Thmthm6.p6.4.m4.1.2.cmml"><mi id="S1.Thmthm6.p6.4.m4.1.2.2" xref="S1.Thmthm6.p6.4.m4.1.2.2.cmml">F</mi><mo id="S1.Thmthm6.p6.4.m4.1.2.1" xref="S1.Thmthm6.p6.4.m4.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.4.m4.1.2.3.2" xref="S1.Thmthm6.p6.4.m4.1.2.cmml"><mo id="S1.Thmthm6.p6.4.m4.1.2.3.2.1" stretchy="false" xref="S1.Thmthm6.p6.4.m4.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.4.m4.1.1" xref="S1.Thmthm6.p6.4.m4.1.1.cmml">𝒜</mi><mo id="S1.Thmthm6.p6.4.m4.1.2.3.2.2" stretchy="false" xref="S1.Thmthm6.p6.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.4.m4.1b"><apply id="S1.Thmthm6.p6.4.m4.1.2.cmml" xref="S1.Thmthm6.p6.4.m4.1.2"><times id="S1.Thmthm6.p6.4.m4.1.2.1.cmml" xref="S1.Thmthm6.p6.4.m4.1.2.1"></times><ci id="S1.Thmthm6.p6.4.m4.1.2.2.cmml" xref="S1.Thmthm6.p6.4.m4.1.2.2">𝐹</ci><ci id="S1.Thmthm6.p6.4.m4.1.1.cmml" xref="S1.Thmthm6.p6.4.m4.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.4.m4.1c">F(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.4.m4.1d">italic_F ( caligraphic_A )</annotation></semantics></math> on the other, as well as the similarly well established action of <math alttext="\mbox{Aut}(F(\cal A))" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.5.m5.2"><semantics id="S1.Thmthm6.p6.5.m5.2a"><mrow id="S1.Thmthm6.p6.5.m5.2.2" xref="S1.Thmthm6.p6.5.m5.2.2.cmml"><mtext id="S1.Thmthm6.p6.5.m5.2.2.3" xref="S1.Thmthm6.p6.5.m5.2.2.3a.cmml">Aut</mtext><mo id="S1.Thmthm6.p6.5.m5.2.2.2" xref="S1.Thmthm6.p6.5.m5.2.2.2.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.5.m5.2.2.1.1" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.cmml"><mo id="S1.Thmthm6.p6.5.m5.2.2.1.1.2" stretchy="false" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.cmml">(</mo><mrow id="S1.Thmthm6.p6.5.m5.2.2.1.1.1" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.cmml"><mi id="S1.Thmthm6.p6.5.m5.2.2.1.1.1.2" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.2.cmml">F</mi><mo id="S1.Thmthm6.p6.5.m5.2.2.1.1.1.1" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.5.m5.2.2.1.1.1.3.2" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.cmml"><mo id="S1.Thmthm6.p6.5.m5.2.2.1.1.1.3.2.1" stretchy="false" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.5.m5.1.1" xref="S1.Thmthm6.p6.5.m5.1.1.cmml">𝒜</mi><mo id="S1.Thmthm6.p6.5.m5.2.2.1.1.1.3.2.2" stretchy="false" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm6.p6.5.m5.2.2.1.1.3" stretchy="false" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.5.m5.2b"><apply id="S1.Thmthm6.p6.5.m5.2.2.cmml" xref="S1.Thmthm6.p6.5.m5.2.2"><times id="S1.Thmthm6.p6.5.m5.2.2.2.cmml" xref="S1.Thmthm6.p6.5.m5.2.2.2"></times><ci id="S1.Thmthm6.p6.5.m5.2.2.3a.cmml" xref="S1.Thmthm6.p6.5.m5.2.2.3"><mtext id="S1.Thmthm6.p6.5.m5.2.2.3.cmml" xref="S1.Thmthm6.p6.5.m5.2.2.3">Aut</mtext></ci><apply id="S1.Thmthm6.p6.5.m5.2.2.1.1.1.cmml" xref="S1.Thmthm6.p6.5.m5.2.2.1.1"><times id="S1.Thmthm6.p6.5.m5.2.2.1.1.1.1.cmml" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.1"></times><ci id="S1.Thmthm6.p6.5.m5.2.2.1.1.1.2.cmml" xref="S1.Thmthm6.p6.5.m5.2.2.1.1.1.2">𝐹</ci><ci id="S1.Thmthm6.p6.5.m5.1.1.cmml" xref="S1.Thmthm6.p6.5.m5.1.1">𝒜</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.5.m5.2c">\mbox{Aut}(F(\cal A))</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.5.m5.2d">Aut ( italic_F ( caligraphic_A ) )</annotation></semantics></math> on the current space <math alttext="\mbox{Curr}(F(\cal A))\cong\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.6.m6.3"><semantics id="S1.Thmthm6.p6.6.m6.3a"><mrow id="S1.Thmthm6.p6.6.m6.3.3" xref="S1.Thmthm6.p6.6.m6.3.3.cmml"><mrow id="S1.Thmthm6.p6.6.m6.2.2.1" xref="S1.Thmthm6.p6.6.m6.2.2.1.cmml"><mtext id="S1.Thmthm6.p6.6.m6.2.2.1.3" xref="S1.Thmthm6.p6.6.m6.2.2.1.3a.cmml">Curr</mtext><mo id="S1.Thmthm6.p6.6.m6.2.2.1.2" xref="S1.Thmthm6.p6.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.6.m6.2.2.1.1.1" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.cmml"><mo id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.2" stretchy="false" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.cmml">(</mo><mrow id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.cmml"><mi id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.2" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.2.cmml">F</mi><mo id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.1" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.3.2" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.cmml"><mo id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.6.m6.1.1" xref="S1.Thmthm6.p6.6.m6.1.1.cmml">𝒜</mi><mo id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.3" stretchy="false" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmthm6.p6.6.m6.3.3.3" xref="S1.Thmthm6.p6.6.m6.3.3.3.cmml">≅</mo><mrow id="S1.Thmthm6.p6.6.m6.3.3.2" xref="S1.Thmthm6.p6.6.m6.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.6.m6.3.3.2.3" xref="S1.Thmthm6.p6.6.m6.3.3.2.3.cmml">ℳ</mi><mo id="S1.Thmthm6.p6.6.m6.3.3.2.2" xref="S1.Thmthm6.p6.6.m6.3.3.2.2.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.6.m6.3.3.2.1.1" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.cmml"><mo id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.2" stretchy="false" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.cmml">(</mo><msup id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.2" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.2.cmml">𝒜</mi><mi id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.3" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.3" stretchy="false" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.6.m6.3b"><apply id="S1.Thmthm6.p6.6.m6.3.3.cmml" xref="S1.Thmthm6.p6.6.m6.3.3"><approx id="S1.Thmthm6.p6.6.m6.3.3.3.cmml" xref="S1.Thmthm6.p6.6.m6.3.3.3"></approx><apply id="S1.Thmthm6.p6.6.m6.2.2.1.cmml" xref="S1.Thmthm6.p6.6.m6.2.2.1"><times id="S1.Thmthm6.p6.6.m6.2.2.1.2.cmml" xref="S1.Thmthm6.p6.6.m6.2.2.1.2"></times><ci id="S1.Thmthm6.p6.6.m6.2.2.1.3a.cmml" xref="S1.Thmthm6.p6.6.m6.2.2.1.3"><mtext id="S1.Thmthm6.p6.6.m6.2.2.1.3.cmml" xref="S1.Thmthm6.p6.6.m6.2.2.1.3">Curr</mtext></ci><apply id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.cmml" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1"><times id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.1.cmml" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.1"></times><ci id="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.2.cmml" xref="S1.Thmthm6.p6.6.m6.2.2.1.1.1.1.2">𝐹</ci><ci id="S1.Thmthm6.p6.6.m6.1.1.cmml" xref="S1.Thmthm6.p6.6.m6.1.1">𝒜</ci></apply></apply><apply id="S1.Thmthm6.p6.6.m6.3.3.2.cmml" xref="S1.Thmthm6.p6.6.m6.3.3.2"><times id="S1.Thmthm6.p6.6.m6.3.3.2.2.cmml" xref="S1.Thmthm6.p6.6.m6.3.3.2.2"></times><ci id="S1.Thmthm6.p6.6.m6.3.3.2.3.cmml" xref="S1.Thmthm6.p6.6.m6.3.3.2.3">ℳ</ci><apply id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.cmml" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1"><csymbol cd="ambiguous" id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.1.cmml" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1">superscript</csymbol><ci id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.2.cmml" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.2">𝒜</ci><ci id="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.3.cmml" xref="S1.Thmthm6.p6.6.m6.3.3.2.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.6.m6.3c">\mbox{Curr}(F(\cal A))\cong\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.6.m6.3d">Curr ( italic_F ( caligraphic_A ) ) ≅ caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib11" title="">11</a>]</cite> for details). Denoting by <math alttext="\mu" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.7.m7.1"><semantics id="S1.Thmthm6.p6.7.m7.1a"><mi id="S1.Thmthm6.p6.7.m7.1.1" xref="S1.Thmthm6.p6.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.7.m7.1b"><ci id="S1.Thmthm6.p6.7.m7.1.1.cmml" xref="S1.Thmthm6.p6.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.7.m7.1d">italic_μ</annotation></semantics></math> also the current on <math alttext="F(\cal A)" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.8.m8.1"><semantics id="S1.Thmthm6.p6.8.m8.1a"><mrow id="S1.Thmthm6.p6.8.m8.1.2" xref="S1.Thmthm6.p6.8.m8.1.2.cmml"><mi id="S1.Thmthm6.p6.8.m8.1.2.2" xref="S1.Thmthm6.p6.8.m8.1.2.2.cmml">F</mi><mo id="S1.Thmthm6.p6.8.m8.1.2.1" xref="S1.Thmthm6.p6.8.m8.1.2.1.cmml">⁢</mo><mrow id="S1.Thmthm6.p6.8.m8.1.2.3.2" xref="S1.Thmthm6.p6.8.m8.1.2.cmml"><mo id="S1.Thmthm6.p6.8.m8.1.2.3.2.1" stretchy="false" xref="S1.Thmthm6.p6.8.m8.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S1.Thmthm6.p6.8.m8.1.1" xref="S1.Thmthm6.p6.8.m8.1.1.cmml">𝒜</mi><mo id="S1.Thmthm6.p6.8.m8.1.2.3.2.2" stretchy="false" xref="S1.Thmthm6.p6.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.8.m8.1b"><apply id="S1.Thmthm6.p6.8.m8.1.2.cmml" xref="S1.Thmthm6.p6.8.m8.1.2"><times id="S1.Thmthm6.p6.8.m8.1.2.1.cmml" xref="S1.Thmthm6.p6.8.m8.1.2.1"></times><ci id="S1.Thmthm6.p6.8.m8.1.2.2.cmml" xref="S1.Thmthm6.p6.8.m8.1.2.2">𝐹</ci><ci id="S1.Thmthm6.p6.8.m8.1.1.cmml" xref="S1.Thmthm6.p6.8.m8.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.8.m8.1c">F(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.8.m8.1d">italic_F ( caligraphic_A )</annotation></semantics></math> defined by the invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S1.Thmthm6.p6.9.m9.1"><semantics id="S1.Thmthm6.p6.9.m9.1a"><mi id="S1.Thmthm6.p6.9.m9.1.1" xref="S1.Thmthm6.p6.9.m9.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.Thmthm6.p6.9.m9.1b"><ci id="S1.Thmthm6.p6.9.m9.1.1.cmml" xref="S1.Thmthm6.p6.9.m9.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmthm6.p6.9.m9.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.Thmthm6.p6.9.m9.1d">italic_μ</annotation></semantics></math>, one has</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\varphi_{\sigma}(\mu)=\sigma M(\mu)\,." class="ltx_Math" display="block" id="S1.Ex8.m1.3"><semantics id="S1.Ex8.m1.3a"><mrow id="S1.Ex8.m1.3.3.1" xref="S1.Ex8.m1.3.3.1.1.cmml"><mrow id="S1.Ex8.m1.3.3.1.1" xref="S1.Ex8.m1.3.3.1.1.cmml"><mrow id="S1.Ex8.m1.3.3.1.1.2" xref="S1.Ex8.m1.3.3.1.1.2.cmml"><msub id="S1.Ex8.m1.3.3.1.1.2.2" xref="S1.Ex8.m1.3.3.1.1.2.2.cmml"><mi id="S1.Ex8.m1.3.3.1.1.2.2.2" xref="S1.Ex8.m1.3.3.1.1.2.2.2.cmml">φ</mi><mi id="S1.Ex8.m1.3.3.1.1.2.2.3" xref="S1.Ex8.m1.3.3.1.1.2.2.3.cmml">σ</mi></msub><mo id="S1.Ex8.m1.3.3.1.1.2.1" xref="S1.Ex8.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S1.Ex8.m1.3.3.1.1.2.3.2" xref="S1.Ex8.m1.3.3.1.1.2.cmml"><mo id="S1.Ex8.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S1.Ex8.m1.3.3.1.1.2.cmml">(</mo><mi id="S1.Ex8.m1.1.1" xref="S1.Ex8.m1.1.1.cmml">μ</mi><mo id="S1.Ex8.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="S1.Ex8.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.Ex8.m1.3.3.1.1.1" xref="S1.Ex8.m1.3.3.1.1.1.cmml">=</mo><mrow id="S1.Ex8.m1.3.3.1.1.3" xref="S1.Ex8.m1.3.3.1.1.3.cmml"><mi id="S1.Ex8.m1.3.3.1.1.3.2" xref="S1.Ex8.m1.3.3.1.1.3.2.cmml">σ</mi><mo id="S1.Ex8.m1.3.3.1.1.3.1" xref="S1.Ex8.m1.3.3.1.1.3.1.cmml">⁢</mo><mi id="S1.Ex8.m1.3.3.1.1.3.3" xref="S1.Ex8.m1.3.3.1.1.3.3.cmml">M</mi><mo id="S1.Ex8.m1.3.3.1.1.3.1a" xref="S1.Ex8.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S1.Ex8.m1.3.3.1.1.3.4.2" xref="S1.Ex8.m1.3.3.1.1.3.cmml"><mo id="S1.Ex8.m1.3.3.1.1.3.4.2.1" stretchy="false" xref="S1.Ex8.m1.3.3.1.1.3.cmml">(</mo><mi id="S1.Ex8.m1.2.2" xref="S1.Ex8.m1.2.2.cmml">μ</mi><mo id="S1.Ex8.m1.3.3.1.1.3.4.2.2" stretchy="false" xref="S1.Ex8.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S1.Ex8.m1.3.3.1.2" lspace="0.170em" xref="S1.Ex8.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex8.m1.3b"><apply id="S1.Ex8.m1.3.3.1.1.cmml" xref="S1.Ex8.m1.3.3.1"><eq id="S1.Ex8.m1.3.3.1.1.1.cmml" xref="S1.Ex8.m1.3.3.1.1.1"></eq><apply id="S1.Ex8.m1.3.3.1.1.2.cmml" xref="S1.Ex8.m1.3.3.1.1.2"><times id="S1.Ex8.m1.3.3.1.1.2.1.cmml" xref="S1.Ex8.m1.3.3.1.1.2.1"></times><apply id="S1.Ex8.m1.3.3.1.1.2.2.cmml" xref="S1.Ex8.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S1.Ex8.m1.3.3.1.1.2.2.1.cmml" xref="S1.Ex8.m1.3.3.1.1.2.2">subscript</csymbol><ci id="S1.Ex8.m1.3.3.1.1.2.2.2.cmml" xref="S1.Ex8.m1.3.3.1.1.2.2.2">𝜑</ci><ci id="S1.Ex8.m1.3.3.1.1.2.2.3.cmml" xref="S1.Ex8.m1.3.3.1.1.2.2.3">𝜎</ci></apply><ci id="S1.Ex8.m1.1.1.cmml" xref="S1.Ex8.m1.1.1">𝜇</ci></apply><apply id="S1.Ex8.m1.3.3.1.1.3.cmml" xref="S1.Ex8.m1.3.3.1.1.3"><times id="S1.Ex8.m1.3.3.1.1.3.1.cmml" xref="S1.Ex8.m1.3.3.1.1.3.1"></times><ci id="S1.Ex8.m1.3.3.1.1.3.2.cmml" xref="S1.Ex8.m1.3.3.1.1.3.2">𝜎</ci><ci id="S1.Ex8.m1.3.3.1.1.3.3.cmml" xref="S1.Ex8.m1.3.3.1.1.3.3">𝑀</ci><ci id="S1.Ex8.m1.2.2.cmml" xref="S1.Ex8.m1.2.2">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex8.m1.3c">\varphi_{\sigma}(\mu)=\sigma M(\mu)\,.</annotation><annotation encoding="application/x-llamapun" id="S1.Ex8.m1.3d">italic_φ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_μ ) = italic_σ italic_M ( italic_μ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para ltx_noindent" id="S1.p12"> <br class="ltx_break"/> <p class="ltx_p" id="S1.p12.1"><span class="ltx_text ltx_font_italic" id="S1.p12.1.1">Acknowledgements:</span> We would like to thank Fabien Durand and Samuel Petite for encouraging remarks and interesting comments, and the two referees, who have sent us very stimulating questions and useful suggestions.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2. </span>Notation and conventions</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1. </span>Standard terminology and well known facts</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1"></p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.10">Throughout this paper we denote by <math alttext="\cal A,\cal B,\cal C,\ldots" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.4"><semantics id="S2.SS1.p2.1.m1.4a"><mrow id="S2.SS1.p2.1.m1.4.5.2" xref="S2.SS1.p2.1.m1.4.5.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml">𝒜</mi><mo id="S2.SS1.p2.1.m1.4.5.2.1" xref="S2.SS1.p2.1.m1.4.5.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.1.m1.2.2" xref="S2.SS1.p2.1.m1.2.2.cmml">ℬ</mi><mo id="S2.SS1.p2.1.m1.4.5.2.2" xref="S2.SS1.p2.1.m1.4.5.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.1.m1.3.3" xref="S2.SS1.p2.1.m1.3.3.cmml">𝒞</mi><mo id="S2.SS1.p2.1.m1.4.5.2.3" xref="S2.SS1.p2.1.m1.4.5.1.cmml">,</mo><mi id="S2.SS1.p2.1.m1.4.4" mathvariant="normal" xref="S2.SS1.p2.1.m1.4.4.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.4b"><list id="S2.SS1.p2.1.m1.4.5.1.cmml" xref="S2.SS1.p2.1.m1.4.5.2"><ci id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">𝒜</ci><ci id="S2.SS1.p2.1.m1.2.2.cmml" xref="S2.SS1.p2.1.m1.2.2">ℬ</ci><ci id="S2.SS1.p2.1.m1.3.3.cmml" xref="S2.SS1.p2.1.m1.3.3">𝒞</ci><ci id="S2.SS1.p2.1.m1.4.4.cmml" xref="S2.SS1.p2.1.m1.4.4">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.4c">\cal A,\cal B,\cal C,\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.4d">caligraphic_A , caligraphic_B , caligraphic_C , …</annotation></semantics></math> finite non-empty sets, called <span class="ltx_text ltx_font_italic" id="S2.SS1.p2.10.1">alphabets</span>. For any such alphabet, say <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">caligraphic_A</annotation></semantics></math>, we denote by <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><msup id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.3.m3.1.1.2" xref="S2.SS1.p2.3.m3.1.1.2.cmml">𝒜</mi><mo id="S2.SS1.p2.3.m3.1.1.3" xref="S2.SS1.p2.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">superscript</csymbol><ci id="S2.SS1.p2.3.m3.1.1.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2">𝒜</ci><times id="S2.SS1.p2.3.m3.1.1.3.cmml" xref="S2.SS1.p2.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the free monoid over the set <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><ci id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">caligraphic_A</annotation></semantics></math>, given by all finite words <math alttext="w=x_{1}x_{2}\ldots x_{n}" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><mrow id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml"><mi id="S2.SS1.p2.5.m5.1.1.2" xref="S2.SS1.p2.5.m5.1.1.2.cmml">w</mi><mo id="S2.SS1.p2.5.m5.1.1.1" xref="S2.SS1.p2.5.m5.1.1.1.cmml">=</mo><mrow id="S2.SS1.p2.5.m5.1.1.3" xref="S2.SS1.p2.5.m5.1.1.3.cmml"><msub id="S2.SS1.p2.5.m5.1.1.3.2" xref="S2.SS1.p2.5.m5.1.1.3.2.cmml"><mi id="S2.SS1.p2.5.m5.1.1.3.2.2" xref="S2.SS1.p2.5.m5.1.1.3.2.2.cmml">x</mi><mn id="S2.SS1.p2.5.m5.1.1.3.2.3" xref="S2.SS1.p2.5.m5.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.SS1.p2.5.m5.1.1.3.1" xref="S2.SS1.p2.5.m5.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p2.5.m5.1.1.3.3" xref="S2.SS1.p2.5.m5.1.1.3.3.cmml"><mi id="S2.SS1.p2.5.m5.1.1.3.3.2" xref="S2.SS1.p2.5.m5.1.1.3.3.2.cmml">x</mi><mn id="S2.SS1.p2.5.m5.1.1.3.3.3" xref="S2.SS1.p2.5.m5.1.1.3.3.3.cmml">2</mn></msub><mo id="S2.SS1.p2.5.m5.1.1.3.1a" xref="S2.SS1.p2.5.m5.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.5.m5.1.1.3.4" mathvariant="normal" xref="S2.SS1.p2.5.m5.1.1.3.4.cmml">…</mi><mo id="S2.SS1.p2.5.m5.1.1.3.1b" xref="S2.SS1.p2.5.m5.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p2.5.m5.1.1.3.5" xref="S2.SS1.p2.5.m5.1.1.3.5.cmml"><mi id="S2.SS1.p2.5.m5.1.1.3.5.2" xref="S2.SS1.p2.5.m5.1.1.3.5.2.cmml">x</mi><mi id="S2.SS1.p2.5.m5.1.1.3.5.3" xref="S2.SS1.p2.5.m5.1.1.3.5.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><apply id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1"><eq id="S2.SS1.p2.5.m5.1.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1.1"></eq><ci id="S2.SS1.p2.5.m5.1.1.2.cmml" xref="S2.SS1.p2.5.m5.1.1.2">𝑤</ci><apply id="S2.SS1.p2.5.m5.1.1.3.cmml" xref="S2.SS1.p2.5.m5.1.1.3"><times id="S2.SS1.p2.5.m5.1.1.3.1.cmml" xref="S2.SS1.p2.5.m5.1.1.3.1"></times><apply id="S2.SS1.p2.5.m5.1.1.3.2.cmml" xref="S2.SS1.p2.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m5.1.1.3.2.1.cmml" xref="S2.SS1.p2.5.m5.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p2.5.m5.1.1.3.2.2.cmml" xref="S2.SS1.p2.5.m5.1.1.3.2.2">𝑥</ci><cn id="S2.SS1.p2.5.m5.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.p2.5.m5.1.1.3.2.3">1</cn></apply><apply id="S2.SS1.p2.5.m5.1.1.3.3.cmml" xref="S2.SS1.p2.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m5.1.1.3.3.1.cmml" xref="S2.SS1.p2.5.m5.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p2.5.m5.1.1.3.3.2.cmml" xref="S2.SS1.p2.5.m5.1.1.3.3.2">𝑥</ci><cn id="S2.SS1.p2.5.m5.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.p2.5.m5.1.1.3.3.3">2</cn></apply><ci id="S2.SS1.p2.5.m5.1.1.3.4.cmml" xref="S2.SS1.p2.5.m5.1.1.3.4">…</ci><apply id="S2.SS1.p2.5.m5.1.1.3.5.cmml" xref="S2.SS1.p2.5.m5.1.1.3.5"><csymbol cd="ambiguous" id="S2.SS1.p2.5.m5.1.1.3.5.1.cmml" xref="S2.SS1.p2.5.m5.1.1.3.5">subscript</csymbol><ci id="S2.SS1.p2.5.m5.1.1.3.5.2.cmml" xref="S2.SS1.p2.5.m5.1.1.3.5.2">𝑥</ci><ci id="S2.SS1.p2.5.m5.1.1.3.5.3.cmml" xref="S2.SS1.p2.5.m5.1.1.3.5.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">w=x_{1}x_{2}\ldots x_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">italic_w = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="x_{i}\in\cal A" class="ltx_Math" display="inline" id="S2.SS1.p2.6.m6.1"><semantics id="S2.SS1.p2.6.m6.1a"><mrow id="S2.SS1.p2.6.m6.1.1" xref="S2.SS1.p2.6.m6.1.1.cmml"><msub id="S2.SS1.p2.6.m6.1.1.2" xref="S2.SS1.p2.6.m6.1.1.2.cmml"><mi id="S2.SS1.p2.6.m6.1.1.2.2" xref="S2.SS1.p2.6.m6.1.1.2.2.cmml">x</mi><mi id="S2.SS1.p2.6.m6.1.1.2.3" xref="S2.SS1.p2.6.m6.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p2.6.m6.1.1.1" xref="S2.SS1.p2.6.m6.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.6.m6.1.1.3" xref="S2.SS1.p2.6.m6.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.6.m6.1b"><apply id="S2.SS1.p2.6.m6.1.1.cmml" xref="S2.SS1.p2.6.m6.1.1"><in id="S2.SS1.p2.6.m6.1.1.1.cmml" xref="S2.SS1.p2.6.m6.1.1.1"></in><apply id="S2.SS1.p2.6.m6.1.1.2.cmml" xref="S2.SS1.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.1.1.2.1.cmml" xref="S2.SS1.p2.6.m6.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.6.m6.1.1.2.2.cmml" xref="S2.SS1.p2.6.m6.1.1.2.2">𝑥</ci><ci id="S2.SS1.p2.6.m6.1.1.2.3.cmml" xref="S2.SS1.p2.6.m6.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p2.6.m6.1.1.3.cmml" xref="S2.SS1.p2.6.m6.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.1c">x_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math>, and equipped with the multiplication defined by concatenation. We denote by <math alttext="|w|:=n" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.1"><semantics id="S2.SS1.p2.7.m7.1a"><mrow id="S2.SS1.p2.7.m7.1.2" xref="S2.SS1.p2.7.m7.1.2.cmml"><mrow id="S2.SS1.p2.7.m7.1.2.2.2" xref="S2.SS1.p2.7.m7.1.2.2.1.cmml"><mo id="S2.SS1.p2.7.m7.1.2.2.2.1" stretchy="false" xref="S2.SS1.p2.7.m7.1.2.2.1.1.cmml">|</mo><mi id="S2.SS1.p2.7.m7.1.1" xref="S2.SS1.p2.7.m7.1.1.cmml">w</mi><mo id="S2.SS1.p2.7.m7.1.2.2.2.2" rspace="0.278em" stretchy="false" xref="S2.SS1.p2.7.m7.1.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS1.p2.7.m7.1.2.1" rspace="0.278em" xref="S2.SS1.p2.7.m7.1.2.1.cmml">:=</mo><mi id="S2.SS1.p2.7.m7.1.2.3" xref="S2.SS1.p2.7.m7.1.2.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m7.1b"><apply id="S2.SS1.p2.7.m7.1.2.cmml" xref="S2.SS1.p2.7.m7.1.2"><csymbol cd="latexml" id="S2.SS1.p2.7.m7.1.2.1.cmml" xref="S2.SS1.p2.7.m7.1.2.1">assign</csymbol><apply id="S2.SS1.p2.7.m7.1.2.2.1.cmml" xref="S2.SS1.p2.7.m7.1.2.2.2"><abs id="S2.SS1.p2.7.m7.1.2.2.1.1.cmml" xref="S2.SS1.p2.7.m7.1.2.2.2.1"></abs><ci id="S2.SS1.p2.7.m7.1.1.cmml" xref="S2.SS1.p2.7.m7.1.1">𝑤</ci></apply><ci id="S2.SS1.p2.7.m7.1.2.3.cmml" xref="S2.SS1.p2.7.m7.1.2.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m7.1c">|w|:=n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.1d">| italic_w | := italic_n</annotation></semantics></math> the <span class="ltx_text ltx_font_italic" id="S2.SS1.p2.10.2">length</span> of any such word. The empty word <math alttext="\varepsilon\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p2.8.m8.1"><semantics id="S2.SS1.p2.8.m8.1a"><mrow id="S2.SS1.p2.8.m8.1.1" xref="S2.SS1.p2.8.m8.1.1.cmml"><mi id="S2.SS1.p2.8.m8.1.1.2" xref="S2.SS1.p2.8.m8.1.1.2.cmml">ε</mi><mo id="S2.SS1.p2.8.m8.1.1.1" xref="S2.SS1.p2.8.m8.1.1.1.cmml">∈</mo><msup id="S2.SS1.p2.8.m8.1.1.3" xref="S2.SS1.p2.8.m8.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.8.m8.1.1.3.2" xref="S2.SS1.p2.8.m8.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p2.8.m8.1.1.3.3" xref="S2.SS1.p2.8.m8.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.8.m8.1b"><apply id="S2.SS1.p2.8.m8.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1"><in id="S2.SS1.p2.8.m8.1.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1.1"></in><ci id="S2.SS1.p2.8.m8.1.1.2.cmml" xref="S2.SS1.p2.8.m8.1.1.2">𝜀</ci><apply id="S2.SS1.p2.8.m8.1.1.3.cmml" xref="S2.SS1.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p2.8.m8.1.1.3.1.cmml" xref="S2.SS1.p2.8.m8.1.1.3">superscript</csymbol><ci id="S2.SS1.p2.8.m8.1.1.3.2.cmml" xref="S2.SS1.p2.8.m8.1.1.3.2">𝒜</ci><times id="S2.SS1.p2.8.m8.1.1.3.3.cmml" xref="S2.SS1.p2.8.m8.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.8.m8.1c">\varepsilon\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m8.1d">italic_ε ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is defined by <math alttext="|\varepsilon|=0" class="ltx_Math" display="inline" id="S2.SS1.p2.9.m9.1"><semantics id="S2.SS1.p2.9.m9.1a"><mrow id="S2.SS1.p2.9.m9.1.2" xref="S2.SS1.p2.9.m9.1.2.cmml"><mrow id="S2.SS1.p2.9.m9.1.2.2.2" xref="S2.SS1.p2.9.m9.1.2.2.1.cmml"><mo id="S2.SS1.p2.9.m9.1.2.2.2.1" stretchy="false" xref="S2.SS1.p2.9.m9.1.2.2.1.1.cmml">|</mo><mi id="S2.SS1.p2.9.m9.1.1" xref="S2.SS1.p2.9.m9.1.1.cmml">ε</mi><mo id="S2.SS1.p2.9.m9.1.2.2.2.2" stretchy="false" xref="S2.SS1.p2.9.m9.1.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS1.p2.9.m9.1.2.1" xref="S2.SS1.p2.9.m9.1.2.1.cmml">=</mo><mn id="S2.SS1.p2.9.m9.1.2.3" xref="S2.SS1.p2.9.m9.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.9.m9.1b"><apply id="S2.SS1.p2.9.m9.1.2.cmml" xref="S2.SS1.p2.9.m9.1.2"><eq id="S2.SS1.p2.9.m9.1.2.1.cmml" xref="S2.SS1.p2.9.m9.1.2.1"></eq><apply id="S2.SS1.p2.9.m9.1.2.2.1.cmml" xref="S2.SS1.p2.9.m9.1.2.2.2"><abs id="S2.SS1.p2.9.m9.1.2.2.1.1.cmml" xref="S2.SS1.p2.9.m9.1.2.2.2.1"></abs><ci id="S2.SS1.p2.9.m9.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1">𝜀</ci></apply><cn id="S2.SS1.p2.9.m9.1.2.3.cmml" type="integer" xref="S2.SS1.p2.9.m9.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.9.m9.1c">|\varepsilon|=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.9.m9.1d">| italic_ε | = 0</annotation></semantics></math>; it is the unit element with respect to the multiplication in <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p2.10.m10.1"><semantics id="S2.SS1.p2.10.m10.1a"><msup id="S2.SS1.p2.10.m10.1.1" xref="S2.SS1.p2.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.10.m10.1.1.2" xref="S2.SS1.p2.10.m10.1.1.2.cmml">𝒜</mi><mo id="S2.SS1.p2.10.m10.1.1.3" xref="S2.SS1.p2.10.m10.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.10.m10.1b"><apply id="S2.SS1.p2.10.m10.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.10.m10.1.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1">superscript</csymbol><ci id="S2.SS1.p2.10.m10.1.1.2.cmml" xref="S2.SS1.p2.10.m10.1.1.2">𝒜</ci><times id="S2.SS1.p2.10.m10.1.1.3.cmml" xref="S2.SS1.p2.10.m10.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.10.m10.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.10.m10.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.5">The elements <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><msub id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2.2" xref="S2.SS1.p3.1.m1.1.1.2.2.cmml">a</mi><mi id="S2.SS1.p3.1.m1.1.1.2.3" xref="S2.SS1.p3.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p3.1.m1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><in id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1.1"></in><apply id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.1.2.1.cmml" xref="S2.SS1.p3.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS1.p3.1.m1.1.1.2.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2.2">𝑎</ci><ci id="S2.SS1.p3.1.m1.1.1.2.3.cmml" xref="S2.SS1.p3.1.m1.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> are called the <span class="ltx_text ltx_font_italic" id="S2.SS1.p3.5.1">letters</span>; they constitute a set of generators of <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><msup id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.2.m2.1.1.2" xref="S2.SS1.p3.2.m2.1.1.2.cmml">𝒜</mi><mo id="S2.SS1.p3.2.m2.1.1.3" xref="S2.SS1.p3.2.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><apply id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.2.m2.1.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">superscript</csymbol><ci id="S2.SS1.p3.2.m2.1.1.2.cmml" xref="S2.SS1.p3.2.m2.1.1.2">𝒜</ci><times id="S2.SS1.p3.2.m2.1.1.3.cmml" xref="S2.SS1.p3.2.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and moreover a basis of the associated free group <math alttext="F(\cal A)" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.1"><semantics id="S2.SS1.p3.3.m3.1a"><mrow id="S2.SS1.p3.3.m3.1.2" xref="S2.SS1.p3.3.m3.1.2.cmml"><mi id="S2.SS1.p3.3.m3.1.2.2" xref="S2.SS1.p3.3.m3.1.2.2.cmml">F</mi><mo id="S2.SS1.p3.3.m3.1.2.1" xref="S2.SS1.p3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.3.m3.1.2.3.2" xref="S2.SS1.p3.3.m3.1.2.cmml"><mo id="S2.SS1.p3.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1.p3.3.m3.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml">𝒜</mi><mo id="S2.SS1.p3.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1.p3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.1b"><apply id="S2.SS1.p3.3.m3.1.2.cmml" xref="S2.SS1.p3.3.m3.1.2"><times id="S2.SS1.p3.3.m3.1.2.1.cmml" xref="S2.SS1.p3.3.m3.1.2.1"></times><ci id="S2.SS1.p3.3.m3.1.2.2.cmml" xref="S2.SS1.p3.3.m3.1.2.2">𝐹</ci><ci id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.1c">F(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.1d">italic_F ( caligraphic_A )</annotation></semantics></math>. Any subset <math alttext="\cal L\subseteq\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m4.1"><semantics id="S2.SS1.p3.4.m4.1a"><mrow id="S2.SS1.p3.4.m4.1.1" xref="S2.SS1.p3.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.4.m4.1.1.2" xref="S2.SS1.p3.4.m4.1.1.2.cmml">ℒ</mi><mo id="S2.SS1.p3.4.m4.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p3.4.m4.1.1.3" xref="S2.SS1.p3.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.4.m4.1.1.3.2" xref="S2.SS1.p3.4.m4.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p3.4.m4.1.1.3.3" xref="S2.SS1.p3.4.m4.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m4.1b"><apply id="S2.SS1.p3.4.m4.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1"><subset id="S2.SS1.p3.4.m4.1.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1.1"></subset><ci id="S2.SS1.p3.4.m4.1.1.2.cmml" xref="S2.SS1.p3.4.m4.1.1.2">ℒ</ci><apply id="S2.SS1.p3.4.m4.1.1.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m4.1.1.3.1.cmml" xref="S2.SS1.p3.4.m4.1.1.3">superscript</csymbol><ci id="S2.SS1.p3.4.m4.1.1.3.2.cmml" xref="S2.SS1.p3.4.m4.1.1.3.2">𝒜</ci><times id="S2.SS1.p3.4.m4.1.1.3.3.cmml" xref="S2.SS1.p3.4.m4.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m4.1c">\cal L\subseteq\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m4.1d">caligraphic_L ⊆ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> (often assumed to be infinite) is called a <span class="ltx_text ltx_font_italic" id="S2.SS1.p3.5.2">language</span> over <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m5.1"><semantics id="S2.SS1.p3.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.5.m5.1.1" xref="S2.SS1.p3.5.m5.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m5.1b"><ci id="S2.SS1.p3.5.m5.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m5.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m5.1d">caligraphic_A</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.10">A <span class="ltx_text ltx_font_italic" id="S2.SS1.p4.10.1">monoid morphism</span> <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><mrow id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml"><mi id="S2.SS1.p4.1.m1.1.1.2" xref="S2.SS1.p4.1.m1.1.1.2.cmml">σ</mi><mo id="S2.SS1.p4.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p4.1.m1.1.1.1.cmml">:</mo><mrow id="S2.SS1.p4.1.m1.1.1.3" xref="S2.SS1.p4.1.m1.1.1.3.cmml"><msup id="S2.SS1.p4.1.m1.1.1.3.2" xref="S2.SS1.p4.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.1.m1.1.1.3.2.2" xref="S2.SS1.p4.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.SS1.p4.1.m1.1.1.3.2.3" xref="S2.SS1.p4.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.SS1.p4.1.m1.1.1.3.1" stretchy="false" xref="S2.SS1.p4.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.SS1.p4.1.m1.1.1.3.3" xref="S2.SS1.p4.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.1.m1.1.1.3.3.2" xref="S2.SS1.p4.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.SS1.p4.1.m1.1.1.3.3.3" xref="S2.SS1.p4.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.1b"><apply id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1"><ci id="S2.SS1.p4.1.m1.1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1.1">:</ci><ci id="S2.SS1.p4.1.m1.1.1.2.cmml" xref="S2.SS1.p4.1.m1.1.1.2">𝜎</ci><apply id="S2.SS1.p4.1.m1.1.1.3.cmml" xref="S2.SS1.p4.1.m1.1.1.3"><ci id="S2.SS1.p4.1.m1.1.1.3.1.cmml" xref="S2.SS1.p4.1.m1.1.1.3.1">→</ci><apply id="S2.SS1.p4.1.m1.1.1.3.2.cmml" xref="S2.SS1.p4.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.1.m1.1.1.3.2.1.cmml" xref="S2.SS1.p4.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p4.1.m1.1.1.3.2.2.cmml" xref="S2.SS1.p4.1.m1.1.1.3.2.2">𝒜</ci><times id="S2.SS1.p4.1.m1.1.1.3.2.3.cmml" xref="S2.SS1.p4.1.m1.1.1.3.2.3"></times></apply><apply id="S2.SS1.p4.1.m1.1.1.3.3.cmml" xref="S2.SS1.p4.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p4.1.m1.1.1.3.3.1.cmml" xref="S2.SS1.p4.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.SS1.p4.1.m1.1.1.3.3.2.cmml" xref="S2.SS1.p4.1.m1.1.1.3.3.2">ℬ</ci><times id="S2.SS1.p4.1.m1.1.1.3.3.3.cmml" xref="S2.SS1.p4.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is well defined by knowing the image word <math alttext="\sigma(a_{i})\in\cal B^{*}" class="ltx_Math" display="inline" id="S2.SS1.p4.2.m2.1"><semantics id="S2.SS1.p4.2.m2.1a"><mrow id="S2.SS1.p4.2.m2.1.1" xref="S2.SS1.p4.2.m2.1.1.cmml"><mrow id="S2.SS1.p4.2.m2.1.1.1" xref="S2.SS1.p4.2.m2.1.1.1.cmml"><mi id="S2.SS1.p4.2.m2.1.1.1.3" xref="S2.SS1.p4.2.m2.1.1.1.3.cmml">σ</mi><mo id="S2.SS1.p4.2.m2.1.1.1.2" xref="S2.SS1.p4.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p4.2.m2.1.1.1.1.1" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p4.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p4.2.m2.1.1.1.1.1.1" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p4.2.m2.1.1.1.1.1.1.2" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.2.cmml">a</mi><mi id="S2.SS1.p4.2.m2.1.1.1.1.1.1.3" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS1.p4.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p4.2.m2.1.1.2" xref="S2.SS1.p4.2.m2.1.1.2.cmml">∈</mo><msup id="S2.SS1.p4.2.m2.1.1.3" xref="S2.SS1.p4.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.2.m2.1.1.3.2" xref="S2.SS1.p4.2.m2.1.1.3.2.cmml">ℬ</mi><mo id="S2.SS1.p4.2.m2.1.1.3.3" xref="S2.SS1.p4.2.m2.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.2.m2.1b"><apply id="S2.SS1.p4.2.m2.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1"><in id="S2.SS1.p4.2.m2.1.1.2.cmml" xref="S2.SS1.p4.2.m2.1.1.2"></in><apply id="S2.SS1.p4.2.m2.1.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1.1"><times id="S2.SS1.p4.2.m2.1.1.1.2.cmml" xref="S2.SS1.p4.2.m2.1.1.1.2"></times><ci id="S2.SS1.p4.2.m2.1.1.1.3.cmml" xref="S2.SS1.p4.2.m2.1.1.1.3">𝜎</ci><apply id="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p4.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.2">𝑎</ci><ci id="S2.SS1.p4.2.m2.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S2.SS1.p4.2.m2.1.1.3.cmml" xref="S2.SS1.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.2.m2.1.1.3.1.cmml" xref="S2.SS1.p4.2.m2.1.1.3">superscript</csymbol><ci id="S2.SS1.p4.2.m2.1.1.3.2.cmml" xref="S2.SS1.p4.2.m2.1.1.3.2">ℬ</ci><times id="S2.SS1.p4.2.m2.1.1.3.3.cmml" xref="S2.SS1.p4.2.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">\sigma(a_{i})\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> for each of the letters <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><mrow id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml"><msub id="S2.SS1.p4.3.m3.1.1.2" xref="S2.SS1.p4.3.m3.1.1.2.cmml"><mi id="S2.SS1.p4.3.m3.1.1.2.2" xref="S2.SS1.p4.3.m3.1.1.2.2.cmml">a</mi><mi id="S2.SS1.p4.3.m3.1.1.2.3" xref="S2.SS1.p4.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p4.3.m3.1.1.1" xref="S2.SS1.p4.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.3.m3.1.1.3" xref="S2.SS1.p4.3.m3.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.1b"><apply id="S2.SS1.p4.3.m3.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1"><in id="S2.SS1.p4.3.m3.1.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1.1"></in><apply id="S2.SS1.p4.3.m3.1.1.2.cmml" xref="S2.SS1.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.3.m3.1.1.2.1.cmml" xref="S2.SS1.p4.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p4.3.m3.1.1.2.2.cmml" xref="S2.SS1.p4.3.m3.1.1.2.2">𝑎</ci><ci id="S2.SS1.p4.3.m3.1.1.2.3.cmml" xref="S2.SS1.p4.3.m3.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p4.3.m3.1.1.3.cmml" xref="S2.SS1.p4.3.m3.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math>. Conversely, each choice of such image words defines a monoid morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS1.p4.4.m4.1"><semantics id="S2.SS1.p4.4.m4.1a"><mi id="S2.SS1.p4.4.m4.1.1" xref="S2.SS1.p4.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.4.m4.1b"><ci id="S2.SS1.p4.4.m4.1.1.cmml" xref="S2.SS1.p4.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.4.m4.1d">italic_σ</annotation></semantics></math> as above. The monoid morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS1.p4.5.m5.1"><semantics id="S2.SS1.p4.5.m5.1a"><mi id="S2.SS1.p4.5.m5.1.1" xref="S2.SS1.p4.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.5.m5.1b"><ci id="S2.SS1.p4.5.m5.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.5.m5.1d">italic_σ</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S2.SS1.p4.10.2">non-erasing</span> if <math alttext="|\sigma(a_{i})|\geq 1" class="ltx_Math" display="inline" id="S2.SS1.p4.6.m6.1"><semantics id="S2.SS1.p4.6.m6.1a"><mrow id="S2.SS1.p4.6.m6.1.1" xref="S2.SS1.p4.6.m6.1.1.cmml"><mrow id="S2.SS1.p4.6.m6.1.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.2.cmml"><mo id="S2.SS1.p4.6.m6.1.1.1.1.2" stretchy="false" xref="S2.SS1.p4.6.m6.1.1.1.2.1.cmml">|</mo><mrow id="S2.SS1.p4.6.m6.1.1.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.1.1.cmml"><mi id="S2.SS1.p4.6.m6.1.1.1.1.1.3" xref="S2.SS1.p4.6.m6.1.1.1.1.1.3.cmml">σ</mi><mo id="S2.SS1.p4.6.m6.1.1.1.1.1.2" xref="S2.SS1.p4.6.m6.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.2" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.3" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p4.6.m6.1.1.1.1.3" stretchy="false" xref="S2.SS1.p4.6.m6.1.1.1.2.1.cmml">|</mo></mrow><mo id="S2.SS1.p4.6.m6.1.1.2" xref="S2.SS1.p4.6.m6.1.1.2.cmml">≥</mo><mn id="S2.SS1.p4.6.m6.1.1.3" xref="S2.SS1.p4.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.6.m6.1b"><apply id="S2.SS1.p4.6.m6.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1"><geq id="S2.SS1.p4.6.m6.1.1.2.cmml" xref="S2.SS1.p4.6.m6.1.1.2"></geq><apply id="S2.SS1.p4.6.m6.1.1.1.2.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1"><abs id="S2.SS1.p4.6.m6.1.1.1.2.1.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1.2"></abs><apply id="S2.SS1.p4.6.m6.1.1.1.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1.1"><times id="S2.SS1.p4.6.m6.1.1.1.1.1.2.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1.1.2"></times><ci id="S2.SS1.p4.6.m6.1.1.1.1.1.3.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1.1.3">𝜎</ci><apply id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p4.6.m6.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply><cn id="S2.SS1.p4.6.m6.1.1.3.cmml" type="integer" xref="S2.SS1.p4.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.6.m6.1c">|\sigma(a_{i})|\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.6.m6.1d">| italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) | ≥ 1</annotation></semantics></math> for each of the letters <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S2.SS1.p4.7.m7.1"><semantics id="S2.SS1.p4.7.m7.1a"><mrow id="S2.SS1.p4.7.m7.1.1" xref="S2.SS1.p4.7.m7.1.1.cmml"><msub id="S2.SS1.p4.7.m7.1.1.2" xref="S2.SS1.p4.7.m7.1.1.2.cmml"><mi id="S2.SS1.p4.7.m7.1.1.2.2" xref="S2.SS1.p4.7.m7.1.1.2.2.cmml">a</mi><mi id="S2.SS1.p4.7.m7.1.1.2.3" xref="S2.SS1.p4.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p4.7.m7.1.1.1" xref="S2.SS1.p4.7.m7.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.7.m7.1.1.3" xref="S2.SS1.p4.7.m7.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.7.m7.1b"><apply id="S2.SS1.p4.7.m7.1.1.cmml" xref="S2.SS1.p4.7.m7.1.1"><in id="S2.SS1.p4.7.m7.1.1.1.cmml" xref="S2.SS1.p4.7.m7.1.1.1"></in><apply id="S2.SS1.p4.7.m7.1.1.2.cmml" xref="S2.SS1.p4.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.7.m7.1.1.2.1.cmml" xref="S2.SS1.p4.7.m7.1.1.2">subscript</csymbol><ci id="S2.SS1.p4.7.m7.1.1.2.2.cmml" xref="S2.SS1.p4.7.m7.1.1.2.2">𝑎</ci><ci id="S2.SS1.p4.7.m7.1.1.2.3.cmml" xref="S2.SS1.p4.7.m7.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p4.7.m7.1.1.3.cmml" xref="S2.SS1.p4.7.m7.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.7.m7.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.7.m7.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math>. In this paper we will only consider non-erasing morphisms. Note that any non-erasing morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.SS1.p4.8.m8.1"><semantics id="S2.SS1.p4.8.m8.1a"><mrow id="S2.SS1.p4.8.m8.1.1" xref="S2.SS1.p4.8.m8.1.1.cmml"><mi id="S2.SS1.p4.8.m8.1.1.2" xref="S2.SS1.p4.8.m8.1.1.2.cmml">σ</mi><mo id="S2.SS1.p4.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p4.8.m8.1.1.1.cmml">:</mo><mrow id="S2.SS1.p4.8.m8.1.1.3" xref="S2.SS1.p4.8.m8.1.1.3.cmml"><msup id="S2.SS1.p4.8.m8.1.1.3.2" xref="S2.SS1.p4.8.m8.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.8.m8.1.1.3.2.2" xref="S2.SS1.p4.8.m8.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.SS1.p4.8.m8.1.1.3.2.3" xref="S2.SS1.p4.8.m8.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.SS1.p4.8.m8.1.1.3.1" stretchy="false" xref="S2.SS1.p4.8.m8.1.1.3.1.cmml">→</mo><msup id="S2.SS1.p4.8.m8.1.1.3.3" xref="S2.SS1.p4.8.m8.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.8.m8.1.1.3.3.2" xref="S2.SS1.p4.8.m8.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.SS1.p4.8.m8.1.1.3.3.3" xref="S2.SS1.p4.8.m8.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.8.m8.1b"><apply id="S2.SS1.p4.8.m8.1.1.cmml" xref="S2.SS1.p4.8.m8.1.1"><ci id="S2.SS1.p4.8.m8.1.1.1.cmml" xref="S2.SS1.p4.8.m8.1.1.1">:</ci><ci id="S2.SS1.p4.8.m8.1.1.2.cmml" xref="S2.SS1.p4.8.m8.1.1.2">𝜎</ci><apply id="S2.SS1.p4.8.m8.1.1.3.cmml" xref="S2.SS1.p4.8.m8.1.1.3"><ci id="S2.SS1.p4.8.m8.1.1.3.1.cmml" xref="S2.SS1.p4.8.m8.1.1.3.1">→</ci><apply id="S2.SS1.p4.8.m8.1.1.3.2.cmml" xref="S2.SS1.p4.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.8.m8.1.1.3.2.1.cmml" xref="S2.SS1.p4.8.m8.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p4.8.m8.1.1.3.2.2.cmml" xref="S2.SS1.p4.8.m8.1.1.3.2.2">𝒜</ci><times id="S2.SS1.p4.8.m8.1.1.3.2.3.cmml" xref="S2.SS1.p4.8.m8.1.1.3.2.3"></times></apply><apply id="S2.SS1.p4.8.m8.1.1.3.3.cmml" xref="S2.SS1.p4.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p4.8.m8.1.1.3.3.1.cmml" xref="S2.SS1.p4.8.m8.1.1.3.3">superscript</csymbol><ci id="S2.SS1.p4.8.m8.1.1.3.3.2.cmml" xref="S2.SS1.p4.8.m8.1.1.3.3.2">ℬ</ci><times id="S2.SS1.p4.8.m8.1.1.3.3.3.cmml" xref="S2.SS1.p4.8.m8.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.8.m8.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.8.m8.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is “finite-to-one”, i.e. for any element <math alttext="w\in\cal B^{*}" class="ltx_Math" display="inline" id="S2.SS1.p4.9.m9.1"><semantics id="S2.SS1.p4.9.m9.1a"><mrow id="S2.SS1.p4.9.m9.1.1" xref="S2.SS1.p4.9.m9.1.1.cmml"><mi id="S2.SS1.p4.9.m9.1.1.2" xref="S2.SS1.p4.9.m9.1.1.2.cmml">w</mi><mo id="S2.SS1.p4.9.m9.1.1.1" xref="S2.SS1.p4.9.m9.1.1.1.cmml">∈</mo><msup id="S2.SS1.p4.9.m9.1.1.3" xref="S2.SS1.p4.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.9.m9.1.1.3.2" xref="S2.SS1.p4.9.m9.1.1.3.2.cmml">ℬ</mi><mo id="S2.SS1.p4.9.m9.1.1.3.3" xref="S2.SS1.p4.9.m9.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.9.m9.1b"><apply id="S2.SS1.p4.9.m9.1.1.cmml" xref="S2.SS1.p4.9.m9.1.1"><in id="S2.SS1.p4.9.m9.1.1.1.cmml" xref="S2.SS1.p4.9.m9.1.1.1"></in><ci id="S2.SS1.p4.9.m9.1.1.2.cmml" xref="S2.SS1.p4.9.m9.1.1.2">𝑤</ci><apply id="S2.SS1.p4.9.m9.1.1.3.cmml" xref="S2.SS1.p4.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.9.m9.1.1.3.1.cmml" xref="S2.SS1.p4.9.m9.1.1.3">superscript</csymbol><ci id="S2.SS1.p4.9.m9.1.1.3.2.cmml" xref="S2.SS1.p4.9.m9.1.1.3.2">ℬ</ci><times id="S2.SS1.p4.9.m9.1.1.3.3.cmml" xref="S2.SS1.p4.9.m9.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.9.m9.1c">w\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.9.m9.1d">italic_w ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the preimage set <math alttext="\sigma^{-1}(w)" class="ltx_Math" display="inline" id="S2.SS1.p4.10.m10.1"><semantics id="S2.SS1.p4.10.m10.1a"><mrow id="S2.SS1.p4.10.m10.1.2" xref="S2.SS1.p4.10.m10.1.2.cmml"><msup id="S2.SS1.p4.10.m10.1.2.2" xref="S2.SS1.p4.10.m10.1.2.2.cmml"><mi id="S2.SS1.p4.10.m10.1.2.2.2" xref="S2.SS1.p4.10.m10.1.2.2.2.cmml">σ</mi><mrow id="S2.SS1.p4.10.m10.1.2.2.3" xref="S2.SS1.p4.10.m10.1.2.2.3.cmml"><mo id="S2.SS1.p4.10.m10.1.2.2.3a" xref="S2.SS1.p4.10.m10.1.2.2.3.cmml">−</mo><mn id="S2.SS1.p4.10.m10.1.2.2.3.2" xref="S2.SS1.p4.10.m10.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS1.p4.10.m10.1.2.1" xref="S2.SS1.p4.10.m10.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p4.10.m10.1.2.3.2" xref="S2.SS1.p4.10.m10.1.2.cmml"><mo id="S2.SS1.p4.10.m10.1.2.3.2.1" stretchy="false" xref="S2.SS1.p4.10.m10.1.2.cmml">(</mo><mi id="S2.SS1.p4.10.m10.1.1" xref="S2.SS1.p4.10.m10.1.1.cmml">w</mi><mo id="S2.SS1.p4.10.m10.1.2.3.2.2" stretchy="false" xref="S2.SS1.p4.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.10.m10.1b"><apply id="S2.SS1.p4.10.m10.1.2.cmml" xref="S2.SS1.p4.10.m10.1.2"><times id="S2.SS1.p4.10.m10.1.2.1.cmml" xref="S2.SS1.p4.10.m10.1.2.1"></times><apply id="S2.SS1.p4.10.m10.1.2.2.cmml" xref="S2.SS1.p4.10.m10.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p4.10.m10.1.2.2.1.cmml" xref="S2.SS1.p4.10.m10.1.2.2">superscript</csymbol><ci id="S2.SS1.p4.10.m10.1.2.2.2.cmml" xref="S2.SS1.p4.10.m10.1.2.2.2">𝜎</ci><apply id="S2.SS1.p4.10.m10.1.2.2.3.cmml" xref="S2.SS1.p4.10.m10.1.2.2.3"><minus id="S2.SS1.p4.10.m10.1.2.2.3.1.cmml" xref="S2.SS1.p4.10.m10.1.2.2.3"></minus><cn id="S2.SS1.p4.10.m10.1.2.2.3.2.cmml" type="integer" xref="S2.SS1.p4.10.m10.1.2.2.3.2">1</cn></apply></apply><ci id="S2.SS1.p4.10.m10.1.1.cmml" xref="S2.SS1.p4.10.m10.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.10.m10.1c">\sigma^{-1}(w)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.10.m10.1d">italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w )</annotation></semantics></math> is finite.</p> </div> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.1">Every monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.1"><semantics id="S2.SS1.p5.1.m1.1a"><mrow id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml"><mi id="S2.SS1.p5.1.m1.1.1.2" xref="S2.SS1.p5.1.m1.1.1.2.cmml">σ</mi><mo id="S2.SS1.p5.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p5.1.m1.1.1.1.cmml">:</mo><mrow id="S2.SS1.p5.1.m1.1.1.3" xref="S2.SS1.p5.1.m1.1.1.3.cmml"><msup id="S2.SS1.p5.1.m1.1.1.3.2" xref="S2.SS1.p5.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p5.1.m1.1.1.3.2.2" xref="S2.SS1.p5.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.SS1.p5.1.m1.1.1.3.2.3" xref="S2.SS1.p5.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.SS1.p5.1.m1.1.1.3.1" stretchy="false" xref="S2.SS1.p5.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.SS1.p5.1.m1.1.1.3.3" xref="S2.SS1.p5.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p5.1.m1.1.1.3.3.2" xref="S2.SS1.p5.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.SS1.p5.1.m1.1.1.3.3.3" xref="S2.SS1.p5.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.1b"><apply id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1"><ci id="S2.SS1.p5.1.m1.1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1.1">:</ci><ci id="S2.SS1.p5.1.m1.1.1.2.cmml" xref="S2.SS1.p5.1.m1.1.1.2">𝜎</ci><apply id="S2.SS1.p5.1.m1.1.1.3.cmml" xref="S2.SS1.p5.1.m1.1.1.3"><ci id="S2.SS1.p5.1.m1.1.1.3.1.cmml" xref="S2.SS1.p5.1.m1.1.1.3.1">→</ci><apply id="S2.SS1.p5.1.m1.1.1.3.2.cmml" xref="S2.SS1.p5.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p5.1.m1.1.1.3.2.1.cmml" xref="S2.SS1.p5.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p5.1.m1.1.1.3.2.2.cmml" xref="S2.SS1.p5.1.m1.1.1.3.2.2">𝒜</ci><times id="S2.SS1.p5.1.m1.1.1.3.2.3.cmml" xref="S2.SS1.p5.1.m1.1.1.3.2.3"></times></apply><apply id="S2.SS1.p5.1.m1.1.1.3.3.cmml" xref="S2.SS1.p5.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p5.1.m1.1.1.3.3.1.cmml" xref="S2.SS1.p5.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.SS1.p5.1.m1.1.1.3.3.2.cmml" xref="S2.SS1.p5.1.m1.1.1.3.3.2">ℬ</ci><times id="S2.SS1.p5.1.m1.1.1.3.3.3.cmml" xref="S2.SS1.p5.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> defines an <span class="ltx_text ltx_font_italic" id="S2.SS1.p5.1.1">incidence matrix</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.1)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="M(\sigma)=(|\sigma(a_{j})|_{b_{i}})_{b_{i}\in\cal B,a_{j}\in\cal A}" class="ltx_Math" display="block" id="S2.E1.m1.4"><semantics id="S2.E1.m1.4a"><mrow id="S2.E1.m1.4.4" xref="S2.E1.m1.4.4.cmml"><mrow id="S2.E1.m1.4.4.3" xref="S2.E1.m1.4.4.3.cmml"><mi id="S2.E1.m1.4.4.3.2" xref="S2.E1.m1.4.4.3.2.cmml">M</mi><mo id="S2.E1.m1.4.4.3.1" xref="S2.E1.m1.4.4.3.1.cmml">⁢</mo><mrow id="S2.E1.m1.4.4.3.3.2" xref="S2.E1.m1.4.4.3.cmml"><mo id="S2.E1.m1.4.4.3.3.2.1" stretchy="false" xref="S2.E1.m1.4.4.3.cmml">(</mo><mi id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml">σ</mi><mo id="S2.E1.m1.4.4.3.3.2.2" stretchy="false" xref="S2.E1.m1.4.4.3.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.4.4.2" xref="S2.E1.m1.4.4.2.cmml">=</mo><msub id="S2.E1.m1.4.4.1" xref="S2.E1.m1.4.4.1.cmml"><mrow id="S2.E1.m1.4.4.1.1.1" xref="S2.E1.m1.4.4.1.1.1.1.cmml"><mo id="S2.E1.m1.4.4.1.1.1.2" stretchy="false" xref="S2.E1.m1.4.4.1.1.1.1.cmml">(</mo><msub id="S2.E1.m1.4.4.1.1.1.1" xref="S2.E1.m1.4.4.1.1.1.1.cmml"><mrow id="S2.E1.m1.4.4.1.1.1.1.1.1" xref="S2.E1.m1.4.4.1.1.1.1.1.2.cmml"><mo id="S2.E1.m1.4.4.1.1.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.4.4.1.1.1.1.1.2.1.cmml">|</mo><mrow id="S2.E1.m1.4.4.1.1.1.1.1.1.1" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.4.4.1.1.1.1.1.1.1.3" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.3.cmml">σ</mi><mo id="S2.E1.m1.4.4.1.1.1.1.1.1.1.2" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E1.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.4.4.1.1.1.1.1.1.3" stretchy="false" xref="S2.E1.m1.4.4.1.1.1.1.1.2.1.cmml">|</mo></mrow><msub id="S2.E1.m1.4.4.1.1.1.1.3" xref="S2.E1.m1.4.4.1.1.1.1.3.cmml"><mi id="S2.E1.m1.4.4.1.1.1.1.3.2" xref="S2.E1.m1.4.4.1.1.1.1.3.2.cmml">b</mi><mi id="S2.E1.m1.4.4.1.1.1.1.3.3" xref="S2.E1.m1.4.4.1.1.1.1.3.3.cmml">i</mi></msub></msub><mo id="S2.E1.m1.4.4.1.1.1.3" stretchy="false" xref="S2.E1.m1.4.4.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.E1.m1.2.2.2.2" xref="S2.E1.m1.2.2.2.3.cmml"><mrow id="S2.E1.m1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.cmml"><msub id="S2.E1.m1.1.1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.1.2.cmml"><mi id="S2.E1.m1.1.1.1.1.1.2.2" xref="S2.E1.m1.1.1.1.1.1.2.2.cmml">b</mi><mi id="S2.E1.m1.1.1.1.1.1.2.3" xref="S2.E1.m1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.E1.m1.1.1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.1.1.1.1.1.3" xref="S2.E1.m1.1.1.1.1.1.3.cmml">ℬ</mi></mrow><mo id="S2.E1.m1.2.2.2.2.3" xref="S2.E1.m1.2.2.2.3a.cmml">,</mo><mrow id="S2.E1.m1.2.2.2.2.2" xref="S2.E1.m1.2.2.2.2.2.cmml"><msub id="S2.E1.m1.2.2.2.2.2.2" xref="S2.E1.m1.2.2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.2.2.2.2.2.2.2" xref="S2.E1.m1.2.2.2.2.2.2.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.2.2.2.2.2.2.3" xref="S2.E1.m1.2.2.2.2.2.2.3.cmml">𝒿</mi></msub><mo id="S2.E1.m1.2.2.2.2.2.1" xref="S2.E1.m1.2.2.2.2.2.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.2.2.2.2.2.3" xref="S2.E1.m1.2.2.2.2.2.3.cmml">𝒜</mi></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.E1.m1.4b"><apply id="S2.E1.m1.4.4.cmml" xref="S2.E1.m1.4.4"><eq id="S2.E1.m1.4.4.2.cmml" xref="S2.E1.m1.4.4.2"></eq><apply id="S2.E1.m1.4.4.3.cmml" xref="S2.E1.m1.4.4.3"><times id="S2.E1.m1.4.4.3.1.cmml" xref="S2.E1.m1.4.4.3.1"></times><ci id="S2.E1.m1.4.4.3.2.cmml" xref="S2.E1.m1.4.4.3.2">𝑀</ci><ci id="S2.E1.m1.3.3.cmml" xref="S2.E1.m1.3.3">𝜎</ci></apply><apply id="S2.E1.m1.4.4.1.cmml" xref="S2.E1.m1.4.4.1"><csymbol cd="ambiguous" id="S2.E1.m1.4.4.1.2.cmml" xref="S2.E1.m1.4.4.1">subscript</csymbol><apply id="S2.E1.m1.4.4.1.1.1.1.cmml" xref="S2.E1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S2.E1.m1.4.4.1.1.1.1.2.cmml" xref="S2.E1.m1.4.4.1.1.1">subscript</csymbol><apply id="S2.E1.m1.4.4.1.1.1.1.1.2.cmml" xref="S2.E1.m1.4.4.1.1.1.1.1.1"><abs 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xref="S2.E1.m1.4.4.1.1.1.1.3.2">𝑏</ci><ci id="S2.E1.m1.4.4.1.1.1.1.3.3.cmml" xref="S2.E1.m1.4.4.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="S2.E1.m1.2.2.2.3.cmml" xref="S2.E1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S2.E1.m1.2.2.2.3a.cmml" xref="S2.E1.m1.2.2.2.2.3">formulae-sequence</csymbol><apply id="S2.E1.m1.1.1.1.1.1.cmml" xref="S2.E1.m1.1.1.1.1.1"><in id="S2.E1.m1.1.1.1.1.1.1.cmml" xref="S2.E1.m1.1.1.1.1.1.1"></in><apply id="S2.E1.m1.1.1.1.1.1.2.cmml" xref="S2.E1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E1.m1.1.1.1.1.1.2.1.cmml" xref="S2.E1.m1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.E1.m1.1.1.1.1.1.2.2.cmml" xref="S2.E1.m1.1.1.1.1.1.2.2">𝑏</ci><ci id="S2.E1.m1.1.1.1.1.1.2.3.cmml" xref="S2.E1.m1.1.1.1.1.1.2.3">𝑖</ci></apply><ci id="S2.E1.m1.1.1.1.1.1.3.cmml" xref="S2.E1.m1.1.1.1.1.1.3">ℬ</ci></apply><apply id="S2.E1.m1.2.2.2.2.2.cmml" xref="S2.E1.m1.2.2.2.2.2"><in id="S2.E1.m1.2.2.2.2.2.1.cmml" xref="S2.E1.m1.2.2.2.2.2.1"></in><apply id="S2.E1.m1.2.2.2.2.2.2.cmml" xref="S2.E1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.E1.m1.2.2.2.2.2.2.1.cmml" xref="S2.E1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="S2.E1.m1.2.2.2.2.2.2.2.cmml" xref="S2.E1.m1.2.2.2.2.2.2.2">𝒶</ci><ci id="S2.E1.m1.2.2.2.2.2.2.3.cmml" xref="S2.E1.m1.2.2.2.2.2.2.3">𝒿</ci></apply><ci id="S2.E1.m1.2.2.2.2.2.3.cmml" xref="S2.E1.m1.2.2.2.2.2.3">𝒜</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E1.m1.4c">M(\sigma)=(|\sigma(a_{j})|_{b_{i}})_{b_{i}\in\cal B,a_{j}\in\cal A}</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.4d">italic_M ( italic_σ ) = ( | italic_σ ( italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_B , caligraphic_a start_POSTSUBSCRIPT caligraphic_j end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p5.7">where <math alttext="|\sigma(a_{j})|_{b_{i}}" class="ltx_Math" display="inline" id="S2.SS1.p5.2.m1.1"><semantics id="S2.SS1.p5.2.m1.1a"><msub id="S2.SS1.p5.2.m1.1.1" xref="S2.SS1.p5.2.m1.1.1.cmml"><mrow id="S2.SS1.p5.2.m1.1.1.1.1" xref="S2.SS1.p5.2.m1.1.1.1.2.cmml"><mo id="S2.SS1.p5.2.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p5.2.m1.1.1.1.2.1.cmml">|</mo><mrow id="S2.SS1.p5.2.m1.1.1.1.1.1" xref="S2.SS1.p5.2.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p5.2.m1.1.1.1.1.1.3" xref="S2.SS1.p5.2.m1.1.1.1.1.1.3.cmml">σ</mi><mo id="S2.SS1.p5.2.m1.1.1.1.1.1.2" xref="S2.SS1.p5.2.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.2" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.3" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p5.2.m1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p5.2.m1.1.1.1.2.1.cmml">|</mo></mrow><msub id="S2.SS1.p5.2.m1.1.1.3" xref="S2.SS1.p5.2.m1.1.1.3.cmml"><mi id="S2.SS1.p5.2.m1.1.1.3.2" xref="S2.SS1.p5.2.m1.1.1.3.2.cmml">b</mi><mi id="S2.SS1.p5.2.m1.1.1.3.3" xref="S2.SS1.p5.2.m1.1.1.3.3.cmml">i</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m1.1b"><apply id="S2.SS1.p5.2.m1.1.1.cmml" xref="S2.SS1.p5.2.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m1.1.1.2.cmml" xref="S2.SS1.p5.2.m1.1.1">subscript</csymbol><apply id="S2.SS1.p5.2.m1.1.1.1.2.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1"><abs id="S2.SS1.p5.2.m1.1.1.1.2.1.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1.2"></abs><apply id="S2.SS1.p5.2.m1.1.1.1.1.1.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1.1"><times id="S2.SS1.p5.2.m1.1.1.1.1.1.2.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1.1.2"></times><ci id="S2.SS1.p5.2.m1.1.1.1.1.1.3.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1.1.3">𝜎</ci><apply id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p5.2.m1.1.1.1.1.1.1.1.1.3">𝑗</ci></apply></apply></apply><apply id="S2.SS1.p5.2.m1.1.1.3.cmml" xref="S2.SS1.p5.2.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m1.1.1.3.1.cmml" xref="S2.SS1.p5.2.m1.1.1.3">subscript</csymbol><ci id="S2.SS1.p5.2.m1.1.1.3.2.cmml" xref="S2.SS1.p5.2.m1.1.1.3.2">𝑏</ci><ci id="S2.SS1.p5.2.m1.1.1.3.3.cmml" xref="S2.SS1.p5.2.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m1.1c">|\sigma(a_{j})|_{b_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m1.1d">| italic_σ ( italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> denotes the number of occurrences of the letter <math alttext="b_{i}\in\cal B" class="ltx_Math" display="inline" id="S2.SS1.p5.3.m2.1"><semantics id="S2.SS1.p5.3.m2.1a"><mrow id="S2.SS1.p5.3.m2.1.1" xref="S2.SS1.p5.3.m2.1.1.cmml"><msub id="S2.SS1.p5.3.m2.1.1.2" xref="S2.SS1.p5.3.m2.1.1.2.cmml"><mi id="S2.SS1.p5.3.m2.1.1.2.2" xref="S2.SS1.p5.3.m2.1.1.2.2.cmml">b</mi><mi id="S2.SS1.p5.3.m2.1.1.2.3" xref="S2.SS1.p5.3.m2.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p5.3.m2.1.1.1" xref="S2.SS1.p5.3.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p5.3.m2.1.1.3" xref="S2.SS1.p5.3.m2.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.3.m2.1b"><apply id="S2.SS1.p5.3.m2.1.1.cmml" xref="S2.SS1.p5.3.m2.1.1"><in id="S2.SS1.p5.3.m2.1.1.1.cmml" xref="S2.SS1.p5.3.m2.1.1.1"></in><apply id="S2.SS1.p5.3.m2.1.1.2.cmml" xref="S2.SS1.p5.3.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p5.3.m2.1.1.2.1.cmml" xref="S2.SS1.p5.3.m2.1.1.2">subscript</csymbol><ci id="S2.SS1.p5.3.m2.1.1.2.2.cmml" xref="S2.SS1.p5.3.m2.1.1.2.2">𝑏</ci><ci id="S2.SS1.p5.3.m2.1.1.2.3.cmml" xref="S2.SS1.p5.3.m2.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p5.3.m2.1.1.3.cmml" xref="S2.SS1.p5.3.m2.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m2.1c">b_{i}\in\cal B</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m2.1d">italic_b start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_B</annotation></semantics></math> in the <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS1.p5.4.m3.1"><semantics id="S2.SS1.p5.4.m3.1a"><mi id="S2.SS1.p5.4.m3.1.1" xref="S2.SS1.p5.4.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.4.m3.1b"><ci id="S2.SS1.p5.4.m3.1.1.cmml" xref="S2.SS1.p5.4.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.4.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.4.m3.1d">italic_σ</annotation></semantics></math>-image of any <math alttext="a_{j}\in\cal A" class="ltx_Math" display="inline" id="S2.SS1.p5.5.m4.1"><semantics id="S2.SS1.p5.5.m4.1a"><mrow id="S2.SS1.p5.5.m4.1.1" xref="S2.SS1.p5.5.m4.1.1.cmml"><msub id="S2.SS1.p5.5.m4.1.1.2" xref="S2.SS1.p5.5.m4.1.1.2.cmml"><mi id="S2.SS1.p5.5.m4.1.1.2.2" xref="S2.SS1.p5.5.m4.1.1.2.2.cmml">a</mi><mi id="S2.SS1.p5.5.m4.1.1.2.3" xref="S2.SS1.p5.5.m4.1.1.2.3.cmml">j</mi></msub><mo id="S2.SS1.p5.5.m4.1.1.1" xref="S2.SS1.p5.5.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p5.5.m4.1.1.3" xref="S2.SS1.p5.5.m4.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.5.m4.1b"><apply id="S2.SS1.p5.5.m4.1.1.cmml" xref="S2.SS1.p5.5.m4.1.1"><in id="S2.SS1.p5.5.m4.1.1.1.cmml" xref="S2.SS1.p5.5.m4.1.1.1"></in><apply id="S2.SS1.p5.5.m4.1.1.2.cmml" xref="S2.SS1.p5.5.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p5.5.m4.1.1.2.1.cmml" xref="S2.SS1.p5.5.m4.1.1.2">subscript</csymbol><ci id="S2.SS1.p5.5.m4.1.1.2.2.cmml" xref="S2.SS1.p5.5.m4.1.1.2.2">𝑎</ci><ci id="S2.SS1.p5.5.m4.1.1.2.3.cmml" xref="S2.SS1.p5.5.m4.1.1.2.3">𝑗</ci></apply><ci id="S2.SS1.p5.5.m4.1.1.3.cmml" xref="S2.SS1.p5.5.m4.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.5.m4.1c">a_{j}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.5.m4.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math>. One easily verifies the formula <math alttext="M(\sigma)=M(\sigma_{2})\cdot M(\sigma_{1})" class="ltx_Math" display="inline" id="S2.SS1.p5.6.m5.3"><semantics id="S2.SS1.p5.6.m5.3a"><mrow id="S2.SS1.p5.6.m5.3.3" xref="S2.SS1.p5.6.m5.3.3.cmml"><mrow id="S2.SS1.p5.6.m5.3.3.4" xref="S2.SS1.p5.6.m5.3.3.4.cmml"><mi id="S2.SS1.p5.6.m5.3.3.4.2" xref="S2.SS1.p5.6.m5.3.3.4.2.cmml">M</mi><mo id="S2.SS1.p5.6.m5.3.3.4.1" xref="S2.SS1.p5.6.m5.3.3.4.1.cmml">⁢</mo><mrow id="S2.SS1.p5.6.m5.3.3.4.3.2" xref="S2.SS1.p5.6.m5.3.3.4.cmml"><mo id="S2.SS1.p5.6.m5.3.3.4.3.2.1" stretchy="false" xref="S2.SS1.p5.6.m5.3.3.4.cmml">(</mo><mi id="S2.SS1.p5.6.m5.1.1" xref="S2.SS1.p5.6.m5.1.1.cmml">σ</mi><mo id="S2.SS1.p5.6.m5.3.3.4.3.2.2" stretchy="false" xref="S2.SS1.p5.6.m5.3.3.4.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p5.6.m5.3.3.3" xref="S2.SS1.p5.6.m5.3.3.3.cmml">=</mo><mrow id="S2.SS1.p5.6.m5.3.3.2" xref="S2.SS1.p5.6.m5.3.3.2.cmml"><mrow id="S2.SS1.p5.6.m5.2.2.1.1" xref="S2.SS1.p5.6.m5.2.2.1.1.cmml"><mrow id="S2.SS1.p5.6.m5.2.2.1.1.1" xref="S2.SS1.p5.6.m5.2.2.1.1.1.cmml"><mi id="S2.SS1.p5.6.m5.2.2.1.1.1.3" xref="S2.SS1.p5.6.m5.2.2.1.1.1.3.cmml">M</mi><mo id="S2.SS1.p5.6.m5.2.2.1.1.1.2" xref="S2.SS1.p5.6.m5.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p5.6.m5.2.2.1.1.1.1.1" xref="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1" xref="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.2" xref="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.2.cmml">σ</mi><mn id="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.3" xref="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p5.6.m5.2.2.1.1.2" rspace="0.222em" xref="S2.SS1.p5.6.m5.2.2.1.1.2.cmml">⋅</mo><mi id="S2.SS1.p5.6.m5.2.2.1.1.3" xref="S2.SS1.p5.6.m5.2.2.1.1.3.cmml">M</mi></mrow><mo id="S2.SS1.p5.6.m5.3.3.2.3" xref="S2.SS1.p5.6.m5.3.3.2.3.cmml">⁢</mo><mrow id="S2.SS1.p5.6.m5.3.3.2.2.1" xref="S2.SS1.p5.6.m5.3.3.2.2.1.1.cmml"><mo id="S2.SS1.p5.6.m5.3.3.2.2.1.2" stretchy="false" xref="S2.SS1.p5.6.m5.3.3.2.2.1.1.cmml">(</mo><msub id="S2.SS1.p5.6.m5.3.3.2.2.1.1" xref="S2.SS1.p5.6.m5.3.3.2.2.1.1.cmml"><mi id="S2.SS1.p5.6.m5.3.3.2.2.1.1.2" xref="S2.SS1.p5.6.m5.3.3.2.2.1.1.2.cmml">σ</mi><mn id="S2.SS1.p5.6.m5.3.3.2.2.1.1.3" xref="S2.SS1.p5.6.m5.3.3.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p5.6.m5.3.3.2.2.1.3" stretchy="false" xref="S2.SS1.p5.6.m5.3.3.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.6.m5.3b"><apply id="S2.SS1.p5.6.m5.3.3.cmml" xref="S2.SS1.p5.6.m5.3.3"><eq id="S2.SS1.p5.6.m5.3.3.3.cmml" xref="S2.SS1.p5.6.m5.3.3.3"></eq><apply id="S2.SS1.p5.6.m5.3.3.4.cmml" xref="S2.SS1.p5.6.m5.3.3.4"><times id="S2.SS1.p5.6.m5.3.3.4.1.cmml" 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xref="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.2">𝜎</ci><cn id="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p5.6.m5.2.2.1.1.1.1.1.1.3">2</cn></apply></apply><ci id="S2.SS1.p5.6.m5.2.2.1.1.3.cmml" xref="S2.SS1.p5.6.m5.2.2.1.1.3">𝑀</ci></apply><apply id="S2.SS1.p5.6.m5.3.3.2.2.1.1.cmml" xref="S2.SS1.p5.6.m5.3.3.2.2.1"><csymbol cd="ambiguous" id="S2.SS1.p5.6.m5.3.3.2.2.1.1.1.cmml" xref="S2.SS1.p5.6.m5.3.3.2.2.1">subscript</csymbol><ci id="S2.SS1.p5.6.m5.3.3.2.2.1.1.2.cmml" xref="S2.SS1.p5.6.m5.3.3.2.2.1.1.2">𝜎</ci><cn id="S2.SS1.p5.6.m5.3.3.2.2.1.1.3.cmml" type="integer" xref="S2.SS1.p5.6.m5.3.3.2.2.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.6.m5.3c">M(\sigma)=M(\sigma_{2})\cdot M(\sigma_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.6.m5.3d">italic_M ( italic_σ ) = italic_M ( italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ⋅ italic_M ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> for any composition of monoid morphisms <math alttext="\sigma=\sigma_{2}\circ\sigma_{1}" class="ltx_Math" display="inline" id="S2.SS1.p5.7.m6.1"><semantics id="S2.SS1.p5.7.m6.1a"><mrow id="S2.SS1.p5.7.m6.1.1" xref="S2.SS1.p5.7.m6.1.1.cmml"><mi id="S2.SS1.p5.7.m6.1.1.2" xref="S2.SS1.p5.7.m6.1.1.2.cmml">σ</mi><mo id="S2.SS1.p5.7.m6.1.1.1" xref="S2.SS1.p5.7.m6.1.1.1.cmml">=</mo><mrow id="S2.SS1.p5.7.m6.1.1.3" xref="S2.SS1.p5.7.m6.1.1.3.cmml"><msub id="S2.SS1.p5.7.m6.1.1.3.2" xref="S2.SS1.p5.7.m6.1.1.3.2.cmml"><mi id="S2.SS1.p5.7.m6.1.1.3.2.2" xref="S2.SS1.p5.7.m6.1.1.3.2.2.cmml">σ</mi><mn id="S2.SS1.p5.7.m6.1.1.3.2.3" xref="S2.SS1.p5.7.m6.1.1.3.2.3.cmml">2</mn></msub><mo id="S2.SS1.p5.7.m6.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p5.7.m6.1.1.3.1.cmml">∘</mo><msub id="S2.SS1.p5.7.m6.1.1.3.3" xref="S2.SS1.p5.7.m6.1.1.3.3.cmml"><mi id="S2.SS1.p5.7.m6.1.1.3.3.2" xref="S2.SS1.p5.7.m6.1.1.3.3.2.cmml">σ</mi><mn id="S2.SS1.p5.7.m6.1.1.3.3.3" xref="S2.SS1.p5.7.m6.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.7.m6.1b"><apply id="S2.SS1.p5.7.m6.1.1.cmml" xref="S2.SS1.p5.7.m6.1.1"><eq id="S2.SS1.p5.7.m6.1.1.1.cmml" xref="S2.SS1.p5.7.m6.1.1.1"></eq><ci id="S2.SS1.p5.7.m6.1.1.2.cmml" xref="S2.SS1.p5.7.m6.1.1.2">𝜎</ci><apply id="S2.SS1.p5.7.m6.1.1.3.cmml" xref="S2.SS1.p5.7.m6.1.1.3"><compose id="S2.SS1.p5.7.m6.1.1.3.1.cmml" xref="S2.SS1.p5.7.m6.1.1.3.1"></compose><apply id="S2.SS1.p5.7.m6.1.1.3.2.cmml" xref="S2.SS1.p5.7.m6.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p5.7.m6.1.1.3.2.1.cmml" xref="S2.SS1.p5.7.m6.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p5.7.m6.1.1.3.2.2.cmml" xref="S2.SS1.p5.7.m6.1.1.3.2.2">𝜎</ci><cn id="S2.SS1.p5.7.m6.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.p5.7.m6.1.1.3.2.3">2</cn></apply><apply id="S2.SS1.p5.7.m6.1.1.3.3.cmml" xref="S2.SS1.p5.7.m6.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p5.7.m6.1.1.3.3.1.cmml" xref="S2.SS1.p5.7.m6.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p5.7.m6.1.1.3.3.2.cmml" xref="S2.SS1.p5.7.m6.1.1.3.3.2">𝜎</ci><cn id="S2.SS1.p5.7.m6.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.p5.7.m6.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.7.m6.1c">\sigma=\sigma_{2}\circ\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.7.m6.1d">italic_σ = italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∘ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p6"> <p class="ltx_p" id="S2.SS1.p6.13">To any alphabet <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.SS1.p6.1.m1.1"><semantics id="S2.SS1.p6.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p6.1.m1.1.1" xref="S2.SS1.p6.1.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.1.m1.1b"><ci id="S2.SS1.p6.1.m1.1.1.cmml" xref="S2.SS1.p6.1.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.1.m1.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.1.m1.1d">caligraphic_A</annotation></semantics></math> there is canonically associated the shift space <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p6.2.m2.1"><semantics id="S2.SS1.p6.2.m2.1a"><msup id="S2.SS1.p6.2.m2.1.1" xref="S2.SS1.p6.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p6.2.m2.1.1.2" xref="S2.SS1.p6.2.m2.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p6.2.m2.1.1.3" xref="S2.SS1.p6.2.m2.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.2.m2.1b"><apply id="S2.SS1.p6.2.m2.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.2.m2.1.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1">superscript</csymbol><ci id="S2.SS1.p6.2.m2.1.1.2.cmml" xref="S2.SS1.p6.2.m2.1.1.2">𝒜</ci><ci id="S2.SS1.p6.2.m2.1.1.3.cmml" xref="S2.SS1.p6.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.2.m2.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.2.m2.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Its elements are written as biinfinite words <math alttext="{\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots" class="ltx_Math" display="inline" id="S2.SS1.p6.3.m3.1"><semantics id="S2.SS1.p6.3.m3.1a"><mrow id="S2.SS1.p6.3.m3.1.1" xref="S2.SS1.p6.3.m3.1.1.cmml"><mi id="S2.SS1.p6.3.m3.1.1.2" xref="S2.SS1.p6.3.m3.1.1.2.cmml">𝐱</mi><mo id="S2.SS1.p6.3.m3.1.1.1" xref="S2.SS1.p6.3.m3.1.1.1.cmml">=</mo><mrow id="S2.SS1.p6.3.m3.1.1.3" xref="S2.SS1.p6.3.m3.1.1.3.cmml"><mi id="S2.SS1.p6.3.m3.1.1.3.2" mathvariant="normal" xref="S2.SS1.p6.3.m3.1.1.3.2.cmml">…</mi><mo id="S2.SS1.p6.3.m3.1.1.3.1" xref="S2.SS1.p6.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p6.3.m3.1.1.3.3" xref="S2.SS1.p6.3.m3.1.1.3.3.cmml"><mi id="S2.SS1.p6.3.m3.1.1.3.3.2" xref="S2.SS1.p6.3.m3.1.1.3.3.2.cmml">x</mi><mrow id="S2.SS1.p6.3.m3.1.1.3.3.3" xref="S2.SS1.p6.3.m3.1.1.3.3.3.cmml"><mi id="S2.SS1.p6.3.m3.1.1.3.3.3.2" xref="S2.SS1.p6.3.m3.1.1.3.3.3.2.cmml">i</mi><mo id="S2.SS1.p6.3.m3.1.1.3.3.3.1" xref="S2.SS1.p6.3.m3.1.1.3.3.3.1.cmml">−</mo><mn id="S2.SS1.p6.3.m3.1.1.3.3.3.3" xref="S2.SS1.p6.3.m3.1.1.3.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p6.3.m3.1.1.3.1a" xref="S2.SS1.p6.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p6.3.m3.1.1.3.4" xref="S2.SS1.p6.3.m3.1.1.3.4.cmml"><mi id="S2.SS1.p6.3.m3.1.1.3.4.2" xref="S2.SS1.p6.3.m3.1.1.3.4.2.cmml">x</mi><mi id="S2.SS1.p6.3.m3.1.1.3.4.3" xref="S2.SS1.p6.3.m3.1.1.3.4.3.cmml">i</mi></msub><mo id="S2.SS1.p6.3.m3.1.1.3.1b" xref="S2.SS1.p6.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p6.3.m3.1.1.3.5" xref="S2.SS1.p6.3.m3.1.1.3.5.cmml"><mi id="S2.SS1.p6.3.m3.1.1.3.5.2" xref="S2.SS1.p6.3.m3.1.1.3.5.2.cmml">x</mi><mrow id="S2.SS1.p6.3.m3.1.1.3.5.3" xref="S2.SS1.p6.3.m3.1.1.3.5.3.cmml"><mi id="S2.SS1.p6.3.m3.1.1.3.5.3.2" xref="S2.SS1.p6.3.m3.1.1.3.5.3.2.cmml">i</mi><mo id="S2.SS1.p6.3.m3.1.1.3.5.3.1" xref="S2.SS1.p6.3.m3.1.1.3.5.3.1.cmml">+</mo><mn id="S2.SS1.p6.3.m3.1.1.3.5.3.3" xref="S2.SS1.p6.3.m3.1.1.3.5.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p6.3.m3.1.1.3.1c" xref="S2.SS1.p6.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p6.3.m3.1.1.3.6" mathvariant="normal" xref="S2.SS1.p6.3.m3.1.1.3.6.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.3.m3.1b"><apply id="S2.SS1.p6.3.m3.1.1.cmml" xref="S2.SS1.p6.3.m3.1.1"><eq id="S2.SS1.p6.3.m3.1.1.1.cmml" xref="S2.SS1.p6.3.m3.1.1.1"></eq><ci id="S2.SS1.p6.3.m3.1.1.2.cmml" xref="S2.SS1.p6.3.m3.1.1.2">𝐱</ci><apply id="S2.SS1.p6.3.m3.1.1.3.cmml" xref="S2.SS1.p6.3.m3.1.1.3"><times id="S2.SS1.p6.3.m3.1.1.3.1.cmml" xref="S2.SS1.p6.3.m3.1.1.3.1"></times><ci id="S2.SS1.p6.3.m3.1.1.3.2.cmml" xref="S2.SS1.p6.3.m3.1.1.3.2">…</ci><apply id="S2.SS1.p6.3.m3.1.1.3.3.cmml" xref="S2.SS1.p6.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p6.3.m3.1.1.3.3.1.cmml" xref="S2.SS1.p6.3.m3.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p6.3.m3.1.1.3.3.2.cmml" xref="S2.SS1.p6.3.m3.1.1.3.3.2">𝑥</ci><apply id="S2.SS1.p6.3.m3.1.1.3.3.3.cmml" xref="S2.SS1.p6.3.m3.1.1.3.3.3"><minus id="S2.SS1.p6.3.m3.1.1.3.3.3.1.cmml" xref="S2.SS1.p6.3.m3.1.1.3.3.3.1"></minus><ci id="S2.SS1.p6.3.m3.1.1.3.3.3.2.cmml" xref="S2.SS1.p6.3.m3.1.1.3.3.3.2">𝑖</ci><cn id="S2.SS1.p6.3.m3.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p6.3.m3.1.1.3.3.3.3">1</cn></apply></apply><apply id="S2.SS1.p6.3.m3.1.1.3.4.cmml" xref="S2.SS1.p6.3.m3.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p6.3.m3.1.1.3.4.1.cmml" xref="S2.SS1.p6.3.m3.1.1.3.4">subscript</csymbol><ci id="S2.SS1.p6.3.m3.1.1.3.4.2.cmml" xref="S2.SS1.p6.3.m3.1.1.3.4.2">𝑥</ci><ci id="S2.SS1.p6.3.m3.1.1.3.4.3.cmml" xref="S2.SS1.p6.3.m3.1.1.3.4.3">𝑖</ci></apply><apply id="S2.SS1.p6.3.m3.1.1.3.5.cmml" xref="S2.SS1.p6.3.m3.1.1.3.5"><csymbol cd="ambiguous" id="S2.SS1.p6.3.m3.1.1.3.5.1.cmml" xref="S2.SS1.p6.3.m3.1.1.3.5">subscript</csymbol><ci id="S2.SS1.p6.3.m3.1.1.3.5.2.cmml" xref="S2.SS1.p6.3.m3.1.1.3.5.2">𝑥</ci><apply id="S2.SS1.p6.3.m3.1.1.3.5.3.cmml" xref="S2.SS1.p6.3.m3.1.1.3.5.3"><plus id="S2.SS1.p6.3.m3.1.1.3.5.3.1.cmml" xref="S2.SS1.p6.3.m3.1.1.3.5.3.1"></plus><ci id="S2.SS1.p6.3.m3.1.1.3.5.3.2.cmml" xref="S2.SS1.p6.3.m3.1.1.3.5.3.2">𝑖</ci><cn id="S2.SS1.p6.3.m3.1.1.3.5.3.3.cmml" type="integer" xref="S2.SS1.p6.3.m3.1.1.3.5.3.3">1</cn></apply></apply><ci id="S2.SS1.p6.3.m3.1.1.3.6.cmml" xref="S2.SS1.p6.3.m3.1.1.3.6">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.3.m3.1c">{\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.3.m3.1d">bold_x = … italic_x start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT …</annotation></semantics></math> with <math alttext="x_{i}\in\cal A" class="ltx_Math" display="inline" id="S2.SS1.p6.4.m4.1"><semantics id="S2.SS1.p6.4.m4.1a"><mrow id="S2.SS1.p6.4.m4.1.1" xref="S2.SS1.p6.4.m4.1.1.cmml"><msub id="S2.SS1.p6.4.m4.1.1.2" xref="S2.SS1.p6.4.m4.1.1.2.cmml"><mi id="S2.SS1.p6.4.m4.1.1.2.2" xref="S2.SS1.p6.4.m4.1.1.2.2.cmml">x</mi><mi id="S2.SS1.p6.4.m4.1.1.2.3" xref="S2.SS1.p6.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p6.4.m4.1.1.1" xref="S2.SS1.p6.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p6.4.m4.1.1.3" xref="S2.SS1.p6.4.m4.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.4.m4.1b"><apply id="S2.SS1.p6.4.m4.1.1.cmml" xref="S2.SS1.p6.4.m4.1.1"><in id="S2.SS1.p6.4.m4.1.1.1.cmml" xref="S2.SS1.p6.4.m4.1.1.1"></in><apply id="S2.SS1.p6.4.m4.1.1.2.cmml" xref="S2.SS1.p6.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p6.4.m4.1.1.2.1.cmml" xref="S2.SS1.p6.4.m4.1.1.2">subscript</csymbol><ci id="S2.SS1.p6.4.m4.1.1.2.2.cmml" xref="S2.SS1.p6.4.m4.1.1.2.2">𝑥</ci><ci id="S2.SS1.p6.4.m4.1.1.2.3.cmml" xref="S2.SS1.p6.4.m4.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p6.4.m4.1.1.3.cmml" xref="S2.SS1.p6.4.m4.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.4.m4.1c">x_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math>. The set <math alttext="\cal L({\bf x})\subseteq\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p6.5.m5.1"><semantics id="S2.SS1.p6.5.m5.1a"><mrow id="S2.SS1.p6.5.m5.1.2" xref="S2.SS1.p6.5.m5.1.2.cmml"><mrow id="S2.SS1.p6.5.m5.1.2.2" xref="S2.SS1.p6.5.m5.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p6.5.m5.1.2.2.2" xref="S2.SS1.p6.5.m5.1.2.2.2.cmml">ℒ</mi><mo id="S2.SS1.p6.5.m5.1.2.2.1" xref="S2.SS1.p6.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p6.5.m5.1.2.2.3.2" xref="S2.SS1.p6.5.m5.1.2.2.cmml"><mo id="S2.SS1.p6.5.m5.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.p6.5.m5.1.2.2.cmml">(</mo><mi id="S2.SS1.p6.5.m5.1.1" xref="S2.SS1.p6.5.m5.1.1.cmml">𝐱</mi><mo id="S2.SS1.p6.5.m5.1.2.2.3.2.2" stretchy="false" xref="S2.SS1.p6.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p6.5.m5.1.2.1" xref="S2.SS1.p6.5.m5.1.2.1.cmml">⊆</mo><msup id="S2.SS1.p6.5.m5.1.2.3" xref="S2.SS1.p6.5.m5.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p6.5.m5.1.2.3.2" xref="S2.SS1.p6.5.m5.1.2.3.2.cmml">𝒜</mi><mo id="S2.SS1.p6.5.m5.1.2.3.3" xref="S2.SS1.p6.5.m5.1.2.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.5.m5.1b"><apply id="S2.SS1.p6.5.m5.1.2.cmml" xref="S2.SS1.p6.5.m5.1.2"><subset id="S2.SS1.p6.5.m5.1.2.1.cmml" xref="S2.SS1.p6.5.m5.1.2.1"></subset><apply id="S2.SS1.p6.5.m5.1.2.2.cmml" xref="S2.SS1.p6.5.m5.1.2.2"><times id="S2.SS1.p6.5.m5.1.2.2.1.cmml" xref="S2.SS1.p6.5.m5.1.2.2.1"></times><ci id="S2.SS1.p6.5.m5.1.2.2.2.cmml" xref="S2.SS1.p6.5.m5.1.2.2.2">ℒ</ci><ci id="S2.SS1.p6.5.m5.1.1.cmml" xref="S2.SS1.p6.5.m5.1.1">𝐱</ci></apply><apply id="S2.SS1.p6.5.m5.1.2.3.cmml" xref="S2.SS1.p6.5.m5.1.2.3"><csymbol cd="ambiguous" id="S2.SS1.p6.5.m5.1.2.3.1.cmml" xref="S2.SS1.p6.5.m5.1.2.3">superscript</csymbol><ci id="S2.SS1.p6.5.m5.1.2.3.2.cmml" xref="S2.SS1.p6.5.m5.1.2.3.2">𝒜</ci><times id="S2.SS1.p6.5.m5.1.2.3.3.cmml" xref="S2.SS1.p6.5.m5.1.2.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.5.m5.1c">\cal L({\bf x})\subseteq\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.5.m5.1d">caligraphic_L ( bold_x ) ⊆ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of all finite subwords (called <span class="ltx_text ltx_font_italic" id="S2.SS1.p6.13.2">factors</span>) <math alttext="{\bf x}_{[k,\ell]}:=x_{k}x_{k+1}\ldots x_{\ell}" class="ltx_Math" display="inline" id="S2.SS1.p6.6.m6.2"><semantics id="S2.SS1.p6.6.m6.2a"><mrow id="S2.SS1.p6.6.m6.2.3" xref="S2.SS1.p6.6.m6.2.3.cmml"><msub id="S2.SS1.p6.6.m6.2.3.2" xref="S2.SS1.p6.6.m6.2.3.2.cmml"><mi id="S2.SS1.p6.6.m6.2.3.2.2" xref="S2.SS1.p6.6.m6.2.3.2.2.cmml">𝐱</mi><mrow id="S2.SS1.p6.6.m6.2.2.2.4" xref="S2.SS1.p6.6.m6.2.2.2.3.cmml"><mo id="S2.SS1.p6.6.m6.2.2.2.4.1" stretchy="false" xref="S2.SS1.p6.6.m6.2.2.2.3.cmml">[</mo><mi id="S2.SS1.p6.6.m6.1.1.1.1" xref="S2.SS1.p6.6.m6.1.1.1.1.cmml">k</mi><mo id="S2.SS1.p6.6.m6.2.2.2.4.2" xref="S2.SS1.p6.6.m6.2.2.2.3.cmml">,</mo><mi id="S2.SS1.p6.6.m6.2.2.2.2" mathvariant="normal" xref="S2.SS1.p6.6.m6.2.2.2.2.cmml">ℓ</mi><mo id="S2.SS1.p6.6.m6.2.2.2.4.3" stretchy="false" xref="S2.SS1.p6.6.m6.2.2.2.3.cmml">]</mo></mrow></msub><mo id="S2.SS1.p6.6.m6.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p6.6.m6.2.3.1.cmml">:=</mo><mrow id="S2.SS1.p6.6.m6.2.3.3" xref="S2.SS1.p6.6.m6.2.3.3.cmml"><msub id="S2.SS1.p6.6.m6.2.3.3.2" xref="S2.SS1.p6.6.m6.2.3.3.2.cmml"><mi id="S2.SS1.p6.6.m6.2.3.3.2.2" xref="S2.SS1.p6.6.m6.2.3.3.2.2.cmml">x</mi><mi id="S2.SS1.p6.6.m6.2.3.3.2.3" xref="S2.SS1.p6.6.m6.2.3.3.2.3.cmml">k</mi></msub><mo id="S2.SS1.p6.6.m6.2.3.3.1" xref="S2.SS1.p6.6.m6.2.3.3.1.cmml">⁢</mo><msub id="S2.SS1.p6.6.m6.2.3.3.3" xref="S2.SS1.p6.6.m6.2.3.3.3.cmml"><mi id="S2.SS1.p6.6.m6.2.3.3.3.2" xref="S2.SS1.p6.6.m6.2.3.3.3.2.cmml">x</mi><mrow id="S2.SS1.p6.6.m6.2.3.3.3.3" xref="S2.SS1.p6.6.m6.2.3.3.3.3.cmml"><mi id="S2.SS1.p6.6.m6.2.3.3.3.3.2" xref="S2.SS1.p6.6.m6.2.3.3.3.3.2.cmml">k</mi><mo id="S2.SS1.p6.6.m6.2.3.3.3.3.1" xref="S2.SS1.p6.6.m6.2.3.3.3.3.1.cmml">+</mo><mn id="S2.SS1.p6.6.m6.2.3.3.3.3.3" xref="S2.SS1.p6.6.m6.2.3.3.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p6.6.m6.2.3.3.1a" xref="S2.SS1.p6.6.m6.2.3.3.1.cmml">⁢</mo><mi id="S2.SS1.p6.6.m6.2.3.3.4" mathvariant="normal" xref="S2.SS1.p6.6.m6.2.3.3.4.cmml">…</mi><mo id="S2.SS1.p6.6.m6.2.3.3.1b" xref="S2.SS1.p6.6.m6.2.3.3.1.cmml">⁢</mo><msub id="S2.SS1.p6.6.m6.2.3.3.5" xref="S2.SS1.p6.6.m6.2.3.3.5.cmml"><mi id="S2.SS1.p6.6.m6.2.3.3.5.2" xref="S2.SS1.p6.6.m6.2.3.3.5.2.cmml">x</mi><mi id="S2.SS1.p6.6.m6.2.3.3.5.3" mathvariant="normal" xref="S2.SS1.p6.6.m6.2.3.3.5.3.cmml">ℓ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.6.m6.2b"><apply id="S2.SS1.p6.6.m6.2.3.cmml" xref="S2.SS1.p6.6.m6.2.3"><csymbol cd="latexml" id="S2.SS1.p6.6.m6.2.3.1.cmml" xref="S2.SS1.p6.6.m6.2.3.1">assign</csymbol><apply id="S2.SS1.p6.6.m6.2.3.2.cmml" xref="S2.SS1.p6.6.m6.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p6.6.m6.2.3.2.1.cmml" xref="S2.SS1.p6.6.m6.2.3.2">subscript</csymbol><ci id="S2.SS1.p6.6.m6.2.3.2.2.cmml" xref="S2.SS1.p6.6.m6.2.3.2.2">𝐱</ci><interval closure="closed" id="S2.SS1.p6.6.m6.2.2.2.3.cmml" xref="S2.SS1.p6.6.m6.2.2.2.4"><ci id="S2.SS1.p6.6.m6.1.1.1.1.cmml" xref="S2.SS1.p6.6.m6.1.1.1.1">𝑘</ci><ci id="S2.SS1.p6.6.m6.2.2.2.2.cmml" xref="S2.SS1.p6.6.m6.2.2.2.2">ℓ</ci></interval></apply><apply id="S2.SS1.p6.6.m6.2.3.3.cmml" xref="S2.SS1.p6.6.m6.2.3.3"><times id="S2.SS1.p6.6.m6.2.3.3.1.cmml" xref="S2.SS1.p6.6.m6.2.3.3.1"></times><apply id="S2.SS1.p6.6.m6.2.3.3.2.cmml" xref="S2.SS1.p6.6.m6.2.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.p6.6.m6.2.3.3.2.1.cmml" xref="S2.SS1.p6.6.m6.2.3.3.2">subscript</csymbol><ci id="S2.SS1.p6.6.m6.2.3.3.2.2.cmml" xref="S2.SS1.p6.6.m6.2.3.3.2.2">𝑥</ci><ci id="S2.SS1.p6.6.m6.2.3.3.2.3.cmml" xref="S2.SS1.p6.6.m6.2.3.3.2.3">𝑘</ci></apply><apply id="S2.SS1.p6.6.m6.2.3.3.3.cmml" xref="S2.SS1.p6.6.m6.2.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p6.6.m6.2.3.3.3.1.cmml" xref="S2.SS1.p6.6.m6.2.3.3.3">subscript</csymbol><ci id="S2.SS1.p6.6.m6.2.3.3.3.2.cmml" xref="S2.SS1.p6.6.m6.2.3.3.3.2">𝑥</ci><apply id="S2.SS1.p6.6.m6.2.3.3.3.3.cmml" xref="S2.SS1.p6.6.m6.2.3.3.3.3"><plus id="S2.SS1.p6.6.m6.2.3.3.3.3.1.cmml" xref="S2.SS1.p6.6.m6.2.3.3.3.3.1"></plus><ci id="S2.SS1.p6.6.m6.2.3.3.3.3.2.cmml" xref="S2.SS1.p6.6.m6.2.3.3.3.3.2">𝑘</ci><cn id="S2.SS1.p6.6.m6.2.3.3.3.3.3.cmml" type="integer" xref="S2.SS1.p6.6.m6.2.3.3.3.3.3">1</cn></apply></apply><ci id="S2.SS1.p6.6.m6.2.3.3.4.cmml" xref="S2.SS1.p6.6.m6.2.3.3.4">…</ci><apply id="S2.SS1.p6.6.m6.2.3.3.5.cmml" xref="S2.SS1.p6.6.m6.2.3.3.5"><csymbol cd="ambiguous" id="S2.SS1.p6.6.m6.2.3.3.5.1.cmml" xref="S2.SS1.p6.6.m6.2.3.3.5">subscript</csymbol><ci id="S2.SS1.p6.6.m6.2.3.3.5.2.cmml" xref="S2.SS1.p6.6.m6.2.3.3.5.2">𝑥</ci><ci id="S2.SS1.p6.6.m6.2.3.3.5.3.cmml" xref="S2.SS1.p6.6.m6.2.3.3.5.3">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.6.m6.2c">{\bf x}_{[k,\ell]}:=x_{k}x_{k+1}\ldots x_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.6.m6.2d">bold_x start_POSTSUBSCRIPT [ italic_k , roman_ℓ ] end_POSTSUBSCRIPT := italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT … italic_x start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> is the <span class="ltx_text ltx_font_italic" id="S2.SS1.p6.7.1">language associated to <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S2.SS1.p6.7.1.m1.1"><semantics id="S2.SS1.p6.7.1.m1.1a"><mi id="S2.SS1.p6.7.1.m1.1.1" xref="S2.SS1.p6.7.1.m1.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.7.1.m1.1b"><ci id="S2.SS1.p6.7.1.m1.1.1.cmml" xref="S2.SS1.p6.7.1.m1.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.7.1.m1.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.7.1.m1.1d">bold_x</annotation></semantics></math></span>. The one-sided infinite <span class="ltx_text ltx_font_italic" id="S2.SS1.p6.13.3">positive half-word</span> <math alttext="x_{1}x_{2}\ldots" class="ltx_Math" display="inline" id="S2.SS1.p6.8.m7.1"><semantics id="S2.SS1.p6.8.m7.1a"><mrow id="S2.SS1.p6.8.m7.1.1" xref="S2.SS1.p6.8.m7.1.1.cmml"><msub id="S2.SS1.p6.8.m7.1.1.2" xref="S2.SS1.p6.8.m7.1.1.2.cmml"><mi id="S2.SS1.p6.8.m7.1.1.2.2" xref="S2.SS1.p6.8.m7.1.1.2.2.cmml">x</mi><mn id="S2.SS1.p6.8.m7.1.1.2.3" xref="S2.SS1.p6.8.m7.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p6.8.m7.1.1.1" xref="S2.SS1.p6.8.m7.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p6.8.m7.1.1.3" xref="S2.SS1.p6.8.m7.1.1.3.cmml"><mi id="S2.SS1.p6.8.m7.1.1.3.2" xref="S2.SS1.p6.8.m7.1.1.3.2.cmml">x</mi><mn id="S2.SS1.p6.8.m7.1.1.3.3" xref="S2.SS1.p6.8.m7.1.1.3.3.cmml">2</mn></msub><mo id="S2.SS1.p6.8.m7.1.1.1a" xref="S2.SS1.p6.8.m7.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p6.8.m7.1.1.4" mathvariant="normal" xref="S2.SS1.p6.8.m7.1.1.4.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.8.m7.1b"><apply id="S2.SS1.p6.8.m7.1.1.cmml" xref="S2.SS1.p6.8.m7.1.1"><times id="S2.SS1.p6.8.m7.1.1.1.cmml" xref="S2.SS1.p6.8.m7.1.1.1"></times><apply id="S2.SS1.p6.8.m7.1.1.2.cmml" xref="S2.SS1.p6.8.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p6.8.m7.1.1.2.1.cmml" xref="S2.SS1.p6.8.m7.1.1.2">subscript</csymbol><ci id="S2.SS1.p6.8.m7.1.1.2.2.cmml" xref="S2.SS1.p6.8.m7.1.1.2.2">𝑥</ci><cn id="S2.SS1.p6.8.m7.1.1.2.3.cmml" type="integer" xref="S2.SS1.p6.8.m7.1.1.2.3">1</cn></apply><apply id="S2.SS1.p6.8.m7.1.1.3.cmml" xref="S2.SS1.p6.8.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p6.8.m7.1.1.3.1.cmml" xref="S2.SS1.p6.8.m7.1.1.3">subscript</csymbol><ci id="S2.SS1.p6.8.m7.1.1.3.2.cmml" xref="S2.SS1.p6.8.m7.1.1.3.2">𝑥</ci><cn id="S2.SS1.p6.8.m7.1.1.3.3.cmml" type="integer" xref="S2.SS1.p6.8.m7.1.1.3.3">2</cn></apply><ci id="S2.SS1.p6.8.m7.1.1.4.cmml" xref="S2.SS1.p6.8.m7.1.1.4">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.8.m7.1c">x_{1}x_{2}\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.8.m7.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT …</annotation></semantics></math> of <math alttext="\bf x" class="ltx_Math" display="inline" id="S2.SS1.p6.9.m8.1"><semantics id="S2.SS1.p6.9.m8.1a"><mi id="S2.SS1.p6.9.m8.1.1" xref="S2.SS1.p6.9.m8.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.9.m8.1b"><ci id="S2.SS1.p6.9.m8.1.1.cmml" xref="S2.SS1.p6.9.m8.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.9.m8.1c">\bf x</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.9.m8.1d">bold_x</annotation></semantics></math> is denoted by <math alttext="{\bf x}_{[1,\infty)}" class="ltx_Math" display="inline" id="S2.SS1.p6.10.m9.2"><semantics id="S2.SS1.p6.10.m9.2a"><msub id="S2.SS1.p6.10.m9.2.3" xref="S2.SS1.p6.10.m9.2.3.cmml"><mi id="S2.SS1.p6.10.m9.2.3.2" xref="S2.SS1.p6.10.m9.2.3.2.cmml">𝐱</mi><mrow id="S2.SS1.p6.10.m9.2.2.2.4" xref="S2.SS1.p6.10.m9.2.2.2.3.cmml"><mo id="S2.SS1.p6.10.m9.2.2.2.4.1" stretchy="false" xref="S2.SS1.p6.10.m9.2.2.2.3.cmml">[</mo><mn id="S2.SS1.p6.10.m9.1.1.1.1" xref="S2.SS1.p6.10.m9.1.1.1.1.cmml">1</mn><mo id="S2.SS1.p6.10.m9.2.2.2.4.2" xref="S2.SS1.p6.10.m9.2.2.2.3.cmml">,</mo><mi id="S2.SS1.p6.10.m9.2.2.2.2" mathvariant="normal" xref="S2.SS1.p6.10.m9.2.2.2.2.cmml">∞</mi><mo id="S2.SS1.p6.10.m9.2.2.2.4.3" stretchy="false" xref="S2.SS1.p6.10.m9.2.2.2.3.cmml">)</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.10.m9.2b"><apply id="S2.SS1.p6.10.m9.2.3.cmml" xref="S2.SS1.p6.10.m9.2.3"><csymbol cd="ambiguous" id="S2.SS1.p6.10.m9.2.3.1.cmml" xref="S2.SS1.p6.10.m9.2.3">subscript</csymbol><ci id="S2.SS1.p6.10.m9.2.3.2.cmml" xref="S2.SS1.p6.10.m9.2.3.2">𝐱</ci><interval closure="closed-open" id="S2.SS1.p6.10.m9.2.2.2.3.cmml" xref="S2.SS1.p6.10.m9.2.2.2.4"><cn id="S2.SS1.p6.10.m9.1.1.1.1.cmml" type="integer" xref="S2.SS1.p6.10.m9.1.1.1.1">1</cn><infinity id="S2.SS1.p6.10.m9.2.2.2.2.cmml" xref="S2.SS1.p6.10.m9.2.2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.10.m9.2c">{\bf x}_{[1,\infty)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.10.m9.2d">bold_x start_POSTSUBSCRIPT [ 1 , ∞ ) end_POSTSUBSCRIPT</annotation></semantics></math>. Similarly, we denote the complementary infinite half-word <math alttext="\ldots x_{-1}x_{0}" class="ltx_Math" display="inline" id="S2.SS1.p6.11.m10.1"><semantics id="S2.SS1.p6.11.m10.1a"><mrow id="S2.SS1.p6.11.m10.1.1" xref="S2.SS1.p6.11.m10.1.1.cmml"><mi id="S2.SS1.p6.11.m10.1.1.2" mathvariant="normal" xref="S2.SS1.p6.11.m10.1.1.2.cmml">…</mi><mo id="S2.SS1.p6.11.m10.1.1.1" xref="S2.SS1.p6.11.m10.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p6.11.m10.1.1.3" xref="S2.SS1.p6.11.m10.1.1.3.cmml"><mi id="S2.SS1.p6.11.m10.1.1.3.2" xref="S2.SS1.p6.11.m10.1.1.3.2.cmml">x</mi><mrow id="S2.SS1.p6.11.m10.1.1.3.3" xref="S2.SS1.p6.11.m10.1.1.3.3.cmml"><mo id="S2.SS1.p6.11.m10.1.1.3.3a" xref="S2.SS1.p6.11.m10.1.1.3.3.cmml">−</mo><mn id="S2.SS1.p6.11.m10.1.1.3.3.2" xref="S2.SS1.p6.11.m10.1.1.3.3.2.cmml">1</mn></mrow></msub><mo id="S2.SS1.p6.11.m10.1.1.1a" xref="S2.SS1.p6.11.m10.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p6.11.m10.1.1.4" xref="S2.SS1.p6.11.m10.1.1.4.cmml"><mi id="S2.SS1.p6.11.m10.1.1.4.2" xref="S2.SS1.p6.11.m10.1.1.4.2.cmml">x</mi><mn id="S2.SS1.p6.11.m10.1.1.4.3" xref="S2.SS1.p6.11.m10.1.1.4.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.11.m10.1b"><apply id="S2.SS1.p6.11.m10.1.1.cmml" xref="S2.SS1.p6.11.m10.1.1"><times id="S2.SS1.p6.11.m10.1.1.1.cmml" xref="S2.SS1.p6.11.m10.1.1.1"></times><ci id="S2.SS1.p6.11.m10.1.1.2.cmml" xref="S2.SS1.p6.11.m10.1.1.2">…</ci><apply id="S2.SS1.p6.11.m10.1.1.3.cmml" xref="S2.SS1.p6.11.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p6.11.m10.1.1.3.1.cmml" xref="S2.SS1.p6.11.m10.1.1.3">subscript</csymbol><ci id="S2.SS1.p6.11.m10.1.1.3.2.cmml" xref="S2.SS1.p6.11.m10.1.1.3.2">𝑥</ci><apply id="S2.SS1.p6.11.m10.1.1.3.3.cmml" xref="S2.SS1.p6.11.m10.1.1.3.3"><minus id="S2.SS1.p6.11.m10.1.1.3.3.1.cmml" xref="S2.SS1.p6.11.m10.1.1.3.3"></minus><cn id="S2.SS1.p6.11.m10.1.1.3.3.2.cmml" type="integer" xref="S2.SS1.p6.11.m10.1.1.3.3.2">1</cn></apply></apply><apply id="S2.SS1.p6.11.m10.1.1.4.cmml" xref="S2.SS1.p6.11.m10.1.1.4"><csymbol cd="ambiguous" id="S2.SS1.p6.11.m10.1.1.4.1.cmml" xref="S2.SS1.p6.11.m10.1.1.4">subscript</csymbol><ci id="S2.SS1.p6.11.m10.1.1.4.2.cmml" xref="S2.SS1.p6.11.m10.1.1.4.2">𝑥</ci><cn id="S2.SS1.p6.11.m10.1.1.4.3.cmml" type="integer" xref="S2.SS1.p6.11.m10.1.1.4.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.11.m10.1c">\ldots x_{-1}x_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.11.m10.1d">… italic_x start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\bf x" class="ltx_Math" display="inline" id="S2.SS1.p6.12.m11.1"><semantics id="S2.SS1.p6.12.m11.1a"><mi id="S2.SS1.p6.12.m11.1.1" xref="S2.SS1.p6.12.m11.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.12.m11.1b"><ci id="S2.SS1.p6.12.m11.1.1.cmml" xref="S2.SS1.p6.12.m11.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.12.m11.1c">\bf x</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.12.m11.1d">bold_x</annotation></semantics></math> by <math alttext="{\bf x}_{(-\infty,0]}" class="ltx_Math" display="inline" id="S2.SS1.p6.13.m12.2"><semantics id="S2.SS1.p6.13.m12.2a"><msub id="S2.SS1.p6.13.m12.2.3" xref="S2.SS1.p6.13.m12.2.3.cmml"><mi id="S2.SS1.p6.13.m12.2.3.2" xref="S2.SS1.p6.13.m12.2.3.2.cmml">𝐱</mi><mrow id="S2.SS1.p6.13.m12.2.2.2.2" xref="S2.SS1.p6.13.m12.2.2.2.3.cmml"><mo id="S2.SS1.p6.13.m12.2.2.2.2.2" stretchy="false" xref="S2.SS1.p6.13.m12.2.2.2.3.cmml">(</mo><mrow id="S2.SS1.p6.13.m12.2.2.2.2.1" xref="S2.SS1.p6.13.m12.2.2.2.2.1.cmml"><mo id="S2.SS1.p6.13.m12.2.2.2.2.1a" xref="S2.SS1.p6.13.m12.2.2.2.2.1.cmml">−</mo><mi id="S2.SS1.p6.13.m12.2.2.2.2.1.2" mathvariant="normal" xref="S2.SS1.p6.13.m12.2.2.2.2.1.2.cmml">∞</mi></mrow><mo id="S2.SS1.p6.13.m12.2.2.2.2.3" xref="S2.SS1.p6.13.m12.2.2.2.3.cmml">,</mo><mn id="S2.SS1.p6.13.m12.1.1.1.1" xref="S2.SS1.p6.13.m12.1.1.1.1.cmml">0</mn><mo id="S2.SS1.p6.13.m12.2.2.2.2.4" stretchy="false" xref="S2.SS1.p6.13.m12.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.13.m12.2b"><apply id="S2.SS1.p6.13.m12.2.3.cmml" xref="S2.SS1.p6.13.m12.2.3"><csymbol cd="ambiguous" id="S2.SS1.p6.13.m12.2.3.1.cmml" xref="S2.SS1.p6.13.m12.2.3">subscript</csymbol><ci id="S2.SS1.p6.13.m12.2.3.2.cmml" xref="S2.SS1.p6.13.m12.2.3.2">𝐱</ci><interval closure="open-closed" id="S2.SS1.p6.13.m12.2.2.2.3.cmml" xref="S2.SS1.p6.13.m12.2.2.2.2"><apply id="S2.SS1.p6.13.m12.2.2.2.2.1.cmml" xref="S2.SS1.p6.13.m12.2.2.2.2.1"><minus id="S2.SS1.p6.13.m12.2.2.2.2.1.1.cmml" xref="S2.SS1.p6.13.m12.2.2.2.2.1"></minus><infinity id="S2.SS1.p6.13.m12.2.2.2.2.1.2.cmml" xref="S2.SS1.p6.13.m12.2.2.2.2.1.2"></infinity></apply><cn id="S2.SS1.p6.13.m12.1.1.1.1.cmml" type="integer" xref="S2.SS1.p6.13.m12.1.1.1.1">0</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.13.m12.2c">{\bf x}_{(-\infty,0]}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.13.m12.2d">bold_x start_POSTSUBSCRIPT ( - ∞ , 0 ] end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p7"> <p class="ltx_p" id="S2.SS1.p7.17">The shift space <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p7.1.m1.1"><semantics id="S2.SS1.p7.1.m1.1a"><msup id="S2.SS1.p7.1.m1.1.1" xref="S2.SS1.p7.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.1.m1.1.1.2" xref="S2.SS1.p7.1.m1.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p7.1.m1.1.1.3" xref="S2.SS1.p7.1.m1.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.1.m1.1b"><apply id="S2.SS1.p7.1.m1.1.1.cmml" xref="S2.SS1.p7.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p7.1.m1.1.1.1.cmml" xref="S2.SS1.p7.1.m1.1.1">superscript</csymbol><ci id="S2.SS1.p7.1.m1.1.1.2.cmml" xref="S2.SS1.p7.1.m1.1.1.2">𝒜</ci><ci id="S2.SS1.p7.1.m1.1.1.3.cmml" xref="S2.SS1.p7.1.m1.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.1.m1.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.1.m1.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is canonically equipped with the <span class="ltx_text ltx_font_italic" id="S2.SS1.p7.17.1">shift operator</span> <math alttext="T:\cal A^{\mathbb{Z}}\to\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p7.2.m2.1"><semantics id="S2.SS1.p7.2.m2.1a"><mrow id="S2.SS1.p7.2.m2.1.1" xref="S2.SS1.p7.2.m2.1.1.cmml"><mi id="S2.SS1.p7.2.m2.1.1.2" xref="S2.SS1.p7.2.m2.1.1.2.cmml">T</mi><mo id="S2.SS1.p7.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p7.2.m2.1.1.1.cmml">:</mo><mrow id="S2.SS1.p7.2.m2.1.1.3" xref="S2.SS1.p7.2.m2.1.1.3.cmml"><msup id="S2.SS1.p7.2.m2.1.1.3.2" xref="S2.SS1.p7.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.2.m2.1.1.3.2.2" xref="S2.SS1.p7.2.m2.1.1.3.2.2.cmml">𝒜</mi><mi id="S2.SS1.p7.2.m2.1.1.3.2.3" xref="S2.SS1.p7.2.m2.1.1.3.2.3.cmml">ℤ</mi></msup><mo id="S2.SS1.p7.2.m2.1.1.3.1" stretchy="false" xref="S2.SS1.p7.2.m2.1.1.3.1.cmml">→</mo><msup id="S2.SS1.p7.2.m2.1.1.3.3" xref="S2.SS1.p7.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.2.m2.1.1.3.3.2" xref="S2.SS1.p7.2.m2.1.1.3.3.2.cmml">𝒜</mi><mi id="S2.SS1.p7.2.m2.1.1.3.3.3" xref="S2.SS1.p7.2.m2.1.1.3.3.3.cmml">ℤ</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.2.m2.1b"><apply id="S2.SS1.p7.2.m2.1.1.cmml" xref="S2.SS1.p7.2.m2.1.1"><ci id="S2.SS1.p7.2.m2.1.1.1.cmml" xref="S2.SS1.p7.2.m2.1.1.1">:</ci><ci id="S2.SS1.p7.2.m2.1.1.2.cmml" xref="S2.SS1.p7.2.m2.1.1.2">𝑇</ci><apply id="S2.SS1.p7.2.m2.1.1.3.cmml" xref="S2.SS1.p7.2.m2.1.1.3"><ci id="S2.SS1.p7.2.m2.1.1.3.1.cmml" xref="S2.SS1.p7.2.m2.1.1.3.1">→</ci><apply id="S2.SS1.p7.2.m2.1.1.3.2.cmml" xref="S2.SS1.p7.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p7.2.m2.1.1.3.2.1.cmml" xref="S2.SS1.p7.2.m2.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p7.2.m2.1.1.3.2.2.cmml" xref="S2.SS1.p7.2.m2.1.1.3.2.2">𝒜</ci><ci id="S2.SS1.p7.2.m2.1.1.3.2.3.cmml" xref="S2.SS1.p7.2.m2.1.1.3.2.3">ℤ</ci></apply><apply id="S2.SS1.p7.2.m2.1.1.3.3.cmml" xref="S2.SS1.p7.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p7.2.m2.1.1.3.3.1.cmml" xref="S2.SS1.p7.2.m2.1.1.3.3">superscript</csymbol><ci id="S2.SS1.p7.2.m2.1.1.3.3.2.cmml" xref="S2.SS1.p7.2.m2.1.1.3.3.2">𝒜</ci><ci id="S2.SS1.p7.2.m2.1.1.3.3.3.cmml" xref="S2.SS1.p7.2.m2.1.1.3.3.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.2.m2.1c">T:\cal A^{\mathbb{Z}}\to\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.2.m2.1d">italic_T : caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> which maps the word <math alttext="{\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots" class="ltx_Math" display="inline" id="S2.SS1.p7.3.m3.1"><semantics id="S2.SS1.p7.3.m3.1a"><mrow id="S2.SS1.p7.3.m3.1.1" xref="S2.SS1.p7.3.m3.1.1.cmml"><mi id="S2.SS1.p7.3.m3.1.1.2" xref="S2.SS1.p7.3.m3.1.1.2.cmml">𝐱</mi><mo id="S2.SS1.p7.3.m3.1.1.1" xref="S2.SS1.p7.3.m3.1.1.1.cmml">=</mo><mrow id="S2.SS1.p7.3.m3.1.1.3" xref="S2.SS1.p7.3.m3.1.1.3.cmml"><mi id="S2.SS1.p7.3.m3.1.1.3.2" mathvariant="normal" xref="S2.SS1.p7.3.m3.1.1.3.2.cmml">…</mi><mo id="S2.SS1.p7.3.m3.1.1.3.1" xref="S2.SS1.p7.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.3.m3.1.1.3.3" xref="S2.SS1.p7.3.m3.1.1.3.3.cmml"><mi id="S2.SS1.p7.3.m3.1.1.3.3.2" xref="S2.SS1.p7.3.m3.1.1.3.3.2.cmml">x</mi><mrow id="S2.SS1.p7.3.m3.1.1.3.3.3" xref="S2.SS1.p7.3.m3.1.1.3.3.3.cmml"><mi id="S2.SS1.p7.3.m3.1.1.3.3.3.2" xref="S2.SS1.p7.3.m3.1.1.3.3.3.2.cmml">i</mi><mo id="S2.SS1.p7.3.m3.1.1.3.3.3.1" xref="S2.SS1.p7.3.m3.1.1.3.3.3.1.cmml">−</mo><mn id="S2.SS1.p7.3.m3.1.1.3.3.3.3" xref="S2.SS1.p7.3.m3.1.1.3.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p7.3.m3.1.1.3.1a" xref="S2.SS1.p7.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.3.m3.1.1.3.4" xref="S2.SS1.p7.3.m3.1.1.3.4.cmml"><mi id="S2.SS1.p7.3.m3.1.1.3.4.2" xref="S2.SS1.p7.3.m3.1.1.3.4.2.cmml">x</mi><mi id="S2.SS1.p7.3.m3.1.1.3.4.3" xref="S2.SS1.p7.3.m3.1.1.3.4.3.cmml">i</mi></msub><mo id="S2.SS1.p7.3.m3.1.1.3.1b" xref="S2.SS1.p7.3.m3.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.3.m3.1.1.3.5" xref="S2.SS1.p7.3.m3.1.1.3.5.cmml"><mi id="S2.SS1.p7.3.m3.1.1.3.5.2" xref="S2.SS1.p7.3.m3.1.1.3.5.2.cmml">x</mi><mrow id="S2.SS1.p7.3.m3.1.1.3.5.3" xref="S2.SS1.p7.3.m3.1.1.3.5.3.cmml"><mi id="S2.SS1.p7.3.m3.1.1.3.5.3.2" xref="S2.SS1.p7.3.m3.1.1.3.5.3.2.cmml">i</mi><mo id="S2.SS1.p7.3.m3.1.1.3.5.3.1" xref="S2.SS1.p7.3.m3.1.1.3.5.3.1.cmml">+</mo><mn id="S2.SS1.p7.3.m3.1.1.3.5.3.3" xref="S2.SS1.p7.3.m3.1.1.3.5.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p7.3.m3.1.1.3.1c" xref="S2.SS1.p7.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p7.3.m3.1.1.3.6" mathvariant="normal" xref="S2.SS1.p7.3.m3.1.1.3.6.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.3.m3.1b"><apply id="S2.SS1.p7.3.m3.1.1.cmml" xref="S2.SS1.p7.3.m3.1.1"><eq id="S2.SS1.p7.3.m3.1.1.1.cmml" xref="S2.SS1.p7.3.m3.1.1.1"></eq><ci id="S2.SS1.p7.3.m3.1.1.2.cmml" xref="S2.SS1.p7.3.m3.1.1.2">𝐱</ci><apply id="S2.SS1.p7.3.m3.1.1.3.cmml" xref="S2.SS1.p7.3.m3.1.1.3"><times id="S2.SS1.p7.3.m3.1.1.3.1.cmml" xref="S2.SS1.p7.3.m3.1.1.3.1"></times><ci id="S2.SS1.p7.3.m3.1.1.3.2.cmml" xref="S2.SS1.p7.3.m3.1.1.3.2">…</ci><apply id="S2.SS1.p7.3.m3.1.1.3.3.cmml" xref="S2.SS1.p7.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p7.3.m3.1.1.3.3.1.cmml" xref="S2.SS1.p7.3.m3.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p7.3.m3.1.1.3.3.2.cmml" xref="S2.SS1.p7.3.m3.1.1.3.3.2">𝑥</ci><apply id="S2.SS1.p7.3.m3.1.1.3.3.3.cmml" xref="S2.SS1.p7.3.m3.1.1.3.3.3"><minus id="S2.SS1.p7.3.m3.1.1.3.3.3.1.cmml" xref="S2.SS1.p7.3.m3.1.1.3.3.3.1"></minus><ci id="S2.SS1.p7.3.m3.1.1.3.3.3.2.cmml" xref="S2.SS1.p7.3.m3.1.1.3.3.3.2">𝑖</ci><cn id="S2.SS1.p7.3.m3.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p7.3.m3.1.1.3.3.3.3">1</cn></apply></apply><apply id="S2.SS1.p7.3.m3.1.1.3.4.cmml" xref="S2.SS1.p7.3.m3.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p7.3.m3.1.1.3.4.1.cmml" xref="S2.SS1.p7.3.m3.1.1.3.4">subscript</csymbol><ci id="S2.SS1.p7.3.m3.1.1.3.4.2.cmml" xref="S2.SS1.p7.3.m3.1.1.3.4.2">𝑥</ci><ci id="S2.SS1.p7.3.m3.1.1.3.4.3.cmml" xref="S2.SS1.p7.3.m3.1.1.3.4.3">𝑖</ci></apply><apply id="S2.SS1.p7.3.m3.1.1.3.5.cmml" xref="S2.SS1.p7.3.m3.1.1.3.5"><csymbol cd="ambiguous" id="S2.SS1.p7.3.m3.1.1.3.5.1.cmml" xref="S2.SS1.p7.3.m3.1.1.3.5">subscript</csymbol><ci id="S2.SS1.p7.3.m3.1.1.3.5.2.cmml" xref="S2.SS1.p7.3.m3.1.1.3.5.2">𝑥</ci><apply id="S2.SS1.p7.3.m3.1.1.3.5.3.cmml" xref="S2.SS1.p7.3.m3.1.1.3.5.3"><plus id="S2.SS1.p7.3.m3.1.1.3.5.3.1.cmml" xref="S2.SS1.p7.3.m3.1.1.3.5.3.1"></plus><ci id="S2.SS1.p7.3.m3.1.1.3.5.3.2.cmml" xref="S2.SS1.p7.3.m3.1.1.3.5.3.2">𝑖</ci><cn id="S2.SS1.p7.3.m3.1.1.3.5.3.3.cmml" type="integer" xref="S2.SS1.p7.3.m3.1.1.3.5.3.3">1</cn></apply></apply><ci id="S2.SS1.p7.3.m3.1.1.3.6.cmml" xref="S2.SS1.p7.3.m3.1.1.3.6">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.3.m3.1c">{\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.3.m3.1d">bold_x = … italic_x start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT …</annotation></semantics></math> to the word <math alttext="{\bf y}=\ldots y_{j-1}y_{j}y_{j+1}\ldots" class="ltx_Math" display="inline" id="S2.SS1.p7.4.m4.1"><semantics id="S2.SS1.p7.4.m4.1a"><mrow id="S2.SS1.p7.4.m4.1.1" xref="S2.SS1.p7.4.m4.1.1.cmml"><mi id="S2.SS1.p7.4.m4.1.1.2" xref="S2.SS1.p7.4.m4.1.1.2.cmml">𝐲</mi><mo id="S2.SS1.p7.4.m4.1.1.1" xref="S2.SS1.p7.4.m4.1.1.1.cmml">=</mo><mrow id="S2.SS1.p7.4.m4.1.1.3" xref="S2.SS1.p7.4.m4.1.1.3.cmml"><mi id="S2.SS1.p7.4.m4.1.1.3.2" mathvariant="normal" xref="S2.SS1.p7.4.m4.1.1.3.2.cmml">…</mi><mo id="S2.SS1.p7.4.m4.1.1.3.1" xref="S2.SS1.p7.4.m4.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.4.m4.1.1.3.3" xref="S2.SS1.p7.4.m4.1.1.3.3.cmml"><mi id="S2.SS1.p7.4.m4.1.1.3.3.2" xref="S2.SS1.p7.4.m4.1.1.3.3.2.cmml">y</mi><mrow id="S2.SS1.p7.4.m4.1.1.3.3.3" xref="S2.SS1.p7.4.m4.1.1.3.3.3.cmml"><mi id="S2.SS1.p7.4.m4.1.1.3.3.3.2" xref="S2.SS1.p7.4.m4.1.1.3.3.3.2.cmml">j</mi><mo id="S2.SS1.p7.4.m4.1.1.3.3.3.1" xref="S2.SS1.p7.4.m4.1.1.3.3.3.1.cmml">−</mo><mn id="S2.SS1.p7.4.m4.1.1.3.3.3.3" xref="S2.SS1.p7.4.m4.1.1.3.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p7.4.m4.1.1.3.1a" xref="S2.SS1.p7.4.m4.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.4.m4.1.1.3.4" xref="S2.SS1.p7.4.m4.1.1.3.4.cmml"><mi id="S2.SS1.p7.4.m4.1.1.3.4.2" xref="S2.SS1.p7.4.m4.1.1.3.4.2.cmml">y</mi><mi id="S2.SS1.p7.4.m4.1.1.3.4.3" xref="S2.SS1.p7.4.m4.1.1.3.4.3.cmml">j</mi></msub><mo id="S2.SS1.p7.4.m4.1.1.3.1b" xref="S2.SS1.p7.4.m4.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.4.m4.1.1.3.5" xref="S2.SS1.p7.4.m4.1.1.3.5.cmml"><mi id="S2.SS1.p7.4.m4.1.1.3.5.2" xref="S2.SS1.p7.4.m4.1.1.3.5.2.cmml">y</mi><mrow id="S2.SS1.p7.4.m4.1.1.3.5.3" xref="S2.SS1.p7.4.m4.1.1.3.5.3.cmml"><mi id="S2.SS1.p7.4.m4.1.1.3.5.3.2" xref="S2.SS1.p7.4.m4.1.1.3.5.3.2.cmml">j</mi><mo id="S2.SS1.p7.4.m4.1.1.3.5.3.1" xref="S2.SS1.p7.4.m4.1.1.3.5.3.1.cmml">+</mo><mn id="S2.SS1.p7.4.m4.1.1.3.5.3.3" xref="S2.SS1.p7.4.m4.1.1.3.5.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p7.4.m4.1.1.3.1c" xref="S2.SS1.p7.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p7.4.m4.1.1.3.6" mathvariant="normal" xref="S2.SS1.p7.4.m4.1.1.3.6.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.4.m4.1b"><apply id="S2.SS1.p7.4.m4.1.1.cmml" xref="S2.SS1.p7.4.m4.1.1"><eq id="S2.SS1.p7.4.m4.1.1.1.cmml" xref="S2.SS1.p7.4.m4.1.1.1"></eq><ci id="S2.SS1.p7.4.m4.1.1.2.cmml" xref="S2.SS1.p7.4.m4.1.1.2">𝐲</ci><apply id="S2.SS1.p7.4.m4.1.1.3.cmml" xref="S2.SS1.p7.4.m4.1.1.3"><times id="S2.SS1.p7.4.m4.1.1.3.1.cmml" xref="S2.SS1.p7.4.m4.1.1.3.1"></times><ci id="S2.SS1.p7.4.m4.1.1.3.2.cmml" xref="S2.SS1.p7.4.m4.1.1.3.2">…</ci><apply id="S2.SS1.p7.4.m4.1.1.3.3.cmml" xref="S2.SS1.p7.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p7.4.m4.1.1.3.3.1.cmml" xref="S2.SS1.p7.4.m4.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p7.4.m4.1.1.3.3.2.cmml" xref="S2.SS1.p7.4.m4.1.1.3.3.2">𝑦</ci><apply id="S2.SS1.p7.4.m4.1.1.3.3.3.cmml" xref="S2.SS1.p7.4.m4.1.1.3.3.3"><minus id="S2.SS1.p7.4.m4.1.1.3.3.3.1.cmml" xref="S2.SS1.p7.4.m4.1.1.3.3.3.1"></minus><ci id="S2.SS1.p7.4.m4.1.1.3.3.3.2.cmml" xref="S2.SS1.p7.4.m4.1.1.3.3.3.2">𝑗</ci><cn id="S2.SS1.p7.4.m4.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p7.4.m4.1.1.3.3.3.3">1</cn></apply></apply><apply id="S2.SS1.p7.4.m4.1.1.3.4.cmml" xref="S2.SS1.p7.4.m4.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p7.4.m4.1.1.3.4.1.cmml" xref="S2.SS1.p7.4.m4.1.1.3.4">subscript</csymbol><ci id="S2.SS1.p7.4.m4.1.1.3.4.2.cmml" xref="S2.SS1.p7.4.m4.1.1.3.4.2">𝑦</ci><ci id="S2.SS1.p7.4.m4.1.1.3.4.3.cmml" xref="S2.SS1.p7.4.m4.1.1.3.4.3">𝑗</ci></apply><apply id="S2.SS1.p7.4.m4.1.1.3.5.cmml" xref="S2.SS1.p7.4.m4.1.1.3.5"><csymbol cd="ambiguous" id="S2.SS1.p7.4.m4.1.1.3.5.1.cmml" xref="S2.SS1.p7.4.m4.1.1.3.5">subscript</csymbol><ci id="S2.SS1.p7.4.m4.1.1.3.5.2.cmml" xref="S2.SS1.p7.4.m4.1.1.3.5.2">𝑦</ci><apply id="S2.SS1.p7.4.m4.1.1.3.5.3.cmml" xref="S2.SS1.p7.4.m4.1.1.3.5.3"><plus id="S2.SS1.p7.4.m4.1.1.3.5.3.1.cmml" xref="S2.SS1.p7.4.m4.1.1.3.5.3.1"></plus><ci id="S2.SS1.p7.4.m4.1.1.3.5.3.2.cmml" xref="S2.SS1.p7.4.m4.1.1.3.5.3.2">𝑗</ci><cn id="S2.SS1.p7.4.m4.1.1.3.5.3.3.cmml" type="integer" xref="S2.SS1.p7.4.m4.1.1.3.5.3.3">1</cn></apply></apply><ci id="S2.SS1.p7.4.m4.1.1.3.6.cmml" xref="S2.SS1.p7.4.m4.1.1.3.6">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.4.m4.1c">{\bf y}=\ldots y_{j-1}y_{j}y_{j+1}\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.4.m4.1d">bold_y = … italic_y start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT …</annotation></semantics></math> given by <math alttext="y_{k}=x_{k+1}" class="ltx_Math" display="inline" id="S2.SS1.p7.5.m5.1"><semantics id="S2.SS1.p7.5.m5.1a"><mrow id="S2.SS1.p7.5.m5.1.1" xref="S2.SS1.p7.5.m5.1.1.cmml"><msub id="S2.SS1.p7.5.m5.1.1.2" xref="S2.SS1.p7.5.m5.1.1.2.cmml"><mi id="S2.SS1.p7.5.m5.1.1.2.2" xref="S2.SS1.p7.5.m5.1.1.2.2.cmml">y</mi><mi id="S2.SS1.p7.5.m5.1.1.2.3" xref="S2.SS1.p7.5.m5.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.p7.5.m5.1.1.1" xref="S2.SS1.p7.5.m5.1.1.1.cmml">=</mo><msub id="S2.SS1.p7.5.m5.1.1.3" xref="S2.SS1.p7.5.m5.1.1.3.cmml"><mi id="S2.SS1.p7.5.m5.1.1.3.2" xref="S2.SS1.p7.5.m5.1.1.3.2.cmml">x</mi><mrow id="S2.SS1.p7.5.m5.1.1.3.3" xref="S2.SS1.p7.5.m5.1.1.3.3.cmml"><mi id="S2.SS1.p7.5.m5.1.1.3.3.2" xref="S2.SS1.p7.5.m5.1.1.3.3.2.cmml">k</mi><mo id="S2.SS1.p7.5.m5.1.1.3.3.1" xref="S2.SS1.p7.5.m5.1.1.3.3.1.cmml">+</mo><mn id="S2.SS1.p7.5.m5.1.1.3.3.3" xref="S2.SS1.p7.5.m5.1.1.3.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.5.m5.1b"><apply id="S2.SS1.p7.5.m5.1.1.cmml" xref="S2.SS1.p7.5.m5.1.1"><eq id="S2.SS1.p7.5.m5.1.1.1.cmml" xref="S2.SS1.p7.5.m5.1.1.1"></eq><apply id="S2.SS1.p7.5.m5.1.1.2.cmml" xref="S2.SS1.p7.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p7.5.m5.1.1.2.1.cmml" xref="S2.SS1.p7.5.m5.1.1.2">subscript</csymbol><ci id="S2.SS1.p7.5.m5.1.1.2.2.cmml" xref="S2.SS1.p7.5.m5.1.1.2.2">𝑦</ci><ci id="S2.SS1.p7.5.m5.1.1.2.3.cmml" xref="S2.SS1.p7.5.m5.1.1.2.3">𝑘</ci></apply><apply id="S2.SS1.p7.5.m5.1.1.3.cmml" xref="S2.SS1.p7.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p7.5.m5.1.1.3.1.cmml" xref="S2.SS1.p7.5.m5.1.1.3">subscript</csymbol><ci id="S2.SS1.p7.5.m5.1.1.3.2.cmml" xref="S2.SS1.p7.5.m5.1.1.3.2">𝑥</ci><apply id="S2.SS1.p7.5.m5.1.1.3.3.cmml" xref="S2.SS1.p7.5.m5.1.1.3.3"><plus id="S2.SS1.p7.5.m5.1.1.3.3.1.cmml" xref="S2.SS1.p7.5.m5.1.1.3.3.1"></plus><ci id="S2.SS1.p7.5.m5.1.1.3.3.2.cmml" xref="S2.SS1.p7.5.m5.1.1.3.3.2">𝑘</ci><cn id="S2.SS1.p7.5.m5.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.p7.5.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.5.m5.1c">y_{k}=x_{k+1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.5.m5.1d">italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math> for all indices <math alttext="k\in\mathbb{Z}" class="ltx_Math" display="inline" id="S2.SS1.p7.6.m6.1"><semantics id="S2.SS1.p7.6.m6.1a"><mrow id="S2.SS1.p7.6.m6.1.1" xref="S2.SS1.p7.6.m6.1.1.cmml"><mi id="S2.SS1.p7.6.m6.1.1.2" xref="S2.SS1.p7.6.m6.1.1.2.cmml">k</mi><mo id="S2.SS1.p7.6.m6.1.1.1" xref="S2.SS1.p7.6.m6.1.1.1.cmml">∈</mo><mi id="S2.SS1.p7.6.m6.1.1.3" xref="S2.SS1.p7.6.m6.1.1.3.cmml">ℤ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.6.m6.1b"><apply id="S2.SS1.p7.6.m6.1.1.cmml" xref="S2.SS1.p7.6.m6.1.1"><in id="S2.SS1.p7.6.m6.1.1.1.cmml" xref="S2.SS1.p7.6.m6.1.1.1"></in><ci id="S2.SS1.p7.6.m6.1.1.2.cmml" xref="S2.SS1.p7.6.m6.1.1.2">𝑘</ci><ci id="S2.SS1.p7.6.m6.1.1.3.cmml" xref="S2.SS1.p7.6.m6.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.6.m6.1c">k\in\mathbb{Z}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.6.m6.1d">italic_k ∈ blackboard_Z</annotation></semantics></math>. Similarly, <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p7.7.m7.1"><semantics id="S2.SS1.p7.7.m7.1a"><msup id="S2.SS1.p7.7.m7.1.1" xref="S2.SS1.p7.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.7.m7.1.1.2" xref="S2.SS1.p7.7.m7.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p7.7.m7.1.1.3" xref="S2.SS1.p7.7.m7.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.7.m7.1b"><apply id="S2.SS1.p7.7.m7.1.1.cmml" xref="S2.SS1.p7.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p7.7.m7.1.1.1.cmml" xref="S2.SS1.p7.7.m7.1.1">superscript</csymbol><ci id="S2.SS1.p7.7.m7.1.1.2.cmml" xref="S2.SS1.p7.7.m7.1.1.2">𝒜</ci><ci id="S2.SS1.p7.7.m7.1.1.3.cmml" xref="S2.SS1.p7.7.m7.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.7.m7.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.7.m7.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is naturally equipped with the product topology (with respect to the discrete topology on <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.SS1.p7.8.m8.1"><semantics id="S2.SS1.p7.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.8.m8.1.1" xref="S2.SS1.p7.8.m8.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.8.m8.1b"><ci id="S2.SS1.p7.8.m8.1.1.cmml" xref="S2.SS1.p7.8.m8.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.8.m8.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.8.m8.1d">caligraphic_A</annotation></semantics></math>), which makes <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p7.9.m9.1"><semantics id="S2.SS1.p7.9.m9.1a"><msup id="S2.SS1.p7.9.m9.1.1" xref="S2.SS1.p7.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.9.m9.1.1.2" xref="S2.SS1.p7.9.m9.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p7.9.m9.1.1.3" xref="S2.SS1.p7.9.m9.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.9.m9.1b"><apply id="S2.SS1.p7.9.m9.1.1.cmml" xref="S2.SS1.p7.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p7.9.m9.1.1.1.cmml" xref="S2.SS1.p7.9.m9.1.1">superscript</csymbol><ci id="S2.SS1.p7.9.m9.1.1.2.cmml" xref="S2.SS1.p7.9.m9.1.1.2">𝒜</ci><ci id="S2.SS1.p7.9.m9.1.1.3.cmml" xref="S2.SS1.p7.9.m9.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.9.m9.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.9.m9.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> into a compact space, indeed a Cantor set (unless <math alttext="\mbox{card}(\cal A)=1" class="ltx_Math" display="inline" id="S2.SS1.p7.10.m10.1"><semantics id="S2.SS1.p7.10.m10.1a"><mrow id="S2.SS1.p7.10.m10.1.2" xref="S2.SS1.p7.10.m10.1.2.cmml"><mrow id="S2.SS1.p7.10.m10.1.2.2" xref="S2.SS1.p7.10.m10.1.2.2.cmml"><mtext id="S2.SS1.p7.10.m10.1.2.2.2" xref="S2.SS1.p7.10.m10.1.2.2.2a.cmml">card</mtext><mo id="S2.SS1.p7.10.m10.1.2.2.1" xref="S2.SS1.p7.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p7.10.m10.1.2.2.3.2" xref="S2.SS1.p7.10.m10.1.2.2.cmml"><mo id="S2.SS1.p7.10.m10.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.p7.10.m10.1.2.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.10.m10.1.1" xref="S2.SS1.p7.10.m10.1.1.cmml">𝒜</mi><mo id="S2.SS1.p7.10.m10.1.2.2.3.2.2" stretchy="false" xref="S2.SS1.p7.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p7.10.m10.1.2.1" xref="S2.SS1.p7.10.m10.1.2.1.cmml">=</mo><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p7.10.m10.1.2.3" mathvariant="script" xref="S2.SS1.p7.10.m10.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.10.m10.1b"><apply id="S2.SS1.p7.10.m10.1.2.cmml" xref="S2.SS1.p7.10.m10.1.2"><eq id="S2.SS1.p7.10.m10.1.2.1.cmml" xref="S2.SS1.p7.10.m10.1.2.1"></eq><apply id="S2.SS1.p7.10.m10.1.2.2.cmml" xref="S2.SS1.p7.10.m10.1.2.2"><times id="S2.SS1.p7.10.m10.1.2.2.1.cmml" xref="S2.SS1.p7.10.m10.1.2.2.1"></times><ci id="S2.SS1.p7.10.m10.1.2.2.2a.cmml" xref="S2.SS1.p7.10.m10.1.2.2.2"><mtext id="S2.SS1.p7.10.m10.1.2.2.2.cmml" xref="S2.SS1.p7.10.m10.1.2.2.2">card</mtext></ci><ci id="S2.SS1.p7.10.m10.1.1.cmml" xref="S2.SS1.p7.10.m10.1.1">𝒜</ci></apply><cn id="S2.SS1.p7.10.m10.1.2.3.cmml" type="integer" xref="S2.SS1.p7.10.m10.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.10.m10.1c">\mbox{card}(\cal A)=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.10.m10.1d">card ( caligraphic_A ) = caligraphic_1</annotation></semantics></math>). For any <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p7.11.m11.1"><semantics id="S2.SS1.p7.11.m11.1a"><mrow id="S2.SS1.p7.11.m11.1.1" xref="S2.SS1.p7.11.m11.1.1.cmml"><mi id="S2.SS1.p7.11.m11.1.1.2" xref="S2.SS1.p7.11.m11.1.1.2.cmml">w</mi><mo id="S2.SS1.p7.11.m11.1.1.1" xref="S2.SS1.p7.11.m11.1.1.1.cmml">∈</mo><msup id="S2.SS1.p7.11.m11.1.1.3" xref="S2.SS1.p7.11.m11.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.11.m11.1.1.3.2" xref="S2.SS1.p7.11.m11.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p7.11.m11.1.1.3.3" xref="S2.SS1.p7.11.m11.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.11.m11.1b"><apply id="S2.SS1.p7.11.m11.1.1.cmml" xref="S2.SS1.p7.11.m11.1.1"><in id="S2.SS1.p7.11.m11.1.1.1.cmml" xref="S2.SS1.p7.11.m11.1.1.1"></in><ci id="S2.SS1.p7.11.m11.1.1.2.cmml" xref="S2.SS1.p7.11.m11.1.1.2">𝑤</ci><apply id="S2.SS1.p7.11.m11.1.1.3.cmml" xref="S2.SS1.p7.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p7.11.m11.1.1.3.1.cmml" xref="S2.SS1.p7.11.m11.1.1.3">superscript</csymbol><ci id="S2.SS1.p7.11.m11.1.1.3.2.cmml" xref="S2.SS1.p7.11.m11.1.1.3.2">𝒜</ci><times id="S2.SS1.p7.11.m11.1.1.3.3.cmml" xref="S2.SS1.p7.11.m11.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.11.m11.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.11.m11.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the <span class="ltx_text ltx_font_italic" id="S2.SS1.p7.17.2">cylinder</span> <math alttext="[w]\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p7.12.m12.1"><semantics id="S2.SS1.p7.12.m12.1a"><mrow id="S2.SS1.p7.12.m12.1.2" xref="S2.SS1.p7.12.m12.1.2.cmml"><mrow id="S2.SS1.p7.12.m12.1.2.2.2" xref="S2.SS1.p7.12.m12.1.2.2.1.cmml"><mo id="S2.SS1.p7.12.m12.1.2.2.2.1" stretchy="false" xref="S2.SS1.p7.12.m12.1.2.2.1.1.cmml">[</mo><mi id="S2.SS1.p7.12.m12.1.1" xref="S2.SS1.p7.12.m12.1.1.cmml">w</mi><mo id="S2.SS1.p7.12.m12.1.2.2.2.2" stretchy="false" xref="S2.SS1.p7.12.m12.1.2.2.1.1.cmml">]</mo></mrow><mo id="S2.SS1.p7.12.m12.1.2.1" xref="S2.SS1.p7.12.m12.1.2.1.cmml">⊆</mo><msup id="S2.SS1.p7.12.m12.1.2.3" xref="S2.SS1.p7.12.m12.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.12.m12.1.2.3.2" xref="S2.SS1.p7.12.m12.1.2.3.2.cmml">𝒜</mi><mi id="S2.SS1.p7.12.m12.1.2.3.3" xref="S2.SS1.p7.12.m12.1.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.12.m12.1b"><apply id="S2.SS1.p7.12.m12.1.2.cmml" xref="S2.SS1.p7.12.m12.1.2"><subset id="S2.SS1.p7.12.m12.1.2.1.cmml" xref="S2.SS1.p7.12.m12.1.2.1"></subset><apply id="S2.SS1.p7.12.m12.1.2.2.1.cmml" xref="S2.SS1.p7.12.m12.1.2.2.2"><csymbol cd="latexml" id="S2.SS1.p7.12.m12.1.2.2.1.1.cmml" xref="S2.SS1.p7.12.m12.1.2.2.2.1">delimited-[]</csymbol><ci id="S2.SS1.p7.12.m12.1.1.cmml" xref="S2.SS1.p7.12.m12.1.1">𝑤</ci></apply><apply id="S2.SS1.p7.12.m12.1.2.3.cmml" xref="S2.SS1.p7.12.m12.1.2.3"><csymbol cd="ambiguous" id="S2.SS1.p7.12.m12.1.2.3.1.cmml" xref="S2.SS1.p7.12.m12.1.2.3">superscript</csymbol><ci id="S2.SS1.p7.12.m12.1.2.3.2.cmml" xref="S2.SS1.p7.12.m12.1.2.3.2">𝒜</ci><ci id="S2.SS1.p7.12.m12.1.2.3.3.cmml" xref="S2.SS1.p7.12.m12.1.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.12.m12.1c">[w]\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.12.m12.1d">[ italic_w ] ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is open and closed (and thus compact); it consists of all biinfinite words <math alttext="{\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots" class="ltx_Math" display="inline" id="S2.SS1.p7.13.m13.1"><semantics id="S2.SS1.p7.13.m13.1a"><mrow id="S2.SS1.p7.13.m13.1.1" xref="S2.SS1.p7.13.m13.1.1.cmml"><mi id="S2.SS1.p7.13.m13.1.1.2" xref="S2.SS1.p7.13.m13.1.1.2.cmml">𝐱</mi><mo id="S2.SS1.p7.13.m13.1.1.1" xref="S2.SS1.p7.13.m13.1.1.1.cmml">=</mo><mrow id="S2.SS1.p7.13.m13.1.1.3" xref="S2.SS1.p7.13.m13.1.1.3.cmml"><mi id="S2.SS1.p7.13.m13.1.1.3.2" mathvariant="normal" xref="S2.SS1.p7.13.m13.1.1.3.2.cmml">…</mi><mo id="S2.SS1.p7.13.m13.1.1.3.1" xref="S2.SS1.p7.13.m13.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.13.m13.1.1.3.3" xref="S2.SS1.p7.13.m13.1.1.3.3.cmml"><mi id="S2.SS1.p7.13.m13.1.1.3.3.2" xref="S2.SS1.p7.13.m13.1.1.3.3.2.cmml">x</mi><mrow id="S2.SS1.p7.13.m13.1.1.3.3.3" xref="S2.SS1.p7.13.m13.1.1.3.3.3.cmml"><mi id="S2.SS1.p7.13.m13.1.1.3.3.3.2" xref="S2.SS1.p7.13.m13.1.1.3.3.3.2.cmml">i</mi><mo id="S2.SS1.p7.13.m13.1.1.3.3.3.1" xref="S2.SS1.p7.13.m13.1.1.3.3.3.1.cmml">−</mo><mn id="S2.SS1.p7.13.m13.1.1.3.3.3.3" xref="S2.SS1.p7.13.m13.1.1.3.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p7.13.m13.1.1.3.1a" xref="S2.SS1.p7.13.m13.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.13.m13.1.1.3.4" xref="S2.SS1.p7.13.m13.1.1.3.4.cmml"><mi id="S2.SS1.p7.13.m13.1.1.3.4.2" xref="S2.SS1.p7.13.m13.1.1.3.4.2.cmml">x</mi><mi id="S2.SS1.p7.13.m13.1.1.3.4.3" xref="S2.SS1.p7.13.m13.1.1.3.4.3.cmml">i</mi></msub><mo id="S2.SS1.p7.13.m13.1.1.3.1b" xref="S2.SS1.p7.13.m13.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p7.13.m13.1.1.3.5" xref="S2.SS1.p7.13.m13.1.1.3.5.cmml"><mi id="S2.SS1.p7.13.m13.1.1.3.5.2" xref="S2.SS1.p7.13.m13.1.1.3.5.2.cmml">x</mi><mrow id="S2.SS1.p7.13.m13.1.1.3.5.3" xref="S2.SS1.p7.13.m13.1.1.3.5.3.cmml"><mi id="S2.SS1.p7.13.m13.1.1.3.5.3.2" xref="S2.SS1.p7.13.m13.1.1.3.5.3.2.cmml">i</mi><mo id="S2.SS1.p7.13.m13.1.1.3.5.3.1" xref="S2.SS1.p7.13.m13.1.1.3.5.3.1.cmml">+</mo><mn id="S2.SS1.p7.13.m13.1.1.3.5.3.3" xref="S2.SS1.p7.13.m13.1.1.3.5.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS1.p7.13.m13.1.1.3.1c" xref="S2.SS1.p7.13.m13.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p7.13.m13.1.1.3.6" mathvariant="normal" xref="S2.SS1.p7.13.m13.1.1.3.6.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.13.m13.1b"><apply id="S2.SS1.p7.13.m13.1.1.cmml" xref="S2.SS1.p7.13.m13.1.1"><eq id="S2.SS1.p7.13.m13.1.1.1.cmml" xref="S2.SS1.p7.13.m13.1.1.1"></eq><ci id="S2.SS1.p7.13.m13.1.1.2.cmml" xref="S2.SS1.p7.13.m13.1.1.2">𝐱</ci><apply id="S2.SS1.p7.13.m13.1.1.3.cmml" xref="S2.SS1.p7.13.m13.1.1.3"><times id="S2.SS1.p7.13.m13.1.1.3.1.cmml" xref="S2.SS1.p7.13.m13.1.1.3.1"></times><ci id="S2.SS1.p7.13.m13.1.1.3.2.cmml" xref="S2.SS1.p7.13.m13.1.1.3.2">…</ci><apply id="S2.SS1.p7.13.m13.1.1.3.3.cmml" xref="S2.SS1.p7.13.m13.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p7.13.m13.1.1.3.3.1.cmml" xref="S2.SS1.p7.13.m13.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p7.13.m13.1.1.3.3.2.cmml" xref="S2.SS1.p7.13.m13.1.1.3.3.2">𝑥</ci><apply id="S2.SS1.p7.13.m13.1.1.3.3.3.cmml" xref="S2.SS1.p7.13.m13.1.1.3.3.3"><minus id="S2.SS1.p7.13.m13.1.1.3.3.3.1.cmml" xref="S2.SS1.p7.13.m13.1.1.3.3.3.1"></minus><ci id="S2.SS1.p7.13.m13.1.1.3.3.3.2.cmml" xref="S2.SS1.p7.13.m13.1.1.3.3.3.2">𝑖</ci><cn id="S2.SS1.p7.13.m13.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p7.13.m13.1.1.3.3.3.3">1</cn></apply></apply><apply id="S2.SS1.p7.13.m13.1.1.3.4.cmml" xref="S2.SS1.p7.13.m13.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS1.p7.13.m13.1.1.3.4.1.cmml" xref="S2.SS1.p7.13.m13.1.1.3.4">subscript</csymbol><ci id="S2.SS1.p7.13.m13.1.1.3.4.2.cmml" xref="S2.SS1.p7.13.m13.1.1.3.4.2">𝑥</ci><ci id="S2.SS1.p7.13.m13.1.1.3.4.3.cmml" xref="S2.SS1.p7.13.m13.1.1.3.4.3">𝑖</ci></apply><apply id="S2.SS1.p7.13.m13.1.1.3.5.cmml" xref="S2.SS1.p7.13.m13.1.1.3.5"><csymbol cd="ambiguous" id="S2.SS1.p7.13.m13.1.1.3.5.1.cmml" xref="S2.SS1.p7.13.m13.1.1.3.5">subscript</csymbol><ci id="S2.SS1.p7.13.m13.1.1.3.5.2.cmml" xref="S2.SS1.p7.13.m13.1.1.3.5.2">𝑥</ci><apply id="S2.SS1.p7.13.m13.1.1.3.5.3.cmml" xref="S2.SS1.p7.13.m13.1.1.3.5.3"><plus id="S2.SS1.p7.13.m13.1.1.3.5.3.1.cmml" xref="S2.SS1.p7.13.m13.1.1.3.5.3.1"></plus><ci id="S2.SS1.p7.13.m13.1.1.3.5.3.2.cmml" xref="S2.SS1.p7.13.m13.1.1.3.5.3.2">𝑖</ci><cn id="S2.SS1.p7.13.m13.1.1.3.5.3.3.cmml" type="integer" xref="S2.SS1.p7.13.m13.1.1.3.5.3.3">1</cn></apply></apply><ci id="S2.SS1.p7.13.m13.1.1.3.6.cmml" xref="S2.SS1.p7.13.m13.1.1.3.6">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.13.m13.1c">{\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.13.m13.1d">bold_x = … italic_x start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT …</annotation></semantics></math> with <math alttext="{\bf x}_{[1,|w|]}=w" class="ltx_Math" display="inline" id="S2.SS1.p7.14.m14.3"><semantics id="S2.SS1.p7.14.m14.3a"><mrow id="S2.SS1.p7.14.m14.3.4" xref="S2.SS1.p7.14.m14.3.4.cmml"><msub id="S2.SS1.p7.14.m14.3.4.2" xref="S2.SS1.p7.14.m14.3.4.2.cmml"><mi id="S2.SS1.p7.14.m14.3.4.2.2" xref="S2.SS1.p7.14.m14.3.4.2.2.cmml">𝐱</mi><mrow id="S2.SS1.p7.14.m14.3.3.3.3" xref="S2.SS1.p7.14.m14.3.3.3.4.cmml"><mo id="S2.SS1.p7.14.m14.3.3.3.3.2" stretchy="false" xref="S2.SS1.p7.14.m14.3.3.3.4.cmml">[</mo><mn id="S2.SS1.p7.14.m14.2.2.2.2" xref="S2.SS1.p7.14.m14.2.2.2.2.cmml">1</mn><mo id="S2.SS1.p7.14.m14.3.3.3.3.3" xref="S2.SS1.p7.14.m14.3.3.3.4.cmml">,</mo><mrow id="S2.SS1.p7.14.m14.3.3.3.3.1.2" xref="S2.SS1.p7.14.m14.3.3.3.3.1.1.cmml"><mo id="S2.SS1.p7.14.m14.3.3.3.3.1.2.1" stretchy="false" xref="S2.SS1.p7.14.m14.3.3.3.3.1.1.1.cmml">|</mo><mi id="S2.SS1.p7.14.m14.1.1.1.1" xref="S2.SS1.p7.14.m14.1.1.1.1.cmml">w</mi><mo id="S2.SS1.p7.14.m14.3.3.3.3.1.2.2" stretchy="false" xref="S2.SS1.p7.14.m14.3.3.3.3.1.1.1.cmml">|</mo></mrow><mo id="S2.SS1.p7.14.m14.3.3.3.3.4" stretchy="false" xref="S2.SS1.p7.14.m14.3.3.3.4.cmml">]</mo></mrow></msub><mo id="S2.SS1.p7.14.m14.3.4.1" xref="S2.SS1.p7.14.m14.3.4.1.cmml">=</mo><mi id="S2.SS1.p7.14.m14.3.4.3" xref="S2.SS1.p7.14.m14.3.4.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.14.m14.3b"><apply id="S2.SS1.p7.14.m14.3.4.cmml" xref="S2.SS1.p7.14.m14.3.4"><eq id="S2.SS1.p7.14.m14.3.4.1.cmml" xref="S2.SS1.p7.14.m14.3.4.1"></eq><apply id="S2.SS1.p7.14.m14.3.4.2.cmml" xref="S2.SS1.p7.14.m14.3.4.2"><csymbol cd="ambiguous" id="S2.SS1.p7.14.m14.3.4.2.1.cmml" xref="S2.SS1.p7.14.m14.3.4.2">subscript</csymbol><ci id="S2.SS1.p7.14.m14.3.4.2.2.cmml" xref="S2.SS1.p7.14.m14.3.4.2.2">𝐱</ci><interval closure="closed" id="S2.SS1.p7.14.m14.3.3.3.4.cmml" xref="S2.SS1.p7.14.m14.3.3.3.3"><cn id="S2.SS1.p7.14.m14.2.2.2.2.cmml" type="integer" xref="S2.SS1.p7.14.m14.2.2.2.2">1</cn><apply id="S2.SS1.p7.14.m14.3.3.3.3.1.1.cmml" xref="S2.SS1.p7.14.m14.3.3.3.3.1.2"><abs id="S2.SS1.p7.14.m14.3.3.3.3.1.1.1.cmml" xref="S2.SS1.p7.14.m14.3.3.3.3.1.2.1"></abs><ci id="S2.SS1.p7.14.m14.1.1.1.1.cmml" xref="S2.SS1.p7.14.m14.1.1.1.1">𝑤</ci></apply></interval></apply><ci id="S2.SS1.p7.14.m14.3.4.3.cmml" xref="S2.SS1.p7.14.m14.3.4.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.14.m14.3c">{\bf x}_{[1,|w|]}=w</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.14.m14.3d">bold_x start_POSTSUBSCRIPT [ 1 , | italic_w | ] end_POSTSUBSCRIPT = italic_w</annotation></semantics></math>. The set of all cylinders and their shift-translates constitutes a basis for the topology on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p7.15.m15.1"><semantics id="S2.SS1.p7.15.m15.1a"><msup id="S2.SS1.p7.15.m15.1.1" xref="S2.SS1.p7.15.m15.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.15.m15.1.1.2" xref="S2.SS1.p7.15.m15.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p7.15.m15.1.1.3" xref="S2.SS1.p7.15.m15.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.15.m15.1b"><apply id="S2.SS1.p7.15.m15.1.1.cmml" xref="S2.SS1.p7.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS1.p7.15.m15.1.1.1.cmml" xref="S2.SS1.p7.15.m15.1.1">superscript</csymbol><ci id="S2.SS1.p7.15.m15.1.1.2.cmml" xref="S2.SS1.p7.15.m15.1.1.2">𝒜</ci><ci id="S2.SS1.p7.15.m15.1.1.3.cmml" xref="S2.SS1.p7.15.m15.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.15.m15.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.15.m15.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. The shift operator <math alttext="T" class="ltx_Math" display="inline" id="S2.SS1.p7.16.m16.1"><semantics id="S2.SS1.p7.16.m16.1a"><mi id="S2.SS1.p7.16.m16.1.1" xref="S2.SS1.p7.16.m16.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.16.m16.1b"><ci id="S2.SS1.p7.16.m16.1.1.cmml" xref="S2.SS1.p7.16.m16.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.16.m16.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.16.m16.1d">italic_T</annotation></semantics></math> acts as homeomorphism on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p7.17.m17.1"><semantics id="S2.SS1.p7.17.m17.1a"><msup id="S2.SS1.p7.17.m17.1.1" xref="S2.SS1.p7.17.m17.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p7.17.m17.1.1.2" xref="S2.SS1.p7.17.m17.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p7.17.m17.1.1.3" xref="S2.SS1.p7.17.m17.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p7.17.m17.1b"><apply id="S2.SS1.p7.17.m17.1.1.cmml" xref="S2.SS1.p7.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS1.p7.17.m17.1.1.1.cmml" xref="S2.SS1.p7.17.m17.1.1">superscript</csymbol><ci id="S2.SS1.p7.17.m17.1.1.2.cmml" xref="S2.SS1.p7.17.m17.1.1.2">𝒜</ci><ci id="S2.SS1.p7.17.m17.1.1.3.cmml" xref="S2.SS1.p7.17.m17.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p7.17.m17.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p7.17.m17.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p8"> <p class="ltx_p" id="S2.SS1.p8.6">A subset <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p8.1.m1.1"><semantics id="S2.SS1.p8.1.m1.1a"><mrow id="S2.SS1.p8.1.m1.1.1" xref="S2.SS1.p8.1.m1.1.1.cmml"><mi id="S2.SS1.p8.1.m1.1.1.2" xref="S2.SS1.p8.1.m1.1.1.2.cmml">X</mi><mo id="S2.SS1.p8.1.m1.1.1.1" xref="S2.SS1.p8.1.m1.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p8.1.m1.1.1.3" xref="S2.SS1.p8.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.1.m1.1.1.3.2" xref="S2.SS1.p8.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p8.1.m1.1.1.3.3" xref="S2.SS1.p8.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.1.m1.1b"><apply id="S2.SS1.p8.1.m1.1.1.cmml" xref="S2.SS1.p8.1.m1.1.1"><subset id="S2.SS1.p8.1.m1.1.1.1.cmml" xref="S2.SS1.p8.1.m1.1.1.1"></subset><ci id="S2.SS1.p8.1.m1.1.1.2.cmml" xref="S2.SS1.p8.1.m1.1.1.2">𝑋</ci><apply id="S2.SS1.p8.1.m1.1.1.3.cmml" xref="S2.SS1.p8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p8.1.m1.1.1.3.1.cmml" xref="S2.SS1.p8.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS1.p8.1.m1.1.1.3.2.cmml" xref="S2.SS1.p8.1.m1.1.1.3.2">𝒜</ci><ci id="S2.SS1.p8.1.m1.1.1.3.3.cmml" xref="S2.SS1.p8.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.1.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.1.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is called a <span class="ltx_text ltx_font_italic" id="S2.SS1.p8.6.2">subshift</span> if <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p8.2.m2.1"><semantics id="S2.SS1.p8.2.m2.1a"><mi id="S2.SS1.p8.2.m2.1.1" xref="S2.SS1.p8.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.2.m2.1b"><ci id="S2.SS1.p8.2.m2.1.1.cmml" xref="S2.SS1.p8.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.2.m2.1d">italic_X</annotation></semantics></math> is non-empty, closed, and if it satisfies <math alttext="T(X)=X" class="ltx_Math" display="inline" id="S2.SS1.p8.3.m3.1"><semantics id="S2.SS1.p8.3.m3.1a"><mrow id="S2.SS1.p8.3.m3.1.2" xref="S2.SS1.p8.3.m3.1.2.cmml"><mrow id="S2.SS1.p8.3.m3.1.2.2" xref="S2.SS1.p8.3.m3.1.2.2.cmml"><mi id="S2.SS1.p8.3.m3.1.2.2.2" xref="S2.SS1.p8.3.m3.1.2.2.2.cmml">T</mi><mo id="S2.SS1.p8.3.m3.1.2.2.1" xref="S2.SS1.p8.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p8.3.m3.1.2.2.3.2" xref="S2.SS1.p8.3.m3.1.2.2.cmml"><mo id="S2.SS1.p8.3.m3.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.p8.3.m3.1.2.2.cmml">(</mo><mi id="S2.SS1.p8.3.m3.1.1" xref="S2.SS1.p8.3.m3.1.1.cmml">X</mi><mo id="S2.SS1.p8.3.m3.1.2.2.3.2.2" stretchy="false" xref="S2.SS1.p8.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p8.3.m3.1.2.1" xref="S2.SS1.p8.3.m3.1.2.1.cmml">=</mo><mi id="S2.SS1.p8.3.m3.1.2.3" xref="S2.SS1.p8.3.m3.1.2.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.3.m3.1b"><apply id="S2.SS1.p8.3.m3.1.2.cmml" xref="S2.SS1.p8.3.m3.1.2"><eq id="S2.SS1.p8.3.m3.1.2.1.cmml" xref="S2.SS1.p8.3.m3.1.2.1"></eq><apply id="S2.SS1.p8.3.m3.1.2.2.cmml" xref="S2.SS1.p8.3.m3.1.2.2"><times id="S2.SS1.p8.3.m3.1.2.2.1.cmml" xref="S2.SS1.p8.3.m3.1.2.2.1"></times><ci id="S2.SS1.p8.3.m3.1.2.2.2.cmml" xref="S2.SS1.p8.3.m3.1.2.2.2">𝑇</ci><ci id="S2.SS1.p8.3.m3.1.1.cmml" xref="S2.SS1.p8.3.m3.1.1">𝑋</ci></apply><ci id="S2.SS1.p8.3.m3.1.2.3.cmml" xref="S2.SS1.p8.3.m3.1.2.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.3.m3.1c">T(X)=X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.3.m3.1d">italic_T ( italic_X ) = italic_X</annotation></semantics></math>. To any infinite language <math alttext="\cal L\subseteq\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p8.4.m4.1"><semantics id="S2.SS1.p8.4.m4.1a"><mrow id="S2.SS1.p8.4.m4.1.1" xref="S2.SS1.p8.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.4.m4.1.1.2" xref="S2.SS1.p8.4.m4.1.1.2.cmml">ℒ</mi><mo id="S2.SS1.p8.4.m4.1.1.1" xref="S2.SS1.p8.4.m4.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p8.4.m4.1.1.3" xref="S2.SS1.p8.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.4.m4.1.1.3.2" xref="S2.SS1.p8.4.m4.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p8.4.m4.1.1.3.3" xref="S2.SS1.p8.4.m4.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.4.m4.1b"><apply id="S2.SS1.p8.4.m4.1.1.cmml" xref="S2.SS1.p8.4.m4.1.1"><subset id="S2.SS1.p8.4.m4.1.1.1.cmml" xref="S2.SS1.p8.4.m4.1.1.1"></subset><ci id="S2.SS1.p8.4.m4.1.1.2.cmml" xref="S2.SS1.p8.4.m4.1.1.2">ℒ</ci><apply id="S2.SS1.p8.4.m4.1.1.3.cmml" xref="S2.SS1.p8.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p8.4.m4.1.1.3.1.cmml" xref="S2.SS1.p8.4.m4.1.1.3">superscript</csymbol><ci id="S2.SS1.p8.4.m4.1.1.3.2.cmml" xref="S2.SS1.p8.4.m4.1.1.3.2">𝒜</ci><times id="S2.SS1.p8.4.m4.1.1.3.3.cmml" xref="S2.SS1.p8.4.m4.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.4.m4.1c">\cal L\subseteq\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.4.m4.1d">caligraphic_L ⊆ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> there is canonically associated the subshift <math alttext="X(\cal L)" class="ltx_Math" display="inline" id="S2.SS1.p8.5.m5.1"><semantics id="S2.SS1.p8.5.m5.1a"><mrow id="S2.SS1.p8.5.m5.1.2" xref="S2.SS1.p8.5.m5.1.2.cmml"><mi id="S2.SS1.p8.5.m5.1.2.2" xref="S2.SS1.p8.5.m5.1.2.2.cmml">X</mi><mo id="S2.SS1.p8.5.m5.1.2.1" xref="S2.SS1.p8.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p8.5.m5.1.2.3.2" xref="S2.SS1.p8.5.m5.1.2.cmml"><mo id="S2.SS1.p8.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS1.p8.5.m5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.5.m5.1.1" xref="S2.SS1.p8.5.m5.1.1.cmml">ℒ</mi><mo id="S2.SS1.p8.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS1.p8.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.5.m5.1b"><apply id="S2.SS1.p8.5.m5.1.2.cmml" xref="S2.SS1.p8.5.m5.1.2"><times id="S2.SS1.p8.5.m5.1.2.1.cmml" xref="S2.SS1.p8.5.m5.1.2.1"></times><ci id="S2.SS1.p8.5.m5.1.2.2.cmml" xref="S2.SS1.p8.5.m5.1.2.2">𝑋</ci><ci id="S2.SS1.p8.5.m5.1.1.cmml" xref="S2.SS1.p8.5.m5.1.1">ℒ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.5.m5.1c">X(\cal L)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.5.m5.1d">italic_X ( caligraphic_L )</annotation></semantics></math> <span class="ltx_text ltx_font_italic" id="S2.SS1.p8.6.1">generated by <math alttext="\cal L" class="ltx_Math" display="inline" id="S2.SS1.p8.6.1.m1.1"><semantics id="S2.SS1.p8.6.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.6.1.m1.1.1" xref="S2.SS1.p8.6.1.m1.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.6.1.m1.1b"><ci id="S2.SS1.p8.6.1.m1.1.1.cmml" xref="S2.SS1.p8.6.1.m1.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.6.1.m1.1c">\cal L</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.6.1.m1.1d">caligraphic_L</annotation></semantics></math></span>: It is defined through</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\bf x}\in X(\cal L)\quad\Longleftrightarrow\quad\cal L({\bf x})\subseteq\cal L% ^{f}\,," class="ltx_Math" display="block" id="S2.Ex1.m1.4"><semantics id="S2.Ex1.m1.4a"><mrow id="S2.Ex1.m1.4.4.1"><mrow id="S2.Ex1.m1.4.4.1.1.2" xref="S2.Ex1.m1.4.4.1.1.3.cmml"><mrow id="S2.Ex1.m1.4.4.1.1.1.1" xref="S2.Ex1.m1.4.4.1.1.1.1.cmml"><mi id="S2.Ex1.m1.4.4.1.1.1.1.3" xref="S2.Ex1.m1.4.4.1.1.1.1.3.cmml">𝐱</mi><mo id="S2.Ex1.m1.4.4.1.1.1.1.2" xref="S2.Ex1.m1.4.4.1.1.1.1.2.cmml">∈</mo><mrow id="S2.Ex1.m1.4.4.1.1.1.1.1.1" xref="S2.Ex1.m1.4.4.1.1.1.1.1.2.cmml"><mrow id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.2" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.2.cmml">X</mi><mo id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.3.2" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml">ℒ</mi><mo id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mspace id="S2.Ex1.m1.4.4.1.1.1.1.1.1.2" width="1em" xref="S2.Ex1.m1.4.4.1.1.1.1.1.2.cmml"></mspace><mo id="S2.Ex1.m1.3.3" stretchy="false" xref="S2.Ex1.m1.3.3.cmml">⟺</mo></mrow></mrow><mspace id="S2.Ex1.m1.4.4.1.1.2.3" width="1em" xref="S2.Ex1.m1.4.4.1.1.3a.cmml"></mspace><mrow id="S2.Ex1.m1.4.4.1.1.2.2" xref="S2.Ex1.m1.4.4.1.1.2.2.cmml"><mrow id="S2.Ex1.m1.4.4.1.1.2.2.2" xref="S2.Ex1.m1.4.4.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex1.m1.4.4.1.1.2.2.2.2" xref="S2.Ex1.m1.4.4.1.1.2.2.2.2.cmml">ℒ</mi><mo id="S2.Ex1.m1.4.4.1.1.2.2.2.1" xref="S2.Ex1.m1.4.4.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.4.4.1.1.2.2.2.3.2" xref="S2.Ex1.m1.4.4.1.1.2.2.2.cmml"><mo id="S2.Ex1.m1.4.4.1.1.2.2.2.3.2.1" stretchy="false" xref="S2.Ex1.m1.4.4.1.1.2.2.2.cmml">(</mo><mi id="S2.Ex1.m1.2.2" xref="S2.Ex1.m1.2.2.cmml">𝐱</mi><mo id="S2.Ex1.m1.4.4.1.1.2.2.2.3.2.2" stretchy="false" xref="S2.Ex1.m1.4.4.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.4.4.1.1.2.2.1" xref="S2.Ex1.m1.4.4.1.1.2.2.1.cmml">⊆</mo><msup id="S2.Ex1.m1.4.4.1.1.2.2.3" xref="S2.Ex1.m1.4.4.1.1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex1.m1.4.4.1.1.2.2.3.2" xref="S2.Ex1.m1.4.4.1.1.2.2.3.2.cmml">ℒ</mi><mi class="ltx_font_mathcaligraphic" id="S2.Ex1.m1.4.4.1.1.2.2.3.3" xref="S2.Ex1.m1.4.4.1.1.2.2.3.3.cmml">𝒻</mi></msup></mrow></mrow><mo id="S2.Ex1.m1.4.4.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex1.m1.4b"><apply id="S2.Ex1.m1.4.4.1.1.3.cmml" xref="S2.Ex1.m1.4.4.1.1.2"><csymbol cd="ambiguous" id="S2.Ex1.m1.4.4.1.1.3a.cmml" xref="S2.Ex1.m1.4.4.1.1.2.3">formulae-sequence</csymbol><apply id="S2.Ex1.m1.4.4.1.1.1.1.cmml" xref="S2.Ex1.m1.4.4.1.1.1.1"><in id="S2.Ex1.m1.4.4.1.1.1.1.2.cmml" xref="S2.Ex1.m1.4.4.1.1.1.1.2"></in><ci id="S2.Ex1.m1.4.4.1.1.1.1.3.cmml" xref="S2.Ex1.m1.4.4.1.1.1.1.3">𝐱</ci><list id="S2.Ex1.m1.4.4.1.1.1.1.1.2.cmml" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1"><apply id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1"><times id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.1"></times><ci id="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex1.m1.4.4.1.1.1.1.1.1.1.2">𝑋</ci><ci id="S2.Ex1.m1.1.1.cmml" xref="S2.Ex1.m1.1.1">ℒ</ci></apply><ci id="S2.Ex1.m1.3.3.cmml" xref="S2.Ex1.m1.3.3">⟺</ci></list></apply><apply id="S2.Ex1.m1.4.4.1.1.2.2.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2"><subset id="S2.Ex1.m1.4.4.1.1.2.2.1.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2.1"></subset><apply id="S2.Ex1.m1.4.4.1.1.2.2.2.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2.2"><times id="S2.Ex1.m1.4.4.1.1.2.2.2.1.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2.2.1"></times><ci id="S2.Ex1.m1.4.4.1.1.2.2.2.2.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2.2.2">ℒ</ci><ci id="S2.Ex1.m1.2.2.cmml" xref="S2.Ex1.m1.2.2">𝐱</ci></apply><apply id="S2.Ex1.m1.4.4.1.1.2.2.3.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2.3"><csymbol cd="ambiguous" id="S2.Ex1.m1.4.4.1.1.2.2.3.1.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2.3">superscript</csymbol><ci id="S2.Ex1.m1.4.4.1.1.2.2.3.2.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2.3.2">ℒ</ci><ci id="S2.Ex1.m1.4.4.1.1.2.2.3.3.cmml" xref="S2.Ex1.m1.4.4.1.1.2.2.3.3">𝒻</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m1.4c">{\bf x}\in X(\cal L)\quad\Longleftrightarrow\quad\cal L({\bf x})\subseteq\cal L% ^{f}\,,</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m1.4d">bold_x ∈ italic_X ( caligraphic_L ) ⟺ caligraphic_L ( bold_x ) ⊆ caligraphic_L start_POSTSUPERSCRIPT caligraphic_f end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p8.13">where <math alttext="\cal L^{f}" class="ltx_Math" display="inline" id="S2.SS1.p8.7.m1.1"><semantics id="S2.SS1.p8.7.m1.1a"><msup id="S2.SS1.p8.7.m1.1.1" xref="S2.SS1.p8.7.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.7.m1.1.1.2" xref="S2.SS1.p8.7.m1.1.1.2.cmml">ℒ</mi><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.7.m1.1.1.3" xref="S2.SS1.p8.7.m1.1.1.3.cmml">𝒻</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.7.m1.1b"><apply id="S2.SS1.p8.7.m1.1.1.cmml" xref="S2.SS1.p8.7.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p8.7.m1.1.1.1.cmml" xref="S2.SS1.p8.7.m1.1.1">superscript</csymbol><ci id="S2.SS1.p8.7.m1.1.1.2.cmml" xref="S2.SS1.p8.7.m1.1.1.2">ℒ</ci><ci id="S2.SS1.p8.7.m1.1.1.3.cmml" xref="S2.SS1.p8.7.m1.1.1.3">𝒻</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.7.m1.1c">\cal L^{f}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.7.m1.1d">caligraphic_L start_POSTSUPERSCRIPT caligraphic_f end_POSTSUPERSCRIPT</annotation></semantics></math> denotes the <span class="ltx_text ltx_font_italic" id="S2.SS1.p8.13.1">factorial closure</span> of <math alttext="\cal L" class="ltx_Math" display="inline" id="S2.SS1.p8.8.m2.1"><semantics id="S2.SS1.p8.8.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.8.m2.1.1" xref="S2.SS1.p8.8.m2.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.8.m2.1b"><ci id="S2.SS1.p8.8.m2.1.1.cmml" xref="S2.SS1.p8.8.m2.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.8.m2.1c">\cal L</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.8.m2.1d">caligraphic_L</annotation></semantics></math>, which is the language obtained from <math alttext="\cal L" class="ltx_Math" display="inline" id="S2.SS1.p8.9.m3.1"><semantics id="S2.SS1.p8.9.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.9.m3.1.1" xref="S2.SS1.p8.9.m3.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.9.m3.1b"><ci id="S2.SS1.p8.9.m3.1.1.cmml" xref="S2.SS1.p8.9.m3.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.9.m3.1c">\cal L</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.9.m3.1d">caligraphic_L</annotation></semantics></math> by adding in all <span class="ltx_text ltx_font_italic" id="S2.SS1.p8.13.2">factors</span> (= subwords) of any word <math alttext="w\in\cal L" class="ltx_Math" display="inline" id="S2.SS1.p8.10.m4.1"><semantics id="S2.SS1.p8.10.m4.1a"><mrow id="S2.SS1.p8.10.m4.1.1" xref="S2.SS1.p8.10.m4.1.1.cmml"><mi id="S2.SS1.p8.10.m4.1.1.2" xref="S2.SS1.p8.10.m4.1.1.2.cmml">w</mi><mo id="S2.SS1.p8.10.m4.1.1.1" xref="S2.SS1.p8.10.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.10.m4.1.1.3" xref="S2.SS1.p8.10.m4.1.1.3.cmml">ℒ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.10.m4.1b"><apply id="S2.SS1.p8.10.m4.1.1.cmml" xref="S2.SS1.p8.10.m4.1.1"><in id="S2.SS1.p8.10.m4.1.1.1.cmml" xref="S2.SS1.p8.10.m4.1.1.1"></in><ci id="S2.SS1.p8.10.m4.1.1.2.cmml" xref="S2.SS1.p8.10.m4.1.1.2">𝑤</ci><ci id="S2.SS1.p8.10.m4.1.1.3.cmml" xref="S2.SS1.p8.10.m4.1.1.3">ℒ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.10.m4.1c">w\in\cal L</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.10.m4.1d">italic_w ∈ caligraphic_L</annotation></semantics></math>. Conversely, every subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p8.11.m5.1"><semantics id="S2.SS1.p8.11.m5.1a"><mrow id="S2.SS1.p8.11.m5.1.1" xref="S2.SS1.p8.11.m5.1.1.cmml"><mi id="S2.SS1.p8.11.m5.1.1.2" xref="S2.SS1.p8.11.m5.1.1.2.cmml">X</mi><mo id="S2.SS1.p8.11.m5.1.1.1" xref="S2.SS1.p8.11.m5.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p8.11.m5.1.1.3" xref="S2.SS1.p8.11.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.11.m5.1.1.3.2" xref="S2.SS1.p8.11.m5.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p8.11.m5.1.1.3.3" xref="S2.SS1.p8.11.m5.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.11.m5.1b"><apply id="S2.SS1.p8.11.m5.1.1.cmml" xref="S2.SS1.p8.11.m5.1.1"><subset id="S2.SS1.p8.11.m5.1.1.1.cmml" xref="S2.SS1.p8.11.m5.1.1.1"></subset><ci id="S2.SS1.p8.11.m5.1.1.2.cmml" xref="S2.SS1.p8.11.m5.1.1.2">𝑋</ci><apply id="S2.SS1.p8.11.m5.1.1.3.cmml" xref="S2.SS1.p8.11.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p8.11.m5.1.1.3.1.cmml" xref="S2.SS1.p8.11.m5.1.1.3">superscript</csymbol><ci id="S2.SS1.p8.11.m5.1.1.3.2.cmml" xref="S2.SS1.p8.11.m5.1.1.3.2">𝒜</ci><ci id="S2.SS1.p8.11.m5.1.1.3.3.cmml" xref="S2.SS1.p8.11.m5.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.11.m5.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.11.m5.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> determines an <span class="ltx_text ltx_font_italic" id="S2.SS1.p8.13.3">associated subshift language</span> <math alttext="\cal L(X):=\bigcup\{\cal L({\bf x})\mid{\bf x}\in X\}" class="ltx_Math" display="inline" id="S2.SS1.p8.12.m6.4"><semantics id="S2.SS1.p8.12.m6.4a"><mrow id="S2.SS1.p8.12.m6.4.4" xref="S2.SS1.p8.12.m6.4.4.cmml"><mrow id="S2.SS1.p8.12.m6.4.4.4" xref="S2.SS1.p8.12.m6.4.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.12.m6.4.4.4.2" xref="S2.SS1.p8.12.m6.4.4.4.2.cmml">ℒ</mi><mo id="S2.SS1.p8.12.m6.4.4.4.1" xref="S2.SS1.p8.12.m6.4.4.4.1.cmml">⁢</mo><mrow id="S2.SS1.p8.12.m6.4.4.4.3.2" xref="S2.SS1.p8.12.m6.4.4.4.cmml"><mo id="S2.SS1.p8.12.m6.4.4.4.3.2.1" stretchy="false" xref="S2.SS1.p8.12.m6.4.4.4.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.12.m6.1.1" xref="S2.SS1.p8.12.m6.1.1.cmml">𝒳</mi><mo id="S2.SS1.p8.12.m6.4.4.4.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS1.p8.12.m6.4.4.4.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p8.12.m6.4.4.3" rspace="0.111em" xref="S2.SS1.p8.12.m6.4.4.3.cmml">:=</mo><mrow id="S2.SS1.p8.12.m6.4.4.2" xref="S2.SS1.p8.12.m6.4.4.2.cmml"><mo id="S2.SS1.p8.12.m6.4.4.2.3" rspace="0em" xref="S2.SS1.p8.12.m6.4.4.2.3.cmml">⋃</mo><mrow id="S2.SS1.p8.12.m6.4.4.2.2.2" xref="S2.SS1.p8.12.m6.4.4.2.2.3.cmml"><mo id="S2.SS1.p8.12.m6.4.4.2.2.2.3" stretchy="false" xref="S2.SS1.p8.12.m6.4.4.2.2.3.1.cmml">{</mo><mrow id="S2.SS1.p8.12.m6.3.3.1.1.1.1" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.12.m6.3.3.1.1.1.1.2" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1.2.cmml">ℒ</mi><mo id="S2.SS1.p8.12.m6.3.3.1.1.1.1.1" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p8.12.m6.3.3.1.1.1.1.3.2" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1.cmml"><mo id="S2.SS1.p8.12.m6.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1.cmml">(</mo><mi id="S2.SS1.p8.12.m6.2.2" xref="S2.SS1.p8.12.m6.2.2.cmml">𝐱</mi><mo id="S2.SS1.p8.12.m6.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo fence="true" id="S2.SS1.p8.12.m6.4.4.2.2.2.4" lspace="0em" rspace="0em" xref="S2.SS1.p8.12.m6.4.4.2.2.3.1.cmml">∣</mo><mrow id="S2.SS1.p8.12.m6.4.4.2.2.2.2" xref="S2.SS1.p8.12.m6.4.4.2.2.2.2.cmml"><mi id="S2.SS1.p8.12.m6.4.4.2.2.2.2.2" xref="S2.SS1.p8.12.m6.4.4.2.2.2.2.2.cmml">𝐱</mi><mo id="S2.SS1.p8.12.m6.4.4.2.2.2.2.1" xref="S2.SS1.p8.12.m6.4.4.2.2.2.2.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.12.m6.4.4.2.2.2.2.3" xref="S2.SS1.p8.12.m6.4.4.2.2.2.2.3.cmml">𝒳</mi></mrow><mo id="S2.SS1.p8.12.m6.4.4.2.2.2.5" stretchy="false" xref="S2.SS1.p8.12.m6.4.4.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.12.m6.4b"><apply id="S2.SS1.p8.12.m6.4.4.cmml" xref="S2.SS1.p8.12.m6.4.4"><csymbol cd="latexml" id="S2.SS1.p8.12.m6.4.4.3.cmml" xref="S2.SS1.p8.12.m6.4.4.3">assign</csymbol><apply id="S2.SS1.p8.12.m6.4.4.4.cmml" xref="S2.SS1.p8.12.m6.4.4.4"><times id="S2.SS1.p8.12.m6.4.4.4.1.cmml" xref="S2.SS1.p8.12.m6.4.4.4.1"></times><ci id="S2.SS1.p8.12.m6.4.4.4.2.cmml" xref="S2.SS1.p8.12.m6.4.4.4.2">ℒ</ci><ci id="S2.SS1.p8.12.m6.1.1.cmml" xref="S2.SS1.p8.12.m6.1.1">𝒳</ci></apply><apply id="S2.SS1.p8.12.m6.4.4.2.cmml" xref="S2.SS1.p8.12.m6.4.4.2"><union id="S2.SS1.p8.12.m6.4.4.2.3.cmml" xref="S2.SS1.p8.12.m6.4.4.2.3"></union><apply id="S2.SS1.p8.12.m6.4.4.2.2.3.cmml" xref="S2.SS1.p8.12.m6.4.4.2.2.2"><csymbol cd="latexml" id="S2.SS1.p8.12.m6.4.4.2.2.3.1.cmml" xref="S2.SS1.p8.12.m6.4.4.2.2.2.3">conditional-set</csymbol><apply id="S2.SS1.p8.12.m6.3.3.1.1.1.1.cmml" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1"><times id="S2.SS1.p8.12.m6.3.3.1.1.1.1.1.cmml" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1.1"></times><ci id="S2.SS1.p8.12.m6.3.3.1.1.1.1.2.cmml" xref="S2.SS1.p8.12.m6.3.3.1.1.1.1.2">ℒ</ci><ci id="S2.SS1.p8.12.m6.2.2.cmml" xref="S2.SS1.p8.12.m6.2.2">𝐱</ci></apply><apply id="S2.SS1.p8.12.m6.4.4.2.2.2.2.cmml" xref="S2.SS1.p8.12.m6.4.4.2.2.2.2"><in id="S2.SS1.p8.12.m6.4.4.2.2.2.2.1.cmml" xref="S2.SS1.p8.12.m6.4.4.2.2.2.2.1"></in><ci id="S2.SS1.p8.12.m6.4.4.2.2.2.2.2.cmml" xref="S2.SS1.p8.12.m6.4.4.2.2.2.2.2">𝐱</ci><ci id="S2.SS1.p8.12.m6.4.4.2.2.2.2.3.cmml" xref="S2.SS1.p8.12.m6.4.4.2.2.2.2.3">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.12.m6.4c">\cal L(X):=\bigcup\{\cal L({\bf x})\mid{\bf x}\in X\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.12.m6.4d">caligraphic_L ( caligraphic_X ) := ⋃ { caligraphic_L ( bold_x ) ∣ bold_x ∈ caligraphic_X }</annotation></semantics></math>, which is infinite and equal to its factorial closure, so that one has <math alttext="X=X(\cal L(X))" class="ltx_Math" display="inline" id="S2.SS1.p8.13.m7.2"><semantics id="S2.SS1.p8.13.m7.2a"><mrow id="S2.SS1.p8.13.m7.2.2" xref="S2.SS1.p8.13.m7.2.2.cmml"><mi id="S2.SS1.p8.13.m7.2.2.3" xref="S2.SS1.p8.13.m7.2.2.3.cmml">X</mi><mo id="S2.SS1.p8.13.m7.2.2.2" xref="S2.SS1.p8.13.m7.2.2.2.cmml">=</mo><mrow id="S2.SS1.p8.13.m7.2.2.1" xref="S2.SS1.p8.13.m7.2.2.1.cmml"><mi id="S2.SS1.p8.13.m7.2.2.1.3" xref="S2.SS1.p8.13.m7.2.2.1.3.cmml">X</mi><mo id="S2.SS1.p8.13.m7.2.2.1.2" xref="S2.SS1.p8.13.m7.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p8.13.m7.2.2.1.1.1" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.cmml"><mo id="S2.SS1.p8.13.m7.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p8.13.m7.2.2.1.1.1.1" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.13.m7.2.2.1.1.1.1.2" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.2.cmml">ℒ</mi><mo id="S2.SS1.p8.13.m7.2.2.1.1.1.1.1" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p8.13.m7.2.2.1.1.1.1.3.2" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.cmml"><mo id="S2.SS1.p8.13.m7.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p8.13.m7.1.1" xref="S2.SS1.p8.13.m7.1.1.cmml">𝒳</mi><mo id="S2.SS1.p8.13.m7.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p8.13.m7.2.2.1.1.1.3" stretchy="false" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p8.13.m7.2b"><apply id="S2.SS1.p8.13.m7.2.2.cmml" xref="S2.SS1.p8.13.m7.2.2"><eq id="S2.SS1.p8.13.m7.2.2.2.cmml" xref="S2.SS1.p8.13.m7.2.2.2"></eq><ci id="S2.SS1.p8.13.m7.2.2.3.cmml" xref="S2.SS1.p8.13.m7.2.2.3">𝑋</ci><apply id="S2.SS1.p8.13.m7.2.2.1.cmml" xref="S2.SS1.p8.13.m7.2.2.1"><times id="S2.SS1.p8.13.m7.2.2.1.2.cmml" xref="S2.SS1.p8.13.m7.2.2.1.2"></times><ci id="S2.SS1.p8.13.m7.2.2.1.3.cmml" xref="S2.SS1.p8.13.m7.2.2.1.3">𝑋</ci><apply id="S2.SS1.p8.13.m7.2.2.1.1.1.1.cmml" xref="S2.SS1.p8.13.m7.2.2.1.1.1"><times id="S2.SS1.p8.13.m7.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.1"></times><ci id="S2.SS1.p8.13.m7.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p8.13.m7.2.2.1.1.1.1.2">ℒ</ci><ci id="S2.SS1.p8.13.m7.1.1.cmml" xref="S2.SS1.p8.13.m7.1.1">𝒳</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p8.13.m7.2c">X=X(\cal L(X))</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p8.13.m7.2d">italic_X = italic_X ( caligraphic_L ( caligraphic_X ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p9"> <p class="ltx_p" id="S2.SS1.p9.3">A subshift <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p9.1.m1.1"><semantics id="S2.SS1.p9.1.m1.1a"><mi id="S2.SS1.p9.1.m1.1.1" xref="S2.SS1.p9.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.1.m1.1b"><ci id="S2.SS1.p9.1.m1.1.1.cmml" xref="S2.SS1.p9.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.1.m1.1d">italic_X</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S2.SS1.p9.3.1">minimal</span> if for any element <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S2.SS1.p9.2.m2.1"><semantics id="S2.SS1.p9.2.m2.1a"><mrow id="S2.SS1.p9.2.m2.1.1" xref="S2.SS1.p9.2.m2.1.1.cmml"><mi id="S2.SS1.p9.2.m2.1.1.2" xref="S2.SS1.p9.2.m2.1.1.2.cmml">𝐱</mi><mo id="S2.SS1.p9.2.m2.1.1.1" xref="S2.SS1.p9.2.m2.1.1.1.cmml">∈</mo><mi id="S2.SS1.p9.2.m2.1.1.3" xref="S2.SS1.p9.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.2.m2.1b"><apply id="S2.SS1.p9.2.m2.1.1.cmml" xref="S2.SS1.p9.2.m2.1.1"><in id="S2.SS1.p9.2.m2.1.1.1.cmml" xref="S2.SS1.p9.2.m2.1.1.1"></in><ci id="S2.SS1.p9.2.m2.1.1.2.cmml" xref="S2.SS1.p9.2.m2.1.1.2">𝐱</ci><ci id="S2.SS1.p9.2.m2.1.1.3.cmml" xref="S2.SS1.p9.2.m2.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.2.m2.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.2.m2.1d">bold_x ∈ italic_X</annotation></semantics></math> the set <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p9.3.m3.1"><semantics id="S2.SS1.p9.3.m3.1a"><mi id="S2.SS1.p9.3.m3.1.1" xref="S2.SS1.p9.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.3.m3.1b"><ci id="S2.SS1.p9.3.m3.1.1.cmml" xref="S2.SS1.p9.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.3.m3.1d">italic_X</annotation></semantics></math> is equal to the closure of the <span class="ltx_text ltx_font_italic" id="S2.SS1.p9.3.2">shift-orbit</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\cal O({\bf x}):=\{T^{n}({\bf x})\mid n\in\mathbb{Z}\}\,." class="ltx_Math" display="block" id="S2.Ex2.m1.3"><semantics id="S2.Ex2.m1.3a"><mrow id="S2.Ex2.m1.3.3.1" xref="S2.Ex2.m1.3.3.1.1.cmml"><mrow id="S2.Ex2.m1.3.3.1.1" xref="S2.Ex2.m1.3.3.1.1.cmml"><mrow id="S2.Ex2.m1.3.3.1.1.4" xref="S2.Ex2.m1.3.3.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex2.m1.3.3.1.1.4.2" xref="S2.Ex2.m1.3.3.1.1.4.2.cmml">𝒪</mi><mo id="S2.Ex2.m1.3.3.1.1.4.1" xref="S2.Ex2.m1.3.3.1.1.4.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.3.3.1.1.4.3.2" xref="S2.Ex2.m1.3.3.1.1.4.cmml"><mo id="S2.Ex2.m1.3.3.1.1.4.3.2.1" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.4.cmml">(</mo><mi id="S2.Ex2.m1.1.1" xref="S2.Ex2.m1.1.1.cmml">𝐱</mi><mo id="S2.Ex2.m1.3.3.1.1.4.3.2.2" rspace="0.278em" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.4.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.3.3.1.1.3" rspace="0.278em" xref="S2.Ex2.m1.3.3.1.1.3.cmml">:=</mo><mrow id="S2.Ex2.m1.3.3.1.1.2.2" xref="S2.Ex2.m1.3.3.1.1.2.3.cmml"><mo id="S2.Ex2.m1.3.3.1.1.2.2.3" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.2.3.1.cmml">{</mo><mrow id="S2.Ex2.m1.3.3.1.1.1.1.1" xref="S2.Ex2.m1.3.3.1.1.1.1.1.cmml"><msup id="S2.Ex2.m1.3.3.1.1.1.1.1.2" xref="S2.Ex2.m1.3.3.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex2.m1.3.3.1.1.1.1.1.2.2" xref="S2.Ex2.m1.3.3.1.1.1.1.1.2.2.cmml">𝒯</mi><mi class="ltx_font_mathcaligraphic" id="S2.Ex2.m1.3.3.1.1.1.1.1.2.3" xref="S2.Ex2.m1.3.3.1.1.1.1.1.2.3.cmml">𝓃</mi></msup><mo id="S2.Ex2.m1.3.3.1.1.1.1.1.1" xref="S2.Ex2.m1.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.3.3.1.1.1.1.1.3.2" xref="S2.Ex2.m1.3.3.1.1.1.1.1.cmml"><mo id="S2.Ex2.m1.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.1.1.1.cmml">(</mo><mi id="S2.Ex2.m1.2.2" xref="S2.Ex2.m1.2.2.cmml">𝐱</mi><mo id="S2.Ex2.m1.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo fence="true" id="S2.Ex2.m1.3.3.1.1.2.2.4" lspace="0em" rspace="0em" xref="S2.Ex2.m1.3.3.1.1.2.3.1.cmml">∣</mo><mrow id="S2.Ex2.m1.3.3.1.1.2.2.2" xref="S2.Ex2.m1.3.3.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex2.m1.3.3.1.1.2.2.2.2" xref="S2.Ex2.m1.3.3.1.1.2.2.2.2.cmml">𝓃</mi><mo id="S2.Ex2.m1.3.3.1.1.2.2.2.1" xref="S2.Ex2.m1.3.3.1.1.2.2.2.1.cmml">∈</mo><mi id="S2.Ex2.m1.3.3.1.1.2.2.2.3" xref="S2.Ex2.m1.3.3.1.1.2.2.2.3.cmml">ℤ</mi></mrow><mo id="S2.Ex2.m1.3.3.1.1.2.2.5" stretchy="false" xref="S2.Ex2.m1.3.3.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S2.Ex2.m1.3.3.1.2" lspace="0.170em" xref="S2.Ex2.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex2.m1.3b"><apply id="S2.Ex2.m1.3.3.1.1.cmml" xref="S2.Ex2.m1.3.3.1"><csymbol cd="latexml" id="S2.Ex2.m1.3.3.1.1.3.cmml" xref="S2.Ex2.m1.3.3.1.1.3">assign</csymbol><apply id="S2.Ex2.m1.3.3.1.1.4.cmml" xref="S2.Ex2.m1.3.3.1.1.4"><times id="S2.Ex2.m1.3.3.1.1.4.1.cmml" xref="S2.Ex2.m1.3.3.1.1.4.1"></times><ci id="S2.Ex2.m1.3.3.1.1.4.2.cmml" xref="S2.Ex2.m1.3.3.1.1.4.2">𝒪</ci><ci id="S2.Ex2.m1.1.1.cmml" xref="S2.Ex2.m1.1.1">𝐱</ci></apply><apply id="S2.Ex2.m1.3.3.1.1.2.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2"><csymbol cd="latexml" id="S2.Ex2.m1.3.3.1.1.2.3.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.3">conditional-set</csymbol><apply id="S2.Ex2.m1.3.3.1.1.1.1.1.cmml" xref="S2.Ex2.m1.3.3.1.1.1.1.1"><times id="S2.Ex2.m1.3.3.1.1.1.1.1.1.cmml" xref="S2.Ex2.m1.3.3.1.1.1.1.1.1"></times><apply id="S2.Ex2.m1.3.3.1.1.1.1.1.2.cmml" xref="S2.Ex2.m1.3.3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex2.m1.3.3.1.1.1.1.1.2.1.cmml" xref="S2.Ex2.m1.3.3.1.1.1.1.1.2">superscript</csymbol><ci id="S2.Ex2.m1.3.3.1.1.1.1.1.2.2.cmml" xref="S2.Ex2.m1.3.3.1.1.1.1.1.2.2">𝒯</ci><ci id="S2.Ex2.m1.3.3.1.1.1.1.1.2.3.cmml" xref="S2.Ex2.m1.3.3.1.1.1.1.1.2.3">𝓃</ci></apply><ci id="S2.Ex2.m1.2.2.cmml" xref="S2.Ex2.m1.2.2">𝐱</ci></apply><apply id="S2.Ex2.m1.3.3.1.1.2.2.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.2"><in id="S2.Ex2.m1.3.3.1.1.2.2.2.1.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.2.1"></in><ci id="S2.Ex2.m1.3.3.1.1.2.2.2.2.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.2.2">𝓃</ci><ci id="S2.Ex2.m1.3.3.1.1.2.2.2.3.cmml" xref="S2.Ex2.m1.3.3.1.1.2.2.2.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2.m1.3c">\cal O({\bf x}):=\{T^{n}({\bf x})\mid n\in\mathbb{Z}\}\,.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2.m1.3d">caligraphic_O ( bold_x ) := { caligraphic_T start_POSTSUPERSCRIPT caligraphic_n end_POSTSUPERSCRIPT ( bold_x ) ∣ caligraphic_n ∈ blackboard_Z } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p9.11">Important particular examples are minimal subshifts that are finite: They consist of a single orbit <math alttext="\cal O({\bf w})" class="ltx_Math" display="inline" id="S2.SS1.p9.4.m1.1"><semantics id="S2.SS1.p9.4.m1.1a"><mrow id="S2.SS1.p9.4.m1.1.2" xref="S2.SS1.p9.4.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p9.4.m1.1.2.2" xref="S2.SS1.p9.4.m1.1.2.2.cmml">𝒪</mi><mo id="S2.SS1.p9.4.m1.1.2.1" xref="S2.SS1.p9.4.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p9.4.m1.1.2.3.2" xref="S2.SS1.p9.4.m1.1.2.cmml"><mo id="S2.SS1.p9.4.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p9.4.m1.1.2.cmml">(</mo><mi id="S2.SS1.p9.4.m1.1.1" xref="S2.SS1.p9.4.m1.1.1.cmml">𝐰</mi><mo id="S2.SS1.p9.4.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p9.4.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.4.m1.1b"><apply id="S2.SS1.p9.4.m1.1.2.cmml" xref="S2.SS1.p9.4.m1.1.2"><times id="S2.SS1.p9.4.m1.1.2.1.cmml" xref="S2.SS1.p9.4.m1.1.2.1"></times><ci id="S2.SS1.p9.4.m1.1.2.2.cmml" xref="S2.SS1.p9.4.m1.1.2.2">𝒪</ci><ci id="S2.SS1.p9.4.m1.1.1.cmml" xref="S2.SS1.p9.4.m1.1.1">𝐰</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.4.m1.1c">\cal O({\bf w})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.4.m1.1d">caligraphic_O ( bold_w )</annotation></semantics></math> that is given by the finitely many shift-translates of a periodic word <math alttext="{\bf w}=\ldots www\ldots" class="ltx_Math" display="inline" id="S2.SS1.p9.5.m2.1"><semantics id="S2.SS1.p9.5.m2.1a"><mrow id="S2.SS1.p9.5.m2.1.1" xref="S2.SS1.p9.5.m2.1.1.cmml"><mi id="S2.SS1.p9.5.m2.1.1.2" xref="S2.SS1.p9.5.m2.1.1.2.cmml">𝐰</mi><mo id="S2.SS1.p9.5.m2.1.1.1" xref="S2.SS1.p9.5.m2.1.1.1.cmml">=</mo><mrow id="S2.SS1.p9.5.m2.1.1.3" xref="S2.SS1.p9.5.m2.1.1.3.cmml"><mi id="S2.SS1.p9.5.m2.1.1.3.2" mathvariant="normal" xref="S2.SS1.p9.5.m2.1.1.3.2.cmml">…</mi><mo id="S2.SS1.p9.5.m2.1.1.3.1" xref="S2.SS1.p9.5.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p9.5.m2.1.1.3.3" xref="S2.SS1.p9.5.m2.1.1.3.3.cmml">w</mi><mo id="S2.SS1.p9.5.m2.1.1.3.1a" xref="S2.SS1.p9.5.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p9.5.m2.1.1.3.4" xref="S2.SS1.p9.5.m2.1.1.3.4.cmml">w</mi><mo id="S2.SS1.p9.5.m2.1.1.3.1b" xref="S2.SS1.p9.5.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p9.5.m2.1.1.3.5" xref="S2.SS1.p9.5.m2.1.1.3.5.cmml">w</mi><mo id="S2.SS1.p9.5.m2.1.1.3.1c" xref="S2.SS1.p9.5.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p9.5.m2.1.1.3.6" mathvariant="normal" xref="S2.SS1.p9.5.m2.1.1.3.6.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.5.m2.1b"><apply id="S2.SS1.p9.5.m2.1.1.cmml" xref="S2.SS1.p9.5.m2.1.1"><eq id="S2.SS1.p9.5.m2.1.1.1.cmml" xref="S2.SS1.p9.5.m2.1.1.1"></eq><ci id="S2.SS1.p9.5.m2.1.1.2.cmml" xref="S2.SS1.p9.5.m2.1.1.2">𝐰</ci><apply id="S2.SS1.p9.5.m2.1.1.3.cmml" xref="S2.SS1.p9.5.m2.1.1.3"><times id="S2.SS1.p9.5.m2.1.1.3.1.cmml" xref="S2.SS1.p9.5.m2.1.1.3.1"></times><ci id="S2.SS1.p9.5.m2.1.1.3.2.cmml" xref="S2.SS1.p9.5.m2.1.1.3.2">…</ci><ci id="S2.SS1.p9.5.m2.1.1.3.3.cmml" xref="S2.SS1.p9.5.m2.1.1.3.3">𝑤</ci><ci id="S2.SS1.p9.5.m2.1.1.3.4.cmml" xref="S2.SS1.p9.5.m2.1.1.3.4">𝑤</ci><ci id="S2.SS1.p9.5.m2.1.1.3.5.cmml" xref="S2.SS1.p9.5.m2.1.1.3.5">𝑤</ci><ci id="S2.SS1.p9.5.m2.1.1.3.6.cmml" xref="S2.SS1.p9.5.m2.1.1.3.6">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.5.m2.1c">{\bf w}=\ldots www\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.5.m2.1d">bold_w = … italic_w italic_w italic_w …</annotation></semantics></math>. To be specific, we fix the indexing of such a periodic word by the requirement that <math alttext="w" class="ltx_Math" display="inline" id="S2.SS1.p9.6.m3.1"><semantics id="S2.SS1.p9.6.m3.1a"><mi id="S2.SS1.p9.6.m3.1.1" xref="S2.SS1.p9.6.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.6.m3.1b"><ci id="S2.SS1.p9.6.m3.1.1.cmml" xref="S2.SS1.p9.6.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.6.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.6.m3.1d">italic_w</annotation></semantics></math> is a prefix of the positive half-word <math alttext="{\bf w}_{[1,\infty)}" class="ltx_Math" display="inline" id="S2.SS1.p9.7.m4.2"><semantics id="S2.SS1.p9.7.m4.2a"><msub id="S2.SS1.p9.7.m4.2.3" xref="S2.SS1.p9.7.m4.2.3.cmml"><mi id="S2.SS1.p9.7.m4.2.3.2" xref="S2.SS1.p9.7.m4.2.3.2.cmml">𝐰</mi><mrow id="S2.SS1.p9.7.m4.2.2.2.4" xref="S2.SS1.p9.7.m4.2.2.2.3.cmml"><mo id="S2.SS1.p9.7.m4.2.2.2.4.1" stretchy="false" xref="S2.SS1.p9.7.m4.2.2.2.3.cmml">[</mo><mn id="S2.SS1.p9.7.m4.1.1.1.1" xref="S2.SS1.p9.7.m4.1.1.1.1.cmml">1</mn><mo id="S2.SS1.p9.7.m4.2.2.2.4.2" xref="S2.SS1.p9.7.m4.2.2.2.3.cmml">,</mo><mi id="S2.SS1.p9.7.m4.2.2.2.2" mathvariant="normal" xref="S2.SS1.p9.7.m4.2.2.2.2.cmml">∞</mi><mo id="S2.SS1.p9.7.m4.2.2.2.4.3" stretchy="false" xref="S2.SS1.p9.7.m4.2.2.2.3.cmml">)</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.7.m4.2b"><apply id="S2.SS1.p9.7.m4.2.3.cmml" xref="S2.SS1.p9.7.m4.2.3"><csymbol cd="ambiguous" id="S2.SS1.p9.7.m4.2.3.1.cmml" xref="S2.SS1.p9.7.m4.2.3">subscript</csymbol><ci id="S2.SS1.p9.7.m4.2.3.2.cmml" xref="S2.SS1.p9.7.m4.2.3.2">𝐰</ci><interval closure="closed-open" id="S2.SS1.p9.7.m4.2.2.2.3.cmml" xref="S2.SS1.p9.7.m4.2.2.2.4"><cn id="S2.SS1.p9.7.m4.1.1.1.1.cmml" type="integer" xref="S2.SS1.p9.7.m4.1.1.1.1">1</cn><infinity id="S2.SS1.p9.7.m4.2.2.2.2.cmml" xref="S2.SS1.p9.7.m4.2.2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.7.m4.2c">{\bf w}_{[1,\infty)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.7.m4.2d">bold_w start_POSTSUBSCRIPT [ 1 , ∞ ) end_POSTSUBSCRIPT</annotation></semantics></math>; in this case <math alttext="{\bf w}" class="ltx_Math" display="inline" id="S2.SS1.p9.8.m5.1"><semantics id="S2.SS1.p9.8.m5.1a"><mi id="S2.SS1.p9.8.m5.1.1" xref="S2.SS1.p9.8.m5.1.1.cmml">𝐰</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.8.m5.1b"><ci id="S2.SS1.p9.8.m5.1.1.cmml" xref="S2.SS1.p9.8.m5.1.1">𝐰</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.8.m5.1c">{\bf w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.8.m5.1d">bold_w</annotation></semantics></math> is denoted by <math alttext="w^{\pm\infty}" class="ltx_Math" display="inline" id="S2.SS1.p9.9.m6.1"><semantics id="S2.SS1.p9.9.m6.1a"><msup id="S2.SS1.p9.9.m6.1.1" xref="S2.SS1.p9.9.m6.1.1.cmml"><mi id="S2.SS1.p9.9.m6.1.1.2" xref="S2.SS1.p9.9.m6.1.1.2.cmml">w</mi><mrow id="S2.SS1.p9.9.m6.1.1.3" xref="S2.SS1.p9.9.m6.1.1.3.cmml"><mo id="S2.SS1.p9.9.m6.1.1.3a" xref="S2.SS1.p9.9.m6.1.1.3.cmml">±</mo><mi id="S2.SS1.p9.9.m6.1.1.3.2" mathvariant="normal" xref="S2.SS1.p9.9.m6.1.1.3.2.cmml">∞</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p9.9.m6.1b"><apply id="S2.SS1.p9.9.m6.1.1.cmml" xref="S2.SS1.p9.9.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p9.9.m6.1.1.1.cmml" xref="S2.SS1.p9.9.m6.1.1">superscript</csymbol><ci id="S2.SS1.p9.9.m6.1.1.2.cmml" xref="S2.SS1.p9.9.m6.1.1.2">𝑤</ci><apply id="S2.SS1.p9.9.m6.1.1.3.cmml" xref="S2.SS1.p9.9.m6.1.1.3"><csymbol cd="latexml" id="S2.SS1.p9.9.m6.1.1.3.1.cmml" xref="S2.SS1.p9.9.m6.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS1.p9.9.m6.1.1.3.2.cmml" xref="S2.SS1.p9.9.m6.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p9.9.m6.1c">w^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.9.m6.1d">italic_w start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>, and we write <math alttext="{\bf w}_{[1,\infty)}=:w^{+\infty}" class="ltx_math_unparsed" display="inline" id="S2.SS1.p9.10.m7.2"><semantics id="S2.SS1.p9.10.m7.2a"><mrow id="S2.SS1.p9.10.m7.2b"><msub id="S2.SS1.p9.10.m7.2.3"><mi id="S2.SS1.p9.10.m7.2.3.2">𝐰</mi><mrow id="S2.SS1.p9.10.m7.2.2.2.4"><mo id="S2.SS1.p9.10.m7.2.2.2.4.1" stretchy="false">[</mo><mn id="S2.SS1.p9.10.m7.1.1.1.1">1</mn><mo id="S2.SS1.p9.10.m7.2.2.2.4.2">,</mo><mi id="S2.SS1.p9.10.m7.2.2.2.2" mathvariant="normal">∞</mi><mo id="S2.SS1.p9.10.m7.2.2.2.4.3" stretchy="false">)</mo></mrow></msub><mo id="S2.SS1.p9.10.m7.2.4" rspace="0em">=</mo><mo id="S2.SS1.p9.10.m7.2.5" rspace="0.278em">:</mo><msup id="S2.SS1.p9.10.m7.2.6"><mi id="S2.SS1.p9.10.m7.2.6.2">w</mi><mrow id="S2.SS1.p9.10.m7.2.6.3"><mo id="S2.SS1.p9.10.m7.2.6.3a">+</mo><mi id="S2.SS1.p9.10.m7.2.6.3.2" mathvariant="normal">∞</mi></mrow></msup></mrow><annotation encoding="application/x-tex" id="S2.SS1.p9.10.m7.2c">{\bf w}_{[1,\infty)}=:w^{+\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.10.m7.2d">bold_w start_POSTSUBSCRIPT [ 1 , ∞ ) end_POSTSUBSCRIPT = : italic_w start_POSTSUPERSCRIPT + ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="{\bf w}_{(-\infty,0]}=:w^{-\infty}" class="ltx_math_unparsed" display="inline" id="S2.SS1.p9.11.m8.2"><semantics id="S2.SS1.p9.11.m8.2a"><mrow id="S2.SS1.p9.11.m8.2b"><msub id="S2.SS1.p9.11.m8.2.3"><mi id="S2.SS1.p9.11.m8.2.3.2">𝐰</mi><mrow id="S2.SS1.p9.11.m8.2.2.2.2"><mo id="S2.SS1.p9.11.m8.2.2.2.2.2" stretchy="false">(</mo><mrow id="S2.SS1.p9.11.m8.2.2.2.2.1"><mo id="S2.SS1.p9.11.m8.2.2.2.2.1a">−</mo><mi id="S2.SS1.p9.11.m8.2.2.2.2.1.2" mathvariant="normal">∞</mi></mrow><mo id="S2.SS1.p9.11.m8.2.2.2.2.3">,</mo><mn id="S2.SS1.p9.11.m8.1.1.1.1">0</mn><mo id="S2.SS1.p9.11.m8.2.2.2.2.4" stretchy="false">]</mo></mrow></msub><mo id="S2.SS1.p9.11.m8.2.4" rspace="0em">=</mo><mo id="S2.SS1.p9.11.m8.2.5" rspace="0.278em">:</mo><msup id="S2.SS1.p9.11.m8.2.6"><mi id="S2.SS1.p9.11.m8.2.6.2">w</mi><mrow id="S2.SS1.p9.11.m8.2.6.3"><mo id="S2.SS1.p9.11.m8.2.6.3a">−</mo><mi id="S2.SS1.p9.11.m8.2.6.3.2" mathvariant="normal">∞</mi></mrow></msup></mrow><annotation encoding="application/x-tex" id="S2.SS1.p9.11.m8.2c">{\bf w}_{(-\infty,0]}=:w^{-\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p9.11.m8.2d">bold_w start_POSTSUBSCRIPT ( - ∞ , 0 ] end_POSTSUBSCRIPT = : italic_w start_POSTSUPERSCRIPT - ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p10"> <p class="ltx_p" id="S2.SS1.p10.6">The space <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.SS1.p10.1.m1.1"><semantics id="S2.SS1.p10.1.m1.1a"><mrow id="S2.SS1.p10.1.m1.1.2" xref="S2.SS1.p10.1.m1.1.2.cmml"><mi id="S2.SS1.p10.1.m1.1.2.2" mathvariant="normal" xref="S2.SS1.p10.1.m1.1.2.2.cmml">Σ</mi><mo id="S2.SS1.p10.1.m1.1.2.1" xref="S2.SS1.p10.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p10.1.m1.1.2.3.2" xref="S2.SS1.p10.1.m1.1.2.cmml"><mo id="S2.SS1.p10.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p10.1.m1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p10.1.m1.1.1" xref="S2.SS1.p10.1.m1.1.1.cmml">𝒜</mi><mo id="S2.SS1.p10.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p10.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p10.1.m1.1b"><apply id="S2.SS1.p10.1.m1.1.2.cmml" xref="S2.SS1.p10.1.m1.1.2"><times id="S2.SS1.p10.1.m1.1.2.1.cmml" xref="S2.SS1.p10.1.m1.1.2.1"></times><ci id="S2.SS1.p10.1.m1.1.2.2.cmml" xref="S2.SS1.p10.1.m1.1.2.2">Σ</ci><ci id="S2.SS1.p10.1.m1.1.1.cmml" xref="S2.SS1.p10.1.m1.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p10.1.m1.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p10.1.m1.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math> of all subshifts <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p10.2.m2.1"><semantics id="S2.SS1.p10.2.m2.1a"><mrow id="S2.SS1.p10.2.m2.1.1" xref="S2.SS1.p10.2.m2.1.1.cmml"><mi id="S2.SS1.p10.2.m2.1.1.2" xref="S2.SS1.p10.2.m2.1.1.2.cmml">X</mi><mo id="S2.SS1.p10.2.m2.1.1.1" xref="S2.SS1.p10.2.m2.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p10.2.m2.1.1.3" xref="S2.SS1.p10.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p10.2.m2.1.1.3.2" xref="S2.SS1.p10.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p10.2.m2.1.1.3.3" xref="S2.SS1.p10.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p10.2.m2.1b"><apply id="S2.SS1.p10.2.m2.1.1.cmml" xref="S2.SS1.p10.2.m2.1.1"><subset id="S2.SS1.p10.2.m2.1.1.1.cmml" xref="S2.SS1.p10.2.m2.1.1.1"></subset><ci id="S2.SS1.p10.2.m2.1.1.2.cmml" xref="S2.SS1.p10.2.m2.1.1.2">𝑋</ci><apply id="S2.SS1.p10.2.m2.1.1.3.cmml" xref="S2.SS1.p10.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p10.2.m2.1.1.3.1.cmml" xref="S2.SS1.p10.2.m2.1.1.3">superscript</csymbol><ci id="S2.SS1.p10.2.m2.1.1.3.2.cmml" xref="S2.SS1.p10.2.m2.1.1.3.2">𝒜</ci><ci id="S2.SS1.p10.2.m2.1.1.3.3.cmml" xref="S2.SS1.p10.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p10.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p10.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is naturally equipped with the partial order given by the inclusion; the minimal elements with respect to this partial order are precisely the minimal subshifts. The shift space <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p10.3.m3.1"><semantics id="S2.SS1.p10.3.m3.1a"><msup id="S2.SS1.p10.3.m3.1.1" xref="S2.SS1.p10.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p10.3.m3.1.1.2" xref="S2.SS1.p10.3.m3.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p10.3.m3.1.1.3" xref="S2.SS1.p10.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p10.3.m3.1b"><apply id="S2.SS1.p10.3.m3.1.1.cmml" xref="S2.SS1.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p10.3.m3.1.1.1.cmml" xref="S2.SS1.p10.3.m3.1.1">superscript</csymbol><ci id="S2.SS1.p10.3.m3.1.1.2.cmml" xref="S2.SS1.p10.3.m3.1.1.2">𝒜</ci><ci id="S2.SS1.p10.3.m3.1.1.3.cmml" xref="S2.SS1.p10.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p10.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p10.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> itself is the only maximal element with respect to this partial order; it is often also called the <span class="ltx_text ltx_font_italic" id="S2.SS1.p10.4.1">full shift over <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.SS1.p10.4.1.m1.1"><semantics id="S2.SS1.p10.4.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p10.4.1.m1.1.1" xref="S2.SS1.p10.4.1.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p10.4.1.m1.1b"><ci id="S2.SS1.p10.4.1.m1.1.1.cmml" xref="S2.SS1.p10.4.1.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p10.4.1.m1.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p10.4.1.m1.1d">caligraphic_A</annotation></semantics></math></span>. Similarly, a subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p10.5.m4.1"><semantics id="S2.SS1.p10.5.m4.1a"><mrow id="S2.SS1.p10.5.m4.1.1" xref="S2.SS1.p10.5.m4.1.1.cmml"><mi id="S2.SS1.p10.5.m4.1.1.2" xref="S2.SS1.p10.5.m4.1.1.2.cmml">X</mi><mo id="S2.SS1.p10.5.m4.1.1.1" xref="S2.SS1.p10.5.m4.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p10.5.m4.1.1.3" xref="S2.SS1.p10.5.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p10.5.m4.1.1.3.2" xref="S2.SS1.p10.5.m4.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p10.5.m4.1.1.3.3" xref="S2.SS1.p10.5.m4.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p10.5.m4.1b"><apply id="S2.SS1.p10.5.m4.1.1.cmml" xref="S2.SS1.p10.5.m4.1.1"><subset id="S2.SS1.p10.5.m4.1.1.1.cmml" xref="S2.SS1.p10.5.m4.1.1.1"></subset><ci id="S2.SS1.p10.5.m4.1.1.2.cmml" xref="S2.SS1.p10.5.m4.1.1.2">𝑋</ci><apply id="S2.SS1.p10.5.m4.1.1.3.cmml" xref="S2.SS1.p10.5.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p10.5.m4.1.1.3.1.cmml" xref="S2.SS1.p10.5.m4.1.1.3">superscript</csymbol><ci id="S2.SS1.p10.5.m4.1.1.3.2.cmml" xref="S2.SS1.p10.5.m4.1.1.3.2">𝒜</ci><ci id="S2.SS1.p10.5.m4.1.1.3.3.cmml" xref="S2.SS1.p10.5.m4.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p10.5.m4.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p10.5.m4.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is sometimes called <span class="ltx_text ltx_font_italic" id="S2.SS1.p10.6.2">a subshift over <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.SS1.p10.6.2.m1.1"><semantics id="S2.SS1.p10.6.2.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p10.6.2.m1.1.1" xref="S2.SS1.p10.6.2.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p10.6.2.m1.1b"><ci id="S2.SS1.p10.6.2.m1.1.1.cmml" xref="S2.SS1.p10.6.2.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p10.6.2.m1.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p10.6.2.m1.1d">caligraphic_A</annotation></semantics></math></span>.</p> </div> <div class="ltx_para" id="S2.SS1.p11"> <p class="ltx_p" id="S2.SS1.p11.6">The space <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.SS1.p11.1.m1.1"><semantics id="S2.SS1.p11.1.m1.1a"><mrow id="S2.SS1.p11.1.m1.1.2" xref="S2.SS1.p11.1.m1.1.2.cmml"><mi id="S2.SS1.p11.1.m1.1.2.2" mathvariant="normal" xref="S2.SS1.p11.1.m1.1.2.2.cmml">Σ</mi><mo id="S2.SS1.p11.1.m1.1.2.1" xref="S2.SS1.p11.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p11.1.m1.1.2.3.2" xref="S2.SS1.p11.1.m1.1.2.cmml"><mo id="S2.SS1.p11.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p11.1.m1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p11.1.m1.1.1" xref="S2.SS1.p11.1.m1.1.1.cmml">𝒜</mi><mo id="S2.SS1.p11.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p11.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p11.1.m1.1b"><apply id="S2.SS1.p11.1.m1.1.2.cmml" xref="S2.SS1.p11.1.m1.1.2"><times id="S2.SS1.p11.1.m1.1.2.1.cmml" xref="S2.SS1.p11.1.m1.1.2.1"></times><ci id="S2.SS1.p11.1.m1.1.2.2.cmml" xref="S2.SS1.p11.1.m1.1.2.2">Σ</ci><ci id="S2.SS1.p11.1.m1.1.1.cmml" xref="S2.SS1.p11.1.m1.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p11.1.m1.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p11.1.m1.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math> is also equipped with a natural topology, inherited from the canonical embedding <math alttext="\Sigma(\cal A)\subseteq\cal P(\cal A^{*})" class="ltx_Math" display="inline" id="S2.SS1.p11.2.m2.2"><semantics id="S2.SS1.p11.2.m2.2a"><mrow id="S2.SS1.p11.2.m2.2.2" xref="S2.SS1.p11.2.m2.2.2.cmml"><mrow id="S2.SS1.p11.2.m2.2.2.3" xref="S2.SS1.p11.2.m2.2.2.3.cmml"><mi id="S2.SS1.p11.2.m2.2.2.3.2" mathvariant="normal" xref="S2.SS1.p11.2.m2.2.2.3.2.cmml">Σ</mi><mo id="S2.SS1.p11.2.m2.2.2.3.1" xref="S2.SS1.p11.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p11.2.m2.2.2.3.3.2" xref="S2.SS1.p11.2.m2.2.2.3.cmml"><mo id="S2.SS1.p11.2.m2.2.2.3.3.2.1" stretchy="false" xref="S2.SS1.p11.2.m2.2.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p11.2.m2.1.1" xref="S2.SS1.p11.2.m2.1.1.cmml">𝒜</mi><mo id="S2.SS1.p11.2.m2.2.2.3.3.2.2" stretchy="false" xref="S2.SS1.p11.2.m2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p11.2.m2.2.2.2" xref="S2.SS1.p11.2.m2.2.2.2.cmml">⊆</mo><mrow id="S2.SS1.p11.2.m2.2.2.1" xref="S2.SS1.p11.2.m2.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p11.2.m2.2.2.1.3" xref="S2.SS1.p11.2.m2.2.2.1.3.cmml">𝒫</mi><mo id="S2.SS1.p11.2.m2.2.2.1.2" xref="S2.SS1.p11.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p11.2.m2.2.2.1.1.1" xref="S2.SS1.p11.2.m2.2.2.1.1.1.1.cmml"><mo 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xref="S2.SS1.p11.2.m2.1.1">𝒜</ci></apply><apply id="S2.SS1.p11.2.m2.2.2.1.cmml" xref="S2.SS1.p11.2.m2.2.2.1"><times id="S2.SS1.p11.2.m2.2.2.1.2.cmml" xref="S2.SS1.p11.2.m2.2.2.1.2"></times><ci id="S2.SS1.p11.2.m2.2.2.1.3.cmml" xref="S2.SS1.p11.2.m2.2.2.1.3">𝒫</ci><apply id="S2.SS1.p11.2.m2.2.2.1.1.1.1.cmml" xref="S2.SS1.p11.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p11.2.m2.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p11.2.m2.2.2.1.1.1">superscript</csymbol><ci id="S2.SS1.p11.2.m2.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p11.2.m2.2.2.1.1.1.1.2">𝒜</ci><times id="S2.SS1.p11.2.m2.2.2.1.1.1.1.3.cmml" xref="S2.SS1.p11.2.m2.2.2.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p11.2.m2.2c">\Sigma(\cal A)\subseteq\cal P(\cal A^{*})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p11.2.m2.2d">roman_Σ ( caligraphic_A ) ⊆ caligraphic_P ( caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math> which is defined by the above bijection between subshifts and their associated languages. Since the topology of the shift space doesn’t play a role in this paper, we will not give details here and refer the reader instead to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib14" title="">14</a>]</cite>. Just for “general interest” we note that the subset of <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.SS1.p11.3.m3.1"><semantics id="S2.SS1.p11.3.m3.1a"><mrow id="S2.SS1.p11.3.m3.1.2" xref="S2.SS1.p11.3.m3.1.2.cmml"><mi id="S2.SS1.p11.3.m3.1.2.2" mathvariant="normal" xref="S2.SS1.p11.3.m3.1.2.2.cmml">Σ</mi><mo id="S2.SS1.p11.3.m3.1.2.1" xref="S2.SS1.p11.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p11.3.m3.1.2.3.2" xref="S2.SS1.p11.3.m3.1.2.cmml"><mo id="S2.SS1.p11.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1.p11.3.m3.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p11.3.m3.1.1" xref="S2.SS1.p11.3.m3.1.1.cmml">𝒜</mi><mo id="S2.SS1.p11.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1.p11.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p11.3.m3.1b"><apply id="S2.SS1.p11.3.m3.1.2.cmml" xref="S2.SS1.p11.3.m3.1.2"><times id="S2.SS1.p11.3.m3.1.2.1.cmml" xref="S2.SS1.p11.3.m3.1.2.1"></times><ci id="S2.SS1.p11.3.m3.1.2.2.cmml" xref="S2.SS1.p11.3.m3.1.2.2">Σ</ci><ci id="S2.SS1.p11.3.m3.1.1.cmml" xref="S2.SS1.p11.3.m3.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p11.3.m3.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p11.3.m3.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math>, which consists of all subshifts that are a union of finitely many shift-orbits, is dense in <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.SS1.p11.4.m4.1"><semantics id="S2.SS1.p11.4.m4.1a"><mrow id="S2.SS1.p11.4.m4.1.2" xref="S2.SS1.p11.4.m4.1.2.cmml"><mi id="S2.SS1.p11.4.m4.1.2.2" mathvariant="normal" xref="S2.SS1.p11.4.m4.1.2.2.cmml">Σ</mi><mo id="S2.SS1.p11.4.m4.1.2.1" xref="S2.SS1.p11.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p11.4.m4.1.2.3.2" xref="S2.SS1.p11.4.m4.1.2.cmml"><mo id="S2.SS1.p11.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS1.p11.4.m4.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p11.4.m4.1.1" xref="S2.SS1.p11.4.m4.1.1.cmml">𝒜</mi><mo id="S2.SS1.p11.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS1.p11.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p11.4.m4.1b"><apply id="S2.SS1.p11.4.m4.1.2.cmml" xref="S2.SS1.p11.4.m4.1.2"><times id="S2.SS1.p11.4.m4.1.2.1.cmml" xref="S2.SS1.p11.4.m4.1.2.1"></times><ci id="S2.SS1.p11.4.m4.1.2.2.cmml" xref="S2.SS1.p11.4.m4.1.2.2">Σ</ci><ci id="S2.SS1.p11.4.m4.1.1.cmml" xref="S2.SS1.p11.4.m4.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p11.4.m4.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p11.4.m4.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math>. More information about the topology on <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.SS1.p11.5.m5.1"><semantics id="S2.SS1.p11.5.m5.1a"><mrow id="S2.SS1.p11.5.m5.1.2" xref="S2.SS1.p11.5.m5.1.2.cmml"><mi id="S2.SS1.p11.5.m5.1.2.2" mathvariant="normal" xref="S2.SS1.p11.5.m5.1.2.2.cmml">Σ</mi><mo id="S2.SS1.p11.5.m5.1.2.1" xref="S2.SS1.p11.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p11.5.m5.1.2.3.2" xref="S2.SS1.p11.5.m5.1.2.cmml"><mo id="S2.SS1.p11.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS1.p11.5.m5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p11.5.m5.1.1" xref="S2.SS1.p11.5.m5.1.1.cmml">𝒜</mi><mo id="S2.SS1.p11.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS1.p11.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p11.5.m5.1b"><apply id="S2.SS1.p11.5.m5.1.2.cmml" xref="S2.SS1.p11.5.m5.1.2"><times id="S2.SS1.p11.5.m5.1.2.1.cmml" xref="S2.SS1.p11.5.m5.1.2.1"></times><ci id="S2.SS1.p11.5.m5.1.2.2.cmml" xref="S2.SS1.p11.5.m5.1.2.2">Σ</ci><ci id="S2.SS1.p11.5.m5.1.1.cmml" xref="S2.SS1.p11.5.m5.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p11.5.m5.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p11.5.m5.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math> can be found in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib14" title="">14</a>]</cite>, where in particular it is shown that <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.SS1.p11.6.m6.1"><semantics id="S2.SS1.p11.6.m6.1a"><mrow id="S2.SS1.p11.6.m6.1.2" xref="S2.SS1.p11.6.m6.1.2.cmml"><mi id="S2.SS1.p11.6.m6.1.2.2" mathvariant="normal" xref="S2.SS1.p11.6.m6.1.2.2.cmml">Σ</mi><mo id="S2.SS1.p11.6.m6.1.2.1" xref="S2.SS1.p11.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p11.6.m6.1.2.3.2" xref="S2.SS1.p11.6.m6.1.2.cmml"><mo id="S2.SS1.p11.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS1.p11.6.m6.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p11.6.m6.1.1" xref="S2.SS1.p11.6.m6.1.1.cmml">𝒜</mi><mo id="S2.SS1.p11.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS1.p11.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p11.6.m6.1b"><apply id="S2.SS1.p11.6.m6.1.2.cmml" xref="S2.SS1.p11.6.m6.1.2"><times id="S2.SS1.p11.6.m6.1.2.1.cmml" xref="S2.SS1.p11.6.m6.1.2.1"></times><ci id="S2.SS1.p11.6.m6.1.2.2.cmml" xref="S2.SS1.p11.6.m6.1.2.2">Σ</ci><ci id="S2.SS1.p11.6.m6.1.1.cmml" xref="S2.SS1.p11.6.m6.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p11.6.m6.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p11.6.m6.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math> is a Pełczyński space.</p> </div> <div class="ltx_para" id="S2.SS1.p12"> <p class="ltx_p" id="S2.SS1.p12.8">An <span class="ltx_text ltx_font_italic" id="S2.SS1.p12.8.1">invariant measure</span> <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p12.1.m1.1"><semantics id="S2.SS1.p12.1.m1.1a"><mi id="S2.SS1.p12.1.m1.1.1" xref="S2.SS1.p12.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.1.m1.1b"><ci id="S2.SS1.p12.1.m1.1.1.cmml" xref="S2.SS1.p12.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.1.m1.1d">italic_μ</annotation></semantics></math> on a subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p12.2.m2.1"><semantics id="S2.SS1.p12.2.m2.1a"><mrow id="S2.SS1.p12.2.m2.1.1" xref="S2.SS1.p12.2.m2.1.1.cmml"><mi id="S2.SS1.p12.2.m2.1.1.2" xref="S2.SS1.p12.2.m2.1.1.2.cmml">X</mi><mo id="S2.SS1.p12.2.m2.1.1.1" xref="S2.SS1.p12.2.m2.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p12.2.m2.1.1.3" xref="S2.SS1.p12.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p12.2.m2.1.1.3.2" xref="S2.SS1.p12.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p12.2.m2.1.1.3.3" xref="S2.SS1.p12.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.2.m2.1b"><apply id="S2.SS1.p12.2.m2.1.1.cmml" xref="S2.SS1.p12.2.m2.1.1"><subset id="S2.SS1.p12.2.m2.1.1.1.cmml" xref="S2.SS1.p12.2.m2.1.1.1"></subset><ci id="S2.SS1.p12.2.m2.1.1.2.cmml" xref="S2.SS1.p12.2.m2.1.1.2">𝑋</ci><apply id="S2.SS1.p12.2.m2.1.1.3.cmml" xref="S2.SS1.p12.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p12.2.m2.1.1.3.1.cmml" xref="S2.SS1.p12.2.m2.1.1.3">superscript</csymbol><ci id="S2.SS1.p12.2.m2.1.1.3.2.cmml" xref="S2.SS1.p12.2.m2.1.1.3.2">𝒜</ci><ci id="S2.SS1.p12.2.m2.1.1.3.3.cmml" xref="S2.SS1.p12.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is a finite Borel measure on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p12.3.m3.1"><semantics id="S2.SS1.p12.3.m3.1a"><msup id="S2.SS1.p12.3.m3.1.1" xref="S2.SS1.p12.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p12.3.m3.1.1.2" xref="S2.SS1.p12.3.m3.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p12.3.m3.1.1.3" xref="S2.SS1.p12.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.3.m3.1b"><apply id="S2.SS1.p12.3.m3.1.1.cmml" xref="S2.SS1.p12.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p12.3.m3.1.1.1.cmml" xref="S2.SS1.p12.3.m3.1.1">superscript</csymbol><ci id="S2.SS1.p12.3.m3.1.1.2.cmml" xref="S2.SS1.p12.3.m3.1.1.2">𝒜</ci><ci id="S2.SS1.p12.3.m3.1.1.3.cmml" xref="S2.SS1.p12.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> with support in <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p12.4.m4.1"><semantics id="S2.SS1.p12.4.m4.1a"><mi id="S2.SS1.p12.4.m4.1.1" xref="S2.SS1.p12.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.4.m4.1b"><ci id="S2.SS1.p12.4.m4.1.1.cmml" xref="S2.SS1.p12.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.4.m4.1d">italic_X</annotation></semantics></math> that is invariant under <math alttext="T" class="ltx_Math" display="inline" id="S2.SS1.p12.5.m5.1"><semantics id="S2.SS1.p12.5.m5.1a"><mi id="S2.SS1.p12.5.m5.1.1" xref="S2.SS1.p12.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.5.m5.1b"><ci id="S2.SS1.p12.5.m5.1.1.cmml" xref="S2.SS1.p12.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.5.m5.1d">italic_T</annotation></semantics></math>, i.e. <math alttext="\mu(T^{-1}(B))=\mu(B)" class="ltx_Math" display="inline" id="S2.SS1.p12.6.m6.3"><semantics id="S2.SS1.p12.6.m6.3a"><mrow id="S2.SS1.p12.6.m6.3.3" xref="S2.SS1.p12.6.m6.3.3.cmml"><mrow id="S2.SS1.p12.6.m6.3.3.1" xref="S2.SS1.p12.6.m6.3.3.1.cmml"><mi id="S2.SS1.p12.6.m6.3.3.1.3" xref="S2.SS1.p12.6.m6.3.3.1.3.cmml">μ</mi><mo id="S2.SS1.p12.6.m6.3.3.1.2" xref="S2.SS1.p12.6.m6.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS1.p12.6.m6.3.3.1.1.1" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.cmml"><mo id="S2.SS1.p12.6.m6.3.3.1.1.1.2" stretchy="false" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p12.6.m6.3.3.1.1.1.1" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.cmml"><msup id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.cmml"><mi id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.2" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.2.cmml">T</mi><mrow id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3.cmml"><mo id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3a" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3.cmml">−</mo><mn id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3.2" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS1.p12.6.m6.3.3.1.1.1.1.1" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p12.6.m6.3.3.1.1.1.1.3.2" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.cmml"><mo id="S2.SS1.p12.6.m6.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.cmml">(</mo><mi id="S2.SS1.p12.6.m6.1.1" xref="S2.SS1.p12.6.m6.1.1.cmml">B</mi><mo id="S2.SS1.p12.6.m6.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p12.6.m6.3.3.1.1.1.3" stretchy="false" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p12.6.m6.3.3.2" xref="S2.SS1.p12.6.m6.3.3.2.cmml">=</mo><mrow id="S2.SS1.p12.6.m6.3.3.3" xref="S2.SS1.p12.6.m6.3.3.3.cmml"><mi id="S2.SS1.p12.6.m6.3.3.3.2" xref="S2.SS1.p12.6.m6.3.3.3.2.cmml">μ</mi><mo id="S2.SS1.p12.6.m6.3.3.3.1" xref="S2.SS1.p12.6.m6.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p12.6.m6.3.3.3.3.2" xref="S2.SS1.p12.6.m6.3.3.3.cmml"><mo id="S2.SS1.p12.6.m6.3.3.3.3.2.1" stretchy="false" xref="S2.SS1.p12.6.m6.3.3.3.cmml">(</mo><mi id="S2.SS1.p12.6.m6.2.2" xref="S2.SS1.p12.6.m6.2.2.cmml">B</mi><mo id="S2.SS1.p12.6.m6.3.3.3.3.2.2" stretchy="false" xref="S2.SS1.p12.6.m6.3.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.6.m6.3b"><apply id="S2.SS1.p12.6.m6.3.3.cmml" xref="S2.SS1.p12.6.m6.3.3"><eq id="S2.SS1.p12.6.m6.3.3.2.cmml" xref="S2.SS1.p12.6.m6.3.3.2"></eq><apply id="S2.SS1.p12.6.m6.3.3.1.cmml" xref="S2.SS1.p12.6.m6.3.3.1"><times id="S2.SS1.p12.6.m6.3.3.1.2.cmml" xref="S2.SS1.p12.6.m6.3.3.1.2"></times><ci id="S2.SS1.p12.6.m6.3.3.1.3.cmml" xref="S2.SS1.p12.6.m6.3.3.1.3">𝜇</ci><apply id="S2.SS1.p12.6.m6.3.3.1.1.1.1.cmml" xref="S2.SS1.p12.6.m6.3.3.1.1.1"><times id="S2.SS1.p12.6.m6.3.3.1.1.1.1.1.cmml" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.1"></times><apply id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.cmml" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.1.cmml" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2">superscript</csymbol><ci id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.2.cmml" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.2">𝑇</ci><apply id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3.cmml" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3"><minus id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3.1.cmml" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3"></minus><cn id="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3.2.cmml" type="integer" xref="S2.SS1.p12.6.m6.3.3.1.1.1.1.2.3.2">1</cn></apply></apply><ci id="S2.SS1.p12.6.m6.1.1.cmml" xref="S2.SS1.p12.6.m6.1.1">𝐵</ci></apply></apply><apply id="S2.SS1.p12.6.m6.3.3.3.cmml" xref="S2.SS1.p12.6.m6.3.3.3"><times id="S2.SS1.p12.6.m6.3.3.3.1.cmml" xref="S2.SS1.p12.6.m6.3.3.3.1"></times><ci id="S2.SS1.p12.6.m6.3.3.3.2.cmml" xref="S2.SS1.p12.6.m6.3.3.3.2">𝜇</ci><ci id="S2.SS1.p12.6.m6.2.2.cmml" xref="S2.SS1.p12.6.m6.2.2">𝐵</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.6.m6.3c">\mu(T^{-1}(B))=\mu(B)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.6.m6.3d">italic_μ ( italic_T start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_B ) ) = italic_μ ( italic_B )</annotation></semantics></math> for every measurable set <math alttext="B\subseteq X" class="ltx_Math" display="inline" id="S2.SS1.p12.7.m7.1"><semantics id="S2.SS1.p12.7.m7.1a"><mrow id="S2.SS1.p12.7.m7.1.1" xref="S2.SS1.p12.7.m7.1.1.cmml"><mi id="S2.SS1.p12.7.m7.1.1.2" xref="S2.SS1.p12.7.m7.1.1.2.cmml">B</mi><mo id="S2.SS1.p12.7.m7.1.1.1" xref="S2.SS1.p12.7.m7.1.1.1.cmml">⊆</mo><mi id="S2.SS1.p12.7.m7.1.1.3" xref="S2.SS1.p12.7.m7.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.7.m7.1b"><apply id="S2.SS1.p12.7.m7.1.1.cmml" xref="S2.SS1.p12.7.m7.1.1"><subset id="S2.SS1.p12.7.m7.1.1.1.cmml" xref="S2.SS1.p12.7.m7.1.1.1"></subset><ci id="S2.SS1.p12.7.m7.1.1.2.cmml" xref="S2.SS1.p12.7.m7.1.1.2">𝐵</ci><ci id="S2.SS1.p12.7.m7.1.1.3.cmml" xref="S2.SS1.p12.7.m7.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.7.m7.1c">B\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.7.m7.1d">italic_B ⊆ italic_X</annotation></semantics></math>. Any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p12.8.m8.1"><semantics id="S2.SS1.p12.8.m8.1a"><mi id="S2.SS1.p12.8.m8.1.1" xref="S2.SS1.p12.8.m8.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.8.m8.1b"><ci id="S2.SS1.p12.8.m8.1.1.cmml" xref="S2.SS1.p12.8.m8.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.8.m8.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.8.m8.1d">italic_μ</annotation></semantics></math> defines a <span class="ltx_text ltx_font_italic" id="S2.SS1.p12.8.2">weight function</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\omega_{\mu}:\cal A^{*}\to\mathbb{R}_{\geq 0},\,\,w\mapsto\mu([w])\,," class="ltx_Math" display="block" id="S2.Ex3.m1.2"><semantics 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xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.cmml"><mi id="S2.Ex3.m1.2.2.1.1.1.1.1.3.2" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.2.cmml">ℝ</mi><mrow id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.cmml"><mi id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.2" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.2.cmml"></mi><mo id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.1" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.3" mathvariant="script" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.3.cmml">0</mn></mrow></msub></mrow><mo id="S2.Ex3.m1.2.2.1.1.2.2.3" rspace="0.497em" xref="S2.Ex3.m1.2.2.1.1.2.3a.cmml">,</mo><mrow id="S2.Ex3.m1.2.2.1.1.2.2.2" xref="S2.Ex3.m1.2.2.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex3.m1.2.2.1.1.2.2.2.3" xref="S2.Ex3.m1.2.2.1.1.2.2.2.3.cmml">𝓌</mi><mo id="S2.Ex3.m1.2.2.1.1.2.2.2.2" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.2.2.2.2.cmml">↦</mo><mrow id="S2.Ex3.m1.2.2.1.1.2.2.2.1" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.cmml"><mi id="S2.Ex3.m1.2.2.1.1.2.2.2.1.3" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.3.cmml">μ</mi><mo id="S2.Ex3.m1.2.2.1.1.2.2.2.1.2" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.2.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.cmml"><mo id="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.2" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.cmml">(</mo><mrow id="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.2" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.1.cmml"><mo id="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.2.1" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml">[</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml">𝓌</mi><mo id="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.2.2" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.3" rspace="0.170em" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><mo id="S2.Ex3.m1.2.2.1.2" xref="S2.Ex3.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml 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xref="S2.Ex3.m1.2.2.1.1.1.1.1.2">superscript</csymbol><ci id="S2.Ex3.m1.2.2.1.1.1.1.1.2.2.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.2.2">𝒜</ci><times id="S2.Ex3.m1.2.2.1.1.1.1.1.2.3.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.2.3"></times></apply><apply id="S2.Ex3.m1.2.2.1.1.1.1.1.3.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex3.m1.2.2.1.1.1.1.1.3.1.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex3.m1.2.2.1.1.1.1.1.3.2.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.2">ℝ</ci><apply id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3"><geq id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.1.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.1"></geq><csymbol cd="latexml" id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.2.cmml" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.2">absent</csymbol><cn id="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.Ex3.m1.2.2.1.1.1.1.1.3.3.3">0</cn></apply></apply></apply><apply id="S2.Ex3.m1.2.2.1.1.2.2.2.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2"><csymbol cd="latexml" id="S2.Ex3.m1.2.2.1.1.2.2.2.2.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.2">maps-to</csymbol><ci id="S2.Ex3.m1.2.2.1.1.2.2.2.3.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.3">𝓌</ci><apply id="S2.Ex3.m1.2.2.1.1.2.2.2.1.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1"><times id="S2.Ex3.m1.2.2.1.1.2.2.2.1.2.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.2"></times><ci id="S2.Ex3.m1.2.2.1.1.2.2.2.1.3.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.3">𝜇</ci><apply id="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.1.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.2"><csymbol cd="latexml" id="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml" xref="S2.Ex3.m1.2.2.1.1.2.2.2.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S2.Ex3.m1.1.1.cmml" xref="S2.Ex3.m1.1.1">𝓌</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.2c">\omega_{\mu}:\cal A^{*}\to\mathbb{R}_{\geq 0},\,\,w\mapsto\mu([w])\,,</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.2d">italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → blackboard_R start_POSTSUBSCRIPT ≥ caligraphic_0 end_POSTSUBSCRIPT , caligraphic_w ↦ italic_μ ( [ caligraphic_w ] ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p12.9">by which we mean any function <math alttext="\omega:\cal A^{*}\to\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S2.SS1.p12.9.m1.1"><semantics id="S2.SS1.p12.9.m1.1a"><mrow id="S2.SS1.p12.9.m1.1.1" xref="S2.SS1.p12.9.m1.1.1.cmml"><mi id="S2.SS1.p12.9.m1.1.1.2" xref="S2.SS1.p12.9.m1.1.1.2.cmml">ω</mi><mo id="S2.SS1.p12.9.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p12.9.m1.1.1.1.cmml">:</mo><mrow id="S2.SS1.p12.9.m1.1.1.3" xref="S2.SS1.p12.9.m1.1.1.3.cmml"><msup id="S2.SS1.p12.9.m1.1.1.3.2" xref="S2.SS1.p12.9.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p12.9.m1.1.1.3.2.2" xref="S2.SS1.p12.9.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.SS1.p12.9.m1.1.1.3.2.3" xref="S2.SS1.p12.9.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.SS1.p12.9.m1.1.1.3.1" stretchy="false" xref="S2.SS1.p12.9.m1.1.1.3.1.cmml">→</mo><msub id="S2.SS1.p12.9.m1.1.1.3.3" xref="S2.SS1.p12.9.m1.1.1.3.3.cmml"><mi id="S2.SS1.p12.9.m1.1.1.3.3.2" xref="S2.SS1.p12.9.m1.1.1.3.3.2.cmml">ℝ</mi><mrow id="S2.SS1.p12.9.m1.1.1.3.3.3" xref="S2.SS1.p12.9.m1.1.1.3.3.3.cmml"><mi id="S2.SS1.p12.9.m1.1.1.3.3.3.2" xref="S2.SS1.p12.9.m1.1.1.3.3.3.2.cmml"></mi><mo id="S2.SS1.p12.9.m1.1.1.3.3.3.1" xref="S2.SS1.p12.9.m1.1.1.3.3.3.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p12.9.m1.1.1.3.3.3.3" mathvariant="script" xref="S2.SS1.p12.9.m1.1.1.3.3.3.3.cmml">0</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.9.m1.1b"><apply id="S2.SS1.p12.9.m1.1.1.cmml" xref="S2.SS1.p12.9.m1.1.1"><ci id="S2.SS1.p12.9.m1.1.1.1.cmml" xref="S2.SS1.p12.9.m1.1.1.1">:</ci><ci id="S2.SS1.p12.9.m1.1.1.2.cmml" xref="S2.SS1.p12.9.m1.1.1.2">𝜔</ci><apply id="S2.SS1.p12.9.m1.1.1.3.cmml" xref="S2.SS1.p12.9.m1.1.1.3"><ci id="S2.SS1.p12.9.m1.1.1.3.1.cmml" xref="S2.SS1.p12.9.m1.1.1.3.1">→</ci><apply id="S2.SS1.p12.9.m1.1.1.3.2.cmml" xref="S2.SS1.p12.9.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p12.9.m1.1.1.3.2.1.cmml" xref="S2.SS1.p12.9.m1.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p12.9.m1.1.1.3.2.2.cmml" xref="S2.SS1.p12.9.m1.1.1.3.2.2">𝒜</ci><times id="S2.SS1.p12.9.m1.1.1.3.2.3.cmml" xref="S2.SS1.p12.9.m1.1.1.3.2.3"></times></apply><apply id="S2.SS1.p12.9.m1.1.1.3.3.cmml" xref="S2.SS1.p12.9.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p12.9.m1.1.1.3.3.1.cmml" xref="S2.SS1.p12.9.m1.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p12.9.m1.1.1.3.3.2.cmml" xref="S2.SS1.p12.9.m1.1.1.3.3.2">ℝ</ci><apply id="S2.SS1.p12.9.m1.1.1.3.3.3.cmml" xref="S2.SS1.p12.9.m1.1.1.3.3.3"><geq id="S2.SS1.p12.9.m1.1.1.3.3.3.1.cmml" xref="S2.SS1.p12.9.m1.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S2.SS1.p12.9.m1.1.1.3.3.3.2.cmml" xref="S2.SS1.p12.9.m1.1.1.3.3.3.2">absent</csymbol><cn id="S2.SS1.p12.9.m1.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p12.9.m1.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.9.m1.1c">\omega:\cal A^{*}\to\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.9.m1.1d">italic_ω : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → blackboard_R start_POSTSUBSCRIPT ≥ caligraphic_0 end_POSTSUBSCRIPT</annotation></semantics></math> that satisfies the <span class="ltx_text ltx_font_italic" id="S2.SS1.p12.9.1">Kirchhoff equalities</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.2)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\omega(w)=\sum_{a_{i}\in\cal A}\omega(a_{i}w)=\sum_{a_{i}\in\cal A}\omega(wa_{% i})" class="ltx_Math" display="block" id="S2.E2.m1.3"><semantics id="S2.E2.m1.3a"><mrow id="S2.E2.m1.3.3" xref="S2.E2.m1.3.3.cmml"><mrow id="S2.E2.m1.3.3.4" xref="S2.E2.m1.3.3.4.cmml"><mi id="S2.E2.m1.3.3.4.2" xref="S2.E2.m1.3.3.4.2.cmml">ω</mi><mo id="S2.E2.m1.3.3.4.1" xref="S2.E2.m1.3.3.4.1.cmml">⁢</mo><mrow id="S2.E2.m1.3.3.4.3.2" xref="S2.E2.m1.3.3.4.cmml"><mo id="S2.E2.m1.3.3.4.3.2.1" stretchy="false" xref="S2.E2.m1.3.3.4.cmml">(</mo><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">w</mi><mo id="S2.E2.m1.3.3.4.3.2.2" stretchy="false" xref="S2.E2.m1.3.3.4.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.3.3.5" rspace="0.111em" xref="S2.E2.m1.3.3.5.cmml">=</mo><mrow id="S2.E2.m1.2.2.1" xref="S2.E2.m1.2.2.1.cmml"><munder id="S2.E2.m1.2.2.1.2" xref="S2.E2.m1.2.2.1.2.cmml"><mo id="S2.E2.m1.2.2.1.2.2" movablelimits="false" xref="S2.E2.m1.2.2.1.2.2.cmml">∑</mo><mrow id="S2.E2.m1.2.2.1.2.3" xref="S2.E2.m1.2.2.1.2.3.cmml"><msub id="S2.E2.m1.2.2.1.2.3.2" xref="S2.E2.m1.2.2.1.2.3.2.cmml"><mi id="S2.E2.m1.2.2.1.2.3.2.2" xref="S2.E2.m1.2.2.1.2.3.2.2.cmml">a</mi><mi id="S2.E2.m1.2.2.1.2.3.2.3" xref="S2.E2.m1.2.2.1.2.3.2.3.cmml">i</mi></msub><mo id="S2.E2.m1.2.2.1.2.3.1" xref="S2.E2.m1.2.2.1.2.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.E2.m1.2.2.1.2.3.3" xref="S2.E2.m1.2.2.1.2.3.3.cmml">𝒜</mi></mrow></munder><mrow id="S2.E2.m1.2.2.1.1" xref="S2.E2.m1.2.2.1.1.cmml"><mi id="S2.E2.m1.2.2.1.1.3" xref="S2.E2.m1.2.2.1.1.3.cmml">ω</mi><mo id="S2.E2.m1.2.2.1.1.2" xref="S2.E2.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.2.2.1.1.1.1" xref="S2.E2.m1.2.2.1.1.1.1.1.cmml"><mo id="S2.E2.m1.2.2.1.1.1.1.2" stretchy="false" xref="S2.E2.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.E2.m1.2.2.1.1.1.1.1" xref="S2.E2.m1.2.2.1.1.1.1.1.cmml"><msub id="S2.E2.m1.2.2.1.1.1.1.1.2" xref="S2.E2.m1.2.2.1.1.1.1.1.2.cmml"><mi id="S2.E2.m1.2.2.1.1.1.1.1.2.2" xref="S2.E2.m1.2.2.1.1.1.1.1.2.2.cmml">a</mi><mi id="S2.E2.m1.2.2.1.1.1.1.1.2.3" xref="S2.E2.m1.2.2.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.E2.m1.2.2.1.1.1.1.1.1" xref="S2.E2.m1.2.2.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E2.m1.2.2.1.1.1.1.1.3" xref="S2.E2.m1.2.2.1.1.1.1.1.3.cmml">w</mi></mrow><mo id="S2.E2.m1.2.2.1.1.1.1.3" stretchy="false" xref="S2.E2.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E2.m1.3.3.6" rspace="0.111em" xref="S2.E2.m1.3.3.6.cmml">=</mo><mrow id="S2.E2.m1.3.3.2" xref="S2.E2.m1.3.3.2.cmml"><munder id="S2.E2.m1.3.3.2.2" xref="S2.E2.m1.3.3.2.2.cmml"><mo id="S2.E2.m1.3.3.2.2.2" movablelimits="false" xref="S2.E2.m1.3.3.2.2.2.cmml">∑</mo><mrow id="S2.E2.m1.3.3.2.2.3" xref="S2.E2.m1.3.3.2.2.3.cmml"><msub id="S2.E2.m1.3.3.2.2.3.2" xref="S2.E2.m1.3.3.2.2.3.2.cmml"><mi id="S2.E2.m1.3.3.2.2.3.2.2" xref="S2.E2.m1.3.3.2.2.3.2.2.cmml">a</mi><mi id="S2.E2.m1.3.3.2.2.3.2.3" xref="S2.E2.m1.3.3.2.2.3.2.3.cmml">i</mi></msub><mo id="S2.E2.m1.3.3.2.2.3.1" xref="S2.E2.m1.3.3.2.2.3.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.E2.m1.3.3.2.2.3.3" xref="S2.E2.m1.3.3.2.2.3.3.cmml">𝒜</mi></mrow></munder><mrow id="S2.E2.m1.3.3.2.1" xref="S2.E2.m1.3.3.2.1.cmml"><mi id="S2.E2.m1.3.3.2.1.3" xref="S2.E2.m1.3.3.2.1.3.cmml">ω</mi><mo id="S2.E2.m1.3.3.2.1.2" xref="S2.E2.m1.3.3.2.1.2.cmml">⁢</mo><mrow id="S2.E2.m1.3.3.2.1.1.1" xref="S2.E2.m1.3.3.2.1.1.1.1.cmml"><mo id="S2.E2.m1.3.3.2.1.1.1.2" stretchy="false" xref="S2.E2.m1.3.3.2.1.1.1.1.cmml">(</mo><mrow id="S2.E2.m1.3.3.2.1.1.1.1" xref="S2.E2.m1.3.3.2.1.1.1.1.cmml"><mi id="S2.E2.m1.3.3.2.1.1.1.1.2" xref="S2.E2.m1.3.3.2.1.1.1.1.2.cmml">w</mi><mo id="S2.E2.m1.3.3.2.1.1.1.1.1" xref="S2.E2.m1.3.3.2.1.1.1.1.1.cmml">⁢</mo><msub id="S2.E2.m1.3.3.2.1.1.1.1.3" xref="S2.E2.m1.3.3.2.1.1.1.1.3.cmml"><mi id="S2.E2.m1.3.3.2.1.1.1.1.3.2" xref="S2.E2.m1.3.3.2.1.1.1.1.3.2.cmml">a</mi><mi id="S2.E2.m1.3.3.2.1.1.1.1.3.3" 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xref="S2.E2.m1.3.3.2.2.3.2"><csymbol cd="ambiguous" id="S2.E2.m1.3.3.2.2.3.2.1.cmml" xref="S2.E2.m1.3.3.2.2.3.2">subscript</csymbol><ci id="S2.E2.m1.3.3.2.2.3.2.2.cmml" xref="S2.E2.m1.3.3.2.2.3.2.2">𝑎</ci><ci id="S2.E2.m1.3.3.2.2.3.2.3.cmml" xref="S2.E2.m1.3.3.2.2.3.2.3">𝑖</ci></apply><ci id="S2.E2.m1.3.3.2.2.3.3.cmml" xref="S2.E2.m1.3.3.2.2.3.3">𝒜</ci></apply></apply><apply id="S2.E2.m1.3.3.2.1.cmml" xref="S2.E2.m1.3.3.2.1"><times id="S2.E2.m1.3.3.2.1.2.cmml" xref="S2.E2.m1.3.3.2.1.2"></times><ci id="S2.E2.m1.3.3.2.1.3.cmml" xref="S2.E2.m1.3.3.2.1.3">𝜔</ci><apply id="S2.E2.m1.3.3.2.1.1.1.1.cmml" xref="S2.E2.m1.3.3.2.1.1.1"><times id="S2.E2.m1.3.3.2.1.1.1.1.1.cmml" xref="S2.E2.m1.3.3.2.1.1.1.1.1"></times><ci id="S2.E2.m1.3.3.2.1.1.1.1.2.cmml" xref="S2.E2.m1.3.3.2.1.1.1.1.2">𝑤</ci><apply id="S2.E2.m1.3.3.2.1.1.1.1.3.cmml" xref="S2.E2.m1.3.3.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E2.m1.3.3.2.1.1.1.1.3.1.cmml" xref="S2.E2.m1.3.3.2.1.1.1.1.3">subscript</csymbol><ci id="S2.E2.m1.3.3.2.1.1.1.1.3.2.cmml" xref="S2.E2.m1.3.3.2.1.1.1.1.3.2">𝑎</ci><ci id="S2.E2.m1.3.3.2.1.1.1.1.3.3.cmml" xref="S2.E2.m1.3.3.2.1.1.1.1.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.3c">\omega(w)=\sum_{a_{i}\in\cal A}\omega(a_{i}w)=\sum_{a_{i}\in\cal A}\omega(wa_{% i})</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.3d">italic_ω ( italic_w ) = ∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT italic_ω ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_w ) = ∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT italic_ω ( italic_w italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p12.17">for all <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p12.10.m1.1"><semantics id="S2.SS1.p12.10.m1.1a"><mrow id="S2.SS1.p12.10.m1.1.1" xref="S2.SS1.p12.10.m1.1.1.cmml"><mi id="S2.SS1.p12.10.m1.1.1.2" xref="S2.SS1.p12.10.m1.1.1.2.cmml">w</mi><mo id="S2.SS1.p12.10.m1.1.1.1" xref="S2.SS1.p12.10.m1.1.1.1.cmml">∈</mo><msup id="S2.SS1.p12.10.m1.1.1.3" xref="S2.SS1.p12.10.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p12.10.m1.1.1.3.2" xref="S2.SS1.p12.10.m1.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p12.10.m1.1.1.3.3" xref="S2.SS1.p12.10.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.10.m1.1b"><apply id="S2.SS1.p12.10.m1.1.1.cmml" xref="S2.SS1.p12.10.m1.1.1"><in id="S2.SS1.p12.10.m1.1.1.1.cmml" xref="S2.SS1.p12.10.m1.1.1.1"></in><ci id="S2.SS1.p12.10.m1.1.1.2.cmml" xref="S2.SS1.p12.10.m1.1.1.2">𝑤</ci><apply id="S2.SS1.p12.10.m1.1.1.3.cmml" xref="S2.SS1.p12.10.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p12.10.m1.1.1.3.1.cmml" xref="S2.SS1.p12.10.m1.1.1.3">superscript</csymbol><ci id="S2.SS1.p12.10.m1.1.1.3.2.cmml" xref="S2.SS1.p12.10.m1.1.1.3.2">𝒜</ci><times id="S2.SS1.p12.10.m1.1.1.3.3.cmml" xref="S2.SS1.p12.10.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.10.m1.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.10.m1.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. Conversely, it is well known (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib1" title="">1</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib16" title="">16</a>]</cite>) that every weight function <math alttext="\omega:\cal A^{*}\to\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S2.SS1.p12.11.m2.1"><semantics id="S2.SS1.p12.11.m2.1a"><mrow id="S2.SS1.p12.11.m2.1.1" xref="S2.SS1.p12.11.m2.1.1.cmml"><mi id="S2.SS1.p12.11.m2.1.1.2" xref="S2.SS1.p12.11.m2.1.1.2.cmml">ω</mi><mo id="S2.SS1.p12.11.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p12.11.m2.1.1.1.cmml">:</mo><mrow id="S2.SS1.p12.11.m2.1.1.3" xref="S2.SS1.p12.11.m2.1.1.3.cmml"><msup id="S2.SS1.p12.11.m2.1.1.3.2" xref="S2.SS1.p12.11.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p12.11.m2.1.1.3.2.2" xref="S2.SS1.p12.11.m2.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.SS1.p12.11.m2.1.1.3.2.3" xref="S2.SS1.p12.11.m2.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.SS1.p12.11.m2.1.1.3.1" stretchy="false" xref="S2.SS1.p12.11.m2.1.1.3.1.cmml">→</mo><msub id="S2.SS1.p12.11.m2.1.1.3.3" xref="S2.SS1.p12.11.m2.1.1.3.3.cmml"><mi id="S2.SS1.p12.11.m2.1.1.3.3.2" xref="S2.SS1.p12.11.m2.1.1.3.3.2.cmml">ℝ</mi><mrow id="S2.SS1.p12.11.m2.1.1.3.3.3" xref="S2.SS1.p12.11.m2.1.1.3.3.3.cmml"><mi id="S2.SS1.p12.11.m2.1.1.3.3.3.2" xref="S2.SS1.p12.11.m2.1.1.3.3.3.2.cmml"></mi><mo id="S2.SS1.p12.11.m2.1.1.3.3.3.1" xref="S2.SS1.p12.11.m2.1.1.3.3.3.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p12.11.m2.1.1.3.3.3.3" mathvariant="script" xref="S2.SS1.p12.11.m2.1.1.3.3.3.3.cmml">0</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.11.m2.1b"><apply id="S2.SS1.p12.11.m2.1.1.cmml" xref="S2.SS1.p12.11.m2.1.1"><ci id="S2.SS1.p12.11.m2.1.1.1.cmml" xref="S2.SS1.p12.11.m2.1.1.1">:</ci><ci id="S2.SS1.p12.11.m2.1.1.2.cmml" xref="S2.SS1.p12.11.m2.1.1.2">𝜔</ci><apply id="S2.SS1.p12.11.m2.1.1.3.cmml" xref="S2.SS1.p12.11.m2.1.1.3"><ci id="S2.SS1.p12.11.m2.1.1.3.1.cmml" xref="S2.SS1.p12.11.m2.1.1.3.1">→</ci><apply id="S2.SS1.p12.11.m2.1.1.3.2.cmml" xref="S2.SS1.p12.11.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p12.11.m2.1.1.3.2.1.cmml" xref="S2.SS1.p12.11.m2.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p12.11.m2.1.1.3.2.2.cmml" xref="S2.SS1.p12.11.m2.1.1.3.2.2">𝒜</ci><times id="S2.SS1.p12.11.m2.1.1.3.2.3.cmml" xref="S2.SS1.p12.11.m2.1.1.3.2.3"></times></apply><apply id="S2.SS1.p12.11.m2.1.1.3.3.cmml" xref="S2.SS1.p12.11.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p12.11.m2.1.1.3.3.1.cmml" xref="S2.SS1.p12.11.m2.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p12.11.m2.1.1.3.3.2.cmml" xref="S2.SS1.p12.11.m2.1.1.3.3.2">ℝ</ci><apply id="S2.SS1.p12.11.m2.1.1.3.3.3.cmml" xref="S2.SS1.p12.11.m2.1.1.3.3.3"><geq id="S2.SS1.p12.11.m2.1.1.3.3.3.1.cmml" xref="S2.SS1.p12.11.m2.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S2.SS1.p12.11.m2.1.1.3.3.3.2.cmml" xref="S2.SS1.p12.11.m2.1.1.3.3.3.2">absent</csymbol><cn id="S2.SS1.p12.11.m2.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p12.11.m2.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.11.m2.1c">\omega:\cal A^{*}\to\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.11.m2.1d">italic_ω : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → blackboard_R start_POSTSUBSCRIPT ≥ caligraphic_0 end_POSTSUBSCRIPT</annotation></semantics></math> defines an invariant measure <math alttext="\mu_{\omega}" class="ltx_Math" display="inline" id="S2.SS1.p12.12.m3.1"><semantics id="S2.SS1.p12.12.m3.1a"><msub id="S2.SS1.p12.12.m3.1.1" xref="S2.SS1.p12.12.m3.1.1.cmml"><mi id="S2.SS1.p12.12.m3.1.1.2" xref="S2.SS1.p12.12.m3.1.1.2.cmml">μ</mi><mi id="S2.SS1.p12.12.m3.1.1.3" xref="S2.SS1.p12.12.m3.1.1.3.cmml">ω</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.12.m3.1b"><apply id="S2.SS1.p12.12.m3.1.1.cmml" xref="S2.SS1.p12.12.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p12.12.m3.1.1.1.cmml" xref="S2.SS1.p12.12.m3.1.1">subscript</csymbol><ci id="S2.SS1.p12.12.m3.1.1.2.cmml" xref="S2.SS1.p12.12.m3.1.1.2">𝜇</ci><ci id="S2.SS1.p12.12.m3.1.1.3.cmml" xref="S2.SS1.p12.12.m3.1.1.3">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.12.m3.1c">\mu_{\omega}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.12.m3.1d">italic_μ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT</annotation></semantics></math> via <math alttext="\mu_{\omega}([w]):=\omega(w)" class="ltx_Math" display="inline" id="S2.SS1.p12.13.m4.3"><semantics id="S2.SS1.p12.13.m4.3a"><mrow id="S2.SS1.p12.13.m4.3.3" xref="S2.SS1.p12.13.m4.3.3.cmml"><mrow id="S2.SS1.p12.13.m4.3.3.1" xref="S2.SS1.p12.13.m4.3.3.1.cmml"><msub id="S2.SS1.p12.13.m4.3.3.1.3" xref="S2.SS1.p12.13.m4.3.3.1.3.cmml"><mi id="S2.SS1.p12.13.m4.3.3.1.3.2" xref="S2.SS1.p12.13.m4.3.3.1.3.2.cmml">μ</mi><mi id="S2.SS1.p12.13.m4.3.3.1.3.3" xref="S2.SS1.p12.13.m4.3.3.1.3.3.cmml">ω</mi></msub><mo id="S2.SS1.p12.13.m4.3.3.1.2" xref="S2.SS1.p12.13.m4.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS1.p12.13.m4.3.3.1.1.1" xref="S2.SS1.p12.13.m4.3.3.1.cmml"><mo id="S2.SS1.p12.13.m4.3.3.1.1.1.2" stretchy="false" xref="S2.SS1.p12.13.m4.3.3.1.cmml">(</mo><mrow id="S2.SS1.p12.13.m4.3.3.1.1.1.1.2" xref="S2.SS1.p12.13.m4.3.3.1.1.1.1.1.cmml"><mo id="S2.SS1.p12.13.m4.3.3.1.1.1.1.2.1" stretchy="false" xref="S2.SS1.p12.13.m4.3.3.1.1.1.1.1.1.cmml">[</mo><mi id="S2.SS1.p12.13.m4.1.1" xref="S2.SS1.p12.13.m4.1.1.cmml">w</mi><mo id="S2.SS1.p12.13.m4.3.3.1.1.1.1.2.2" stretchy="false" xref="S2.SS1.p12.13.m4.3.3.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S2.SS1.p12.13.m4.3.3.1.1.1.3" rspace="0.278em" stretchy="false" xref="S2.SS1.p12.13.m4.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p12.13.m4.3.3.2" rspace="0.278em" xref="S2.SS1.p12.13.m4.3.3.2.cmml">:=</mo><mrow id="S2.SS1.p12.13.m4.3.3.3" xref="S2.SS1.p12.13.m4.3.3.3.cmml"><mi id="S2.SS1.p12.13.m4.3.3.3.2" xref="S2.SS1.p12.13.m4.3.3.3.2.cmml">ω</mi><mo id="S2.SS1.p12.13.m4.3.3.3.1" xref="S2.SS1.p12.13.m4.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p12.13.m4.3.3.3.3.2" xref="S2.SS1.p12.13.m4.3.3.3.cmml"><mo id="S2.SS1.p12.13.m4.3.3.3.3.2.1" stretchy="false" xref="S2.SS1.p12.13.m4.3.3.3.cmml">(</mo><mi id="S2.SS1.p12.13.m4.2.2" xref="S2.SS1.p12.13.m4.2.2.cmml">w</mi><mo id="S2.SS1.p12.13.m4.3.3.3.3.2.2" stretchy="false" xref="S2.SS1.p12.13.m4.3.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.13.m4.3b"><apply id="S2.SS1.p12.13.m4.3.3.cmml" xref="S2.SS1.p12.13.m4.3.3"><csymbol cd="latexml" id="S2.SS1.p12.13.m4.3.3.2.cmml" xref="S2.SS1.p12.13.m4.3.3.2">assign</csymbol><apply id="S2.SS1.p12.13.m4.3.3.1.cmml" xref="S2.SS1.p12.13.m4.3.3.1"><times id="S2.SS1.p12.13.m4.3.3.1.2.cmml" xref="S2.SS1.p12.13.m4.3.3.1.2"></times><apply id="S2.SS1.p12.13.m4.3.3.1.3.cmml" xref="S2.SS1.p12.13.m4.3.3.1.3"><csymbol cd="ambiguous" id="S2.SS1.p12.13.m4.3.3.1.3.1.cmml" xref="S2.SS1.p12.13.m4.3.3.1.3">subscript</csymbol><ci id="S2.SS1.p12.13.m4.3.3.1.3.2.cmml" xref="S2.SS1.p12.13.m4.3.3.1.3.2">𝜇</ci><ci id="S2.SS1.p12.13.m4.3.3.1.3.3.cmml" xref="S2.SS1.p12.13.m4.3.3.1.3.3">𝜔</ci></apply><apply id="S2.SS1.p12.13.m4.3.3.1.1.1.1.1.cmml" xref="S2.SS1.p12.13.m4.3.3.1.1.1.1.2"><csymbol cd="latexml" id="S2.SS1.p12.13.m4.3.3.1.1.1.1.1.1.cmml" xref="S2.SS1.p12.13.m4.3.3.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S2.SS1.p12.13.m4.1.1.cmml" xref="S2.SS1.p12.13.m4.1.1">𝑤</ci></apply></apply><apply id="S2.SS1.p12.13.m4.3.3.3.cmml" xref="S2.SS1.p12.13.m4.3.3.3"><times id="S2.SS1.p12.13.m4.3.3.3.1.cmml" xref="S2.SS1.p12.13.m4.3.3.3.1"></times><ci id="S2.SS1.p12.13.m4.3.3.3.2.cmml" xref="S2.SS1.p12.13.m4.3.3.3.2">𝜔</ci><ci id="S2.SS1.p12.13.m4.2.2.cmml" xref="S2.SS1.p12.13.m4.2.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.13.m4.3c">\mu_{\omega}([w]):=\omega(w)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.13.m4.3d">italic_μ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT ( [ italic_w ] ) := italic_ω ( italic_w )</annotation></semantics></math> for all <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p12.14.m5.1"><semantics id="S2.SS1.p12.14.m5.1a"><mrow id="S2.SS1.p12.14.m5.1.1" xref="S2.SS1.p12.14.m5.1.1.cmml"><mi id="S2.SS1.p12.14.m5.1.1.2" xref="S2.SS1.p12.14.m5.1.1.2.cmml">w</mi><mo id="S2.SS1.p12.14.m5.1.1.1" xref="S2.SS1.p12.14.m5.1.1.1.cmml">∈</mo><msup id="S2.SS1.p12.14.m5.1.1.3" xref="S2.SS1.p12.14.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p12.14.m5.1.1.3.2" xref="S2.SS1.p12.14.m5.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p12.14.m5.1.1.3.3" xref="S2.SS1.p12.14.m5.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.14.m5.1b"><apply id="S2.SS1.p12.14.m5.1.1.cmml" xref="S2.SS1.p12.14.m5.1.1"><in id="S2.SS1.p12.14.m5.1.1.1.cmml" xref="S2.SS1.p12.14.m5.1.1.1"></in><ci id="S2.SS1.p12.14.m5.1.1.2.cmml" xref="S2.SS1.p12.14.m5.1.1.2">𝑤</ci><apply id="S2.SS1.p12.14.m5.1.1.3.cmml" xref="S2.SS1.p12.14.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p12.14.m5.1.1.3.1.cmml" xref="S2.SS1.p12.14.m5.1.1.3">superscript</csymbol><ci id="S2.SS1.p12.14.m5.1.1.3.2.cmml" xref="S2.SS1.p12.14.m5.1.1.3.2">𝒜</ci><times id="S2.SS1.p12.14.m5.1.1.3.3.cmml" xref="S2.SS1.p12.14.m5.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.14.m5.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.14.m5.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. This gives <math alttext="\mu_{\omega_{\mu}}=\mu" class="ltx_Math" display="inline" id="S2.SS1.p12.15.m6.1"><semantics id="S2.SS1.p12.15.m6.1a"><mrow id="S2.SS1.p12.15.m6.1.1" xref="S2.SS1.p12.15.m6.1.1.cmml"><msub id="S2.SS1.p12.15.m6.1.1.2" xref="S2.SS1.p12.15.m6.1.1.2.cmml"><mi id="S2.SS1.p12.15.m6.1.1.2.2" xref="S2.SS1.p12.15.m6.1.1.2.2.cmml">μ</mi><msub id="S2.SS1.p12.15.m6.1.1.2.3" xref="S2.SS1.p12.15.m6.1.1.2.3.cmml"><mi id="S2.SS1.p12.15.m6.1.1.2.3.2" xref="S2.SS1.p12.15.m6.1.1.2.3.2.cmml">ω</mi><mi id="S2.SS1.p12.15.m6.1.1.2.3.3" xref="S2.SS1.p12.15.m6.1.1.2.3.3.cmml">μ</mi></msub></msub><mo id="S2.SS1.p12.15.m6.1.1.1" xref="S2.SS1.p12.15.m6.1.1.1.cmml">=</mo><mi id="S2.SS1.p12.15.m6.1.1.3" xref="S2.SS1.p12.15.m6.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.15.m6.1b"><apply id="S2.SS1.p12.15.m6.1.1.cmml" xref="S2.SS1.p12.15.m6.1.1"><eq id="S2.SS1.p12.15.m6.1.1.1.cmml" xref="S2.SS1.p12.15.m6.1.1.1"></eq><apply id="S2.SS1.p12.15.m6.1.1.2.cmml" xref="S2.SS1.p12.15.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p12.15.m6.1.1.2.1.cmml" xref="S2.SS1.p12.15.m6.1.1.2">subscript</csymbol><ci id="S2.SS1.p12.15.m6.1.1.2.2.cmml" xref="S2.SS1.p12.15.m6.1.1.2.2">𝜇</ci><apply id="S2.SS1.p12.15.m6.1.1.2.3.cmml" xref="S2.SS1.p12.15.m6.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS1.p12.15.m6.1.1.2.3.1.cmml" xref="S2.SS1.p12.15.m6.1.1.2.3">subscript</csymbol><ci id="S2.SS1.p12.15.m6.1.1.2.3.2.cmml" xref="S2.SS1.p12.15.m6.1.1.2.3.2">𝜔</ci><ci id="S2.SS1.p12.15.m6.1.1.2.3.3.cmml" xref="S2.SS1.p12.15.m6.1.1.2.3.3">𝜇</ci></apply></apply><ci id="S2.SS1.p12.15.m6.1.1.3.cmml" xref="S2.SS1.p12.15.m6.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.15.m6.1c">\mu_{\omega_{\mu}}=\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.15.m6.1d">italic_μ start_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_μ</annotation></semantics></math> and <math alttext="\omega_{\mu_{\omega}}=\omega" class="ltx_Math" display="inline" id="S2.SS1.p12.16.m7.1"><semantics id="S2.SS1.p12.16.m7.1a"><mrow id="S2.SS1.p12.16.m7.1.1" xref="S2.SS1.p12.16.m7.1.1.cmml"><msub id="S2.SS1.p12.16.m7.1.1.2" xref="S2.SS1.p12.16.m7.1.1.2.cmml"><mi id="S2.SS1.p12.16.m7.1.1.2.2" xref="S2.SS1.p12.16.m7.1.1.2.2.cmml">ω</mi><msub id="S2.SS1.p12.16.m7.1.1.2.3" xref="S2.SS1.p12.16.m7.1.1.2.3.cmml"><mi id="S2.SS1.p12.16.m7.1.1.2.3.2" xref="S2.SS1.p12.16.m7.1.1.2.3.2.cmml">μ</mi><mi id="S2.SS1.p12.16.m7.1.1.2.3.3" xref="S2.SS1.p12.16.m7.1.1.2.3.3.cmml">ω</mi></msub></msub><mo id="S2.SS1.p12.16.m7.1.1.1" xref="S2.SS1.p12.16.m7.1.1.1.cmml">=</mo><mi id="S2.SS1.p12.16.m7.1.1.3" xref="S2.SS1.p12.16.m7.1.1.3.cmml">ω</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.16.m7.1b"><apply id="S2.SS1.p12.16.m7.1.1.cmml" xref="S2.SS1.p12.16.m7.1.1"><eq id="S2.SS1.p12.16.m7.1.1.1.cmml" xref="S2.SS1.p12.16.m7.1.1.1"></eq><apply id="S2.SS1.p12.16.m7.1.1.2.cmml" xref="S2.SS1.p12.16.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p12.16.m7.1.1.2.1.cmml" xref="S2.SS1.p12.16.m7.1.1.2">subscript</csymbol><ci id="S2.SS1.p12.16.m7.1.1.2.2.cmml" xref="S2.SS1.p12.16.m7.1.1.2.2">𝜔</ci><apply id="S2.SS1.p12.16.m7.1.1.2.3.cmml" xref="S2.SS1.p12.16.m7.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS1.p12.16.m7.1.1.2.3.1.cmml" xref="S2.SS1.p12.16.m7.1.1.2.3">subscript</csymbol><ci id="S2.SS1.p12.16.m7.1.1.2.3.2.cmml" xref="S2.SS1.p12.16.m7.1.1.2.3.2">𝜇</ci><ci id="S2.SS1.p12.16.m7.1.1.2.3.3.cmml" xref="S2.SS1.p12.16.m7.1.1.2.3.3">𝜔</ci></apply></apply><ci id="S2.SS1.p12.16.m7.1.1.3.cmml" xref="S2.SS1.p12.16.m7.1.1.3">𝜔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.16.m7.1c">\omega_{\mu_{\omega}}=\omega</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.16.m7.1d">italic_ω start_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_ω end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_ω</annotation></semantics></math>, and hence a bijective relation between invariant measures and weight functions. This bijection respects the addition and the multiplication with scalars <math alttext="\lambda\in\mathbb{R}_{\geq 0}\," class="ltx_Math" display="inline" id="S2.SS1.p12.17.m8.1"><semantics id="S2.SS1.p12.17.m8.1a"><mrow id="S2.SS1.p12.17.m8.1.1" xref="S2.SS1.p12.17.m8.1.1.cmml"><mi id="S2.SS1.p12.17.m8.1.1.2" xref="S2.SS1.p12.17.m8.1.1.2.cmml">λ</mi><mo id="S2.SS1.p12.17.m8.1.1.1" xref="S2.SS1.p12.17.m8.1.1.1.cmml">∈</mo><msub id="S2.SS1.p12.17.m8.1.1.3" xref="S2.SS1.p12.17.m8.1.1.3.cmml"><mi id="S2.SS1.p12.17.m8.1.1.3.2" xref="S2.SS1.p12.17.m8.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS1.p12.17.m8.1.1.3.3" xref="S2.SS1.p12.17.m8.1.1.3.3.cmml"><mi id="S2.SS1.p12.17.m8.1.1.3.3.2" xref="S2.SS1.p12.17.m8.1.1.3.3.2.cmml"></mi><mo id="S2.SS1.p12.17.m8.1.1.3.3.1" xref="S2.SS1.p12.17.m8.1.1.3.3.1.cmml">≥</mo><mn id="S2.SS1.p12.17.m8.1.1.3.3.3" xref="S2.SS1.p12.17.m8.1.1.3.3.3.cmml">0</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p12.17.m8.1b"><apply id="S2.SS1.p12.17.m8.1.1.cmml" xref="S2.SS1.p12.17.m8.1.1"><in id="S2.SS1.p12.17.m8.1.1.1.cmml" xref="S2.SS1.p12.17.m8.1.1.1"></in><ci id="S2.SS1.p12.17.m8.1.1.2.cmml" xref="S2.SS1.p12.17.m8.1.1.2">𝜆</ci><apply id="S2.SS1.p12.17.m8.1.1.3.cmml" xref="S2.SS1.p12.17.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p12.17.m8.1.1.3.1.cmml" xref="S2.SS1.p12.17.m8.1.1.3">subscript</csymbol><ci id="S2.SS1.p12.17.m8.1.1.3.2.cmml" xref="S2.SS1.p12.17.m8.1.1.3.2">ℝ</ci><apply id="S2.SS1.p12.17.m8.1.1.3.3.cmml" xref="S2.SS1.p12.17.m8.1.1.3.3"><geq id="S2.SS1.p12.17.m8.1.1.3.3.1.cmml" xref="S2.SS1.p12.17.m8.1.1.3.3.1"></geq><csymbol cd="latexml" id="S2.SS1.p12.17.m8.1.1.3.3.2.cmml" xref="S2.SS1.p12.17.m8.1.1.3.3.2">absent</csymbol><cn id="S2.SS1.p12.17.m8.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.p12.17.m8.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p12.17.m8.1c">\lambda\in\mathbb{R}_{\geq 0}\,</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p12.17.m8.1d">italic_λ ∈ blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math>, both of which are naturally defined for invariant measures as well as for weight functions. We thus simplify our notation by writing</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.3)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu(w):=\mu([w])=\omega_{\mu}(w)\,." class="ltx_Math" display="block" id="S2.E3.m1.4"><semantics id="S2.E3.m1.4a"><mrow id="S2.E3.m1.4.4.1" xref="S2.E3.m1.4.4.1.1.cmml"><mrow id="S2.E3.m1.4.4.1.1" xref="S2.E3.m1.4.4.1.1.cmml"><mrow id="S2.E3.m1.4.4.1.1.3" xref="S2.E3.m1.4.4.1.1.3.cmml"><mi id="S2.E3.m1.4.4.1.1.3.2" xref="S2.E3.m1.4.4.1.1.3.2.cmml">μ</mi><mo id="S2.E3.m1.4.4.1.1.3.1" xref="S2.E3.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.3.3.2" xref="S2.E3.m1.4.4.1.1.3.cmml"><mo id="S2.E3.m1.4.4.1.1.3.3.2.1" stretchy="false" 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id="S2.E3.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="S2.E3.m1.4.4.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.4.4.1.1.5" xref="S2.E3.m1.4.4.1.1.5.cmml">=</mo><mrow id="S2.E3.m1.4.4.1.1.6" xref="S2.E3.m1.4.4.1.1.6.cmml"><msub id="S2.E3.m1.4.4.1.1.6.2" xref="S2.E3.m1.4.4.1.1.6.2.cmml"><mi id="S2.E3.m1.4.4.1.1.6.2.2" xref="S2.E3.m1.4.4.1.1.6.2.2.cmml">ω</mi><mi id="S2.E3.m1.4.4.1.1.6.2.3" xref="S2.E3.m1.4.4.1.1.6.2.3.cmml">μ</mi></msub><mo id="S2.E3.m1.4.4.1.1.6.1" xref="S2.E3.m1.4.4.1.1.6.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.6.3.2" xref="S2.E3.m1.4.4.1.1.6.cmml"><mo id="S2.E3.m1.4.4.1.1.6.3.2.1" stretchy="false" xref="S2.E3.m1.4.4.1.1.6.cmml">(</mo><mi id="S2.E3.m1.3.3" xref="S2.E3.m1.3.3.cmml">w</mi><mo id="S2.E3.m1.4.4.1.1.6.3.2.2" stretchy="false" xref="S2.E3.m1.4.4.1.1.6.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E3.m1.4.4.1.2" lspace="0.170em" xref="S2.E3.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m1.4b"><apply 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encoding="application/x-llamapun" id="S2.E3.m1.4d">italic_μ ( italic_w ) := italic_μ ( [ italic_w ] ) = italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_w ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS1.p13"> <p class="ltx_p" id="S2.SS1.p13.10">An invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p13.1.m1.1"><semantics id="S2.SS1.p13.1.m1.1a"><mi id="S2.SS1.p13.1.m1.1.1" xref="S2.SS1.p13.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.1.m1.1b"><ci id="S2.SS1.p13.1.m1.1.1.cmml" xref="S2.SS1.p13.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.1.m1.1d">italic_μ</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S2.SS1.p13.10.1">ergodic</span> if <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p13.2.m2.1"><semantics id="S2.SS1.p13.2.m2.1a"><mi id="S2.SS1.p13.2.m2.1.1" xref="S2.SS1.p13.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.2.m2.1b"><ci id="S2.SS1.p13.2.m2.1.1.cmml" xref="S2.SS1.p13.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.2.m2.1d">italic_μ</annotation></semantics></math> can not be written in any non-trivial way as sum <math alttext="\mu_{1}+\mu_{2}" class="ltx_Math" display="inline" id="S2.SS1.p13.3.m3.1"><semantics id="S2.SS1.p13.3.m3.1a"><mrow id="S2.SS1.p13.3.m3.1.1" xref="S2.SS1.p13.3.m3.1.1.cmml"><msub id="S2.SS1.p13.3.m3.1.1.2" xref="S2.SS1.p13.3.m3.1.1.2.cmml"><mi id="S2.SS1.p13.3.m3.1.1.2.2" xref="S2.SS1.p13.3.m3.1.1.2.2.cmml">μ</mi><mn id="S2.SS1.p13.3.m3.1.1.2.3" xref="S2.SS1.p13.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p13.3.m3.1.1.1" xref="S2.SS1.p13.3.m3.1.1.1.cmml">+</mo><msub id="S2.SS1.p13.3.m3.1.1.3" xref="S2.SS1.p13.3.m3.1.1.3.cmml"><mi id="S2.SS1.p13.3.m3.1.1.3.2" xref="S2.SS1.p13.3.m3.1.1.3.2.cmml">μ</mi><mn id="S2.SS1.p13.3.m3.1.1.3.3" xref="S2.SS1.p13.3.m3.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.3.m3.1b"><apply id="S2.SS1.p13.3.m3.1.1.cmml" xref="S2.SS1.p13.3.m3.1.1"><plus id="S2.SS1.p13.3.m3.1.1.1.cmml" xref="S2.SS1.p13.3.m3.1.1.1"></plus><apply id="S2.SS1.p13.3.m3.1.1.2.cmml" xref="S2.SS1.p13.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p13.3.m3.1.1.2.1.cmml" xref="S2.SS1.p13.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p13.3.m3.1.1.2.2.cmml" xref="S2.SS1.p13.3.m3.1.1.2.2">𝜇</ci><cn id="S2.SS1.p13.3.m3.1.1.2.3.cmml" type="integer" xref="S2.SS1.p13.3.m3.1.1.2.3">1</cn></apply><apply id="S2.SS1.p13.3.m3.1.1.3.cmml" xref="S2.SS1.p13.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p13.3.m3.1.1.3.1.cmml" xref="S2.SS1.p13.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS1.p13.3.m3.1.1.3.2.cmml" xref="S2.SS1.p13.3.m3.1.1.3.2">𝜇</ci><cn id="S2.SS1.p13.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS1.p13.3.m3.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.3.m3.1c">\mu_{1}+\mu_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.3.m3.1d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> of two invariant measures <math alttext="\mu_{1}" class="ltx_Math" display="inline" id="S2.SS1.p13.4.m4.1"><semantics id="S2.SS1.p13.4.m4.1a"><msub id="S2.SS1.p13.4.m4.1.1" xref="S2.SS1.p13.4.m4.1.1.cmml"><mi id="S2.SS1.p13.4.m4.1.1.2" xref="S2.SS1.p13.4.m4.1.1.2.cmml">μ</mi><mn id="S2.SS1.p13.4.m4.1.1.3" xref="S2.SS1.p13.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.4.m4.1b"><apply id="S2.SS1.p13.4.m4.1.1.cmml" xref="S2.SS1.p13.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p13.4.m4.1.1.1.cmml" xref="S2.SS1.p13.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p13.4.m4.1.1.2.cmml" xref="S2.SS1.p13.4.m4.1.1.2">𝜇</ci><cn id="S2.SS1.p13.4.m4.1.1.3.cmml" type="integer" xref="S2.SS1.p13.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.4.m4.1c">\mu_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.4.m4.1d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mu_{2}" class="ltx_Math" display="inline" id="S2.SS1.p13.5.m5.1"><semantics id="S2.SS1.p13.5.m5.1a"><msub id="S2.SS1.p13.5.m5.1.1" xref="S2.SS1.p13.5.m5.1.1.cmml"><mi id="S2.SS1.p13.5.m5.1.1.2" xref="S2.SS1.p13.5.m5.1.1.2.cmml">μ</mi><mn id="S2.SS1.p13.5.m5.1.1.3" xref="S2.SS1.p13.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.5.m5.1b"><apply id="S2.SS1.p13.5.m5.1.1.cmml" xref="S2.SS1.p13.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p13.5.m5.1.1.1.cmml" xref="S2.SS1.p13.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p13.5.m5.1.1.2.cmml" xref="S2.SS1.p13.5.m5.1.1.2">𝜇</ci><cn id="S2.SS1.p13.5.m5.1.1.3.cmml" type="integer" xref="S2.SS1.p13.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.5.m5.1c">\mu_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.5.m5.1d">italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> (i.e. <math alttext="\mu_{1}\neq 0\neq\mu_{2}" class="ltx_Math" display="inline" id="S2.SS1.p13.6.m6.1"><semantics id="S2.SS1.p13.6.m6.1a"><mrow id="S2.SS1.p13.6.m6.1.1" xref="S2.SS1.p13.6.m6.1.1.cmml"><msub id="S2.SS1.p13.6.m6.1.1.2" xref="S2.SS1.p13.6.m6.1.1.2.cmml"><mi id="S2.SS1.p13.6.m6.1.1.2.2" xref="S2.SS1.p13.6.m6.1.1.2.2.cmml">μ</mi><mn id="S2.SS1.p13.6.m6.1.1.2.3" xref="S2.SS1.p13.6.m6.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p13.6.m6.1.1.3" xref="S2.SS1.p13.6.m6.1.1.3.cmml">≠</mo><mn id="S2.SS1.p13.6.m6.1.1.4" xref="S2.SS1.p13.6.m6.1.1.4.cmml">0</mn><mo id="S2.SS1.p13.6.m6.1.1.5" xref="S2.SS1.p13.6.m6.1.1.5.cmml">≠</mo><msub id="S2.SS1.p13.6.m6.1.1.6" xref="S2.SS1.p13.6.m6.1.1.6.cmml"><mi id="S2.SS1.p13.6.m6.1.1.6.2" xref="S2.SS1.p13.6.m6.1.1.6.2.cmml">μ</mi><mn id="S2.SS1.p13.6.m6.1.1.6.3" xref="S2.SS1.p13.6.m6.1.1.6.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.6.m6.1b"><apply id="S2.SS1.p13.6.m6.1.1.cmml" xref="S2.SS1.p13.6.m6.1.1"><and id="S2.SS1.p13.6.m6.1.1a.cmml" xref="S2.SS1.p13.6.m6.1.1"></and><apply id="S2.SS1.p13.6.m6.1.1b.cmml" xref="S2.SS1.p13.6.m6.1.1"><neq id="S2.SS1.p13.6.m6.1.1.3.cmml" xref="S2.SS1.p13.6.m6.1.1.3"></neq><apply id="S2.SS1.p13.6.m6.1.1.2.cmml" xref="S2.SS1.p13.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p13.6.m6.1.1.2.1.cmml" xref="S2.SS1.p13.6.m6.1.1.2">subscript</csymbol><ci id="S2.SS1.p13.6.m6.1.1.2.2.cmml" xref="S2.SS1.p13.6.m6.1.1.2.2">𝜇</ci><cn id="S2.SS1.p13.6.m6.1.1.2.3.cmml" type="integer" xref="S2.SS1.p13.6.m6.1.1.2.3">1</cn></apply><cn id="S2.SS1.p13.6.m6.1.1.4.cmml" type="integer" xref="S2.SS1.p13.6.m6.1.1.4">0</cn></apply><apply id="S2.SS1.p13.6.m6.1.1c.cmml" xref="S2.SS1.p13.6.m6.1.1"><neq id="S2.SS1.p13.6.m6.1.1.5.cmml" xref="S2.SS1.p13.6.m6.1.1.5"></neq><share href="https://arxiv.org/html/2211.11234v4#S2.SS1.p13.6.m6.1.1.4.cmml" id="S2.SS1.p13.6.m6.1.1d.cmml" xref="S2.SS1.p13.6.m6.1.1"></share><apply id="S2.SS1.p13.6.m6.1.1.6.cmml" xref="S2.SS1.p13.6.m6.1.1.6"><csymbol cd="ambiguous" id="S2.SS1.p13.6.m6.1.1.6.1.cmml" xref="S2.SS1.p13.6.m6.1.1.6">subscript</csymbol><ci id="S2.SS1.p13.6.m6.1.1.6.2.cmml" xref="S2.SS1.p13.6.m6.1.1.6.2">𝜇</ci><cn id="S2.SS1.p13.6.m6.1.1.6.3.cmml" type="integer" xref="S2.SS1.p13.6.m6.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.6.m6.1c">\mu_{1}\neq 0\neq\mu_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.6.m6.1d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ 0 ≠ italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mu_{1}\neq\lambda\mu_{2}" class="ltx_Math" display="inline" id="S2.SS1.p13.7.m7.1"><semantics id="S2.SS1.p13.7.m7.1a"><mrow id="S2.SS1.p13.7.m7.1.1" xref="S2.SS1.p13.7.m7.1.1.cmml"><msub id="S2.SS1.p13.7.m7.1.1.2" xref="S2.SS1.p13.7.m7.1.1.2.cmml"><mi id="S2.SS1.p13.7.m7.1.1.2.2" xref="S2.SS1.p13.7.m7.1.1.2.2.cmml">μ</mi><mn id="S2.SS1.p13.7.m7.1.1.2.3" xref="S2.SS1.p13.7.m7.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p13.7.m7.1.1.1" xref="S2.SS1.p13.7.m7.1.1.1.cmml">≠</mo><mrow id="S2.SS1.p13.7.m7.1.1.3" xref="S2.SS1.p13.7.m7.1.1.3.cmml"><mi id="S2.SS1.p13.7.m7.1.1.3.2" xref="S2.SS1.p13.7.m7.1.1.3.2.cmml">λ</mi><mo id="S2.SS1.p13.7.m7.1.1.3.1" xref="S2.SS1.p13.7.m7.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p13.7.m7.1.1.3.3" xref="S2.SS1.p13.7.m7.1.1.3.3.cmml"><mi id="S2.SS1.p13.7.m7.1.1.3.3.2" xref="S2.SS1.p13.7.m7.1.1.3.3.2.cmml">μ</mi><mn id="S2.SS1.p13.7.m7.1.1.3.3.3" xref="S2.SS1.p13.7.m7.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.7.m7.1b"><apply id="S2.SS1.p13.7.m7.1.1.cmml" xref="S2.SS1.p13.7.m7.1.1"><neq id="S2.SS1.p13.7.m7.1.1.1.cmml" xref="S2.SS1.p13.7.m7.1.1.1"></neq><apply id="S2.SS1.p13.7.m7.1.1.2.cmml" xref="S2.SS1.p13.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p13.7.m7.1.1.2.1.cmml" xref="S2.SS1.p13.7.m7.1.1.2">subscript</csymbol><ci id="S2.SS1.p13.7.m7.1.1.2.2.cmml" xref="S2.SS1.p13.7.m7.1.1.2.2">𝜇</ci><cn id="S2.SS1.p13.7.m7.1.1.2.3.cmml" type="integer" xref="S2.SS1.p13.7.m7.1.1.2.3">1</cn></apply><apply id="S2.SS1.p13.7.m7.1.1.3.cmml" xref="S2.SS1.p13.7.m7.1.1.3"><times id="S2.SS1.p13.7.m7.1.1.3.1.cmml" xref="S2.SS1.p13.7.m7.1.1.3.1"></times><ci id="S2.SS1.p13.7.m7.1.1.3.2.cmml" xref="S2.SS1.p13.7.m7.1.1.3.2">𝜆</ci><apply id="S2.SS1.p13.7.m7.1.1.3.3.cmml" xref="S2.SS1.p13.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p13.7.m7.1.1.3.3.1.cmml" xref="S2.SS1.p13.7.m7.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p13.7.m7.1.1.3.3.2.cmml" xref="S2.SS1.p13.7.m7.1.1.3.3.2">𝜇</ci><cn id="S2.SS1.p13.7.m7.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.p13.7.m7.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.7.m7.1c">\mu_{1}\neq\lambda\mu_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.7.m7.1d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≠ italic_λ italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> for any <math alttext="\lambda\in\mathbb{R}_{>0}" class="ltx_Math" display="inline" id="S2.SS1.p13.8.m8.1"><semantics id="S2.SS1.p13.8.m8.1a"><mrow id="S2.SS1.p13.8.m8.1.1" xref="S2.SS1.p13.8.m8.1.1.cmml"><mi id="S2.SS1.p13.8.m8.1.1.2" xref="S2.SS1.p13.8.m8.1.1.2.cmml">λ</mi><mo id="S2.SS1.p13.8.m8.1.1.1" xref="S2.SS1.p13.8.m8.1.1.1.cmml">∈</mo><msub id="S2.SS1.p13.8.m8.1.1.3" xref="S2.SS1.p13.8.m8.1.1.3.cmml"><mi id="S2.SS1.p13.8.m8.1.1.3.2" xref="S2.SS1.p13.8.m8.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS1.p13.8.m8.1.1.3.3" xref="S2.SS1.p13.8.m8.1.1.3.3.cmml"><mi id="S2.SS1.p13.8.m8.1.1.3.3.2" xref="S2.SS1.p13.8.m8.1.1.3.3.2.cmml"></mi><mo id="S2.SS1.p13.8.m8.1.1.3.3.1" xref="S2.SS1.p13.8.m8.1.1.3.3.1.cmml">></mo><mn id="S2.SS1.p13.8.m8.1.1.3.3.3" xref="S2.SS1.p13.8.m8.1.1.3.3.3.cmml">0</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.8.m8.1b"><apply id="S2.SS1.p13.8.m8.1.1.cmml" xref="S2.SS1.p13.8.m8.1.1"><in id="S2.SS1.p13.8.m8.1.1.1.cmml" xref="S2.SS1.p13.8.m8.1.1.1"></in><ci id="S2.SS1.p13.8.m8.1.1.2.cmml" xref="S2.SS1.p13.8.m8.1.1.2">𝜆</ci><apply id="S2.SS1.p13.8.m8.1.1.3.cmml" xref="S2.SS1.p13.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p13.8.m8.1.1.3.1.cmml" xref="S2.SS1.p13.8.m8.1.1.3">subscript</csymbol><ci id="S2.SS1.p13.8.m8.1.1.3.2.cmml" xref="S2.SS1.p13.8.m8.1.1.3.2">ℝ</ci><apply id="S2.SS1.p13.8.m8.1.1.3.3.cmml" xref="S2.SS1.p13.8.m8.1.1.3.3"><gt id="S2.SS1.p13.8.m8.1.1.3.3.1.cmml" xref="S2.SS1.p13.8.m8.1.1.3.3.1"></gt><csymbol cd="latexml" id="S2.SS1.p13.8.m8.1.1.3.3.2.cmml" xref="S2.SS1.p13.8.m8.1.1.3.3.2">absent</csymbol><cn id="S2.SS1.p13.8.m8.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.p13.8.m8.1.1.3.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.8.m8.1c">\lambda\in\mathbb{R}_{>0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.8.m8.1d">italic_λ ∈ blackboard_R start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT</annotation></semantics></math>). An invariant measure is called a <span class="ltx_text ltx_font_italic" id="S2.SS1.p13.10.2">probability measure</span> if <math alttext="\mu(X)=1" class="ltx_Math" display="inline" id="S2.SS1.p13.9.m9.1"><semantics id="S2.SS1.p13.9.m9.1a"><mrow id="S2.SS1.p13.9.m9.1.2" xref="S2.SS1.p13.9.m9.1.2.cmml"><mrow id="S2.SS1.p13.9.m9.1.2.2" xref="S2.SS1.p13.9.m9.1.2.2.cmml"><mi id="S2.SS1.p13.9.m9.1.2.2.2" xref="S2.SS1.p13.9.m9.1.2.2.2.cmml">μ</mi><mo id="S2.SS1.p13.9.m9.1.2.2.1" xref="S2.SS1.p13.9.m9.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p13.9.m9.1.2.2.3.2" xref="S2.SS1.p13.9.m9.1.2.2.cmml"><mo id="S2.SS1.p13.9.m9.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.p13.9.m9.1.2.2.cmml">(</mo><mi id="S2.SS1.p13.9.m9.1.1" xref="S2.SS1.p13.9.m9.1.1.cmml">X</mi><mo id="S2.SS1.p13.9.m9.1.2.2.3.2.2" stretchy="false" xref="S2.SS1.p13.9.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p13.9.m9.1.2.1" xref="S2.SS1.p13.9.m9.1.2.1.cmml">=</mo><mn id="S2.SS1.p13.9.m9.1.2.3" xref="S2.SS1.p13.9.m9.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.9.m9.1b"><apply id="S2.SS1.p13.9.m9.1.2.cmml" xref="S2.SS1.p13.9.m9.1.2"><eq id="S2.SS1.p13.9.m9.1.2.1.cmml" xref="S2.SS1.p13.9.m9.1.2.1"></eq><apply id="S2.SS1.p13.9.m9.1.2.2.cmml" xref="S2.SS1.p13.9.m9.1.2.2"><times id="S2.SS1.p13.9.m9.1.2.2.1.cmml" xref="S2.SS1.p13.9.m9.1.2.2.1"></times><ci id="S2.SS1.p13.9.m9.1.2.2.2.cmml" xref="S2.SS1.p13.9.m9.1.2.2.2">𝜇</ci><ci id="S2.SS1.p13.9.m9.1.1.cmml" xref="S2.SS1.p13.9.m9.1.1">𝑋</ci></apply><cn id="S2.SS1.p13.9.m9.1.2.3.cmml" type="integer" xref="S2.SS1.p13.9.m9.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.9.m9.1c">\mu(X)=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.9.m9.1d">italic_μ ( italic_X ) = 1</annotation></semantics></math>, which is equivalent to <math alttext="\underset{a_{i}\in\cal A}{\sum}\mu([a_{i}])=1" class="ltx_Math" display="inline" id="S2.SS1.p13.10.m10.1"><semantics id="S2.SS1.p13.10.m10.1a"><mrow id="S2.SS1.p13.10.m10.1.1" xref="S2.SS1.p13.10.m10.1.1.cmml"><mrow id="S2.SS1.p13.10.m10.1.1.1" xref="S2.SS1.p13.10.m10.1.1.1.cmml"><munder accentunder="true" id="S2.SS1.p13.10.m10.1.1.1.3" xref="S2.SS1.p13.10.m10.1.1.1.3.cmml"><mo id="S2.SS1.p13.10.m10.1.1.1.3.2" xref="S2.SS1.p13.10.m10.1.1.1.3.2.cmml">∑</mo><mrow id="S2.SS1.p13.10.m10.1.1.1.3.1" xref="S2.SS1.p13.10.m10.1.1.1.3.1.cmml"><msub id="S2.SS1.p13.10.m10.1.1.1.3.1.2" xref="S2.SS1.p13.10.m10.1.1.1.3.1.2.cmml"><mi id="S2.SS1.p13.10.m10.1.1.1.3.1.2.2" xref="S2.SS1.p13.10.m10.1.1.1.3.1.2.2.cmml">a</mi><mi id="S2.SS1.p13.10.m10.1.1.1.3.1.2.3" xref="S2.SS1.p13.10.m10.1.1.1.3.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p13.10.m10.1.1.1.3.1.1" xref="S2.SS1.p13.10.m10.1.1.1.3.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p13.10.m10.1.1.1.3.1.3" xref="S2.SS1.p13.10.m10.1.1.1.3.1.3.cmml">𝒜</mi></mrow></munder><mo id="S2.SS1.p13.10.m10.1.1.1.2" xref="S2.SS1.p13.10.m10.1.1.1.2.cmml">⁢</mo><mi id="S2.SS1.p13.10.m10.1.1.1.4" xref="S2.SS1.p13.10.m10.1.1.1.4.cmml">μ</mi><mo id="S2.SS1.p13.10.m10.1.1.1.2a" xref="S2.SS1.p13.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p13.10.m10.1.1.1.1.1" xref="S2.SS1.p13.10.m10.1.1.1.cmml"><mo id="S2.SS1.p13.10.m10.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p13.10.m10.1.1.1.cmml">(</mo><mrow id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.2.cmml"><mo id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.2.1.cmml">[</mo><msub id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.2" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.3" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.3.cmml">𝒾</mi></msub><mo id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S2.SS1.p13.10.m10.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p13.10.m10.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p13.10.m10.1.1.2" xref="S2.SS1.p13.10.m10.1.1.2.cmml">=</mo><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p13.10.m10.1.1.3" mathvariant="script" xref="S2.SS1.p13.10.m10.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p13.10.m10.1b"><apply id="S2.SS1.p13.10.m10.1.1.cmml" xref="S2.SS1.p13.10.m10.1.1"><eq id="S2.SS1.p13.10.m10.1.1.2.cmml" xref="S2.SS1.p13.10.m10.1.1.2"></eq><apply id="S2.SS1.p13.10.m10.1.1.1.cmml" xref="S2.SS1.p13.10.m10.1.1.1"><times id="S2.SS1.p13.10.m10.1.1.1.2.cmml" xref="S2.SS1.p13.10.m10.1.1.1.2"></times><apply id="S2.SS1.p13.10.m10.1.1.1.3.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3"><apply id="S2.SS1.p13.10.m10.1.1.1.3.1.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3.1"><in id="S2.SS1.p13.10.m10.1.1.1.3.1.1.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3.1.1"></in><apply id="S2.SS1.p13.10.m10.1.1.1.3.1.2.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3.1.2"><csymbol cd="ambiguous" id="S2.SS1.p13.10.m10.1.1.1.3.1.2.1.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3.1.2">subscript</csymbol><ci id="S2.SS1.p13.10.m10.1.1.1.3.1.2.2.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3.1.2.2">𝑎</ci><ci id="S2.SS1.p13.10.m10.1.1.1.3.1.2.3.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p13.10.m10.1.1.1.3.1.3.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3.1.3">𝒜</ci></apply><sum id="S2.SS1.p13.10.m10.1.1.1.3.2.cmml" xref="S2.SS1.p13.10.m10.1.1.1.3.2"></sum></apply><ci id="S2.SS1.p13.10.m10.1.1.1.4.cmml" xref="S2.SS1.p13.10.m10.1.1.1.4">𝜇</ci><apply id="S2.SS1.p13.10.m10.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p13.10.m10.1.1.1.1.1.1.2.1.cmml" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.2">𝒶</ci><ci id="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p13.10.m10.1.1.1.1.1.1.1.1.3">𝒾</ci></apply></apply></apply><cn id="S2.SS1.p13.10.m10.1.1.3.cmml" type="integer" xref="S2.SS1.p13.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p13.10.m10.1c">\underset{a_{i}\in\cal A}{\sum}\mu([a_{i}])=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p13.10.m10.1d">start_UNDERACCENT italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A end_UNDERACCENT start_ARG ∑ end_ARG italic_μ ( [ caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ] ) = caligraphic_1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p14"> <p class="ltx_p" id="S2.SS1.p14.8">We denote by <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p14.1.m1.1"><semantics id="S2.SS1.p14.1.m1.1a"><mrow id="S2.SS1.p14.1.m1.1.2" xref="S2.SS1.p14.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.1.m1.1.2.2" xref="S2.SS1.p14.1.m1.1.2.2.cmml">ℳ</mi><mo id="S2.SS1.p14.1.m1.1.2.1" xref="S2.SS1.p14.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p14.1.m1.1.2.3.2" xref="S2.SS1.p14.1.m1.1.2.cmml"><mo id="S2.SS1.p14.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p14.1.m1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.1.m1.1.1" xref="S2.SS1.p14.1.m1.1.1.cmml">𝒳</mi><mo id="S2.SS1.p14.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p14.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p14.1.m1.1b"><apply id="S2.SS1.p14.1.m1.1.2.cmml" xref="S2.SS1.p14.1.m1.1.2"><times id="S2.SS1.p14.1.m1.1.2.1.cmml" xref="S2.SS1.p14.1.m1.1.2.1"></times><ci id="S2.SS1.p14.1.m1.1.2.2.cmml" xref="S2.SS1.p14.1.m1.1.2.2">ℳ</ci><ci id="S2.SS1.p14.1.m1.1.1.cmml" xref="S2.SS1.p14.1.m1.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p14.1.m1.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p14.1.m1.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> the set of invariant measures on <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p14.2.m2.1"><semantics id="S2.SS1.p14.2.m2.1a"><mi id="S2.SS1.p14.2.m2.1.1" xref="S2.SS1.p14.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p14.2.m2.1b"><ci id="S2.SS1.p14.2.m2.1.1.cmml" xref="S2.SS1.p14.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p14.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p14.2.m2.1d">italic_X</annotation></semantics></math>, and by <math alttext="\cal M_{1}(X)\subseteq\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p14.3.m3.2"><semantics id="S2.SS1.p14.3.m3.2a"><mrow id="S2.SS1.p14.3.m3.2.3" xref="S2.SS1.p14.3.m3.2.3.cmml"><mrow id="S2.SS1.p14.3.m3.2.3.2" xref="S2.SS1.p14.3.m3.2.3.2.cmml"><msub id="S2.SS1.p14.3.m3.2.3.2.2" xref="S2.SS1.p14.3.m3.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.3.m3.2.3.2.2.2" xref="S2.SS1.p14.3.m3.2.3.2.2.2.cmml">ℳ</mi><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p14.3.m3.2.3.2.2.3" mathvariant="script" xref="S2.SS1.p14.3.m3.2.3.2.2.3.cmml">1</mn></msub><mo id="S2.SS1.p14.3.m3.2.3.2.1" xref="S2.SS1.p14.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS1.p14.3.m3.2.3.2.3.2" xref="S2.SS1.p14.3.m3.2.3.2.cmml"><mo id="S2.SS1.p14.3.m3.2.3.2.3.2.1" stretchy="false" xref="S2.SS1.p14.3.m3.2.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.3.m3.1.1" xref="S2.SS1.p14.3.m3.1.1.cmml">𝒳</mi><mo id="S2.SS1.p14.3.m3.2.3.2.3.2.2" stretchy="false" xref="S2.SS1.p14.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p14.3.m3.2.3.1" xref="S2.SS1.p14.3.m3.2.3.1.cmml">⊆</mo><mrow id="S2.SS1.p14.3.m3.2.3.3" xref="S2.SS1.p14.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.3.m3.2.3.3.2" xref="S2.SS1.p14.3.m3.2.3.3.2.cmml">ℳ</mi><mo id="S2.SS1.p14.3.m3.2.3.3.1" xref="S2.SS1.p14.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p14.3.m3.2.3.3.3.2" xref="S2.SS1.p14.3.m3.2.3.3.cmml"><mo id="S2.SS1.p14.3.m3.2.3.3.3.2.1" stretchy="false" xref="S2.SS1.p14.3.m3.2.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.3.m3.2.2" xref="S2.SS1.p14.3.m3.2.2.cmml">𝒳</mi><mo id="S2.SS1.p14.3.m3.2.3.3.3.2.2" stretchy="false" xref="S2.SS1.p14.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p14.3.m3.2b"><apply id="S2.SS1.p14.3.m3.2.3.cmml" xref="S2.SS1.p14.3.m3.2.3"><subset id="S2.SS1.p14.3.m3.2.3.1.cmml" xref="S2.SS1.p14.3.m3.2.3.1"></subset><apply id="S2.SS1.p14.3.m3.2.3.2.cmml" xref="S2.SS1.p14.3.m3.2.3.2"><times id="S2.SS1.p14.3.m3.2.3.2.1.cmml" xref="S2.SS1.p14.3.m3.2.3.2.1"></times><apply id="S2.SS1.p14.3.m3.2.3.2.2.cmml" xref="S2.SS1.p14.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p14.3.m3.2.3.2.2.1.cmml" xref="S2.SS1.p14.3.m3.2.3.2.2">subscript</csymbol><ci id="S2.SS1.p14.3.m3.2.3.2.2.2.cmml" xref="S2.SS1.p14.3.m3.2.3.2.2.2">ℳ</ci><cn id="S2.SS1.p14.3.m3.2.3.2.2.3.cmml" type="integer" xref="S2.SS1.p14.3.m3.2.3.2.2.3">1</cn></apply><ci id="S2.SS1.p14.3.m3.1.1.cmml" xref="S2.SS1.p14.3.m3.1.1">𝒳</ci></apply><apply id="S2.SS1.p14.3.m3.2.3.3.cmml" xref="S2.SS1.p14.3.m3.2.3.3"><times id="S2.SS1.p14.3.m3.2.3.3.1.cmml" xref="S2.SS1.p14.3.m3.2.3.3.1"></times><ci id="S2.SS1.p14.3.m3.2.3.3.2.cmml" xref="S2.SS1.p14.3.m3.2.3.3.2">ℳ</ci><ci id="S2.SS1.p14.3.m3.2.2.cmml" xref="S2.SS1.p14.3.m3.2.2">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p14.3.m3.2c">\cal M_{1}(X)\subseteq\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p14.3.m3.2d">caligraphic_M start_POSTSUBSCRIPT caligraphic_1 end_POSTSUBSCRIPT ( caligraphic_X ) ⊆ caligraphic_M ( caligraphic_X )</annotation></semantics></math> the subset of probability measures. The set <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p14.4.m4.1"><semantics id="S2.SS1.p14.4.m4.1a"><mrow id="S2.SS1.p14.4.m4.1.2" xref="S2.SS1.p14.4.m4.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.4.m4.1.2.2" xref="S2.SS1.p14.4.m4.1.2.2.cmml">ℳ</mi><mo id="S2.SS1.p14.4.m4.1.2.1" xref="S2.SS1.p14.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p14.4.m4.1.2.3.2" xref="S2.SS1.p14.4.m4.1.2.cmml"><mo id="S2.SS1.p14.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS1.p14.4.m4.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.4.m4.1.1" xref="S2.SS1.p14.4.m4.1.1.cmml">𝒳</mi><mo id="S2.SS1.p14.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS1.p14.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p14.4.m4.1b"><apply id="S2.SS1.p14.4.m4.1.2.cmml" xref="S2.SS1.p14.4.m4.1.2"><times id="S2.SS1.p14.4.m4.1.2.1.cmml" xref="S2.SS1.p14.4.m4.1.2.1"></times><ci id="S2.SS1.p14.4.m4.1.2.2.cmml" xref="S2.SS1.p14.4.m4.1.2.2">ℳ</ci><ci id="S2.SS1.p14.4.m4.1.1.cmml" xref="S2.SS1.p14.4.m4.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p14.4.m4.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p14.4.m4.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> is naturally equipped with the <span class="ltx_text ltx_font_italic" id="S2.SS1.p14.5.1">weak<sup class="ltx_sup" id="S2.SS1.p14.5.1.1"><span class="ltx_text ltx_font_upright" id="S2.SS1.p14.5.1.1.1">∗</span></sup>-topology</span> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib1" title="">1</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib11" title="">11</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib16" title="">16</a>]</cite>). Using the above bijection to the set of weight functions this topology turns out to be equivalent to the topology inherited from the canonical embedding of <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p14.6.m5.1"><semantics id="S2.SS1.p14.6.m5.1a"><mrow id="S2.SS1.p14.6.m5.1.2" xref="S2.SS1.p14.6.m5.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.6.m5.1.2.2" xref="S2.SS1.p14.6.m5.1.2.2.cmml">ℳ</mi><mo id="S2.SS1.p14.6.m5.1.2.1" xref="S2.SS1.p14.6.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p14.6.m5.1.2.3.2" xref="S2.SS1.p14.6.m5.1.2.cmml"><mo id="S2.SS1.p14.6.m5.1.2.3.2.1" stretchy="false" xref="S2.SS1.p14.6.m5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.6.m5.1.1" xref="S2.SS1.p14.6.m5.1.1.cmml">𝒳</mi><mo id="S2.SS1.p14.6.m5.1.2.3.2.2" stretchy="false" xref="S2.SS1.p14.6.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p14.6.m5.1b"><apply id="S2.SS1.p14.6.m5.1.2.cmml" xref="S2.SS1.p14.6.m5.1.2"><times id="S2.SS1.p14.6.m5.1.2.1.cmml" xref="S2.SS1.p14.6.m5.1.2.1"></times><ci id="S2.SS1.p14.6.m5.1.2.2.cmml" xref="S2.SS1.p14.6.m5.1.2.2">ℳ</ci><ci id="S2.SS1.p14.6.m5.1.1.cmml" xref="S2.SS1.p14.6.m5.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p14.6.m5.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p14.6.m5.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> into the product space <math alttext="\mathbb{R}^{\cal A^{*}}" class="ltx_Math" display="inline" id="S2.SS1.p14.7.m6.1"><semantics id="S2.SS1.p14.7.m6.1a"><msup id="S2.SS1.p14.7.m6.1.1" xref="S2.SS1.p14.7.m6.1.1.cmml"><mi id="S2.SS1.p14.7.m6.1.1.2" xref="S2.SS1.p14.7.m6.1.1.2.cmml">ℝ</mi><msup id="S2.SS1.p14.7.m6.1.1.3" xref="S2.SS1.p14.7.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.7.m6.1.1.3.2" xref="S2.SS1.p14.7.m6.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p14.7.m6.1.1.3.3" xref="S2.SS1.p14.7.m6.1.1.3.3.cmml">∗</mo></msup></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p14.7.m6.1b"><apply id="S2.SS1.p14.7.m6.1.1.cmml" xref="S2.SS1.p14.7.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p14.7.m6.1.1.1.cmml" xref="S2.SS1.p14.7.m6.1.1">superscript</csymbol><ci id="S2.SS1.p14.7.m6.1.1.2.cmml" xref="S2.SS1.p14.7.m6.1.1.2">ℝ</ci><apply id="S2.SS1.p14.7.m6.1.1.3.cmml" xref="S2.SS1.p14.7.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p14.7.m6.1.1.3.1.cmml" xref="S2.SS1.p14.7.m6.1.1.3">superscript</csymbol><ci id="S2.SS1.p14.7.m6.1.1.3.2.cmml" xref="S2.SS1.p14.7.m6.1.1.3.2">𝒜</ci><times id="S2.SS1.p14.7.m6.1.1.3.3.cmml" xref="S2.SS1.p14.7.m6.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p14.7.m6.1c">\mathbb{R}^{\cal A^{*}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p14.7.m6.1d">blackboard_R start_POSTSUPERSCRIPT caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> given by <math alttext="\mu\mapsto(\mu(w))_{w\in\cal A^{*}}" class="ltx_Math" display="inline" id="S2.SS1.p14.8.m7.2"><semantics id="S2.SS1.p14.8.m7.2a"><mrow id="S2.SS1.p14.8.m7.2.2" xref="S2.SS1.p14.8.m7.2.2.cmml"><mi id="S2.SS1.p14.8.m7.2.2.3" xref="S2.SS1.p14.8.m7.2.2.3.cmml">μ</mi><mo id="S2.SS1.p14.8.m7.2.2.2" stretchy="false" xref="S2.SS1.p14.8.m7.2.2.2.cmml">↦</mo><msub id="S2.SS1.p14.8.m7.2.2.1" xref="S2.SS1.p14.8.m7.2.2.1.cmml"><mrow id="S2.SS1.p14.8.m7.2.2.1.1.1" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.cmml"><mo id="S2.SS1.p14.8.m7.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p14.8.m7.2.2.1.1.1.1" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.cmml"><mi id="S2.SS1.p14.8.m7.2.2.1.1.1.1.2" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.2.cmml">μ</mi><mo id="S2.SS1.p14.8.m7.2.2.1.1.1.1.1" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p14.8.m7.2.2.1.1.1.1.3.2" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.cmml"><mo id="S2.SS1.p14.8.m7.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.cmml">(</mo><mi id="S2.SS1.p14.8.m7.1.1" xref="S2.SS1.p14.8.m7.1.1.cmml">w</mi><mo id="S2.SS1.p14.8.m7.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p14.8.m7.2.2.1.1.1.3" stretchy="false" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS1.p14.8.m7.2.2.1.3" xref="S2.SS1.p14.8.m7.2.2.1.3.cmml"><mi id="S2.SS1.p14.8.m7.2.2.1.3.2" xref="S2.SS1.p14.8.m7.2.2.1.3.2.cmml">w</mi><mo id="S2.SS1.p14.8.m7.2.2.1.3.1" xref="S2.SS1.p14.8.m7.2.2.1.3.1.cmml">∈</mo><msup id="S2.SS1.p14.8.m7.2.2.1.3.3" xref="S2.SS1.p14.8.m7.2.2.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p14.8.m7.2.2.1.3.3.2" xref="S2.SS1.p14.8.m7.2.2.1.3.3.2.cmml">𝒜</mi><mo id="S2.SS1.p14.8.m7.2.2.1.3.3.3" xref="S2.SS1.p14.8.m7.2.2.1.3.3.3.cmml">∗</mo></msup></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p14.8.m7.2b"><apply id="S2.SS1.p14.8.m7.2.2.cmml" xref="S2.SS1.p14.8.m7.2.2"><csymbol cd="latexml" id="S2.SS1.p14.8.m7.2.2.2.cmml" xref="S2.SS1.p14.8.m7.2.2.2">maps-to</csymbol><ci id="S2.SS1.p14.8.m7.2.2.3.cmml" xref="S2.SS1.p14.8.m7.2.2.3">𝜇</ci><apply id="S2.SS1.p14.8.m7.2.2.1.cmml" xref="S2.SS1.p14.8.m7.2.2.1"><csymbol cd="ambiguous" id="S2.SS1.p14.8.m7.2.2.1.2.cmml" xref="S2.SS1.p14.8.m7.2.2.1">subscript</csymbol><apply id="S2.SS1.p14.8.m7.2.2.1.1.1.1.cmml" xref="S2.SS1.p14.8.m7.2.2.1.1.1"><times id="S2.SS1.p14.8.m7.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.1"></times><ci id="S2.SS1.p14.8.m7.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p14.8.m7.2.2.1.1.1.1.2">𝜇</ci><ci id="S2.SS1.p14.8.m7.1.1.cmml" xref="S2.SS1.p14.8.m7.1.1">𝑤</ci></apply><apply id="S2.SS1.p14.8.m7.2.2.1.3.cmml" xref="S2.SS1.p14.8.m7.2.2.1.3"><in id="S2.SS1.p14.8.m7.2.2.1.3.1.cmml" xref="S2.SS1.p14.8.m7.2.2.1.3.1"></in><ci id="S2.SS1.p14.8.m7.2.2.1.3.2.cmml" xref="S2.SS1.p14.8.m7.2.2.1.3.2">𝑤</ci><apply id="S2.SS1.p14.8.m7.2.2.1.3.3.cmml" xref="S2.SS1.p14.8.m7.2.2.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p14.8.m7.2.2.1.3.3.1.cmml" xref="S2.SS1.p14.8.m7.2.2.1.3.3">superscript</csymbol><ci id="S2.SS1.p14.8.m7.2.2.1.3.3.2.cmml" xref="S2.SS1.p14.8.m7.2.2.1.3.3.2">𝒜</ci><times id="S2.SS1.p14.8.m7.2.2.1.3.3.3.cmml" xref="S2.SS1.p14.8.m7.2.2.1.3.3.3"></times></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p14.8.m7.2c">\mu\mapsto(\mu(w))_{w\in\cal A^{*}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p14.8.m7.2d">italic_μ ↦ ( italic_μ ( italic_w ) ) start_POSTSUBSCRIPT italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p15"> <p class="ltx_p" id="S2.SS1.p15.6">It follows that the set <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p15.1.m1.1"><semantics id="S2.SS1.p15.1.m1.1a"><mrow id="S2.SS1.p15.1.m1.1.2" xref="S2.SS1.p15.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.1.m1.1.2.2" xref="S2.SS1.p15.1.m1.1.2.2.cmml">ℳ</mi><mo id="S2.SS1.p15.1.m1.1.2.1" xref="S2.SS1.p15.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p15.1.m1.1.2.3.2" xref="S2.SS1.p15.1.m1.1.2.cmml"><mo id="S2.SS1.p15.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p15.1.m1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.1.m1.1.1" xref="S2.SS1.p15.1.m1.1.1.cmml">𝒳</mi><mo id="S2.SS1.p15.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p15.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p15.1.m1.1b"><apply id="S2.SS1.p15.1.m1.1.2.cmml" xref="S2.SS1.p15.1.m1.1.2"><times id="S2.SS1.p15.1.m1.1.2.1.cmml" xref="S2.SS1.p15.1.m1.1.2.1"></times><ci id="S2.SS1.p15.1.m1.1.2.2.cmml" xref="S2.SS1.p15.1.m1.1.2.2">ℳ</ci><ci id="S2.SS1.p15.1.m1.1.1.cmml" xref="S2.SS1.p15.1.m1.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p15.1.m1.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p15.1.m1.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> is a convex linear cone which is naturally embedded into the non-negative cone of the infinite dimensional vector space <math alttext="\mathbb{R}^{\cal A^{*}}" class="ltx_Math" display="inline" id="S2.SS1.p15.2.m2.1"><semantics id="S2.SS1.p15.2.m2.1a"><msup id="S2.SS1.p15.2.m2.1.1" xref="S2.SS1.p15.2.m2.1.1.cmml"><mi id="S2.SS1.p15.2.m2.1.1.2" xref="S2.SS1.p15.2.m2.1.1.2.cmml">ℝ</mi><msup id="S2.SS1.p15.2.m2.1.1.3" xref="S2.SS1.p15.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.2.m2.1.1.3.2" xref="S2.SS1.p15.2.m2.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p15.2.m2.1.1.3.3" xref="S2.SS1.p15.2.m2.1.1.3.3.cmml">∗</mo></msup></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p15.2.m2.1b"><apply id="S2.SS1.p15.2.m2.1.1.cmml" xref="S2.SS1.p15.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p15.2.m2.1.1.1.cmml" xref="S2.SS1.p15.2.m2.1.1">superscript</csymbol><ci id="S2.SS1.p15.2.m2.1.1.2.cmml" xref="S2.SS1.p15.2.m2.1.1.2">ℝ</ci><apply id="S2.SS1.p15.2.m2.1.1.3.cmml" xref="S2.SS1.p15.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p15.2.m2.1.1.3.1.cmml" xref="S2.SS1.p15.2.m2.1.1.3">superscript</csymbol><ci id="S2.SS1.p15.2.m2.1.1.3.2.cmml" xref="S2.SS1.p15.2.m2.1.1.3.2">𝒜</ci><times id="S2.SS1.p15.2.m2.1.1.3.3.cmml" xref="S2.SS1.p15.2.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p15.2.m2.1c">\mathbb{R}^{\cal A^{*}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p15.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. The cone <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p15.3.m3.1"><semantics id="S2.SS1.p15.3.m3.1a"><mrow id="S2.SS1.p15.3.m3.1.2" xref="S2.SS1.p15.3.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.3.m3.1.2.2" xref="S2.SS1.p15.3.m3.1.2.2.cmml">ℳ</mi><mo id="S2.SS1.p15.3.m3.1.2.1" xref="S2.SS1.p15.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p15.3.m3.1.2.3.2" xref="S2.SS1.p15.3.m3.1.2.cmml"><mo id="S2.SS1.p15.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1.p15.3.m3.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.3.m3.1.1" xref="S2.SS1.p15.3.m3.1.1.cmml">𝒳</mi><mo id="S2.SS1.p15.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1.p15.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p15.3.m3.1b"><apply id="S2.SS1.p15.3.m3.1.2.cmml" xref="S2.SS1.p15.3.m3.1.2"><times id="S2.SS1.p15.3.m3.1.2.1.cmml" xref="S2.SS1.p15.3.m3.1.2.1"></times><ci id="S2.SS1.p15.3.m3.1.2.2.cmml" xref="S2.SS1.p15.3.m3.1.2.2">ℳ</ci><ci id="S2.SS1.p15.3.m3.1.1.cmml" xref="S2.SS1.p15.3.m3.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p15.3.m3.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p15.3.m3.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> is closed, and the extremal vectors of <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p15.4.m4.1"><semantics id="S2.SS1.p15.4.m4.1a"><mrow id="S2.SS1.p15.4.m4.1.2" xref="S2.SS1.p15.4.m4.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.4.m4.1.2.2" xref="S2.SS1.p15.4.m4.1.2.2.cmml">ℳ</mi><mo id="S2.SS1.p15.4.m4.1.2.1" xref="S2.SS1.p15.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p15.4.m4.1.2.3.2" xref="S2.SS1.p15.4.m4.1.2.cmml"><mo id="S2.SS1.p15.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS1.p15.4.m4.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.4.m4.1.1" xref="S2.SS1.p15.4.m4.1.1.cmml">𝒳</mi><mo id="S2.SS1.p15.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS1.p15.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p15.4.m4.1b"><apply id="S2.SS1.p15.4.m4.1.2.cmml" xref="S2.SS1.p15.4.m4.1.2"><times id="S2.SS1.p15.4.m4.1.2.1.cmml" xref="S2.SS1.p15.4.m4.1.2.1"></times><ci id="S2.SS1.p15.4.m4.1.2.2.cmml" xref="S2.SS1.p15.4.m4.1.2.2">ℳ</ci><ci id="S2.SS1.p15.4.m4.1.1.cmml" xref="S2.SS1.p15.4.m4.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p15.4.m4.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p15.4.m4.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> are in 1-1 relation with the ergodic measures on <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p15.5.m5.1"><semantics id="S2.SS1.p15.5.m5.1a"><mi id="S2.SS1.p15.5.m5.1.1" xref="S2.SS1.p15.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p15.5.m5.1b"><ci id="S2.SS1.p15.5.m5.1.1.cmml" xref="S2.SS1.p15.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p15.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p15.5.m5.1d">italic_X</annotation></semantics></math>. Furthermore, <math alttext="\cal M_{1}(X)" class="ltx_Math" display="inline" id="S2.SS1.p15.6.m6.1"><semantics id="S2.SS1.p15.6.m6.1a"><mrow id="S2.SS1.p15.6.m6.1.2" xref="S2.SS1.p15.6.m6.1.2.cmml"><msub id="S2.SS1.p15.6.m6.1.2.2" xref="S2.SS1.p15.6.m6.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.6.m6.1.2.2.2" xref="S2.SS1.p15.6.m6.1.2.2.2.cmml">ℳ</mi><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p15.6.m6.1.2.2.3" mathvariant="script" xref="S2.SS1.p15.6.m6.1.2.2.3.cmml">1</mn></msub><mo id="S2.SS1.p15.6.m6.1.2.1" xref="S2.SS1.p15.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p15.6.m6.1.2.3.2" xref="S2.SS1.p15.6.m6.1.2.cmml"><mo id="S2.SS1.p15.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS1.p15.6.m6.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p15.6.m6.1.1" xref="S2.SS1.p15.6.m6.1.1.cmml">𝒳</mi><mo id="S2.SS1.p15.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS1.p15.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p15.6.m6.1b"><apply id="S2.SS1.p15.6.m6.1.2.cmml" xref="S2.SS1.p15.6.m6.1.2"><times id="S2.SS1.p15.6.m6.1.2.1.cmml" xref="S2.SS1.p15.6.m6.1.2.1"></times><apply id="S2.SS1.p15.6.m6.1.2.2.cmml" xref="S2.SS1.p15.6.m6.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p15.6.m6.1.2.2.1.cmml" xref="S2.SS1.p15.6.m6.1.2.2">subscript</csymbol><ci id="S2.SS1.p15.6.m6.1.2.2.2.cmml" xref="S2.SS1.p15.6.m6.1.2.2.2">ℳ</ci><cn id="S2.SS1.p15.6.m6.1.2.2.3.cmml" type="integer" xref="S2.SS1.p15.6.m6.1.2.2.3">1</cn></apply><ci id="S2.SS1.p15.6.m6.1.1.cmml" xref="S2.SS1.p15.6.m6.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p15.6.m6.1c">\cal M_{1}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p15.6.m6.1d">caligraphic_M start_POSTSUBSCRIPT caligraphic_1 end_POSTSUBSCRIPT ( caligraphic_X )</annotation></semantics></math> is compact, and it is the closed convex hull of its extremal points.</p> </div> <div class="ltx_para" id="S2.SS1.p16"> <p class="ltx_p" id="S2.SS1.p16.4">It is well known (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib1" title="">1</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib16" title="">16</a>]</cite>) that for any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p16.1.m1.1"><semantics id="S2.SS1.p16.1.m1.1a"><mrow id="S2.SS1.p16.1.m1.1.1" xref="S2.SS1.p16.1.m1.1.1.cmml"><mi id="S2.SS1.p16.1.m1.1.1.2" xref="S2.SS1.p16.1.m1.1.1.2.cmml">X</mi><mo id="S2.SS1.p16.1.m1.1.1.1" xref="S2.SS1.p16.1.m1.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p16.1.m1.1.1.3" xref="S2.SS1.p16.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p16.1.m1.1.1.3.2" xref="S2.SS1.p16.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p16.1.m1.1.1.3.3" xref="S2.SS1.p16.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p16.1.m1.1b"><apply id="S2.SS1.p16.1.m1.1.1.cmml" xref="S2.SS1.p16.1.m1.1.1"><subset id="S2.SS1.p16.1.m1.1.1.1.cmml" xref="S2.SS1.p16.1.m1.1.1.1"></subset><ci id="S2.SS1.p16.1.m1.1.1.2.cmml" xref="S2.SS1.p16.1.m1.1.1.2">𝑋</ci><apply id="S2.SS1.p16.1.m1.1.1.3.cmml" xref="S2.SS1.p16.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p16.1.m1.1.1.3.1.cmml" xref="S2.SS1.p16.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS1.p16.1.m1.1.1.3.2.cmml" xref="S2.SS1.p16.1.m1.1.1.3.2">𝒜</ci><ci id="S2.SS1.p16.1.m1.1.1.3.3.cmml" xref="S2.SS1.p16.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p16.1.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p16.1.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> the set <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p16.2.m2.1"><semantics id="S2.SS1.p16.2.m2.1a"><mrow id="S2.SS1.p16.2.m2.1.2" xref="S2.SS1.p16.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p16.2.m2.1.2.2" xref="S2.SS1.p16.2.m2.1.2.2.cmml">ℳ</mi><mo id="S2.SS1.p16.2.m2.1.2.1" xref="S2.SS1.p16.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p16.2.m2.1.2.3.2" xref="S2.SS1.p16.2.m2.1.2.cmml"><mo id="S2.SS1.p16.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS1.p16.2.m2.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p16.2.m2.1.1" xref="S2.SS1.p16.2.m2.1.1.cmml">𝒳</mi><mo id="S2.SS1.p16.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS1.p16.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p16.2.m2.1b"><apply id="S2.SS1.p16.2.m2.1.2.cmml" xref="S2.SS1.p16.2.m2.1.2"><times id="S2.SS1.p16.2.m2.1.2.1.cmml" xref="S2.SS1.p16.2.m2.1.2.1"></times><ci id="S2.SS1.p16.2.m2.1.2.2.cmml" xref="S2.SS1.p16.2.m2.1.2.2">ℳ</ci><ci id="S2.SS1.p16.2.m2.1.1.cmml" xref="S2.SS1.p16.2.m2.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p16.2.m2.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p16.2.m2.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> of invariant measures is not empty. If <math alttext="\cal M_{1}(X)" class="ltx_Math" display="inline" id="S2.SS1.p16.3.m3.1"><semantics id="S2.SS1.p16.3.m3.1a"><mrow id="S2.SS1.p16.3.m3.1.2" xref="S2.SS1.p16.3.m3.1.2.cmml"><msub id="S2.SS1.p16.3.m3.1.2.2" xref="S2.SS1.p16.3.m3.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p16.3.m3.1.2.2.2" xref="S2.SS1.p16.3.m3.1.2.2.2.cmml">ℳ</mi><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p16.3.m3.1.2.2.3" mathvariant="script" xref="S2.SS1.p16.3.m3.1.2.2.3.cmml">1</mn></msub><mo id="S2.SS1.p16.3.m3.1.2.1" xref="S2.SS1.p16.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p16.3.m3.1.2.3.2" xref="S2.SS1.p16.3.m3.1.2.cmml"><mo id="S2.SS1.p16.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1.p16.3.m3.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p16.3.m3.1.1" xref="S2.SS1.p16.3.m3.1.1.cmml">𝒳</mi><mo id="S2.SS1.p16.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1.p16.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p16.3.m3.1b"><apply id="S2.SS1.p16.3.m3.1.2.cmml" xref="S2.SS1.p16.3.m3.1.2"><times id="S2.SS1.p16.3.m3.1.2.1.cmml" xref="S2.SS1.p16.3.m3.1.2.1"></times><apply id="S2.SS1.p16.3.m3.1.2.2.cmml" xref="S2.SS1.p16.3.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p16.3.m3.1.2.2.1.cmml" xref="S2.SS1.p16.3.m3.1.2.2">subscript</csymbol><ci id="S2.SS1.p16.3.m3.1.2.2.2.cmml" xref="S2.SS1.p16.3.m3.1.2.2.2">ℳ</ci><cn id="S2.SS1.p16.3.m3.1.2.2.3.cmml" type="integer" xref="S2.SS1.p16.3.m3.1.2.2.3">1</cn></apply><ci id="S2.SS1.p16.3.m3.1.1.cmml" xref="S2.SS1.p16.3.m3.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p16.3.m3.1c">\cal M_{1}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p16.3.m3.1d">caligraphic_M start_POSTSUBSCRIPT caligraphic_1 end_POSTSUBSCRIPT ( caligraphic_X )</annotation></semantics></math> consists of a single point (which then must be ergodic), then <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p16.4.m4.1"><semantics id="S2.SS1.p16.4.m4.1a"><mi id="S2.SS1.p16.4.m4.1.1" xref="S2.SS1.p16.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p16.4.m4.1b"><ci id="S2.SS1.p16.4.m4.1.1.cmml" xref="S2.SS1.p16.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p16.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p16.4.m4.1d">italic_X</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S2.SS1.p16.4.1">uniquely ergodic</span>.</p> </div> <div class="ltx_para" id="S2.SS1.p17"> <p class="ltx_p" id="S2.SS1.p17.4">An important class of ergodic invariant measures on the full shift <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p17.1.m1.1"><semantics id="S2.SS1.p17.1.m1.1a"><msup id="S2.SS1.p17.1.m1.1.1" xref="S2.SS1.p17.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.1.m1.1.1.2" xref="S2.SS1.p17.1.m1.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p17.1.m1.1.1.3" xref="S2.SS1.p17.1.m1.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.1.m1.1b"><apply id="S2.SS1.p17.1.m1.1.1.cmml" xref="S2.SS1.p17.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p17.1.m1.1.1.1.cmml" xref="S2.SS1.p17.1.m1.1.1">superscript</csymbol><ci id="S2.SS1.p17.1.m1.1.1.2.cmml" xref="S2.SS1.p17.1.m1.1.1.2">𝒜</ci><ci id="S2.SS1.p17.1.m1.1.1.3.cmml" xref="S2.SS1.p17.1.m1.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.1.m1.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.1.m1.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is given by the <span class="ltx_text ltx_font_italic" id="S2.SS1.p17.4.1">characteristic measures</span> <math alttext="\mu_{w}" class="ltx_Math" display="inline" id="S2.SS1.p17.2.m2.1"><semantics id="S2.SS1.p17.2.m2.1a"><msub id="S2.SS1.p17.2.m2.1.1" xref="S2.SS1.p17.2.m2.1.1.cmml"><mi id="S2.SS1.p17.2.m2.1.1.2" xref="S2.SS1.p17.2.m2.1.1.2.cmml">μ</mi><mi id="S2.SS1.p17.2.m2.1.1.3" xref="S2.SS1.p17.2.m2.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.2.m2.1b"><apply id="S2.SS1.p17.2.m2.1.1.cmml" xref="S2.SS1.p17.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p17.2.m2.1.1.1.cmml" xref="S2.SS1.p17.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p17.2.m2.1.1.2.cmml" xref="S2.SS1.p17.2.m2.1.1.2">𝜇</ci><ci id="S2.SS1.p17.2.m2.1.1.3.cmml" xref="S2.SS1.p17.2.m2.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.2.m2.1c">\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.2.m2.1d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math>, for any non-empty <math alttext="w\in\cal A^{*}\smallsetminus\{\varepsilon\}" class="ltx_Math" display="inline" id="S2.SS1.p17.3.m3.1"><semantics id="S2.SS1.p17.3.m3.1a"><mrow id="S2.SS1.p17.3.m3.1.2" xref="S2.SS1.p17.3.m3.1.2.cmml"><mi id="S2.SS1.p17.3.m3.1.2.2" xref="S2.SS1.p17.3.m3.1.2.2.cmml">w</mi><mo id="S2.SS1.p17.3.m3.1.2.1" xref="S2.SS1.p17.3.m3.1.2.1.cmml">∈</mo><mrow id="S2.SS1.p17.3.m3.1.2.3" xref="S2.SS1.p17.3.m3.1.2.3.cmml"><msup id="S2.SS1.p17.3.m3.1.2.3.2" xref="S2.SS1.p17.3.m3.1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.3.m3.1.2.3.2.2" xref="S2.SS1.p17.3.m3.1.2.3.2.2.cmml">𝒜</mi><mo id="S2.SS1.p17.3.m3.1.2.3.2.3" xref="S2.SS1.p17.3.m3.1.2.3.2.3.cmml">∗</mo></msup><mo id="S2.SS1.p17.3.m3.1.2.3.1" xref="S2.SS1.p17.3.m3.1.2.3.1.cmml">∖</mo><mrow id="S2.SS1.p17.3.m3.1.2.3.3.2" xref="S2.SS1.p17.3.m3.1.2.3.3.1.cmml"><mo id="S2.SS1.p17.3.m3.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p17.3.m3.1.2.3.3.1.cmml">{</mo><mi id="S2.SS1.p17.3.m3.1.1" xref="S2.SS1.p17.3.m3.1.1.cmml">ε</mi><mo id="S2.SS1.p17.3.m3.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.p17.3.m3.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.3.m3.1b"><apply id="S2.SS1.p17.3.m3.1.2.cmml" xref="S2.SS1.p17.3.m3.1.2"><in id="S2.SS1.p17.3.m3.1.2.1.cmml" xref="S2.SS1.p17.3.m3.1.2.1"></in><ci id="S2.SS1.p17.3.m3.1.2.2.cmml" xref="S2.SS1.p17.3.m3.1.2.2">𝑤</ci><apply id="S2.SS1.p17.3.m3.1.2.3.cmml" xref="S2.SS1.p17.3.m3.1.2.3"><setdiff id="S2.SS1.p17.3.m3.1.2.3.1.cmml" xref="S2.SS1.p17.3.m3.1.2.3.1"></setdiff><apply id="S2.SS1.p17.3.m3.1.2.3.2.cmml" xref="S2.SS1.p17.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p17.3.m3.1.2.3.2.1.cmml" xref="S2.SS1.p17.3.m3.1.2.3.2">superscript</csymbol><ci id="S2.SS1.p17.3.m3.1.2.3.2.2.cmml" xref="S2.SS1.p17.3.m3.1.2.3.2.2">𝒜</ci><times id="S2.SS1.p17.3.m3.1.2.3.2.3.cmml" xref="S2.SS1.p17.3.m3.1.2.3.2.3"></times></apply><set id="S2.SS1.p17.3.m3.1.2.3.3.1.cmml" xref="S2.SS1.p17.3.m3.1.2.3.3.2"><ci id="S2.SS1.p17.3.m3.1.1.cmml" xref="S2.SS1.p17.3.m3.1.1">𝜀</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.3.m3.1c">w\in\cal A^{*}\smallsetminus\{\varepsilon\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.3.m3.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∖ { italic_ε }</annotation></semantics></math>, which are defined as follows: If <math alttext="w" class="ltx_Math" display="inline" id="S2.SS1.p17.4.m4.1"><semantics id="S2.SS1.p17.4.m4.1a"><mi id="S2.SS1.p17.4.m4.1.1" xref="S2.SS1.p17.4.m4.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.4.m4.1b"><ci id="S2.SS1.p17.4.m4.1.1.cmml" xref="S2.SS1.p17.4.m4.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.4.m4.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.4.m4.1d">italic_w</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S2.SS1.p17.4.2">not a proper power</span>, i.e.</p> <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="0"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.4)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><span class="ltx_text ltx_markedasmath" id="S2.E4.m1.3.3.3"><math alttext="w\neq w_{0}^{r}" class="ltx_Math" display="inline" id="S2.E4.m1.1.1.1.m1.1"><semantics id="S2.E4.m1.1.1.1.m1.1a"><mrow id="S2.E4.m1.1.1.1.m1.1.1" xref="S2.E4.m1.1.1.1.m1.1.1.cmml"><mi id="S2.E4.m1.1.1.1.m1.1.1.2" xref="S2.E4.m1.1.1.1.m1.1.1.2.cmml">w</mi><mo id="S2.E4.m1.1.1.1.m1.1.1.1" xref="S2.E4.m1.1.1.1.m1.1.1.1.cmml">≠</mo><msubsup id="S2.E4.m1.1.1.1.m1.1.1.3" xref="S2.E4.m1.1.1.1.m1.1.1.3.cmml"><mi id="S2.E4.m1.1.1.1.m1.1.1.3.2.2" xref="S2.E4.m1.1.1.1.m1.1.1.3.2.2.cmml">w</mi><mn id="S2.E4.m1.1.1.1.m1.1.1.3.2.3" xref="S2.E4.m1.1.1.1.m1.1.1.3.2.3.cmml">0</mn><mi id="S2.E4.m1.1.1.1.m1.1.1.3.3" xref="S2.E4.m1.1.1.1.m1.1.1.3.3.cmml">r</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.E4.m1.1.1.1.m1.1b"><apply id="S2.E4.m1.1.1.1.m1.1.1.cmml" xref="S2.E4.m1.1.1.1.m1.1.1"><neq id="S2.E4.m1.1.1.1.m1.1.1.1.cmml" xref="S2.E4.m1.1.1.1.m1.1.1.1"></neq><ci id="S2.E4.m1.1.1.1.m1.1.1.2.cmml" xref="S2.E4.m1.1.1.1.m1.1.1.2">𝑤</ci><apply id="S2.E4.m1.1.1.1.m1.1.1.3.cmml" xref="S2.E4.m1.1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.E4.m1.1.1.1.m1.1.1.3.1.cmml" xref="S2.E4.m1.1.1.1.m1.1.1.3">superscript</csymbol><apply id="S2.E4.m1.1.1.1.m1.1.1.3.2.cmml" xref="S2.E4.m1.1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.E4.m1.1.1.1.m1.1.1.3.2.1.cmml" xref="S2.E4.m1.1.1.1.m1.1.1.3">subscript</csymbol><ci id="S2.E4.m1.1.1.1.m1.1.1.3.2.2.cmml" xref="S2.E4.m1.1.1.1.m1.1.1.3.2.2">𝑤</ci><cn id="S2.E4.m1.1.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S2.E4.m1.1.1.1.m1.1.1.3.2.3">0</cn></apply><ci id="S2.E4.m1.1.1.1.m1.1.1.3.3.cmml" xref="S2.E4.m1.1.1.1.m1.1.1.3.3">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.1.1.1.m1.1c">w\neq w_{0}^{r}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.1.1.1.m1.1d">italic_w ≠ italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> for any <math alttext="w_{0}\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.E4.m1.2.2.2.m2.1"><semantics id="S2.E4.m1.2.2.2.m2.1a"><mrow id="S2.E4.m1.2.2.2.m2.1.1" xref="S2.E4.m1.2.2.2.m2.1.1.cmml"><msub id="S2.E4.m1.2.2.2.m2.1.1.2" xref="S2.E4.m1.2.2.2.m2.1.1.2.cmml"><mi id="S2.E4.m1.2.2.2.m2.1.1.2.2" xref="S2.E4.m1.2.2.2.m2.1.1.2.2.cmml">w</mi><mn id="S2.E4.m1.2.2.2.m2.1.1.2.3" xref="S2.E4.m1.2.2.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S2.E4.m1.2.2.2.m2.1.1.1" xref="S2.E4.m1.2.2.2.m2.1.1.1.cmml">∈</mo><msup id="S2.E4.m1.2.2.2.m2.1.1.3" xref="S2.E4.m1.2.2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E4.m1.2.2.2.m2.1.1.3.2" xref="S2.E4.m1.2.2.2.m2.1.1.3.2.cmml">𝒜</mi><mo id="S2.E4.m1.2.2.2.m2.1.1.3.3" xref="S2.E4.m1.2.2.2.m2.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.E4.m1.2.2.2.m2.1b"><apply id="S2.E4.m1.2.2.2.m2.1.1.cmml" xref="S2.E4.m1.2.2.2.m2.1.1"><in id="S2.E4.m1.2.2.2.m2.1.1.1.cmml" xref="S2.E4.m1.2.2.2.m2.1.1.1"></in><apply id="S2.E4.m1.2.2.2.m2.1.1.2.cmml" xref="S2.E4.m1.2.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.E4.m1.2.2.2.m2.1.1.2.1.cmml" xref="S2.E4.m1.2.2.2.m2.1.1.2">subscript</csymbol><ci id="S2.E4.m1.2.2.2.m2.1.1.2.2.cmml" xref="S2.E4.m1.2.2.2.m2.1.1.2.2">𝑤</ci><cn id="S2.E4.m1.2.2.2.m2.1.1.2.3.cmml" type="integer" xref="S2.E4.m1.2.2.2.m2.1.1.2.3">0</cn></apply><apply id="S2.E4.m1.2.2.2.m2.1.1.3.cmml" xref="S2.E4.m1.2.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.E4.m1.2.2.2.m2.1.1.3.1.cmml" xref="S2.E4.m1.2.2.2.m2.1.1.3">superscript</csymbol><ci id="S2.E4.m1.2.2.2.m2.1.1.3.2.cmml" xref="S2.E4.m1.2.2.2.m2.1.1.3.2">𝒜</ci><times id="S2.E4.m1.2.2.2.m2.1.1.3.3.cmml" xref="S2.E4.m1.2.2.2.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.2.2.2.m2.1c">w_{0}\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.2.2.2.m2.1d">italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and any integer <math alttext="r\geq 2" class="ltx_Math" display="inline" id="S2.E4.m1.3.3.3.m3.1"><semantics id="S2.E4.m1.3.3.3.m3.1a"><mrow id="S2.E4.m1.3.3.3.m3.1.1" xref="S2.E4.m1.3.3.3.m3.1.1.cmml"><mi id="S2.E4.m1.3.3.3.m3.1.1.2" xref="S2.E4.m1.3.3.3.m3.1.1.2.cmml">r</mi><mo id="S2.E4.m1.3.3.3.m3.1.1.1" xref="S2.E4.m1.3.3.3.m3.1.1.1.cmml">≥</mo><mn id="S2.E4.m1.3.3.3.m3.1.1.3" xref="S2.E4.m1.3.3.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.E4.m1.3.3.3.m3.1b"><apply id="S2.E4.m1.3.3.3.m3.1.1.cmml" xref="S2.E4.m1.3.3.3.m3.1.1"><geq id="S2.E4.m1.3.3.3.m3.1.1.1.cmml" xref="S2.E4.m1.3.3.3.m3.1.1.1"></geq><ci id="S2.E4.m1.3.3.3.m3.1.1.2.cmml" xref="S2.E4.m1.3.3.3.m3.1.1.2">𝑟</ci><cn id="S2.E4.m1.3.3.3.m3.1.1.3.cmml" type="integer" xref="S2.E4.m1.3.3.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.3.3.3.m3.1c">r\geq 2</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.3.3.3.m3.1d">italic_r ≥ 2</annotation></semantics></math>,</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p17.9">then <math alttext="\mu_{w}" class="ltx_Math" display="inline" id="S2.SS1.p17.5.m1.1"><semantics id="S2.SS1.p17.5.m1.1a"><msub id="S2.SS1.p17.5.m1.1.1" xref="S2.SS1.p17.5.m1.1.1.cmml"><mi id="S2.SS1.p17.5.m1.1.1.2" xref="S2.SS1.p17.5.m1.1.1.2.cmml">μ</mi><mi id="S2.SS1.p17.5.m1.1.1.3" xref="S2.SS1.p17.5.m1.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.5.m1.1b"><apply id="S2.SS1.p17.5.m1.1.1.cmml" xref="S2.SS1.p17.5.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p17.5.m1.1.1.1.cmml" xref="S2.SS1.p17.5.m1.1.1">subscript</csymbol><ci id="S2.SS1.p17.5.m1.1.1.2.cmml" xref="S2.SS1.p17.5.m1.1.1.2">𝜇</ci><ci id="S2.SS1.p17.5.m1.1.1.3.cmml" xref="S2.SS1.p17.5.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.5.m1.1c">\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.5.m1.1d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> simply counts for any measurable set <math alttext="B\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p17.6.m2.1"><semantics id="S2.SS1.p17.6.m2.1a"><mrow id="S2.SS1.p17.6.m2.1.1" xref="S2.SS1.p17.6.m2.1.1.cmml"><mi id="S2.SS1.p17.6.m2.1.1.2" xref="S2.SS1.p17.6.m2.1.1.2.cmml">B</mi><mo id="S2.SS1.p17.6.m2.1.1.1" xref="S2.SS1.p17.6.m2.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p17.6.m2.1.1.3" xref="S2.SS1.p17.6.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.6.m2.1.1.3.2" xref="S2.SS1.p17.6.m2.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p17.6.m2.1.1.3.3" xref="S2.SS1.p17.6.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.6.m2.1b"><apply id="S2.SS1.p17.6.m2.1.1.cmml" xref="S2.SS1.p17.6.m2.1.1"><subset id="S2.SS1.p17.6.m2.1.1.1.cmml" xref="S2.SS1.p17.6.m2.1.1.1"></subset><ci id="S2.SS1.p17.6.m2.1.1.2.cmml" xref="S2.SS1.p17.6.m2.1.1.2">𝐵</ci><apply id="S2.SS1.p17.6.m2.1.1.3.cmml" xref="S2.SS1.p17.6.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p17.6.m2.1.1.3.1.cmml" xref="S2.SS1.p17.6.m2.1.1.3">superscript</csymbol><ci id="S2.SS1.p17.6.m2.1.1.3.2.cmml" xref="S2.SS1.p17.6.m2.1.1.3.2">𝒜</ci><ci id="S2.SS1.p17.6.m2.1.1.3.3.cmml" xref="S2.SS1.p17.6.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.6.m2.1c">B\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.6.m2.1d">italic_B ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> the number of intersections of <math alttext="B" class="ltx_Math" display="inline" id="S2.SS1.p17.7.m3.1"><semantics id="S2.SS1.p17.7.m3.1a"><mi id="S2.SS1.p17.7.m3.1.1" xref="S2.SS1.p17.7.m3.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.7.m3.1b"><ci id="S2.SS1.p17.7.m3.1.1.cmml" xref="S2.SS1.p17.7.m3.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.7.m3.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.7.m3.1d">italic_B</annotation></semantics></math> with the minimal finite subshift <math alttext="\cal O(w^{\pm\infty})" class="ltx_Math" display="inline" id="S2.SS1.p17.8.m4.1"><semantics id="S2.SS1.p17.8.m4.1a"><mrow id="S2.SS1.p17.8.m4.1.1" xref="S2.SS1.p17.8.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.8.m4.1.1.3" xref="S2.SS1.p17.8.m4.1.1.3.cmml">𝒪</mi><mo id="S2.SS1.p17.8.m4.1.1.2" xref="S2.SS1.p17.8.m4.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p17.8.m4.1.1.1.1" xref="S2.SS1.p17.8.m4.1.1.1.1.1.cmml"><mo id="S2.SS1.p17.8.m4.1.1.1.1.2" stretchy="false" xref="S2.SS1.p17.8.m4.1.1.1.1.1.cmml">(</mo><msup id="S2.SS1.p17.8.m4.1.1.1.1.1" xref="S2.SS1.p17.8.m4.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.8.m4.1.1.1.1.1.2" xref="S2.SS1.p17.8.m4.1.1.1.1.1.2.cmml">𝓌</mi><mrow id="S2.SS1.p17.8.m4.1.1.1.1.1.3" xref="S2.SS1.p17.8.m4.1.1.1.1.1.3.cmml"><mo id="S2.SS1.p17.8.m4.1.1.1.1.1.3a" xref="S2.SS1.p17.8.m4.1.1.1.1.1.3.cmml">±</mo><mi id="S2.SS1.p17.8.m4.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.SS1.p17.8.m4.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.SS1.p17.8.m4.1.1.1.1.3" stretchy="false" xref="S2.SS1.p17.8.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.8.m4.1b"><apply id="S2.SS1.p17.8.m4.1.1.cmml" xref="S2.SS1.p17.8.m4.1.1"><times id="S2.SS1.p17.8.m4.1.1.2.cmml" xref="S2.SS1.p17.8.m4.1.1.2"></times><ci id="S2.SS1.p17.8.m4.1.1.3.cmml" xref="S2.SS1.p17.8.m4.1.1.3">𝒪</ci><apply id="S2.SS1.p17.8.m4.1.1.1.1.1.cmml" xref="S2.SS1.p17.8.m4.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p17.8.m4.1.1.1.1.1.1.cmml" xref="S2.SS1.p17.8.m4.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p17.8.m4.1.1.1.1.1.2.cmml" xref="S2.SS1.p17.8.m4.1.1.1.1.1.2">𝓌</ci><apply id="S2.SS1.p17.8.m4.1.1.1.1.1.3.cmml" xref="S2.SS1.p17.8.m4.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p17.8.m4.1.1.1.1.1.3.1.cmml" xref="S2.SS1.p17.8.m4.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS1.p17.8.m4.1.1.1.1.1.3.2.cmml" xref="S2.SS1.p17.8.m4.1.1.1.1.1.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.8.m4.1c">\cal O(w^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.8.m4.1d">caligraphic_O ( caligraphic_w start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math> associated to <math alttext="w\," class="ltx_Math" display="inline" id="S2.SS1.p17.9.m5.1"><semantics id="S2.SS1.p17.9.m5.1a"><mi id="S2.SS1.p17.9.m5.1.1" xref="S2.SS1.p17.9.m5.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.9.m5.1b"><ci id="S2.SS1.p17.9.m5.1.1.cmml" xref="S2.SS1.p17.9.m5.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.9.m5.1c">w\,</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.9.m5.1d">italic_w</annotation></semantics></math>, giving the equality</p> <table class="ltx_equation ltx_eqn_table" id="S2.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.5)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu_{w}(B)\,\,:=\,\,\mbox{card}(B\cap\cal O(w^{\pm\infty}))\,." class="ltx_Math" display="block" id="S2.E5.m1.2"><semantics id="S2.E5.m1.2a"><mrow id="S2.E5.m1.2.2.1" xref="S2.E5.m1.2.2.1.1.cmml"><mrow id="S2.E5.m1.2.2.1.1" xref="S2.E5.m1.2.2.1.1.cmml"><mrow id="S2.E5.m1.2.2.1.1.3" xref="S2.E5.m1.2.2.1.1.3.cmml"><msub id="S2.E5.m1.2.2.1.1.3.2" xref="S2.E5.m1.2.2.1.1.3.2.cmml"><mi id="S2.E5.m1.2.2.1.1.3.2.2" xref="S2.E5.m1.2.2.1.1.3.2.2.cmml">μ</mi><mi id="S2.E5.m1.2.2.1.1.3.2.3" xref="S2.E5.m1.2.2.1.1.3.2.3.cmml">w</mi></msub><mo id="S2.E5.m1.2.2.1.1.3.1" xref="S2.E5.m1.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S2.E5.m1.2.2.1.1.3.3.2" xref="S2.E5.m1.2.2.1.1.3.cmml"><mo id="S2.E5.m1.2.2.1.1.3.3.2.1" stretchy="false" xref="S2.E5.m1.2.2.1.1.3.cmml">(</mo><mi id="S2.E5.m1.1.1" xref="S2.E5.m1.1.1.cmml">B</mi><mo id="S2.E5.m1.2.2.1.1.3.3.2.2" rspace="0.608em" stretchy="false" xref="S2.E5.m1.2.2.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E5.m1.2.2.1.1.2" rspace="0.608em" xref="S2.E5.m1.2.2.1.1.2.cmml">:=</mo><mrow id="S2.E5.m1.2.2.1.1.1" xref="S2.E5.m1.2.2.1.1.1.cmml"><mtext id="S2.E5.m1.2.2.1.1.1.3" xref="S2.E5.m1.2.2.1.1.1.3a.cmml">card</mtext><mo id="S2.E5.m1.2.2.1.1.1.2" xref="S2.E5.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.E5.m1.2.2.1.1.1.1.1" xref="S2.E5.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S2.E5.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.E5.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E5.m1.2.2.1.1.1.1.1.1" xref="S2.E5.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S2.E5.m1.2.2.1.1.1.1.1.1.3" xref="S2.E5.m1.2.2.1.1.1.1.1.1.3.cmml">B</mi><mo id="S2.E5.m1.2.2.1.1.1.1.1.1.2" xref="S2.E5.m1.2.2.1.1.1.1.1.1.2.cmml">∩</mo><mrow id="S2.E5.m1.2.2.1.1.1.1.1.1.1" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E5.m1.2.2.1.1.1.1.1.1.1.3" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.3.cmml">𝒪</mi><mo id="S2.E5.m1.2.2.1.1.1.1.1.1.1.2" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">𝓌</mi><mrow id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3a" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml">±</mo><mi id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E5.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.E5.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E5.m1.2.2.1.2" lspace="0.170em" xref="S2.E5.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E5.m1.2b"><apply id="S2.E5.m1.2.2.1.1.cmml" xref="S2.E5.m1.2.2.1"><csymbol cd="latexml" id="S2.E5.m1.2.2.1.1.2.cmml" xref="S2.E5.m1.2.2.1.1.2">assign</csymbol><apply id="S2.E5.m1.2.2.1.1.3.cmml" xref="S2.E5.m1.2.2.1.1.3"><times id="S2.E5.m1.2.2.1.1.3.1.cmml" xref="S2.E5.m1.2.2.1.1.3.1"></times><apply id="S2.E5.m1.2.2.1.1.3.2.cmml" xref="S2.E5.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S2.E5.m1.2.2.1.1.3.2.1.cmml" xref="S2.E5.m1.2.2.1.1.3.2">subscript</csymbol><ci id="S2.E5.m1.2.2.1.1.3.2.2.cmml" xref="S2.E5.m1.2.2.1.1.3.2.2">𝜇</ci><ci id="S2.E5.m1.2.2.1.1.3.2.3.cmml" xref="S2.E5.m1.2.2.1.1.3.2.3">𝑤</ci></apply><ci id="S2.E5.m1.1.1.cmml" xref="S2.E5.m1.1.1">𝐵</ci></apply><apply id="S2.E5.m1.2.2.1.1.1.cmml" xref="S2.E5.m1.2.2.1.1.1"><times id="S2.E5.m1.2.2.1.1.1.2.cmml" xref="S2.E5.m1.2.2.1.1.1.2"></times><ci id="S2.E5.m1.2.2.1.1.1.3a.cmml" xref="S2.E5.m1.2.2.1.1.1.3"><mtext id="S2.E5.m1.2.2.1.1.1.3.cmml" xref="S2.E5.m1.2.2.1.1.1.3">card</mtext></ci><apply id="S2.E5.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1"><intersect id="S2.E5.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.2"></intersect><ci id="S2.E5.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.3">𝐵</ci><apply id="S2.E5.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1"><times id="S2.E5.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.2"></times><ci id="S2.E5.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.3">𝒪</ci><apply id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.2">𝓌</ci><apply id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.E5.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2"></infinity></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.2c">\mu_{w}(B)\,\,:=\,\,\mbox{card}(B\cap\cal O(w^{\pm\infty}))\,.</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.2d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_B ) := card ( italic_B ∩ caligraphic_O ( caligraphic_w start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p17.13">If on the other hand <math alttext="w={w_{0}}^{r}" class="ltx_Math" display="inline" id="S2.SS1.p17.10.m1.1"><semantics id="S2.SS1.p17.10.m1.1a"><mrow id="S2.SS1.p17.10.m1.1.1" xref="S2.SS1.p17.10.m1.1.1.cmml"><mi id="S2.SS1.p17.10.m1.1.1.2" xref="S2.SS1.p17.10.m1.1.1.2.cmml">w</mi><mo id="S2.SS1.p17.10.m1.1.1.1" xref="S2.SS1.p17.10.m1.1.1.1.cmml">=</mo><mmultiscripts id="S2.SS1.p17.10.m1.1.1.3" xref="S2.SS1.p17.10.m1.1.1.3.cmml"><mi id="S2.SS1.p17.10.m1.1.1.3.2.2" xref="S2.SS1.p17.10.m1.1.1.3.2.2.cmml">w</mi><mn id="S2.SS1.p17.10.m1.1.1.3.2.3" xref="S2.SS1.p17.10.m1.1.1.3.2.3.cmml">0</mn><mrow id="S2.SS1.p17.10.m1.1.1.3a" xref="S2.SS1.p17.10.m1.1.1.3.cmml"></mrow><mrow id="S2.SS1.p17.10.m1.1.1.3b" xref="S2.SS1.p17.10.m1.1.1.3.cmml"></mrow><mi id="S2.SS1.p17.10.m1.1.1.3.3" xref="S2.SS1.p17.10.m1.1.1.3.3.cmml">r</mi></mmultiscripts></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.10.m1.1b"><apply id="S2.SS1.p17.10.m1.1.1.cmml" xref="S2.SS1.p17.10.m1.1.1"><eq id="S2.SS1.p17.10.m1.1.1.1.cmml" xref="S2.SS1.p17.10.m1.1.1.1"></eq><ci id="S2.SS1.p17.10.m1.1.1.2.cmml" xref="S2.SS1.p17.10.m1.1.1.2">𝑤</ci><apply id="S2.SS1.p17.10.m1.1.1.3.cmml" xref="S2.SS1.p17.10.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p17.10.m1.1.1.3.1.cmml" xref="S2.SS1.p17.10.m1.1.1.3">superscript</csymbol><apply id="S2.SS1.p17.10.m1.1.1.3.2.cmml" xref="S2.SS1.p17.10.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p17.10.m1.1.1.3.2.1.cmml" xref="S2.SS1.p17.10.m1.1.1.3">subscript</csymbol><ci id="S2.SS1.p17.10.m1.1.1.3.2.2.cmml" xref="S2.SS1.p17.10.m1.1.1.3.2.2">𝑤</ci><cn id="S2.SS1.p17.10.m1.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.p17.10.m1.1.1.3.2.3">0</cn></apply><ci id="S2.SS1.p17.10.m1.1.1.3.3.cmml" xref="S2.SS1.p17.10.m1.1.1.3.3">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.10.m1.1c">w={w_{0}}^{r}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.10.m1.1d">italic_w = italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> for some <math alttext="w_{0}\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p17.11.m2.1"><semantics id="S2.SS1.p17.11.m2.1a"><mrow id="S2.SS1.p17.11.m2.1.1" xref="S2.SS1.p17.11.m2.1.1.cmml"><msub id="S2.SS1.p17.11.m2.1.1.2" xref="S2.SS1.p17.11.m2.1.1.2.cmml"><mi id="S2.SS1.p17.11.m2.1.1.2.2" xref="S2.SS1.p17.11.m2.1.1.2.2.cmml">w</mi><mn id="S2.SS1.p17.11.m2.1.1.2.3" xref="S2.SS1.p17.11.m2.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS1.p17.11.m2.1.1.1" xref="S2.SS1.p17.11.m2.1.1.1.cmml">∈</mo><msup id="S2.SS1.p17.11.m2.1.1.3" xref="S2.SS1.p17.11.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.11.m2.1.1.3.2" xref="S2.SS1.p17.11.m2.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p17.11.m2.1.1.3.3" xref="S2.SS1.p17.11.m2.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.11.m2.1b"><apply id="S2.SS1.p17.11.m2.1.1.cmml" xref="S2.SS1.p17.11.m2.1.1"><in id="S2.SS1.p17.11.m2.1.1.1.cmml" xref="S2.SS1.p17.11.m2.1.1.1"></in><apply id="S2.SS1.p17.11.m2.1.1.2.cmml" xref="S2.SS1.p17.11.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p17.11.m2.1.1.2.1.cmml" xref="S2.SS1.p17.11.m2.1.1.2">subscript</csymbol><ci id="S2.SS1.p17.11.m2.1.1.2.2.cmml" xref="S2.SS1.p17.11.m2.1.1.2.2">𝑤</ci><cn id="S2.SS1.p17.11.m2.1.1.2.3.cmml" type="integer" xref="S2.SS1.p17.11.m2.1.1.2.3">0</cn></apply><apply id="S2.SS1.p17.11.m2.1.1.3.cmml" xref="S2.SS1.p17.11.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p17.11.m2.1.1.3.1.cmml" xref="S2.SS1.p17.11.m2.1.1.3">superscript</csymbol><ci id="S2.SS1.p17.11.m2.1.1.3.2.cmml" xref="S2.SS1.p17.11.m2.1.1.3.2">𝒜</ci><times id="S2.SS1.p17.11.m2.1.1.3.3.cmml" xref="S2.SS1.p17.11.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.11.m2.1c">w_{0}\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.11.m2.1d">italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and some integer <math alttext="r\geq 2" class="ltx_Math" display="inline" id="S2.SS1.p17.12.m3.1"><semantics id="S2.SS1.p17.12.m3.1a"><mrow id="S2.SS1.p17.12.m3.1.1" xref="S2.SS1.p17.12.m3.1.1.cmml"><mi id="S2.SS1.p17.12.m3.1.1.2" xref="S2.SS1.p17.12.m3.1.1.2.cmml">r</mi><mo id="S2.SS1.p17.12.m3.1.1.1" xref="S2.SS1.p17.12.m3.1.1.1.cmml">≥</mo><mn id="S2.SS1.p17.12.m3.1.1.3" xref="S2.SS1.p17.12.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.12.m3.1b"><apply id="S2.SS1.p17.12.m3.1.1.cmml" xref="S2.SS1.p17.12.m3.1.1"><geq id="S2.SS1.p17.12.m3.1.1.1.cmml" xref="S2.SS1.p17.12.m3.1.1.1"></geq><ci id="S2.SS1.p17.12.m3.1.1.2.cmml" xref="S2.SS1.p17.12.m3.1.1.2">𝑟</ci><cn id="S2.SS1.p17.12.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p17.12.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.12.m3.1c">r\geq 2</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.12.m3.1d">italic_r ≥ 2</annotation></semantics></math>, where <math alttext="w_{0}" class="ltx_Math" display="inline" id="S2.SS1.p17.13.m4.1"><semantics id="S2.SS1.p17.13.m4.1a"><msub id="S2.SS1.p17.13.m4.1.1" xref="S2.SS1.p17.13.m4.1.1.cmml"><mi id="S2.SS1.p17.13.m4.1.1.2" xref="S2.SS1.p17.13.m4.1.1.2.cmml">w</mi><mn id="S2.SS1.p17.13.m4.1.1.3" xref="S2.SS1.p17.13.m4.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.13.m4.1b"><apply id="S2.SS1.p17.13.m4.1.1.cmml" xref="S2.SS1.p17.13.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p17.13.m4.1.1.1.cmml" xref="S2.SS1.p17.13.m4.1.1">subscript</csymbol><ci id="S2.SS1.p17.13.m4.1.1.2.cmml" xref="S2.SS1.p17.13.m4.1.1.2">𝑤</ci><cn id="S2.SS1.p17.13.m4.1.1.3.cmml" type="integer" xref="S2.SS1.p17.13.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.13.m4.1c">w_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.13.m4.1d">italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is assumed not to be a proper power, one sets</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu_{w}\,\,:=\,\,r\cdot\mu_{w_{0}}\,." class="ltx_Math" display="block" id="S2.Ex4.m1.1"><semantics id="S2.Ex4.m1.1a"><mrow id="S2.Ex4.m1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.cmml"><mrow id="S2.Ex4.m1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.cmml"><msub id="S2.Ex4.m1.1.1.1.1.2" xref="S2.Ex4.m1.1.1.1.1.2.cmml"><mi id="S2.Ex4.m1.1.1.1.1.2.2" xref="S2.Ex4.m1.1.1.1.1.2.2.cmml">μ</mi><mi id="S2.Ex4.m1.1.1.1.1.2.3" xref="S2.Ex4.m1.1.1.1.1.2.3.cmml">w</mi></msub><mo id="S2.Ex4.m1.1.1.1.1.1" lspace="0.608em" rspace="0.608em" xref="S2.Ex4.m1.1.1.1.1.1.cmml">:=</mo><mrow id="S2.Ex4.m1.1.1.1.1.3" xref="S2.Ex4.m1.1.1.1.1.3.cmml"><mi id="S2.Ex4.m1.1.1.1.1.3.2" xref="S2.Ex4.m1.1.1.1.1.3.2.cmml">r</mi><mo id="S2.Ex4.m1.1.1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex4.m1.1.1.1.1.3.1.cmml">⋅</mo><msub id="S2.Ex4.m1.1.1.1.1.3.3" xref="S2.Ex4.m1.1.1.1.1.3.3.cmml"><mi id="S2.Ex4.m1.1.1.1.1.3.3.2" xref="S2.Ex4.m1.1.1.1.1.3.3.2.cmml">μ</mi><msub id="S2.Ex4.m1.1.1.1.1.3.3.3" xref="S2.Ex4.m1.1.1.1.1.3.3.3.cmml"><mi id="S2.Ex4.m1.1.1.1.1.3.3.3.2" xref="S2.Ex4.m1.1.1.1.1.3.3.3.2.cmml">w</mi><mn id="S2.Ex4.m1.1.1.1.1.3.3.3.3" xref="S2.Ex4.m1.1.1.1.1.3.3.3.3.cmml">0</mn></msub></msub></mrow></mrow><mo id="S2.Ex4.m1.1.1.1.2" lspace="0em" xref="S2.Ex4.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex4.m1.1b"><apply id="S2.Ex4.m1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1"><csymbol cd="latexml" id="S2.Ex4.m1.1.1.1.1.1.cmml" xref="S2.Ex4.m1.1.1.1.1.1">assign</csymbol><apply id="S2.Ex4.m1.1.1.1.1.2.cmml" xref="S2.Ex4.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex4.m1.1.1.1.1.2.1.cmml" xref="S2.Ex4.m1.1.1.1.1.2">subscript</csymbol><ci id="S2.Ex4.m1.1.1.1.1.2.2.cmml" xref="S2.Ex4.m1.1.1.1.1.2.2">𝜇</ci><ci id="S2.Ex4.m1.1.1.1.1.2.3.cmml" xref="S2.Ex4.m1.1.1.1.1.2.3">𝑤</ci></apply><apply id="S2.Ex4.m1.1.1.1.1.3.cmml" xref="S2.Ex4.m1.1.1.1.1.3"><ci id="S2.Ex4.m1.1.1.1.1.3.1.cmml" xref="S2.Ex4.m1.1.1.1.1.3.1">⋅</ci><ci id="S2.Ex4.m1.1.1.1.1.3.2.cmml" xref="S2.Ex4.m1.1.1.1.1.3.2">𝑟</ci><apply id="S2.Ex4.m1.1.1.1.1.3.3.cmml" xref="S2.Ex4.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Ex4.m1.1.1.1.1.3.3.1.cmml" xref="S2.Ex4.m1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.Ex4.m1.1.1.1.1.3.3.2.cmml" xref="S2.Ex4.m1.1.1.1.1.3.3.2">𝜇</ci><apply id="S2.Ex4.m1.1.1.1.1.3.3.3.cmml" xref="S2.Ex4.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.Ex4.m1.1.1.1.1.3.3.3.1.cmml" xref="S2.Ex4.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="S2.Ex4.m1.1.1.1.1.3.3.3.2.cmml" xref="S2.Ex4.m1.1.1.1.1.3.3.3.2">𝑤</ci><cn id="S2.Ex4.m1.1.1.1.1.3.3.3.3.cmml" type="integer" xref="S2.Ex4.m1.1.1.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m1.1c">\mu_{w}\,\,:=\,\,r\cdot\mu_{w_{0}}\,.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m1.1d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT := italic_r ⋅ italic_μ start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p17.18">In either case, it follows that <math alttext="\frac{1}{|w|}\mu_{w}" class="ltx_Math" display="inline" id="S2.SS1.p17.14.m1.1"><semantics id="S2.SS1.p17.14.m1.1a"><mrow id="S2.SS1.p17.14.m1.1.2" xref="S2.SS1.p17.14.m1.1.2.cmml"><mfrac id="S2.SS1.p17.14.m1.1.1" xref="S2.SS1.p17.14.m1.1.1.cmml"><mn id="S2.SS1.p17.14.m1.1.1.3" xref="S2.SS1.p17.14.m1.1.1.3.cmml">1</mn><mrow id="S2.SS1.p17.14.m1.1.1.1.3" xref="S2.SS1.p17.14.m1.1.1.1.2.cmml"><mo id="S2.SS1.p17.14.m1.1.1.1.3.1" stretchy="false" xref="S2.SS1.p17.14.m1.1.1.1.2.1.cmml">|</mo><mi id="S2.SS1.p17.14.m1.1.1.1.1" xref="S2.SS1.p17.14.m1.1.1.1.1.cmml">w</mi><mo id="S2.SS1.p17.14.m1.1.1.1.3.2" stretchy="false" xref="S2.SS1.p17.14.m1.1.1.1.2.1.cmml">|</mo></mrow></mfrac><mo id="S2.SS1.p17.14.m1.1.2.1" xref="S2.SS1.p17.14.m1.1.2.1.cmml">⁢</mo><msub id="S2.SS1.p17.14.m1.1.2.2" xref="S2.SS1.p17.14.m1.1.2.2.cmml"><mi id="S2.SS1.p17.14.m1.1.2.2.2" xref="S2.SS1.p17.14.m1.1.2.2.2.cmml">μ</mi><mi id="S2.SS1.p17.14.m1.1.2.2.3" xref="S2.SS1.p17.14.m1.1.2.2.3.cmml">w</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.14.m1.1b"><apply id="S2.SS1.p17.14.m1.1.2.cmml" xref="S2.SS1.p17.14.m1.1.2"><times id="S2.SS1.p17.14.m1.1.2.1.cmml" xref="S2.SS1.p17.14.m1.1.2.1"></times><apply id="S2.SS1.p17.14.m1.1.1.cmml" xref="S2.SS1.p17.14.m1.1.1"><divide id="S2.SS1.p17.14.m1.1.1.2.cmml" xref="S2.SS1.p17.14.m1.1.1"></divide><cn id="S2.SS1.p17.14.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p17.14.m1.1.1.3">1</cn><apply id="S2.SS1.p17.14.m1.1.1.1.2.cmml" xref="S2.SS1.p17.14.m1.1.1.1.3"><abs id="S2.SS1.p17.14.m1.1.1.1.2.1.cmml" xref="S2.SS1.p17.14.m1.1.1.1.3.1"></abs><ci id="S2.SS1.p17.14.m1.1.1.1.1.cmml" xref="S2.SS1.p17.14.m1.1.1.1.1">𝑤</ci></apply></apply><apply id="S2.SS1.p17.14.m1.1.2.2.cmml" xref="S2.SS1.p17.14.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p17.14.m1.1.2.2.1.cmml" xref="S2.SS1.p17.14.m1.1.2.2">subscript</csymbol><ci id="S2.SS1.p17.14.m1.1.2.2.2.cmml" xref="S2.SS1.p17.14.m1.1.2.2.2">𝜇</ci><ci id="S2.SS1.p17.14.m1.1.2.2.3.cmml" xref="S2.SS1.p17.14.m1.1.2.2.3">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.14.m1.1c">\frac{1}{|w|}\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.14.m1.1d">divide start_ARG 1 end_ARG start_ARG | italic_w | end_ARG italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> is a probability measure. The set of weighted characteristic measures <math alttext="\lambda\,\mu_{w}" class="ltx_Math" display="inline" id="S2.SS1.p17.15.m2.1"><semantics id="S2.SS1.p17.15.m2.1a"><mrow id="S2.SS1.p17.15.m2.1.1" xref="S2.SS1.p17.15.m2.1.1.cmml"><mi id="S2.SS1.p17.15.m2.1.1.2" xref="S2.SS1.p17.15.m2.1.1.2.cmml">λ</mi><mo id="S2.SS1.p17.15.m2.1.1.1" lspace="0.170em" xref="S2.SS1.p17.15.m2.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p17.15.m2.1.1.3" xref="S2.SS1.p17.15.m2.1.1.3.cmml"><mi id="S2.SS1.p17.15.m2.1.1.3.2" xref="S2.SS1.p17.15.m2.1.1.3.2.cmml">μ</mi><mi id="S2.SS1.p17.15.m2.1.1.3.3" xref="S2.SS1.p17.15.m2.1.1.3.3.cmml">w</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.15.m2.1b"><apply id="S2.SS1.p17.15.m2.1.1.cmml" xref="S2.SS1.p17.15.m2.1.1"><times id="S2.SS1.p17.15.m2.1.1.1.cmml" xref="S2.SS1.p17.15.m2.1.1.1"></times><ci id="S2.SS1.p17.15.m2.1.1.2.cmml" xref="S2.SS1.p17.15.m2.1.1.2">𝜆</ci><apply id="S2.SS1.p17.15.m2.1.1.3.cmml" xref="S2.SS1.p17.15.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p17.15.m2.1.1.3.1.cmml" xref="S2.SS1.p17.15.m2.1.1.3">subscript</csymbol><ci id="S2.SS1.p17.15.m2.1.1.3.2.cmml" xref="S2.SS1.p17.15.m2.1.1.3.2">𝜇</ci><ci id="S2.SS1.p17.15.m2.1.1.3.3.cmml" xref="S2.SS1.p17.15.m2.1.1.3.3">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.15.m2.1c">\lambda\,\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.15.m2.1d">italic_λ italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> (for any <math alttext="\lambda>0" class="ltx_Math" display="inline" id="S2.SS1.p17.16.m3.1"><semantics id="S2.SS1.p17.16.m3.1a"><mrow id="S2.SS1.p17.16.m3.1.1" xref="S2.SS1.p17.16.m3.1.1.cmml"><mi id="S2.SS1.p17.16.m3.1.1.2" xref="S2.SS1.p17.16.m3.1.1.2.cmml">λ</mi><mo id="S2.SS1.p17.16.m3.1.1.1" xref="S2.SS1.p17.16.m3.1.1.1.cmml">></mo><mn id="S2.SS1.p17.16.m3.1.1.3" xref="S2.SS1.p17.16.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.16.m3.1b"><apply id="S2.SS1.p17.16.m3.1.1.cmml" xref="S2.SS1.p17.16.m3.1.1"><gt id="S2.SS1.p17.16.m3.1.1.1.cmml" xref="S2.SS1.p17.16.m3.1.1.1"></gt><ci id="S2.SS1.p17.16.m3.1.1.2.cmml" xref="S2.SS1.p17.16.m3.1.1.2">𝜆</ci><cn id="S2.SS1.p17.16.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p17.16.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.16.m3.1c">\lambda>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.16.m3.1d">italic_λ > 0</annotation></semantics></math> and any <math alttext="w\in\cal A^{*}\smallsetminus\{\varepsilon\}" class="ltx_Math" display="inline" id="S2.SS1.p17.17.m4.1"><semantics id="S2.SS1.p17.17.m4.1a"><mrow id="S2.SS1.p17.17.m4.1.2" xref="S2.SS1.p17.17.m4.1.2.cmml"><mi id="S2.SS1.p17.17.m4.1.2.2" xref="S2.SS1.p17.17.m4.1.2.2.cmml">w</mi><mo id="S2.SS1.p17.17.m4.1.2.1" xref="S2.SS1.p17.17.m4.1.2.1.cmml">∈</mo><mrow id="S2.SS1.p17.17.m4.1.2.3" xref="S2.SS1.p17.17.m4.1.2.3.cmml"><msup id="S2.SS1.p17.17.m4.1.2.3.2" xref="S2.SS1.p17.17.m4.1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.17.m4.1.2.3.2.2" xref="S2.SS1.p17.17.m4.1.2.3.2.2.cmml">𝒜</mi><mo id="S2.SS1.p17.17.m4.1.2.3.2.3" xref="S2.SS1.p17.17.m4.1.2.3.2.3.cmml">∗</mo></msup><mo id="S2.SS1.p17.17.m4.1.2.3.1" xref="S2.SS1.p17.17.m4.1.2.3.1.cmml">∖</mo><mrow id="S2.SS1.p17.17.m4.1.2.3.3.2" xref="S2.SS1.p17.17.m4.1.2.3.3.1.cmml"><mo id="S2.SS1.p17.17.m4.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p17.17.m4.1.2.3.3.1.cmml">{</mo><mi id="S2.SS1.p17.17.m4.1.1" xref="S2.SS1.p17.17.m4.1.1.cmml">ε</mi><mo id="S2.SS1.p17.17.m4.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.p17.17.m4.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.17.m4.1b"><apply id="S2.SS1.p17.17.m4.1.2.cmml" xref="S2.SS1.p17.17.m4.1.2"><in id="S2.SS1.p17.17.m4.1.2.1.cmml" xref="S2.SS1.p17.17.m4.1.2.1"></in><ci id="S2.SS1.p17.17.m4.1.2.2.cmml" xref="S2.SS1.p17.17.m4.1.2.2">𝑤</ci><apply id="S2.SS1.p17.17.m4.1.2.3.cmml" xref="S2.SS1.p17.17.m4.1.2.3"><setdiff id="S2.SS1.p17.17.m4.1.2.3.1.cmml" xref="S2.SS1.p17.17.m4.1.2.3.1"></setdiff><apply id="S2.SS1.p17.17.m4.1.2.3.2.cmml" xref="S2.SS1.p17.17.m4.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p17.17.m4.1.2.3.2.1.cmml" xref="S2.SS1.p17.17.m4.1.2.3.2">superscript</csymbol><ci id="S2.SS1.p17.17.m4.1.2.3.2.2.cmml" xref="S2.SS1.p17.17.m4.1.2.3.2.2">𝒜</ci><times id="S2.SS1.p17.17.m4.1.2.3.2.3.cmml" xref="S2.SS1.p17.17.m4.1.2.3.2.3"></times></apply><set id="S2.SS1.p17.17.m4.1.2.3.3.1.cmml" xref="S2.SS1.p17.17.m4.1.2.3.3.2"><ci id="S2.SS1.p17.17.m4.1.1.cmml" xref="S2.SS1.p17.17.m4.1.1">𝜀</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.17.m4.1c">w\in\cal A^{*}\smallsetminus\{\varepsilon\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.17.m4.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∖ { italic_ε }</annotation></semantics></math>) is known to be dense in <math alttext="\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S2.SS1.p17.18.m5.1"><semantics id="S2.SS1.p17.18.m5.1a"><mrow id="S2.SS1.p17.18.m5.1.1" xref="S2.SS1.p17.18.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.18.m5.1.1.3" xref="S2.SS1.p17.18.m5.1.1.3.cmml">ℳ</mi><mo id="S2.SS1.p17.18.m5.1.1.2" xref="S2.SS1.p17.18.m5.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p17.18.m5.1.1.1.1" xref="S2.SS1.p17.18.m5.1.1.1.1.1.cmml"><mo id="S2.SS1.p17.18.m5.1.1.1.1.2" stretchy="false" xref="S2.SS1.p17.18.m5.1.1.1.1.1.cmml">(</mo><msup id="S2.SS1.p17.18.m5.1.1.1.1.1" xref="S2.SS1.p17.18.m5.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p17.18.m5.1.1.1.1.1.2" xref="S2.SS1.p17.18.m5.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p17.18.m5.1.1.1.1.1.3" xref="S2.SS1.p17.18.m5.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S2.SS1.p17.18.m5.1.1.1.1.3" stretchy="false" xref="S2.SS1.p17.18.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p17.18.m5.1b"><apply id="S2.SS1.p17.18.m5.1.1.cmml" xref="S2.SS1.p17.18.m5.1.1"><times id="S2.SS1.p17.18.m5.1.1.2.cmml" xref="S2.SS1.p17.18.m5.1.1.2"></times><ci id="S2.SS1.p17.18.m5.1.1.3.cmml" xref="S2.SS1.p17.18.m5.1.1.3">ℳ</ci><apply id="S2.SS1.p17.18.m5.1.1.1.1.1.cmml" xref="S2.SS1.p17.18.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p17.18.m5.1.1.1.1.1.1.cmml" xref="S2.SS1.p17.18.m5.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p17.18.m5.1.1.1.1.1.2.cmml" xref="S2.SS1.p17.18.m5.1.1.1.1.1.2">𝒜</ci><ci id="S2.SS1.p17.18.m5.1.1.1.1.1.3.cmml" xref="S2.SS1.p17.18.m5.1.1.1.1.1.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p17.18.m5.1c">\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p17.18.m5.1d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib1" title="">1</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib11" title="">11</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib16" title="">16</a>]</cite>).</p> </div> <div class="ltx_para" id="S2.SS1.p18"> <p class="ltx_p" id="S2.SS1.p18.3">It is well known (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib1" title="">1</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib16" title="">16</a>]</cite>) and easy to verify that the support <math alttext="X_{\mu}:=\mbox{Supp}(\mu)" class="ltx_Math" display="inline" id="S2.SS1.p18.1.m1.1"><semantics id="S2.SS1.p18.1.m1.1a"><mrow id="S2.SS1.p18.1.m1.1.2" xref="S2.SS1.p18.1.m1.1.2.cmml"><msub id="S2.SS1.p18.1.m1.1.2.2" xref="S2.SS1.p18.1.m1.1.2.2.cmml"><mi id="S2.SS1.p18.1.m1.1.2.2.2" xref="S2.SS1.p18.1.m1.1.2.2.2.cmml">X</mi><mi id="S2.SS1.p18.1.m1.1.2.2.3" xref="S2.SS1.p18.1.m1.1.2.2.3.cmml">μ</mi></msub><mo id="S2.SS1.p18.1.m1.1.2.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p18.1.m1.1.2.1.cmml">:=</mo><mrow id="S2.SS1.p18.1.m1.1.2.3" xref="S2.SS1.p18.1.m1.1.2.3.cmml"><mtext id="S2.SS1.p18.1.m1.1.2.3.2" xref="S2.SS1.p18.1.m1.1.2.3.2a.cmml">Supp</mtext><mo id="S2.SS1.p18.1.m1.1.2.3.1" xref="S2.SS1.p18.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p18.1.m1.1.2.3.3.2" xref="S2.SS1.p18.1.m1.1.2.3.cmml"><mo id="S2.SS1.p18.1.m1.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p18.1.m1.1.2.3.cmml">(</mo><mi id="S2.SS1.p18.1.m1.1.1" xref="S2.SS1.p18.1.m1.1.1.cmml">μ</mi><mo id="S2.SS1.p18.1.m1.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.p18.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.1.m1.1b"><apply id="S2.SS1.p18.1.m1.1.2.cmml" xref="S2.SS1.p18.1.m1.1.2"><csymbol cd="latexml" id="S2.SS1.p18.1.m1.1.2.1.cmml" xref="S2.SS1.p18.1.m1.1.2.1">assign</csymbol><apply id="S2.SS1.p18.1.m1.1.2.2.cmml" xref="S2.SS1.p18.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p18.1.m1.1.2.2.1.cmml" xref="S2.SS1.p18.1.m1.1.2.2">subscript</csymbol><ci id="S2.SS1.p18.1.m1.1.2.2.2.cmml" xref="S2.SS1.p18.1.m1.1.2.2.2">𝑋</ci><ci id="S2.SS1.p18.1.m1.1.2.2.3.cmml" xref="S2.SS1.p18.1.m1.1.2.2.3">𝜇</ci></apply><apply id="S2.SS1.p18.1.m1.1.2.3.cmml" xref="S2.SS1.p18.1.m1.1.2.3"><times id="S2.SS1.p18.1.m1.1.2.3.1.cmml" xref="S2.SS1.p18.1.m1.1.2.3.1"></times><ci id="S2.SS1.p18.1.m1.1.2.3.2a.cmml" xref="S2.SS1.p18.1.m1.1.2.3.2"><mtext id="S2.SS1.p18.1.m1.1.2.3.2.cmml" xref="S2.SS1.p18.1.m1.1.2.3.2">Supp</mtext></ci><ci id="S2.SS1.p18.1.m1.1.1.cmml" xref="S2.SS1.p18.1.m1.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.1.m1.1c">X_{\mu}:=\mbox{Supp}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.1.m1.1d">italic_X start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT := Supp ( italic_μ )</annotation></semantics></math> of any non-zero invariant measure <math alttext="\mu\in\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S2.SS1.p18.2.m2.1"><semantics id="S2.SS1.p18.2.m2.1a"><mrow id="S2.SS1.p18.2.m2.1.1" xref="S2.SS1.p18.2.m2.1.1.cmml"><mi id="S2.SS1.p18.2.m2.1.1.3" xref="S2.SS1.p18.2.m2.1.1.3.cmml">μ</mi><mo id="S2.SS1.p18.2.m2.1.1.2" xref="S2.SS1.p18.2.m2.1.1.2.cmml">∈</mo><mrow id="S2.SS1.p18.2.m2.1.1.1" xref="S2.SS1.p18.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.2.m2.1.1.1.3" xref="S2.SS1.p18.2.m2.1.1.1.3.cmml">ℳ</mi><mo id="S2.SS1.p18.2.m2.1.1.1.2" xref="S2.SS1.p18.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p18.2.m2.1.1.1.1.1" xref="S2.SS1.p18.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p18.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p18.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS1.p18.2.m2.1.1.1.1.1.1" xref="S2.SS1.p18.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.2.m2.1.1.1.1.1.1.2" xref="S2.SS1.p18.2.m2.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p18.2.m2.1.1.1.1.1.1.3" xref="S2.SS1.p18.2.m2.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S2.SS1.p18.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p18.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.2.m2.1b"><apply id="S2.SS1.p18.2.m2.1.1.cmml" xref="S2.SS1.p18.2.m2.1.1"><in id="S2.SS1.p18.2.m2.1.1.2.cmml" xref="S2.SS1.p18.2.m2.1.1.2"></in><ci id="S2.SS1.p18.2.m2.1.1.3.cmml" xref="S2.SS1.p18.2.m2.1.1.3">𝜇</ci><apply id="S2.SS1.p18.2.m2.1.1.1.cmml" xref="S2.SS1.p18.2.m2.1.1.1"><times id="S2.SS1.p18.2.m2.1.1.1.2.cmml" xref="S2.SS1.p18.2.m2.1.1.1.2"></times><ci id="S2.SS1.p18.2.m2.1.1.1.3.cmml" xref="S2.SS1.p18.2.m2.1.1.1.3">ℳ</ci><apply id="S2.SS1.p18.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p18.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p18.2.m2.1.1.1.1.1.1.2">𝒜</ci><ci id="S2.SS1.p18.2.m2.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p18.2.m2.1.1.1.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.2.m2.1c">\mu\in\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.2.m2.1d">italic_μ ∈ caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> is a subshift. Its language <math alttext="\cal L(X_{\mu})" class="ltx_Math" display="inline" id="S2.SS1.p18.3.m3.1"><semantics id="S2.SS1.p18.3.m3.1a"><mrow id="S2.SS1.p18.3.m3.1.1" xref="S2.SS1.p18.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.3.m3.1.1.3" xref="S2.SS1.p18.3.m3.1.1.3.cmml">ℒ</mi><mo id="S2.SS1.p18.3.m3.1.1.2" xref="S2.SS1.p18.3.m3.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p18.3.m3.1.1.1.1" xref="S2.SS1.p18.3.m3.1.1.1.1.1.cmml"><mo id="S2.SS1.p18.3.m3.1.1.1.1.2" stretchy="false" xref="S2.SS1.p18.3.m3.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p18.3.m3.1.1.1.1.1" xref="S2.SS1.p18.3.m3.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.3.m3.1.1.1.1.1.2" xref="S2.SS1.p18.3.m3.1.1.1.1.1.2.cmml">𝒳</mi><mi id="S2.SS1.p18.3.m3.1.1.1.1.1.3" xref="S2.SS1.p18.3.m3.1.1.1.1.1.3.cmml">μ</mi></msub><mo id="S2.SS1.p18.3.m3.1.1.1.1.3" stretchy="false" xref="S2.SS1.p18.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.3.m3.1b"><apply id="S2.SS1.p18.3.m3.1.1.cmml" xref="S2.SS1.p18.3.m3.1.1"><times id="S2.SS1.p18.3.m3.1.1.2.cmml" xref="S2.SS1.p18.3.m3.1.1.2"></times><ci id="S2.SS1.p18.3.m3.1.1.3.cmml" xref="S2.SS1.p18.3.m3.1.1.3">ℒ</ci><apply id="S2.SS1.p18.3.m3.1.1.1.1.1.cmml" xref="S2.SS1.p18.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.3.m3.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p18.3.m3.1.1.1.1.1.2.cmml" xref="S2.SS1.p18.3.m3.1.1.1.1.1.2">𝒳</ci><ci id="S2.SS1.p18.3.m3.1.1.1.1.1.3.cmml" xref="S2.SS1.p18.3.m3.1.1.1.1.1.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.3.m3.1c">\cal L(X_{\mu})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.3.m3.1d">caligraphic_L ( caligraphic_X start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT )</annotation></semantics></math> is given by</p> <table class="ltx_equation ltx_eqn_table" id="S2.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.6)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w\in\cal L(X_{\mu})\quad\Longleftrightarrow\quad\mu(w)>0" class="ltx_Math" display="block" id="S2.E6.m1.4"><semantics id="S2.E6.m1.4a"><mrow id="S2.E6.m1.4.4.2" xref="S2.E6.m1.4.4.3.cmml"><mrow id="S2.E6.m1.3.3.1.1" xref="S2.E6.m1.3.3.1.1.cmml"><mi id="S2.E6.m1.3.3.1.1.3" xref="S2.E6.m1.3.3.1.1.3.cmml">w</mi><mo id="S2.E6.m1.3.3.1.1.2" xref="S2.E6.m1.3.3.1.1.2.cmml">∈</mo><mrow id="S2.E6.m1.3.3.1.1.1.1" xref="S2.E6.m1.3.3.1.1.1.2.cmml"><mrow id="S2.E6.m1.3.3.1.1.1.1.1" xref="S2.E6.m1.3.3.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E6.m1.3.3.1.1.1.1.1.3" xref="S2.E6.m1.3.3.1.1.1.1.1.3.cmml">ℒ</mi><mo id="S2.E6.m1.3.3.1.1.1.1.1.2" xref="S2.E6.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E6.m1.3.3.1.1.1.1.1.1.1" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E6.m1.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E6.m1.3.3.1.1.1.1.1.1.1.1" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.2" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.2.cmml">𝒳</mi><mi id="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.3" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.3.cmml">μ</mi></msub><mo id="S2.E6.m1.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mspace id="S2.E6.m1.3.3.1.1.1.1.2" width="1em" xref="S2.E6.m1.3.3.1.1.1.2.cmml"></mspace><mo id="S2.E6.m1.2.2" stretchy="false" xref="S2.E6.m1.2.2.cmml">⟺</mo></mrow></mrow><mspace id="S2.E6.m1.4.4.2.3" width="1em" xref="S2.E6.m1.4.4.3a.cmml"></mspace><mrow id="S2.E6.m1.4.4.2.2" xref="S2.E6.m1.4.4.2.2.cmml"><mrow id="S2.E6.m1.4.4.2.2.2" xref="S2.E6.m1.4.4.2.2.2.cmml"><mi id="S2.E6.m1.4.4.2.2.2.2" xref="S2.E6.m1.4.4.2.2.2.2.cmml">μ</mi><mo id="S2.E6.m1.4.4.2.2.2.1" xref="S2.E6.m1.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S2.E6.m1.4.4.2.2.2.3.2" xref="S2.E6.m1.4.4.2.2.2.cmml"><mo id="S2.E6.m1.4.4.2.2.2.3.2.1" stretchy="false" xref="S2.E6.m1.4.4.2.2.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.E6.m1.1.1" xref="S2.E6.m1.1.1.cmml">𝓌</mi><mo id="S2.E6.m1.4.4.2.2.2.3.2.2" stretchy="false" xref="S2.E6.m1.4.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.E6.m1.4.4.2.2.1" xref="S2.E6.m1.4.4.2.2.1.cmml">></mo><mn class="ltx_font_mathcaligraphic" id="S2.E6.m1.4.4.2.2.3" mathvariant="script" xref="S2.E6.m1.4.4.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E6.m1.4b"><apply id="S2.E6.m1.4.4.3.cmml" xref="S2.E6.m1.4.4.2"><csymbol cd="ambiguous" id="S2.E6.m1.4.4.3a.cmml" xref="S2.E6.m1.4.4.2.3">formulae-sequence</csymbol><apply id="S2.E6.m1.3.3.1.1.cmml" xref="S2.E6.m1.3.3.1.1"><in id="S2.E6.m1.3.3.1.1.2.cmml" xref="S2.E6.m1.3.3.1.1.2"></in><ci id="S2.E6.m1.3.3.1.1.3.cmml" xref="S2.E6.m1.3.3.1.1.3">𝑤</ci><list id="S2.E6.m1.3.3.1.1.1.2.cmml" xref="S2.E6.m1.3.3.1.1.1.1"><apply id="S2.E6.m1.3.3.1.1.1.1.1.cmml" xref="S2.E6.m1.3.3.1.1.1.1.1"><times id="S2.E6.m1.3.3.1.1.1.1.1.2.cmml" xref="S2.E6.m1.3.3.1.1.1.1.1.2"></times><ci id="S2.E6.m1.3.3.1.1.1.1.1.3.cmml" xref="S2.E6.m1.3.3.1.1.1.1.1.3">ℒ</ci><apply id="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.2">𝒳</ci><ci id="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E6.m1.3.3.1.1.1.1.1.1.1.1.3">𝜇</ci></apply></apply><ci id="S2.E6.m1.2.2.cmml" xref="S2.E6.m1.2.2">⟺</ci></list></apply><apply id="S2.E6.m1.4.4.2.2.cmml" xref="S2.E6.m1.4.4.2.2"><gt id="S2.E6.m1.4.4.2.2.1.cmml" xref="S2.E6.m1.4.4.2.2.1"></gt><apply id="S2.E6.m1.4.4.2.2.2.cmml" xref="S2.E6.m1.4.4.2.2.2"><times id="S2.E6.m1.4.4.2.2.2.1.cmml" xref="S2.E6.m1.4.4.2.2.2.1"></times><ci id="S2.E6.m1.4.4.2.2.2.2.cmml" xref="S2.E6.m1.4.4.2.2.2.2">𝜇</ci><ci id="S2.E6.m1.1.1.cmml" xref="S2.E6.m1.1.1">𝓌</ci></apply><cn id="S2.E6.m1.4.4.2.2.3.cmml" type="integer" xref="S2.E6.m1.4.4.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.4c">w\in\cal L(X_{\mu})\quad\Longleftrightarrow\quad\mu(w)>0</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.4d">italic_w ∈ caligraphic_L ( caligraphic_X start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ) ⟺ italic_μ ( caligraphic_w ) > caligraphic_0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p18.8">for any <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS1.p18.4.m1.1"><semantics id="S2.SS1.p18.4.m1.1a"><mrow id="S2.SS1.p18.4.m1.1.1" xref="S2.SS1.p18.4.m1.1.1.cmml"><mi id="S2.SS1.p18.4.m1.1.1.2" xref="S2.SS1.p18.4.m1.1.1.2.cmml">w</mi><mo id="S2.SS1.p18.4.m1.1.1.1" xref="S2.SS1.p18.4.m1.1.1.1.cmml">∈</mo><msup id="S2.SS1.p18.4.m1.1.1.3" xref="S2.SS1.p18.4.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.4.m1.1.1.3.2" xref="S2.SS1.p18.4.m1.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS1.p18.4.m1.1.1.3.3" xref="S2.SS1.p18.4.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.4.m1.1b"><apply id="S2.SS1.p18.4.m1.1.1.cmml" xref="S2.SS1.p18.4.m1.1.1"><in id="S2.SS1.p18.4.m1.1.1.1.cmml" xref="S2.SS1.p18.4.m1.1.1.1"></in><ci id="S2.SS1.p18.4.m1.1.1.2.cmml" xref="S2.SS1.p18.4.m1.1.1.2">𝑤</ci><apply id="S2.SS1.p18.4.m1.1.1.3.cmml" xref="S2.SS1.p18.4.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p18.4.m1.1.1.3.1.cmml" xref="S2.SS1.p18.4.m1.1.1.3">superscript</csymbol><ci id="S2.SS1.p18.4.m1.1.1.3.2.cmml" xref="S2.SS1.p18.4.m1.1.1.3.2">𝒜</ci><times id="S2.SS1.p18.4.m1.1.1.3.3.cmml" xref="S2.SS1.p18.4.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.4.m1.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.4.m1.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. If <math alttext="X_{\mu}" class="ltx_Math" display="inline" id="S2.SS1.p18.5.m2.1"><semantics id="S2.SS1.p18.5.m2.1a"><msub id="S2.SS1.p18.5.m2.1.1" xref="S2.SS1.p18.5.m2.1.1.cmml"><mi id="S2.SS1.p18.5.m2.1.1.2" xref="S2.SS1.p18.5.m2.1.1.2.cmml">X</mi><mi id="S2.SS1.p18.5.m2.1.1.3" xref="S2.SS1.p18.5.m2.1.1.3.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.5.m2.1b"><apply id="S2.SS1.p18.5.m2.1.1.cmml" xref="S2.SS1.p18.5.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.5.m2.1.1.1.cmml" xref="S2.SS1.p18.5.m2.1.1">subscript</csymbol><ci id="S2.SS1.p18.5.m2.1.1.2.cmml" xref="S2.SS1.p18.5.m2.1.1.2">𝑋</ci><ci id="S2.SS1.p18.5.m2.1.1.3.cmml" xref="S2.SS1.p18.5.m2.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.5.m2.1c">X_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.5.m2.1d">italic_X start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is uniquely ergodic, then <math alttext="X_{\mu}" class="ltx_Math" display="inline" id="S2.SS1.p18.6.m3.1"><semantics id="S2.SS1.p18.6.m3.1a"><msub id="S2.SS1.p18.6.m3.1.1" xref="S2.SS1.p18.6.m3.1.1.cmml"><mi id="S2.SS1.p18.6.m3.1.1.2" xref="S2.SS1.p18.6.m3.1.1.2.cmml">X</mi><mi id="S2.SS1.p18.6.m3.1.1.3" xref="S2.SS1.p18.6.m3.1.1.3.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.6.m3.1b"><apply id="S2.SS1.p18.6.m3.1.1.cmml" xref="S2.SS1.p18.6.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.6.m3.1.1.1.cmml" xref="S2.SS1.p18.6.m3.1.1">subscript</csymbol><ci id="S2.SS1.p18.6.m3.1.1.2.cmml" xref="S2.SS1.p18.6.m3.1.1.2">𝑋</ci><ci id="S2.SS1.p18.6.m3.1.1.3.cmml" xref="S2.SS1.p18.6.m3.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.6.m3.1c">X_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.6.m3.1d">italic_X start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is minimal, but the converse is famously wrong (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib12" title="">12</a>]</cite>). For any characteristic measure <math alttext="\mu_{w}" class="ltx_Math" display="inline" id="S2.SS1.p18.7.m4.1"><semantics id="S2.SS1.p18.7.m4.1a"><msub id="S2.SS1.p18.7.m4.1.1" xref="S2.SS1.p18.7.m4.1.1.cmml"><mi id="S2.SS1.p18.7.m4.1.1.2" xref="S2.SS1.p18.7.m4.1.1.2.cmml">μ</mi><mi id="S2.SS1.p18.7.m4.1.1.3" xref="S2.SS1.p18.7.m4.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.7.m4.1b"><apply id="S2.SS1.p18.7.m4.1.1.cmml" xref="S2.SS1.p18.7.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.7.m4.1.1.1.cmml" xref="S2.SS1.p18.7.m4.1.1">subscript</csymbol><ci id="S2.SS1.p18.7.m4.1.1.2.cmml" xref="S2.SS1.p18.7.m4.1.1.2">𝜇</ci><ci id="S2.SS1.p18.7.m4.1.1.3.cmml" xref="S2.SS1.p18.7.m4.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.7.m4.1c">\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.7.m4.1d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> one has <math alttext="\mbox{Supp}(\mu_{w})=\cal O(w^{\pm\infty})" class="ltx_Math" display="inline" id="S2.SS1.p18.8.m5.2"><semantics id="S2.SS1.p18.8.m5.2a"><mrow id="S2.SS1.p18.8.m5.2.2" xref="S2.SS1.p18.8.m5.2.2.cmml"><mrow id="S2.SS1.p18.8.m5.1.1.1" xref="S2.SS1.p18.8.m5.1.1.1.cmml"><mtext id="S2.SS1.p18.8.m5.1.1.1.3" xref="S2.SS1.p18.8.m5.1.1.1.3a.cmml">Supp</mtext><mo id="S2.SS1.p18.8.m5.1.1.1.2" xref="S2.SS1.p18.8.m5.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p18.8.m5.1.1.1.1.1" xref="S2.SS1.p18.8.m5.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p18.8.m5.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p18.8.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p18.8.m5.1.1.1.1.1.1" xref="S2.SS1.p18.8.m5.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p18.8.m5.1.1.1.1.1.1.2" xref="S2.SS1.p18.8.m5.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S2.SS1.p18.8.m5.1.1.1.1.1.1.3" xref="S2.SS1.p18.8.m5.1.1.1.1.1.1.3.cmml">w</mi></msub><mo id="S2.SS1.p18.8.m5.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p18.8.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p18.8.m5.2.2.3" xref="S2.SS1.p18.8.m5.2.2.3.cmml">=</mo><mrow id="S2.SS1.p18.8.m5.2.2.2" xref="S2.SS1.p18.8.m5.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.8.m5.2.2.2.3" xref="S2.SS1.p18.8.m5.2.2.2.3.cmml">𝒪</mi><mo id="S2.SS1.p18.8.m5.2.2.2.2" xref="S2.SS1.p18.8.m5.2.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p18.8.m5.2.2.2.1.1" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.cmml"><mo id="S2.SS1.p18.8.m5.2.2.2.1.1.2" stretchy="false" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.cmml">(</mo><msup id="S2.SS1.p18.8.m5.2.2.2.1.1.1" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.8.m5.2.2.2.1.1.1.2" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.2.cmml">𝓌</mi><mrow id="S2.SS1.p18.8.m5.2.2.2.1.1.1.3" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.3.cmml"><mo id="S2.SS1.p18.8.m5.2.2.2.1.1.1.3a" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.3.cmml">±</mo><mi id="S2.SS1.p18.8.m5.2.2.2.1.1.1.3.2" mathvariant="normal" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.SS1.p18.8.m5.2.2.2.1.1.3" stretchy="false" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.8.m5.2b"><apply id="S2.SS1.p18.8.m5.2.2.cmml" xref="S2.SS1.p18.8.m5.2.2"><eq id="S2.SS1.p18.8.m5.2.2.3.cmml" xref="S2.SS1.p18.8.m5.2.2.3"></eq><apply id="S2.SS1.p18.8.m5.1.1.1.cmml" xref="S2.SS1.p18.8.m5.1.1.1"><times id="S2.SS1.p18.8.m5.1.1.1.2.cmml" xref="S2.SS1.p18.8.m5.1.1.1.2"></times><ci id="S2.SS1.p18.8.m5.1.1.1.3a.cmml" xref="S2.SS1.p18.8.m5.1.1.1.3"><mtext id="S2.SS1.p18.8.m5.1.1.1.3.cmml" xref="S2.SS1.p18.8.m5.1.1.1.3">Supp</mtext></ci><apply id="S2.SS1.p18.8.m5.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.8.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.8.m5.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.8.m5.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p18.8.m5.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p18.8.m5.1.1.1.1.1.1.2">𝜇</ci><ci id="S2.SS1.p18.8.m5.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p18.8.m5.1.1.1.1.1.1.3">𝑤</ci></apply></apply><apply id="S2.SS1.p18.8.m5.2.2.2.cmml" xref="S2.SS1.p18.8.m5.2.2.2"><times id="S2.SS1.p18.8.m5.2.2.2.2.cmml" xref="S2.SS1.p18.8.m5.2.2.2.2"></times><ci id="S2.SS1.p18.8.m5.2.2.2.3.cmml" xref="S2.SS1.p18.8.m5.2.2.2.3">𝒪</ci><apply id="S2.SS1.p18.8.m5.2.2.2.1.1.1.cmml" xref="S2.SS1.p18.8.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.8.m5.2.2.2.1.1.1.1.cmml" xref="S2.SS1.p18.8.m5.2.2.2.1.1">superscript</csymbol><ci id="S2.SS1.p18.8.m5.2.2.2.1.1.1.2.cmml" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.2">𝓌</ci><apply id="S2.SS1.p18.8.m5.2.2.2.1.1.1.3.cmml" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p18.8.m5.2.2.2.1.1.1.3.1.cmml" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS1.p18.8.m5.2.2.2.1.1.1.3.2.cmml" xref="S2.SS1.p18.8.m5.2.2.2.1.1.1.3.2"></infinity></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.8.m5.2c">\mbox{Supp}(\mu_{w})=\cal O(w^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.8.m5.2d">Supp ( italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ) = caligraphic_O ( caligraphic_w start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math>. The support map</p> <table class="ltx_equation ltx_eqn_table" id="S2.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.7)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mbox{Supp}:\cal M(\cal A^{\mathbb{Z}})\smallsetminus\{0\}\to\Sigma(\cal A),\,% \,\mu\mapsto X_{\mu}" class="ltx_Math" display="block" id="S2.E7.m1.4"><semantics id="S2.E7.m1.4a"><mrow id="S2.E7.m1.4.4" xref="S2.E7.m1.4.4.cmml"><mtext id="S2.E7.m1.4.4.4" xref="S2.E7.m1.4.4.4a.cmml">Supp</mtext><mo id="S2.E7.m1.4.4.3" lspace="0.278em" rspace="0.278em" xref="S2.E7.m1.4.4.3.cmml">:</mo><mrow id="S2.E7.m1.4.4.2.2" xref="S2.E7.m1.4.4.2.3.cmml"><mrow id="S2.E7.m1.3.3.1.1.1" xref="S2.E7.m1.3.3.1.1.1.cmml"><mrow id="S2.E7.m1.3.3.1.1.1.1" xref="S2.E7.m1.3.3.1.1.1.1.cmml"><mrow id="S2.E7.m1.3.3.1.1.1.1.1" xref="S2.E7.m1.3.3.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E7.m1.3.3.1.1.1.1.1.3" xref="S2.E7.m1.3.3.1.1.1.1.1.3.cmml">ℳ</mi><mo id="S2.E7.m1.3.3.1.1.1.1.1.2" xref="S2.E7.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E7.m1.3.3.1.1.1.1.1.1.1" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E7.m1.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S2.E7.m1.3.3.1.1.1.1.1.1.1.1" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.2" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.3" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S2.E7.m1.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E7.m1.3.3.1.1.1.1.2" xref="S2.E7.m1.3.3.1.1.1.1.2.cmml">∖</mo><mrow id="S2.E7.m1.3.3.1.1.1.1.3.2" xref="S2.E7.m1.3.3.1.1.1.1.3.1.cmml"><mo id="S2.E7.m1.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.E7.m1.3.3.1.1.1.1.3.1.cmml">{</mo><mn class="ltx_font_mathcaligraphic" id="S2.E7.m1.1.1" mathvariant="script" xref="S2.E7.m1.1.1.cmml">0</mn><mo id="S2.E7.m1.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.E7.m1.3.3.1.1.1.1.3.1.cmml">}</mo></mrow></mrow><mo id="S2.E7.m1.3.3.1.1.1.2" stretchy="false" xref="S2.E7.m1.3.3.1.1.1.2.cmml">→</mo><mrow id="S2.E7.m1.3.3.1.1.1.3" xref="S2.E7.m1.3.3.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E7.m1.3.3.1.1.1.3.2" mathvariant="script" xref="S2.E7.m1.3.3.1.1.1.3.2.cmml">Σ</mi><mo id="S2.E7.m1.3.3.1.1.1.3.1" xref="S2.E7.m1.3.3.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.E7.m1.3.3.1.1.1.3.3.2" xref="S2.E7.m1.3.3.1.1.1.3.cmml"><mo id="S2.E7.m1.3.3.1.1.1.3.3.2.1" stretchy="false" xref="S2.E7.m1.3.3.1.1.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.E7.m1.2.2" xref="S2.E7.m1.2.2.cmml">𝒜</mi><mo id="S2.E7.m1.3.3.1.1.1.3.3.2.2" stretchy="false" xref="S2.E7.m1.3.3.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E7.m1.4.4.2.2.3" rspace="0.497em" xref="S2.E7.m1.4.4.2.3a.cmml">,</mo><mrow id="S2.E7.m1.4.4.2.2.2" xref="S2.E7.m1.4.4.2.2.2.cmml"><mi id="S2.E7.m1.4.4.2.2.2.2" xref="S2.E7.m1.4.4.2.2.2.2.cmml">μ</mi><mo id="S2.E7.m1.4.4.2.2.2.1" stretchy="false" xref="S2.E7.m1.4.4.2.2.2.1.cmml">↦</mo><msub id="S2.E7.m1.4.4.2.2.2.3" xref="S2.E7.m1.4.4.2.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E7.m1.4.4.2.2.2.3.2" xref="S2.E7.m1.4.4.2.2.2.3.2.cmml">𝒳</mi><mi id="S2.E7.m1.4.4.2.2.2.3.3" xref="S2.E7.m1.4.4.2.2.2.3.3.cmml">μ</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E7.m1.4b"><apply id="S2.E7.m1.4.4.cmml" xref="S2.E7.m1.4.4"><ci id="S2.E7.m1.4.4.3.cmml" xref="S2.E7.m1.4.4.3">:</ci><ci id="S2.E7.m1.4.4.4a.cmml" xref="S2.E7.m1.4.4.4"><mtext id="S2.E7.m1.4.4.4.cmml" xref="S2.E7.m1.4.4.4">Supp</mtext></ci><apply id="S2.E7.m1.4.4.2.3.cmml" xref="S2.E7.m1.4.4.2.2"><csymbol cd="ambiguous" id="S2.E7.m1.4.4.2.3a.cmml" xref="S2.E7.m1.4.4.2.2.3">formulae-sequence</csymbol><apply id="S2.E7.m1.3.3.1.1.1.cmml" xref="S2.E7.m1.3.3.1.1.1"><ci id="S2.E7.m1.3.3.1.1.1.2.cmml" xref="S2.E7.m1.3.3.1.1.1.2">→</ci><apply id="S2.E7.m1.3.3.1.1.1.1.cmml" xref="S2.E7.m1.3.3.1.1.1.1"><setdiff id="S2.E7.m1.3.3.1.1.1.1.2.cmml" xref="S2.E7.m1.3.3.1.1.1.1.2"></setdiff><apply id="S2.E7.m1.3.3.1.1.1.1.1.cmml" xref="S2.E7.m1.3.3.1.1.1.1.1"><times id="S2.E7.m1.3.3.1.1.1.1.1.2.cmml" xref="S2.E7.m1.3.3.1.1.1.1.1.2"></times><ci id="S2.E7.m1.3.3.1.1.1.1.1.3.cmml" xref="S2.E7.m1.3.3.1.1.1.1.1.3">ℳ</ci><apply id="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.1.cmml" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1">superscript</csymbol><ci id="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.2.cmml" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.2">𝒜</ci><ci id="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.3.cmml" xref="S2.E7.m1.3.3.1.1.1.1.1.1.1.1.3">ℤ</ci></apply></apply><set id="S2.E7.m1.3.3.1.1.1.1.3.1.cmml" xref="S2.E7.m1.3.3.1.1.1.1.3.2"><cn id="S2.E7.m1.1.1.cmml" type="integer" xref="S2.E7.m1.1.1">0</cn></set></apply><apply id="S2.E7.m1.3.3.1.1.1.3.cmml" xref="S2.E7.m1.3.3.1.1.1.3"><times id="S2.E7.m1.3.3.1.1.1.3.1.cmml" xref="S2.E7.m1.3.3.1.1.1.3.1"></times><ci id="S2.E7.m1.3.3.1.1.1.3.2.cmml" xref="S2.E7.m1.3.3.1.1.1.3.2">script-Σ</ci><ci id="S2.E7.m1.2.2.cmml" xref="S2.E7.m1.2.2">𝒜</ci></apply></apply><apply id="S2.E7.m1.4.4.2.2.2.cmml" xref="S2.E7.m1.4.4.2.2.2"><csymbol cd="latexml" id="S2.E7.m1.4.4.2.2.2.1.cmml" xref="S2.E7.m1.4.4.2.2.2.1">maps-to</csymbol><ci id="S2.E7.m1.4.4.2.2.2.2.cmml" xref="S2.E7.m1.4.4.2.2.2.2">𝜇</ci><apply id="S2.E7.m1.4.4.2.2.2.3.cmml" xref="S2.E7.m1.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S2.E7.m1.4.4.2.2.2.3.1.cmml" xref="S2.E7.m1.4.4.2.2.2.3">subscript</csymbol><ci id="S2.E7.m1.4.4.2.2.2.3.2.cmml" xref="S2.E7.m1.4.4.2.2.2.3.2">𝒳</ci><ci id="S2.E7.m1.4.4.2.2.2.3.3.cmml" xref="S2.E7.m1.4.4.2.2.2.3.3">𝜇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m1.4c">\mbox{Supp}:\cal M(\cal A^{\mathbb{Z}})\smallsetminus\{0\}\to\Sigma(\cal A),\,% \,\mu\mapsto X_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.4d">Supp : caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) ∖ { caligraphic_0 } → caligraphic_Σ ( caligraphic_A ) , italic_μ ↦ caligraphic_X start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p18.16">has some nice natural properties: for example, if <math alttext="\mu_{1},\mu_{2}\in\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S2.SS1.p18.9.m1.3"><semantics id="S2.SS1.p18.9.m1.3a"><mrow id="S2.SS1.p18.9.m1.3.3" xref="S2.SS1.p18.9.m1.3.3.cmml"><mrow id="S2.SS1.p18.9.m1.2.2.2.2" xref="S2.SS1.p18.9.m1.2.2.2.3.cmml"><msub id="S2.SS1.p18.9.m1.1.1.1.1.1" xref="S2.SS1.p18.9.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p18.9.m1.1.1.1.1.1.2" xref="S2.SS1.p18.9.m1.1.1.1.1.1.2.cmml">μ</mi><mn id="S2.SS1.p18.9.m1.1.1.1.1.1.3" xref="S2.SS1.p18.9.m1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p18.9.m1.2.2.2.2.3" xref="S2.SS1.p18.9.m1.2.2.2.3.cmml">,</mo><msub id="S2.SS1.p18.9.m1.2.2.2.2.2" xref="S2.SS1.p18.9.m1.2.2.2.2.2.cmml"><mi id="S2.SS1.p18.9.m1.2.2.2.2.2.2" xref="S2.SS1.p18.9.m1.2.2.2.2.2.2.cmml">μ</mi><mn id="S2.SS1.p18.9.m1.2.2.2.2.2.3" xref="S2.SS1.p18.9.m1.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S2.SS1.p18.9.m1.3.3.4" xref="S2.SS1.p18.9.m1.3.3.4.cmml">∈</mo><mrow id="S2.SS1.p18.9.m1.3.3.3" xref="S2.SS1.p18.9.m1.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.9.m1.3.3.3.3" xref="S2.SS1.p18.9.m1.3.3.3.3.cmml">ℳ</mi><mo id="S2.SS1.p18.9.m1.3.3.3.2" xref="S2.SS1.p18.9.m1.3.3.3.2.cmml">⁢</mo><mrow id="S2.SS1.p18.9.m1.3.3.3.1.1" xref="S2.SS1.p18.9.m1.3.3.3.1.1.1.cmml"><mo id="S2.SS1.p18.9.m1.3.3.3.1.1.2" stretchy="false" xref="S2.SS1.p18.9.m1.3.3.3.1.1.1.cmml">(</mo><msup id="S2.SS1.p18.9.m1.3.3.3.1.1.1" xref="S2.SS1.p18.9.m1.3.3.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.9.m1.3.3.3.1.1.1.2" xref="S2.SS1.p18.9.m1.3.3.3.1.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p18.9.m1.3.3.3.1.1.1.3" xref="S2.SS1.p18.9.m1.3.3.3.1.1.1.3.cmml">ℤ</mi></msup><mo id="S2.SS1.p18.9.m1.3.3.3.1.1.3" stretchy="false" xref="S2.SS1.p18.9.m1.3.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.9.m1.3b"><apply id="S2.SS1.p18.9.m1.3.3.cmml" xref="S2.SS1.p18.9.m1.3.3"><in id="S2.SS1.p18.9.m1.3.3.4.cmml" xref="S2.SS1.p18.9.m1.3.3.4"></in><list id="S2.SS1.p18.9.m1.2.2.2.3.cmml" xref="S2.SS1.p18.9.m1.2.2.2.2"><apply id="S2.SS1.p18.9.m1.1.1.1.1.1.cmml" xref="S2.SS1.p18.9.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.9.m1.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.9.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p18.9.m1.1.1.1.1.1.2.cmml" xref="S2.SS1.p18.9.m1.1.1.1.1.1.2">𝜇</ci><cn id="S2.SS1.p18.9.m1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p18.9.m1.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS1.p18.9.m1.2.2.2.2.2.cmml" xref="S2.SS1.p18.9.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p18.9.m1.2.2.2.2.2.1.cmml" xref="S2.SS1.p18.9.m1.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p18.9.m1.2.2.2.2.2.2.cmml" xref="S2.SS1.p18.9.m1.2.2.2.2.2.2">𝜇</ci><cn id="S2.SS1.p18.9.m1.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p18.9.m1.2.2.2.2.2.3">2</cn></apply></list><apply id="S2.SS1.p18.9.m1.3.3.3.cmml" xref="S2.SS1.p18.9.m1.3.3.3"><times id="S2.SS1.p18.9.m1.3.3.3.2.cmml" xref="S2.SS1.p18.9.m1.3.3.3.2"></times><ci id="S2.SS1.p18.9.m1.3.3.3.3.cmml" xref="S2.SS1.p18.9.m1.3.3.3.3">ℳ</ci><apply id="S2.SS1.p18.9.m1.3.3.3.1.1.1.cmml" xref="S2.SS1.p18.9.m1.3.3.3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.9.m1.3.3.3.1.1.1.1.cmml" xref="S2.SS1.p18.9.m1.3.3.3.1.1">superscript</csymbol><ci id="S2.SS1.p18.9.m1.3.3.3.1.1.1.2.cmml" xref="S2.SS1.p18.9.m1.3.3.3.1.1.1.2">𝒜</ci><ci id="S2.SS1.p18.9.m1.3.3.3.1.1.1.3.cmml" xref="S2.SS1.p18.9.m1.3.3.3.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.9.m1.3c">\mu_{1},\mu_{2}\in\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.9.m1.3d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\lambda_{1}>0\,,\,\,\lambda_{2}>0" class="ltx_Math" display="inline" id="S2.SS1.p18.10.m2.2"><semantics id="S2.SS1.p18.10.m2.2a"><mrow id="S2.SS1.p18.10.m2.2.2.2" xref="S2.SS1.p18.10.m2.2.2.3.cmml"><mrow id="S2.SS1.p18.10.m2.1.1.1.1" xref="S2.SS1.p18.10.m2.1.1.1.1.cmml"><msub id="S2.SS1.p18.10.m2.1.1.1.1.2" xref="S2.SS1.p18.10.m2.1.1.1.1.2.cmml"><mi id="S2.SS1.p18.10.m2.1.1.1.1.2.2" xref="S2.SS1.p18.10.m2.1.1.1.1.2.2.cmml">λ</mi><mn id="S2.SS1.p18.10.m2.1.1.1.1.2.3" xref="S2.SS1.p18.10.m2.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p18.10.m2.1.1.1.1.1" xref="S2.SS1.p18.10.m2.1.1.1.1.1.cmml">></mo><mn id="S2.SS1.p18.10.m2.1.1.1.1.3" xref="S2.SS1.p18.10.m2.1.1.1.1.3.cmml">0</mn></mrow><mo id="S2.SS1.p18.10.m2.2.2.2.3" lspace="0.170em" rspace="0.497em" xref="S2.SS1.p18.10.m2.2.2.3a.cmml">,</mo><mrow id="S2.SS1.p18.10.m2.2.2.2.2" xref="S2.SS1.p18.10.m2.2.2.2.2.cmml"><msub id="S2.SS1.p18.10.m2.2.2.2.2.2" xref="S2.SS1.p18.10.m2.2.2.2.2.2.cmml"><mi id="S2.SS1.p18.10.m2.2.2.2.2.2.2" xref="S2.SS1.p18.10.m2.2.2.2.2.2.2.cmml">λ</mi><mn id="S2.SS1.p18.10.m2.2.2.2.2.2.3" xref="S2.SS1.p18.10.m2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS1.p18.10.m2.2.2.2.2.1" xref="S2.SS1.p18.10.m2.2.2.2.2.1.cmml">></mo><mn id="S2.SS1.p18.10.m2.2.2.2.2.3" xref="S2.SS1.p18.10.m2.2.2.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.10.m2.2b"><apply id="S2.SS1.p18.10.m2.2.2.3.cmml" xref="S2.SS1.p18.10.m2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p18.10.m2.2.2.3a.cmml" xref="S2.SS1.p18.10.m2.2.2.2.3">formulae-sequence</csymbol><apply id="S2.SS1.p18.10.m2.1.1.1.1.cmml" xref="S2.SS1.p18.10.m2.1.1.1.1"><gt id="S2.SS1.p18.10.m2.1.1.1.1.1.cmml" xref="S2.SS1.p18.10.m2.1.1.1.1.1"></gt><apply id="S2.SS1.p18.10.m2.1.1.1.1.2.cmml" xref="S2.SS1.p18.10.m2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p18.10.m2.1.1.1.1.2.1.cmml" xref="S2.SS1.p18.10.m2.1.1.1.1.2">subscript</csymbol><ci id="S2.SS1.p18.10.m2.1.1.1.1.2.2.cmml" xref="S2.SS1.p18.10.m2.1.1.1.1.2.2">𝜆</ci><cn id="S2.SS1.p18.10.m2.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS1.p18.10.m2.1.1.1.1.2.3">1</cn></apply><cn id="S2.SS1.p18.10.m2.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p18.10.m2.1.1.1.1.3">0</cn></apply><apply id="S2.SS1.p18.10.m2.2.2.2.2.cmml" xref="S2.SS1.p18.10.m2.2.2.2.2"><gt id="S2.SS1.p18.10.m2.2.2.2.2.1.cmml" xref="S2.SS1.p18.10.m2.2.2.2.2.1"></gt><apply id="S2.SS1.p18.10.m2.2.2.2.2.2.cmml" xref="S2.SS1.p18.10.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p18.10.m2.2.2.2.2.2.1.cmml" xref="S2.SS1.p18.10.m2.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p18.10.m2.2.2.2.2.2.2.cmml" xref="S2.SS1.p18.10.m2.2.2.2.2.2.2">𝜆</ci><cn id="S2.SS1.p18.10.m2.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p18.10.m2.2.2.2.2.2.3">2</cn></apply><cn id="S2.SS1.p18.10.m2.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p18.10.m2.2.2.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.10.m2.2c">\lambda_{1}>0\,,\,\,\lambda_{2}>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.10.m2.2d">italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0 , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT > 0</annotation></semantics></math> are given, then for <math alttext="\mu=\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}" class="ltx_Math" display="inline" id="S2.SS1.p18.11.m3.1"><semantics id="S2.SS1.p18.11.m3.1a"><mrow id="S2.SS1.p18.11.m3.1.1" xref="S2.SS1.p18.11.m3.1.1.cmml"><mi id="S2.SS1.p18.11.m3.1.1.2" xref="S2.SS1.p18.11.m3.1.1.2.cmml">μ</mi><mo id="S2.SS1.p18.11.m3.1.1.1" xref="S2.SS1.p18.11.m3.1.1.1.cmml">=</mo><mrow id="S2.SS1.p18.11.m3.1.1.3" xref="S2.SS1.p18.11.m3.1.1.3.cmml"><mrow id="S2.SS1.p18.11.m3.1.1.3.2" xref="S2.SS1.p18.11.m3.1.1.3.2.cmml"><msub id="S2.SS1.p18.11.m3.1.1.3.2.2" xref="S2.SS1.p18.11.m3.1.1.3.2.2.cmml"><mi id="S2.SS1.p18.11.m3.1.1.3.2.2.2" xref="S2.SS1.p18.11.m3.1.1.3.2.2.2.cmml">λ</mi><mn id="S2.SS1.p18.11.m3.1.1.3.2.2.3" xref="S2.SS1.p18.11.m3.1.1.3.2.2.3.cmml">1</mn></msub><mo id="S2.SS1.p18.11.m3.1.1.3.2.1" xref="S2.SS1.p18.11.m3.1.1.3.2.1.cmml">⁢</mo><msub id="S2.SS1.p18.11.m3.1.1.3.2.3" xref="S2.SS1.p18.11.m3.1.1.3.2.3.cmml"><mi id="S2.SS1.p18.11.m3.1.1.3.2.3.2" xref="S2.SS1.p18.11.m3.1.1.3.2.3.2.cmml">μ</mi><mn id="S2.SS1.p18.11.m3.1.1.3.2.3.3" xref="S2.SS1.p18.11.m3.1.1.3.2.3.3.cmml">1</mn></msub></mrow><mo id="S2.SS1.p18.11.m3.1.1.3.1" xref="S2.SS1.p18.11.m3.1.1.3.1.cmml">+</mo><mrow id="S2.SS1.p18.11.m3.1.1.3.3" xref="S2.SS1.p18.11.m3.1.1.3.3.cmml"><msub id="S2.SS1.p18.11.m3.1.1.3.3.2" xref="S2.SS1.p18.11.m3.1.1.3.3.2.cmml"><mi id="S2.SS1.p18.11.m3.1.1.3.3.2.2" xref="S2.SS1.p18.11.m3.1.1.3.3.2.2.cmml">λ</mi><mn id="S2.SS1.p18.11.m3.1.1.3.3.2.3" xref="S2.SS1.p18.11.m3.1.1.3.3.2.3.cmml">2</mn></msub><mo id="S2.SS1.p18.11.m3.1.1.3.3.1" xref="S2.SS1.p18.11.m3.1.1.3.3.1.cmml">⁢</mo><msub id="S2.SS1.p18.11.m3.1.1.3.3.3" xref="S2.SS1.p18.11.m3.1.1.3.3.3.cmml"><mi id="S2.SS1.p18.11.m3.1.1.3.3.3.2" xref="S2.SS1.p18.11.m3.1.1.3.3.3.2.cmml">μ</mi><mn id="S2.SS1.p18.11.m3.1.1.3.3.3.3" xref="S2.SS1.p18.11.m3.1.1.3.3.3.3.cmml">2</mn></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.11.m3.1b"><apply id="S2.SS1.p18.11.m3.1.1.cmml" xref="S2.SS1.p18.11.m3.1.1"><eq id="S2.SS1.p18.11.m3.1.1.1.cmml" xref="S2.SS1.p18.11.m3.1.1.1"></eq><ci id="S2.SS1.p18.11.m3.1.1.2.cmml" xref="S2.SS1.p18.11.m3.1.1.2">𝜇</ci><apply id="S2.SS1.p18.11.m3.1.1.3.cmml" xref="S2.SS1.p18.11.m3.1.1.3"><plus id="S2.SS1.p18.11.m3.1.1.3.1.cmml" xref="S2.SS1.p18.11.m3.1.1.3.1"></plus><apply id="S2.SS1.p18.11.m3.1.1.3.2.cmml" xref="S2.SS1.p18.11.m3.1.1.3.2"><times id="S2.SS1.p18.11.m3.1.1.3.2.1.cmml" xref="S2.SS1.p18.11.m3.1.1.3.2.1"></times><apply id="S2.SS1.p18.11.m3.1.1.3.2.2.cmml" xref="S2.SS1.p18.11.m3.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p18.11.m3.1.1.3.2.2.1.cmml" xref="S2.SS1.p18.11.m3.1.1.3.2.2">subscript</csymbol><ci id="S2.SS1.p18.11.m3.1.1.3.2.2.2.cmml" xref="S2.SS1.p18.11.m3.1.1.3.2.2.2">𝜆</ci><cn id="S2.SS1.p18.11.m3.1.1.3.2.2.3.cmml" type="integer" xref="S2.SS1.p18.11.m3.1.1.3.2.2.3">1</cn></apply><apply id="S2.SS1.p18.11.m3.1.1.3.2.3.cmml" xref="S2.SS1.p18.11.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS1.p18.11.m3.1.1.3.2.3.1.cmml" xref="S2.SS1.p18.11.m3.1.1.3.2.3">subscript</csymbol><ci id="S2.SS1.p18.11.m3.1.1.3.2.3.2.cmml" xref="S2.SS1.p18.11.m3.1.1.3.2.3.2">𝜇</ci><cn id="S2.SS1.p18.11.m3.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS1.p18.11.m3.1.1.3.2.3.3">1</cn></apply></apply><apply id="S2.SS1.p18.11.m3.1.1.3.3.cmml" xref="S2.SS1.p18.11.m3.1.1.3.3"><times id="S2.SS1.p18.11.m3.1.1.3.3.1.cmml" xref="S2.SS1.p18.11.m3.1.1.3.3.1"></times><apply id="S2.SS1.p18.11.m3.1.1.3.3.2.cmml" xref="S2.SS1.p18.11.m3.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.p18.11.m3.1.1.3.3.2.1.cmml" xref="S2.SS1.p18.11.m3.1.1.3.3.2">subscript</csymbol><ci id="S2.SS1.p18.11.m3.1.1.3.3.2.2.cmml" xref="S2.SS1.p18.11.m3.1.1.3.3.2.2">𝜆</ci><cn id="S2.SS1.p18.11.m3.1.1.3.3.2.3.cmml" type="integer" xref="S2.SS1.p18.11.m3.1.1.3.3.2.3">2</cn></apply><apply id="S2.SS1.p18.11.m3.1.1.3.3.3.cmml" xref="S2.SS1.p18.11.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p18.11.m3.1.1.3.3.3.1.cmml" xref="S2.SS1.p18.11.m3.1.1.3.3.3">subscript</csymbol><ci id="S2.SS1.p18.11.m3.1.1.3.3.3.2.cmml" xref="S2.SS1.p18.11.m3.1.1.3.3.3.2">𝜇</ci><cn id="S2.SS1.p18.11.m3.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p18.11.m3.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.11.m3.1c">\mu=\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.11.m3.1d">italic_μ = italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> one has <math alttext="X_{\mu}=X_{\mu_{1}}\cup X_{\mu_{2}}" class="ltx_Math" display="inline" id="S2.SS1.p18.12.m4.1"><semantics id="S2.SS1.p18.12.m4.1a"><mrow id="S2.SS1.p18.12.m4.1.1" xref="S2.SS1.p18.12.m4.1.1.cmml"><msub id="S2.SS1.p18.12.m4.1.1.2" xref="S2.SS1.p18.12.m4.1.1.2.cmml"><mi id="S2.SS1.p18.12.m4.1.1.2.2" xref="S2.SS1.p18.12.m4.1.1.2.2.cmml">X</mi><mi id="S2.SS1.p18.12.m4.1.1.2.3" xref="S2.SS1.p18.12.m4.1.1.2.3.cmml">μ</mi></msub><mo id="S2.SS1.p18.12.m4.1.1.1" xref="S2.SS1.p18.12.m4.1.1.1.cmml">=</mo><mrow id="S2.SS1.p18.12.m4.1.1.3" xref="S2.SS1.p18.12.m4.1.1.3.cmml"><msub id="S2.SS1.p18.12.m4.1.1.3.2" xref="S2.SS1.p18.12.m4.1.1.3.2.cmml"><mi id="S2.SS1.p18.12.m4.1.1.3.2.2" xref="S2.SS1.p18.12.m4.1.1.3.2.2.cmml">X</mi><msub id="S2.SS1.p18.12.m4.1.1.3.2.3" xref="S2.SS1.p18.12.m4.1.1.3.2.3.cmml"><mi id="S2.SS1.p18.12.m4.1.1.3.2.3.2" xref="S2.SS1.p18.12.m4.1.1.3.2.3.2.cmml">μ</mi><mn id="S2.SS1.p18.12.m4.1.1.3.2.3.3" xref="S2.SS1.p18.12.m4.1.1.3.2.3.3.cmml">1</mn></msub></msub><mo id="S2.SS1.p18.12.m4.1.1.3.1" xref="S2.SS1.p18.12.m4.1.1.3.1.cmml">∪</mo><msub id="S2.SS1.p18.12.m4.1.1.3.3" xref="S2.SS1.p18.12.m4.1.1.3.3.cmml"><mi id="S2.SS1.p18.12.m4.1.1.3.3.2" xref="S2.SS1.p18.12.m4.1.1.3.3.2.cmml">X</mi><msub id="S2.SS1.p18.12.m4.1.1.3.3.3" xref="S2.SS1.p18.12.m4.1.1.3.3.3.cmml"><mi id="S2.SS1.p18.12.m4.1.1.3.3.3.2" xref="S2.SS1.p18.12.m4.1.1.3.3.3.2.cmml">μ</mi><mn id="S2.SS1.p18.12.m4.1.1.3.3.3.3" xref="S2.SS1.p18.12.m4.1.1.3.3.3.3.cmml">2</mn></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.12.m4.1b"><apply id="S2.SS1.p18.12.m4.1.1.cmml" xref="S2.SS1.p18.12.m4.1.1"><eq id="S2.SS1.p18.12.m4.1.1.1.cmml" xref="S2.SS1.p18.12.m4.1.1.1"></eq><apply id="S2.SS1.p18.12.m4.1.1.2.cmml" xref="S2.SS1.p18.12.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p18.12.m4.1.1.2.1.cmml" xref="S2.SS1.p18.12.m4.1.1.2">subscript</csymbol><ci id="S2.SS1.p18.12.m4.1.1.2.2.cmml" xref="S2.SS1.p18.12.m4.1.1.2.2">𝑋</ci><ci id="S2.SS1.p18.12.m4.1.1.2.3.cmml" xref="S2.SS1.p18.12.m4.1.1.2.3">𝜇</ci></apply><apply id="S2.SS1.p18.12.m4.1.1.3.cmml" xref="S2.SS1.p18.12.m4.1.1.3"><union id="S2.SS1.p18.12.m4.1.1.3.1.cmml" xref="S2.SS1.p18.12.m4.1.1.3.1"></union><apply id="S2.SS1.p18.12.m4.1.1.3.2.cmml" xref="S2.SS1.p18.12.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p18.12.m4.1.1.3.2.1.cmml" xref="S2.SS1.p18.12.m4.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p18.12.m4.1.1.3.2.2.cmml" xref="S2.SS1.p18.12.m4.1.1.3.2.2">𝑋</ci><apply id="S2.SS1.p18.12.m4.1.1.3.2.3.cmml" xref="S2.SS1.p18.12.m4.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS1.p18.12.m4.1.1.3.2.3.1.cmml" xref="S2.SS1.p18.12.m4.1.1.3.2.3">subscript</csymbol><ci id="S2.SS1.p18.12.m4.1.1.3.2.3.2.cmml" xref="S2.SS1.p18.12.m4.1.1.3.2.3.2">𝜇</ci><cn id="S2.SS1.p18.12.m4.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS1.p18.12.m4.1.1.3.2.3.3">1</cn></apply></apply><apply id="S2.SS1.p18.12.m4.1.1.3.3.cmml" xref="S2.SS1.p18.12.m4.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p18.12.m4.1.1.3.3.1.cmml" xref="S2.SS1.p18.12.m4.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p18.12.m4.1.1.3.3.2.cmml" xref="S2.SS1.p18.12.m4.1.1.3.3.2">𝑋</ci><apply id="S2.SS1.p18.12.m4.1.1.3.3.3.cmml" xref="S2.SS1.p18.12.m4.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p18.12.m4.1.1.3.3.3.1.cmml" xref="S2.SS1.p18.12.m4.1.1.3.3.3">subscript</csymbol><ci id="S2.SS1.p18.12.m4.1.1.3.3.3.2.cmml" xref="S2.SS1.p18.12.m4.1.1.3.3.3.2">𝜇</ci><cn id="S2.SS1.p18.12.m4.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p18.12.m4.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.12.m4.1c">X_{\mu}=X_{\mu_{1}}\cup X_{\mu_{2}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.12.m4.1d">italic_X start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = italic_X start_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∪ italic_X start_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. Also, every minimal subshift is the support of some measure <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p18.13.m5.1"><semantics id="S2.SS1.p18.13.m5.1a"><mi id="S2.SS1.p18.13.m5.1.1" xref="S2.SS1.p18.13.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.13.m5.1b"><ci id="S2.SS1.p18.13.m5.1.1.cmml" xref="S2.SS1.p18.13.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.13.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.13.m5.1d">italic_μ</annotation></semantics></math>, but (by a variety of reasons) there are non-minimal subshifts that are not in the image of the map <span class="ltx_text ltx_markedasmath" id="S2.SS1.p18.16.1">Supp</span>. An example is given by the subshift that consists of the three orbits <math alttext="\cal O(a^{\pm\infty}),\cal O(b^{\pm\infty})" class="ltx_Math" display="inline" id="S2.SS1.p18.15.m7.2"><semantics id="S2.SS1.p18.15.m7.2a"><mrow id="S2.SS1.p18.15.m7.2.2.2" xref="S2.SS1.p18.15.m7.2.2.3.cmml"><mrow id="S2.SS1.p18.15.m7.1.1.1.1" xref="S2.SS1.p18.15.m7.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.15.m7.1.1.1.1.3" xref="S2.SS1.p18.15.m7.1.1.1.1.3.cmml">𝒪</mi><mo id="S2.SS1.p18.15.m7.1.1.1.1.2" xref="S2.SS1.p18.15.m7.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p18.15.m7.1.1.1.1.1.1" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p18.15.m7.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.2" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.2.cmml">𝒶</mi><mrow id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3.cmml"><mo id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3a" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3.cmml">±</mo><mi id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.SS1.p18.15.m7.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p18.15.m7.2.2.2.3" xref="S2.SS1.p18.15.m7.2.2.3.cmml">,</mo><mrow id="S2.SS1.p18.15.m7.2.2.2.2" xref="S2.SS1.p18.15.m7.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.15.m7.2.2.2.2.3" xref="S2.SS1.p18.15.m7.2.2.2.2.3.cmml">𝒪</mi><mo id="S2.SS1.p18.15.m7.2.2.2.2.2" xref="S2.SS1.p18.15.m7.2.2.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p18.15.m7.2.2.2.2.1.1" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.cmml"><mo id="S2.SS1.p18.15.m7.2.2.2.2.1.1.2" stretchy="false" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.cmml">(</mo><msup id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.2" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.2.cmml">𝒷</mi><mrow id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3.cmml"><mo id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3a" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3.cmml">±</mo><mi id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3.2" mathvariant="normal" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.SS1.p18.15.m7.2.2.2.2.1.1.3" stretchy="false" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.15.m7.2b"><list id="S2.SS1.p18.15.m7.2.2.3.cmml" xref="S2.SS1.p18.15.m7.2.2.2"><apply id="S2.SS1.p18.15.m7.1.1.1.1.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1"><times id="S2.SS1.p18.15.m7.1.1.1.1.2.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1.2"></times><ci id="S2.SS1.p18.15.m7.1.1.1.1.3.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1.3">𝒪</ci><apply id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.2">𝒶</ci><apply id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3.1.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3.2.cmml" xref="S2.SS1.p18.15.m7.1.1.1.1.1.1.1.3.2"></infinity></apply></apply></apply><apply id="S2.SS1.p18.15.m7.2.2.2.2.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2"><times id="S2.SS1.p18.15.m7.2.2.2.2.2.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2.2"></times><ci id="S2.SS1.p18.15.m7.2.2.2.2.3.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2.3">𝒪</ci><apply id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.1.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1">superscript</csymbol><ci id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.2.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.2">𝒷</ci><apply id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3.1.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3.2.cmml" xref="S2.SS1.p18.15.m7.2.2.2.2.1.1.1.3.2"></infinity></apply></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.15.m7.2c">\cal O(a^{\pm\infty}),\cal O(b^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.15.m7.2d">caligraphic_O ( caligraphic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) , caligraphic_O ( caligraphic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\cal O(\ldots aaabbb\ldots)" class="ltx_Math" display="inline" id="S2.SS1.p18.16.m8.1"><semantics id="S2.SS1.p18.16.m8.1a"><mrow id="S2.SS1.p18.16.m8.1.1" xref="S2.SS1.p18.16.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.16.m8.1.1.3" xref="S2.SS1.p18.16.m8.1.1.3.cmml">𝒪</mi><mo id="S2.SS1.p18.16.m8.1.1.2" xref="S2.SS1.p18.16.m8.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p18.16.m8.1.1.1.1" xref="S2.SS1.p18.16.m8.1.1.1.1.1.cmml"><mo id="S2.SS1.p18.16.m8.1.1.1.1.2" stretchy="false" xref="S2.SS1.p18.16.m8.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p18.16.m8.1.1.1.1.1" xref="S2.SS1.p18.16.m8.1.1.1.1.1.cmml"><mi id="S2.SS1.p18.16.m8.1.1.1.1.1.2" mathvariant="normal" xref="S2.SS1.p18.16.m8.1.1.1.1.1.2.cmml">…</mi><mo id="S2.SS1.p18.16.m8.1.1.1.1.1.1" xref="S2.SS1.p18.16.m8.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.16.m8.1.1.1.1.1.3" xref="S2.SS1.p18.16.m8.1.1.1.1.1.3.cmml">𝒶</mi><mo id="S2.SS1.p18.16.m8.1.1.1.1.1.1a" xref="S2.SS1.p18.16.m8.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.16.m8.1.1.1.1.1.4" xref="S2.SS1.p18.16.m8.1.1.1.1.1.4.cmml">𝒶</mi><mo id="S2.SS1.p18.16.m8.1.1.1.1.1.1b" xref="S2.SS1.p18.16.m8.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.16.m8.1.1.1.1.1.5" xref="S2.SS1.p18.16.m8.1.1.1.1.1.5.cmml">𝒶</mi><mo id="S2.SS1.p18.16.m8.1.1.1.1.1.1c" xref="S2.SS1.p18.16.m8.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.16.m8.1.1.1.1.1.6" xref="S2.SS1.p18.16.m8.1.1.1.1.1.6.cmml">𝒷</mi><mo id="S2.SS1.p18.16.m8.1.1.1.1.1.1d" xref="S2.SS1.p18.16.m8.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.16.m8.1.1.1.1.1.7" xref="S2.SS1.p18.16.m8.1.1.1.1.1.7.cmml">𝒷</mi><mo id="S2.SS1.p18.16.m8.1.1.1.1.1.1e" xref="S2.SS1.p18.16.m8.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p18.16.m8.1.1.1.1.1.8" xref="S2.SS1.p18.16.m8.1.1.1.1.1.8.cmml">𝒷</mi><mo id="S2.SS1.p18.16.m8.1.1.1.1.1.1f" xref="S2.SS1.p18.16.m8.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p18.16.m8.1.1.1.1.1.9" mathvariant="normal" xref="S2.SS1.p18.16.m8.1.1.1.1.1.9.cmml">…</mi></mrow><mo id="S2.SS1.p18.16.m8.1.1.1.1.3" stretchy="false" xref="S2.SS1.p18.16.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p18.16.m8.1b"><apply id="S2.SS1.p18.16.m8.1.1.cmml" xref="S2.SS1.p18.16.m8.1.1"><times id="S2.SS1.p18.16.m8.1.1.2.cmml" xref="S2.SS1.p18.16.m8.1.1.2"></times><ci id="S2.SS1.p18.16.m8.1.1.3.cmml" xref="S2.SS1.p18.16.m8.1.1.3">𝒪</ci><apply id="S2.SS1.p18.16.m8.1.1.1.1.1.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1"><times id="S2.SS1.p18.16.m8.1.1.1.1.1.1.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.1"></times><ci id="S2.SS1.p18.16.m8.1.1.1.1.1.2.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.2">…</ci><ci id="S2.SS1.p18.16.m8.1.1.1.1.1.3.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.3">𝒶</ci><ci id="S2.SS1.p18.16.m8.1.1.1.1.1.4.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.4">𝒶</ci><ci id="S2.SS1.p18.16.m8.1.1.1.1.1.5.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.5">𝒶</ci><ci id="S2.SS1.p18.16.m8.1.1.1.1.1.6.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.6">𝒷</ci><ci id="S2.SS1.p18.16.m8.1.1.1.1.1.7.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.7">𝒷</ci><ci id="S2.SS1.p18.16.m8.1.1.1.1.1.8.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.8">𝒷</ci><ci id="S2.SS1.p18.16.m8.1.1.1.1.1.9.cmml" xref="S2.SS1.p18.16.m8.1.1.1.1.1.9">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p18.16.m8.1c">\cal O(\ldots aaabbb\ldots)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p18.16.m8.1d">caligraphic_O ( … caligraphic_a caligraphic_a caligraphic_a caligraphic_b caligraphic_b caligraphic_b … )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p19"> <p class="ltx_p" id="S2.SS1.p19.7">Unfortunately, however, there is no topology on <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.SS1.p19.1.m1.1"><semantics id="S2.SS1.p19.1.m1.1a"><mrow id="S2.SS1.p19.1.m1.1.2" xref="S2.SS1.p19.1.m1.1.2.cmml"><mi id="S2.SS1.p19.1.m1.1.2.2" mathvariant="normal" xref="S2.SS1.p19.1.m1.1.2.2.cmml">Σ</mi><mo id="S2.SS1.p19.1.m1.1.2.1" xref="S2.SS1.p19.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p19.1.m1.1.2.3.2" xref="S2.SS1.p19.1.m1.1.2.cmml"><mo id="S2.SS1.p19.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p19.1.m1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p19.1.m1.1.1" xref="S2.SS1.p19.1.m1.1.1.cmml">𝒜</mi><mo id="S2.SS1.p19.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p19.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p19.1.m1.1b"><apply id="S2.SS1.p19.1.m1.1.2.cmml" xref="S2.SS1.p19.1.m1.1.2"><times id="S2.SS1.p19.1.m1.1.2.1.cmml" xref="S2.SS1.p19.1.m1.1.2.1"></times><ci id="S2.SS1.p19.1.m1.1.2.2.cmml" xref="S2.SS1.p19.1.m1.1.2.2">Σ</ci><ci id="S2.SS1.p19.1.m1.1.1.cmml" xref="S2.SS1.p19.1.m1.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p19.1.m1.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p19.1.m1.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math> which is at the same time Hausdorff and for which the support map is continuous: For any <math alttext="a_{1},a_{2}\in\cal A" class="ltx_Math" display="inline" id="S2.SS1.p19.2.m2.2"><semantics id="S2.SS1.p19.2.m2.2a"><mrow id="S2.SS1.p19.2.m2.2.2" xref="S2.SS1.p19.2.m2.2.2.cmml"><mrow id="S2.SS1.p19.2.m2.2.2.2.2" xref="S2.SS1.p19.2.m2.2.2.2.3.cmml"><msub id="S2.SS1.p19.2.m2.1.1.1.1.1" xref="S2.SS1.p19.2.m2.1.1.1.1.1.cmml"><mi id="S2.SS1.p19.2.m2.1.1.1.1.1.2" xref="S2.SS1.p19.2.m2.1.1.1.1.1.2.cmml">a</mi><mn id="S2.SS1.p19.2.m2.1.1.1.1.1.3" xref="S2.SS1.p19.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p19.2.m2.2.2.2.2.3" xref="S2.SS1.p19.2.m2.2.2.2.3.cmml">,</mo><msub id="S2.SS1.p19.2.m2.2.2.2.2.2" xref="S2.SS1.p19.2.m2.2.2.2.2.2.cmml"><mi id="S2.SS1.p19.2.m2.2.2.2.2.2.2" xref="S2.SS1.p19.2.m2.2.2.2.2.2.2.cmml">a</mi><mn id="S2.SS1.p19.2.m2.2.2.2.2.2.3" xref="S2.SS1.p19.2.m2.2.2.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S2.SS1.p19.2.m2.2.2.3" xref="S2.SS1.p19.2.m2.2.2.3.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p19.2.m2.2.2.4" xref="S2.SS1.p19.2.m2.2.2.4.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p19.2.m2.2b"><apply id="S2.SS1.p19.2.m2.2.2.cmml" xref="S2.SS1.p19.2.m2.2.2"><in id="S2.SS1.p19.2.m2.2.2.3.cmml" xref="S2.SS1.p19.2.m2.2.2.3"></in><list id="S2.SS1.p19.2.m2.2.2.2.3.cmml" xref="S2.SS1.p19.2.m2.2.2.2.2"><apply id="S2.SS1.p19.2.m2.1.1.1.1.1.cmml" xref="S2.SS1.p19.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p19.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS1.p19.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p19.2.m2.1.1.1.1.1.2.cmml" xref="S2.SS1.p19.2.m2.1.1.1.1.1.2">𝑎</ci><cn id="S2.SS1.p19.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p19.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS1.p19.2.m2.2.2.2.2.2.cmml" xref="S2.SS1.p19.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p19.2.m2.2.2.2.2.2.1.cmml" xref="S2.SS1.p19.2.m2.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p19.2.m2.2.2.2.2.2.2.cmml" xref="S2.SS1.p19.2.m2.2.2.2.2.2.2">𝑎</ci><cn id="S2.SS1.p19.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p19.2.m2.2.2.2.2.2.3">2</cn></apply></list><ci id="S2.SS1.p19.2.m2.2.2.4.cmml" xref="S2.SS1.p19.2.m2.2.2.4">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p19.2.m2.2c">a_{1},a_{2}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p19.2.m2.2d">italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> and <math alttext="\mu_{n}:=\frac{1}{n}\mu_{a_{1}}+\mu_{a_{2}}" class="ltx_Math" display="inline" id="S2.SS1.p19.3.m3.1"><semantics id="S2.SS1.p19.3.m3.1a"><mrow id="S2.SS1.p19.3.m3.1.1" xref="S2.SS1.p19.3.m3.1.1.cmml"><msub id="S2.SS1.p19.3.m3.1.1.2" xref="S2.SS1.p19.3.m3.1.1.2.cmml"><mi id="S2.SS1.p19.3.m3.1.1.2.2" xref="S2.SS1.p19.3.m3.1.1.2.2.cmml">μ</mi><mi id="S2.SS1.p19.3.m3.1.1.2.3" xref="S2.SS1.p19.3.m3.1.1.2.3.cmml">n</mi></msub><mo id="S2.SS1.p19.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p19.3.m3.1.1.1.cmml">:=</mo><mrow id="S2.SS1.p19.3.m3.1.1.3" xref="S2.SS1.p19.3.m3.1.1.3.cmml"><mrow id="S2.SS1.p19.3.m3.1.1.3.2" xref="S2.SS1.p19.3.m3.1.1.3.2.cmml"><mfrac id="S2.SS1.p19.3.m3.1.1.3.2.2" xref="S2.SS1.p19.3.m3.1.1.3.2.2.cmml"><mn id="S2.SS1.p19.3.m3.1.1.3.2.2.2" xref="S2.SS1.p19.3.m3.1.1.3.2.2.2.cmml">1</mn><mi id="S2.SS1.p19.3.m3.1.1.3.2.2.3" xref="S2.SS1.p19.3.m3.1.1.3.2.2.3.cmml">n</mi></mfrac><mo id="S2.SS1.p19.3.m3.1.1.3.2.1" xref="S2.SS1.p19.3.m3.1.1.3.2.1.cmml">⁢</mo><msub id="S2.SS1.p19.3.m3.1.1.3.2.3" xref="S2.SS1.p19.3.m3.1.1.3.2.3.cmml"><mi id="S2.SS1.p19.3.m3.1.1.3.2.3.2" xref="S2.SS1.p19.3.m3.1.1.3.2.3.2.cmml">μ</mi><msub id="S2.SS1.p19.3.m3.1.1.3.2.3.3" xref="S2.SS1.p19.3.m3.1.1.3.2.3.3.cmml"><mi id="S2.SS1.p19.3.m3.1.1.3.2.3.3.2" xref="S2.SS1.p19.3.m3.1.1.3.2.3.3.2.cmml">a</mi><mn id="S2.SS1.p19.3.m3.1.1.3.2.3.3.3" xref="S2.SS1.p19.3.m3.1.1.3.2.3.3.3.cmml">1</mn></msub></msub></mrow><mo id="S2.SS1.p19.3.m3.1.1.3.1" xref="S2.SS1.p19.3.m3.1.1.3.1.cmml">+</mo><msub id="S2.SS1.p19.3.m3.1.1.3.3" xref="S2.SS1.p19.3.m3.1.1.3.3.cmml"><mi id="S2.SS1.p19.3.m3.1.1.3.3.2" xref="S2.SS1.p19.3.m3.1.1.3.3.2.cmml">μ</mi><msub id="S2.SS1.p19.3.m3.1.1.3.3.3" xref="S2.SS1.p19.3.m3.1.1.3.3.3.cmml"><mi id="S2.SS1.p19.3.m3.1.1.3.3.3.2" xref="S2.SS1.p19.3.m3.1.1.3.3.3.2.cmml">a</mi><mn id="S2.SS1.p19.3.m3.1.1.3.3.3.3" xref="S2.SS1.p19.3.m3.1.1.3.3.3.3.cmml">2</mn></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p19.3.m3.1b"><apply id="S2.SS1.p19.3.m3.1.1.cmml" xref="S2.SS1.p19.3.m3.1.1"><csymbol cd="latexml" id="S2.SS1.p19.3.m3.1.1.1.cmml" xref="S2.SS1.p19.3.m3.1.1.1">assign</csymbol><apply id="S2.SS1.p19.3.m3.1.1.2.cmml" xref="S2.SS1.p19.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p19.3.m3.1.1.2.1.cmml" xref="S2.SS1.p19.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p19.3.m3.1.1.2.2.cmml" xref="S2.SS1.p19.3.m3.1.1.2.2">𝜇</ci><ci id="S2.SS1.p19.3.m3.1.1.2.3.cmml" xref="S2.SS1.p19.3.m3.1.1.2.3">𝑛</ci></apply><apply id="S2.SS1.p19.3.m3.1.1.3.cmml" xref="S2.SS1.p19.3.m3.1.1.3"><plus id="S2.SS1.p19.3.m3.1.1.3.1.cmml" xref="S2.SS1.p19.3.m3.1.1.3.1"></plus><apply id="S2.SS1.p19.3.m3.1.1.3.2.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2"><times id="S2.SS1.p19.3.m3.1.1.3.2.1.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.1"></times><apply id="S2.SS1.p19.3.m3.1.1.3.2.2.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.2"><divide id="S2.SS1.p19.3.m3.1.1.3.2.2.1.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.2"></divide><cn id="S2.SS1.p19.3.m3.1.1.3.2.2.2.cmml" type="integer" xref="S2.SS1.p19.3.m3.1.1.3.2.2.2">1</cn><ci id="S2.SS1.p19.3.m3.1.1.3.2.2.3.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.2.3">𝑛</ci></apply><apply id="S2.SS1.p19.3.m3.1.1.3.2.3.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS1.p19.3.m3.1.1.3.2.3.1.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.3">subscript</csymbol><ci id="S2.SS1.p19.3.m3.1.1.3.2.3.2.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.3.2">𝜇</ci><apply id="S2.SS1.p19.3.m3.1.1.3.2.3.3.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.3.3"><csymbol cd="ambiguous" id="S2.SS1.p19.3.m3.1.1.3.2.3.3.1.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.3.3">subscript</csymbol><ci id="S2.SS1.p19.3.m3.1.1.3.2.3.3.2.cmml" xref="S2.SS1.p19.3.m3.1.1.3.2.3.3.2">𝑎</ci><cn id="S2.SS1.p19.3.m3.1.1.3.2.3.3.3.cmml" type="integer" xref="S2.SS1.p19.3.m3.1.1.3.2.3.3.3">1</cn></apply></apply></apply><apply id="S2.SS1.p19.3.m3.1.1.3.3.cmml" xref="S2.SS1.p19.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p19.3.m3.1.1.3.3.1.cmml" xref="S2.SS1.p19.3.m3.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p19.3.m3.1.1.3.3.2.cmml" xref="S2.SS1.p19.3.m3.1.1.3.3.2">𝜇</ci><apply id="S2.SS1.p19.3.m3.1.1.3.3.3.cmml" xref="S2.SS1.p19.3.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p19.3.m3.1.1.3.3.3.1.cmml" xref="S2.SS1.p19.3.m3.1.1.3.3.3">subscript</csymbol><ci id="S2.SS1.p19.3.m3.1.1.3.3.3.2.cmml" xref="S2.SS1.p19.3.m3.1.1.3.3.3.2">𝑎</ci><cn id="S2.SS1.p19.3.m3.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p19.3.m3.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p19.3.m3.1c">\mu_{n}:=\frac{1}{n}\mu_{a_{1}}+\mu_{a_{2}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p19.3.m3.1d">italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT := divide start_ARG 1 end_ARG start_ARG italic_n end_ARG italic_μ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_μ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for any <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="S2.SS1.p19.4.m4.1"><semantics id="S2.SS1.p19.4.m4.1a"><mrow id="S2.SS1.p19.4.m4.1.1" xref="S2.SS1.p19.4.m4.1.1.cmml"><mi id="S2.SS1.p19.4.m4.1.1.2" xref="S2.SS1.p19.4.m4.1.1.2.cmml">n</mi><mo id="S2.SS1.p19.4.m4.1.1.1" xref="S2.SS1.p19.4.m4.1.1.1.cmml">∈</mo><mi id="S2.SS1.p19.4.m4.1.1.3" xref="S2.SS1.p19.4.m4.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p19.4.m4.1b"><apply id="S2.SS1.p19.4.m4.1.1.cmml" xref="S2.SS1.p19.4.m4.1.1"><in id="S2.SS1.p19.4.m4.1.1.1.cmml" xref="S2.SS1.p19.4.m4.1.1.1"></in><ci id="S2.SS1.p19.4.m4.1.1.2.cmml" xref="S2.SS1.p19.4.m4.1.1.2">𝑛</ci><ci id="S2.SS1.p19.4.m4.1.1.3.cmml" xref="S2.SS1.p19.4.m4.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p19.4.m4.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p19.4.m4.1d">italic_n ∈ blackboard_N</annotation></semantics></math> we have <math alttext="\mbox{Supp}(\mu_{n})=\cal O(a_{1}^{\pm\infty})\cup\cal O(a_{2}^{\pm\infty})" class="ltx_Math" display="inline" id="S2.SS1.p19.5.m5.3"><semantics id="S2.SS1.p19.5.m5.3a"><mrow id="S2.SS1.p19.5.m5.3.3" xref="S2.SS1.p19.5.m5.3.3.cmml"><mrow id="S2.SS1.p19.5.m5.1.1.1" xref="S2.SS1.p19.5.m5.1.1.1.cmml"><mtext id="S2.SS1.p19.5.m5.1.1.1.3" xref="S2.SS1.p19.5.m5.1.1.1.3a.cmml">Supp</mtext><mo id="S2.SS1.p19.5.m5.1.1.1.2" xref="S2.SS1.p19.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p19.5.m5.1.1.1.1.1" xref="S2.SS1.p19.5.m5.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p19.5.m5.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p19.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p19.5.m5.1.1.1.1.1.1" xref="S2.SS1.p19.5.m5.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p19.5.m5.1.1.1.1.1.1.2" xref="S2.SS1.p19.5.m5.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S2.SS1.p19.5.m5.1.1.1.1.1.1.3" xref="S2.SS1.p19.5.m5.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.SS1.p19.5.m5.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p19.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p19.5.m5.3.3.4" xref="S2.SS1.p19.5.m5.3.3.4.cmml">=</mo><mrow id="S2.SS1.p19.5.m5.3.3.3" xref="S2.SS1.p19.5.m5.3.3.3.cmml"><mrow id="S2.SS1.p19.5.m5.2.2.2.1" xref="S2.SS1.p19.5.m5.2.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p19.5.m5.2.2.2.1.3" xref="S2.SS1.p19.5.m5.2.2.2.1.3.cmml">𝒪</mi><mo id="S2.SS1.p19.5.m5.2.2.2.1.2" xref="S2.SS1.p19.5.m5.2.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p19.5.m5.2.2.2.1.1.1" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.cmml"><mo id="S2.SS1.p19.5.m5.2.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.cmml">(</mo><msubsup id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.2" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.2.cmml">𝒶</mi><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.3" mathvariant="script" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.3.cmml">1</mn><mrow id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3.cmml"><mo id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3a" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3.cmml">±</mo><mi id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3.2" mathvariant="normal" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3.2.cmml">∞</mi></mrow></msubsup><mo id="S2.SS1.p19.5.m5.2.2.2.1.1.1.3" stretchy="false" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p19.5.m5.3.3.3.3" xref="S2.SS1.p19.5.m5.3.3.3.3.cmml">∪</mo><mrow id="S2.SS1.p19.5.m5.3.3.3.2" xref="S2.SS1.p19.5.m5.3.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p19.5.m5.3.3.3.2.3" xref="S2.SS1.p19.5.m5.3.3.3.2.3.cmml">𝒪</mi><mo id="S2.SS1.p19.5.m5.3.3.3.2.2" xref="S2.SS1.p19.5.m5.3.3.3.2.2.cmml">⁢</mo><mrow id="S2.SS1.p19.5.m5.3.3.3.2.1.1" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.cmml"><mo id="S2.SS1.p19.5.m5.3.3.3.2.1.1.2" stretchy="false" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.cmml">(</mo><msubsup id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.2" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.2.cmml">𝒶</mi><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.3" mathvariant="script" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.3.cmml">2</mn><mrow id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3.cmml"><mo id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3a" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3.cmml">±</mo><mi id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3.2" mathvariant="normal" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3.2.cmml">∞</mi></mrow></msubsup><mo id="S2.SS1.p19.5.m5.3.3.3.2.1.1.3" stretchy="false" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p19.5.m5.3b"><apply id="S2.SS1.p19.5.m5.3.3.cmml" xref="S2.SS1.p19.5.m5.3.3"><eq id="S2.SS1.p19.5.m5.3.3.4.cmml" xref="S2.SS1.p19.5.m5.3.3.4"></eq><apply id="S2.SS1.p19.5.m5.1.1.1.cmml" xref="S2.SS1.p19.5.m5.1.1.1"><times id="S2.SS1.p19.5.m5.1.1.1.2.cmml" xref="S2.SS1.p19.5.m5.1.1.1.2"></times><ci id="S2.SS1.p19.5.m5.1.1.1.3a.cmml" xref="S2.SS1.p19.5.m5.1.1.1.3"><mtext id="S2.SS1.p19.5.m5.1.1.1.3.cmml" xref="S2.SS1.p19.5.m5.1.1.1.3">Supp</mtext></ci><apply id="S2.SS1.p19.5.m5.1.1.1.1.1.1.cmml" xref="S2.SS1.p19.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p19.5.m5.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p19.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p19.5.m5.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p19.5.m5.1.1.1.1.1.1.2">𝜇</ci><ci id="S2.SS1.p19.5.m5.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p19.5.m5.1.1.1.1.1.1.3">𝑛</ci></apply></apply><apply id="S2.SS1.p19.5.m5.3.3.3.cmml" xref="S2.SS1.p19.5.m5.3.3.3"><union id="S2.SS1.p19.5.m5.3.3.3.3.cmml" xref="S2.SS1.p19.5.m5.3.3.3.3"></union><apply id="S2.SS1.p19.5.m5.2.2.2.1.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1"><times id="S2.SS1.p19.5.m5.2.2.2.1.2.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.2"></times><ci id="S2.SS1.p19.5.m5.2.2.2.1.3.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.3">𝒪</ci><apply id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1">superscript</csymbol><apply id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.1.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.2.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.2">𝒶</ci><cn id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.2.3">1</cn></apply><apply id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3.1.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3.2.cmml" xref="S2.SS1.p19.5.m5.2.2.2.1.1.1.1.3.2"></infinity></apply></apply></apply><apply id="S2.SS1.p19.5.m5.3.3.3.2.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2"><times id="S2.SS1.p19.5.m5.3.3.3.2.2.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.2"></times><ci id="S2.SS1.p19.5.m5.3.3.3.2.3.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.3">𝒪</ci><apply id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.1.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1">superscript</csymbol><apply id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.1.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1">subscript</csymbol><ci id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.2.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.2">𝒶</ci><cn id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.3.cmml" type="integer" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.2.3">2</cn></apply><apply id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3.1.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3.2.cmml" xref="S2.SS1.p19.5.m5.3.3.3.2.1.1.1.3.2"></infinity></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p19.5.m5.3c">\mbox{Supp}(\mu_{n})=\cal O(a_{1}^{\pm\infty})\cup\cal O(a_{2}^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p19.5.m5.3d">Supp ( italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = caligraphic_O ( caligraphic_a start_POSTSUBSCRIPT caligraphic_1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) ∪ caligraphic_O ( caligraphic_a start_POSTSUBSCRIPT caligraphic_2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math> but <math alttext="\lim\mu_{n}=\mu_{a_{2}}" class="ltx_Math" display="inline" id="S2.SS1.p19.6.m6.1"><semantics id="S2.SS1.p19.6.m6.1a"><mrow id="S2.SS1.p19.6.m6.1.1" xref="S2.SS1.p19.6.m6.1.1.cmml"><mrow id="S2.SS1.p19.6.m6.1.1.2" xref="S2.SS1.p19.6.m6.1.1.2.cmml"><mo id="S2.SS1.p19.6.m6.1.1.2.1" rspace="0.167em" xref="S2.SS1.p19.6.m6.1.1.2.1.cmml">lim</mo><msub id="S2.SS1.p19.6.m6.1.1.2.2" xref="S2.SS1.p19.6.m6.1.1.2.2.cmml"><mi id="S2.SS1.p19.6.m6.1.1.2.2.2" xref="S2.SS1.p19.6.m6.1.1.2.2.2.cmml">μ</mi><mi id="S2.SS1.p19.6.m6.1.1.2.2.3" xref="S2.SS1.p19.6.m6.1.1.2.2.3.cmml">n</mi></msub></mrow><mo id="S2.SS1.p19.6.m6.1.1.1" xref="S2.SS1.p19.6.m6.1.1.1.cmml">=</mo><msub id="S2.SS1.p19.6.m6.1.1.3" xref="S2.SS1.p19.6.m6.1.1.3.cmml"><mi id="S2.SS1.p19.6.m6.1.1.3.2" xref="S2.SS1.p19.6.m6.1.1.3.2.cmml">μ</mi><msub id="S2.SS1.p19.6.m6.1.1.3.3" xref="S2.SS1.p19.6.m6.1.1.3.3.cmml"><mi id="S2.SS1.p19.6.m6.1.1.3.3.2" xref="S2.SS1.p19.6.m6.1.1.3.3.2.cmml">a</mi><mn id="S2.SS1.p19.6.m6.1.1.3.3.3" xref="S2.SS1.p19.6.m6.1.1.3.3.3.cmml">2</mn></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p19.6.m6.1b"><apply id="S2.SS1.p19.6.m6.1.1.cmml" xref="S2.SS1.p19.6.m6.1.1"><eq id="S2.SS1.p19.6.m6.1.1.1.cmml" xref="S2.SS1.p19.6.m6.1.1.1"></eq><apply id="S2.SS1.p19.6.m6.1.1.2.cmml" xref="S2.SS1.p19.6.m6.1.1.2"><limit id="S2.SS1.p19.6.m6.1.1.2.1.cmml" xref="S2.SS1.p19.6.m6.1.1.2.1"></limit><apply id="S2.SS1.p19.6.m6.1.1.2.2.cmml" xref="S2.SS1.p19.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p19.6.m6.1.1.2.2.1.cmml" xref="S2.SS1.p19.6.m6.1.1.2.2">subscript</csymbol><ci id="S2.SS1.p19.6.m6.1.1.2.2.2.cmml" xref="S2.SS1.p19.6.m6.1.1.2.2.2">𝜇</ci><ci id="S2.SS1.p19.6.m6.1.1.2.2.3.cmml" xref="S2.SS1.p19.6.m6.1.1.2.2.3">𝑛</ci></apply></apply><apply id="S2.SS1.p19.6.m6.1.1.3.cmml" xref="S2.SS1.p19.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p19.6.m6.1.1.3.1.cmml" xref="S2.SS1.p19.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS1.p19.6.m6.1.1.3.2.cmml" xref="S2.SS1.p19.6.m6.1.1.3.2">𝜇</ci><apply id="S2.SS1.p19.6.m6.1.1.3.3.cmml" xref="S2.SS1.p19.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p19.6.m6.1.1.3.3.1.cmml" xref="S2.SS1.p19.6.m6.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p19.6.m6.1.1.3.3.2.cmml" xref="S2.SS1.p19.6.m6.1.1.3.3.2">𝑎</ci><cn id="S2.SS1.p19.6.m6.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.p19.6.m6.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p19.6.m6.1c">\lim\mu_{n}=\mu_{a_{2}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p19.6.m6.1d">roman_lim italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_μ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and thus <math alttext="\mbox{Supp}(\lim\mu_{n})=\cal O(a_{2}^{\pm\infty})" class="ltx_Math" display="inline" id="S2.SS1.p19.7.m7.2"><semantics id="S2.SS1.p19.7.m7.2a"><mrow id="S2.SS1.p19.7.m7.2.2" xref="S2.SS1.p19.7.m7.2.2.cmml"><mrow id="S2.SS1.p19.7.m7.1.1.1" xref="S2.SS1.p19.7.m7.1.1.1.cmml"><mtext id="S2.SS1.p19.7.m7.1.1.1.3" xref="S2.SS1.p19.7.m7.1.1.1.3a.cmml">Supp</mtext><mo id="S2.SS1.p19.7.m7.1.1.1.2" xref="S2.SS1.p19.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p19.7.m7.1.1.1.1.1" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p19.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p19.7.m7.1.1.1.1.1.1" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p19.7.m7.1.1.1.1.1.1.1" lspace="0em" rspace="0.167em" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.1.cmml">lim</mo><msub id="S2.SS1.p19.7.m7.1.1.1.1.1.1.2" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.cmml"><mi id="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.2" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.2.cmml">μ</mi><mi id="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.3" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.3.cmml">n</mi></msub></mrow><mo id="S2.SS1.p19.7.m7.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p19.7.m7.2.2.3" xref="S2.SS1.p19.7.m7.2.2.3.cmml">=</mo><mrow id="S2.SS1.p19.7.m7.2.2.2" xref="S2.SS1.p19.7.m7.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p19.7.m7.2.2.2.3" xref="S2.SS1.p19.7.m7.2.2.2.3.cmml">𝒪</mi><mo id="S2.SS1.p19.7.m7.2.2.2.2" xref="S2.SS1.p19.7.m7.2.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p19.7.m7.2.2.2.1.1" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.cmml"><mo id="S2.SS1.p19.7.m7.2.2.2.1.1.2" stretchy="false" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.cmml">(</mo><msubsup id="S2.SS1.p19.7.m7.2.2.2.1.1.1" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.2" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.2.cmml">𝒶</mi><mn class="ltx_font_mathcaligraphic" id="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.3" mathvariant="script" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.3.cmml">2</mn><mrow id="S2.SS1.p19.7.m7.2.2.2.1.1.1.3" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.3.cmml"><mo id="S2.SS1.p19.7.m7.2.2.2.1.1.1.3a" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.3.cmml">±</mo><mi id="S2.SS1.p19.7.m7.2.2.2.1.1.1.3.2" mathvariant="normal" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.3.2.cmml">∞</mi></mrow></msubsup><mo id="S2.SS1.p19.7.m7.2.2.2.1.1.3" stretchy="false" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p19.7.m7.2b"><apply id="S2.SS1.p19.7.m7.2.2.cmml" xref="S2.SS1.p19.7.m7.2.2"><eq id="S2.SS1.p19.7.m7.2.2.3.cmml" xref="S2.SS1.p19.7.m7.2.2.3"></eq><apply id="S2.SS1.p19.7.m7.1.1.1.cmml" xref="S2.SS1.p19.7.m7.1.1.1"><times id="S2.SS1.p19.7.m7.1.1.1.2.cmml" xref="S2.SS1.p19.7.m7.1.1.1.2"></times><ci id="S2.SS1.p19.7.m7.1.1.1.3a.cmml" xref="S2.SS1.p19.7.m7.1.1.1.3"><mtext id="S2.SS1.p19.7.m7.1.1.1.3.cmml" xref="S2.SS1.p19.7.m7.1.1.1.3">Supp</mtext></ci><apply id="S2.SS1.p19.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS1.p19.7.m7.1.1.1.1.1"><limit id="S2.SS1.p19.7.m7.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.1"></limit><apply id="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.1.cmml" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.2.cmml" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.2">𝜇</ci><ci id="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.3.cmml" xref="S2.SS1.p19.7.m7.1.1.1.1.1.1.2.3">𝑛</ci></apply></apply></apply><apply id="S2.SS1.p19.7.m7.2.2.2.cmml" xref="S2.SS1.p19.7.m7.2.2.2"><times id="S2.SS1.p19.7.m7.2.2.2.2.cmml" xref="S2.SS1.p19.7.m7.2.2.2.2"></times><ci id="S2.SS1.p19.7.m7.2.2.2.3.cmml" xref="S2.SS1.p19.7.m7.2.2.2.3">𝒪</ci><apply id="S2.SS1.p19.7.m7.2.2.2.1.1.1.cmml" xref="S2.SS1.p19.7.m7.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p19.7.m7.2.2.2.1.1.1.1.cmml" xref="S2.SS1.p19.7.m7.2.2.2.1.1">superscript</csymbol><apply id="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.cmml" xref="S2.SS1.p19.7.m7.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.1.cmml" xref="S2.SS1.p19.7.m7.2.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.2.cmml" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.2">𝒶</ci><cn id="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.3.cmml" type="integer" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.2.3">2</cn></apply><apply id="S2.SS1.p19.7.m7.2.2.2.1.1.1.3.cmml" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p19.7.m7.2.2.2.1.1.1.3.1.cmml" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.SS1.p19.7.m7.2.2.2.1.1.1.3.2.cmml" xref="S2.SS1.p19.7.m7.2.2.2.1.1.1.3.2"></infinity></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p19.7.m7.2c">\mbox{Supp}(\lim\mu_{n})=\cal O(a_{2}^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p19.7.m7.2d">Supp ( roman_lim italic_μ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = caligraphic_O ( caligraphic_a start_POSTSUBSCRIPT caligraphic_2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p20"> <p class="ltx_p" id="S2.SS1.p20.11">An invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p20.1.m1.1"><semantics id="S2.SS1.p20.1.m1.1a"><mi id="S2.SS1.p20.1.m1.1.1" xref="S2.SS1.p20.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.1.m1.1b"><ci id="S2.SS1.p20.1.m1.1.1.cmml" xref="S2.SS1.p20.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.1.m1.1d">italic_μ</annotation></semantics></math> on some subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p20.2.m2.1"><semantics id="S2.SS1.p20.2.m2.1a"><mrow id="S2.SS1.p20.2.m2.1.1" xref="S2.SS1.p20.2.m2.1.1.cmml"><mi id="S2.SS1.p20.2.m2.1.1.2" xref="S2.SS1.p20.2.m2.1.1.2.cmml">X</mi><mo id="S2.SS1.p20.2.m2.1.1.1" xref="S2.SS1.p20.2.m2.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p20.2.m2.1.1.3" xref="S2.SS1.p20.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.2.m2.1.1.3.2" xref="S2.SS1.p20.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p20.2.m2.1.1.3.3" xref="S2.SS1.p20.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.2.m2.1b"><apply id="S2.SS1.p20.2.m2.1.1.cmml" xref="S2.SS1.p20.2.m2.1.1"><subset id="S2.SS1.p20.2.m2.1.1.1.cmml" xref="S2.SS1.p20.2.m2.1.1.1"></subset><ci id="S2.SS1.p20.2.m2.1.1.2.cmml" xref="S2.SS1.p20.2.m2.1.1.2">𝑋</ci><apply id="S2.SS1.p20.2.m2.1.1.3.cmml" xref="S2.SS1.p20.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p20.2.m2.1.1.3.1.cmml" xref="S2.SS1.p20.2.m2.1.1.3">superscript</csymbol><ci id="S2.SS1.p20.2.m2.1.1.3.2.cmml" xref="S2.SS1.p20.2.m2.1.1.3.2">𝒜</ci><ci id="S2.SS1.p20.2.m2.1.1.3.3.cmml" xref="S2.SS1.p20.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> extends canonically to an invariant measure on all of <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p20.3.m3.1"><semantics id="S2.SS1.p20.3.m3.1a"><msup id="S2.SS1.p20.3.m3.1.1" xref="S2.SS1.p20.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.3.m3.1.1.2" xref="S2.SS1.p20.3.m3.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p20.3.m3.1.1.3" xref="S2.SS1.p20.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.3.m3.1b"><apply id="S2.SS1.p20.3.m3.1.1.cmml" xref="S2.SS1.p20.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p20.3.m3.1.1.1.cmml" xref="S2.SS1.p20.3.m3.1.1">superscript</csymbol><ci id="S2.SS1.p20.3.m3.1.1.2.cmml" xref="S2.SS1.p20.3.m3.1.1.2">𝒜</ci><ci id="S2.SS1.p20.3.m3.1.1.3.cmml" xref="S2.SS1.p20.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. For the corresponding weight functions this extension is obtained by simply declaring <math alttext="\omega_{\mu}(w)=0" class="ltx_Math" display="inline" id="S2.SS1.p20.4.m4.1"><semantics id="S2.SS1.p20.4.m4.1a"><mrow id="S2.SS1.p20.4.m4.1.2" xref="S2.SS1.p20.4.m4.1.2.cmml"><mrow id="S2.SS1.p20.4.m4.1.2.2" xref="S2.SS1.p20.4.m4.1.2.2.cmml"><msub id="S2.SS1.p20.4.m4.1.2.2.2" xref="S2.SS1.p20.4.m4.1.2.2.2.cmml"><mi id="S2.SS1.p20.4.m4.1.2.2.2.2" xref="S2.SS1.p20.4.m4.1.2.2.2.2.cmml">ω</mi><mi id="S2.SS1.p20.4.m4.1.2.2.2.3" xref="S2.SS1.p20.4.m4.1.2.2.2.3.cmml">μ</mi></msub><mo id="S2.SS1.p20.4.m4.1.2.2.1" xref="S2.SS1.p20.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p20.4.m4.1.2.2.3.2" xref="S2.SS1.p20.4.m4.1.2.2.cmml"><mo id="S2.SS1.p20.4.m4.1.2.2.3.2.1" stretchy="false" xref="S2.SS1.p20.4.m4.1.2.2.cmml">(</mo><mi id="S2.SS1.p20.4.m4.1.1" xref="S2.SS1.p20.4.m4.1.1.cmml">w</mi><mo id="S2.SS1.p20.4.m4.1.2.2.3.2.2" stretchy="false" xref="S2.SS1.p20.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p20.4.m4.1.2.1" xref="S2.SS1.p20.4.m4.1.2.1.cmml">=</mo><mn id="S2.SS1.p20.4.m4.1.2.3" xref="S2.SS1.p20.4.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.4.m4.1b"><apply id="S2.SS1.p20.4.m4.1.2.cmml" xref="S2.SS1.p20.4.m4.1.2"><eq id="S2.SS1.p20.4.m4.1.2.1.cmml" xref="S2.SS1.p20.4.m4.1.2.1"></eq><apply id="S2.SS1.p20.4.m4.1.2.2.cmml" xref="S2.SS1.p20.4.m4.1.2.2"><times id="S2.SS1.p20.4.m4.1.2.2.1.cmml" xref="S2.SS1.p20.4.m4.1.2.2.1"></times><apply id="S2.SS1.p20.4.m4.1.2.2.2.cmml" xref="S2.SS1.p20.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p20.4.m4.1.2.2.2.1.cmml" xref="S2.SS1.p20.4.m4.1.2.2.2">subscript</csymbol><ci id="S2.SS1.p20.4.m4.1.2.2.2.2.cmml" xref="S2.SS1.p20.4.m4.1.2.2.2.2">𝜔</ci><ci id="S2.SS1.p20.4.m4.1.2.2.2.3.cmml" xref="S2.SS1.p20.4.m4.1.2.2.2.3">𝜇</ci></apply><ci id="S2.SS1.p20.4.m4.1.1.cmml" xref="S2.SS1.p20.4.m4.1.1">𝑤</ci></apply><cn id="S2.SS1.p20.4.m4.1.2.3.cmml" type="integer" xref="S2.SS1.p20.4.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.4.m4.1c">\omega_{\mu}(w)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.4.m4.1d">italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_w ) = 0</annotation></semantics></math> for any <math alttext="w\notin\cal L(X)" class="ltx_Math" display="inline" id="S2.SS1.p20.5.m5.1"><semantics id="S2.SS1.p20.5.m5.1a"><mrow id="S2.SS1.p20.5.m5.1.2" xref="S2.SS1.p20.5.m5.1.2.cmml"><mi id="S2.SS1.p20.5.m5.1.2.2" xref="S2.SS1.p20.5.m5.1.2.2.cmml">w</mi><mo id="S2.SS1.p20.5.m5.1.2.1" xref="S2.SS1.p20.5.m5.1.2.1.cmml">∉</mo><mrow id="S2.SS1.p20.5.m5.1.2.3" xref="S2.SS1.p20.5.m5.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.5.m5.1.2.3.2" xref="S2.SS1.p20.5.m5.1.2.3.2.cmml">ℒ</mi><mo id="S2.SS1.p20.5.m5.1.2.3.1" xref="S2.SS1.p20.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p20.5.m5.1.2.3.3.2" xref="S2.SS1.p20.5.m5.1.2.3.cmml"><mo id="S2.SS1.p20.5.m5.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p20.5.m5.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.5.m5.1.1" xref="S2.SS1.p20.5.m5.1.1.cmml">𝒳</mi><mo id="S2.SS1.p20.5.m5.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.p20.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.5.m5.1b"><apply id="S2.SS1.p20.5.m5.1.2.cmml" xref="S2.SS1.p20.5.m5.1.2"><notin id="S2.SS1.p20.5.m5.1.2.1.cmml" xref="S2.SS1.p20.5.m5.1.2.1"></notin><ci id="S2.SS1.p20.5.m5.1.2.2.cmml" xref="S2.SS1.p20.5.m5.1.2.2">𝑤</ci><apply id="S2.SS1.p20.5.m5.1.2.3.cmml" xref="S2.SS1.p20.5.m5.1.2.3"><times id="S2.SS1.p20.5.m5.1.2.3.1.cmml" xref="S2.SS1.p20.5.m5.1.2.3.1"></times><ci id="S2.SS1.p20.5.m5.1.2.3.2.cmml" xref="S2.SS1.p20.5.m5.1.2.3.2">ℒ</ci><ci id="S2.SS1.p20.5.m5.1.1.cmml" xref="S2.SS1.p20.5.m5.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.5.m5.1c">w\notin\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.5.m5.1d">italic_w ∉ caligraphic_L ( caligraphic_X )</annotation></semantics></math>. We will notationally not distinguish between <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p20.6.m6.1"><semantics id="S2.SS1.p20.6.m6.1a"><mi id="S2.SS1.p20.6.m6.1.1" xref="S2.SS1.p20.6.m6.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.6.m6.1b"><ci id="S2.SS1.p20.6.m6.1.1.cmml" xref="S2.SS1.p20.6.m6.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.6.m6.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.6.m6.1d">italic_μ</annotation></semantics></math> and its canonical extension, or conversely, between <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p20.7.m7.1"><semantics id="S2.SS1.p20.7.m7.1a"><mi id="S2.SS1.p20.7.m7.1.1" xref="S2.SS1.p20.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.7.m7.1b"><ci id="S2.SS1.p20.7.m7.1.1.cmml" xref="S2.SS1.p20.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.7.m7.1d">italic_μ</annotation></semantics></math> and the restriction of <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p20.8.m8.1"><semantics id="S2.SS1.p20.8.m8.1a"><mi id="S2.SS1.p20.8.m8.1.1" xref="S2.SS1.p20.8.m8.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.8.m8.1b"><ci id="S2.SS1.p20.8.m8.1.1.cmml" xref="S2.SS1.p20.8.m8.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.8.m8.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.8.m8.1d">italic_μ</annotation></semantics></math> to its support. In particular, for any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS1.p20.9.m9.1"><semantics id="S2.SS1.p20.9.m9.1a"><mrow id="S2.SS1.p20.9.m9.1.1" xref="S2.SS1.p20.9.m9.1.1.cmml"><mi id="S2.SS1.p20.9.m9.1.1.2" xref="S2.SS1.p20.9.m9.1.1.2.cmml">X</mi><mo id="S2.SS1.p20.9.m9.1.1.1" xref="S2.SS1.p20.9.m9.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p20.9.m9.1.1.3" xref="S2.SS1.p20.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.9.m9.1.1.3.2" xref="S2.SS1.p20.9.m9.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS1.p20.9.m9.1.1.3.3" xref="S2.SS1.p20.9.m9.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.9.m9.1b"><apply id="S2.SS1.p20.9.m9.1.1.cmml" xref="S2.SS1.p20.9.m9.1.1"><subset id="S2.SS1.p20.9.m9.1.1.1.cmml" xref="S2.SS1.p20.9.m9.1.1.1"></subset><ci id="S2.SS1.p20.9.m9.1.1.2.cmml" xref="S2.SS1.p20.9.m9.1.1.2">𝑋</ci><apply id="S2.SS1.p20.9.m9.1.1.3.cmml" xref="S2.SS1.p20.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p20.9.m9.1.1.3.1.cmml" xref="S2.SS1.p20.9.m9.1.1.3">superscript</csymbol><ci id="S2.SS1.p20.9.m9.1.1.3.2.cmml" xref="S2.SS1.p20.9.m9.1.1.3.2">𝒜</ci><ci id="S2.SS1.p20.9.m9.1.1.3.3.cmml" xref="S2.SS1.p20.9.m9.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.9.m9.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.9.m9.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> we understand <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S2.SS1.p20.10.m10.1"><semantics id="S2.SS1.p20.10.m10.1a"><mrow id="S2.SS1.p20.10.m10.1.2" xref="S2.SS1.p20.10.m10.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.10.m10.1.2.2" xref="S2.SS1.p20.10.m10.1.2.2.cmml">ℳ</mi><mo id="S2.SS1.p20.10.m10.1.2.1" xref="S2.SS1.p20.10.m10.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p20.10.m10.1.2.3.2" xref="S2.SS1.p20.10.m10.1.2.cmml"><mo id="S2.SS1.p20.10.m10.1.2.3.2.1" stretchy="false" xref="S2.SS1.p20.10.m10.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.10.m10.1.1" xref="S2.SS1.p20.10.m10.1.1.cmml">𝒳</mi><mo id="S2.SS1.p20.10.m10.1.2.3.2.2" stretchy="false" xref="S2.SS1.p20.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.10.m10.1b"><apply id="S2.SS1.p20.10.m10.1.2.cmml" xref="S2.SS1.p20.10.m10.1.2"><times id="S2.SS1.p20.10.m10.1.2.1.cmml" xref="S2.SS1.p20.10.m10.1.2.1"></times><ci id="S2.SS1.p20.10.m10.1.2.2.cmml" xref="S2.SS1.p20.10.m10.1.2.2">ℳ</ci><ci id="S2.SS1.p20.10.m10.1.1.cmml" xref="S2.SS1.p20.10.m10.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.10.m10.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.10.m10.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> as canonical subset of <math alttext="\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S2.SS1.p20.11.m11.1"><semantics id="S2.SS1.p20.11.m11.1a"><mrow id="S2.SS1.p20.11.m11.1.1" xref="S2.SS1.p20.11.m11.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.11.m11.1.1.3" xref="S2.SS1.p20.11.m11.1.1.3.cmml">ℳ</mi><mo id="S2.SS1.p20.11.m11.1.1.2" xref="S2.SS1.p20.11.m11.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p20.11.m11.1.1.1.1" xref="S2.SS1.p20.11.m11.1.1.1.1.1.cmml"><mo id="S2.SS1.p20.11.m11.1.1.1.1.2" stretchy="false" xref="S2.SS1.p20.11.m11.1.1.1.1.1.cmml">(</mo><msup id="S2.SS1.p20.11.m11.1.1.1.1.1" xref="S2.SS1.p20.11.m11.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p20.11.m11.1.1.1.1.1.2" xref="S2.SS1.p20.11.m11.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S2.SS1.p20.11.m11.1.1.1.1.1.3" xref="S2.SS1.p20.11.m11.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S2.SS1.p20.11.m11.1.1.1.1.3" stretchy="false" xref="S2.SS1.p20.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p20.11.m11.1b"><apply id="S2.SS1.p20.11.m11.1.1.cmml" xref="S2.SS1.p20.11.m11.1.1"><times id="S2.SS1.p20.11.m11.1.1.2.cmml" xref="S2.SS1.p20.11.m11.1.1.2"></times><ci id="S2.SS1.p20.11.m11.1.1.3.cmml" xref="S2.SS1.p20.11.m11.1.1.3">ℳ</ci><apply id="S2.SS1.p20.11.m11.1.1.1.1.1.cmml" xref="S2.SS1.p20.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p20.11.m11.1.1.1.1.1.1.cmml" xref="S2.SS1.p20.11.m11.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p20.11.m11.1.1.1.1.1.2.cmml" xref="S2.SS1.p20.11.m11.1.1.1.1.1.2">𝒜</ci><ci id="S2.SS1.p20.11.m11.1.1.1.1.1.3.cmml" xref="S2.SS1.p20.11.m11.1.1.1.1.1.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p20.11.m11.1c">\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p20.11.m11.1d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2. </span>“Not so standard” basic facts and terminology</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1"></p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.1">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mrow id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml"><mi id="S2.SS2.p2.1.m1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.2.cmml">σ</mi><mo id="S2.SS2.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S2.SS2.p2.1.m1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.3.cmml"><msup id="S2.SS2.p2.1.m1.1.1.3.2" xref="S2.SS2.p2.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.1.m1.1.1.3.2.2" xref="S2.SS2.p2.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.SS2.p2.1.m1.1.1.3.2.3" xref="S2.SS2.p2.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.SS2.p2.1.m1.1.1.3.1" stretchy="false" xref="S2.SS2.p2.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.SS2.p2.1.m1.1.1.3.3" xref="S2.SS2.p2.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.1.m1.1.1.3.3.2" xref="S2.SS2.p2.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.SS2.p2.1.m1.1.1.3.3.3" xref="S2.SS2.p2.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1"><ci id="S2.SS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1.1">:</ci><ci id="S2.SS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2">𝜎</ci><apply id="S2.SS2.p2.1.m1.1.1.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3"><ci id="S2.SS2.p2.1.m1.1.1.3.1.cmml" xref="S2.SS2.p2.1.m1.1.1.3.1">→</ci><apply id="S2.SS2.p2.1.m1.1.1.3.2.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.3.2.1.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.3.2.2.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2.2">𝒜</ci><times id="S2.SS2.p2.1.m1.1.1.3.2.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2.3"></times></apply><apply id="S2.SS2.p2.1.m1.1.1.3.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.3.3.1.cmml" xref="S2.SS2.p2.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.3.3.2.cmml" xref="S2.SS2.p2.1.m1.1.1.3.3.2">ℬ</ci><times id="S2.SS2.p2.1.m1.1.1.3.3.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> a non-erasing monoid morphism. Then there is a canonically induced map</p> <table class="ltx_equation ltx_eqn_table" id="S2.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.8)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal B^{\mathbb{Z}}" class="ltx_Math" display="block" id="S2.E8.m1.1"><semantics id="S2.E8.m1.1a"><mrow id="S2.E8.m1.1.1" xref="S2.E8.m1.1.1.cmml"><msup id="S2.E8.m1.1.1.2" xref="S2.E8.m1.1.1.2.cmml"><mi id="S2.E8.m1.1.1.2.2" xref="S2.E8.m1.1.1.2.2.cmml">σ</mi><mi id="S2.E8.m1.1.1.2.3" xref="S2.E8.m1.1.1.2.3.cmml">ℤ</mi></msup><mo id="S2.E8.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.E8.m1.1.1.1.cmml">:</mo><mrow id="S2.E8.m1.1.1.3" xref="S2.E8.m1.1.1.3.cmml"><msup id="S2.E8.m1.1.1.3.2" xref="S2.E8.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E8.m1.1.1.3.2.2" xref="S2.E8.m1.1.1.3.2.2.cmml">𝒜</mi><mi id="S2.E8.m1.1.1.3.2.3" xref="S2.E8.m1.1.1.3.2.3.cmml">ℤ</mi></msup><mo id="S2.E8.m1.1.1.3.1" stretchy="false" xref="S2.E8.m1.1.1.3.1.cmml">→</mo><msup id="S2.E8.m1.1.1.3.3" xref="S2.E8.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E8.m1.1.1.3.3.2" xref="S2.E8.m1.1.1.3.3.2.cmml">ℬ</mi><mi id="S2.E8.m1.1.1.3.3.3" xref="S2.E8.m1.1.1.3.3.3.cmml">ℤ</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E8.m1.1b"><apply id="S2.E8.m1.1.1.cmml" xref="S2.E8.m1.1.1"><ci id="S2.E8.m1.1.1.1.cmml" xref="S2.E8.m1.1.1.1">:</ci><apply id="S2.E8.m1.1.1.2.cmml" xref="S2.E8.m1.1.1.2"><csymbol cd="ambiguous" id="S2.E8.m1.1.1.2.1.cmml" xref="S2.E8.m1.1.1.2">superscript</csymbol><ci id="S2.E8.m1.1.1.2.2.cmml" xref="S2.E8.m1.1.1.2.2">𝜎</ci><ci id="S2.E8.m1.1.1.2.3.cmml" xref="S2.E8.m1.1.1.2.3">ℤ</ci></apply><apply id="S2.E8.m1.1.1.3.cmml" xref="S2.E8.m1.1.1.3"><ci id="S2.E8.m1.1.1.3.1.cmml" xref="S2.E8.m1.1.1.3.1">→</ci><apply id="S2.E8.m1.1.1.3.2.cmml" xref="S2.E8.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.E8.m1.1.1.3.2.1.cmml" xref="S2.E8.m1.1.1.3.2">superscript</csymbol><ci id="S2.E8.m1.1.1.3.2.2.cmml" xref="S2.E8.m1.1.1.3.2.2">𝒜</ci><ci id="S2.E8.m1.1.1.3.2.3.cmml" xref="S2.E8.m1.1.1.3.2.3">ℤ</ci></apply><apply id="S2.E8.m1.1.1.3.3.cmml" xref="S2.E8.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.E8.m1.1.1.3.3.1.cmml" xref="S2.E8.m1.1.1.3.3">superscript</csymbol><ci id="S2.E8.m1.1.1.3.3.2.cmml" xref="S2.E8.m1.1.1.3.3.2">ℬ</ci><ci id="S2.E8.m1.1.1.3.3.3.cmml" xref="S2.E8.m1.1.1.3.3.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.1c">\sigma^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT : caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.4">that maps a biinfinite word <math alttext="{\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots\in\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m1.1"><semantics id="S2.SS2.p2.2.m1.1a"><mrow id="S2.SS2.p2.2.m1.1.1" xref="S2.SS2.p2.2.m1.1.1.cmml"><mi id="S2.SS2.p2.2.m1.1.1.2" xref="S2.SS2.p2.2.m1.1.1.2.cmml">𝐱</mi><mo id="S2.SS2.p2.2.m1.1.1.3" xref="S2.SS2.p2.2.m1.1.1.3.cmml">=</mo><mrow id="S2.SS2.p2.2.m1.1.1.4" xref="S2.SS2.p2.2.m1.1.1.4.cmml"><mi id="S2.SS2.p2.2.m1.1.1.4.2" mathvariant="normal" xref="S2.SS2.p2.2.m1.1.1.4.2.cmml">…</mi><mo id="S2.SS2.p2.2.m1.1.1.4.1" xref="S2.SS2.p2.2.m1.1.1.4.1.cmml">⁢</mo><msub id="S2.SS2.p2.2.m1.1.1.4.3" xref="S2.SS2.p2.2.m1.1.1.4.3.cmml"><mi id="S2.SS2.p2.2.m1.1.1.4.3.2" 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xref="S2.SS2.p2.2.m1.1.1.4.6">…</ci></apply></apply><apply id="S2.SS2.p2.2.m1.1.1c.cmml" xref="S2.SS2.p2.2.m1.1.1"><in id="S2.SS2.p2.2.m1.1.1.5.cmml" xref="S2.SS2.p2.2.m1.1.1.5"></in><share href="https://arxiv.org/html/2211.11234v4#S2.SS2.p2.2.m1.1.1.4.cmml" id="S2.SS2.p2.2.m1.1.1d.cmml" xref="S2.SS2.p2.2.m1.1.1"></share><apply id="S2.SS2.p2.2.m1.1.1.6.cmml" xref="S2.SS2.p2.2.m1.1.1.6"><csymbol cd="ambiguous" id="S2.SS2.p2.2.m1.1.1.6.1.cmml" xref="S2.SS2.p2.2.m1.1.1.6">superscript</csymbol><ci id="S2.SS2.p2.2.m1.1.1.6.2.cmml" xref="S2.SS2.p2.2.m1.1.1.6.2">𝒜</ci><ci id="S2.SS2.p2.2.m1.1.1.6.3.cmml" xref="S2.SS2.p2.2.m1.1.1.6.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m1.1c">{\bf x}=\ldots x_{i-1}x_{i}x_{i+1}\ldots\in\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m1.1d">bold_x = … italic_x start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT … ∈ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> to the biinfinite word <math alttext="{\bf y}=\ldots y_{j-1}y_{j}y_{j+1}\ldots\in\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m2.1"><semantics id="S2.SS2.p2.3.m2.1a"><mrow id="S2.SS2.p2.3.m2.1.1" xref="S2.SS2.p2.3.m2.1.1.cmml"><mi id="S2.SS2.p2.3.m2.1.1.2" xref="S2.SS2.p2.3.m2.1.1.2.cmml">𝐲</mi><mo id="S2.SS2.p2.3.m2.1.1.3" xref="S2.SS2.p2.3.m2.1.1.3.cmml">=</mo><mrow id="S2.SS2.p2.3.m2.1.1.4" xref="S2.SS2.p2.3.m2.1.1.4.cmml"><mi id="S2.SS2.p2.3.m2.1.1.4.2" mathvariant="normal" xref="S2.SS2.p2.3.m2.1.1.4.2.cmml">…</mi><mo id="S2.SS2.p2.3.m2.1.1.4.1" xref="S2.SS2.p2.3.m2.1.1.4.1.cmml">⁢</mo><msub id="S2.SS2.p2.3.m2.1.1.4.3" xref="S2.SS2.p2.3.m2.1.1.4.3.cmml"><mi id="S2.SS2.p2.3.m2.1.1.4.3.2" xref="S2.SS2.p2.3.m2.1.1.4.3.2.cmml">y</mi><mrow id="S2.SS2.p2.3.m2.1.1.4.3.3" xref="S2.SS2.p2.3.m2.1.1.4.3.3.cmml"><mi id="S2.SS2.p2.3.m2.1.1.4.3.3.2" xref="S2.SS2.p2.3.m2.1.1.4.3.3.2.cmml">j</mi><mo id="S2.SS2.p2.3.m2.1.1.4.3.3.1" xref="S2.SS2.p2.3.m2.1.1.4.3.3.1.cmml">−</mo><mn id="S2.SS2.p2.3.m2.1.1.4.3.3.3" xref="S2.SS2.p2.3.m2.1.1.4.3.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS2.p2.3.m2.1.1.4.1a" xref="S2.SS2.p2.3.m2.1.1.4.1.cmml">⁢</mo><msub id="S2.SS2.p2.3.m2.1.1.4.4" xref="S2.SS2.p2.3.m2.1.1.4.4.cmml"><mi id="S2.SS2.p2.3.m2.1.1.4.4.2" xref="S2.SS2.p2.3.m2.1.1.4.4.2.cmml">y</mi><mi id="S2.SS2.p2.3.m2.1.1.4.4.3" xref="S2.SS2.p2.3.m2.1.1.4.4.3.cmml">j</mi></msub><mo id="S2.SS2.p2.3.m2.1.1.4.1b" xref="S2.SS2.p2.3.m2.1.1.4.1.cmml">⁢</mo><msub id="S2.SS2.p2.3.m2.1.1.4.5" xref="S2.SS2.p2.3.m2.1.1.4.5.cmml"><mi id="S2.SS2.p2.3.m2.1.1.4.5.2" xref="S2.SS2.p2.3.m2.1.1.4.5.2.cmml">y</mi><mrow id="S2.SS2.p2.3.m2.1.1.4.5.3" xref="S2.SS2.p2.3.m2.1.1.4.5.3.cmml"><mi id="S2.SS2.p2.3.m2.1.1.4.5.3.2" xref="S2.SS2.p2.3.m2.1.1.4.5.3.2.cmml">j</mi><mo id="S2.SS2.p2.3.m2.1.1.4.5.3.1" xref="S2.SS2.p2.3.m2.1.1.4.5.3.1.cmml">+</mo><mn id="S2.SS2.p2.3.m2.1.1.4.5.3.3" xref="S2.SS2.p2.3.m2.1.1.4.5.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS2.p2.3.m2.1.1.4.1c" xref="S2.SS2.p2.3.m2.1.1.4.1.cmml">⁢</mo><mi id="S2.SS2.p2.3.m2.1.1.4.6" mathvariant="normal" xref="S2.SS2.p2.3.m2.1.1.4.6.cmml">…</mi></mrow><mo id="S2.SS2.p2.3.m2.1.1.5" xref="S2.SS2.p2.3.m2.1.1.5.cmml">∈</mo><msup id="S2.SS2.p2.3.m2.1.1.6" xref="S2.SS2.p2.3.m2.1.1.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.3.m2.1.1.6.2" xref="S2.SS2.p2.3.m2.1.1.6.2.cmml">ℬ</mi><mi id="S2.SS2.p2.3.m2.1.1.6.3" xref="S2.SS2.p2.3.m2.1.1.6.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m2.1b"><apply id="S2.SS2.p2.3.m2.1.1.cmml" xref="S2.SS2.p2.3.m2.1.1"><and id="S2.SS2.p2.3.m2.1.1a.cmml" xref="S2.SS2.p2.3.m2.1.1"></and><apply id="S2.SS2.p2.3.m2.1.1b.cmml" xref="S2.SS2.p2.3.m2.1.1"><eq id="S2.SS2.p2.3.m2.1.1.3.cmml" xref="S2.SS2.p2.3.m2.1.1.3"></eq><ci id="S2.SS2.p2.3.m2.1.1.2.cmml" xref="S2.SS2.p2.3.m2.1.1.2">𝐲</ci><apply id="S2.SS2.p2.3.m2.1.1.4.cmml" xref="S2.SS2.p2.3.m2.1.1.4"><times id="S2.SS2.p2.3.m2.1.1.4.1.cmml" xref="S2.SS2.p2.3.m2.1.1.4.1"></times><ci id="S2.SS2.p2.3.m2.1.1.4.2.cmml" xref="S2.SS2.p2.3.m2.1.1.4.2">…</ci><apply id="S2.SS2.p2.3.m2.1.1.4.3.cmml" xref="S2.SS2.p2.3.m2.1.1.4.3"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m2.1.1.4.3.1.cmml" xref="S2.SS2.p2.3.m2.1.1.4.3">subscript</csymbol><ci id="S2.SS2.p2.3.m2.1.1.4.3.2.cmml" xref="S2.SS2.p2.3.m2.1.1.4.3.2">𝑦</ci><apply id="S2.SS2.p2.3.m2.1.1.4.3.3.cmml" xref="S2.SS2.p2.3.m2.1.1.4.3.3"><minus id="S2.SS2.p2.3.m2.1.1.4.3.3.1.cmml" xref="S2.SS2.p2.3.m2.1.1.4.3.3.1"></minus><ci id="S2.SS2.p2.3.m2.1.1.4.3.3.2.cmml" xref="S2.SS2.p2.3.m2.1.1.4.3.3.2">𝑗</ci><cn id="S2.SS2.p2.3.m2.1.1.4.3.3.3.cmml" type="integer" xref="S2.SS2.p2.3.m2.1.1.4.3.3.3">1</cn></apply></apply><apply id="S2.SS2.p2.3.m2.1.1.4.4.cmml" xref="S2.SS2.p2.3.m2.1.1.4.4"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m2.1.1.4.4.1.cmml" xref="S2.SS2.p2.3.m2.1.1.4.4">subscript</csymbol><ci id="S2.SS2.p2.3.m2.1.1.4.4.2.cmml" xref="S2.SS2.p2.3.m2.1.1.4.4.2">𝑦</ci><ci id="S2.SS2.p2.3.m2.1.1.4.4.3.cmml" xref="S2.SS2.p2.3.m2.1.1.4.4.3">𝑗</ci></apply><apply id="S2.SS2.p2.3.m2.1.1.4.5.cmml" xref="S2.SS2.p2.3.m2.1.1.4.5"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m2.1.1.4.5.1.cmml" xref="S2.SS2.p2.3.m2.1.1.4.5">subscript</csymbol><ci id="S2.SS2.p2.3.m2.1.1.4.5.2.cmml" xref="S2.SS2.p2.3.m2.1.1.4.5.2">𝑦</ci><apply id="S2.SS2.p2.3.m2.1.1.4.5.3.cmml" xref="S2.SS2.p2.3.m2.1.1.4.5.3"><plus id="S2.SS2.p2.3.m2.1.1.4.5.3.1.cmml" xref="S2.SS2.p2.3.m2.1.1.4.5.3.1"></plus><ci id="S2.SS2.p2.3.m2.1.1.4.5.3.2.cmml" xref="S2.SS2.p2.3.m2.1.1.4.5.3.2">𝑗</ci><cn id="S2.SS2.p2.3.m2.1.1.4.5.3.3.cmml" type="integer" xref="S2.SS2.p2.3.m2.1.1.4.5.3.3">1</cn></apply></apply><ci id="S2.SS2.p2.3.m2.1.1.4.6.cmml" xref="S2.SS2.p2.3.m2.1.1.4.6">…</ci></apply></apply><apply id="S2.SS2.p2.3.m2.1.1c.cmml" xref="S2.SS2.p2.3.m2.1.1"><in id="S2.SS2.p2.3.m2.1.1.5.cmml" xref="S2.SS2.p2.3.m2.1.1.5"></in><share href="https://arxiv.org/html/2211.11234v4#S2.SS2.p2.3.m2.1.1.4.cmml" id="S2.SS2.p2.3.m2.1.1d.cmml" xref="S2.SS2.p2.3.m2.1.1"></share><apply id="S2.SS2.p2.3.m2.1.1.6.cmml" xref="S2.SS2.p2.3.m2.1.1.6"><csymbol cd="ambiguous" id="S2.SS2.p2.3.m2.1.1.6.1.cmml" xref="S2.SS2.p2.3.m2.1.1.6">superscript</csymbol><ci id="S2.SS2.p2.3.m2.1.1.6.2.cmml" xref="S2.SS2.p2.3.m2.1.1.6.2">ℬ</ci><ci id="S2.SS2.p2.3.m2.1.1.6.3.cmml" xref="S2.SS2.p2.3.m2.1.1.6.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m2.1c">{\bf y}=\ldots y_{j-1}y_{j}y_{j+1}\ldots\in\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m2.1d">bold_y = … italic_y start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT … ∈ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> obtained from concatenating the <math alttext="\sigma(x_{i})" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m3.1"><semantics id="S2.SS2.p2.4.m3.1a"><mrow id="S2.SS2.p2.4.m3.1.1" xref="S2.SS2.p2.4.m3.1.1.cmml"><mi id="S2.SS2.p2.4.m3.1.1.3" xref="S2.SS2.p2.4.m3.1.1.3.cmml">σ</mi><mo id="S2.SS2.p2.4.m3.1.1.2" xref="S2.SS2.p2.4.m3.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.4.m3.1.1.1.1" xref="S2.SS2.p2.4.m3.1.1.1.1.1.cmml"><mo id="S2.SS2.p2.4.m3.1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.4.m3.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p2.4.m3.1.1.1.1.1" xref="S2.SS2.p2.4.m3.1.1.1.1.1.cmml"><mi id="S2.SS2.p2.4.m3.1.1.1.1.1.2" xref="S2.SS2.p2.4.m3.1.1.1.1.1.2.cmml">x</mi><mi id="S2.SS2.p2.4.m3.1.1.1.1.1.3" xref="S2.SS2.p2.4.m3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS2.p2.4.m3.1.1.1.1.3" stretchy="false" xref="S2.SS2.p2.4.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m3.1b"><apply id="S2.SS2.p2.4.m3.1.1.cmml" xref="S2.SS2.p2.4.m3.1.1"><times id="S2.SS2.p2.4.m3.1.1.2.cmml" xref="S2.SS2.p2.4.m3.1.1.2"></times><ci id="S2.SS2.p2.4.m3.1.1.3.cmml" xref="S2.SS2.p2.4.m3.1.1.3">𝜎</ci><apply id="S2.SS2.p2.4.m3.1.1.1.1.1.cmml" xref="S2.SS2.p2.4.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m3.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.4.m3.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p2.4.m3.1.1.1.1.1.2.cmml" xref="S2.SS2.p2.4.m3.1.1.1.1.1.2">𝑥</ci><ci id="S2.SS2.p2.4.m3.1.1.1.1.1.3.cmml" xref="S2.SS2.p2.4.m3.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m3.1c">\sigma(x_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m3.1d">italic_σ ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> in the obvious way, starting with the convention</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma(x_{1})=y_{1}\ldots y_{|\sigma(x_{1})|}\,." class="ltx_Math" display="block" id="S2.Ex5.m1.2"><semantics id="S2.Ex5.m1.2a"><mrow id="S2.Ex5.m1.2.2.1" xref="S2.Ex5.m1.2.2.1.1.cmml"><mrow id="S2.Ex5.m1.2.2.1.1" xref="S2.Ex5.m1.2.2.1.1.cmml"><mrow id="S2.Ex5.m1.2.2.1.1.1" xref="S2.Ex5.m1.2.2.1.1.1.cmml"><mi id="S2.Ex5.m1.2.2.1.1.1.3" xref="S2.Ex5.m1.2.2.1.1.1.3.cmml">σ</mi><mo id="S2.Ex5.m1.2.2.1.1.1.2" xref="S2.Ex5.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex5.m1.2.2.1.1.1.1.1" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S2.Ex5.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex5.m1.2.2.1.1.1.1.1.1" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S2.Ex5.m1.2.2.1.1.1.1.1.1.2" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.2.cmml">x</mi><mn id="S2.Ex5.m1.2.2.1.1.1.1.1.1.3" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex5.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex5.m1.2.2.1.1.2" xref="S2.Ex5.m1.2.2.1.1.2.cmml">=</mo><mrow id="S2.Ex5.m1.2.2.1.1.3" xref="S2.Ex5.m1.2.2.1.1.3.cmml"><msub id="S2.Ex5.m1.2.2.1.1.3.2" xref="S2.Ex5.m1.2.2.1.1.3.2.cmml"><mi id="S2.Ex5.m1.2.2.1.1.3.2.2" xref="S2.Ex5.m1.2.2.1.1.3.2.2.cmml">y</mi><mn id="S2.Ex5.m1.2.2.1.1.3.2.3" xref="S2.Ex5.m1.2.2.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.Ex5.m1.2.2.1.1.3.1" xref="S2.Ex5.m1.2.2.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex5.m1.2.2.1.1.3.3" mathvariant="normal" xref="S2.Ex5.m1.2.2.1.1.3.3.cmml">…</mi><mo id="S2.Ex5.m1.2.2.1.1.3.1a" xref="S2.Ex5.m1.2.2.1.1.3.1.cmml">⁢</mo><msub id="S2.Ex5.m1.2.2.1.1.3.4" xref="S2.Ex5.m1.2.2.1.1.3.4.cmml"><mi id="S2.Ex5.m1.2.2.1.1.3.4.2" xref="S2.Ex5.m1.2.2.1.1.3.4.2.cmml">y</mi><mrow id="S2.Ex5.m1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.2.cmml"><mo id="S2.Ex5.m1.1.1.1.1.2" stretchy="false" xref="S2.Ex5.m1.1.1.1.2.1.cmml">|</mo><mrow id="S2.Ex5.m1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.cmml"><mi id="S2.Ex5.m1.1.1.1.1.1.3" xref="S2.Ex5.m1.1.1.1.1.1.3.cmml">σ</mi><mo id="S2.Ex5.m1.1.1.1.1.1.2" xref="S2.Ex5.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex5.m1.1.1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex5.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex5.m1.1.1.1.1.1.1.1.1" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex5.m1.1.1.1.1.1.1.1.1.2" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mn id="S2.Ex5.m1.1.1.1.1.1.1.1.1.3" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex5.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex5.m1.1.1.1.1.3" stretchy="false" xref="S2.Ex5.m1.1.1.1.2.1.cmml">|</mo></mrow></msub></mrow></mrow><mo id="S2.Ex5.m1.2.2.1.2" lspace="0em" xref="S2.Ex5.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex5.m1.2b"><apply id="S2.Ex5.m1.2.2.1.1.cmml" xref="S2.Ex5.m1.2.2.1"><eq id="S2.Ex5.m1.2.2.1.1.2.cmml" xref="S2.Ex5.m1.2.2.1.1.2"></eq><apply id="S2.Ex5.m1.2.2.1.1.1.cmml" xref="S2.Ex5.m1.2.2.1.1.1"><times id="S2.Ex5.m1.2.2.1.1.1.2.cmml" xref="S2.Ex5.m1.2.2.1.1.1.2"></times><ci id="S2.Ex5.m1.2.2.1.1.1.3.cmml" xref="S2.Ex5.m1.2.2.1.1.1.3">𝜎</ci><apply id="S2.Ex5.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex5.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S2.Ex5.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.2">𝑥</ci><cn id="S2.Ex5.m1.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Ex5.m1.2.2.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S2.Ex5.m1.2.2.1.1.3.cmml" xref="S2.Ex5.m1.2.2.1.1.3"><times id="S2.Ex5.m1.2.2.1.1.3.1.cmml" xref="S2.Ex5.m1.2.2.1.1.3.1"></times><apply id="S2.Ex5.m1.2.2.1.1.3.2.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S2.Ex5.m1.2.2.1.1.3.2.1.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2">subscript</csymbol><ci id="S2.Ex5.m1.2.2.1.1.3.2.2.cmml" xref="S2.Ex5.m1.2.2.1.1.3.2.2">𝑦</ci><cn id="S2.Ex5.m1.2.2.1.1.3.2.3.cmml" type="integer" xref="S2.Ex5.m1.2.2.1.1.3.2.3">1</cn></apply><ci id="S2.Ex5.m1.2.2.1.1.3.3.cmml" xref="S2.Ex5.m1.2.2.1.1.3.3">…</ci><apply id="S2.Ex5.m1.2.2.1.1.3.4.cmml" xref="S2.Ex5.m1.2.2.1.1.3.4"><csymbol cd="ambiguous" id="S2.Ex5.m1.2.2.1.1.3.4.1.cmml" xref="S2.Ex5.m1.2.2.1.1.3.4">subscript</csymbol><ci id="S2.Ex5.m1.2.2.1.1.3.4.2.cmml" xref="S2.Ex5.m1.2.2.1.1.3.4.2">𝑦</ci><apply id="S2.Ex5.m1.1.1.1.2.cmml" xref="S2.Ex5.m1.1.1.1.1"><abs id="S2.Ex5.m1.1.1.1.2.1.cmml" xref="S2.Ex5.m1.1.1.1.1.2"></abs><apply id="S2.Ex5.m1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.1.1.1.1.1"><times id="S2.Ex5.m1.1.1.1.1.1.2.cmml" xref="S2.Ex5.m1.1.1.1.1.1.2"></times><ci id="S2.Ex5.m1.1.1.1.1.1.3.cmml" xref="S2.Ex5.m1.1.1.1.1.1.3">𝜎</ci><apply id="S2.Ex5.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex5.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex5.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.Ex5.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.2">𝑥</ci><cn id="S2.Ex5.m1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Ex5.m1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5.m1.2c">\sigma(x_{1})=y_{1}\ldots y_{|\sigma(x_{1})|}\,.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.2d">italic_σ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_y start_POSTSUBSCRIPT | italic_σ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) | end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.13">This map <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m1.1"><semantics id="S2.SS2.p2.5.m1.1a"><msup id="S2.SS2.p2.5.m1.1.1" xref="S2.SS2.p2.5.m1.1.1.cmml"><mi id="S2.SS2.p2.5.m1.1.1.2" xref="S2.SS2.p2.5.m1.1.1.2.cmml">σ</mi><mi id="S2.SS2.p2.5.m1.1.1.3" xref="S2.SS2.p2.5.m1.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m1.1b"><apply id="S2.SS2.p2.5.m1.1.1.cmml" xref="S2.SS2.p2.5.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m1.1.1.1.cmml" xref="S2.SS2.p2.5.m1.1.1">superscript</csymbol><ci id="S2.SS2.p2.5.m1.1.1.2.cmml" xref="S2.SS2.p2.5.m1.1.1.2">𝜎</ci><ci id="S2.SS2.p2.5.m1.1.1.3.cmml" xref="S2.SS2.p2.5.m1.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m1.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m1.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> will in general not inherit any of the properties of <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m2.1"><semantics id="S2.SS2.p2.6.m2.1a"><mi id="S2.SS2.p2.6.m2.1.1" xref="S2.SS2.p2.6.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m2.1b"><ci id="S2.SS2.p2.6.m2.1.1.cmml" xref="S2.SS2.p2.6.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m2.1d">italic_σ</annotation></semantics></math> (for instance, <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m3.1"><semantics id="S2.SS2.p2.7.m3.1a"><msup id="S2.SS2.p2.7.m3.1.1" xref="S2.SS2.p2.7.m3.1.1.cmml"><mi id="S2.SS2.p2.7.m3.1.1.2" xref="S2.SS2.p2.7.m3.1.1.2.cmml">σ</mi><mi id="S2.SS2.p2.7.m3.1.1.3" xref="S2.SS2.p2.7.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m3.1b"><apply id="S2.SS2.p2.7.m3.1.1.cmml" xref="S2.SS2.p2.7.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.7.m3.1.1.1.cmml" xref="S2.SS2.p2.7.m3.1.1">superscript</csymbol><ci id="S2.SS2.p2.7.m3.1.1.2.cmml" xref="S2.SS2.p2.7.m3.1.1.2">𝜎</ci><ci id="S2.SS2.p2.7.m3.1.1.3.cmml" xref="S2.SS2.p2.7.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m3.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m3.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is almost never surjective and almost never finite-to-one). However, it satisfies <math alttext="\sigma^{\mathbb{Z}}(T({\bf x}))=T^{k}({\bf y})" class="ltx_Math" display="inline" id="S2.SS2.p2.8.m4.3"><semantics id="S2.SS2.p2.8.m4.3a"><mrow id="S2.SS2.p2.8.m4.3.3" xref="S2.SS2.p2.8.m4.3.3.cmml"><mrow id="S2.SS2.p2.8.m4.3.3.1" xref="S2.SS2.p2.8.m4.3.3.1.cmml"><msup id="S2.SS2.p2.8.m4.3.3.1.3" xref="S2.SS2.p2.8.m4.3.3.1.3.cmml"><mi id="S2.SS2.p2.8.m4.3.3.1.3.2" xref="S2.SS2.p2.8.m4.3.3.1.3.2.cmml">σ</mi><mi id="S2.SS2.p2.8.m4.3.3.1.3.3" xref="S2.SS2.p2.8.m4.3.3.1.3.3.cmml">ℤ</mi></msup><mo id="S2.SS2.p2.8.m4.3.3.1.2" xref="S2.SS2.p2.8.m4.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.8.m4.3.3.1.1.1" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.cmml"><mo id="S2.SS2.p2.8.m4.3.3.1.1.1.2" stretchy="false" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p2.8.m4.3.3.1.1.1.1" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.cmml"><mi id="S2.SS2.p2.8.m4.3.3.1.1.1.1.2" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.2.cmml">T</mi><mo id="S2.SS2.p2.8.m4.3.3.1.1.1.1.1" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS2.p2.8.m4.3.3.1.1.1.1.3.2" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.cmml"><mo id="S2.SS2.p2.8.m4.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.cmml">(</mo><mi id="S2.SS2.p2.8.m4.1.1" xref="S2.SS2.p2.8.m4.1.1.cmml">𝐱</mi><mo id="S2.SS2.p2.8.m4.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.8.m4.3.3.1.1.1.3" stretchy="false" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.8.m4.3.3.2" xref="S2.SS2.p2.8.m4.3.3.2.cmml">=</mo><mrow id="S2.SS2.p2.8.m4.3.3.3" xref="S2.SS2.p2.8.m4.3.3.3.cmml"><msup id="S2.SS2.p2.8.m4.3.3.3.2" xref="S2.SS2.p2.8.m4.3.3.3.2.cmml"><mi id="S2.SS2.p2.8.m4.3.3.3.2.2" xref="S2.SS2.p2.8.m4.3.3.3.2.2.cmml">T</mi><mi id="S2.SS2.p2.8.m4.3.3.3.2.3" xref="S2.SS2.p2.8.m4.3.3.3.2.3.cmml">k</mi></msup><mo id="S2.SS2.p2.8.m4.3.3.3.1" xref="S2.SS2.p2.8.m4.3.3.3.1.cmml">⁢</mo><mrow id="S2.SS2.p2.8.m4.3.3.3.3.2" xref="S2.SS2.p2.8.m4.3.3.3.cmml"><mo id="S2.SS2.p2.8.m4.3.3.3.3.2.1" stretchy="false" xref="S2.SS2.p2.8.m4.3.3.3.cmml">(</mo><mi id="S2.SS2.p2.8.m4.2.2" xref="S2.SS2.p2.8.m4.2.2.cmml">𝐲</mi><mo id="S2.SS2.p2.8.m4.3.3.3.3.2.2" stretchy="false" xref="S2.SS2.p2.8.m4.3.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.8.m4.3b"><apply id="S2.SS2.p2.8.m4.3.3.cmml" xref="S2.SS2.p2.8.m4.3.3"><eq id="S2.SS2.p2.8.m4.3.3.2.cmml" xref="S2.SS2.p2.8.m4.3.3.2"></eq><apply id="S2.SS2.p2.8.m4.3.3.1.cmml" xref="S2.SS2.p2.8.m4.3.3.1"><times id="S2.SS2.p2.8.m4.3.3.1.2.cmml" xref="S2.SS2.p2.8.m4.3.3.1.2"></times><apply id="S2.SS2.p2.8.m4.3.3.1.3.cmml" xref="S2.SS2.p2.8.m4.3.3.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.8.m4.3.3.1.3.1.cmml" xref="S2.SS2.p2.8.m4.3.3.1.3">superscript</csymbol><ci id="S2.SS2.p2.8.m4.3.3.1.3.2.cmml" xref="S2.SS2.p2.8.m4.3.3.1.3.2">𝜎</ci><ci id="S2.SS2.p2.8.m4.3.3.1.3.3.cmml" xref="S2.SS2.p2.8.m4.3.3.1.3.3">ℤ</ci></apply><apply id="S2.SS2.p2.8.m4.3.3.1.1.1.1.cmml" xref="S2.SS2.p2.8.m4.3.3.1.1.1"><times id="S2.SS2.p2.8.m4.3.3.1.1.1.1.1.cmml" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.1"></times><ci id="S2.SS2.p2.8.m4.3.3.1.1.1.1.2.cmml" xref="S2.SS2.p2.8.m4.3.3.1.1.1.1.2">𝑇</ci><ci id="S2.SS2.p2.8.m4.1.1.cmml" xref="S2.SS2.p2.8.m4.1.1">𝐱</ci></apply></apply><apply id="S2.SS2.p2.8.m4.3.3.3.cmml" xref="S2.SS2.p2.8.m4.3.3.3"><times id="S2.SS2.p2.8.m4.3.3.3.1.cmml" xref="S2.SS2.p2.8.m4.3.3.3.1"></times><apply id="S2.SS2.p2.8.m4.3.3.3.2.cmml" xref="S2.SS2.p2.8.m4.3.3.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.8.m4.3.3.3.2.1.cmml" xref="S2.SS2.p2.8.m4.3.3.3.2">superscript</csymbol><ci id="S2.SS2.p2.8.m4.3.3.3.2.2.cmml" xref="S2.SS2.p2.8.m4.3.3.3.2.2">𝑇</ci><ci id="S2.SS2.p2.8.m4.3.3.3.2.3.cmml" xref="S2.SS2.p2.8.m4.3.3.3.2.3">𝑘</ci></apply><ci id="S2.SS2.p2.8.m4.2.2.cmml" xref="S2.SS2.p2.8.m4.2.2">𝐲</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.8.m4.3c">\sigma^{\mathbb{Z}}(T({\bf x}))=T^{k}({\bf y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.8.m4.3d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_T ( bold_x ) ) = italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( bold_y )</annotation></semantics></math> for suitable <math alttext="k\geq 0" class="ltx_Math" display="inline" id="S2.SS2.p2.9.m5.1"><semantics id="S2.SS2.p2.9.m5.1a"><mrow id="S2.SS2.p2.9.m5.1.1" xref="S2.SS2.p2.9.m5.1.1.cmml"><mi id="S2.SS2.p2.9.m5.1.1.2" xref="S2.SS2.p2.9.m5.1.1.2.cmml">k</mi><mo id="S2.SS2.p2.9.m5.1.1.1" xref="S2.SS2.p2.9.m5.1.1.1.cmml">≥</mo><mn id="S2.SS2.p2.9.m5.1.1.3" xref="S2.SS2.p2.9.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.9.m5.1b"><apply id="S2.SS2.p2.9.m5.1.1.cmml" xref="S2.SS2.p2.9.m5.1.1"><geq id="S2.SS2.p2.9.m5.1.1.1.cmml" xref="S2.SS2.p2.9.m5.1.1.1"></geq><ci id="S2.SS2.p2.9.m5.1.1.2.cmml" xref="S2.SS2.p2.9.m5.1.1.2">𝑘</ci><cn id="S2.SS2.p2.9.m5.1.1.3.cmml" type="integer" xref="S2.SS2.p2.9.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.9.m5.1c">k\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.9.m5.1d">italic_k ≥ 0</annotation></semantics></math> depending on <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S2.SS2.p2.10.m6.1"><semantics id="S2.SS2.p2.10.m6.1a"><mi id="S2.SS2.p2.10.m6.1.1" xref="S2.SS2.p2.10.m6.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.10.m6.1b"><ci id="S2.SS2.p2.10.m6.1.1.cmml" xref="S2.SS2.p2.10.m6.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.10.m6.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.10.m6.1d">bold_x</annotation></semantics></math>, so that the shift-orbit <math alttext="\cal O({\bf x})" class="ltx_Math" display="inline" id="S2.SS2.p2.11.m7.1"><semantics id="S2.SS2.p2.11.m7.1a"><mrow id="S2.SS2.p2.11.m7.1.2" xref="S2.SS2.p2.11.m7.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.11.m7.1.2.2" xref="S2.SS2.p2.11.m7.1.2.2.cmml">𝒪</mi><mo id="S2.SS2.p2.11.m7.1.2.1" xref="S2.SS2.p2.11.m7.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.11.m7.1.2.3.2" xref="S2.SS2.p2.11.m7.1.2.cmml"><mo id="S2.SS2.p2.11.m7.1.2.3.2.1" stretchy="false" xref="S2.SS2.p2.11.m7.1.2.cmml">(</mo><mi id="S2.SS2.p2.11.m7.1.1" xref="S2.SS2.p2.11.m7.1.1.cmml">𝐱</mi><mo id="S2.SS2.p2.11.m7.1.2.3.2.2" stretchy="false" xref="S2.SS2.p2.11.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.11.m7.1b"><apply id="S2.SS2.p2.11.m7.1.2.cmml" xref="S2.SS2.p2.11.m7.1.2"><times id="S2.SS2.p2.11.m7.1.2.1.cmml" xref="S2.SS2.p2.11.m7.1.2.1"></times><ci id="S2.SS2.p2.11.m7.1.2.2.cmml" xref="S2.SS2.p2.11.m7.1.2.2">𝒪</ci><ci id="S2.SS2.p2.11.m7.1.1.cmml" xref="S2.SS2.p2.11.m7.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.11.m7.1c">\cal O({\bf x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.11.m7.1d">caligraphic_O ( bold_x )</annotation></semantics></math> has a well defined image shift-orbit <math alttext="\cal O({\bf y})" class="ltx_Math" display="inline" id="S2.SS2.p2.12.m8.1"><semantics id="S2.SS2.p2.12.m8.1a"><mrow id="S2.SS2.p2.12.m8.1.2" xref="S2.SS2.p2.12.m8.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.12.m8.1.2.2" xref="S2.SS2.p2.12.m8.1.2.2.cmml">𝒪</mi><mo id="S2.SS2.p2.12.m8.1.2.1" xref="S2.SS2.p2.12.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.12.m8.1.2.3.2" xref="S2.SS2.p2.12.m8.1.2.cmml"><mo id="S2.SS2.p2.12.m8.1.2.3.2.1" stretchy="false" xref="S2.SS2.p2.12.m8.1.2.cmml">(</mo><mi id="S2.SS2.p2.12.m8.1.1" xref="S2.SS2.p2.12.m8.1.1.cmml">𝐲</mi><mo id="S2.SS2.p2.12.m8.1.2.3.2.2" stretchy="false" xref="S2.SS2.p2.12.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.12.m8.1b"><apply id="S2.SS2.p2.12.m8.1.2.cmml" xref="S2.SS2.p2.12.m8.1.2"><times id="S2.SS2.p2.12.m8.1.2.1.cmml" xref="S2.SS2.p2.12.m8.1.2.1"></times><ci id="S2.SS2.p2.12.m8.1.2.2.cmml" xref="S2.SS2.p2.12.m8.1.2.2">𝒪</ci><ci id="S2.SS2.p2.12.m8.1.1.cmml" xref="S2.SS2.p2.12.m8.1.1">𝐲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.12.m8.1c">\cal O({\bf y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.12.m8.1d">caligraphic_O ( bold_y )</annotation></semantics></math>. Hence <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p2.13.m9.1"><semantics id="S2.SS2.p2.13.m9.1a"><msup id="S2.SS2.p2.13.m9.1.1" xref="S2.SS2.p2.13.m9.1.1.cmml"><mi id="S2.SS2.p2.13.m9.1.1.2" xref="S2.SS2.p2.13.m9.1.1.2.cmml">σ</mi><mi id="S2.SS2.p2.13.m9.1.1.3" xref="S2.SS2.p2.13.m9.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.13.m9.1b"><apply id="S2.SS2.p2.13.m9.1.1.cmml" xref="S2.SS2.p2.13.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.13.m9.1.1.1.cmml" xref="S2.SS2.p2.13.m9.1.1">superscript</csymbol><ci id="S2.SS2.p2.13.m9.1.1.2.cmml" xref="S2.SS2.p2.13.m9.1.1.2">𝜎</ci><ci id="S2.SS2.p2.13.m9.1.1.3.cmml" xref="S2.SS2.p2.13.m9.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.13.m9.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.13.m9.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> induces a map</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{T}:\cal A^{\mathbb{Z}}/\langle T\rangle\to\cal B^{\mathbb{Z}}/\langle T% \rangle,\,\,\cal O({\bf x})\mapsto\cal O(\sigma^{\mathbb{Z}}({\bf x}))" class="ltx_Math" display="block" id="S2.Ex6.m1.6"><semantics id="S2.Ex6.m1.6a"><mrow id="S2.Ex6.m1.6.6" xref="S2.Ex6.m1.6.6.cmml"><msup id="S2.Ex6.m1.6.6.4" xref="S2.Ex6.m1.6.6.4.cmml"><mi id="S2.Ex6.m1.6.6.4.2" xref="S2.Ex6.m1.6.6.4.2.cmml">σ</mi><mi id="S2.Ex6.m1.6.6.4.3" xref="S2.Ex6.m1.6.6.4.3.cmml">T</mi></msup><mo id="S2.Ex6.m1.6.6.3" lspace="0.278em" rspace="0.278em" xref="S2.Ex6.m1.6.6.3.cmml">:</mo><mrow id="S2.Ex6.m1.6.6.2.2" xref="S2.Ex6.m1.6.6.2.3.cmml"><mrow id="S2.Ex6.m1.5.5.1.1.1" xref="S2.Ex6.m1.5.5.1.1.1.cmml"><mrow 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follows straight from the definitions that both induced maps <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm1.p1.1.m1.1"><semantics id="S2.Thmthm1.p1.1.m1.1a"><msup id="S2.Thmthm1.p1.1.m1.1.1" xref="S2.Thmthm1.p1.1.m1.1.1.cmml"><mi id="S2.Thmthm1.p1.1.m1.1.1.2" xref="S2.Thmthm1.p1.1.m1.1.1.2.cmml">σ</mi><mi id="S2.Thmthm1.p1.1.m1.1.1.3" xref="S2.Thmthm1.p1.1.m1.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm1.p1.1.m1.1b"><apply id="S2.Thmthm1.p1.1.m1.1.1.cmml" xref="S2.Thmthm1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm1.p1.1.m1.1.1.1.cmml" xref="S2.Thmthm1.p1.1.m1.1.1">superscript</csymbol><ci id="S2.Thmthm1.p1.1.m1.1.1.2.cmml" xref="S2.Thmthm1.p1.1.m1.1.1.2">𝜎</ci><ci id="S2.Thmthm1.p1.1.m1.1.1.3.cmml" xref="S2.Thmthm1.p1.1.m1.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm1.p1.1.m1.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm1.p1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\sigma^{T}" class="ltx_Math" display="inline" id="S2.Thmthm1.p1.2.m2.1"><semantics id="S2.Thmthm1.p1.2.m2.1a"><msup id="S2.Thmthm1.p1.2.m2.1.1" xref="S2.Thmthm1.p1.2.m2.1.1.cmml"><mi id="S2.Thmthm1.p1.2.m2.1.1.2" xref="S2.Thmthm1.p1.2.m2.1.1.2.cmml">σ</mi><mi id="S2.Thmthm1.p1.2.m2.1.1.3" xref="S2.Thmthm1.p1.2.m2.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm1.p1.2.m2.1b"><apply id="S2.Thmthm1.p1.2.m2.1.1.cmml" xref="S2.Thmthm1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.Thmthm1.p1.2.m2.1.1.1.cmml" xref="S2.Thmthm1.p1.2.m2.1.1">superscript</csymbol><ci id="S2.Thmthm1.p1.2.m2.1.1.2.cmml" xref="S2.Thmthm1.p1.2.m2.1.1.2">𝜎</ci><ci id="S2.Thmthm1.p1.2.m2.1.1.3.cmml" xref="S2.Thmthm1.p1.2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm1.p1.2.m2.1c">\sigma^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm1.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> behave functorially: in particular, for any two non-erasing monoid morphisms <math alttext="\sigma_{1}:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.Thmthm1.p1.3.m3.1"><semantics id="S2.Thmthm1.p1.3.m3.1a"><mrow id="S2.Thmthm1.p1.3.m3.1.1" xref="S2.Thmthm1.p1.3.m3.1.1.cmml"><msub id="S2.Thmthm1.p1.3.m3.1.1.2" xref="S2.Thmthm1.p1.3.m3.1.1.2.cmml"><mi id="S2.Thmthm1.p1.3.m3.1.1.2.2" xref="S2.Thmthm1.p1.3.m3.1.1.2.2.cmml">σ</mi><mn id="S2.Thmthm1.p1.3.m3.1.1.2.3" xref="S2.Thmthm1.p1.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S2.Thmthm1.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm1.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S2.Thmthm1.p1.3.m3.1.1.3" xref="S2.Thmthm1.p1.3.m3.1.1.3.cmml"><msup id="S2.Thmthm1.p1.3.m3.1.1.3.2" xref="S2.Thmthm1.p1.3.m3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm1.p1.3.m3.1.1.3.2.2" xref="S2.Thmthm1.p1.3.m3.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.Thmthm1.p1.3.m3.1.1.3.2.3" xref="S2.Thmthm1.p1.3.m3.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm1.p1.3.m3.1.1.3.1" stretchy="false" xref="S2.Thmthm1.p1.3.m3.1.1.3.1.cmml">→</mo><msup id="S2.Thmthm1.p1.3.m3.1.1.3.3" xref="S2.Thmthm1.p1.3.m3.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm1.p1.3.m3.1.1.3.3.2" xref="S2.Thmthm1.p1.3.m3.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.Thmthm1.p1.3.m3.1.1.3.3.3" xref="S2.Thmthm1.p1.3.m3.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm1.p1.3.m3.1b"><apply id="S2.Thmthm1.p1.3.m3.1.1.cmml" xref="S2.Thmthm1.p1.3.m3.1.1"><ci id="S2.Thmthm1.p1.3.m3.1.1.1.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.1">:</ci><apply id="S2.Thmthm1.p1.3.m3.1.1.2.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.Thmthm1.p1.3.m3.1.1.2.1.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.Thmthm1.p1.3.m3.1.1.2.2.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.2.2">𝜎</ci><cn id="S2.Thmthm1.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S2.Thmthm1.p1.3.m3.1.1.2.3">1</cn></apply><apply id="S2.Thmthm1.p1.3.m3.1.1.3.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3"><ci id="S2.Thmthm1.p1.3.m3.1.1.3.1.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.1">→</ci><apply id="S2.Thmthm1.p1.3.m3.1.1.3.2.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmthm1.p1.3.m3.1.1.3.2.1.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S2.Thmthm1.p1.3.m3.1.1.3.2.2.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.2.2">𝒜</ci><times id="S2.Thmthm1.p1.3.m3.1.1.3.2.3.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.2.3"></times></apply><apply id="S2.Thmthm1.p1.3.m3.1.1.3.3.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmthm1.p1.3.m3.1.1.3.3.1.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.3">superscript</csymbol><ci id="S2.Thmthm1.p1.3.m3.1.1.3.3.2.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.3.2">ℬ</ci><times id="S2.Thmthm1.p1.3.m3.1.1.3.3.3.cmml" xref="S2.Thmthm1.p1.3.m3.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm1.p1.3.m3.1c">\sigma_{1}:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm1.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{2}:\cal B^{*}\to\cal C^{*}" class="ltx_Math" display="inline" id="S2.Thmthm1.p1.4.m4.1"><semantics id="S2.Thmthm1.p1.4.m4.1a"><mrow id="S2.Thmthm1.p1.4.m4.1.1" xref="S2.Thmthm1.p1.4.m4.1.1.cmml"><msub id="S2.Thmthm1.p1.4.m4.1.1.2" xref="S2.Thmthm1.p1.4.m4.1.1.2.cmml"><mi id="S2.Thmthm1.p1.4.m4.1.1.2.2" xref="S2.Thmthm1.p1.4.m4.1.1.2.2.cmml">σ</mi><mn id="S2.Thmthm1.p1.4.m4.1.1.2.3" xref="S2.Thmthm1.p1.4.m4.1.1.2.3.cmml">2</mn></msub><mo id="S2.Thmthm1.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm1.p1.4.m4.1.1.1.cmml">:</mo><mrow id="S2.Thmthm1.p1.4.m4.1.1.3" xref="S2.Thmthm1.p1.4.m4.1.1.3.cmml"><msup id="S2.Thmthm1.p1.4.m4.1.1.3.2" xref="S2.Thmthm1.p1.4.m4.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm1.p1.4.m4.1.1.3.2.2" xref="S2.Thmthm1.p1.4.m4.1.1.3.2.2.cmml">ℬ</mi><mo id="S2.Thmthm1.p1.4.m4.1.1.3.2.3" xref="S2.Thmthm1.p1.4.m4.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm1.p1.4.m4.1.1.3.1" stretchy="false" xref="S2.Thmthm1.p1.4.m4.1.1.3.1.cmml">→</mo><msup id="S2.Thmthm1.p1.4.m4.1.1.3.3" xref="S2.Thmthm1.p1.4.m4.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm1.p1.4.m4.1.1.3.3.2" xref="S2.Thmthm1.p1.4.m4.1.1.3.3.2.cmml">𝒞</mi><mo id="S2.Thmthm1.p1.4.m4.1.1.3.3.3" xref="S2.Thmthm1.p1.4.m4.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm1.p1.4.m4.1b"><apply id="S2.Thmthm1.p1.4.m4.1.1.cmml" xref="S2.Thmthm1.p1.4.m4.1.1"><ci id="S2.Thmthm1.p1.4.m4.1.1.1.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.1">:</ci><apply id="S2.Thmthm1.p1.4.m4.1.1.2.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.Thmthm1.p1.4.m4.1.1.2.1.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.2">subscript</csymbol><ci id="S2.Thmthm1.p1.4.m4.1.1.2.2.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.2.2">𝜎</ci><cn id="S2.Thmthm1.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S2.Thmthm1.p1.4.m4.1.1.2.3">2</cn></apply><apply id="S2.Thmthm1.p1.4.m4.1.1.3.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3"><ci id="S2.Thmthm1.p1.4.m4.1.1.3.1.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.1">→</ci><apply id="S2.Thmthm1.p1.4.m4.1.1.3.2.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmthm1.p1.4.m4.1.1.3.2.1.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.2">superscript</csymbol><ci id="S2.Thmthm1.p1.4.m4.1.1.3.2.2.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.2.2">ℬ</ci><times id="S2.Thmthm1.p1.4.m4.1.1.3.2.3.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.2.3"></times></apply><apply id="S2.Thmthm1.p1.4.m4.1.1.3.3.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmthm1.p1.4.m4.1.1.3.3.1.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.3">superscript</csymbol><ci id="S2.Thmthm1.p1.4.m4.1.1.3.3.2.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.3.2">𝒞</ci><times id="S2.Thmthm1.p1.4.m4.1.1.3.3.3.cmml" xref="S2.Thmthm1.p1.4.m4.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm1.p1.4.m4.1c">\sigma_{2}:\cal B^{*}\to\cal C^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm1.p1.4.m4.1d">italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> we have</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(\sigma_{2}\circ\sigma_{1})^{\mathbb{Z}}=\sigma_{2}^{\mathbb{Z}}\circ\sigma_{1% }^{\mathbb{Z}}\quad\text{and}\quad(\sigma_{2}\circ\sigma_{1})^{T}=\sigma_{2}^{% T}\circ\sigma_{1}^{T}\,." class="ltx_Math" display="block" id="S2.Ex7.m1.2"><semantics id="S2.Ex7.m1.2a"><mrow id="S2.Ex7.m1.2.2.1"><mrow id="S2.Ex7.m1.2.2.1.1.2" xref="S2.Ex7.m1.2.2.1.1.3.cmml"><mrow id="S2.Ex7.m1.2.2.1.1.1.1" xref="S2.Ex7.m1.2.2.1.1.1.1.cmml"><msup id="S2.Ex7.m1.2.2.1.1.1.1.1" xref="S2.Ex7.m1.2.2.1.1.1.1.1.cmml"><mrow id="S2.Ex7.m1.2.2.1.1.1.1.1.1.1" xref="S2.Ex7.m1.2.2.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex7.m1.2.2.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex7.m1.2.2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex7.m1.2.2.1.1.1.1.1.1.1.1" xref="S2.Ex7.m1.2.2.1.1.1.1.1.1.1.1.cmml"><msub id="S2.Ex7.m1.2.2.1.1.1.1.1.1.1.1.2" xref="S2.Ex7.m1.2.2.1.1.1.1.1.1.1.1.2.cmml"><mi 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xref="S2.Ex7.m1.2.2.1.1.2.2.3.3">subscript</csymbol><ci id="S2.Ex7.m1.2.2.1.1.2.2.3.3.2.2.cmml" xref="S2.Ex7.m1.2.2.1.1.2.2.3.3.2.2">𝜎</ci><cn id="S2.Ex7.m1.2.2.1.1.2.2.3.3.2.3.cmml" type="integer" xref="S2.Ex7.m1.2.2.1.1.2.2.3.3.2.3">1</cn></apply><ci id="S2.Ex7.m1.2.2.1.1.2.2.3.3.3.cmml" xref="S2.Ex7.m1.2.2.1.1.2.2.3.3.3">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex7.m1.2c">(\sigma_{2}\circ\sigma_{1})^{\mathbb{Z}}=\sigma_{2}^{\mathbb{Z}}\circ\sigma_{1% }^{\mathbb{Z}}\quad\text{and}\quad(\sigma_{2}\circ\sigma_{1})^{T}=\sigma_{2}^{% T}\circ\sigma_{1}^{T}\,.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m1.2d">( italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∘ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT = italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ∘ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT and ( italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∘ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∘ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.1">We now come to a third map induced by any non-erasing monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><mrow id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml">σ</mi><mo id="S2.SS2.p3.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p3.1.m1.1.1.1.cmml">:</mo><mrow id="S2.SS2.p3.1.m1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.3.cmml"><msup id="S2.SS2.p3.1.m1.1.1.3.2" xref="S2.SS2.p3.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.1.m1.1.1.3.2.2" xref="S2.SS2.p3.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.SS2.p3.1.m1.1.1.3.2.3" xref="S2.SS2.p3.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.SS2.p3.1.m1.1.1.3.1" stretchy="false" xref="S2.SS2.p3.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.SS2.p3.1.m1.1.1.3.3" xref="S2.SS2.p3.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.1.m1.1.1.3.3.2" xref="S2.SS2.p3.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.SS2.p3.1.m1.1.1.3.3.3" xref="S2.SS2.p3.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><apply id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"><ci id="S2.SS2.p3.1.m1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1.1">:</ci><ci id="S2.SS2.p3.1.m1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1.2">𝜎</ci><apply id="S2.SS2.p3.1.m1.1.1.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3"><ci id="S2.SS2.p3.1.m1.1.1.3.1.cmml" xref="S2.SS2.p3.1.m1.1.1.3.1">→</ci><apply id="S2.SS2.p3.1.m1.1.1.3.2.cmml" xref="S2.SS2.p3.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.1.1.3.2.1.cmml" xref="S2.SS2.p3.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.SS2.p3.1.m1.1.1.3.2.2.cmml" xref="S2.SS2.p3.1.m1.1.1.3.2.2">𝒜</ci><times id="S2.SS2.p3.1.m1.1.1.3.2.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3.2.3"></times></apply><apply id="S2.SS2.p3.1.m1.1.1.3.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.1.1.3.3.1.cmml" xref="S2.SS2.p3.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.SS2.p3.1.m1.1.1.3.3.2.cmml" xref="S2.SS2.p3.1.m1.1.1.3.3.2">ℬ</ci><times id="S2.SS2.p3.1.m1.1.1.3.3.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, namely the map</p> <table class="ltx_equation ltx_eqn_table" id="S2.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.9)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{\Sigma}:\Sigma(\cal A)\to\Sigma(\cal B)\,,\,\,X\mapsto\sigma^{\Sigma}(X)" class="ltx_Math" display="block" id="S2.E9.m1.5"><semantics id="S2.E9.m1.5a"><mrow id="S2.E9.m1.5.5" xref="S2.E9.m1.5.5.cmml"><msup id="S2.E9.m1.5.5.4" xref="S2.E9.m1.5.5.4.cmml"><mi id="S2.E9.m1.5.5.4.2" xref="S2.E9.m1.5.5.4.2.cmml">σ</mi><mi id="S2.E9.m1.5.5.4.3" mathvariant="normal" xref="S2.E9.m1.5.5.4.3.cmml">Σ</mi></msup><mo id="S2.E9.m1.5.5.3" lspace="0.278em" rspace="0.278em" xref="S2.E9.m1.5.5.3.cmml">:</mo><mrow id="S2.E9.m1.5.5.2.2" xref="S2.E9.m1.5.5.2.3.cmml"><mrow id="S2.E9.m1.4.4.1.1.1" xref="S2.E9.m1.4.4.1.1.1.cmml"><mrow id="S2.E9.m1.4.4.1.1.1.2" xref="S2.E9.m1.4.4.1.1.1.2.cmml"><mi id="S2.E9.m1.4.4.1.1.1.2.2" mathvariant="normal" xref="S2.E9.m1.4.4.1.1.1.2.2.cmml">Σ</mi><mo id="S2.E9.m1.4.4.1.1.1.2.1" xref="S2.E9.m1.4.4.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.E9.m1.4.4.1.1.1.2.3.2" xref="S2.E9.m1.4.4.1.1.1.2.cmml"><mo id="S2.E9.m1.4.4.1.1.1.2.3.2.1" stretchy="false" xref="S2.E9.m1.4.4.1.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.1.1" xref="S2.E9.m1.1.1.cmml">𝒜</mi><mo id="S2.E9.m1.4.4.1.1.1.2.3.2.2" stretchy="false" xref="S2.E9.m1.4.4.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E9.m1.4.4.1.1.1.1" stretchy="false" xref="S2.E9.m1.4.4.1.1.1.1.cmml">→</mo><mrow id="S2.E9.m1.4.4.1.1.1.3" xref="S2.E9.m1.4.4.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.4.4.1.1.1.3.2" mathvariant="script" xref="S2.E9.m1.4.4.1.1.1.3.2.cmml">Σ</mi><mo id="S2.E9.m1.4.4.1.1.1.3.1" xref="S2.E9.m1.4.4.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.E9.m1.4.4.1.1.1.3.3.2" xref="S2.E9.m1.4.4.1.1.1.3.cmml"><mo id="S2.E9.m1.4.4.1.1.1.3.3.2.1" stretchy="false" xref="S2.E9.m1.4.4.1.1.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.2.2" xref="S2.E9.m1.2.2.cmml">ℬ</mi><mo id="S2.E9.m1.4.4.1.1.1.3.3.2.2" rspace="0.170em" stretchy="false" xref="S2.E9.m1.4.4.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E9.m1.5.5.2.2.3" rspace="0.497em" xref="S2.E9.m1.5.5.2.3a.cmml">,</mo><mrow id="S2.E9.m1.5.5.2.2.2" xref="S2.E9.m1.5.5.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.5.5.2.2.2.2" xref="S2.E9.m1.5.5.2.2.2.2.cmml">𝒳</mi><mo id="S2.E9.m1.5.5.2.2.2.1" stretchy="false" xref="S2.E9.m1.5.5.2.2.2.1.cmml">↦</mo><mrow id="S2.E9.m1.5.5.2.2.2.3" xref="S2.E9.m1.5.5.2.2.2.3.cmml"><msup id="S2.E9.m1.5.5.2.2.2.3.2" xref="S2.E9.m1.5.5.2.2.2.3.2.cmml"><mi id="S2.E9.m1.5.5.2.2.2.3.2.2" xref="S2.E9.m1.5.5.2.2.2.3.2.2.cmml">σ</mi><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.5.5.2.2.2.3.2.3" mathvariant="script" xref="S2.E9.m1.5.5.2.2.2.3.2.3.cmml">Σ</mi></msup><mo id="S2.E9.m1.5.5.2.2.2.3.1" xref="S2.E9.m1.5.5.2.2.2.3.1.cmml">⁢</mo><mrow id="S2.E9.m1.5.5.2.2.2.3.3.2" xref="S2.E9.m1.5.5.2.2.2.3.cmml"><mo id="S2.E9.m1.5.5.2.2.2.3.3.2.1" stretchy="false" xref="S2.E9.m1.5.5.2.2.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.E9.m1.3.3" xref="S2.E9.m1.3.3.cmml">𝒳</mi><mo id="S2.E9.m1.5.5.2.2.2.3.3.2.2" stretchy="false" xref="S2.E9.m1.5.5.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E9.m1.5b"><apply id="S2.E9.m1.5.5.cmml" xref="S2.E9.m1.5.5"><ci id="S2.E9.m1.5.5.3.cmml" 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xref="S2.E9.m1.4.4.1.1.1.3.1"></times><ci id="S2.E9.m1.4.4.1.1.1.3.2.cmml" xref="S2.E9.m1.4.4.1.1.1.3.2">script-Σ</ci><ci id="S2.E9.m1.2.2.cmml" xref="S2.E9.m1.2.2">ℬ</ci></apply></apply><apply id="S2.E9.m1.5.5.2.2.2.cmml" xref="S2.E9.m1.5.5.2.2.2"><csymbol cd="latexml" id="S2.E9.m1.5.5.2.2.2.1.cmml" xref="S2.E9.m1.5.5.2.2.2.1">maps-to</csymbol><ci id="S2.E9.m1.5.5.2.2.2.2.cmml" xref="S2.E9.m1.5.5.2.2.2.2">𝒳</ci><apply id="S2.E9.m1.5.5.2.2.2.3.cmml" xref="S2.E9.m1.5.5.2.2.2.3"><times id="S2.E9.m1.5.5.2.2.2.3.1.cmml" xref="S2.E9.m1.5.5.2.2.2.3.1"></times><apply id="S2.E9.m1.5.5.2.2.2.3.2.cmml" xref="S2.E9.m1.5.5.2.2.2.3.2"><csymbol cd="ambiguous" id="S2.E9.m1.5.5.2.2.2.3.2.1.cmml" xref="S2.E9.m1.5.5.2.2.2.3.2">superscript</csymbol><ci id="S2.E9.m1.5.5.2.2.2.3.2.2.cmml" xref="S2.E9.m1.5.5.2.2.2.3.2.2">𝜎</ci><ci id="S2.E9.m1.5.5.2.2.2.3.2.3.cmml" xref="S2.E9.m1.5.5.2.2.2.3.2.3">script-Σ</ci></apply><ci id="S2.E9.m1.3.3.cmml" xref="S2.E9.m1.3.3">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E9.m1.5c">\sigma^{\Sigma}:\Sigma(\cal A)\to\Sigma(\cal B)\,,\,\,X\mapsto\sigma^{\Sigma}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.5d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT : roman_Σ ( caligraphic_A ) → caligraphic_Σ ( caligraphic_B ) , caligraphic_X ↦ italic_σ start_POSTSUPERSCRIPT caligraphic_Σ end_POSTSUPERSCRIPT ( caligraphic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p3.8">from the space <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m1.1"><semantics id="S2.SS2.p3.2.m1.1a"><mrow id="S2.SS2.p3.2.m1.1.2" xref="S2.SS2.p3.2.m1.1.2.cmml"><mi id="S2.SS2.p3.2.m1.1.2.2" mathvariant="normal" xref="S2.SS2.p3.2.m1.1.2.2.cmml">Σ</mi><mo id="S2.SS2.p3.2.m1.1.2.1" xref="S2.SS2.p3.2.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p3.2.m1.1.2.3.2" xref="S2.SS2.p3.2.m1.1.2.cmml"><mo id="S2.SS2.p3.2.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.p3.2.m1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.2.m1.1.1" xref="S2.SS2.p3.2.m1.1.1.cmml">𝒜</mi><mo id="S2.SS2.p3.2.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.p3.2.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m1.1b"><apply id="S2.SS2.p3.2.m1.1.2.cmml" xref="S2.SS2.p3.2.m1.1.2"><times id="S2.SS2.p3.2.m1.1.2.1.cmml" xref="S2.SS2.p3.2.m1.1.2.1"></times><ci id="S2.SS2.p3.2.m1.1.2.2.cmml" xref="S2.SS2.p3.2.m1.1.2.2">Σ</ci><ci id="S2.SS2.p3.2.m1.1.1.cmml" xref="S2.SS2.p3.2.m1.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m1.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m1.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math> of subshifts <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m2.1"><semantics id="S2.SS2.p3.3.m2.1a"><mrow id="S2.SS2.p3.3.m2.1.1" xref="S2.SS2.p3.3.m2.1.1.cmml"><mi id="S2.SS2.p3.3.m2.1.1.2" xref="S2.SS2.p3.3.m2.1.1.2.cmml">X</mi><mo id="S2.SS2.p3.3.m2.1.1.1" xref="S2.SS2.p3.3.m2.1.1.1.cmml">⊆</mo><msup id="S2.SS2.p3.3.m2.1.1.3" xref="S2.SS2.p3.3.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.3.m2.1.1.3.2" xref="S2.SS2.p3.3.m2.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS2.p3.3.m2.1.1.3.3" xref="S2.SS2.p3.3.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.3.m2.1b"><apply id="S2.SS2.p3.3.m2.1.1.cmml" xref="S2.SS2.p3.3.m2.1.1"><subset id="S2.SS2.p3.3.m2.1.1.1.cmml" xref="S2.SS2.p3.3.m2.1.1.1"></subset><ci id="S2.SS2.p3.3.m2.1.1.2.cmml" xref="S2.SS2.p3.3.m2.1.1.2">𝑋</ci><apply id="S2.SS2.p3.3.m2.1.1.3.cmml" xref="S2.SS2.p3.3.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.3.m2.1.1.3.1.cmml" xref="S2.SS2.p3.3.m2.1.1.3">superscript</csymbol><ci id="S2.SS2.p3.3.m2.1.1.3.2.cmml" xref="S2.SS2.p3.3.m2.1.1.3.2">𝒜</ci><ci id="S2.SS2.p3.3.m2.1.1.3.3.cmml" xref="S2.SS2.p3.3.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.3.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.3.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> to the space <math alttext="\Sigma(\cal B)" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m3.1"><semantics id="S2.SS2.p3.4.m3.1a"><mrow id="S2.SS2.p3.4.m3.1.2" xref="S2.SS2.p3.4.m3.1.2.cmml"><mi id="S2.SS2.p3.4.m3.1.2.2" mathvariant="normal" xref="S2.SS2.p3.4.m3.1.2.2.cmml">Σ</mi><mo id="S2.SS2.p3.4.m3.1.2.1" xref="S2.SS2.p3.4.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p3.4.m3.1.2.3.2" xref="S2.SS2.p3.4.m3.1.2.cmml"><mo id="S2.SS2.p3.4.m3.1.2.3.2.1" stretchy="false" xref="S2.SS2.p3.4.m3.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.4.m3.1.1" xref="S2.SS2.p3.4.m3.1.1.cmml">ℬ</mi><mo id="S2.SS2.p3.4.m3.1.2.3.2.2" stretchy="false" xref="S2.SS2.p3.4.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.4.m3.1b"><apply id="S2.SS2.p3.4.m3.1.2.cmml" xref="S2.SS2.p3.4.m3.1.2"><times id="S2.SS2.p3.4.m3.1.2.1.cmml" xref="S2.SS2.p3.4.m3.1.2.1"></times><ci id="S2.SS2.p3.4.m3.1.2.2.cmml" xref="S2.SS2.p3.4.m3.1.2.2">Σ</ci><ci id="S2.SS2.p3.4.m3.1.1.cmml" xref="S2.SS2.p3.4.m3.1.1">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.4.m3.1c">\Sigma(\cal B)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.4.m3.1d">roman_Σ ( caligraphic_B )</annotation></semantics></math> of subshifts <math alttext="Y\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p3.5.m4.1"><semantics id="S2.SS2.p3.5.m4.1a"><mrow id="S2.SS2.p3.5.m4.1.1" xref="S2.SS2.p3.5.m4.1.1.cmml"><mi id="S2.SS2.p3.5.m4.1.1.2" xref="S2.SS2.p3.5.m4.1.1.2.cmml">Y</mi><mo id="S2.SS2.p3.5.m4.1.1.1" xref="S2.SS2.p3.5.m4.1.1.1.cmml">⊆</mo><msup id="S2.SS2.p3.5.m4.1.1.3" xref="S2.SS2.p3.5.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.5.m4.1.1.3.2" xref="S2.SS2.p3.5.m4.1.1.3.2.cmml">ℬ</mi><mi id="S2.SS2.p3.5.m4.1.1.3.3" xref="S2.SS2.p3.5.m4.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.5.m4.1b"><apply id="S2.SS2.p3.5.m4.1.1.cmml" xref="S2.SS2.p3.5.m4.1.1"><subset id="S2.SS2.p3.5.m4.1.1.1.cmml" xref="S2.SS2.p3.5.m4.1.1.1"></subset><ci id="S2.SS2.p3.5.m4.1.1.2.cmml" xref="S2.SS2.p3.5.m4.1.1.2">𝑌</ci><apply id="S2.SS2.p3.5.m4.1.1.3.cmml" xref="S2.SS2.p3.5.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.5.m4.1.1.3.1.cmml" xref="S2.SS2.p3.5.m4.1.1.3">superscript</csymbol><ci id="S2.SS2.p3.5.m4.1.1.3.2.cmml" xref="S2.SS2.p3.5.m4.1.1.3.2">ℬ</ci><ci id="S2.SS2.p3.5.m4.1.1.3.3.cmml" xref="S2.SS2.p3.5.m4.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.5.m4.1c">Y\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.5.m4.1d">italic_Y ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. We know of three natural ways to define the image <math alttext="\sigma^{\Sigma}(X)\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p3.6.m5.1"><semantics id="S2.SS2.p3.6.m5.1a"><mrow id="S2.SS2.p3.6.m5.1.2" xref="S2.SS2.p3.6.m5.1.2.cmml"><mrow id="S2.SS2.p3.6.m5.1.2.2" xref="S2.SS2.p3.6.m5.1.2.2.cmml"><msup id="S2.SS2.p3.6.m5.1.2.2.2" xref="S2.SS2.p3.6.m5.1.2.2.2.cmml"><mi id="S2.SS2.p3.6.m5.1.2.2.2.2" xref="S2.SS2.p3.6.m5.1.2.2.2.2.cmml">σ</mi><mi id="S2.SS2.p3.6.m5.1.2.2.2.3" mathvariant="normal" xref="S2.SS2.p3.6.m5.1.2.2.2.3.cmml">Σ</mi></msup><mo id="S2.SS2.p3.6.m5.1.2.2.1" xref="S2.SS2.p3.6.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p3.6.m5.1.2.2.3.2" xref="S2.SS2.p3.6.m5.1.2.2.cmml"><mo id="S2.SS2.p3.6.m5.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p3.6.m5.1.2.2.cmml">(</mo><mi id="S2.SS2.p3.6.m5.1.1" xref="S2.SS2.p3.6.m5.1.1.cmml">X</mi><mo id="S2.SS2.p3.6.m5.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p3.6.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p3.6.m5.1.2.1" xref="S2.SS2.p3.6.m5.1.2.1.cmml">⊆</mo><msup id="S2.SS2.p3.6.m5.1.2.3" xref="S2.SS2.p3.6.m5.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.6.m5.1.2.3.2" xref="S2.SS2.p3.6.m5.1.2.3.2.cmml">ℬ</mi><mi id="S2.SS2.p3.6.m5.1.2.3.3" xref="S2.SS2.p3.6.m5.1.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.6.m5.1b"><apply id="S2.SS2.p3.6.m5.1.2.cmml" xref="S2.SS2.p3.6.m5.1.2"><subset id="S2.SS2.p3.6.m5.1.2.1.cmml" xref="S2.SS2.p3.6.m5.1.2.1"></subset><apply id="S2.SS2.p3.6.m5.1.2.2.cmml" xref="S2.SS2.p3.6.m5.1.2.2"><times id="S2.SS2.p3.6.m5.1.2.2.1.cmml" xref="S2.SS2.p3.6.m5.1.2.2.1"></times><apply id="S2.SS2.p3.6.m5.1.2.2.2.cmml" xref="S2.SS2.p3.6.m5.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p3.6.m5.1.2.2.2.1.cmml" xref="S2.SS2.p3.6.m5.1.2.2.2">superscript</csymbol><ci id="S2.SS2.p3.6.m5.1.2.2.2.2.cmml" xref="S2.SS2.p3.6.m5.1.2.2.2.2">𝜎</ci><ci id="S2.SS2.p3.6.m5.1.2.2.2.3.cmml" xref="S2.SS2.p3.6.m5.1.2.2.2.3">Σ</ci></apply><ci id="S2.SS2.p3.6.m5.1.1.cmml" xref="S2.SS2.p3.6.m5.1.1">𝑋</ci></apply><apply id="S2.SS2.p3.6.m5.1.2.3.cmml" xref="S2.SS2.p3.6.m5.1.2.3"><csymbol cd="ambiguous" id="S2.SS2.p3.6.m5.1.2.3.1.cmml" xref="S2.SS2.p3.6.m5.1.2.3">superscript</csymbol><ci id="S2.SS2.p3.6.m5.1.2.3.2.cmml" xref="S2.SS2.p3.6.m5.1.2.3.2">ℬ</ci><ci id="S2.SS2.p3.6.m5.1.2.3.3.cmml" xref="S2.SS2.p3.6.m5.1.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.6.m5.1c">\sigma^{\Sigma}(X)\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.6.m5.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT ( italic_X ) ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> under this map, for any given subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p3.7.m6.1"><semantics id="S2.SS2.p3.7.m6.1a"><mrow id="S2.SS2.p3.7.m6.1.1" xref="S2.SS2.p3.7.m6.1.1.cmml"><mi id="S2.SS2.p3.7.m6.1.1.2" xref="S2.SS2.p3.7.m6.1.1.2.cmml">X</mi><mo id="S2.SS2.p3.7.m6.1.1.1" xref="S2.SS2.p3.7.m6.1.1.1.cmml">⊆</mo><msup id="S2.SS2.p3.7.m6.1.1.3" xref="S2.SS2.p3.7.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.7.m6.1.1.3.2" xref="S2.SS2.p3.7.m6.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS2.p3.7.m6.1.1.3.3" xref="S2.SS2.p3.7.m6.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.7.m6.1b"><apply id="S2.SS2.p3.7.m6.1.1.cmml" xref="S2.SS2.p3.7.m6.1.1"><subset id="S2.SS2.p3.7.m6.1.1.1.cmml" xref="S2.SS2.p3.7.m6.1.1.1"></subset><ci id="S2.SS2.p3.7.m6.1.1.2.cmml" xref="S2.SS2.p3.7.m6.1.1.2">𝑋</ci><apply id="S2.SS2.p3.7.m6.1.1.3.cmml" xref="S2.SS2.p3.7.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.7.m6.1.1.3.1.cmml" xref="S2.SS2.p3.7.m6.1.1.3">superscript</csymbol><ci id="S2.SS2.p3.7.m6.1.1.3.2.cmml" xref="S2.SS2.p3.7.m6.1.1.3.2">𝒜</ci><ci id="S2.SS2.p3.7.m6.1.1.3.3.cmml" xref="S2.SS2.p3.7.m6.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.7.m6.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.7.m6.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, listed below as follows. Notice that here the assumption on <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS2.p3.8.m7.1"><semantics id="S2.SS2.p3.8.m7.1a"><mi id="S2.SS2.p3.8.m7.1.1" xref="S2.SS2.p3.8.m7.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.8.m7.1b"><ci id="S2.SS2.p3.8.m7.1.1.cmml" xref="S2.SS2.p3.8.m7.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.8.m7.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.8.m7.1d">italic_σ</annotation></semantics></math> to be non-erasing is necessary.</p> </div> <div class="ltx_theorem ltx_theorem_defnrem" id="S2.Thmthm2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm2.1.1.1">Definition-Remark 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm2.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm2.p1"> <p class="ltx_p" id="S2.Thmthm2.p1.3">If <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.Thmthm2.p1.1.m1.1"><semantics id="S2.Thmthm2.p1.1.m1.1a"><mi id="S2.Thmthm2.p1.1.m1.1.1" xref="S2.Thmthm2.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm2.p1.1.m1.1b"><ci id="S2.Thmthm2.p1.1.m1.1.1.cmml" xref="S2.Thmthm2.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm2.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm2.p1.1.m1.1d">italic_σ</annotation></semantics></math> is non-erasing, then the following three definitions of the <span class="ltx_text ltx_font_italic" id="S2.Thmthm2.p1.3.1">image subshift</span> <math alttext="Y:=\sigma^{\Sigma}(X)" class="ltx_Math" display="inline" id="S2.Thmthm2.p1.2.m2.1"><semantics id="S2.Thmthm2.p1.2.m2.1a"><mrow id="S2.Thmthm2.p1.2.m2.1.2" xref="S2.Thmthm2.p1.2.m2.1.2.cmml"><mi id="S2.Thmthm2.p1.2.m2.1.2.2" xref="S2.Thmthm2.p1.2.m2.1.2.2.cmml">Y</mi><mo id="S2.Thmthm2.p1.2.m2.1.2.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm2.p1.2.m2.1.2.1.cmml">:=</mo><mrow id="S2.Thmthm2.p1.2.m2.1.2.3" xref="S2.Thmthm2.p1.2.m2.1.2.3.cmml"><msup id="S2.Thmthm2.p1.2.m2.1.2.3.2" xref="S2.Thmthm2.p1.2.m2.1.2.3.2.cmml"><mi id="S2.Thmthm2.p1.2.m2.1.2.3.2.2" xref="S2.Thmthm2.p1.2.m2.1.2.3.2.2.cmml">σ</mi><mi id="S2.Thmthm2.p1.2.m2.1.2.3.2.3" mathvariant="normal" xref="S2.Thmthm2.p1.2.m2.1.2.3.2.3.cmml">Σ</mi></msup><mo id="S2.Thmthm2.p1.2.m2.1.2.3.1" xref="S2.Thmthm2.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmthm2.p1.2.m2.1.2.3.3.2" xref="S2.Thmthm2.p1.2.m2.1.2.3.cmml"><mo id="S2.Thmthm2.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S2.Thmthm2.p1.2.m2.1.2.3.cmml">(</mo><mi id="S2.Thmthm2.p1.2.m2.1.1" xref="S2.Thmthm2.p1.2.m2.1.1.cmml">X</mi><mo id="S2.Thmthm2.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S2.Thmthm2.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm2.p1.2.m2.1b"><apply id="S2.Thmthm2.p1.2.m2.1.2.cmml" xref="S2.Thmthm2.p1.2.m2.1.2"><csymbol cd="latexml" id="S2.Thmthm2.p1.2.m2.1.2.1.cmml" xref="S2.Thmthm2.p1.2.m2.1.2.1">assign</csymbol><ci id="S2.Thmthm2.p1.2.m2.1.2.2.cmml" xref="S2.Thmthm2.p1.2.m2.1.2.2">𝑌</ci><apply id="S2.Thmthm2.p1.2.m2.1.2.3.cmml" xref="S2.Thmthm2.p1.2.m2.1.2.3"><times id="S2.Thmthm2.p1.2.m2.1.2.3.1.cmml" xref="S2.Thmthm2.p1.2.m2.1.2.3.1"></times><apply id="S2.Thmthm2.p1.2.m2.1.2.3.2.cmml" xref="S2.Thmthm2.p1.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S2.Thmthm2.p1.2.m2.1.2.3.2.1.cmml" xref="S2.Thmthm2.p1.2.m2.1.2.3.2">superscript</csymbol><ci id="S2.Thmthm2.p1.2.m2.1.2.3.2.2.cmml" xref="S2.Thmthm2.p1.2.m2.1.2.3.2.2">𝜎</ci><ci id="S2.Thmthm2.p1.2.m2.1.2.3.2.3.cmml" xref="S2.Thmthm2.p1.2.m2.1.2.3.2.3">Σ</ci></apply><ci id="S2.Thmthm2.p1.2.m2.1.1.cmml" xref="S2.Thmthm2.p1.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm2.p1.2.m2.1c">Y:=\sigma^{\Sigma}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm2.p1.2.m2.1d">italic_Y := italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math>, for simplicity usually denoted by <math alttext="Y=\sigma(X)" class="ltx_Math" display="inline" id="S2.Thmthm2.p1.3.m3.1"><semantics id="S2.Thmthm2.p1.3.m3.1a"><mrow id="S2.Thmthm2.p1.3.m3.1.2" xref="S2.Thmthm2.p1.3.m3.1.2.cmml"><mi id="S2.Thmthm2.p1.3.m3.1.2.2" xref="S2.Thmthm2.p1.3.m3.1.2.2.cmml">Y</mi><mo id="S2.Thmthm2.p1.3.m3.1.2.1" xref="S2.Thmthm2.p1.3.m3.1.2.1.cmml">=</mo><mrow id="S2.Thmthm2.p1.3.m3.1.2.3" xref="S2.Thmthm2.p1.3.m3.1.2.3.cmml"><mi id="S2.Thmthm2.p1.3.m3.1.2.3.2" xref="S2.Thmthm2.p1.3.m3.1.2.3.2.cmml">σ</mi><mo id="S2.Thmthm2.p1.3.m3.1.2.3.1" xref="S2.Thmthm2.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmthm2.p1.3.m3.1.2.3.3.2" xref="S2.Thmthm2.p1.3.m3.1.2.3.cmml"><mo id="S2.Thmthm2.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S2.Thmthm2.p1.3.m3.1.2.3.cmml">(</mo><mi id="S2.Thmthm2.p1.3.m3.1.1" xref="S2.Thmthm2.p1.3.m3.1.1.cmml">X</mi><mo id="S2.Thmthm2.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S2.Thmthm2.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm2.p1.3.m3.1b"><apply id="S2.Thmthm2.p1.3.m3.1.2.cmml" xref="S2.Thmthm2.p1.3.m3.1.2"><eq id="S2.Thmthm2.p1.3.m3.1.2.1.cmml" xref="S2.Thmthm2.p1.3.m3.1.2.1"></eq><ci id="S2.Thmthm2.p1.3.m3.1.2.2.cmml" xref="S2.Thmthm2.p1.3.m3.1.2.2">𝑌</ci><apply id="S2.Thmthm2.p1.3.m3.1.2.3.cmml" xref="S2.Thmthm2.p1.3.m3.1.2.3"><times id="S2.Thmthm2.p1.3.m3.1.2.3.1.cmml" xref="S2.Thmthm2.p1.3.m3.1.2.3.1"></times><ci id="S2.Thmthm2.p1.3.m3.1.2.3.2.cmml" xref="S2.Thmthm2.p1.3.m3.1.2.3.2">𝜎</ci><ci id="S2.Thmthm2.p1.3.m3.1.1.cmml" xref="S2.Thmthm2.p1.3.m3.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm2.p1.3.m3.1c">Y=\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm2.p1.3.m3.1d">italic_Y = italic_σ ( italic_X )</annotation></semantics></math>, are equivalent:</p> <ol class="ltx_enumerate" id="S2.I1"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.2"><math alttext="Y" class="ltx_Math" display="inline" id="S2.I1.i1.p1.1.m1.1"><semantics id="S2.I1.i1.p1.1.m1.1a"><mi id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><ci id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.1d">italic_Y</annotation></semantics></math> is the intersection of all subshifts that contain the set <math alttext="\sigma^{\mathbb{Z}}(X)" class="ltx_Math" display="inline" id="S2.I1.i1.p1.2.m2.1"><semantics id="S2.I1.i1.p1.2.m2.1a"><mrow id="S2.I1.i1.p1.2.m2.1.2" xref="S2.I1.i1.p1.2.m2.1.2.cmml"><msup id="S2.I1.i1.p1.2.m2.1.2.2" xref="S2.I1.i1.p1.2.m2.1.2.2.cmml"><mi id="S2.I1.i1.p1.2.m2.1.2.2.2" xref="S2.I1.i1.p1.2.m2.1.2.2.2.cmml">σ</mi><mi id="S2.I1.i1.p1.2.m2.1.2.2.3" xref="S2.I1.i1.p1.2.m2.1.2.2.3.cmml">ℤ</mi></msup><mo id="S2.I1.i1.p1.2.m2.1.2.1" xref="S2.I1.i1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.I1.i1.p1.2.m2.1.2.3.2" xref="S2.I1.i1.p1.2.m2.1.2.cmml"><mo id="S2.I1.i1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.I1.i1.p1.2.m2.1.2.cmml">(</mo><mi id="S2.I1.i1.p1.2.m2.1.1" xref="S2.I1.i1.p1.2.m2.1.1.cmml">X</mi><mo id="S2.I1.i1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.I1.i1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.2.m2.1b"><apply id="S2.I1.i1.p1.2.m2.1.2.cmml" xref="S2.I1.i1.p1.2.m2.1.2"><times id="S2.I1.i1.p1.2.m2.1.2.1.cmml" xref="S2.I1.i1.p1.2.m2.1.2.1"></times><apply id="S2.I1.i1.p1.2.m2.1.2.2.cmml" xref="S2.I1.i1.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.I1.i1.p1.2.m2.1.2.2.1.cmml" xref="S2.I1.i1.p1.2.m2.1.2.2">superscript</csymbol><ci id="S2.I1.i1.p1.2.m2.1.2.2.2.cmml" xref="S2.I1.i1.p1.2.m2.1.2.2.2">𝜎</ci><ci id="S2.I1.i1.p1.2.m2.1.2.2.3.cmml" xref="S2.I1.i1.p1.2.m2.1.2.2.3">ℤ</ci></apply><ci id="S2.I1.i1.p1.2.m2.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.2.m2.1c">\sigma^{\mathbb{Z}}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> (which in general is not shift-invariant and hence not a subshift itself).</p> </div> </li> <li class="ltx_item" id="S2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.3"><math alttext="Y" class="ltx_Math" display="inline" id="S2.I1.i2.p1.1.m1.1"><semantics id="S2.I1.i2.p1.1.m1.1a"><mi id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.1b"><ci id="S2.I1.i2.p1.1.m1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.1.m1.1d">italic_Y</annotation></semantics></math> is the closure of the union of all image orbits <math alttext="\sigma^{T}(\cal O({\bf x}))" class="ltx_Math" display="inline" id="S2.I1.i2.p1.2.m2.2"><semantics id="S2.I1.i2.p1.2.m2.2a"><mrow id="S2.I1.i2.p1.2.m2.2.2" xref="S2.I1.i2.p1.2.m2.2.2.cmml"><msup id="S2.I1.i2.p1.2.m2.2.2.3" xref="S2.I1.i2.p1.2.m2.2.2.3.cmml"><mi id="S2.I1.i2.p1.2.m2.2.2.3.2" xref="S2.I1.i2.p1.2.m2.2.2.3.2.cmml">σ</mi><mi id="S2.I1.i2.p1.2.m2.2.2.3.3" xref="S2.I1.i2.p1.2.m2.2.2.3.3.cmml">T</mi></msup><mo id="S2.I1.i2.p1.2.m2.2.2.2" xref="S2.I1.i2.p1.2.m2.2.2.2.cmml">⁢</mo><mrow id="S2.I1.i2.p1.2.m2.2.2.1.1" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.cmml"><mo id="S2.I1.i2.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.cmml">(</mo><mrow id="S2.I1.i2.p1.2.m2.2.2.1.1.1" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I1.i2.p1.2.m2.2.2.1.1.1.2" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.2.cmml">𝒪</mi><mo id="S2.I1.i2.p1.2.m2.2.2.1.1.1.1" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.I1.i2.p1.2.m2.2.2.1.1.1.3.2" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.cmml"><mo id="S2.I1.i2.p1.2.m2.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.cmml">(</mo><mi id="S2.I1.i2.p1.2.m2.1.1" xref="S2.I1.i2.p1.2.m2.1.1.cmml">𝐱</mi><mo id="S2.I1.i2.p1.2.m2.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I1.i2.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.2b"><apply id="S2.I1.i2.p1.2.m2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2"><times id="S2.I1.i2.p1.2.m2.2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2.2"></times><apply id="S2.I1.i2.p1.2.m2.2.2.3.cmml" xref="S2.I1.i2.p1.2.m2.2.2.3"><csymbol cd="ambiguous" id="S2.I1.i2.p1.2.m2.2.2.3.1.cmml" xref="S2.I1.i2.p1.2.m2.2.2.3">superscript</csymbol><ci id="S2.I1.i2.p1.2.m2.2.2.3.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2.3.2">𝜎</ci><ci id="S2.I1.i2.p1.2.m2.2.2.3.3.cmml" xref="S2.I1.i2.p1.2.m2.2.2.3.3">𝑇</ci></apply><apply id="S2.I1.i2.p1.2.m2.2.2.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.2.2.1.1"><times id="S2.I1.i2.p1.2.m2.2.2.1.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.1"></times><ci id="S2.I1.i2.p1.2.m2.2.2.1.1.1.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2.1.1.1.2">𝒪</ci><ci id="S2.I1.i2.p1.2.m2.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.2c">\sigma^{T}(\cal O({\bf x}))</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.2.m2.2d">italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( caligraphic_O ( bold_x ) )</annotation></semantics></math>, for any <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S2.I1.i2.p1.3.m3.1"><semantics id="S2.I1.i2.p1.3.m3.1a"><mrow id="S2.I1.i2.p1.3.m3.1.1" xref="S2.I1.i2.p1.3.m3.1.1.cmml"><mi id="S2.I1.i2.p1.3.m3.1.1.2" xref="S2.I1.i2.p1.3.m3.1.1.2.cmml">𝐱</mi><mo id="S2.I1.i2.p1.3.m3.1.1.1" xref="S2.I1.i2.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S2.I1.i2.p1.3.m3.1.1.3" xref="S2.I1.i2.p1.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.3.m3.1b"><apply id="S2.I1.i2.p1.3.m3.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1"><in id="S2.I1.i2.p1.3.m3.1.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1.1"></in><ci id="S2.I1.i2.p1.3.m3.1.1.2.cmml" xref="S2.I1.i2.p1.3.m3.1.1.2">𝐱</ci><ci id="S2.I1.i2.p1.3.m3.1.1.3.cmml" xref="S2.I1.i2.p1.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.3.m3.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.3.m3.1d">bold_x ∈ italic_X</annotation></semantics></math>. In fact (see Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm4" title="Lemma 2.4. ‣ 2.2. “Not so standard” basic facts and terminology ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.4</span></a> below), taking the closure in the previous sentence can be omitted.</p> </div> </li> <li class="ltx_item" id="S2.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S2.I1.i3.p1"> <p class="ltx_p" id="S2.I1.i3.p1.6"><math alttext="Y" class="ltx_Math" display="inline" id="S2.I1.i3.p1.1.m1.1"><semantics id="S2.I1.i3.p1.1.m1.1a"><mi id="S2.I1.i3.p1.1.m1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.1.m1.1b"><ci id="S2.I1.i3.p1.1.m1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.1.m1.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.1.m1.1d">italic_Y</annotation></semantics></math> is the subshift generated by the language <math alttext="\sigma(\cal L(X))" class="ltx_Math" display="inline" id="S2.I1.i3.p1.2.m2.2"><semantics id="S2.I1.i3.p1.2.m2.2a"><mrow id="S2.I1.i3.p1.2.m2.2.2" xref="S2.I1.i3.p1.2.m2.2.2.cmml"><mi id="S2.I1.i3.p1.2.m2.2.2.3" xref="S2.I1.i3.p1.2.m2.2.2.3.cmml">σ</mi><mo id="S2.I1.i3.p1.2.m2.2.2.2" xref="S2.I1.i3.p1.2.m2.2.2.2.cmml">⁢</mo><mrow id="S2.I1.i3.p1.2.m2.2.2.1.1" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.cmml"><mo id="S2.I1.i3.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.cmml">(</mo><mrow id="S2.I1.i3.p1.2.m2.2.2.1.1.1" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I1.i3.p1.2.m2.2.2.1.1.1.2" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.2.cmml">ℒ</mi><mo id="S2.I1.i3.p1.2.m2.2.2.1.1.1.1" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.I1.i3.p1.2.m2.2.2.1.1.1.3.2" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.cmml"><mo id="S2.I1.i3.p1.2.m2.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I1.i3.p1.2.m2.1.1" xref="S2.I1.i3.p1.2.m2.1.1.cmml">𝒳</mi><mo id="S2.I1.i3.p1.2.m2.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I1.i3.p1.2.m2.2.2.1.1.3" stretchy="false" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.2.m2.2b"><apply id="S2.I1.i3.p1.2.m2.2.2.cmml" xref="S2.I1.i3.p1.2.m2.2.2"><times id="S2.I1.i3.p1.2.m2.2.2.2.cmml" xref="S2.I1.i3.p1.2.m2.2.2.2"></times><ci id="S2.I1.i3.p1.2.m2.2.2.3.cmml" xref="S2.I1.i3.p1.2.m2.2.2.3">𝜎</ci><apply id="S2.I1.i3.p1.2.m2.2.2.1.1.1.cmml" xref="S2.I1.i3.p1.2.m2.2.2.1.1"><times id="S2.I1.i3.p1.2.m2.2.2.1.1.1.1.cmml" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.1"></times><ci id="S2.I1.i3.p1.2.m2.2.2.1.1.1.2.cmml" xref="S2.I1.i3.p1.2.m2.2.2.1.1.1.2">ℒ</ci><ci id="S2.I1.i3.p1.2.m2.1.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.2.m2.2c">\sigma(\cal L(X))</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.2.m2.2d">italic_σ ( caligraphic_L ( caligraphic_X ) )</annotation></semantics></math>. Thus <math alttext="Y" class="ltx_Math" display="inline" id="S2.I1.i3.p1.3.m3.1"><semantics id="S2.I1.i3.p1.3.m3.1a"><mi id="S2.I1.i3.p1.3.m3.1.1" xref="S2.I1.i3.p1.3.m3.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.3.m3.1b"><ci id="S2.I1.i3.p1.3.m3.1.1.cmml" xref="S2.I1.i3.p1.3.m3.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.3.m3.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.3.m3.1d">italic_Y</annotation></semantics></math> consists of all biinfinite words <math alttext="{\bf y}\in\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.I1.i3.p1.4.m4.1"><semantics id="S2.I1.i3.p1.4.m4.1a"><mrow id="S2.I1.i3.p1.4.m4.1.1" xref="S2.I1.i3.p1.4.m4.1.1.cmml"><mi id="S2.I1.i3.p1.4.m4.1.1.2" xref="S2.I1.i3.p1.4.m4.1.1.2.cmml">𝐲</mi><mo id="S2.I1.i3.p1.4.m4.1.1.1" xref="S2.I1.i3.p1.4.m4.1.1.1.cmml">∈</mo><msup id="S2.I1.i3.p1.4.m4.1.1.3" xref="S2.I1.i3.p1.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I1.i3.p1.4.m4.1.1.3.2" xref="S2.I1.i3.p1.4.m4.1.1.3.2.cmml">ℬ</mi><mi id="S2.I1.i3.p1.4.m4.1.1.3.3" xref="S2.I1.i3.p1.4.m4.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.4.m4.1b"><apply id="S2.I1.i3.p1.4.m4.1.1.cmml" xref="S2.I1.i3.p1.4.m4.1.1"><in id="S2.I1.i3.p1.4.m4.1.1.1.cmml" xref="S2.I1.i3.p1.4.m4.1.1.1"></in><ci id="S2.I1.i3.p1.4.m4.1.1.2.cmml" xref="S2.I1.i3.p1.4.m4.1.1.2">𝐲</ci><apply id="S2.I1.i3.p1.4.m4.1.1.3.cmml" xref="S2.I1.i3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.I1.i3.p1.4.m4.1.1.3.1.cmml" xref="S2.I1.i3.p1.4.m4.1.1.3">superscript</csymbol><ci id="S2.I1.i3.p1.4.m4.1.1.3.2.cmml" xref="S2.I1.i3.p1.4.m4.1.1.3.2">ℬ</ci><ci id="S2.I1.i3.p1.4.m4.1.1.3.3.cmml" xref="S2.I1.i3.p1.4.m4.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.4.m4.1c">{\bf y}\in\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.4.m4.1d">bold_y ∈ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> with the property that every finite factor of <math alttext="\bf y" class="ltx_Math" display="inline" id="S2.I1.i3.p1.5.m5.1"><semantics id="S2.I1.i3.p1.5.m5.1a"><mi id="S2.I1.i3.p1.5.m5.1.1" xref="S2.I1.i3.p1.5.m5.1.1.cmml">𝐲</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.5.m5.1b"><ci id="S2.I1.i3.p1.5.m5.1.1.cmml" xref="S2.I1.i3.p1.5.m5.1.1">𝐲</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.5.m5.1c">\bf y</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.5.m5.1d">bold_y</annotation></semantics></math> is also a factor of some word in <math alttext="\sigma(\cal L(X))" class="ltx_Math" display="inline" id="S2.I1.i3.p1.6.m6.2"><semantics id="S2.I1.i3.p1.6.m6.2a"><mrow id="S2.I1.i3.p1.6.m6.2.2" xref="S2.I1.i3.p1.6.m6.2.2.cmml"><mi id="S2.I1.i3.p1.6.m6.2.2.3" xref="S2.I1.i3.p1.6.m6.2.2.3.cmml">σ</mi><mo id="S2.I1.i3.p1.6.m6.2.2.2" xref="S2.I1.i3.p1.6.m6.2.2.2.cmml">⁢</mo><mrow id="S2.I1.i3.p1.6.m6.2.2.1.1" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.cmml"><mo id="S2.I1.i3.p1.6.m6.2.2.1.1.2" stretchy="false" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.cmml">(</mo><mrow id="S2.I1.i3.p1.6.m6.2.2.1.1.1" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I1.i3.p1.6.m6.2.2.1.1.1.2" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.2.cmml">ℒ</mi><mo id="S2.I1.i3.p1.6.m6.2.2.1.1.1.1" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.I1.i3.p1.6.m6.2.2.1.1.1.3.2" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.cmml"><mo id="S2.I1.i3.p1.6.m6.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I1.i3.p1.6.m6.1.1" xref="S2.I1.i3.p1.6.m6.1.1.cmml">𝒳</mi><mo id="S2.I1.i3.p1.6.m6.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I1.i3.p1.6.m6.2.2.1.1.3" stretchy="false" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.6.m6.2b"><apply id="S2.I1.i3.p1.6.m6.2.2.cmml" xref="S2.I1.i3.p1.6.m6.2.2"><times id="S2.I1.i3.p1.6.m6.2.2.2.cmml" xref="S2.I1.i3.p1.6.m6.2.2.2"></times><ci id="S2.I1.i3.p1.6.m6.2.2.3.cmml" xref="S2.I1.i3.p1.6.m6.2.2.3">𝜎</ci><apply id="S2.I1.i3.p1.6.m6.2.2.1.1.1.cmml" xref="S2.I1.i3.p1.6.m6.2.2.1.1"><times id="S2.I1.i3.p1.6.m6.2.2.1.1.1.1.cmml" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.1"></times><ci id="S2.I1.i3.p1.6.m6.2.2.1.1.1.2.cmml" xref="S2.I1.i3.p1.6.m6.2.2.1.1.1.2">ℒ</ci><ci id="S2.I1.i3.p1.6.m6.1.1.cmml" xref="S2.I1.i3.p1.6.m6.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.6.m6.2c">\sigma(\cal L(X))</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.6.m6.2d">italic_σ ( caligraphic_L ( caligraphic_X ) )</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S2.Thmthm2.p1.4">In particular, the map <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm2.p1.4.m1.1"><semantics id="S2.Thmthm2.p1.4.m1.1a"><msup id="S2.Thmthm2.p1.4.m1.1.1" xref="S2.Thmthm2.p1.4.m1.1.1.cmml"><mi id="S2.Thmthm2.p1.4.m1.1.1.2" xref="S2.Thmthm2.p1.4.m1.1.1.2.cmml">σ</mi><mi id="S2.Thmthm2.p1.4.m1.1.1.3" xref="S2.Thmthm2.p1.4.m1.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm2.p1.4.m1.1b"><apply id="S2.Thmthm2.p1.4.m1.1.1.cmml" xref="S2.Thmthm2.p1.4.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm2.p1.4.m1.1.1.1.cmml" xref="S2.Thmthm2.p1.4.m1.1.1">superscript</csymbol><ci id="S2.Thmthm2.p1.4.m1.1.1.2.cmml" xref="S2.Thmthm2.p1.4.m1.1.1.2">𝜎</ci><ci id="S2.Thmthm2.p1.4.m1.1.1.3.cmml" xref="S2.Thmthm2.p1.4.m1.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm2.p1.4.m1.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm2.p1.4.m1.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> restricts/co-restricts to a map</p> <table class="ltx_equation ltx_eqn_table" id="S2.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.10)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{\mathbb{Z}}_{X}:X\to\sigma^{\mathbb{Z}}(X)\subseteq\sigma(X)\,," class="ltx_Math" display="block" id="S2.E10.m1.3"><semantics id="S2.E10.m1.3a"><mrow id="S2.E10.m1.3.3.1" xref="S2.E10.m1.3.3.1.1.cmml"><mrow id="S2.E10.m1.3.3.1.1" xref="S2.E10.m1.3.3.1.1.cmml"><msubsup id="S2.E10.m1.3.3.1.1.2" xref="S2.E10.m1.3.3.1.1.2.cmml"><mi id="S2.E10.m1.3.3.1.1.2.2.2" xref="S2.E10.m1.3.3.1.1.2.2.2.cmml">σ</mi><mi id="S2.E10.m1.3.3.1.1.2.3" xref="S2.E10.m1.3.3.1.1.2.3.cmml">X</mi><mi id="S2.E10.m1.3.3.1.1.2.2.3" xref="S2.E10.m1.3.3.1.1.2.2.3.cmml">ℤ</mi></msubsup><mo id="S2.E10.m1.3.3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.E10.m1.3.3.1.1.1.cmml">:</mo><mrow id="S2.E10.m1.3.3.1.1.3" xref="S2.E10.m1.3.3.1.1.3.cmml"><mi id="S2.E10.m1.3.3.1.1.3.2" xref="S2.E10.m1.3.3.1.1.3.2.cmml">X</mi><mo id="S2.E10.m1.3.3.1.1.3.3" stretchy="false" xref="S2.E10.m1.3.3.1.1.3.3.cmml">→</mo><mrow id="S2.E10.m1.3.3.1.1.3.4" xref="S2.E10.m1.3.3.1.1.3.4.cmml"><msup id="S2.E10.m1.3.3.1.1.3.4.2" xref="S2.E10.m1.3.3.1.1.3.4.2.cmml"><mi id="S2.E10.m1.3.3.1.1.3.4.2.2" xref="S2.E10.m1.3.3.1.1.3.4.2.2.cmml">σ</mi><mi id="S2.E10.m1.3.3.1.1.3.4.2.3" xref="S2.E10.m1.3.3.1.1.3.4.2.3.cmml">ℤ</mi></msup><mo id="S2.E10.m1.3.3.1.1.3.4.1" xref="S2.E10.m1.3.3.1.1.3.4.1.cmml">⁢</mo><mrow id="S2.E10.m1.3.3.1.1.3.4.3.2" xref="S2.E10.m1.3.3.1.1.3.4.cmml"><mo id="S2.E10.m1.3.3.1.1.3.4.3.2.1" stretchy="false" xref="S2.E10.m1.3.3.1.1.3.4.cmml">(</mo><mi id="S2.E10.m1.1.1" xref="S2.E10.m1.1.1.cmml">X</mi><mo id="S2.E10.m1.3.3.1.1.3.4.3.2.2" stretchy="false" xref="S2.E10.m1.3.3.1.1.3.4.cmml">)</mo></mrow></mrow><mo id="S2.E10.m1.3.3.1.1.3.5" xref="S2.E10.m1.3.3.1.1.3.5.cmml">⊆</mo><mrow id="S2.E10.m1.3.3.1.1.3.6" xref="S2.E10.m1.3.3.1.1.3.6.cmml"><mi id="S2.E10.m1.3.3.1.1.3.6.2" xref="S2.E10.m1.3.3.1.1.3.6.2.cmml">σ</mi><mo id="S2.E10.m1.3.3.1.1.3.6.1" xref="S2.E10.m1.3.3.1.1.3.6.1.cmml">⁢</mo><mrow id="S2.E10.m1.3.3.1.1.3.6.3.2" xref="S2.E10.m1.3.3.1.1.3.6.cmml"><mo id="S2.E10.m1.3.3.1.1.3.6.3.2.1" stretchy="false" xref="S2.E10.m1.3.3.1.1.3.6.cmml">(</mo><mi id="S2.E10.m1.2.2" xref="S2.E10.m1.2.2.cmml">X</mi><mo id="S2.E10.m1.3.3.1.1.3.6.3.2.2" rspace="0.170em" stretchy="false" xref="S2.E10.m1.3.3.1.1.3.6.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E10.m1.3.3.1.2" xref="S2.E10.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E10.m1.3b"><apply id="S2.E10.m1.3.3.1.1.cmml" xref="S2.E10.m1.3.3.1"><ci id="S2.E10.m1.3.3.1.1.1.cmml" xref="S2.E10.m1.3.3.1.1.1">:</ci><apply id="S2.E10.m1.3.3.1.1.2.cmml" xref="S2.E10.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S2.E10.m1.3.3.1.1.2.1.cmml" xref="S2.E10.m1.3.3.1.1.2">subscript</csymbol><apply id="S2.E10.m1.3.3.1.1.2.2.cmml" xref="S2.E10.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="S2.E10.m1.3.3.1.1.2.2.1.cmml" xref="S2.E10.m1.3.3.1.1.2">superscript</csymbol><ci id="S2.E10.m1.3.3.1.1.2.2.2.cmml" xref="S2.E10.m1.3.3.1.1.2.2.2">𝜎</ci><ci id="S2.E10.m1.3.3.1.1.2.2.3.cmml" xref="S2.E10.m1.3.3.1.1.2.2.3">ℤ</ci></apply><ci id="S2.E10.m1.3.3.1.1.2.3.cmml" xref="S2.E10.m1.3.3.1.1.2.3">𝑋</ci></apply><apply id="S2.E10.m1.3.3.1.1.3.cmml" xref="S2.E10.m1.3.3.1.1.3"><and id="S2.E10.m1.3.3.1.1.3a.cmml" xref="S2.E10.m1.3.3.1.1.3"></and><apply id="S2.E10.m1.3.3.1.1.3b.cmml" xref="S2.E10.m1.3.3.1.1.3"><ci id="S2.E10.m1.3.3.1.1.3.3.cmml" xref="S2.E10.m1.3.3.1.1.3.3">→</ci><ci id="S2.E10.m1.3.3.1.1.3.2.cmml" xref="S2.E10.m1.3.3.1.1.3.2">𝑋</ci><apply id="S2.E10.m1.3.3.1.1.3.4.cmml" xref="S2.E10.m1.3.3.1.1.3.4"><times id="S2.E10.m1.3.3.1.1.3.4.1.cmml" xref="S2.E10.m1.3.3.1.1.3.4.1"></times><apply id="S2.E10.m1.3.3.1.1.3.4.2.cmml" xref="S2.E10.m1.3.3.1.1.3.4.2"><csymbol cd="ambiguous" id="S2.E10.m1.3.3.1.1.3.4.2.1.cmml" xref="S2.E10.m1.3.3.1.1.3.4.2">superscript</csymbol><ci id="S2.E10.m1.3.3.1.1.3.4.2.2.cmml" xref="S2.E10.m1.3.3.1.1.3.4.2.2">𝜎</ci><ci id="S2.E10.m1.3.3.1.1.3.4.2.3.cmml" xref="S2.E10.m1.3.3.1.1.3.4.2.3">ℤ</ci></apply><ci id="S2.E10.m1.1.1.cmml" xref="S2.E10.m1.1.1">𝑋</ci></apply></apply><apply id="S2.E10.m1.3.3.1.1.3c.cmml" xref="S2.E10.m1.3.3.1.1.3"><subset id="S2.E10.m1.3.3.1.1.3.5.cmml" xref="S2.E10.m1.3.3.1.1.3.5"></subset><share href="https://arxiv.org/html/2211.11234v4#S2.E10.m1.3.3.1.1.3.4.cmml" id="S2.E10.m1.3.3.1.1.3d.cmml" xref="S2.E10.m1.3.3.1.1.3"></share><apply id="S2.E10.m1.3.3.1.1.3.6.cmml" xref="S2.E10.m1.3.3.1.1.3.6"><times id="S2.E10.m1.3.3.1.1.3.6.1.cmml" xref="S2.E10.m1.3.3.1.1.3.6.1"></times><ci id="S2.E10.m1.3.3.1.1.3.6.2.cmml" xref="S2.E10.m1.3.3.1.1.3.6.2">𝜎</ci><ci id="S2.E10.m1.2.2.cmml" xref="S2.E10.m1.2.2">𝑋</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.3c">\sigma^{\mathbb{Z}}_{X}:X\to\sigma^{\mathbb{Z}}(X)\subseteq\sigma(X)\,,</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.3d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT : italic_X → italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_X ) ⊆ italic_σ ( italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmthm2.p1.6">where the inclusion <math alttext="\sigma^{\mathbb{Z}}(X)\subseteq\sigma(X)" class="ltx_Math" display="inline" id="S2.Thmthm2.p1.5.m1.2"><semantics id="S2.Thmthm2.p1.5.m1.2a"><mrow id="S2.Thmthm2.p1.5.m1.2.3" xref="S2.Thmthm2.p1.5.m1.2.3.cmml"><mrow id="S2.Thmthm2.p1.5.m1.2.3.2" xref="S2.Thmthm2.p1.5.m1.2.3.2.cmml"><msup id="S2.Thmthm2.p1.5.m1.2.3.2.2" xref="S2.Thmthm2.p1.5.m1.2.3.2.2.cmml"><mi id="S2.Thmthm2.p1.5.m1.2.3.2.2.2" xref="S2.Thmthm2.p1.5.m1.2.3.2.2.2.cmml">σ</mi><mi id="S2.Thmthm2.p1.5.m1.2.3.2.2.3" xref="S2.Thmthm2.p1.5.m1.2.3.2.2.3.cmml">ℤ</mi></msup><mo id="S2.Thmthm2.p1.5.m1.2.3.2.1" xref="S2.Thmthm2.p1.5.m1.2.3.2.1.cmml">⁢</mo><mrow id="S2.Thmthm2.p1.5.m1.2.3.2.3.2" xref="S2.Thmthm2.p1.5.m1.2.3.2.cmml"><mo id="S2.Thmthm2.p1.5.m1.2.3.2.3.2.1" stretchy="false" xref="S2.Thmthm2.p1.5.m1.2.3.2.cmml">(</mo><mi id="S2.Thmthm2.p1.5.m1.1.1" xref="S2.Thmthm2.p1.5.m1.1.1.cmml">X</mi><mo id="S2.Thmthm2.p1.5.m1.2.3.2.3.2.2" stretchy="false" xref="S2.Thmthm2.p1.5.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm2.p1.5.m1.2.3.1" xref="S2.Thmthm2.p1.5.m1.2.3.1.cmml">⊆</mo><mrow id="S2.Thmthm2.p1.5.m1.2.3.3" xref="S2.Thmthm2.p1.5.m1.2.3.3.cmml"><mi id="S2.Thmthm2.p1.5.m1.2.3.3.2" xref="S2.Thmthm2.p1.5.m1.2.3.3.2.cmml">σ</mi><mo id="S2.Thmthm2.p1.5.m1.2.3.3.1" xref="S2.Thmthm2.p1.5.m1.2.3.3.1.cmml">⁢</mo><mrow id="S2.Thmthm2.p1.5.m1.2.3.3.3.2" xref="S2.Thmthm2.p1.5.m1.2.3.3.cmml"><mo id="S2.Thmthm2.p1.5.m1.2.3.3.3.2.1" stretchy="false" xref="S2.Thmthm2.p1.5.m1.2.3.3.cmml">(</mo><mi id="S2.Thmthm2.p1.5.m1.2.2" xref="S2.Thmthm2.p1.5.m1.2.2.cmml">X</mi><mo id="S2.Thmthm2.p1.5.m1.2.3.3.3.2.2" stretchy="false" xref="S2.Thmthm2.p1.5.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm2.p1.5.m1.2b"><apply id="S2.Thmthm2.p1.5.m1.2.3.cmml" xref="S2.Thmthm2.p1.5.m1.2.3"><subset id="S2.Thmthm2.p1.5.m1.2.3.1.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.1"></subset><apply id="S2.Thmthm2.p1.5.m1.2.3.2.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.2"><times id="S2.Thmthm2.p1.5.m1.2.3.2.1.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.2.1"></times><apply id="S2.Thmthm2.p1.5.m1.2.3.2.2.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.2.2"><csymbol cd="ambiguous" id="S2.Thmthm2.p1.5.m1.2.3.2.2.1.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.2.2">superscript</csymbol><ci id="S2.Thmthm2.p1.5.m1.2.3.2.2.2.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.2.2.2">𝜎</ci><ci id="S2.Thmthm2.p1.5.m1.2.3.2.2.3.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.2.2.3">ℤ</ci></apply><ci id="S2.Thmthm2.p1.5.m1.1.1.cmml" xref="S2.Thmthm2.p1.5.m1.1.1">𝑋</ci></apply><apply id="S2.Thmthm2.p1.5.m1.2.3.3.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.3"><times id="S2.Thmthm2.p1.5.m1.2.3.3.1.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.3.1"></times><ci id="S2.Thmthm2.p1.5.m1.2.3.3.2.cmml" xref="S2.Thmthm2.p1.5.m1.2.3.3.2">𝜎</ci><ci id="S2.Thmthm2.p1.5.m1.2.2.cmml" xref="S2.Thmthm2.p1.5.m1.2.2">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm2.p1.5.m1.2c">\sigma^{\mathbb{Z}}(X)\subseteq\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm2.p1.5.m1.2d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_X ) ⊆ italic_σ ( italic_X )</annotation></semantics></math> is in general not an equality, due to our convention <math alttext="\sigma(X)=\sigma^{\Sigma}(X)" class="ltx_Math" display="inline" id="S2.Thmthm2.p1.6.m2.2"><semantics id="S2.Thmthm2.p1.6.m2.2a"><mrow id="S2.Thmthm2.p1.6.m2.2.3" xref="S2.Thmthm2.p1.6.m2.2.3.cmml"><mrow id="S2.Thmthm2.p1.6.m2.2.3.2" xref="S2.Thmthm2.p1.6.m2.2.3.2.cmml"><mi id="S2.Thmthm2.p1.6.m2.2.3.2.2" xref="S2.Thmthm2.p1.6.m2.2.3.2.2.cmml">σ</mi><mo id="S2.Thmthm2.p1.6.m2.2.3.2.1" xref="S2.Thmthm2.p1.6.m2.2.3.2.1.cmml">⁢</mo><mrow id="S2.Thmthm2.p1.6.m2.2.3.2.3.2" xref="S2.Thmthm2.p1.6.m2.2.3.2.cmml"><mo id="S2.Thmthm2.p1.6.m2.2.3.2.3.2.1" stretchy="false" xref="S2.Thmthm2.p1.6.m2.2.3.2.cmml">(</mo><mi id="S2.Thmthm2.p1.6.m2.1.1" xref="S2.Thmthm2.p1.6.m2.1.1.cmml">X</mi><mo id="S2.Thmthm2.p1.6.m2.2.3.2.3.2.2" stretchy="false" xref="S2.Thmthm2.p1.6.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm2.p1.6.m2.2.3.1" xref="S2.Thmthm2.p1.6.m2.2.3.1.cmml">=</mo><mrow id="S2.Thmthm2.p1.6.m2.2.3.3" xref="S2.Thmthm2.p1.6.m2.2.3.3.cmml"><msup id="S2.Thmthm2.p1.6.m2.2.3.3.2" xref="S2.Thmthm2.p1.6.m2.2.3.3.2.cmml"><mi id="S2.Thmthm2.p1.6.m2.2.3.3.2.2" xref="S2.Thmthm2.p1.6.m2.2.3.3.2.2.cmml">σ</mi><mi id="S2.Thmthm2.p1.6.m2.2.3.3.2.3" mathvariant="normal" xref="S2.Thmthm2.p1.6.m2.2.3.3.2.3.cmml">Σ</mi></msup><mo id="S2.Thmthm2.p1.6.m2.2.3.3.1" xref="S2.Thmthm2.p1.6.m2.2.3.3.1.cmml">⁢</mo><mrow id="S2.Thmthm2.p1.6.m2.2.3.3.3.2" xref="S2.Thmthm2.p1.6.m2.2.3.3.cmml"><mo id="S2.Thmthm2.p1.6.m2.2.3.3.3.2.1" stretchy="false" xref="S2.Thmthm2.p1.6.m2.2.3.3.cmml">(</mo><mi id="S2.Thmthm2.p1.6.m2.2.2" xref="S2.Thmthm2.p1.6.m2.2.2.cmml">X</mi><mo id="S2.Thmthm2.p1.6.m2.2.3.3.3.2.2" stretchy="false" xref="S2.Thmthm2.p1.6.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm2.p1.6.m2.2b"><apply id="S2.Thmthm2.p1.6.m2.2.3.cmml" xref="S2.Thmthm2.p1.6.m2.2.3"><eq id="S2.Thmthm2.p1.6.m2.2.3.1.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.1"></eq><apply id="S2.Thmthm2.p1.6.m2.2.3.2.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.2"><times id="S2.Thmthm2.p1.6.m2.2.3.2.1.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.2.1"></times><ci id="S2.Thmthm2.p1.6.m2.2.3.2.2.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.2.2">𝜎</ci><ci id="S2.Thmthm2.p1.6.m2.1.1.cmml" xref="S2.Thmthm2.p1.6.m2.1.1">𝑋</ci></apply><apply id="S2.Thmthm2.p1.6.m2.2.3.3.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.3"><times id="S2.Thmthm2.p1.6.m2.2.3.3.1.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.3.1"></times><apply id="S2.Thmthm2.p1.6.m2.2.3.3.2.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.3.2"><csymbol cd="ambiguous" id="S2.Thmthm2.p1.6.m2.2.3.3.2.1.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.3.2">superscript</csymbol><ci id="S2.Thmthm2.p1.6.m2.2.3.3.2.2.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.3.2.2">𝜎</ci><ci id="S2.Thmthm2.p1.6.m2.2.3.3.2.3.cmml" xref="S2.Thmthm2.p1.6.m2.2.3.3.2.3">Σ</ci></apply><ci id="S2.Thmthm2.p1.6.m2.2.2.cmml" xref="S2.Thmthm2.p1.6.m2.2.2">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm2.p1.6.m2.2c">\sigma(X)=\sigma^{\Sigma}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm2.p1.6.m2.2d">italic_σ ( italic_X ) = italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.1">The proof of the equivalence of the statements (1) - (3) above is straight forward and hence left here to the reader (except for the part proved below in Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm4" title="Lemma 2.4. ‣ 2.2. “Not so standard” basic facts and terminology ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.4</span></a>). We do however illustrate the terms used above by making them explicit in the following special case:</p> </div> <div class="ltx_theorem ltx_theorem_example" id="S2.Thmthm3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm3.1.1.1">Example 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm3.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm3.p1"> <p class="ltx_p" id="S2.Thmthm3.p1.3">Let <math alttext="\cal A=\{a,b\}" class="ltx_Math" display="inline" id="S2.Thmthm3.p1.1.m1.2"><semantics id="S2.Thmthm3.p1.1.m1.2a"><mrow id="S2.Thmthm3.p1.1.m1.2.3" xref="S2.Thmthm3.p1.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm3.p1.1.m1.2.3.2" xref="S2.Thmthm3.p1.1.m1.2.3.2.cmml">𝒜</mi><mo id="S2.Thmthm3.p1.1.m1.2.3.1" xref="S2.Thmthm3.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.Thmthm3.p1.1.m1.2.3.3.2" xref="S2.Thmthm3.p1.1.m1.2.3.3.1.cmml"><mo id="S2.Thmthm3.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmthm3.p1.1.m1.2.3.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm3.p1.1.m1.1.1" xref="S2.Thmthm3.p1.1.m1.1.1.cmml">𝒶</mi><mo id="S2.Thmthm3.p1.1.m1.2.3.3.2.2" xref="S2.Thmthm3.p1.1.m1.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm3.p1.1.m1.2.2" xref="S2.Thmthm3.p1.1.m1.2.2.cmml">𝒷</mi><mo id="S2.Thmthm3.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmthm3.p1.1.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm3.p1.1.m1.2b"><apply id="S2.Thmthm3.p1.1.m1.2.3.cmml" xref="S2.Thmthm3.p1.1.m1.2.3"><eq id="S2.Thmthm3.p1.1.m1.2.3.1.cmml" xref="S2.Thmthm3.p1.1.m1.2.3.1"></eq><ci id="S2.Thmthm3.p1.1.m1.2.3.2.cmml" xref="S2.Thmthm3.p1.1.m1.2.3.2">𝒜</ci><set id="S2.Thmthm3.p1.1.m1.2.3.3.1.cmml" xref="S2.Thmthm3.p1.1.m1.2.3.3.2"><ci id="S2.Thmthm3.p1.1.m1.1.1.cmml" xref="S2.Thmthm3.p1.1.m1.1.1">𝒶</ci><ci id="S2.Thmthm3.p1.1.m1.2.2.cmml" xref="S2.Thmthm3.p1.1.m1.2.2">𝒷</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm3.p1.1.m1.2c">\cal A=\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm3.p1.1.m1.2d">caligraphic_A = { caligraphic_a , caligraphic_b }</annotation></semantics></math> and <math alttext="\cal B=\{c,d\}" class="ltx_Math" display="inline" id="S2.Thmthm3.p1.2.m2.2"><semantics id="S2.Thmthm3.p1.2.m2.2a"><mrow id="S2.Thmthm3.p1.2.m2.2.3" xref="S2.Thmthm3.p1.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm3.p1.2.m2.2.3.2" xref="S2.Thmthm3.p1.2.m2.2.3.2.cmml">ℬ</mi><mo id="S2.Thmthm3.p1.2.m2.2.3.1" xref="S2.Thmthm3.p1.2.m2.2.3.1.cmml">=</mo><mrow id="S2.Thmthm3.p1.2.m2.2.3.3.2" xref="S2.Thmthm3.p1.2.m2.2.3.3.1.cmml"><mo id="S2.Thmthm3.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S2.Thmthm3.p1.2.m2.2.3.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm3.p1.2.m2.1.1" xref="S2.Thmthm3.p1.2.m2.1.1.cmml">𝒸</mi><mo id="S2.Thmthm3.p1.2.m2.2.3.3.2.2" xref="S2.Thmthm3.p1.2.m2.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm3.p1.2.m2.2.2" xref="S2.Thmthm3.p1.2.m2.2.2.cmml">𝒹</mi><mo id="S2.Thmthm3.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S2.Thmthm3.p1.2.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm3.p1.2.m2.2b"><apply id="S2.Thmthm3.p1.2.m2.2.3.cmml" xref="S2.Thmthm3.p1.2.m2.2.3"><eq id="S2.Thmthm3.p1.2.m2.2.3.1.cmml" xref="S2.Thmthm3.p1.2.m2.2.3.1"></eq><ci id="S2.Thmthm3.p1.2.m2.2.3.2.cmml" xref="S2.Thmthm3.p1.2.m2.2.3.2">ℬ</ci><set id="S2.Thmthm3.p1.2.m2.2.3.3.1.cmml" xref="S2.Thmthm3.p1.2.m2.2.3.3.2"><ci id="S2.Thmthm3.p1.2.m2.1.1.cmml" xref="S2.Thmthm3.p1.2.m2.1.1">𝒸</ci><ci id="S2.Thmthm3.p1.2.m2.2.2.cmml" xref="S2.Thmthm3.p1.2.m2.2.2">𝒹</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm3.p1.2.m2.2c">\cal B=\{c,d\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm3.p1.2.m2.2d">caligraphic_B = { caligraphic_c , caligraphic_d }</annotation></semantics></math>, and define <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.Thmthm3.p1.3.m3.1"><semantics id="S2.Thmthm3.p1.3.m3.1a"><mrow id="S2.Thmthm3.p1.3.m3.1.1" xref="S2.Thmthm3.p1.3.m3.1.1.cmml"><mi id="S2.Thmthm3.p1.3.m3.1.1.2" xref="S2.Thmthm3.p1.3.m3.1.1.2.cmml">σ</mi><mo id="S2.Thmthm3.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm3.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S2.Thmthm3.p1.3.m3.1.1.3" xref="S2.Thmthm3.p1.3.m3.1.1.3.cmml"><msup id="S2.Thmthm3.p1.3.m3.1.1.3.2" xref="S2.Thmthm3.p1.3.m3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm3.p1.3.m3.1.1.3.2.2" xref="S2.Thmthm3.p1.3.m3.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.Thmthm3.p1.3.m3.1.1.3.2.3" xref="S2.Thmthm3.p1.3.m3.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm3.p1.3.m3.1.1.3.1" stretchy="false" xref="S2.Thmthm3.p1.3.m3.1.1.3.1.cmml">→</mo><msup id="S2.Thmthm3.p1.3.m3.1.1.3.3" xref="S2.Thmthm3.p1.3.m3.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm3.p1.3.m3.1.1.3.3.2" xref="S2.Thmthm3.p1.3.m3.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.Thmthm3.p1.3.m3.1.1.3.3.3" xref="S2.Thmthm3.p1.3.m3.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm3.p1.3.m3.1b"><apply id="S2.Thmthm3.p1.3.m3.1.1.cmml" xref="S2.Thmthm3.p1.3.m3.1.1"><ci id="S2.Thmthm3.p1.3.m3.1.1.1.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.1">:</ci><ci id="S2.Thmthm3.p1.3.m3.1.1.2.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.2">𝜎</ci><apply id="S2.Thmthm3.p1.3.m3.1.1.3.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3"><ci id="S2.Thmthm3.p1.3.m3.1.1.3.1.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.1">→</ci><apply id="S2.Thmthm3.p1.3.m3.1.1.3.2.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmthm3.p1.3.m3.1.1.3.2.1.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S2.Thmthm3.p1.3.m3.1.1.3.2.2.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.2.2">𝒜</ci><times id="S2.Thmthm3.p1.3.m3.1.1.3.2.3.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.2.3"></times></apply><apply id="S2.Thmthm3.p1.3.m3.1.1.3.3.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmthm3.p1.3.m3.1.1.3.3.1.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.3">superscript</csymbol><ci id="S2.Thmthm3.p1.3.m3.1.1.3.3.2.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.3.2">ℬ</ci><times id="S2.Thmthm3.p1.3.m3.1.1.3.3.3.cmml" xref="S2.Thmthm3.p1.3.m3.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm3.p1.3.m3.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm3.p1.3.m3.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> via</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="a\mapsto(cd)^{2}=cdcd\,,\,\,b\mapsto(cd)^{3}=cdcdcd\,." class="ltx_Math" display="block" id="S2.Ex8.m1.1"><semantics id="S2.Ex8.m1.1a"><mrow id="S2.Ex8.m1.1.1.1"><mrow id="S2.Ex8.m1.1.1.1.1.2" xref="S2.Ex8.m1.1.1.1.1.3.cmml"><mrow id="S2.Ex8.m1.1.1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.1.1.cmml"><mi id="S2.Ex8.m1.1.1.1.1.1.1.3" xref="S2.Ex8.m1.1.1.1.1.1.1.3.cmml">a</mi><mo id="S2.Ex8.m1.1.1.1.1.1.1.4" stretchy="false" xref="S2.Ex8.m1.1.1.1.1.1.1.4.cmml">↦</mo><msup id="S2.Ex8.m1.1.1.1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.1.1.1.cmml"><mrow id="S2.Ex8.m1.1.1.1.1.1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">c</mi><mo id="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex8.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S2.Ex8.m1.1.1.1.1.1.1.1.3" xref="S2.Ex8.m1.1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="S2.Ex8.m1.1.1.1.1.1.1.5" xref="S2.Ex8.m1.1.1.1.1.1.1.5.cmml">=</mo><mrow id="S2.Ex8.m1.1.1.1.1.1.1.6" xref="S2.Ex8.m1.1.1.1.1.1.1.6.cmml"><mi id="S2.Ex8.m1.1.1.1.1.1.1.6.2" xref="S2.Ex8.m1.1.1.1.1.1.1.6.2.cmml">c</mi><mo id="S2.Ex8.m1.1.1.1.1.1.1.6.1" xref="S2.Ex8.m1.1.1.1.1.1.1.6.1.cmml">⁢</mo><mi id="S2.Ex8.m1.1.1.1.1.1.1.6.3" xref="S2.Ex8.m1.1.1.1.1.1.1.6.3.cmml">d</mi><mo id="S2.Ex8.m1.1.1.1.1.1.1.6.1a" xref="S2.Ex8.m1.1.1.1.1.1.1.6.1.cmml">⁢</mo><mi id="S2.Ex8.m1.1.1.1.1.1.1.6.4" xref="S2.Ex8.m1.1.1.1.1.1.1.6.4.cmml">c</mi><mo id="S2.Ex8.m1.1.1.1.1.1.1.6.1b" xref="S2.Ex8.m1.1.1.1.1.1.1.6.1.cmml">⁢</mo><mi id="S2.Ex8.m1.1.1.1.1.1.1.6.5" xref="S2.Ex8.m1.1.1.1.1.1.1.6.5.cmml">d</mi></mrow></mrow><mo id="S2.Ex8.m1.1.1.1.1.2.3" lspace="0.170em" rspace="0.497em" xref="S2.Ex8.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S2.Ex8.m1.1.1.1.1.2.2" xref="S2.Ex8.m1.1.1.1.1.2.2.cmml"><mi id="S2.Ex8.m1.1.1.1.1.2.2.3" xref="S2.Ex8.m1.1.1.1.1.2.2.3.cmml">b</mi><mo id="S2.Ex8.m1.1.1.1.1.2.2.4" stretchy="false" xref="S2.Ex8.m1.1.1.1.1.2.2.4.cmml">↦</mo><msup id="S2.Ex8.m1.1.1.1.1.2.2.1" xref="S2.Ex8.m1.1.1.1.1.2.2.1.cmml"><mrow id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.cmml"><mo id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.2" stretchy="false" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.cmml"><mi id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.2" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.2.cmml">c</mi><mo id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.1" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.3" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.3.cmml">d</mi></mrow><mo id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.3" stretchy="false" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.cmml">)</mo></mrow><mn id="S2.Ex8.m1.1.1.1.1.2.2.1.3" xref="S2.Ex8.m1.1.1.1.1.2.2.1.3.cmml">3</mn></msup><mo id="S2.Ex8.m1.1.1.1.1.2.2.5" xref="S2.Ex8.m1.1.1.1.1.2.2.5.cmml">=</mo><mrow id="S2.Ex8.m1.1.1.1.1.2.2.6" xref="S2.Ex8.m1.1.1.1.1.2.2.6.cmml"><mi id="S2.Ex8.m1.1.1.1.1.2.2.6.2" xref="S2.Ex8.m1.1.1.1.1.2.2.6.2.cmml">c</mi><mo id="S2.Ex8.m1.1.1.1.1.2.2.6.1" xref="S2.Ex8.m1.1.1.1.1.2.2.6.1.cmml">⁢</mo><mi id="S2.Ex8.m1.1.1.1.1.2.2.6.3" xref="S2.Ex8.m1.1.1.1.1.2.2.6.3.cmml">d</mi><mo id="S2.Ex8.m1.1.1.1.1.2.2.6.1a" xref="S2.Ex8.m1.1.1.1.1.2.2.6.1.cmml">⁢</mo><mi id="S2.Ex8.m1.1.1.1.1.2.2.6.4" xref="S2.Ex8.m1.1.1.1.1.2.2.6.4.cmml">c</mi><mo id="S2.Ex8.m1.1.1.1.1.2.2.6.1b" xref="S2.Ex8.m1.1.1.1.1.2.2.6.1.cmml">⁢</mo><mi 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xref="S2.Ex8.m1.1.1.1.1.2.2.3">𝑏</ci><apply id="S2.Ex8.m1.1.1.1.1.2.2.1.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2.1"><csymbol cd="ambiguous" id="S2.Ex8.m1.1.1.1.1.2.2.1.2.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2.1">superscript</csymbol><apply id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1"><times id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.1.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.1"></times><ci id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.2.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.2">𝑐</ci><ci id="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.3.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2.1.1.1.1.3">𝑑</ci></apply><cn id="S2.Ex8.m1.1.1.1.1.2.2.1.3.cmml" type="integer" xref="S2.Ex8.m1.1.1.1.1.2.2.1.3">3</cn></apply></apply><apply id="S2.Ex8.m1.1.1.1.1.2.2c.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2"><eq id="S2.Ex8.m1.1.1.1.1.2.2.5.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S2.Ex8.m1.1.1.1.1.2.2.1.cmml" id="S2.Ex8.m1.1.1.1.1.2.2d.cmml" xref="S2.Ex8.m1.1.1.1.1.2.2"></share><apply 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italic_d ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT = italic_c italic_d italic_c italic_d italic_c italic_d .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmthm3.p1.4">We set <math alttext="X:=\{a^{\pm\infty},\,b^{\pm\infty}\}" class="ltx_Math" display="inline" id="S2.Thmthm3.p1.4.m1.2"><semantics id="S2.Thmthm3.p1.4.m1.2a"><mrow id="S2.Thmthm3.p1.4.m1.2.2" xref="S2.Thmthm3.p1.4.m1.2.2.cmml"><mi id="S2.Thmthm3.p1.4.m1.2.2.4" xref="S2.Thmthm3.p1.4.m1.2.2.4.cmml">X</mi><mo id="S2.Thmthm3.p1.4.m1.2.2.3" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm3.p1.4.m1.2.2.3.cmml">:=</mo><mrow id="S2.Thmthm3.p1.4.m1.2.2.2.2" xref="S2.Thmthm3.p1.4.m1.2.2.2.3.cmml"><mo id="S2.Thmthm3.p1.4.m1.2.2.2.2.3" stretchy="false" xref="S2.Thmthm3.p1.4.m1.2.2.2.3.cmml">{</mo><msup id="S2.Thmthm3.p1.4.m1.1.1.1.1.1" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.cmml"><mi id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.2" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.2.cmml">a</mi><mrow id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3.cmml"><mo id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3a" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3.cmml">±</mo><mi id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm3.p1.4.m1.2.2.2.2.4" rspace="0.337em" xref="S2.Thmthm3.p1.4.m1.2.2.2.3.cmml">,</mo><msup id="S2.Thmthm3.p1.4.m1.2.2.2.2.2" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.cmml"><mi id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.2" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.2.cmml">b</mi><mrow id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3.cmml"><mo id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3a" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3.cmml">±</mo><mi id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3.2" mathvariant="normal" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm3.p1.4.m1.2.2.2.2.5" stretchy="false" xref="S2.Thmthm3.p1.4.m1.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm3.p1.4.m1.2b"><apply id="S2.Thmthm3.p1.4.m1.2.2.cmml" xref="S2.Thmthm3.p1.4.m1.2.2"><csymbol cd="latexml" id="S2.Thmthm3.p1.4.m1.2.2.3.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.3">assign</csymbol><ci id="S2.Thmthm3.p1.4.m1.2.2.4.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.4">𝑋</ci><set id="S2.Thmthm3.p1.4.m1.2.2.2.3.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.2.2"><apply id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.cmml" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.1.cmml" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1">superscript</csymbol><ci id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.2.cmml" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.2">𝑎</ci><apply id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3.cmml" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm3.p1.4.m1.1.1.1.1.1.3.2"></infinity></apply></apply><apply id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.1.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2">superscript</csymbol><ci id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.2.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.2">𝑏</ci><apply id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3"><csymbol cd="latexml" id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3.1.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3">plus-or-minus</csymbol><infinity id="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3.2.cmml" xref="S2.Thmthm3.p1.4.m1.2.2.2.2.2.3.2"></infinity></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm3.p1.4.m1.2c">X:=\{a^{\pm\infty},\,b^{\pm\infty}\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm3.p1.4.m1.2d">italic_X := { italic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT }</annotation></semantics></math> and obtain:</p> <ul class="ltx_itemize" id="S2.I2"> <li class="ltx_item" id="S2.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i1.p1"> <p class="ltx_p" id="S2.I2.i1.p1.1"><math alttext="\sigma^{\mathbb{Z}}(X)=\{(cd)^{\pm\infty}\}" class="ltx_Math" display="inline" id="S2.I2.i1.p1.1.m1.2"><semantics id="S2.I2.i1.p1.1.m1.2a"><mrow id="S2.I2.i1.p1.1.m1.2.2" xref="S2.I2.i1.p1.1.m1.2.2.cmml"><mrow id="S2.I2.i1.p1.1.m1.2.2.3" xref="S2.I2.i1.p1.1.m1.2.2.3.cmml"><msup id="S2.I2.i1.p1.1.m1.2.2.3.2" xref="S2.I2.i1.p1.1.m1.2.2.3.2.cmml"><mi id="S2.I2.i1.p1.1.m1.2.2.3.2.2" xref="S2.I2.i1.p1.1.m1.2.2.3.2.2.cmml">σ</mi><mi id="S2.I2.i1.p1.1.m1.2.2.3.2.3" xref="S2.I2.i1.p1.1.m1.2.2.3.2.3.cmml">ℤ</mi></msup><mo id="S2.I2.i1.p1.1.m1.2.2.3.1" xref="S2.I2.i1.p1.1.m1.2.2.3.1.cmml">⁢</mo><mrow id="S2.I2.i1.p1.1.m1.2.2.3.3.2" xref="S2.I2.i1.p1.1.m1.2.2.3.cmml"><mo id="S2.I2.i1.p1.1.m1.2.2.3.3.2.1" stretchy="false" xref="S2.I2.i1.p1.1.m1.2.2.3.cmml">(</mo><mi id="S2.I2.i1.p1.1.m1.1.1" xref="S2.I2.i1.p1.1.m1.1.1.cmml">X</mi><mo id="S2.I2.i1.p1.1.m1.2.2.3.3.2.2" stretchy="false" xref="S2.I2.i1.p1.1.m1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.I2.i1.p1.1.m1.2.2.2" xref="S2.I2.i1.p1.1.m1.2.2.2.cmml">=</mo><mrow id="S2.I2.i1.p1.1.m1.2.2.1.1" xref="S2.I2.i1.p1.1.m1.2.2.1.2.cmml"><mo id="S2.I2.i1.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S2.I2.i1.p1.1.m1.2.2.1.2.cmml">{</mo><msup id="S2.I2.i1.p1.1.m1.2.2.1.1.1" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.cmml"><mrow id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml">c</mi><mo id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.1" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.I2.i1.p1.1.m1.2.2.1.1.1.3" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.3.cmml"><mo id="S2.I2.i1.p1.1.m1.2.2.1.1.1.3a" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.3.cmml">±</mo><mi id="S2.I2.i1.p1.1.m1.2.2.1.1.1.3.2" mathvariant="normal" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.I2.i1.p1.1.m1.2.2.1.1.3" stretchy="false" xref="S2.I2.i1.p1.1.m1.2.2.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i1.p1.1.m1.2b"><apply id="S2.I2.i1.p1.1.m1.2.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2"><eq id="S2.I2.i1.p1.1.m1.2.2.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2.2"></eq><apply id="S2.I2.i1.p1.1.m1.2.2.3.cmml" xref="S2.I2.i1.p1.1.m1.2.2.3"><times id="S2.I2.i1.p1.1.m1.2.2.3.1.cmml" xref="S2.I2.i1.p1.1.m1.2.2.3.1"></times><apply id="S2.I2.i1.p1.1.m1.2.2.3.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2.3.2"><csymbol cd="ambiguous" id="S2.I2.i1.p1.1.m1.2.2.3.2.1.cmml" xref="S2.I2.i1.p1.1.m1.2.2.3.2">superscript</csymbol><ci id="S2.I2.i1.p1.1.m1.2.2.3.2.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2.3.2.2">𝜎</ci><ci id="S2.I2.i1.p1.1.m1.2.2.3.2.3.cmml" xref="S2.I2.i1.p1.1.m1.2.2.3.2.3">ℤ</ci></apply><ci id="S2.I2.i1.p1.1.m1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.1.1">𝑋</ci></apply><set id="S2.I2.i1.p1.1.m1.2.2.1.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1"><apply id="S2.I2.i1.p1.1.m1.2.2.1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.I2.i1.p1.1.m1.2.2.1.1.1.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1">superscript</csymbol><apply id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1"><times id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.1"></times><ci id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2">𝑐</ci><ci id="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3">𝑑</ci></apply><apply id="S2.I2.i1.p1.1.m1.2.2.1.1.1.3.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.3"><csymbol cd="latexml" id="S2.I2.i1.p1.1.m1.2.2.1.1.1.3.1.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.I2.i1.p1.1.m1.2.2.1.1.1.3.2.cmml" xref="S2.I2.i1.p1.1.m1.2.2.1.1.1.3.2"></infinity></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i1.p1.1.m1.2c">\sigma^{\mathbb{Z}}(X)=\{(cd)^{\pm\infty}\}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i1.p1.1.m1.2d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_X ) = { ( italic_c italic_d ) start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT }</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S2.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i2.p1"> <p class="ltx_p" id="S2.I2.i2.p1.1"><math alttext="\sigma^{\Sigma}(X)=\{(cd)^{\pm\infty},(dc)^{\pm\infty}\}" class="ltx_Math" display="inline" id="S2.I2.i2.p1.1.m1.3"><semantics id="S2.I2.i2.p1.1.m1.3a"><mrow id="S2.I2.i2.p1.1.m1.3.3" xref="S2.I2.i2.p1.1.m1.3.3.cmml"><mrow id="S2.I2.i2.p1.1.m1.3.3.4" xref="S2.I2.i2.p1.1.m1.3.3.4.cmml"><msup id="S2.I2.i2.p1.1.m1.3.3.4.2" xref="S2.I2.i2.p1.1.m1.3.3.4.2.cmml"><mi id="S2.I2.i2.p1.1.m1.3.3.4.2.2" xref="S2.I2.i2.p1.1.m1.3.3.4.2.2.cmml">σ</mi><mi id="S2.I2.i2.p1.1.m1.3.3.4.2.3" mathvariant="normal" xref="S2.I2.i2.p1.1.m1.3.3.4.2.3.cmml">Σ</mi></msup><mo id="S2.I2.i2.p1.1.m1.3.3.4.1" xref="S2.I2.i2.p1.1.m1.3.3.4.1.cmml">⁢</mo><mrow id="S2.I2.i2.p1.1.m1.3.3.4.3.2" xref="S2.I2.i2.p1.1.m1.3.3.4.cmml"><mo id="S2.I2.i2.p1.1.m1.3.3.4.3.2.1" stretchy="false" xref="S2.I2.i2.p1.1.m1.3.3.4.cmml">(</mo><mi id="S2.I2.i2.p1.1.m1.1.1" xref="S2.I2.i2.p1.1.m1.1.1.cmml">X</mi><mo id="S2.I2.i2.p1.1.m1.3.3.4.3.2.2" stretchy="false" xref="S2.I2.i2.p1.1.m1.3.3.4.cmml">)</mo></mrow></mrow><mo id="S2.I2.i2.p1.1.m1.3.3.3" xref="S2.I2.i2.p1.1.m1.3.3.3.cmml">=</mo><mrow id="S2.I2.i2.p1.1.m1.3.3.2.2" xref="S2.I2.i2.p1.1.m1.3.3.2.3.cmml"><mo id="S2.I2.i2.p1.1.m1.3.3.2.2.3" stretchy="false" xref="S2.I2.i2.p1.1.m1.3.3.2.3.cmml">{</mo><msup id="S2.I2.i2.p1.1.m1.2.2.1.1.1" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.cmml"><mrow id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.2" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml">c</mi><mo id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.1" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.I2.i2.p1.1.m1.2.2.1.1.1.3" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.3.cmml"><mo id="S2.I2.i2.p1.1.m1.2.2.1.1.1.3a" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.3.cmml">±</mo><mi id="S2.I2.i2.p1.1.m1.2.2.1.1.1.3.2" mathvariant="normal" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.I2.i2.p1.1.m1.3.3.2.2.4" xref="S2.I2.i2.p1.1.m1.3.3.2.3.cmml">,</mo><msup id="S2.I2.i2.p1.1.m1.3.3.2.2.2" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.cmml"><mrow id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.cmml"><mo id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.2" stretchy="false" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.cmml"><mi id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.2" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.2.cmml">d</mi><mo id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.1" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.3" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.3.cmml">c</mi></mrow><mo id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.3" stretchy="false" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S2.I2.i2.p1.1.m1.3.3.2.2.2.3" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.3.cmml"><mo id="S2.I2.i2.p1.1.m1.3.3.2.2.2.3a" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.3.cmml">±</mo><mi id="S2.I2.i2.p1.1.m1.3.3.2.2.2.3.2" mathvariant="normal" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.3.2.cmml">∞</mi></mrow></msup><mo id="S2.I2.i2.p1.1.m1.3.3.2.2.5" stretchy="false" xref="S2.I2.i2.p1.1.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i2.p1.1.m1.3b"><apply id="S2.I2.i2.p1.1.m1.3.3.cmml" xref="S2.I2.i2.p1.1.m1.3.3"><eq id="S2.I2.i2.p1.1.m1.3.3.3.cmml" xref="S2.I2.i2.p1.1.m1.3.3.3"></eq><apply id="S2.I2.i2.p1.1.m1.3.3.4.cmml" xref="S2.I2.i2.p1.1.m1.3.3.4"><times id="S2.I2.i2.p1.1.m1.3.3.4.1.cmml" xref="S2.I2.i2.p1.1.m1.3.3.4.1"></times><apply id="S2.I2.i2.p1.1.m1.3.3.4.2.cmml" xref="S2.I2.i2.p1.1.m1.3.3.4.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.1.m1.3.3.4.2.1.cmml" xref="S2.I2.i2.p1.1.m1.3.3.4.2">superscript</csymbol><ci id="S2.I2.i2.p1.1.m1.3.3.4.2.2.cmml" xref="S2.I2.i2.p1.1.m1.3.3.4.2.2">𝜎</ci><ci id="S2.I2.i2.p1.1.m1.3.3.4.2.3.cmml" xref="S2.I2.i2.p1.1.m1.3.3.4.2.3">Σ</ci></apply><ci id="S2.I2.i2.p1.1.m1.1.1.cmml" xref="S2.I2.i2.p1.1.m1.1.1">𝑋</ci></apply><set id="S2.I2.i2.p1.1.m1.3.3.2.3.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2"><apply id="S2.I2.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.I2.i2.p1.1.m1.2.2.1.1.1.2.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1">superscript</csymbol><apply id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1"><times id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.1"></times><ci id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.2">𝑐</ci><ci id="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3">𝑑</ci></apply><apply id="S2.I2.i2.p1.1.m1.2.2.1.1.1.3.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.3"><csymbol cd="latexml" id="S2.I2.i2.p1.1.m1.2.2.1.1.1.3.1.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.I2.i2.p1.1.m1.2.2.1.1.1.3.2.cmml" xref="S2.I2.i2.p1.1.m1.2.2.1.1.1.3.2"></infinity></apply></apply><apply id="S2.I2.i2.p1.1.m1.3.3.2.2.2.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.1.m1.3.3.2.2.2.2.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2">superscript</csymbol><apply id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1"><times id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.1.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.1"></times><ci id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.2.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.2">𝑑</ci><ci id="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.3.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.1.1.1.3">𝑐</ci></apply><apply id="S2.I2.i2.p1.1.m1.3.3.2.2.2.3.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.3"><csymbol cd="latexml" id="S2.I2.i2.p1.1.m1.3.3.2.2.2.3.1.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.3">plus-or-minus</csymbol><infinity id="S2.I2.i2.p1.1.m1.3.3.2.2.2.3.2.cmml" xref="S2.I2.i2.p1.1.m1.3.3.2.2.2.3.2"></infinity></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i2.p1.1.m1.3c">\sigma^{\Sigma}(X)=\{(cd)^{\pm\infty},(dc)^{\pm\infty}\}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i2.p1.1.m1.3d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT ( italic_X ) = { ( italic_c italic_d ) start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT , ( italic_d italic_c ) start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT }</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S2.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i3.p1"> <p class="ltx_p" id="S2.I2.i3.p1.1"><math alttext="\sigma^{T}(\cal O(a^{\pm\infty}))=\sigma^{T}(\cal O(b^{\pm\infty}))=\cal O((cd% )^{\pm\infty})=\cal O((dc)^{\pm\infty})" class="ltx_Math" display="inline" id="S2.I2.i3.p1.1.m1.4"><semantics id="S2.I2.i3.p1.1.m1.4a"><mrow id="S2.I2.i3.p1.1.m1.4.4" xref="S2.I2.i3.p1.1.m1.4.4.cmml"><mrow id="S2.I2.i3.p1.1.m1.1.1.1" xref="S2.I2.i3.p1.1.m1.1.1.1.cmml"><msup id="S2.I2.i3.p1.1.m1.1.1.1.3" xref="S2.I2.i3.p1.1.m1.1.1.1.3.cmml"><mi id="S2.I2.i3.p1.1.m1.1.1.1.3.2" xref="S2.I2.i3.p1.1.m1.1.1.1.3.2.cmml">σ</mi><mi id="S2.I2.i3.p1.1.m1.1.1.1.3.3" xref="S2.I2.i3.p1.1.m1.1.1.1.3.3.cmml">T</mi></msup><mo id="S2.I2.i3.p1.1.m1.1.1.1.2" xref="S2.I2.i3.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.I2.i3.p1.1.m1.1.1.1.1.1" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S2.I2.i3.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.I2.i3.p1.1.m1.1.1.1.1.1.1" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.3" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.3.cmml">𝒪</mi><mo id="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.2" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.1" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.1.2" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml">𝒶</mi><mrow id="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.1.3" xref="S2.I2.i3.p1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mo 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id="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.1.1.1.2.cmml" xref="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.1.1.1.2">𝒹</ci><ci id="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.1.1.1.3.cmml" xref="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.1.1.1.3">𝒸</ci></apply><apply id="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.3.cmml" xref="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.3"><csymbol cd="latexml" id="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.3.1.cmml" xref="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.3.2.cmml" xref="S2.I2.i3.p1.1.m1.4.4.4.1.1.1.3.2"></infinity></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i3.p1.1.m1.4c">\sigma^{T}(\cal O(a^{\pm\infty}))=\sigma^{T}(\cal O(b^{\pm\infty}))=\cal O((cd% )^{\pm\infty})=\cal O((dc)^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i3.p1.1.m1.4d">italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( caligraphic_O ( caligraphic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) ) = italic_σ start_POSTSUPERSCRIPT caligraphic_T end_POSTSUPERSCRIPT ( caligraphic_O ( caligraphic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) ) = caligraphic_O ( ( caligraphic_c caligraphic_d ) start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) = caligraphic_O ( ( caligraphic_d caligraphic_c ) start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S2.I2.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i4.p1"> <p class="ltx_p" id="S2.I2.i4.p1.1"><math alttext="\cal L(X)=\{a^{k},b^{k}\mid k\geq 0\}" class="ltx_Math" display="inline" id="S2.I2.i4.p1.1.m1.3"><semantics id="S2.I2.i4.p1.1.m1.3a"><mrow id="S2.I2.i4.p1.1.m1.3.3" xref="S2.I2.i4.p1.1.m1.3.3.cmml"><mrow id="S2.I2.i4.p1.1.m1.3.3.4" xref="S2.I2.i4.p1.1.m1.3.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i4.p1.1.m1.3.3.4.2" xref="S2.I2.i4.p1.1.m1.3.3.4.2.cmml">ℒ</mi><mo id="S2.I2.i4.p1.1.m1.3.3.4.1" xref="S2.I2.i4.p1.1.m1.3.3.4.1.cmml">⁢</mo><mrow id="S2.I2.i4.p1.1.m1.3.3.4.3.2" xref="S2.I2.i4.p1.1.m1.3.3.4.cmml"><mo id="S2.I2.i4.p1.1.m1.3.3.4.3.2.1" stretchy="false" xref="S2.I2.i4.p1.1.m1.3.3.4.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I2.i4.p1.1.m1.1.1" xref="S2.I2.i4.p1.1.m1.1.1.cmml">𝒳</mi><mo id="S2.I2.i4.p1.1.m1.3.3.4.3.2.2" stretchy="false" xref="S2.I2.i4.p1.1.m1.3.3.4.cmml">)</mo></mrow></mrow><mo id="S2.I2.i4.p1.1.m1.3.3.3" xref="S2.I2.i4.p1.1.m1.3.3.3.cmml">=</mo><mrow id="S2.I2.i4.p1.1.m1.3.3.2.2" xref="S2.I2.i4.p1.1.m1.3.3.2.3.cmml"><mo id="S2.I2.i4.p1.1.m1.3.3.2.2.3" stretchy="false" xref="S2.I2.i4.p1.1.m1.3.3.2.3.1.cmml">{</mo><mrow id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.3.cmml"><msup id="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.2" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.3" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.3.cmml">𝓀</mi></msup><mo id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.3" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.3.cmml">,</mo><msup id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.2" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.2.cmml">𝒷</mi><mi class="ltx_font_mathcaligraphic" id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.3" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.3.cmml">𝓀</mi></msup></mrow><mo fence="true" id="S2.I2.i4.p1.1.m1.3.3.2.2.4" lspace="0em" rspace="0em" xref="S2.I2.i4.p1.1.m1.3.3.2.3.1.cmml">∣</mo><mrow id="S2.I2.i4.p1.1.m1.3.3.2.2.2" xref="S2.I2.i4.p1.1.m1.3.3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i4.p1.1.m1.3.3.2.2.2.2" xref="S2.I2.i4.p1.1.m1.3.3.2.2.2.2.cmml">𝓀</mi><mo id="S2.I2.i4.p1.1.m1.3.3.2.2.2.1" xref="S2.I2.i4.p1.1.m1.3.3.2.2.2.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" 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xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.2"><apply id="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1">superscript</csymbol><ci id="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.2.cmml" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.2">𝒶</ci><ci id="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.3.cmml" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.1.1.3">𝓀</ci></apply><apply id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.cmml" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.1.cmml" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2">superscript</csymbol><ci id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.2.cmml" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.2">𝒷</ci><ci id="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.3.cmml" xref="S2.I2.i4.p1.1.m1.2.2.1.1.1.2.2.3">𝓀</ci></apply></list><apply id="S2.I2.i4.p1.1.m1.3.3.2.2.2.cmml" xref="S2.I2.i4.p1.1.m1.3.3.2.2.2"><geq id="S2.I2.i4.p1.1.m1.3.3.2.2.2.1.cmml" xref="S2.I2.i4.p1.1.m1.3.3.2.2.2.1"></geq><ci id="S2.I2.i4.p1.1.m1.3.3.2.2.2.2.cmml" xref="S2.I2.i4.p1.1.m1.3.3.2.2.2.2">𝓀</ci><cn id="S2.I2.i4.p1.1.m1.3.3.2.2.2.3.cmml" type="integer" xref="S2.I2.i4.p1.1.m1.3.3.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i4.p1.1.m1.3c">\cal L(X)=\{a^{k},b^{k}\mid k\geq 0\}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i4.p1.1.m1.3d">caligraphic_L ( caligraphic_X ) = { caligraphic_a start_POSTSUPERSCRIPT caligraphic_k end_POSTSUPERSCRIPT , caligraphic_b start_POSTSUPERSCRIPT caligraphic_k end_POSTSUPERSCRIPT ∣ caligraphic_k ≥ caligraphic_0 }</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S2.I2.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i5.p1"> <p class="ltx_p" id="S2.I2.i5.p1.1"><math alttext="\sigma(\cal L(X))=\{(cd)^{k}\mid k\geq 2\}" class="ltx_Math" display="inline" id="S2.I2.i5.p1.1.m1.4"><semantics id="S2.I2.i5.p1.1.m1.4a"><mrow id="S2.I2.i5.p1.1.m1.4.4" xref="S2.I2.i5.p1.1.m1.4.4.cmml"><mrow id="S2.I2.i5.p1.1.m1.2.2.1" xref="S2.I2.i5.p1.1.m1.2.2.1.cmml"><mi id="S2.I2.i5.p1.1.m1.2.2.1.3" xref="S2.I2.i5.p1.1.m1.2.2.1.3.cmml">σ</mi><mo id="S2.I2.i5.p1.1.m1.2.2.1.2" xref="S2.I2.i5.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.I2.i5.p1.1.m1.2.2.1.1.1" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S2.I2.i5.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.2" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.2.cmml">ℒ</mi><mo id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.1" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.3.2" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I2.i5.p1.1.m1.1.1" xref="S2.I2.i5.p1.1.m1.1.1.cmml">𝒳</mi><mo id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I2.i5.p1.1.m1.2.2.1.1.1.3" stretchy="false" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I2.i5.p1.1.m1.4.4.4" xref="S2.I2.i5.p1.1.m1.4.4.4.cmml">=</mo><mrow id="S2.I2.i5.p1.1.m1.4.4.3.2" xref="S2.I2.i5.p1.1.m1.4.4.3.3.cmml"><mo id="S2.I2.i5.p1.1.m1.4.4.3.2.3" stretchy="false" xref="S2.I2.i5.p1.1.m1.4.4.3.3.1.cmml">{</mo><msup id="S2.I2.i5.p1.1.m1.3.3.2.1.1" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.cmml"><mrow id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.cmml"><mo id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.2" stretchy="false" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.2" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.2.cmml">𝒸</mi><mo id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.1" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.3" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.3.cmml">𝒹</mi></mrow><mo id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.3" stretchy="false" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.cmml">)</mo></mrow><mi class="ltx_font_mathcaligraphic" id="S2.I2.i5.p1.1.m1.3.3.2.1.1.3" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.3.cmml">𝓀</mi></msup><mo fence="true" id="S2.I2.i5.p1.1.m1.4.4.3.2.4" lspace="0em" rspace="0em" xref="S2.I2.i5.p1.1.m1.4.4.3.3.1.cmml">∣</mo><mrow id="S2.I2.i5.p1.1.m1.4.4.3.2.2" xref="S2.I2.i5.p1.1.m1.4.4.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i5.p1.1.m1.4.4.3.2.2.2" xref="S2.I2.i5.p1.1.m1.4.4.3.2.2.2.cmml">𝓀</mi><mo id="S2.I2.i5.p1.1.m1.4.4.3.2.2.1" xref="S2.I2.i5.p1.1.m1.4.4.3.2.2.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S2.I2.i5.p1.1.m1.4.4.3.2.2.3" mathvariant="script" xref="S2.I2.i5.p1.1.m1.4.4.3.2.2.3.cmml">2</mn></mrow><mo id="S2.I2.i5.p1.1.m1.4.4.3.2.5" stretchy="false" xref="S2.I2.i5.p1.1.m1.4.4.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i5.p1.1.m1.4b"><apply id="S2.I2.i5.p1.1.m1.4.4.cmml" xref="S2.I2.i5.p1.1.m1.4.4"><eq id="S2.I2.i5.p1.1.m1.4.4.4.cmml" xref="S2.I2.i5.p1.1.m1.4.4.4"></eq><apply id="S2.I2.i5.p1.1.m1.2.2.1.cmml" xref="S2.I2.i5.p1.1.m1.2.2.1"><times id="S2.I2.i5.p1.1.m1.2.2.1.2.cmml" xref="S2.I2.i5.p1.1.m1.2.2.1.2"></times><ci id="S2.I2.i5.p1.1.m1.2.2.1.3.cmml" xref="S2.I2.i5.p1.1.m1.2.2.1.3">𝜎</ci><apply id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.cmml" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1"><times id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.1"></times><ci id="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="S2.I2.i5.p1.1.m1.2.2.1.1.1.1.2">ℒ</ci><ci id="S2.I2.i5.p1.1.m1.1.1.cmml" xref="S2.I2.i5.p1.1.m1.1.1">𝒳</ci></apply></apply><apply id="S2.I2.i5.p1.1.m1.4.4.3.3.cmml" xref="S2.I2.i5.p1.1.m1.4.4.3.2"><csymbol cd="latexml" id="S2.I2.i5.p1.1.m1.4.4.3.3.1.cmml" xref="S2.I2.i5.p1.1.m1.4.4.3.2.3">conditional-set</csymbol><apply id="S2.I2.i5.p1.1.m1.3.3.2.1.1.cmml" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S2.I2.i5.p1.1.m1.3.3.2.1.1.2.cmml" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1">superscript</csymbol><apply id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.cmml" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1"><times id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.1.cmml" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.1"></times><ci id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.2.cmml" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.2">𝒸</ci><ci id="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.3.cmml" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.1.1.1.3">𝒹</ci></apply><ci id="S2.I2.i5.p1.1.m1.3.3.2.1.1.3.cmml" xref="S2.I2.i5.p1.1.m1.3.3.2.1.1.3">𝓀</ci></apply><apply id="S2.I2.i5.p1.1.m1.4.4.3.2.2.cmml" xref="S2.I2.i5.p1.1.m1.4.4.3.2.2"><geq id="S2.I2.i5.p1.1.m1.4.4.3.2.2.1.cmml" xref="S2.I2.i5.p1.1.m1.4.4.3.2.2.1"></geq><ci id="S2.I2.i5.p1.1.m1.4.4.3.2.2.2.cmml" xref="S2.I2.i5.p1.1.m1.4.4.3.2.2.2">𝓀</ci><cn id="S2.I2.i5.p1.1.m1.4.4.3.2.2.3.cmml" type="integer" xref="S2.I2.i5.p1.1.m1.4.4.3.2.2.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i5.p1.1.m1.4c">\sigma(\cal L(X))=\{(cd)^{k}\mid k\geq 2\}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i5.p1.1.m1.4d">italic_σ ( caligraphic_L ( caligraphic_X ) ) = { ( caligraphic_c caligraphic_d ) start_POSTSUPERSCRIPT caligraphic_k end_POSTSUPERSCRIPT ∣ caligraphic_k ≥ caligraphic_2 }</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S2.I2.i6" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I2.i6.p1"> <p class="ltx_p" id="S2.I2.i6.p1.1"><math alttext="\cal L(\sigma(X))=\{(cd)^{k},(dc)^{k},(cd)^{k}c,(dc)^{k}d\mid k\geq 0\}\,." class="ltx_Math" display="inline" id="S2.I2.i6.p1.1.m1.2"><semantics id="S2.I2.i6.p1.1.m1.2a"><mrow id="S2.I2.i6.p1.1.m1.2.2.1" xref="S2.I2.i6.p1.1.m1.2.2.1.1.cmml"><mrow id="S2.I2.i6.p1.1.m1.2.2.1.1" xref="S2.I2.i6.p1.1.m1.2.2.1.1.cmml"><mrow id="S2.I2.i6.p1.1.m1.2.2.1.1.1" xref="S2.I2.i6.p1.1.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I2.i6.p1.1.m1.2.2.1.1.1.3" xref="S2.I2.i6.p1.1.m1.2.2.1.1.1.3.cmml">ℒ</mi><mo id="S2.I2.i6.p1.1.m1.2.2.1.1.1.2" xref="S2.I2.i6.p1.1.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.I2.i6.p1.1.m1.2.2.1.1.1.1.1" xref="S2.I2.i6.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S2.I2.i6.p1.1.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.I2.i6.p1.1.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.I2.i6.p1.1.m1.2.2.1.1.1.1.1.1" xref="S2.I2.i6.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S2.I2.i6.p1.1.m1.2.2.1.1.1.1.1.1.2" 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xref="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.1.1"><times id="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.1.1.1.1.cmml" xref="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.1.1.1.1"></times><ci id="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.1.1.1.2.cmml" xref="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.1.1.1.2">𝒹</ci><ci id="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.1.1.1.3.cmml" xref="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.1.1.1.3">𝒸</ci></apply><ci id="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.3.cmml" xref="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.1.3">𝓀</ci></apply><ci id="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.3.cmml" xref="S2.I2.i6.p1.1.m1.2.2.1.1.2.1.1.4.4.3">𝒹</ci></apply></list><apply id="S2.I2.i6.p1.1.m1.2.2.1.1.3.2.2.cmml" xref="S2.I2.i6.p1.1.m1.2.2.1.1.3.2.2"><geq id="S2.I2.i6.p1.1.m1.2.2.1.1.3.2.2.1.cmml" xref="S2.I2.i6.p1.1.m1.2.2.1.1.3.2.2.1"></geq><ci id="S2.I2.i6.p1.1.m1.2.2.1.1.3.2.2.2.cmml" xref="S2.I2.i6.p1.1.m1.2.2.1.1.3.2.2.2">𝓀</ci><cn id="S2.I2.i6.p1.1.m1.2.2.1.1.3.2.2.3.cmml" type="integer" xref="S2.I2.i6.p1.1.m1.2.2.1.1.3.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i6.p1.1.m1.2c">\cal L(\sigma(X))=\{(cd)^{k},(dc)^{k},(cd)^{k}c,(dc)^{k}d\mid k\geq 0\}\,.</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i6.p1.1.m1.2d">caligraphic_L ( italic_σ ( caligraphic_X ) ) = { ( caligraphic_c caligraphic_d ) start_POSTSUPERSCRIPT caligraphic_k end_POSTSUPERSCRIPT , ( caligraphic_d caligraphic_c ) start_POSTSUPERSCRIPT caligraphic_k end_POSTSUPERSCRIPT , ( caligraphic_c caligraphic_d ) start_POSTSUPERSCRIPT caligraphic_k end_POSTSUPERSCRIPT caligraphic_c , ( caligraphic_d caligraphic_c ) start_POSTSUPERSCRIPT caligraphic_k end_POSTSUPERSCRIPT caligraphic_d ∣ caligraphic_k ≥ caligraphic_0 } .</annotation></semantics></math></p> </div> </li> </ul> </div> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.6">The following basic fact is used below in the proofs of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm2" title="Lemma 5.2. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.2</span></a> and of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm1" title="Lemma 6.1. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.1</span></a>; since we don’t know a reference for it, we include here a proof. Please note that the compactness of <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.1"><semantics id="S2.SS2.p5.1.m1.1a"><msup id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p5.1.m1.1.1.2" xref="S2.SS2.p5.1.m1.1.1.2.cmml">𝒜</mi><mi id="S2.SS2.p5.1.m1.1.1.3" xref="S2.SS2.p5.1.m1.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.1b"><apply id="S2.SS2.p5.1.m1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1">superscript</csymbol><ci id="S2.SS2.p5.1.m1.1.1.2.cmml" xref="S2.SS2.p5.1.m1.1.1.2">𝒜</ci><ci id="S2.SS2.p5.1.m1.1.1.3.cmml" xref="S2.SS2.p5.1.m1.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and the continuity of the map <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.1"><semantics id="S2.SS2.p5.2.m2.1a"><msup id="S2.SS2.p5.2.m2.1.1" xref="S2.SS2.p5.2.m2.1.1.cmml"><mi id="S2.SS2.p5.2.m2.1.1.2" xref="S2.SS2.p5.2.m2.1.1.2.cmml">σ</mi><mi id="S2.SS2.p5.2.m2.1.1.3" xref="S2.SS2.p5.2.m2.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.2.m2.1b"><apply id="S2.SS2.p5.2.m2.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.2.m2.1.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1">superscript</csymbol><ci id="S2.SS2.p5.2.m2.1.1.2.cmml" xref="S2.SS2.p5.2.m2.1.1.2">𝜎</ci><ci id="S2.SS2.p5.2.m2.1.1.3.cmml" xref="S2.SS2.p5.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.2.m2.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.2.m2.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> yield directly that for any subshift <math alttext="X" class="ltx_Math" display="inline" id="S2.SS2.p5.3.m3.1"><semantics id="S2.SS2.p5.3.m3.1a"><mi id="S2.SS2.p5.3.m3.1.1" xref="S2.SS2.p5.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.1b"><ci id="S2.SS2.p5.3.m3.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.1d">italic_X</annotation></semantics></math> the set <math alttext="\sigma^{\mathbb{Z}}(X)" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.1"><semantics id="S2.SS2.p5.4.m4.1a"><mrow id="S2.SS2.p5.4.m4.1.2" xref="S2.SS2.p5.4.m4.1.2.cmml"><msup id="S2.SS2.p5.4.m4.1.2.2" xref="S2.SS2.p5.4.m4.1.2.2.cmml"><mi id="S2.SS2.p5.4.m4.1.2.2.2" xref="S2.SS2.p5.4.m4.1.2.2.2.cmml">σ</mi><mi id="S2.SS2.p5.4.m4.1.2.2.3" xref="S2.SS2.p5.4.m4.1.2.2.3.cmml">ℤ</mi></msup><mo id="S2.SS2.p5.4.m4.1.2.1" xref="S2.SS2.p5.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p5.4.m4.1.2.3.2" xref="S2.SS2.p5.4.m4.1.2.cmml"><mo id="S2.SS2.p5.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS2.p5.4.m4.1.2.cmml">(</mo><mi id="S2.SS2.p5.4.m4.1.1" xref="S2.SS2.p5.4.m4.1.1.cmml">X</mi><mo id="S2.SS2.p5.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS2.p5.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.4.m4.1b"><apply id="S2.SS2.p5.4.m4.1.2.cmml" xref="S2.SS2.p5.4.m4.1.2"><times id="S2.SS2.p5.4.m4.1.2.1.cmml" xref="S2.SS2.p5.4.m4.1.2.1"></times><apply id="S2.SS2.p5.4.m4.1.2.2.cmml" xref="S2.SS2.p5.4.m4.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p5.4.m4.1.2.2.1.cmml" xref="S2.SS2.p5.4.m4.1.2.2">superscript</csymbol><ci id="S2.SS2.p5.4.m4.1.2.2.2.cmml" xref="S2.SS2.p5.4.m4.1.2.2.2">𝜎</ci><ci id="S2.SS2.p5.4.m4.1.2.2.3.cmml" xref="S2.SS2.p5.4.m4.1.2.2.3">ℤ</ci></apply><ci id="S2.SS2.p5.4.m4.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.1c">\sigma^{\mathbb{Z}}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> is closed. However, since <math alttext="\sigma^{\mathbb{Z}}(X)" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m5.1"><semantics id="S2.SS2.p5.5.m5.1a"><mrow id="S2.SS2.p5.5.m5.1.2" xref="S2.SS2.p5.5.m5.1.2.cmml"><msup id="S2.SS2.p5.5.m5.1.2.2" xref="S2.SS2.p5.5.m5.1.2.2.cmml"><mi id="S2.SS2.p5.5.m5.1.2.2.2" xref="S2.SS2.p5.5.m5.1.2.2.2.cmml">σ</mi><mi id="S2.SS2.p5.5.m5.1.2.2.3" xref="S2.SS2.p5.5.m5.1.2.2.3.cmml">ℤ</mi></msup><mo id="S2.SS2.p5.5.m5.1.2.1" xref="S2.SS2.p5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p5.5.m5.1.2.3.2" xref="S2.SS2.p5.5.m5.1.2.cmml"><mo id="S2.SS2.p5.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS2.p5.5.m5.1.2.cmml">(</mo><mi id="S2.SS2.p5.5.m5.1.1" xref="S2.SS2.p5.5.m5.1.1.cmml">X</mi><mo id="S2.SS2.p5.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS2.p5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m5.1b"><apply id="S2.SS2.p5.5.m5.1.2.cmml" xref="S2.SS2.p5.5.m5.1.2"><times id="S2.SS2.p5.5.m5.1.2.1.cmml" xref="S2.SS2.p5.5.m5.1.2.1"></times><apply id="S2.SS2.p5.5.m5.1.2.2.cmml" xref="S2.SS2.p5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.1.2.2.1.cmml" xref="S2.SS2.p5.5.m5.1.2.2">superscript</csymbol><ci id="S2.SS2.p5.5.m5.1.2.2.2.cmml" xref="S2.SS2.p5.5.m5.1.2.2.2">𝜎</ci><ci id="S2.SS2.p5.5.m5.1.2.2.3.cmml" xref="S2.SS2.p5.5.m5.1.2.2.3">ℤ</ci></apply><ci id="S2.SS2.p5.5.m5.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m5.1c">\sigma^{\mathbb{Z}}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m5.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> is in general not shift-invariant, this doesn’t imply directly that the set <math alttext="Y" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m6.1"><semantics id="S2.SS2.p5.6.m6.1a"><mi id="S2.SS2.p5.6.m6.1.1" xref="S2.SS2.p5.6.m6.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m6.1b"><ci id="S2.SS2.p5.6.m6.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m6.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m6.1d">italic_Y</annotation></semantics></math> in the lemma below is closed.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="S2.Thmthm4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm4.1.1.1">Lemma 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm4.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm4.p1"> <p class="ltx_p" id="S2.Thmthm4.p1.6"><span class="ltx_text ltx_font_italic" id="S2.Thmthm4.p1.6.6">(1) For any non-erasing monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.Thmthm4.p1.1.1.m1.1"><semantics id="S2.Thmthm4.p1.1.1.m1.1a"><mrow id="S2.Thmthm4.p1.1.1.m1.1.1" xref="S2.Thmthm4.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmthm4.p1.1.1.m1.1.1.2" xref="S2.Thmthm4.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S2.Thmthm4.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm4.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.Thmthm4.p1.1.1.m1.1.1.3" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.cmml"><msup id="S2.Thmthm4.p1.1.1.m1.1.1.3.2" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm4.p1.1.1.m1.1.1.3.2.2" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.Thmthm4.p1.1.1.m1.1.1.3.2.3" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm4.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.Thmthm4.p1.1.1.m1.1.1.3.3" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm4.p1.1.1.m1.1.1.3.3.2" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.Thmthm4.p1.1.1.m1.1.1.3.3.3" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm4.p1.1.1.m1.1b"><apply id="S2.Thmthm4.p1.1.1.m1.1.1.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1"><ci id="S2.Thmthm4.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.1">:</ci><ci id="S2.Thmthm4.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S2.Thmthm4.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3"><ci id="S2.Thmthm4.p1.1.1.m1.1.1.3.1.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.1">→</ci><apply id="S2.Thmthm4.p1.1.1.m1.1.1.3.2.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmthm4.p1.1.1.m1.1.1.3.2.1.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.Thmthm4.p1.1.1.m1.1.1.3.2.2.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S2.Thmthm4.p1.1.1.m1.1.1.3.2.3.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S2.Thmthm4.p1.1.1.m1.1.1.3.3.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmthm4.p1.1.1.m1.1.1.3.3.1.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.Thmthm4.p1.1.1.m1.1.1.3.3.2.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S2.Thmthm4.p1.1.1.m1.1.1.3.3.3.cmml" xref="S2.Thmthm4.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm4.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm4.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm4.p1.2.2.m2.1"><semantics id="S2.Thmthm4.p1.2.2.m2.1a"><mrow id="S2.Thmthm4.p1.2.2.m2.1.1" xref="S2.Thmthm4.p1.2.2.m2.1.1.cmml"><mi id="S2.Thmthm4.p1.2.2.m2.1.1.2" xref="S2.Thmthm4.p1.2.2.m2.1.1.2.cmml">X</mi><mo id="S2.Thmthm4.p1.2.2.m2.1.1.1" xref="S2.Thmthm4.p1.2.2.m2.1.1.1.cmml">⊆</mo><msup id="S2.Thmthm4.p1.2.2.m2.1.1.3" xref="S2.Thmthm4.p1.2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm4.p1.2.2.m2.1.1.3.2" xref="S2.Thmthm4.p1.2.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S2.Thmthm4.p1.2.2.m2.1.1.3.3" xref="S2.Thmthm4.p1.2.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm4.p1.2.2.m2.1b"><apply id="S2.Thmthm4.p1.2.2.m2.1.1.cmml" xref="S2.Thmthm4.p1.2.2.m2.1.1"><subset id="S2.Thmthm4.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmthm4.p1.2.2.m2.1.1.1"></subset><ci id="S2.Thmthm4.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmthm4.p1.2.2.m2.1.1.2">𝑋</ci><apply id="S2.Thmthm4.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmthm4.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm4.p1.2.2.m2.1.1.3.1.cmml" xref="S2.Thmthm4.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S2.Thmthm4.p1.2.2.m2.1.1.3.2.cmml" xref="S2.Thmthm4.p1.2.2.m2.1.1.3.2">𝒜</ci><ci id="S2.Thmthm4.p1.2.2.m2.1.1.3.3.cmml" xref="S2.Thmthm4.p1.2.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm4.p1.2.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm4.p1.2.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> the union <math alttext="Y" class="ltx_Math" display="inline" id="S2.Thmthm4.p1.3.3.m3.1"><semantics id="S2.Thmthm4.p1.3.3.m3.1a"><mi id="S2.Thmthm4.p1.3.3.m3.1.1" xref="S2.Thmthm4.p1.3.3.m3.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm4.p1.3.3.m3.1b"><ci id="S2.Thmthm4.p1.3.3.m3.1.1.cmml" xref="S2.Thmthm4.p1.3.3.m3.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm4.p1.3.3.m3.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm4.p1.3.3.m3.1d">italic_Y</annotation></semantics></math> of all image orbits <math alttext="\sigma^{T}(\cal O({\bf x}))" class="ltx_Math" display="inline" id="S2.Thmthm4.p1.4.4.m4.2"><semantics id="S2.Thmthm4.p1.4.4.m4.2a"><mrow id="S2.Thmthm4.p1.4.4.m4.2.2" xref="S2.Thmthm4.p1.4.4.m4.2.2.cmml"><msup id="S2.Thmthm4.p1.4.4.m4.2.2.3" xref="S2.Thmthm4.p1.4.4.m4.2.2.3.cmml"><mi id="S2.Thmthm4.p1.4.4.m4.2.2.3.2" xref="S2.Thmthm4.p1.4.4.m4.2.2.3.2.cmml">σ</mi><mi id="S2.Thmthm4.p1.4.4.m4.2.2.3.3" xref="S2.Thmthm4.p1.4.4.m4.2.2.3.3.cmml">T</mi></msup><mo id="S2.Thmthm4.p1.4.4.m4.2.2.2" xref="S2.Thmthm4.p1.4.4.m4.2.2.2.cmml">⁢</mo><mrow id="S2.Thmthm4.p1.4.4.m4.2.2.1.1" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.cmml"><mo id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.2" stretchy="false" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.cmml">(</mo><mrow id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.2" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.2.cmml">𝒪</mi><mo id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.1" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.3.2" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.cmml"><mo id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.cmml">(</mo><mi id="S2.Thmthm4.p1.4.4.m4.1.1" xref="S2.Thmthm4.p1.4.4.m4.1.1.cmml">𝐱</mi><mo id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.3" stretchy="false" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm4.p1.4.4.m4.2b"><apply id="S2.Thmthm4.p1.4.4.m4.2.2.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2"><times id="S2.Thmthm4.p1.4.4.m4.2.2.2.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2.2"></times><apply id="S2.Thmthm4.p1.4.4.m4.2.2.3.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2.3"><csymbol cd="ambiguous" id="S2.Thmthm4.p1.4.4.m4.2.2.3.1.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2.3">superscript</csymbol><ci id="S2.Thmthm4.p1.4.4.m4.2.2.3.2.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2.3.2">𝜎</ci><ci id="S2.Thmthm4.p1.4.4.m4.2.2.3.3.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2.3.3">𝑇</ci></apply><apply id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1"><times id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.1.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.1"></times><ci id="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.2.cmml" xref="S2.Thmthm4.p1.4.4.m4.2.2.1.1.1.2">𝒪</ci><ci id="S2.Thmthm4.p1.4.4.m4.1.1.cmml" xref="S2.Thmthm4.p1.4.4.m4.1.1">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm4.p1.4.4.m4.2c">\sigma^{T}(\cal O({\bf x}))</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm4.p1.4.4.m4.2d">italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( caligraphic_O ( bold_x ) )</annotation></semantics></math>, for any <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S2.Thmthm4.p1.5.5.m5.1"><semantics id="S2.Thmthm4.p1.5.5.m5.1a"><mrow id="S2.Thmthm4.p1.5.5.m5.1.1" xref="S2.Thmthm4.p1.5.5.m5.1.1.cmml"><mi id="S2.Thmthm4.p1.5.5.m5.1.1.2" xref="S2.Thmthm4.p1.5.5.m5.1.1.2.cmml">𝐱</mi><mo id="S2.Thmthm4.p1.5.5.m5.1.1.1" xref="S2.Thmthm4.p1.5.5.m5.1.1.1.cmml">∈</mo><mi id="S2.Thmthm4.p1.5.5.m5.1.1.3" xref="S2.Thmthm4.p1.5.5.m5.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm4.p1.5.5.m5.1b"><apply id="S2.Thmthm4.p1.5.5.m5.1.1.cmml" xref="S2.Thmthm4.p1.5.5.m5.1.1"><in id="S2.Thmthm4.p1.5.5.m5.1.1.1.cmml" xref="S2.Thmthm4.p1.5.5.m5.1.1.1"></in><ci id="S2.Thmthm4.p1.5.5.m5.1.1.2.cmml" xref="S2.Thmthm4.p1.5.5.m5.1.1.2">𝐱</ci><ci id="S2.Thmthm4.p1.5.5.m5.1.1.3.cmml" xref="S2.Thmthm4.p1.5.5.m5.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm4.p1.5.5.m5.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm4.p1.5.5.m5.1d">bold_x ∈ italic_X</annotation></semantics></math>, is a closed subset of <math alttext="\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm4.p1.6.6.m6.1"><semantics id="S2.Thmthm4.p1.6.6.m6.1a"><msup id="S2.Thmthm4.p1.6.6.m6.1.1" xref="S2.Thmthm4.p1.6.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm4.p1.6.6.m6.1.1.2" xref="S2.Thmthm4.p1.6.6.m6.1.1.2.cmml">ℬ</mi><mi id="S2.Thmthm4.p1.6.6.m6.1.1.3" xref="S2.Thmthm4.p1.6.6.m6.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm4.p1.6.6.m6.1b"><apply id="S2.Thmthm4.p1.6.6.m6.1.1.cmml" xref="S2.Thmthm4.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S2.Thmthm4.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmthm4.p1.6.6.m6.1.1">superscript</csymbol><ci id="S2.Thmthm4.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmthm4.p1.6.6.m6.1.1.2">ℬ</ci><ci id="S2.Thmthm4.p1.6.6.m6.1.1.3.cmml" xref="S2.Thmthm4.p1.6.6.m6.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm4.p1.6.6.m6.1c">\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm4.p1.6.6.m6.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para ltx_noindent" id="S2.Thmthm4.p2"> <p class="ltx_p" id="S2.Thmthm4.p2.2"><span class="ltx_text ltx_font_italic" id="S2.Thmthm4.p2.2.1">(2) In particular, the map </span><math alttext="\sigma_{X}^{T}:X/\langle T\rangle\to\sigma(X)/\langle T\rangle" class="ltx_Math" display="inline" id="S2.Thmthm4.p2.1.m1.3"><semantics id="S2.Thmthm4.p2.1.m1.3a"><mrow id="S2.Thmthm4.p2.1.m1.3.4" xref="S2.Thmthm4.p2.1.m1.3.4.cmml"><msubsup id="S2.Thmthm4.p2.1.m1.3.4.2" xref="S2.Thmthm4.p2.1.m1.3.4.2.cmml"><mi id="S2.Thmthm4.p2.1.m1.3.4.2.2.2" xref="S2.Thmthm4.p2.1.m1.3.4.2.2.2.cmml">σ</mi><mi id="S2.Thmthm4.p2.1.m1.3.4.2.2.3" xref="S2.Thmthm4.p2.1.m1.3.4.2.2.3.cmml">X</mi><mi id="S2.Thmthm4.p2.1.m1.3.4.2.3" xref="S2.Thmthm4.p2.1.m1.3.4.2.3.cmml">T</mi></msubsup><mo id="S2.Thmthm4.p2.1.m1.3.4.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm4.p2.1.m1.3.4.1.cmml">:</mo><mrow id="S2.Thmthm4.p2.1.m1.3.4.3" xref="S2.Thmthm4.p2.1.m1.3.4.3.cmml"><mrow id="S2.Thmthm4.p2.1.m1.3.4.3.2" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.cmml"><mi id="S2.Thmthm4.p2.1.m1.3.4.3.2.2" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.2.cmml">X</mi><mo id="S2.Thmthm4.p2.1.m1.3.4.3.2.1" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.1.cmml">/</mo><mrow id="S2.Thmthm4.p2.1.m1.3.4.3.2.3.2" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.3.1.cmml"><mo id="S2.Thmthm4.p2.1.m1.3.4.3.2.3.2.1" stretchy="false" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.3.1.1.cmml">⟨</mo><mi id="S2.Thmthm4.p2.1.m1.1.1" xref="S2.Thmthm4.p2.1.m1.1.1.cmml">T</mi><mo id="S2.Thmthm4.p2.1.m1.3.4.3.2.3.2.2" stretchy="false" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.3.1.1.cmml">⟩</mo></mrow></mrow><mo id="S2.Thmthm4.p2.1.m1.3.4.3.1" stretchy="false" xref="S2.Thmthm4.p2.1.m1.3.4.3.1.cmml">→</mo><mrow id="S2.Thmthm4.p2.1.m1.3.4.3.3" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.cmml"><mrow id="S2.Thmthm4.p2.1.m1.3.4.3.3.2" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2.cmml"><mi id="S2.Thmthm4.p2.1.m1.3.4.3.3.2.2" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2.2.cmml">σ</mi><mo id="S2.Thmthm4.p2.1.m1.3.4.3.3.2.1" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2.1.cmml">⁢</mo><mrow id="S2.Thmthm4.p2.1.m1.3.4.3.3.2.3.2" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2.cmml"><mo id="S2.Thmthm4.p2.1.m1.3.4.3.3.2.3.2.1" stretchy="false" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2.cmml">(</mo><mi id="S2.Thmthm4.p2.1.m1.2.2" xref="S2.Thmthm4.p2.1.m1.2.2.cmml">X</mi><mo id="S2.Thmthm4.p2.1.m1.3.4.3.3.2.3.2.2" stretchy="false" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm4.p2.1.m1.3.4.3.3.1" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.1.cmml">/</mo><mrow id="S2.Thmthm4.p2.1.m1.3.4.3.3.3.2" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.3.1.cmml"><mo id="S2.Thmthm4.p2.1.m1.3.4.3.3.3.2.1" stretchy="false" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.3.1.1.cmml">⟨</mo><mi id="S2.Thmthm4.p2.1.m1.3.3" xref="S2.Thmthm4.p2.1.m1.3.3.cmml">T</mi><mo id="S2.Thmthm4.p2.1.m1.3.4.3.3.3.2.2" stretchy="false" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.3.1.1.cmml">⟩</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm4.p2.1.m1.3b"><apply id="S2.Thmthm4.p2.1.m1.3.4.cmml" xref="S2.Thmthm4.p2.1.m1.3.4"><ci id="S2.Thmthm4.p2.1.m1.3.4.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.1">:</ci><apply id="S2.Thmthm4.p2.1.m1.3.4.2.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.2"><csymbol cd="ambiguous" id="S2.Thmthm4.p2.1.m1.3.4.2.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.2">superscript</csymbol><apply id="S2.Thmthm4.p2.1.m1.3.4.2.2.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.2"><csymbol cd="ambiguous" id="S2.Thmthm4.p2.1.m1.3.4.2.2.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.2">subscript</csymbol><ci id="S2.Thmthm4.p2.1.m1.3.4.2.2.2.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.2.2.2">𝜎</ci><ci id="S2.Thmthm4.p2.1.m1.3.4.2.2.3.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.2.2.3">𝑋</ci></apply><ci id="S2.Thmthm4.p2.1.m1.3.4.2.3.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.2.3">𝑇</ci></apply><apply id="S2.Thmthm4.p2.1.m1.3.4.3.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3"><ci id="S2.Thmthm4.p2.1.m1.3.4.3.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.1">→</ci><apply id="S2.Thmthm4.p2.1.m1.3.4.3.2.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.2"><divide id="S2.Thmthm4.p2.1.m1.3.4.3.2.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.1"></divide><ci id="S2.Thmthm4.p2.1.m1.3.4.3.2.2.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.2">𝑋</ci><apply id="S2.Thmthm4.p2.1.m1.3.4.3.2.3.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.3.2"><csymbol cd="latexml" id="S2.Thmthm4.p2.1.m1.3.4.3.2.3.1.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.2.3.2.1">delimited-⟨⟩</csymbol><ci id="S2.Thmthm4.p2.1.m1.1.1.cmml" xref="S2.Thmthm4.p2.1.m1.1.1">𝑇</ci></apply></apply><apply id="S2.Thmthm4.p2.1.m1.3.4.3.3.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.3"><divide id="S2.Thmthm4.p2.1.m1.3.4.3.3.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.1"></divide><apply id="S2.Thmthm4.p2.1.m1.3.4.3.3.2.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2"><times id="S2.Thmthm4.p2.1.m1.3.4.3.3.2.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2.1"></times><ci id="S2.Thmthm4.p2.1.m1.3.4.3.3.2.2.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.2.2">𝜎</ci><ci id="S2.Thmthm4.p2.1.m1.2.2.cmml" xref="S2.Thmthm4.p2.1.m1.2.2">𝑋</ci></apply><apply id="S2.Thmthm4.p2.1.m1.3.4.3.3.3.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.3.2"><csymbol cd="latexml" id="S2.Thmthm4.p2.1.m1.3.4.3.3.3.1.1.cmml" xref="S2.Thmthm4.p2.1.m1.3.4.3.3.3.2.1">delimited-⟨⟩</csymbol><ci id="S2.Thmthm4.p2.1.m1.3.3.cmml" xref="S2.Thmthm4.p2.1.m1.3.3">𝑇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm4.p2.1.m1.3c">\sigma_{X}^{T}:X/\langle T\rangle\to\sigma(X)/\langle T\rangle</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm4.p2.1.m1.3d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT : italic_X / ⟨ italic_T ⟩ → italic_σ ( italic_X ) / ⟨ italic_T ⟩</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.Thmthm4.p2.2.2"> induced by the map </span><math alttext="\sigma^{T}" class="ltx_Math" display="inline" id="S2.Thmthm4.p2.2.m2.1"><semantics id="S2.Thmthm4.p2.2.m2.1a"><msup id="S2.Thmthm4.p2.2.m2.1.1" xref="S2.Thmthm4.p2.2.m2.1.1.cmml"><mi id="S2.Thmthm4.p2.2.m2.1.1.2" xref="S2.Thmthm4.p2.2.m2.1.1.2.cmml">σ</mi><mi id="S2.Thmthm4.p2.2.m2.1.1.3" xref="S2.Thmthm4.p2.2.m2.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm4.p2.2.m2.1b"><apply id="S2.Thmthm4.p2.2.m2.1.1.cmml" xref="S2.Thmthm4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.Thmthm4.p2.2.m2.1.1.1.cmml" xref="S2.Thmthm4.p2.2.m2.1.1">superscript</csymbol><ci id="S2.Thmthm4.p2.2.m2.1.1.2.cmml" xref="S2.Thmthm4.p2.2.m2.1.1.2">𝜎</ci><ci id="S2.Thmthm4.p2.2.m2.1.1.3.cmml" xref="S2.Thmthm4.p2.2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm4.p2.2.m2.1c">\sigma^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm4.p2.2.m2.1d">italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.Thmthm4.p2.2.3"> is surjective.</span></p> </div> </div> <div class="ltx_proof" id="S2.SS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS2.1.p1"> <p class="ltx_p" id="S2.SS2.1.p1.7">We need to show the following fact:</p> <ol class="ltx_enumerate" id="S2.I3"> <li class="ltx_item" id="S2.I3.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(#)</span> <div class="ltx_para" id="S2.I3.ix1.p1"> <p class="ltx_p" id="S2.I3.ix1.p1.9">For any integer <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.1.m1.1"><semantics id="S2.I3.ix1.p1.1.m1.1a"><mrow id="S2.I3.ix1.p1.1.m1.1.1" xref="S2.I3.ix1.p1.1.m1.1.1.cmml"><mi id="S2.I3.ix1.p1.1.m1.1.1.2" xref="S2.I3.ix1.p1.1.m1.1.1.2.cmml">n</mi><mo id="S2.I3.ix1.p1.1.m1.1.1.1" xref="S2.I3.ix1.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S2.I3.ix1.p1.1.m1.1.1.3" xref="S2.I3.ix1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.1.m1.1b"><apply id="S2.I3.ix1.p1.1.m1.1.1.cmml" xref="S2.I3.ix1.p1.1.m1.1.1"><geq id="S2.I3.ix1.p1.1.m1.1.1.1.cmml" xref="S2.I3.ix1.p1.1.m1.1.1.1"></geq><ci id="S2.I3.ix1.p1.1.m1.1.1.2.cmml" xref="S2.I3.ix1.p1.1.m1.1.1.2">𝑛</ci><cn id="S2.I3.ix1.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.I3.ix1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.1.m1.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.1.m1.1d">italic_n ≥ 0</annotation></semantics></math> let <math alttext="{\bf x}(n)\in X" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.2.m2.1"><semantics id="S2.I3.ix1.p1.2.m2.1a"><mrow id="S2.I3.ix1.p1.2.m2.1.2" xref="S2.I3.ix1.p1.2.m2.1.2.cmml"><mrow id="S2.I3.ix1.p1.2.m2.1.2.2" xref="S2.I3.ix1.p1.2.m2.1.2.2.cmml"><mi id="S2.I3.ix1.p1.2.m2.1.2.2.2" xref="S2.I3.ix1.p1.2.m2.1.2.2.2.cmml">𝐱</mi><mo id="S2.I3.ix1.p1.2.m2.1.2.2.1" xref="S2.I3.ix1.p1.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.2.m2.1.2.2.3.2" xref="S2.I3.ix1.p1.2.m2.1.2.2.cmml"><mo id="S2.I3.ix1.p1.2.m2.1.2.2.3.2.1" stretchy="false" xref="S2.I3.ix1.p1.2.m2.1.2.2.cmml">(</mo><mi id="S2.I3.ix1.p1.2.m2.1.1" xref="S2.I3.ix1.p1.2.m2.1.1.cmml">n</mi><mo id="S2.I3.ix1.p1.2.m2.1.2.2.3.2.2" stretchy="false" xref="S2.I3.ix1.p1.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.I3.ix1.p1.2.m2.1.2.1" xref="S2.I3.ix1.p1.2.m2.1.2.1.cmml">∈</mo><mi id="S2.I3.ix1.p1.2.m2.1.2.3" xref="S2.I3.ix1.p1.2.m2.1.2.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.2.m2.1b"><apply id="S2.I3.ix1.p1.2.m2.1.2.cmml" xref="S2.I3.ix1.p1.2.m2.1.2"><in id="S2.I3.ix1.p1.2.m2.1.2.1.cmml" xref="S2.I3.ix1.p1.2.m2.1.2.1"></in><apply id="S2.I3.ix1.p1.2.m2.1.2.2.cmml" xref="S2.I3.ix1.p1.2.m2.1.2.2"><times id="S2.I3.ix1.p1.2.m2.1.2.2.1.cmml" xref="S2.I3.ix1.p1.2.m2.1.2.2.1"></times><ci id="S2.I3.ix1.p1.2.m2.1.2.2.2.cmml" xref="S2.I3.ix1.p1.2.m2.1.2.2.2">𝐱</ci><ci id="S2.I3.ix1.p1.2.m2.1.1.cmml" xref="S2.I3.ix1.p1.2.m2.1.1">𝑛</ci></apply><ci id="S2.I3.ix1.p1.2.m2.1.2.3.cmml" xref="S2.I3.ix1.p1.2.m2.1.2.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.2.m2.1c">{\bf x}(n)\in X</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.2.m2.1d">bold_x ( italic_n ) ∈ italic_X</annotation></semantics></math> be a biinfinite word with image <math alttext="{\bf y}(n):=\sigma^{\mathbb{Z}}({\bf x}(n))\in Y" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.3.m3.3"><semantics id="S2.I3.ix1.p1.3.m3.3a"><mrow id="S2.I3.ix1.p1.3.m3.3.3" xref="S2.I3.ix1.p1.3.m3.3.3.cmml"><mrow id="S2.I3.ix1.p1.3.m3.3.3.3" xref="S2.I3.ix1.p1.3.m3.3.3.3.cmml"><mi id="S2.I3.ix1.p1.3.m3.3.3.3.2" xref="S2.I3.ix1.p1.3.m3.3.3.3.2.cmml">𝐲</mi><mo id="S2.I3.ix1.p1.3.m3.3.3.3.1" xref="S2.I3.ix1.p1.3.m3.3.3.3.1.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.3.m3.3.3.3.3.2" xref="S2.I3.ix1.p1.3.m3.3.3.3.cmml"><mo id="S2.I3.ix1.p1.3.m3.3.3.3.3.2.1" stretchy="false" xref="S2.I3.ix1.p1.3.m3.3.3.3.cmml">(</mo><mi id="S2.I3.ix1.p1.3.m3.1.1" xref="S2.I3.ix1.p1.3.m3.1.1.cmml">n</mi><mo id="S2.I3.ix1.p1.3.m3.3.3.3.3.2.2" rspace="0.278em" stretchy="false" xref="S2.I3.ix1.p1.3.m3.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.I3.ix1.p1.3.m3.3.3.4" rspace="0.278em" xref="S2.I3.ix1.p1.3.m3.3.3.4.cmml">:=</mo><mrow id="S2.I3.ix1.p1.3.m3.3.3.1" xref="S2.I3.ix1.p1.3.m3.3.3.1.cmml"><msup id="S2.I3.ix1.p1.3.m3.3.3.1.3" xref="S2.I3.ix1.p1.3.m3.3.3.1.3.cmml"><mi id="S2.I3.ix1.p1.3.m3.3.3.1.3.2" xref="S2.I3.ix1.p1.3.m3.3.3.1.3.2.cmml">σ</mi><mi id="S2.I3.ix1.p1.3.m3.3.3.1.3.3" xref="S2.I3.ix1.p1.3.m3.3.3.1.3.3.cmml">ℤ</mi></msup><mo id="S2.I3.ix1.p1.3.m3.3.3.1.2" xref="S2.I3.ix1.p1.3.m3.3.3.1.2.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.3.m3.3.3.1.1.1" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.cmml"><mo id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.2" stretchy="false" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.cmml"><mi id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.2" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.2.cmml">𝐱</mi><mo id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.1" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.3.2" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.cmml"><mo id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.cmml">(</mo><mi id="S2.I3.ix1.p1.3.m3.2.2" xref="S2.I3.ix1.p1.3.m3.2.2.cmml">n</mi><mo id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.3" stretchy="false" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I3.ix1.p1.3.m3.3.3.5" xref="S2.I3.ix1.p1.3.m3.3.3.5.cmml">∈</mo><mi id="S2.I3.ix1.p1.3.m3.3.3.6" xref="S2.I3.ix1.p1.3.m3.3.3.6.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.3.m3.3b"><apply id="S2.I3.ix1.p1.3.m3.3.3.cmml" xref="S2.I3.ix1.p1.3.m3.3.3"><and id="S2.I3.ix1.p1.3.m3.3.3a.cmml" xref="S2.I3.ix1.p1.3.m3.3.3"></and><apply id="S2.I3.ix1.p1.3.m3.3.3b.cmml" xref="S2.I3.ix1.p1.3.m3.3.3"><csymbol cd="latexml" id="S2.I3.ix1.p1.3.m3.3.3.4.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.4">assign</csymbol><apply id="S2.I3.ix1.p1.3.m3.3.3.3.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.3"><times id="S2.I3.ix1.p1.3.m3.3.3.3.1.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.3.1"></times><ci id="S2.I3.ix1.p1.3.m3.3.3.3.2.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.3.2">𝐲</ci><ci id="S2.I3.ix1.p1.3.m3.1.1.cmml" xref="S2.I3.ix1.p1.3.m3.1.1">𝑛</ci></apply><apply id="S2.I3.ix1.p1.3.m3.3.3.1.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1"><times id="S2.I3.ix1.p1.3.m3.3.3.1.2.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1.2"></times><apply id="S2.I3.ix1.p1.3.m3.3.3.1.3.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1.3"><csymbol cd="ambiguous" id="S2.I3.ix1.p1.3.m3.3.3.1.3.1.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1.3">superscript</csymbol><ci id="S2.I3.ix1.p1.3.m3.3.3.1.3.2.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1.3.2">𝜎</ci><ci id="S2.I3.ix1.p1.3.m3.3.3.1.3.3.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1.3.3">ℤ</ci></apply><apply id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1"><times id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.1.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.1"></times><ci id="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.2.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.1.1.1.1.2">𝐱</ci><ci id="S2.I3.ix1.p1.3.m3.2.2.cmml" xref="S2.I3.ix1.p1.3.m3.2.2">𝑛</ci></apply></apply></apply><apply id="S2.I3.ix1.p1.3.m3.3.3c.cmml" xref="S2.I3.ix1.p1.3.m3.3.3"><in id="S2.I3.ix1.p1.3.m3.3.3.5.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.5"></in><share href="https://arxiv.org/html/2211.11234v4#S2.I3.ix1.p1.3.m3.3.3.1.cmml" id="S2.I3.ix1.p1.3.m3.3.3d.cmml" xref="S2.I3.ix1.p1.3.m3.3.3"></share><ci id="S2.I3.ix1.p1.3.m3.3.3.6.cmml" xref="S2.I3.ix1.p1.3.m3.3.3.6">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.3.m3.3c">{\bf y}(n):=\sigma^{\mathbb{Z}}({\bf x}(n))\in Y</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.3.m3.3d">bold_y ( italic_n ) := italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x ( italic_n ) ) ∈ italic_Y</annotation></semantics></math>, and assume that for suitable shift exponents <math alttext="k(n)\in\mathbb{Z}" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.4.m4.1"><semantics id="S2.I3.ix1.p1.4.m4.1a"><mrow id="S2.I3.ix1.p1.4.m4.1.2" xref="S2.I3.ix1.p1.4.m4.1.2.cmml"><mrow id="S2.I3.ix1.p1.4.m4.1.2.2" xref="S2.I3.ix1.p1.4.m4.1.2.2.cmml"><mi id="S2.I3.ix1.p1.4.m4.1.2.2.2" xref="S2.I3.ix1.p1.4.m4.1.2.2.2.cmml">k</mi><mo id="S2.I3.ix1.p1.4.m4.1.2.2.1" xref="S2.I3.ix1.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.4.m4.1.2.2.3.2" xref="S2.I3.ix1.p1.4.m4.1.2.2.cmml"><mo id="S2.I3.ix1.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="S2.I3.ix1.p1.4.m4.1.2.2.cmml">(</mo><mi id="S2.I3.ix1.p1.4.m4.1.1" xref="S2.I3.ix1.p1.4.m4.1.1.cmml">n</mi><mo id="S2.I3.ix1.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="S2.I3.ix1.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.I3.ix1.p1.4.m4.1.2.1" xref="S2.I3.ix1.p1.4.m4.1.2.1.cmml">∈</mo><mi id="S2.I3.ix1.p1.4.m4.1.2.3" xref="S2.I3.ix1.p1.4.m4.1.2.3.cmml">ℤ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.4.m4.1b"><apply id="S2.I3.ix1.p1.4.m4.1.2.cmml" xref="S2.I3.ix1.p1.4.m4.1.2"><in id="S2.I3.ix1.p1.4.m4.1.2.1.cmml" xref="S2.I3.ix1.p1.4.m4.1.2.1"></in><apply id="S2.I3.ix1.p1.4.m4.1.2.2.cmml" xref="S2.I3.ix1.p1.4.m4.1.2.2"><times id="S2.I3.ix1.p1.4.m4.1.2.2.1.cmml" xref="S2.I3.ix1.p1.4.m4.1.2.2.1"></times><ci id="S2.I3.ix1.p1.4.m4.1.2.2.2.cmml" xref="S2.I3.ix1.p1.4.m4.1.2.2.2">𝑘</ci><ci id="S2.I3.ix1.p1.4.m4.1.1.cmml" xref="S2.I3.ix1.p1.4.m4.1.1">𝑛</ci></apply><ci id="S2.I3.ix1.p1.4.m4.1.2.3.cmml" xref="S2.I3.ix1.p1.4.m4.1.2.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.4.m4.1c">k(n)\in\mathbb{Z}</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.4.m4.1d">italic_k ( italic_n ) ∈ blackboard_Z</annotation></semantics></math> the sequence of the biinfinite words <math alttext="T^{k(n)}({\bf y}(n))" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.5.m5.3"><semantics id="S2.I3.ix1.p1.5.m5.3a"><mrow id="S2.I3.ix1.p1.5.m5.3.3" xref="S2.I3.ix1.p1.5.m5.3.3.cmml"><msup id="S2.I3.ix1.p1.5.m5.3.3.3" xref="S2.I3.ix1.p1.5.m5.3.3.3.cmml"><mi id="S2.I3.ix1.p1.5.m5.3.3.3.2" xref="S2.I3.ix1.p1.5.m5.3.3.3.2.cmml">T</mi><mrow id="S2.I3.ix1.p1.5.m5.1.1.1" xref="S2.I3.ix1.p1.5.m5.1.1.1.cmml"><mi id="S2.I3.ix1.p1.5.m5.1.1.1.3" xref="S2.I3.ix1.p1.5.m5.1.1.1.3.cmml">k</mi><mo id="S2.I3.ix1.p1.5.m5.1.1.1.2" xref="S2.I3.ix1.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.5.m5.1.1.1.4.2" xref="S2.I3.ix1.p1.5.m5.1.1.1.cmml"><mo id="S2.I3.ix1.p1.5.m5.1.1.1.4.2.1" stretchy="false" xref="S2.I3.ix1.p1.5.m5.1.1.1.cmml">(</mo><mi id="S2.I3.ix1.p1.5.m5.1.1.1.1" xref="S2.I3.ix1.p1.5.m5.1.1.1.1.cmml">n</mi><mo id="S2.I3.ix1.p1.5.m5.1.1.1.4.2.2" stretchy="false" xref="S2.I3.ix1.p1.5.m5.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S2.I3.ix1.p1.5.m5.3.3.2" xref="S2.I3.ix1.p1.5.m5.3.3.2.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.5.m5.3.3.1.1" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.cmml"><mo id="S2.I3.ix1.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.cmml">(</mo><mrow id="S2.I3.ix1.p1.5.m5.3.3.1.1.1" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.cmml"><mi id="S2.I3.ix1.p1.5.m5.3.3.1.1.1.2" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.2.cmml">𝐲</mi><mo id="S2.I3.ix1.p1.5.m5.3.3.1.1.1.1" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.5.m5.3.3.1.1.1.3.2" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.cmml"><mo id="S2.I3.ix1.p1.5.m5.3.3.1.1.1.3.2.1" stretchy="false" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.cmml">(</mo><mi id="S2.I3.ix1.p1.5.m5.2.2" xref="S2.I3.ix1.p1.5.m5.2.2.cmml">n</mi><mo id="S2.I3.ix1.p1.5.m5.3.3.1.1.1.3.2.2" stretchy="false" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I3.ix1.p1.5.m5.3.3.1.1.3" stretchy="false" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.5.m5.3b"><apply id="S2.I3.ix1.p1.5.m5.3.3.cmml" xref="S2.I3.ix1.p1.5.m5.3.3"><times id="S2.I3.ix1.p1.5.m5.3.3.2.cmml" xref="S2.I3.ix1.p1.5.m5.3.3.2"></times><apply id="S2.I3.ix1.p1.5.m5.3.3.3.cmml" xref="S2.I3.ix1.p1.5.m5.3.3.3"><csymbol cd="ambiguous" id="S2.I3.ix1.p1.5.m5.3.3.3.1.cmml" xref="S2.I3.ix1.p1.5.m5.3.3.3">superscript</csymbol><ci id="S2.I3.ix1.p1.5.m5.3.3.3.2.cmml" xref="S2.I3.ix1.p1.5.m5.3.3.3.2">𝑇</ci><apply id="S2.I3.ix1.p1.5.m5.1.1.1.cmml" xref="S2.I3.ix1.p1.5.m5.1.1.1"><times id="S2.I3.ix1.p1.5.m5.1.1.1.2.cmml" xref="S2.I3.ix1.p1.5.m5.1.1.1.2"></times><ci id="S2.I3.ix1.p1.5.m5.1.1.1.3.cmml" xref="S2.I3.ix1.p1.5.m5.1.1.1.3">𝑘</ci><ci id="S2.I3.ix1.p1.5.m5.1.1.1.1.cmml" xref="S2.I3.ix1.p1.5.m5.1.1.1.1">𝑛</ci></apply></apply><apply id="S2.I3.ix1.p1.5.m5.3.3.1.1.1.cmml" xref="S2.I3.ix1.p1.5.m5.3.3.1.1"><times id="S2.I3.ix1.p1.5.m5.3.3.1.1.1.1.cmml" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.1"></times><ci id="S2.I3.ix1.p1.5.m5.3.3.1.1.1.2.cmml" xref="S2.I3.ix1.p1.5.m5.3.3.1.1.1.2">𝐲</ci><ci id="S2.I3.ix1.p1.5.m5.2.2.cmml" xref="S2.I3.ix1.p1.5.m5.2.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.5.m5.3c">T^{k(n)}({\bf y}(n))</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.5.m5.3d">italic_T start_POSTSUPERSCRIPT italic_k ( italic_n ) end_POSTSUPERSCRIPT ( bold_y ( italic_n ) )</annotation></semantics></math> converges to some <math alttext="{\bf y}\in Y" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.6.m6.1"><semantics id="S2.I3.ix1.p1.6.m6.1a"><mrow id="S2.I3.ix1.p1.6.m6.1.1" xref="S2.I3.ix1.p1.6.m6.1.1.cmml"><mi id="S2.I3.ix1.p1.6.m6.1.1.2" xref="S2.I3.ix1.p1.6.m6.1.1.2.cmml">𝐲</mi><mo id="S2.I3.ix1.p1.6.m6.1.1.1" xref="S2.I3.ix1.p1.6.m6.1.1.1.cmml">∈</mo><mi id="S2.I3.ix1.p1.6.m6.1.1.3" xref="S2.I3.ix1.p1.6.m6.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.6.m6.1b"><apply id="S2.I3.ix1.p1.6.m6.1.1.cmml" xref="S2.I3.ix1.p1.6.m6.1.1"><in id="S2.I3.ix1.p1.6.m6.1.1.1.cmml" xref="S2.I3.ix1.p1.6.m6.1.1.1"></in><ci id="S2.I3.ix1.p1.6.m6.1.1.2.cmml" xref="S2.I3.ix1.p1.6.m6.1.1.2">𝐲</ci><ci id="S2.I3.ix1.p1.6.m6.1.1.3.cmml" xref="S2.I3.ix1.p1.6.m6.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.6.m6.1c">{\bf y}\in Y</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.6.m6.1d">bold_y ∈ italic_Y</annotation></semantics></math>. Then there exists a biinfinite word <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.7.m7.1"><semantics id="S2.I3.ix1.p1.7.m7.1a"><mrow id="S2.I3.ix1.p1.7.m7.1.1" xref="S2.I3.ix1.p1.7.m7.1.1.cmml"><mi id="S2.I3.ix1.p1.7.m7.1.1.2" xref="S2.I3.ix1.p1.7.m7.1.1.2.cmml">𝐱</mi><mo id="S2.I3.ix1.p1.7.m7.1.1.1" xref="S2.I3.ix1.p1.7.m7.1.1.1.cmml">∈</mo><mi id="S2.I3.ix1.p1.7.m7.1.1.3" xref="S2.I3.ix1.p1.7.m7.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.7.m7.1b"><apply id="S2.I3.ix1.p1.7.m7.1.1.cmml" xref="S2.I3.ix1.p1.7.m7.1.1"><in id="S2.I3.ix1.p1.7.m7.1.1.1.cmml" xref="S2.I3.ix1.p1.7.m7.1.1.1"></in><ci id="S2.I3.ix1.p1.7.m7.1.1.2.cmml" xref="S2.I3.ix1.p1.7.m7.1.1.2">𝐱</ci><ci id="S2.I3.ix1.p1.7.m7.1.1.3.cmml" xref="S2.I3.ix1.p1.7.m7.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.7.m7.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.7.m7.1d">bold_x ∈ italic_X</annotation></semantics></math> and a subsequence <math alttext="({\bf x}(n_{m}))_{m\in\mathbb{N}}" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.8.m8.1"><semantics id="S2.I3.ix1.p1.8.m8.1a"><msub id="S2.I3.ix1.p1.8.m8.1.1" xref="S2.I3.ix1.p1.8.m8.1.1.cmml"><mrow id="S2.I3.ix1.p1.8.m8.1.1.1.1" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.cmml"><mo id="S2.I3.ix1.p1.8.m8.1.1.1.1.2" stretchy="false" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.cmml">(</mo><mrow id="S2.I3.ix1.p1.8.m8.1.1.1.1.1" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.cmml"><mi id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.3" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.3.cmml">𝐱</mi><mo id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.2" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.cmml"><mo id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.cmml"><mi id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.2" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.2.cmml">n</mi><mi id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.3" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I3.ix1.p1.8.m8.1.1.1.1.3" stretchy="false" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.I3.ix1.p1.8.m8.1.1.3" xref="S2.I3.ix1.p1.8.m8.1.1.3.cmml"><mi id="S2.I3.ix1.p1.8.m8.1.1.3.2" xref="S2.I3.ix1.p1.8.m8.1.1.3.2.cmml">m</mi><mo id="S2.I3.ix1.p1.8.m8.1.1.3.1" xref="S2.I3.ix1.p1.8.m8.1.1.3.1.cmml">∈</mo><mi id="S2.I3.ix1.p1.8.m8.1.1.3.3" xref="S2.I3.ix1.p1.8.m8.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.8.m8.1b"><apply id="S2.I3.ix1.p1.8.m8.1.1.cmml" xref="S2.I3.ix1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.I3.ix1.p1.8.m8.1.1.2.cmml" xref="S2.I3.ix1.p1.8.m8.1.1">subscript</csymbol><apply id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.1.1"><times id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.2.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.2"></times><ci id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.3.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.3">𝐱</ci><apply id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.1.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.2.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.2">𝑛</ci><ci id="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.3.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.1.1.1.1.1.1.3">𝑚</ci></apply></apply><apply id="S2.I3.ix1.p1.8.m8.1.1.3.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.3"><in id="S2.I3.ix1.p1.8.m8.1.1.3.1.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.3.1"></in><ci id="S2.I3.ix1.p1.8.m8.1.1.3.2.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.3.2">𝑚</ci><ci id="S2.I3.ix1.p1.8.m8.1.1.3.3.cmml" xref="S2.I3.ix1.p1.8.m8.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.8.m8.1c">({\bf x}(n_{m}))_{m\in\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.8.m8.1d">( bold_x ( italic_n start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_m ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> such that for a suitable set of integer exponents <math alttext="\ell_{m}\in\mathbb{Z}" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.9.m9.1"><semantics id="S2.I3.ix1.p1.9.m9.1a"><mrow id="S2.I3.ix1.p1.9.m9.1.1" xref="S2.I3.ix1.p1.9.m9.1.1.cmml"><msub id="S2.I3.ix1.p1.9.m9.1.1.2" xref="S2.I3.ix1.p1.9.m9.1.1.2.cmml"><mi id="S2.I3.ix1.p1.9.m9.1.1.2.2" mathvariant="normal" xref="S2.I3.ix1.p1.9.m9.1.1.2.2.cmml">ℓ</mi><mi id="S2.I3.ix1.p1.9.m9.1.1.2.3" xref="S2.I3.ix1.p1.9.m9.1.1.2.3.cmml">m</mi></msub><mo id="S2.I3.ix1.p1.9.m9.1.1.1" xref="S2.I3.ix1.p1.9.m9.1.1.1.cmml">∈</mo><mi id="S2.I3.ix1.p1.9.m9.1.1.3" xref="S2.I3.ix1.p1.9.m9.1.1.3.cmml">ℤ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.9.m9.1b"><apply id="S2.I3.ix1.p1.9.m9.1.1.cmml" xref="S2.I3.ix1.p1.9.m9.1.1"><in id="S2.I3.ix1.p1.9.m9.1.1.1.cmml" xref="S2.I3.ix1.p1.9.m9.1.1.1"></in><apply id="S2.I3.ix1.p1.9.m9.1.1.2.cmml" xref="S2.I3.ix1.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.I3.ix1.p1.9.m9.1.1.2.1.cmml" xref="S2.I3.ix1.p1.9.m9.1.1.2">subscript</csymbol><ci id="S2.I3.ix1.p1.9.m9.1.1.2.2.cmml" xref="S2.I3.ix1.p1.9.m9.1.1.2.2">ℓ</ci><ci id="S2.I3.ix1.p1.9.m9.1.1.2.3.cmml" xref="S2.I3.ix1.p1.9.m9.1.1.2.3">𝑚</ci></apply><ci id="S2.I3.ix1.p1.9.m9.1.1.3.cmml" xref="S2.I3.ix1.p1.9.m9.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.9.m9.1c">\ell_{m}\in\mathbb{Z}</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.9.m9.1d">roman_ℓ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ blackboard_Z</annotation></semantics></math> one has</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\bf x}=\lim_{m\to\infty}T^{\ell_{m}}({\bf x}(n_{m}))" class="ltx_Math" display="block" id="S2.Ex9.m1.1"><semantics id="S2.Ex9.m1.1a"><mrow id="S2.Ex9.m1.1.1" xref="S2.Ex9.m1.1.1.cmml"><mi id="S2.Ex9.m1.1.1.3" xref="S2.Ex9.m1.1.1.3.cmml">𝐱</mi><mo id="S2.Ex9.m1.1.1.2" rspace="0.1389em" xref="S2.Ex9.m1.1.1.2.cmml">=</mo><mrow id="S2.Ex9.m1.1.1.1" xref="S2.Ex9.m1.1.1.1.cmml"><munder id="S2.Ex9.m1.1.1.1.2" xref="S2.Ex9.m1.1.1.1.2.cmml"><mo id="S2.Ex9.m1.1.1.1.2.2" lspace="0.1389em" movablelimits="false" rspace="0.167em" xref="S2.Ex9.m1.1.1.1.2.2.cmml">lim</mo><mrow id="S2.Ex9.m1.1.1.1.2.3" xref="S2.Ex9.m1.1.1.1.2.3.cmml"><mi id="S2.Ex9.m1.1.1.1.2.3.2" xref="S2.Ex9.m1.1.1.1.2.3.2.cmml">m</mi><mo id="S2.Ex9.m1.1.1.1.2.3.1" stretchy="false" xref="S2.Ex9.m1.1.1.1.2.3.1.cmml">→</mo><mi id="S2.Ex9.m1.1.1.1.2.3.3" mathvariant="normal" xref="S2.Ex9.m1.1.1.1.2.3.3.cmml">∞</mi></mrow></munder><mrow id="S2.Ex9.m1.1.1.1.1" xref="S2.Ex9.m1.1.1.1.1.cmml"><msup id="S2.Ex9.m1.1.1.1.1.3" xref="S2.Ex9.m1.1.1.1.1.3.cmml"><mi id="S2.Ex9.m1.1.1.1.1.3.2" xref="S2.Ex9.m1.1.1.1.1.3.2.cmml">T</mi><msub id="S2.Ex9.m1.1.1.1.1.3.3" xref="S2.Ex9.m1.1.1.1.1.3.3.cmml"><mi id="S2.Ex9.m1.1.1.1.1.3.3.2" mathvariant="normal" xref="S2.Ex9.m1.1.1.1.1.3.3.2.cmml">ℓ</mi><mi id="S2.Ex9.m1.1.1.1.1.3.3.3" xref="S2.Ex9.m1.1.1.1.1.3.3.3.cmml">m</mi></msub></msup><mo id="S2.Ex9.m1.1.1.1.1.2" xref="S2.Ex9.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex9.m1.1.1.1.1.1.1" xref="S2.Ex9.m1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex9.m1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex9.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex9.m1.1.1.1.1.1.1.1" xref="S2.Ex9.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex9.m1.1.1.1.1.1.1.1.3" xref="S2.Ex9.m1.1.1.1.1.1.1.1.3.cmml">𝐱</mi><mo id="S2.Ex9.m1.1.1.1.1.1.1.1.2" xref="S2.Ex9.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">n</mi><mi id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex9.m1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex9.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex9.m1.1b"><apply id="S2.Ex9.m1.1.1.cmml" xref="S2.Ex9.m1.1.1"><eq id="S2.Ex9.m1.1.1.2.cmml" xref="S2.Ex9.m1.1.1.2"></eq><ci id="S2.Ex9.m1.1.1.3.cmml" xref="S2.Ex9.m1.1.1.3">𝐱</ci><apply id="S2.Ex9.m1.1.1.1.cmml" xref="S2.Ex9.m1.1.1.1"><apply id="S2.Ex9.m1.1.1.1.2.cmml" xref="S2.Ex9.m1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex9.m1.1.1.1.2.1.cmml" xref="S2.Ex9.m1.1.1.1.2">subscript</csymbol><limit id="S2.Ex9.m1.1.1.1.2.2.cmml" xref="S2.Ex9.m1.1.1.1.2.2"></limit><apply id="S2.Ex9.m1.1.1.1.2.3.cmml" xref="S2.Ex9.m1.1.1.1.2.3"><ci id="S2.Ex9.m1.1.1.1.2.3.1.cmml" xref="S2.Ex9.m1.1.1.1.2.3.1">→</ci><ci id="S2.Ex9.m1.1.1.1.2.3.2.cmml" xref="S2.Ex9.m1.1.1.1.2.3.2">𝑚</ci><infinity id="S2.Ex9.m1.1.1.1.2.3.3.cmml" xref="S2.Ex9.m1.1.1.1.2.3.3"></infinity></apply></apply><apply id="S2.Ex9.m1.1.1.1.1.cmml" xref="S2.Ex9.m1.1.1.1.1"><times id="S2.Ex9.m1.1.1.1.1.2.cmml" xref="S2.Ex9.m1.1.1.1.1.2"></times><apply id="S2.Ex9.m1.1.1.1.1.3.cmml" xref="S2.Ex9.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex9.m1.1.1.1.1.3.1.cmml" xref="S2.Ex9.m1.1.1.1.1.3">superscript</csymbol><ci id="S2.Ex9.m1.1.1.1.1.3.2.cmml" xref="S2.Ex9.m1.1.1.1.1.3.2">𝑇</ci><apply id="S2.Ex9.m1.1.1.1.1.3.3.cmml" xref="S2.Ex9.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Ex9.m1.1.1.1.1.3.3.1.cmml" xref="S2.Ex9.m1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.Ex9.m1.1.1.1.1.3.3.2.cmml" xref="S2.Ex9.m1.1.1.1.1.3.3.2">ℓ</ci><ci id="S2.Ex9.m1.1.1.1.1.3.3.3.cmml" xref="S2.Ex9.m1.1.1.1.1.3.3.3">𝑚</ci></apply></apply><apply id="S2.Ex9.m1.1.1.1.1.1.1.1.cmml" xref="S2.Ex9.m1.1.1.1.1.1.1"><times id="S2.Ex9.m1.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex9.m1.1.1.1.1.1.1.1.2"></times><ci id="S2.Ex9.m1.1.1.1.1.1.1.1.3.cmml" xref="S2.Ex9.m1.1.1.1.1.1.1.1.3">𝐱</ci><apply id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.2">𝑛</ci><ci id="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.Ex9.m1.1.1.1.1.1.1.1.1.1.1.3">𝑚</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex9.m1.1c">{\bf x}=\lim_{m\to\infty}T^{\ell_{m}}({\bf x}(n_{m}))</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m1.1d">bold_x = roman_lim start_POSTSUBSCRIPT italic_m → ∞ end_POSTSUBSCRIPT italic_T start_POSTSUPERSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( bold_x ( italic_n start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.I3.ix1.p1.11">as well as <math alttext="\sigma^{\mathbb{Z}}({\bf x})=T^{k}({\bf y})" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.10.m1.2"><semantics id="S2.I3.ix1.p1.10.m1.2a"><mrow id="S2.I3.ix1.p1.10.m1.2.3" xref="S2.I3.ix1.p1.10.m1.2.3.cmml"><mrow id="S2.I3.ix1.p1.10.m1.2.3.2" xref="S2.I3.ix1.p1.10.m1.2.3.2.cmml"><msup id="S2.I3.ix1.p1.10.m1.2.3.2.2" xref="S2.I3.ix1.p1.10.m1.2.3.2.2.cmml"><mi id="S2.I3.ix1.p1.10.m1.2.3.2.2.2" xref="S2.I3.ix1.p1.10.m1.2.3.2.2.2.cmml">σ</mi><mi id="S2.I3.ix1.p1.10.m1.2.3.2.2.3" xref="S2.I3.ix1.p1.10.m1.2.3.2.2.3.cmml">ℤ</mi></msup><mo id="S2.I3.ix1.p1.10.m1.2.3.2.1" xref="S2.I3.ix1.p1.10.m1.2.3.2.1.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.10.m1.2.3.2.3.2" xref="S2.I3.ix1.p1.10.m1.2.3.2.cmml"><mo id="S2.I3.ix1.p1.10.m1.2.3.2.3.2.1" stretchy="false" xref="S2.I3.ix1.p1.10.m1.2.3.2.cmml">(</mo><mi id="S2.I3.ix1.p1.10.m1.1.1" xref="S2.I3.ix1.p1.10.m1.1.1.cmml">𝐱</mi><mo id="S2.I3.ix1.p1.10.m1.2.3.2.3.2.2" stretchy="false" xref="S2.I3.ix1.p1.10.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.I3.ix1.p1.10.m1.2.3.1" xref="S2.I3.ix1.p1.10.m1.2.3.1.cmml">=</mo><mrow id="S2.I3.ix1.p1.10.m1.2.3.3" xref="S2.I3.ix1.p1.10.m1.2.3.3.cmml"><msup id="S2.I3.ix1.p1.10.m1.2.3.3.2" xref="S2.I3.ix1.p1.10.m1.2.3.3.2.cmml"><mi id="S2.I3.ix1.p1.10.m1.2.3.3.2.2" xref="S2.I3.ix1.p1.10.m1.2.3.3.2.2.cmml">T</mi><mi id="S2.I3.ix1.p1.10.m1.2.3.3.2.3" xref="S2.I3.ix1.p1.10.m1.2.3.3.2.3.cmml">k</mi></msup><mo id="S2.I3.ix1.p1.10.m1.2.3.3.1" xref="S2.I3.ix1.p1.10.m1.2.3.3.1.cmml">⁢</mo><mrow id="S2.I3.ix1.p1.10.m1.2.3.3.3.2" xref="S2.I3.ix1.p1.10.m1.2.3.3.cmml"><mo id="S2.I3.ix1.p1.10.m1.2.3.3.3.2.1" stretchy="false" xref="S2.I3.ix1.p1.10.m1.2.3.3.cmml">(</mo><mi id="S2.I3.ix1.p1.10.m1.2.2" xref="S2.I3.ix1.p1.10.m1.2.2.cmml">𝐲</mi><mo id="S2.I3.ix1.p1.10.m1.2.3.3.3.2.2" stretchy="false" xref="S2.I3.ix1.p1.10.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.10.m1.2b"><apply id="S2.I3.ix1.p1.10.m1.2.3.cmml" xref="S2.I3.ix1.p1.10.m1.2.3"><eq id="S2.I3.ix1.p1.10.m1.2.3.1.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.1"></eq><apply id="S2.I3.ix1.p1.10.m1.2.3.2.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.2"><times id="S2.I3.ix1.p1.10.m1.2.3.2.1.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.2.1"></times><apply id="S2.I3.ix1.p1.10.m1.2.3.2.2.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.2.2"><csymbol cd="ambiguous" id="S2.I3.ix1.p1.10.m1.2.3.2.2.1.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.2.2">superscript</csymbol><ci id="S2.I3.ix1.p1.10.m1.2.3.2.2.2.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.2.2.2">𝜎</ci><ci id="S2.I3.ix1.p1.10.m1.2.3.2.2.3.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.2.2.3">ℤ</ci></apply><ci id="S2.I3.ix1.p1.10.m1.1.1.cmml" xref="S2.I3.ix1.p1.10.m1.1.1">𝐱</ci></apply><apply id="S2.I3.ix1.p1.10.m1.2.3.3.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.3"><times id="S2.I3.ix1.p1.10.m1.2.3.3.1.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.3.1"></times><apply id="S2.I3.ix1.p1.10.m1.2.3.3.2.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.3.2"><csymbol cd="ambiguous" id="S2.I3.ix1.p1.10.m1.2.3.3.2.1.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.3.2">superscript</csymbol><ci id="S2.I3.ix1.p1.10.m1.2.3.3.2.2.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.3.2.2">𝑇</ci><ci id="S2.I3.ix1.p1.10.m1.2.3.3.2.3.cmml" xref="S2.I3.ix1.p1.10.m1.2.3.3.2.3">𝑘</ci></apply><ci id="S2.I3.ix1.p1.10.m1.2.2.cmml" xref="S2.I3.ix1.p1.10.m1.2.2">𝐲</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.10.m1.2c">\sigma^{\mathbb{Z}}({\bf x})=T^{k}({\bf y})</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.10.m1.2d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x ) = italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( bold_y )</annotation></semantics></math> for some <math alttext="k\in\mathbb{Z}" class="ltx_Math" display="inline" id="S2.I3.ix1.p1.11.m2.1"><semantics id="S2.I3.ix1.p1.11.m2.1a"><mrow id="S2.I3.ix1.p1.11.m2.1.1" xref="S2.I3.ix1.p1.11.m2.1.1.cmml"><mi id="S2.I3.ix1.p1.11.m2.1.1.2" xref="S2.I3.ix1.p1.11.m2.1.1.2.cmml">k</mi><mo id="S2.I3.ix1.p1.11.m2.1.1.1" xref="S2.I3.ix1.p1.11.m2.1.1.1.cmml">∈</mo><mi id="S2.I3.ix1.p1.11.m2.1.1.3" xref="S2.I3.ix1.p1.11.m2.1.1.3.cmml">ℤ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I3.ix1.p1.11.m2.1b"><apply id="S2.I3.ix1.p1.11.m2.1.1.cmml" xref="S2.I3.ix1.p1.11.m2.1.1"><in id="S2.I3.ix1.p1.11.m2.1.1.1.cmml" xref="S2.I3.ix1.p1.11.m2.1.1.1"></in><ci id="S2.I3.ix1.p1.11.m2.1.1.2.cmml" xref="S2.I3.ix1.p1.11.m2.1.1.2">𝑘</ci><ci id="S2.I3.ix1.p1.11.m2.1.1.3.cmml" xref="S2.I3.ix1.p1.11.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I3.ix1.p1.11.m2.1c">k\in\mathbb{Z}</annotation><annotation encoding="application/x-llamapun" id="S2.I3.ix1.p1.11.m2.1d">italic_k ∈ blackboard_Z</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S2.SS2.1.p1.6">In order to prove (#) we first observe that without loss of generality we can replace any <math alttext="{\bf x}(n)" class="ltx_Math" display="inline" id="S2.SS2.1.p1.1.m1.1"><semantics id="S2.SS2.1.p1.1.m1.1a"><mrow id="S2.SS2.1.p1.1.m1.1.2" xref="S2.SS2.1.p1.1.m1.1.2.cmml"><mi id="S2.SS2.1.p1.1.m1.1.2.2" xref="S2.SS2.1.p1.1.m1.1.2.2.cmml">𝐱</mi><mo id="S2.SS2.1.p1.1.m1.1.2.1" xref="S2.SS2.1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.1.p1.1.m1.1.2.3.2" xref="S2.SS2.1.p1.1.m1.1.2.cmml"><mo id="S2.SS2.1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.1.p1.1.m1.1.2.cmml">(</mo><mi id="S2.SS2.1.p1.1.m1.1.1" xref="S2.SS2.1.p1.1.m1.1.1.cmml">n</mi><mo id="S2.SS2.1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.1.p1.1.m1.1b"><apply id="S2.SS2.1.p1.1.m1.1.2.cmml" xref="S2.SS2.1.p1.1.m1.1.2"><times id="S2.SS2.1.p1.1.m1.1.2.1.cmml" xref="S2.SS2.1.p1.1.m1.1.2.1"></times><ci id="S2.SS2.1.p1.1.m1.1.2.2.cmml" xref="S2.SS2.1.p1.1.m1.1.2.2">𝐱</ci><ci id="S2.SS2.1.p1.1.m1.1.1.cmml" xref="S2.SS2.1.p1.1.m1.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.1.p1.1.m1.1c">{\bf x}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.1.p1.1.m1.1d">bold_x ( italic_n )</annotation></semantics></math> by some shift-translate and thus achieve that the set of exponents <math alttext="k(n)" class="ltx_Math" display="inline" id="S2.SS2.1.p1.2.m2.1"><semantics id="S2.SS2.1.p1.2.m2.1a"><mrow id="S2.SS2.1.p1.2.m2.1.2" xref="S2.SS2.1.p1.2.m2.1.2.cmml"><mi id="S2.SS2.1.p1.2.m2.1.2.2" xref="S2.SS2.1.p1.2.m2.1.2.2.cmml">k</mi><mo id="S2.SS2.1.p1.2.m2.1.2.1" xref="S2.SS2.1.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.1.p1.2.m2.1.2.3.2" xref="S2.SS2.1.p1.2.m2.1.2.cmml"><mo id="S2.SS2.1.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS2.1.p1.2.m2.1.2.cmml">(</mo><mi id="S2.SS2.1.p1.2.m2.1.1" xref="S2.SS2.1.p1.2.m2.1.1.cmml">n</mi><mo id="S2.SS2.1.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS2.1.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.1.p1.2.m2.1b"><apply id="S2.SS2.1.p1.2.m2.1.2.cmml" xref="S2.SS2.1.p1.2.m2.1.2"><times id="S2.SS2.1.p1.2.m2.1.2.1.cmml" xref="S2.SS2.1.p1.2.m2.1.2.1"></times><ci id="S2.SS2.1.p1.2.m2.1.2.2.cmml" xref="S2.SS2.1.p1.2.m2.1.2.2">𝑘</ci><ci id="S2.SS2.1.p1.2.m2.1.1.cmml" xref="S2.SS2.1.p1.2.m2.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.1.p1.2.m2.1c">k(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.1.p1.2.m2.1d">italic_k ( italic_n )</annotation></semantics></math> is bounded, and indeed, that <math alttext="0\leq k(n)<|\sigma(x_{0})(n)|" class="ltx_Math" display="inline" id="S2.SS2.1.p1.3.m3.3"><semantics id="S2.SS2.1.p1.3.m3.3a"><mrow id="S2.SS2.1.p1.3.m3.3.3" xref="S2.SS2.1.p1.3.m3.3.3.cmml"><mn id="S2.SS2.1.p1.3.m3.3.3.3" xref="S2.SS2.1.p1.3.m3.3.3.3.cmml">0</mn><mo id="S2.SS2.1.p1.3.m3.3.3.4" xref="S2.SS2.1.p1.3.m3.3.3.4.cmml">≤</mo><mrow id="S2.SS2.1.p1.3.m3.3.3.5" xref="S2.SS2.1.p1.3.m3.3.3.5.cmml"><mi id="S2.SS2.1.p1.3.m3.3.3.5.2" xref="S2.SS2.1.p1.3.m3.3.3.5.2.cmml">k</mi><mo id="S2.SS2.1.p1.3.m3.3.3.5.1" xref="S2.SS2.1.p1.3.m3.3.3.5.1.cmml">⁢</mo><mrow id="S2.SS2.1.p1.3.m3.3.3.5.3.2" xref="S2.SS2.1.p1.3.m3.3.3.5.cmml"><mo 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xref="S2.SS2.1.p1.3.m3.3.3"></share><apply id="S2.SS2.1.p1.3.m3.3.3.1.2.cmml" xref="S2.SS2.1.p1.3.m3.3.3.1.1"><abs id="S2.SS2.1.p1.3.m3.3.3.1.2.1.cmml" xref="S2.SS2.1.p1.3.m3.3.3.1.1.2"></abs><apply id="S2.SS2.1.p1.3.m3.3.3.1.1.1.cmml" xref="S2.SS2.1.p1.3.m3.3.3.1.1.1"><times id="S2.SS2.1.p1.3.m3.3.3.1.1.1.2.cmml" xref="S2.SS2.1.p1.3.m3.3.3.1.1.1.2"></times><ci id="S2.SS2.1.p1.3.m3.3.3.1.1.1.3.cmml" xref="S2.SS2.1.p1.3.m3.3.3.1.1.1.3">𝜎</ci><apply id="S2.SS2.1.p1.3.m3.3.3.1.1.1.1.1.1.cmml" xref="S2.SS2.1.p1.3.m3.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.1.p1.3.m3.3.3.1.1.1.1.1.1.1.cmml" xref="S2.SS2.1.p1.3.m3.3.3.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.1.p1.3.m3.3.3.1.1.1.1.1.1.2.cmml" xref="S2.SS2.1.p1.3.m3.3.3.1.1.1.1.1.1.2">𝑥</ci><cn id="S2.SS2.1.p1.3.m3.3.3.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.1.p1.3.m3.3.3.1.1.1.1.1.1.3">0</cn></apply><ci id="S2.SS2.1.p1.3.m3.2.2.cmml" xref="S2.SS2.1.p1.3.m3.2.2">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.1.p1.3.m3.3c">0\leq k(n)<|\sigma(x_{0})(n)|</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.1.p1.3.m3.3d">0 ≤ italic_k ( italic_n ) < | italic_σ ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ( italic_n ) |</annotation></semantics></math>, where <math alttext="x_{0}(n)\in\cal A" class="ltx_Math" display="inline" id="S2.SS2.1.p1.4.m4.1"><semantics id="S2.SS2.1.p1.4.m4.1a"><mrow id="S2.SS2.1.p1.4.m4.1.2" xref="S2.SS2.1.p1.4.m4.1.2.cmml"><mrow id="S2.SS2.1.p1.4.m4.1.2.2" xref="S2.SS2.1.p1.4.m4.1.2.2.cmml"><msub id="S2.SS2.1.p1.4.m4.1.2.2.2" xref="S2.SS2.1.p1.4.m4.1.2.2.2.cmml"><mi id="S2.SS2.1.p1.4.m4.1.2.2.2.2" xref="S2.SS2.1.p1.4.m4.1.2.2.2.2.cmml">x</mi><mn id="S2.SS2.1.p1.4.m4.1.2.2.2.3" xref="S2.SS2.1.p1.4.m4.1.2.2.2.3.cmml">0</mn></msub><mo id="S2.SS2.1.p1.4.m4.1.2.2.1" xref="S2.SS2.1.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow 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xref="S2.SS2.1.p1.5.m5.4.5.3.cmml">(</mo><mi id="S2.SS2.1.p1.5.m5.4.4" xref="S2.SS2.1.p1.5.m5.4.4.cmml">n</mi><mo id="S2.SS2.1.p1.5.m5.4.5.3.8.2.2" stretchy="false" xref="S2.SS2.1.p1.5.m5.4.5.3.cmml">)</mo></mrow><mo id="S2.SS2.1.p1.5.m5.4.5.3.1f" xref="S2.SS2.1.p1.5.m5.4.5.3.1.cmml">⁢</mo><mi id="S2.SS2.1.p1.5.m5.4.5.3.9" mathvariant="normal" xref="S2.SS2.1.p1.5.m5.4.5.3.9.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.1.p1.5.m5.4b"><apply id="S2.SS2.1.p1.5.m5.4.5.cmml" xref="S2.SS2.1.p1.5.m5.4.5"><eq id="S2.SS2.1.p1.5.m5.4.5.1.cmml" xref="S2.SS2.1.p1.5.m5.4.5.1"></eq><apply id="S2.SS2.1.p1.5.m5.4.5.2.cmml" xref="S2.SS2.1.p1.5.m5.4.5.2"><times id="S2.SS2.1.p1.5.m5.4.5.2.1.cmml" xref="S2.SS2.1.p1.5.m5.4.5.2.1"></times><ci id="S2.SS2.1.p1.5.m5.4.5.2.2.cmml" xref="S2.SS2.1.p1.5.m5.4.5.2.2">𝐱</ci><ci id="S2.SS2.1.p1.5.m5.1.1.cmml" xref="S2.SS2.1.p1.5.m5.1.1">𝑛</ci></apply><apply id="S2.SS2.1.p1.5.m5.4.5.3.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3"><times id="S2.SS2.1.p1.5.m5.4.5.3.1.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.1"></times><ci id="S2.SS2.1.p1.5.m5.4.5.3.2.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.2">…</ci><apply id="S2.SS2.1.p1.5.m5.4.5.3.3.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.3"><csymbol cd="ambiguous" id="S2.SS2.1.p1.5.m5.4.5.3.3.1.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.3">subscript</csymbol><ci id="S2.SS2.1.p1.5.m5.4.5.3.3.2.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.3.2">𝑥</ci><apply id="S2.SS2.1.p1.5.m5.4.5.3.3.3.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.3.3"><minus id="S2.SS2.1.p1.5.m5.4.5.3.3.3.1.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.3.3"></minus><cn id="S2.SS2.1.p1.5.m5.4.5.3.3.3.2.cmml" type="integer" xref="S2.SS2.1.p1.5.m5.4.5.3.3.3.2">1</cn></apply></apply><ci id="S2.SS2.1.p1.5.m5.2.2.cmml" xref="S2.SS2.1.p1.5.m5.2.2">𝑛</ci><apply id="S2.SS2.1.p1.5.m5.4.5.3.5.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.5"><csymbol cd="ambiguous" id="S2.SS2.1.p1.5.m5.4.5.3.5.1.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.5">subscript</csymbol><ci id="S2.SS2.1.p1.5.m5.4.5.3.5.2.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.5.2">𝑥</ci><cn id="S2.SS2.1.p1.5.m5.4.5.3.5.3.cmml" type="integer" xref="S2.SS2.1.p1.5.m5.4.5.3.5.3">0</cn></apply><ci id="S2.SS2.1.p1.5.m5.3.3.cmml" xref="S2.SS2.1.p1.5.m5.3.3">𝑛</ci><apply id="S2.SS2.1.p1.5.m5.4.5.3.7.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.7"><csymbol cd="ambiguous" id="S2.SS2.1.p1.5.m5.4.5.3.7.1.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.7">subscript</csymbol><ci id="S2.SS2.1.p1.5.m5.4.5.3.7.2.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.7.2">𝑥</ci><cn id="S2.SS2.1.p1.5.m5.4.5.3.7.3.cmml" type="integer" xref="S2.SS2.1.p1.5.m5.4.5.3.7.3">1</cn></apply><ci id="S2.SS2.1.p1.5.m5.4.4.cmml" xref="S2.SS2.1.p1.5.m5.4.4">𝑛</ci><ci id="S2.SS2.1.p1.5.m5.4.5.3.9.cmml" xref="S2.SS2.1.p1.5.m5.4.5.3.9">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.1.p1.5.m5.4c">{\bf x}(n)=\ldots x_{-1}(n)x_{0}(n)x_{1}(n)\ldots</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.1.p1.5.m5.4d">bold_x ( italic_n ) = … italic_x start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ( italic_n ) italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_n ) italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_n ) …</annotation></semantics></math> for any <math alttext="n\in\mathbb{N}" class="ltx_Math" display="inline" id="S2.SS2.1.p1.6.m6.1"><semantics id="S2.SS2.1.p1.6.m6.1a"><mrow id="S2.SS2.1.p1.6.m6.1.1" xref="S2.SS2.1.p1.6.m6.1.1.cmml"><mi id="S2.SS2.1.p1.6.m6.1.1.2" xref="S2.SS2.1.p1.6.m6.1.1.2.cmml">n</mi><mo id="S2.SS2.1.p1.6.m6.1.1.1" xref="S2.SS2.1.p1.6.m6.1.1.1.cmml">∈</mo><mi id="S2.SS2.1.p1.6.m6.1.1.3" xref="S2.SS2.1.p1.6.m6.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.1.p1.6.m6.1b"><apply id="S2.SS2.1.p1.6.m6.1.1.cmml" xref="S2.SS2.1.p1.6.m6.1.1"><in id="S2.SS2.1.p1.6.m6.1.1.1.cmml" xref="S2.SS2.1.p1.6.m6.1.1.1"></in><ci id="S2.SS2.1.p1.6.m6.1.1.2.cmml" xref="S2.SS2.1.p1.6.m6.1.1.2">𝑛</ci><ci id="S2.SS2.1.p1.6.m6.1.1.3.cmml" xref="S2.SS2.1.p1.6.m6.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.1.p1.6.m6.1c">n\in\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.1.p1.6.m6.1d">italic_n ∈ blackboard_N</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.2.p2"> <p class="ltx_p" id="S2.SS2.2.p2.18">Hence, by extracting a subsequence of the <math alttext="{\bf x}(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.1.m1.1"><semantics id="S2.SS2.2.p2.1.m1.1a"><mrow id="S2.SS2.2.p2.1.m1.1.2" xref="S2.SS2.2.p2.1.m1.1.2.cmml"><mi id="S2.SS2.2.p2.1.m1.1.2.2" xref="S2.SS2.2.p2.1.m1.1.2.2.cmml">𝐱</mi><mo id="S2.SS2.2.p2.1.m1.1.2.1" xref="S2.SS2.2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.1.m1.1.2.3.2" xref="S2.SS2.2.p2.1.m1.1.2.cmml"><mo id="S2.SS2.2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.2.p2.1.m1.1.2.cmml">(</mo><mi id="S2.SS2.2.p2.1.m1.1.1" xref="S2.SS2.2.p2.1.m1.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.1.m1.1b"><apply id="S2.SS2.2.p2.1.m1.1.2.cmml" xref="S2.SS2.2.p2.1.m1.1.2"><times id="S2.SS2.2.p2.1.m1.1.2.1.cmml" xref="S2.SS2.2.p2.1.m1.1.2.1"></times><ci id="S2.SS2.2.p2.1.m1.1.2.2.cmml" xref="S2.SS2.2.p2.1.m1.1.2.2">𝐱</ci><ci id="S2.SS2.2.p2.1.m1.1.1.cmml" xref="S2.SS2.2.p2.1.m1.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.1.m1.1c">{\bf x}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.1.m1.1d">bold_x ( italic_n )</annotation></semantics></math>, we can achieve that the letter <math alttext="x_{0}=x_{0}(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.2.m2.1"><semantics id="S2.SS2.2.p2.2.m2.1a"><mrow id="S2.SS2.2.p2.2.m2.1.2" xref="S2.SS2.2.p2.2.m2.1.2.cmml"><msub id="S2.SS2.2.p2.2.m2.1.2.2" xref="S2.SS2.2.p2.2.m2.1.2.2.cmml"><mi id="S2.SS2.2.p2.2.m2.1.2.2.2" xref="S2.SS2.2.p2.2.m2.1.2.2.2.cmml">x</mi><mn id="S2.SS2.2.p2.2.m2.1.2.2.3" xref="S2.SS2.2.p2.2.m2.1.2.2.3.cmml">0</mn></msub><mo id="S2.SS2.2.p2.2.m2.1.2.1" xref="S2.SS2.2.p2.2.m2.1.2.1.cmml">=</mo><mrow id="S2.SS2.2.p2.2.m2.1.2.3" xref="S2.SS2.2.p2.2.m2.1.2.3.cmml"><msub id="S2.SS2.2.p2.2.m2.1.2.3.2" xref="S2.SS2.2.p2.2.m2.1.2.3.2.cmml"><mi id="S2.SS2.2.p2.2.m2.1.2.3.2.2" xref="S2.SS2.2.p2.2.m2.1.2.3.2.2.cmml">x</mi><mn id="S2.SS2.2.p2.2.m2.1.2.3.2.3" xref="S2.SS2.2.p2.2.m2.1.2.3.2.3.cmml">0</mn></msub><mo id="S2.SS2.2.p2.2.m2.1.2.3.1" xref="S2.SS2.2.p2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.2.m2.1.2.3.3.2" xref="S2.SS2.2.p2.2.m2.1.2.3.cmml"><mo id="S2.SS2.2.p2.2.m2.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.2.p2.2.m2.1.2.3.cmml">(</mo><mi id="S2.SS2.2.p2.2.m2.1.1" xref="S2.SS2.2.p2.2.m2.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.2.m2.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.2.p2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.2.m2.1b"><apply id="S2.SS2.2.p2.2.m2.1.2.cmml" xref="S2.SS2.2.p2.2.m2.1.2"><eq id="S2.SS2.2.p2.2.m2.1.2.1.cmml" xref="S2.SS2.2.p2.2.m2.1.2.1"></eq><apply id="S2.SS2.2.p2.2.m2.1.2.2.cmml" xref="S2.SS2.2.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.2.p2.2.m2.1.2.2.1.cmml" xref="S2.SS2.2.p2.2.m2.1.2.2">subscript</csymbol><ci id="S2.SS2.2.p2.2.m2.1.2.2.2.cmml" xref="S2.SS2.2.p2.2.m2.1.2.2.2">𝑥</ci><cn id="S2.SS2.2.p2.2.m2.1.2.2.3.cmml" type="integer" xref="S2.SS2.2.p2.2.m2.1.2.2.3">0</cn></apply><apply id="S2.SS2.2.p2.2.m2.1.2.3.cmml" xref="S2.SS2.2.p2.2.m2.1.2.3"><times id="S2.SS2.2.p2.2.m2.1.2.3.1.cmml" xref="S2.SS2.2.p2.2.m2.1.2.3.1"></times><apply id="S2.SS2.2.p2.2.m2.1.2.3.2.cmml" xref="S2.SS2.2.p2.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.2.p2.2.m2.1.2.3.2.1.cmml" xref="S2.SS2.2.p2.2.m2.1.2.3.2">subscript</csymbol><ci id="S2.SS2.2.p2.2.m2.1.2.3.2.2.cmml" xref="S2.SS2.2.p2.2.m2.1.2.3.2.2">𝑥</ci><cn id="S2.SS2.2.p2.2.m2.1.2.3.2.3.cmml" type="integer" xref="S2.SS2.2.p2.2.m2.1.2.3.2.3">0</cn></apply><ci id="S2.SS2.2.p2.2.m2.1.1.cmml" xref="S2.SS2.2.p2.2.m2.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.2.m2.1c">x_{0}=x_{0}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.2.m2.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_n )</annotation></semantics></math> as well as the exponent <math alttext="k=k(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.3.m3.1"><semantics id="S2.SS2.2.p2.3.m3.1a"><mrow id="S2.SS2.2.p2.3.m3.1.2" xref="S2.SS2.2.p2.3.m3.1.2.cmml"><mi id="S2.SS2.2.p2.3.m3.1.2.2" xref="S2.SS2.2.p2.3.m3.1.2.2.cmml">k</mi><mo id="S2.SS2.2.p2.3.m3.1.2.1" xref="S2.SS2.2.p2.3.m3.1.2.1.cmml">=</mo><mrow id="S2.SS2.2.p2.3.m3.1.2.3" xref="S2.SS2.2.p2.3.m3.1.2.3.cmml"><mi id="S2.SS2.2.p2.3.m3.1.2.3.2" xref="S2.SS2.2.p2.3.m3.1.2.3.2.cmml">k</mi><mo id="S2.SS2.2.p2.3.m3.1.2.3.1" xref="S2.SS2.2.p2.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.3.m3.1.2.3.3.2" xref="S2.SS2.2.p2.3.m3.1.2.3.cmml"><mo id="S2.SS2.2.p2.3.m3.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.2.p2.3.m3.1.2.3.cmml">(</mo><mi id="S2.SS2.2.p2.3.m3.1.1" xref="S2.SS2.2.p2.3.m3.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.3.m3.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.2.p2.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.3.m3.1b"><apply id="S2.SS2.2.p2.3.m3.1.2.cmml" xref="S2.SS2.2.p2.3.m3.1.2"><eq id="S2.SS2.2.p2.3.m3.1.2.1.cmml" xref="S2.SS2.2.p2.3.m3.1.2.1"></eq><ci id="S2.SS2.2.p2.3.m3.1.2.2.cmml" xref="S2.SS2.2.p2.3.m3.1.2.2">𝑘</ci><apply id="S2.SS2.2.p2.3.m3.1.2.3.cmml" xref="S2.SS2.2.p2.3.m3.1.2.3"><times id="S2.SS2.2.p2.3.m3.1.2.3.1.cmml" xref="S2.SS2.2.p2.3.m3.1.2.3.1"></times><ci id="S2.SS2.2.p2.3.m3.1.2.3.2.cmml" xref="S2.SS2.2.p2.3.m3.1.2.3.2">𝑘</ci><ci id="S2.SS2.2.p2.3.m3.1.1.cmml" xref="S2.SS2.2.p2.3.m3.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.3.m3.1c">k=k(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.3.m3.1d">italic_k = italic_k ( italic_n )</annotation></semantics></math> is independent of <math alttext="n" class="ltx_Math" display="inline" id="S2.SS2.2.p2.4.m4.1"><semantics id="S2.SS2.2.p2.4.m4.1a"><mi id="S2.SS2.2.p2.4.m4.1.1" xref="S2.SS2.2.p2.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.4.m4.1b"><ci id="S2.SS2.2.p2.4.m4.1.1.cmml" xref="S2.SS2.2.p2.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.4.m4.1d">italic_n</annotation></semantics></math>. We now use the assumption that the <math alttext="T^{k}({\bf y}(n))" class="ltx_Math" display="inline" id="S2.SS2.2.p2.5.m5.2"><semantics id="S2.SS2.2.p2.5.m5.2a"><mrow id="S2.SS2.2.p2.5.m5.2.2" xref="S2.SS2.2.p2.5.m5.2.2.cmml"><msup id="S2.SS2.2.p2.5.m5.2.2.3" xref="S2.SS2.2.p2.5.m5.2.2.3.cmml"><mi id="S2.SS2.2.p2.5.m5.2.2.3.2" xref="S2.SS2.2.p2.5.m5.2.2.3.2.cmml">T</mi><mi id="S2.SS2.2.p2.5.m5.2.2.3.3" xref="S2.SS2.2.p2.5.m5.2.2.3.3.cmml">k</mi></msup><mo id="S2.SS2.2.p2.5.m5.2.2.2" xref="S2.SS2.2.p2.5.m5.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.2.p2.5.m5.2.2.1.1" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.cmml"><mo id="S2.SS2.2.p2.5.m5.2.2.1.1.2" stretchy="false" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS2.2.p2.5.m5.2.2.1.1.1" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.cmml"><mi id="S2.SS2.2.p2.5.m5.2.2.1.1.1.2" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.2.cmml">𝐲</mi><mo id="S2.SS2.2.p2.5.m5.2.2.1.1.1.1" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.5.m5.2.2.1.1.1.3.2" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.cmml"><mo id="S2.SS2.2.p2.5.m5.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.cmml">(</mo><mi id="S2.SS2.2.p2.5.m5.1.1" xref="S2.SS2.2.p2.5.m5.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.5.m5.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.2.p2.5.m5.2.2.1.1.3" stretchy="false" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.5.m5.2b"><apply id="S2.SS2.2.p2.5.m5.2.2.cmml" xref="S2.SS2.2.p2.5.m5.2.2"><times id="S2.SS2.2.p2.5.m5.2.2.2.cmml" xref="S2.SS2.2.p2.5.m5.2.2.2"></times><apply id="S2.SS2.2.p2.5.m5.2.2.3.cmml" xref="S2.SS2.2.p2.5.m5.2.2.3"><csymbol cd="ambiguous" id="S2.SS2.2.p2.5.m5.2.2.3.1.cmml" xref="S2.SS2.2.p2.5.m5.2.2.3">superscript</csymbol><ci id="S2.SS2.2.p2.5.m5.2.2.3.2.cmml" xref="S2.SS2.2.p2.5.m5.2.2.3.2">𝑇</ci><ci id="S2.SS2.2.p2.5.m5.2.2.3.3.cmml" xref="S2.SS2.2.p2.5.m5.2.2.3.3">𝑘</ci></apply><apply id="S2.SS2.2.p2.5.m5.2.2.1.1.1.cmml" xref="S2.SS2.2.p2.5.m5.2.2.1.1"><times id="S2.SS2.2.p2.5.m5.2.2.1.1.1.1.cmml" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.1"></times><ci id="S2.SS2.2.p2.5.m5.2.2.1.1.1.2.cmml" xref="S2.SS2.2.p2.5.m5.2.2.1.1.1.2">𝐲</ci><ci id="S2.SS2.2.p2.5.m5.1.1.cmml" xref="S2.SS2.2.p2.5.m5.1.1">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.5.m5.2c">T^{k}({\bf y}(n))</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.5.m5.2d">italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( bold_y ( italic_n ) )</annotation></semantics></math> converge, in order to extract again a subsequence of the previous subsequence of the <math alttext="{\bf x}(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.6.m6.1"><semantics id="S2.SS2.2.p2.6.m6.1a"><mrow id="S2.SS2.2.p2.6.m6.1.2" xref="S2.SS2.2.p2.6.m6.1.2.cmml"><mi id="S2.SS2.2.p2.6.m6.1.2.2" xref="S2.SS2.2.p2.6.m6.1.2.2.cmml">𝐱</mi><mo id="S2.SS2.2.p2.6.m6.1.2.1" xref="S2.SS2.2.p2.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.6.m6.1.2.3.2" xref="S2.SS2.2.p2.6.m6.1.2.cmml"><mo id="S2.SS2.2.p2.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS2.2.p2.6.m6.1.2.cmml">(</mo><mi id="S2.SS2.2.p2.6.m6.1.1" xref="S2.SS2.2.p2.6.m6.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS2.2.p2.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.6.m6.1b"><apply id="S2.SS2.2.p2.6.m6.1.2.cmml" xref="S2.SS2.2.p2.6.m6.1.2"><times id="S2.SS2.2.p2.6.m6.1.2.1.cmml" xref="S2.SS2.2.p2.6.m6.1.2.1"></times><ci id="S2.SS2.2.p2.6.m6.1.2.2.cmml" xref="S2.SS2.2.p2.6.m6.1.2.2">𝐱</ci><ci id="S2.SS2.2.p2.6.m6.1.1.cmml" xref="S2.SS2.2.p2.6.m6.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.6.m6.1c">{\bf x}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.6.m6.1d">bold_x ( italic_n )</annotation></semantics></math> in order to ensure that the letters <math alttext="x_{-1}(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.7.m7.1"><semantics id="S2.SS2.2.p2.7.m7.1a"><mrow id="S2.SS2.2.p2.7.m7.1.2" xref="S2.SS2.2.p2.7.m7.1.2.cmml"><msub id="S2.SS2.2.p2.7.m7.1.2.2" xref="S2.SS2.2.p2.7.m7.1.2.2.cmml"><mi id="S2.SS2.2.p2.7.m7.1.2.2.2" xref="S2.SS2.2.p2.7.m7.1.2.2.2.cmml">x</mi><mrow id="S2.SS2.2.p2.7.m7.1.2.2.3" xref="S2.SS2.2.p2.7.m7.1.2.2.3.cmml"><mo id="S2.SS2.2.p2.7.m7.1.2.2.3a" xref="S2.SS2.2.p2.7.m7.1.2.2.3.cmml">−</mo><mn id="S2.SS2.2.p2.7.m7.1.2.2.3.2" xref="S2.SS2.2.p2.7.m7.1.2.2.3.2.cmml">1</mn></mrow></msub><mo id="S2.SS2.2.p2.7.m7.1.2.1" xref="S2.SS2.2.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.7.m7.1.2.3.2" xref="S2.SS2.2.p2.7.m7.1.2.cmml"><mo id="S2.SS2.2.p2.7.m7.1.2.3.2.1" stretchy="false" xref="S2.SS2.2.p2.7.m7.1.2.cmml">(</mo><mi id="S2.SS2.2.p2.7.m7.1.1" xref="S2.SS2.2.p2.7.m7.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.7.m7.1.2.3.2.2" stretchy="false" xref="S2.SS2.2.p2.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.7.m7.1b"><apply id="S2.SS2.2.p2.7.m7.1.2.cmml" xref="S2.SS2.2.p2.7.m7.1.2"><times id="S2.SS2.2.p2.7.m7.1.2.1.cmml" xref="S2.SS2.2.p2.7.m7.1.2.1"></times><apply id="S2.SS2.2.p2.7.m7.1.2.2.cmml" xref="S2.SS2.2.p2.7.m7.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.2.p2.7.m7.1.2.2.1.cmml" xref="S2.SS2.2.p2.7.m7.1.2.2">subscript</csymbol><ci id="S2.SS2.2.p2.7.m7.1.2.2.2.cmml" xref="S2.SS2.2.p2.7.m7.1.2.2.2">𝑥</ci><apply id="S2.SS2.2.p2.7.m7.1.2.2.3.cmml" xref="S2.SS2.2.p2.7.m7.1.2.2.3"><minus id="S2.SS2.2.p2.7.m7.1.2.2.3.1.cmml" xref="S2.SS2.2.p2.7.m7.1.2.2.3"></minus><cn id="S2.SS2.2.p2.7.m7.1.2.2.3.2.cmml" type="integer" xref="S2.SS2.2.p2.7.m7.1.2.2.3.2">1</cn></apply></apply><ci id="S2.SS2.2.p2.7.m7.1.1.cmml" xref="S2.SS2.2.p2.7.m7.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.7.m7.1c">x_{-1}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.7.m7.1d">italic_x start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT ( italic_n )</annotation></semantics></math> and <math alttext="x_{1}(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.8.m8.1"><semantics id="S2.SS2.2.p2.8.m8.1a"><mrow id="S2.SS2.2.p2.8.m8.1.2" xref="S2.SS2.2.p2.8.m8.1.2.cmml"><msub id="S2.SS2.2.p2.8.m8.1.2.2" xref="S2.SS2.2.p2.8.m8.1.2.2.cmml"><mi id="S2.SS2.2.p2.8.m8.1.2.2.2" xref="S2.SS2.2.p2.8.m8.1.2.2.2.cmml">x</mi><mn id="S2.SS2.2.p2.8.m8.1.2.2.3" xref="S2.SS2.2.p2.8.m8.1.2.2.3.cmml">1</mn></msub><mo id="S2.SS2.2.p2.8.m8.1.2.1" xref="S2.SS2.2.p2.8.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.8.m8.1.2.3.2" xref="S2.SS2.2.p2.8.m8.1.2.cmml"><mo id="S2.SS2.2.p2.8.m8.1.2.3.2.1" stretchy="false" xref="S2.SS2.2.p2.8.m8.1.2.cmml">(</mo><mi id="S2.SS2.2.p2.8.m8.1.1" xref="S2.SS2.2.p2.8.m8.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.8.m8.1.2.3.2.2" stretchy="false" xref="S2.SS2.2.p2.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.8.m8.1b"><apply id="S2.SS2.2.p2.8.m8.1.2.cmml" xref="S2.SS2.2.p2.8.m8.1.2"><times id="S2.SS2.2.p2.8.m8.1.2.1.cmml" xref="S2.SS2.2.p2.8.m8.1.2.1"></times><apply id="S2.SS2.2.p2.8.m8.1.2.2.cmml" xref="S2.SS2.2.p2.8.m8.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.2.p2.8.m8.1.2.2.1.cmml" xref="S2.SS2.2.p2.8.m8.1.2.2">subscript</csymbol><ci id="S2.SS2.2.p2.8.m8.1.2.2.2.cmml" xref="S2.SS2.2.p2.8.m8.1.2.2.2">𝑥</ci><cn id="S2.SS2.2.p2.8.m8.1.2.2.3.cmml" type="integer" xref="S2.SS2.2.p2.8.m8.1.2.2.3">1</cn></apply><ci id="S2.SS2.2.p2.8.m8.1.1.cmml" xref="S2.SS2.2.p2.8.m8.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.8.m8.1c">x_{1}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.8.m8.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_n )</annotation></semantics></math> adjacent to <math alttext="x_{0}(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.9.m9.1"><semantics id="S2.SS2.2.p2.9.m9.1a"><mrow id="S2.SS2.2.p2.9.m9.1.2" xref="S2.SS2.2.p2.9.m9.1.2.cmml"><msub id="S2.SS2.2.p2.9.m9.1.2.2" xref="S2.SS2.2.p2.9.m9.1.2.2.cmml"><mi id="S2.SS2.2.p2.9.m9.1.2.2.2" xref="S2.SS2.2.p2.9.m9.1.2.2.2.cmml">x</mi><mn id="S2.SS2.2.p2.9.m9.1.2.2.3" xref="S2.SS2.2.p2.9.m9.1.2.2.3.cmml">0</mn></msub><mo id="S2.SS2.2.p2.9.m9.1.2.1" xref="S2.SS2.2.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.9.m9.1.2.3.2" xref="S2.SS2.2.p2.9.m9.1.2.cmml"><mo id="S2.SS2.2.p2.9.m9.1.2.3.2.1" stretchy="false" xref="S2.SS2.2.p2.9.m9.1.2.cmml">(</mo><mi id="S2.SS2.2.p2.9.m9.1.1" xref="S2.SS2.2.p2.9.m9.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.9.m9.1.2.3.2.2" stretchy="false" xref="S2.SS2.2.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.9.m9.1b"><apply id="S2.SS2.2.p2.9.m9.1.2.cmml" xref="S2.SS2.2.p2.9.m9.1.2"><times id="S2.SS2.2.p2.9.m9.1.2.1.cmml" xref="S2.SS2.2.p2.9.m9.1.2.1"></times><apply id="S2.SS2.2.p2.9.m9.1.2.2.cmml" xref="S2.SS2.2.p2.9.m9.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.2.p2.9.m9.1.2.2.1.cmml" xref="S2.SS2.2.p2.9.m9.1.2.2">subscript</csymbol><ci id="S2.SS2.2.p2.9.m9.1.2.2.2.cmml" xref="S2.SS2.2.p2.9.m9.1.2.2.2">𝑥</ci><cn id="S2.SS2.2.p2.9.m9.1.2.2.3.cmml" type="integer" xref="S2.SS2.2.p2.9.m9.1.2.2.3">0</cn></apply><ci id="S2.SS2.2.p2.9.m9.1.1.cmml" xref="S2.SS2.2.p2.9.m9.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.9.m9.1c">x_{0}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.9.m9.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_n )</annotation></semantics></math> on <math alttext="{\bf x}(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.10.m10.1"><semantics id="S2.SS2.2.p2.10.m10.1a"><mrow id="S2.SS2.2.p2.10.m10.1.2" xref="S2.SS2.2.p2.10.m10.1.2.cmml"><mi id="S2.SS2.2.p2.10.m10.1.2.2" xref="S2.SS2.2.p2.10.m10.1.2.2.cmml">𝐱</mi><mo id="S2.SS2.2.p2.10.m10.1.2.1" xref="S2.SS2.2.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.10.m10.1.2.3.2" xref="S2.SS2.2.p2.10.m10.1.2.cmml"><mo id="S2.SS2.2.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S2.SS2.2.p2.10.m10.1.2.cmml">(</mo><mi id="S2.SS2.2.p2.10.m10.1.1" xref="S2.SS2.2.p2.10.m10.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S2.SS2.2.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.10.m10.1b"><apply id="S2.SS2.2.p2.10.m10.1.2.cmml" xref="S2.SS2.2.p2.10.m10.1.2"><times id="S2.SS2.2.p2.10.m10.1.2.1.cmml" xref="S2.SS2.2.p2.10.m10.1.2.1"></times><ci id="S2.SS2.2.p2.10.m10.1.2.2.cmml" xref="S2.SS2.2.p2.10.m10.1.2.2">𝐱</ci><ci id="S2.SS2.2.p2.10.m10.1.1.cmml" xref="S2.SS2.2.p2.10.m10.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.10.m10.1c">{\bf x}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.10.m10.1d">bold_x ( italic_n )</annotation></semantics></math> are also independent of <math alttext="n" class="ltx_Math" display="inline" id="S2.SS2.2.p2.11.m11.1"><semantics id="S2.SS2.2.p2.11.m11.1a"><mi id="S2.SS2.2.p2.11.m11.1.1" xref="S2.SS2.2.p2.11.m11.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.11.m11.1b"><ci id="S2.SS2.2.p2.11.m11.1.1.cmml" xref="S2.SS2.2.p2.11.m11.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.11.m11.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.11.m11.1d">italic_n</annotation></semantics></math>. We iteratively proceed in this manner and extract finally a diagonal subsequence that defines a biinfinite word <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S2.SS2.2.p2.12.m12.1"><semantics id="S2.SS2.2.p2.12.m12.1a"><mi id="S2.SS2.2.p2.12.m12.1.1" xref="S2.SS2.2.p2.12.m12.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.12.m12.1b"><ci id="S2.SS2.2.p2.12.m12.1.1.cmml" xref="S2.SS2.2.p2.12.m12.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.12.m12.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.12.m12.1d">bold_x</annotation></semantics></math> which is the limit of some subsequence <math alttext="({\bf x}(n_{m}))_{m\in\mathbb{N}}" class="ltx_Math" display="inline" id="S2.SS2.2.p2.13.m13.1"><semantics id="S2.SS2.2.p2.13.m13.1a"><msub id="S2.SS2.2.p2.13.m13.1.1" xref="S2.SS2.2.p2.13.m13.1.1.cmml"><mrow id="S2.SS2.2.p2.13.m13.1.1.1.1" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.cmml"><mo id="S2.SS2.2.p2.13.m13.1.1.1.1.2" stretchy="false" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.2.p2.13.m13.1.1.1.1.1" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.cmml"><mi id="S2.SS2.2.p2.13.m13.1.1.1.1.1.3" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.3.cmml">𝐱</mi><mo id="S2.SS2.2.p2.13.m13.1.1.1.1.1.2" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.cmml"><mo id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.2" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.2.cmml">n</mi><mi id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.3" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.2.p2.13.m13.1.1.1.1.3" stretchy="false" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS2.2.p2.13.m13.1.1.3" xref="S2.SS2.2.p2.13.m13.1.1.3.cmml"><mi id="S2.SS2.2.p2.13.m13.1.1.3.2" xref="S2.SS2.2.p2.13.m13.1.1.3.2.cmml">m</mi><mo id="S2.SS2.2.p2.13.m13.1.1.3.1" xref="S2.SS2.2.p2.13.m13.1.1.3.1.cmml">∈</mo><mi id="S2.SS2.2.p2.13.m13.1.1.3.3" xref="S2.SS2.2.p2.13.m13.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.13.m13.1b"><apply id="S2.SS2.2.p2.13.m13.1.1.cmml" xref="S2.SS2.2.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S2.SS2.2.p2.13.m13.1.1.2.cmml" xref="S2.SS2.2.p2.13.m13.1.1">subscript</csymbol><apply id="S2.SS2.2.p2.13.m13.1.1.1.1.1.cmml" xref="S2.SS2.2.p2.13.m13.1.1.1.1"><times id="S2.SS2.2.p2.13.m13.1.1.1.1.1.2.cmml" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.2"></times><ci id="S2.SS2.2.p2.13.m13.1.1.1.1.1.3.cmml" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.3">𝐱</ci><apply id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.cmml" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.2">𝑛</ci><ci id="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS2.2.p2.13.m13.1.1.1.1.1.1.1.1.3">𝑚</ci></apply></apply><apply id="S2.SS2.2.p2.13.m13.1.1.3.cmml" xref="S2.SS2.2.p2.13.m13.1.1.3"><in id="S2.SS2.2.p2.13.m13.1.1.3.1.cmml" xref="S2.SS2.2.p2.13.m13.1.1.3.1"></in><ci id="S2.SS2.2.p2.13.m13.1.1.3.2.cmml" xref="S2.SS2.2.p2.13.m13.1.1.3.2">𝑚</ci><ci id="S2.SS2.2.p2.13.m13.1.1.3.3.cmml" xref="S2.SS2.2.p2.13.m13.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.13.m13.1c">({\bf x}(n_{m}))_{m\in\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.13.m13.1d">( bold_x ( italic_n start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_m ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> of the <math alttext="{\bf x}(n)" class="ltx_Math" display="inline" id="S2.SS2.2.p2.14.m14.1"><semantics id="S2.SS2.2.p2.14.m14.1a"><mrow id="S2.SS2.2.p2.14.m14.1.2" xref="S2.SS2.2.p2.14.m14.1.2.cmml"><mi id="S2.SS2.2.p2.14.m14.1.2.2" xref="S2.SS2.2.p2.14.m14.1.2.2.cmml">𝐱</mi><mo id="S2.SS2.2.p2.14.m14.1.2.1" xref="S2.SS2.2.p2.14.m14.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.14.m14.1.2.3.2" xref="S2.SS2.2.p2.14.m14.1.2.cmml"><mo id="S2.SS2.2.p2.14.m14.1.2.3.2.1" stretchy="false" xref="S2.SS2.2.p2.14.m14.1.2.cmml">(</mo><mi id="S2.SS2.2.p2.14.m14.1.1" xref="S2.SS2.2.p2.14.m14.1.1.cmml">n</mi><mo id="S2.SS2.2.p2.14.m14.1.2.3.2.2" stretchy="false" xref="S2.SS2.2.p2.14.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.14.m14.1b"><apply id="S2.SS2.2.p2.14.m14.1.2.cmml" xref="S2.SS2.2.p2.14.m14.1.2"><times id="S2.SS2.2.p2.14.m14.1.2.1.cmml" xref="S2.SS2.2.p2.14.m14.1.2.1"></times><ci id="S2.SS2.2.p2.14.m14.1.2.2.cmml" xref="S2.SS2.2.p2.14.m14.1.2.2">𝐱</ci><ci id="S2.SS2.2.p2.14.m14.1.1.cmml" xref="S2.SS2.2.p2.14.m14.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.14.m14.1c">{\bf x}(n)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.14.m14.1d">bold_x ( italic_n )</annotation></semantics></math>. From our construction and the definition of the map <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS2.2.p2.15.m15.1"><semantics id="S2.SS2.2.p2.15.m15.1a"><msup id="S2.SS2.2.p2.15.m15.1.1" xref="S2.SS2.2.p2.15.m15.1.1.cmml"><mi id="S2.SS2.2.p2.15.m15.1.1.2" xref="S2.SS2.2.p2.15.m15.1.1.2.cmml">σ</mi><mi id="S2.SS2.2.p2.15.m15.1.1.3" xref="S2.SS2.2.p2.15.m15.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.15.m15.1b"><apply id="S2.SS2.2.p2.15.m15.1.1.cmml" xref="S2.SS2.2.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS2.2.p2.15.m15.1.1.1.cmml" xref="S2.SS2.2.p2.15.m15.1.1">superscript</csymbol><ci id="S2.SS2.2.p2.15.m15.1.1.2.cmml" xref="S2.SS2.2.p2.15.m15.1.1.2">𝜎</ci><ci id="S2.SS2.2.p2.15.m15.1.1.3.cmml" xref="S2.SS2.2.p2.15.m15.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.15.m15.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.15.m15.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> we see directly that <math alttext="\sigma^{\mathbb{Z}}({\bf x})=T^{-k}({\bf y})" class="ltx_Math" display="inline" id="S2.SS2.2.p2.16.m16.2"><semantics id="S2.SS2.2.p2.16.m16.2a"><mrow id="S2.SS2.2.p2.16.m16.2.3" xref="S2.SS2.2.p2.16.m16.2.3.cmml"><mrow id="S2.SS2.2.p2.16.m16.2.3.2" xref="S2.SS2.2.p2.16.m16.2.3.2.cmml"><msup id="S2.SS2.2.p2.16.m16.2.3.2.2" xref="S2.SS2.2.p2.16.m16.2.3.2.2.cmml"><mi id="S2.SS2.2.p2.16.m16.2.3.2.2.2" xref="S2.SS2.2.p2.16.m16.2.3.2.2.2.cmml">σ</mi><mi id="S2.SS2.2.p2.16.m16.2.3.2.2.3" xref="S2.SS2.2.p2.16.m16.2.3.2.2.3.cmml">ℤ</mi></msup><mo id="S2.SS2.2.p2.16.m16.2.3.2.1" xref="S2.SS2.2.p2.16.m16.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.16.m16.2.3.2.3.2" xref="S2.SS2.2.p2.16.m16.2.3.2.cmml"><mo id="S2.SS2.2.p2.16.m16.2.3.2.3.2.1" stretchy="false" xref="S2.SS2.2.p2.16.m16.2.3.2.cmml">(</mo><mi id="S2.SS2.2.p2.16.m16.1.1" xref="S2.SS2.2.p2.16.m16.1.1.cmml">𝐱</mi><mo id="S2.SS2.2.p2.16.m16.2.3.2.3.2.2" stretchy="false" xref="S2.SS2.2.p2.16.m16.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.2.p2.16.m16.2.3.1" xref="S2.SS2.2.p2.16.m16.2.3.1.cmml">=</mo><mrow id="S2.SS2.2.p2.16.m16.2.3.3" xref="S2.SS2.2.p2.16.m16.2.3.3.cmml"><msup id="S2.SS2.2.p2.16.m16.2.3.3.2" xref="S2.SS2.2.p2.16.m16.2.3.3.2.cmml"><mi id="S2.SS2.2.p2.16.m16.2.3.3.2.2" xref="S2.SS2.2.p2.16.m16.2.3.3.2.2.cmml">T</mi><mrow id="S2.SS2.2.p2.16.m16.2.3.3.2.3" xref="S2.SS2.2.p2.16.m16.2.3.3.2.3.cmml"><mo id="S2.SS2.2.p2.16.m16.2.3.3.2.3a" xref="S2.SS2.2.p2.16.m16.2.3.3.2.3.cmml">−</mo><mi id="S2.SS2.2.p2.16.m16.2.3.3.2.3.2" xref="S2.SS2.2.p2.16.m16.2.3.3.2.3.2.cmml">k</mi></mrow></msup><mo id="S2.SS2.2.p2.16.m16.2.3.3.1" xref="S2.SS2.2.p2.16.m16.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS2.2.p2.16.m16.2.3.3.3.2" xref="S2.SS2.2.p2.16.m16.2.3.3.cmml"><mo id="S2.SS2.2.p2.16.m16.2.3.3.3.2.1" stretchy="false" xref="S2.SS2.2.p2.16.m16.2.3.3.cmml">(</mo><mi id="S2.SS2.2.p2.16.m16.2.2" xref="S2.SS2.2.p2.16.m16.2.2.cmml">𝐲</mi><mo id="S2.SS2.2.p2.16.m16.2.3.3.3.2.2" stretchy="false" xref="S2.SS2.2.p2.16.m16.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.16.m16.2b"><apply id="S2.SS2.2.p2.16.m16.2.3.cmml" xref="S2.SS2.2.p2.16.m16.2.3"><eq id="S2.SS2.2.p2.16.m16.2.3.1.cmml" xref="S2.SS2.2.p2.16.m16.2.3.1"></eq><apply id="S2.SS2.2.p2.16.m16.2.3.2.cmml" xref="S2.SS2.2.p2.16.m16.2.3.2"><times id="S2.SS2.2.p2.16.m16.2.3.2.1.cmml" xref="S2.SS2.2.p2.16.m16.2.3.2.1"></times><apply id="S2.SS2.2.p2.16.m16.2.3.2.2.cmml" xref="S2.SS2.2.p2.16.m16.2.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.2.p2.16.m16.2.3.2.2.1.cmml" xref="S2.SS2.2.p2.16.m16.2.3.2.2">superscript</csymbol><ci id="S2.SS2.2.p2.16.m16.2.3.2.2.2.cmml" xref="S2.SS2.2.p2.16.m16.2.3.2.2.2">𝜎</ci><ci id="S2.SS2.2.p2.16.m16.2.3.2.2.3.cmml" xref="S2.SS2.2.p2.16.m16.2.3.2.2.3">ℤ</ci></apply><ci id="S2.SS2.2.p2.16.m16.1.1.cmml" xref="S2.SS2.2.p2.16.m16.1.1">𝐱</ci></apply><apply id="S2.SS2.2.p2.16.m16.2.3.3.cmml" xref="S2.SS2.2.p2.16.m16.2.3.3"><times id="S2.SS2.2.p2.16.m16.2.3.3.1.cmml" xref="S2.SS2.2.p2.16.m16.2.3.3.1"></times><apply id="S2.SS2.2.p2.16.m16.2.3.3.2.cmml" xref="S2.SS2.2.p2.16.m16.2.3.3.2"><csymbol cd="ambiguous" id="S2.SS2.2.p2.16.m16.2.3.3.2.1.cmml" xref="S2.SS2.2.p2.16.m16.2.3.3.2">superscript</csymbol><ci id="S2.SS2.2.p2.16.m16.2.3.3.2.2.cmml" xref="S2.SS2.2.p2.16.m16.2.3.3.2.2">𝑇</ci><apply id="S2.SS2.2.p2.16.m16.2.3.3.2.3.cmml" xref="S2.SS2.2.p2.16.m16.2.3.3.2.3"><minus id="S2.SS2.2.p2.16.m16.2.3.3.2.3.1.cmml" xref="S2.SS2.2.p2.16.m16.2.3.3.2.3"></minus><ci id="S2.SS2.2.p2.16.m16.2.3.3.2.3.2.cmml" xref="S2.SS2.2.p2.16.m16.2.3.3.2.3.2">𝑘</ci></apply></apply><ci id="S2.SS2.2.p2.16.m16.2.2.cmml" xref="S2.SS2.2.p2.16.m16.2.2">𝐲</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.16.m16.2c">\sigma^{\mathbb{Z}}({\bf x})=T^{-k}({\bf y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.16.m16.2d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x ) = italic_T start_POSTSUPERSCRIPT - italic_k end_POSTSUPERSCRIPT ( bold_y )</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S2.SS2.2.p2.17.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S2.SS2.2.p2.17.1.m1.1"><semantics id="S2.SS2.2.p2.17.1.m1.1a"><mo id="S2.SS2.2.p2.17.1.m1.1.1" xref="S2.SS2.2.p2.17.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.17.1.m1.1b"><csymbol cd="latexml" id="S2.SS2.2.p2.17.1.m1.1.1.cmml" xref="S2.SS2.2.p2.17.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.17.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.17.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S2.SS2.2.p2.18.m17.1"><semantics id="S2.SS2.2.p2.18.m17.1a"><mo id="S2.SS2.2.p2.18.m17.1.1" xref="S2.SS2.2.p2.18.m17.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.2.p2.18.m17.1b"><csymbol cd="latexml" id="S2.SS2.2.p2.18.m17.1.1.cmml" xref="S2.SS2.2.p2.18.m17.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.2.p2.18.m17.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.2.p2.18.m17.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S2.Thmthm5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm5.1.1.1">Remark 2.5</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm5.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm5.p1"> <p class="ltx_p" id="S2.Thmthm5.p1.1">The concept of an image subshift given in Definition-Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm2" title="Definition-Remark 2.2. ‣ 2.2. “Not so standard” basic facts and terminology ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.2</span></a> is a very natural one; it is used frequently, in particular in the <math alttext="S" class="ltx_Math" display="inline" id="S2.Thmthm5.p1.1.m1.1"><semantics id="S2.Thmthm5.p1.1.m1.1a"><mi id="S2.Thmthm5.p1.1.m1.1.1" xref="S2.Thmthm5.p1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm5.p1.1.m1.1b"><ci id="S2.Thmthm5.p1.1.m1.1.1.cmml" xref="S2.Thmthm5.p1.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm5.p1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm5.p1.1.m1.1d">italic_S</annotation></semantics></math>-adic context. A systematic treatment, however, doesn’t seem to be available anywhere. One derives easily from the above definitions the following properties of the map (see (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E9" title="In 2.2. “Not so standard” basic facts and terminology ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.9</span></a>))</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{\Sigma}:\Sigma(\cal A)\to\Sigma(\cal B)" class="ltx_Math" display="block" id="S2.Ex10.m1.2"><semantics id="S2.Ex10.m1.2a"><mrow id="S2.Ex10.m1.2.3" xref="S2.Ex10.m1.2.3.cmml"><msup id="S2.Ex10.m1.2.3.2" xref="S2.Ex10.m1.2.3.2.cmml"><mi id="S2.Ex10.m1.2.3.2.2" xref="S2.Ex10.m1.2.3.2.2.cmml">σ</mi><mi id="S2.Ex10.m1.2.3.2.3" mathvariant="normal" xref="S2.Ex10.m1.2.3.2.3.cmml">Σ</mi></msup><mo id="S2.Ex10.m1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.Ex10.m1.2.3.1.cmml">:</mo><mrow id="S2.Ex10.m1.2.3.3" xref="S2.Ex10.m1.2.3.3.cmml"><mrow id="S2.Ex10.m1.2.3.3.2" xref="S2.Ex10.m1.2.3.3.2.cmml"><mi id="S2.Ex10.m1.2.3.3.2.2" mathvariant="normal" xref="S2.Ex10.m1.2.3.3.2.2.cmml">Σ</mi><mo id="S2.Ex10.m1.2.3.3.2.1" xref="S2.Ex10.m1.2.3.3.2.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.2.3.3.2.3.2" xref="S2.Ex10.m1.2.3.3.2.cmml"><mo id="S2.Ex10.m1.2.3.3.2.3.2.1" stretchy="false" xref="S2.Ex10.m1.2.3.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex10.m1.1.1" xref="S2.Ex10.m1.1.1.cmml">𝒜</mi><mo id="S2.Ex10.m1.2.3.3.2.3.2.2" stretchy="false" xref="S2.Ex10.m1.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex10.m1.2.3.3.1" stretchy="false" xref="S2.Ex10.m1.2.3.3.1.cmml">→</mo><mrow id="S2.Ex10.m1.2.3.3.3" xref="S2.Ex10.m1.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex10.m1.2.3.3.3.2" mathvariant="script" xref="S2.Ex10.m1.2.3.3.3.2.cmml">Σ</mi><mo id="S2.Ex10.m1.2.3.3.3.1" xref="S2.Ex10.m1.2.3.3.3.1.cmml">⁢</mo><mrow id="S2.Ex10.m1.2.3.3.3.3.2" xref="S2.Ex10.m1.2.3.3.3.cmml"><mo id="S2.Ex10.m1.2.3.3.3.3.2.1" stretchy="false" xref="S2.Ex10.m1.2.3.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex10.m1.2.2" xref="S2.Ex10.m1.2.2.cmml">ℬ</mi><mo id="S2.Ex10.m1.2.3.3.3.3.2.2" stretchy="false" xref="S2.Ex10.m1.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex10.m1.2b"><apply id="S2.Ex10.m1.2.3.cmml" xref="S2.Ex10.m1.2.3"><ci id="S2.Ex10.m1.2.3.1.cmml" xref="S2.Ex10.m1.2.3.1">:</ci><apply id="S2.Ex10.m1.2.3.2.cmml" xref="S2.Ex10.m1.2.3.2"><csymbol cd="ambiguous" id="S2.Ex10.m1.2.3.2.1.cmml" xref="S2.Ex10.m1.2.3.2">superscript</csymbol><ci id="S2.Ex10.m1.2.3.2.2.cmml" xref="S2.Ex10.m1.2.3.2.2">𝜎</ci><ci id="S2.Ex10.m1.2.3.2.3.cmml" xref="S2.Ex10.m1.2.3.2.3">Σ</ci></apply><apply id="S2.Ex10.m1.2.3.3.cmml" xref="S2.Ex10.m1.2.3.3"><ci id="S2.Ex10.m1.2.3.3.1.cmml" xref="S2.Ex10.m1.2.3.3.1">→</ci><apply id="S2.Ex10.m1.2.3.3.2.cmml" xref="S2.Ex10.m1.2.3.3.2"><times id="S2.Ex10.m1.2.3.3.2.1.cmml" xref="S2.Ex10.m1.2.3.3.2.1"></times><ci id="S2.Ex10.m1.2.3.3.2.2.cmml" xref="S2.Ex10.m1.2.3.3.2.2">Σ</ci><ci id="S2.Ex10.m1.1.1.cmml" xref="S2.Ex10.m1.1.1">𝒜</ci></apply><apply id="S2.Ex10.m1.2.3.3.3.cmml" xref="S2.Ex10.m1.2.3.3.3"><times id="S2.Ex10.m1.2.3.3.3.1.cmml" xref="S2.Ex10.m1.2.3.3.3.1"></times><ci id="S2.Ex10.m1.2.3.3.3.2.cmml" xref="S2.Ex10.m1.2.3.3.3.2">script-Σ</ci><ci id="S2.Ex10.m1.2.2.cmml" xref="S2.Ex10.m1.2.2">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex10.m1.2c">\sigma^{\Sigma}:\Sigma(\cal A)\to\Sigma(\cal B)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex10.m1.2d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT : roman_Σ ( caligraphic_A ) → caligraphic_Σ ( caligraphic_B )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmthm5.p1.6">on the subshift spaces over <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.Thmthm5.p1.2.m1.1"><semantics id="S2.Thmthm5.p1.2.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm5.p1.2.m1.1.1" xref="S2.Thmthm5.p1.2.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm5.p1.2.m1.1b"><ci id="S2.Thmthm5.p1.2.m1.1.1.cmml" xref="S2.Thmthm5.p1.2.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm5.p1.2.m1.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm5.p1.2.m1.1d">caligraphic_A</annotation></semantics></math> and <math alttext="\cal B" class="ltx_Math" display="inline" id="S2.Thmthm5.p1.3.m2.1"><semantics id="S2.Thmthm5.p1.3.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm5.p1.3.m2.1.1" xref="S2.Thmthm5.p1.3.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm5.p1.3.m2.1b"><ci id="S2.Thmthm5.p1.3.m2.1.1.cmml" xref="S2.Thmthm5.p1.3.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm5.p1.3.m2.1c">\cal B</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm5.p1.3.m2.1d">caligraphic_B</annotation></semantics></math> respectively. We recall that for simplicity we allow ourselves to denote the image of a subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm5.p1.4.m3.1"><semantics id="S2.Thmthm5.p1.4.m3.1a"><mrow id="S2.Thmthm5.p1.4.m3.1.1" xref="S2.Thmthm5.p1.4.m3.1.1.cmml"><mi id="S2.Thmthm5.p1.4.m3.1.1.2" xref="S2.Thmthm5.p1.4.m3.1.1.2.cmml">X</mi><mo id="S2.Thmthm5.p1.4.m3.1.1.1" xref="S2.Thmthm5.p1.4.m3.1.1.1.cmml">⊆</mo><msup id="S2.Thmthm5.p1.4.m3.1.1.3" xref="S2.Thmthm5.p1.4.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm5.p1.4.m3.1.1.3.2" xref="S2.Thmthm5.p1.4.m3.1.1.3.2.cmml">𝒜</mi><mi id="S2.Thmthm5.p1.4.m3.1.1.3.3" xref="S2.Thmthm5.p1.4.m3.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm5.p1.4.m3.1b"><apply id="S2.Thmthm5.p1.4.m3.1.1.cmml" xref="S2.Thmthm5.p1.4.m3.1.1"><subset id="S2.Thmthm5.p1.4.m3.1.1.1.cmml" xref="S2.Thmthm5.p1.4.m3.1.1.1"></subset><ci id="S2.Thmthm5.p1.4.m3.1.1.2.cmml" xref="S2.Thmthm5.p1.4.m3.1.1.2">𝑋</ci><apply id="S2.Thmthm5.p1.4.m3.1.1.3.cmml" xref="S2.Thmthm5.p1.4.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm5.p1.4.m3.1.1.3.1.cmml" xref="S2.Thmthm5.p1.4.m3.1.1.3">superscript</csymbol><ci id="S2.Thmthm5.p1.4.m3.1.1.3.2.cmml" xref="S2.Thmthm5.p1.4.m3.1.1.3.2">𝒜</ci><ci id="S2.Thmthm5.p1.4.m3.1.1.3.3.cmml" xref="S2.Thmthm5.p1.4.m3.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm5.p1.4.m3.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm5.p1.4.m3.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> by <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S2.Thmthm5.p1.5.m4.1"><semantics id="S2.Thmthm5.p1.5.m4.1a"><mrow id="S2.Thmthm5.p1.5.m4.1.2" xref="S2.Thmthm5.p1.5.m4.1.2.cmml"><mi id="S2.Thmthm5.p1.5.m4.1.2.2" xref="S2.Thmthm5.p1.5.m4.1.2.2.cmml">σ</mi><mo id="S2.Thmthm5.p1.5.m4.1.2.1" xref="S2.Thmthm5.p1.5.m4.1.2.1.cmml">⁢</mo><mrow id="S2.Thmthm5.p1.5.m4.1.2.3.2" xref="S2.Thmthm5.p1.5.m4.1.2.cmml"><mo id="S2.Thmthm5.p1.5.m4.1.2.3.2.1" stretchy="false" xref="S2.Thmthm5.p1.5.m4.1.2.cmml">(</mo><mi id="S2.Thmthm5.p1.5.m4.1.1" xref="S2.Thmthm5.p1.5.m4.1.1.cmml">X</mi><mo id="S2.Thmthm5.p1.5.m4.1.2.3.2.2" stretchy="false" xref="S2.Thmthm5.p1.5.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm5.p1.5.m4.1b"><apply id="S2.Thmthm5.p1.5.m4.1.2.cmml" xref="S2.Thmthm5.p1.5.m4.1.2"><times id="S2.Thmthm5.p1.5.m4.1.2.1.cmml" xref="S2.Thmthm5.p1.5.m4.1.2.1"></times><ci id="S2.Thmthm5.p1.5.m4.1.2.2.cmml" xref="S2.Thmthm5.p1.5.m4.1.2.2">𝜎</ci><ci id="S2.Thmthm5.p1.5.m4.1.1.cmml" xref="S2.Thmthm5.p1.5.m4.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm5.p1.5.m4.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm5.p1.5.m4.1d">italic_σ ( italic_X )</annotation></semantics></math> rather than by <math alttext="\sigma^{\Sigma}(X)" class="ltx_Math" display="inline" id="S2.Thmthm5.p1.6.m5.1"><semantics id="S2.Thmthm5.p1.6.m5.1a"><mrow id="S2.Thmthm5.p1.6.m5.1.2" xref="S2.Thmthm5.p1.6.m5.1.2.cmml"><msup id="S2.Thmthm5.p1.6.m5.1.2.2" xref="S2.Thmthm5.p1.6.m5.1.2.2.cmml"><mi id="S2.Thmthm5.p1.6.m5.1.2.2.2" xref="S2.Thmthm5.p1.6.m5.1.2.2.2.cmml">σ</mi><mi id="S2.Thmthm5.p1.6.m5.1.2.2.3" mathvariant="normal" xref="S2.Thmthm5.p1.6.m5.1.2.2.3.cmml">Σ</mi></msup><mo id="S2.Thmthm5.p1.6.m5.1.2.1" xref="S2.Thmthm5.p1.6.m5.1.2.1.cmml">⁢</mo><mrow id="S2.Thmthm5.p1.6.m5.1.2.3.2" xref="S2.Thmthm5.p1.6.m5.1.2.cmml"><mo id="S2.Thmthm5.p1.6.m5.1.2.3.2.1" stretchy="false" xref="S2.Thmthm5.p1.6.m5.1.2.cmml">(</mo><mi id="S2.Thmthm5.p1.6.m5.1.1" xref="S2.Thmthm5.p1.6.m5.1.1.cmml">X</mi><mo id="S2.Thmthm5.p1.6.m5.1.2.3.2.2" stretchy="false" xref="S2.Thmthm5.p1.6.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm5.p1.6.m5.1b"><apply id="S2.Thmthm5.p1.6.m5.1.2.cmml" xref="S2.Thmthm5.p1.6.m5.1.2"><times id="S2.Thmthm5.p1.6.m5.1.2.1.cmml" xref="S2.Thmthm5.p1.6.m5.1.2.1"></times><apply id="S2.Thmthm5.p1.6.m5.1.2.2.cmml" xref="S2.Thmthm5.p1.6.m5.1.2.2"><csymbol cd="ambiguous" id="S2.Thmthm5.p1.6.m5.1.2.2.1.cmml" xref="S2.Thmthm5.p1.6.m5.1.2.2">superscript</csymbol><ci id="S2.Thmthm5.p1.6.m5.1.2.2.2.cmml" xref="S2.Thmthm5.p1.6.m5.1.2.2.2">𝜎</ci><ci id="S2.Thmthm5.p1.6.m5.1.2.2.3.cmml" xref="S2.Thmthm5.p1.6.m5.1.2.2.3">Σ</ci></apply><ci id="S2.Thmthm5.p1.6.m5.1.1.cmml" xref="S2.Thmthm5.p1.6.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm5.p1.6.m5.1c">\sigma^{\Sigma}(X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm5.p1.6.m5.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math>.</p> <ol class="ltx_enumerate" id="S2.I4"> <li class="ltx_item" id="S2.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S2.I4.i1.p1"> <p class="ltx_p" id="S2.I4.i1.p1.2">The map <math alttext="\sigma^{\Sigma}" class="ltx_Math" display="inline" id="S2.I4.i1.p1.1.m1.1"><semantics id="S2.I4.i1.p1.1.m1.1a"><msup id="S2.I4.i1.p1.1.m1.1.1" xref="S2.I4.i1.p1.1.m1.1.1.cmml"><mi id="S2.I4.i1.p1.1.m1.1.1.2" xref="S2.I4.i1.p1.1.m1.1.1.2.cmml">σ</mi><mi id="S2.I4.i1.p1.1.m1.1.1.3" mathvariant="normal" xref="S2.I4.i1.p1.1.m1.1.1.3.cmml">Σ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.I4.i1.p1.1.m1.1b"><apply id="S2.I4.i1.p1.1.m1.1.1.cmml" xref="S2.I4.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.I4.i1.p1.1.m1.1.1.1.cmml" xref="S2.I4.i1.p1.1.m1.1.1">superscript</csymbol><ci id="S2.I4.i1.p1.1.m1.1.1.2.cmml" xref="S2.I4.i1.p1.1.m1.1.1.2">𝜎</ci><ci id="S2.I4.i1.p1.1.m1.1.1.3.cmml" xref="S2.I4.i1.p1.1.m1.1.1.3">Σ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i1.p1.1.m1.1c">\sigma^{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i1.p1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT</annotation></semantics></math> is functorial. In particular, for any second morphism <math alttext="\sigma^{\prime}:\cal B^{*}\to\cal C^{*}" class="ltx_Math" display="inline" id="S2.I4.i1.p1.2.m2.1"><semantics id="S2.I4.i1.p1.2.m2.1a"><mrow id="S2.I4.i1.p1.2.m2.1.1" xref="S2.I4.i1.p1.2.m2.1.1.cmml"><msup id="S2.I4.i1.p1.2.m2.1.1.2" xref="S2.I4.i1.p1.2.m2.1.1.2.cmml"><mi id="S2.I4.i1.p1.2.m2.1.1.2.2" xref="S2.I4.i1.p1.2.m2.1.1.2.2.cmml">σ</mi><mo id="S2.I4.i1.p1.2.m2.1.1.2.3" xref="S2.I4.i1.p1.2.m2.1.1.2.3.cmml">′</mo></msup><mo id="S2.I4.i1.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.I4.i1.p1.2.m2.1.1.1.cmml">:</mo><mrow id="S2.I4.i1.p1.2.m2.1.1.3" xref="S2.I4.i1.p1.2.m2.1.1.3.cmml"><msup id="S2.I4.i1.p1.2.m2.1.1.3.2" xref="S2.I4.i1.p1.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i1.p1.2.m2.1.1.3.2.2" xref="S2.I4.i1.p1.2.m2.1.1.3.2.2.cmml">ℬ</mi><mo id="S2.I4.i1.p1.2.m2.1.1.3.2.3" xref="S2.I4.i1.p1.2.m2.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.I4.i1.p1.2.m2.1.1.3.1" stretchy="false" xref="S2.I4.i1.p1.2.m2.1.1.3.1.cmml">→</mo><msup id="S2.I4.i1.p1.2.m2.1.1.3.3" xref="S2.I4.i1.p1.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i1.p1.2.m2.1.1.3.3.2" xref="S2.I4.i1.p1.2.m2.1.1.3.3.2.cmml">𝒞</mi><mo id="S2.I4.i1.p1.2.m2.1.1.3.3.3" xref="S2.I4.i1.p1.2.m2.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i1.p1.2.m2.1b"><apply id="S2.I4.i1.p1.2.m2.1.1.cmml" xref="S2.I4.i1.p1.2.m2.1.1"><ci id="S2.I4.i1.p1.2.m2.1.1.1.cmml" xref="S2.I4.i1.p1.2.m2.1.1.1">:</ci><apply id="S2.I4.i1.p1.2.m2.1.1.2.cmml" xref="S2.I4.i1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I4.i1.p1.2.m2.1.1.2.1.cmml" xref="S2.I4.i1.p1.2.m2.1.1.2">superscript</csymbol><ci id="S2.I4.i1.p1.2.m2.1.1.2.2.cmml" xref="S2.I4.i1.p1.2.m2.1.1.2.2">𝜎</ci><ci id="S2.I4.i1.p1.2.m2.1.1.2.3.cmml" xref="S2.I4.i1.p1.2.m2.1.1.2.3">′</ci></apply><apply id="S2.I4.i1.p1.2.m2.1.1.3.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3"><ci id="S2.I4.i1.p1.2.m2.1.1.3.1.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.1">→</ci><apply id="S2.I4.i1.p1.2.m2.1.1.3.2.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.I4.i1.p1.2.m2.1.1.3.2.1.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S2.I4.i1.p1.2.m2.1.1.3.2.2.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.2.2">ℬ</ci><times id="S2.I4.i1.p1.2.m2.1.1.3.2.3.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.2.3"></times></apply><apply id="S2.I4.i1.p1.2.m2.1.1.3.3.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S2.I4.i1.p1.2.m2.1.1.3.3.1.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.3">superscript</csymbol><ci id="S2.I4.i1.p1.2.m2.1.1.3.3.2.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.3.2">𝒞</ci><times id="S2.I4.i1.p1.2.m2.1.1.3.3.3.cmml" xref="S2.I4.i1.p1.2.m2.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i1.p1.2.m2.1c">\sigma^{\prime}:\cal B^{*}\to\cal C^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i1.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> one has</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{\prime}(\sigma(X))=\sigma^{\prime}\circ\sigma(X)\,." class="ltx_Math" display="block" id="S2.Ex11.m1.3"><semantics id="S2.Ex11.m1.3a"><mrow id="S2.Ex11.m1.3.3.1" xref="S2.Ex11.m1.3.3.1.1.cmml"><mrow id="S2.Ex11.m1.3.3.1.1" xref="S2.Ex11.m1.3.3.1.1.cmml"><mrow id="S2.Ex11.m1.3.3.1.1.1" xref="S2.Ex11.m1.3.3.1.1.1.cmml"><msup id="S2.Ex11.m1.3.3.1.1.1.3" xref="S2.Ex11.m1.3.3.1.1.1.3.cmml"><mi id="S2.Ex11.m1.3.3.1.1.1.3.2" xref="S2.Ex11.m1.3.3.1.1.1.3.2.cmml">σ</mi><mo id="S2.Ex11.m1.3.3.1.1.1.3.3" xref="S2.Ex11.m1.3.3.1.1.1.3.3.cmml">′</mo></msup><mo id="S2.Ex11.m1.3.3.1.1.1.2" xref="S2.Ex11.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex11.m1.3.3.1.1.1.1.1" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S2.Ex11.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex11.m1.3.3.1.1.1.1.1.1" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S2.Ex11.m1.3.3.1.1.1.1.1.1.2" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S2.Ex11.m1.3.3.1.1.1.1.1.1.1" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex11.m1.3.3.1.1.1.1.1.1.3.2" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S2.Ex11.m1.3.3.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mi id="S2.Ex11.m1.1.1" xref="S2.Ex11.m1.1.1.cmml">X</mi><mo id="S2.Ex11.m1.3.3.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex11.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex11.m1.3.3.1.1.2" xref="S2.Ex11.m1.3.3.1.1.2.cmml">=</mo><mrow id="S2.Ex11.m1.3.3.1.1.3" xref="S2.Ex11.m1.3.3.1.1.3.cmml"><mrow id="S2.Ex11.m1.3.3.1.1.3.2" xref="S2.Ex11.m1.3.3.1.1.3.2.cmml"><msup id="S2.Ex11.m1.3.3.1.1.3.2.2" xref="S2.Ex11.m1.3.3.1.1.3.2.2.cmml"><mi id="S2.Ex11.m1.3.3.1.1.3.2.2.2" xref="S2.Ex11.m1.3.3.1.1.3.2.2.2.cmml">σ</mi><mo id="S2.Ex11.m1.3.3.1.1.3.2.2.3" xref="S2.Ex11.m1.3.3.1.1.3.2.2.3.cmml">′</mo></msup><mo id="S2.Ex11.m1.3.3.1.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.Ex11.m1.3.3.1.1.3.2.1.cmml">∘</mo><mi id="S2.Ex11.m1.3.3.1.1.3.2.3" xref="S2.Ex11.m1.3.3.1.1.3.2.3.cmml">σ</mi></mrow><mo id="S2.Ex11.m1.3.3.1.1.3.1" xref="S2.Ex11.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex11.m1.3.3.1.1.3.3.2" xref="S2.Ex11.m1.3.3.1.1.3.cmml"><mo id="S2.Ex11.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S2.Ex11.m1.3.3.1.1.3.cmml">(</mo><mi id="S2.Ex11.m1.2.2" xref="S2.Ex11.m1.2.2.cmml">X</mi><mo id="S2.Ex11.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="S2.Ex11.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex11.m1.3.3.1.2" lspace="0.170em" xref="S2.Ex11.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex11.m1.3b"><apply id="S2.Ex11.m1.3.3.1.1.cmml" xref="S2.Ex11.m1.3.3.1"><eq id="S2.Ex11.m1.3.3.1.1.2.cmml" xref="S2.Ex11.m1.3.3.1.1.2"></eq><apply id="S2.Ex11.m1.3.3.1.1.1.cmml" xref="S2.Ex11.m1.3.3.1.1.1"><times id="S2.Ex11.m1.3.3.1.1.1.2.cmml" xref="S2.Ex11.m1.3.3.1.1.1.2"></times><apply id="S2.Ex11.m1.3.3.1.1.1.3.cmml" xref="S2.Ex11.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex11.m1.3.3.1.1.1.3.1.cmml" xref="S2.Ex11.m1.3.3.1.1.1.3">superscript</csymbol><ci id="S2.Ex11.m1.3.3.1.1.1.3.2.cmml" xref="S2.Ex11.m1.3.3.1.1.1.3.2">𝜎</ci><ci id="S2.Ex11.m1.3.3.1.1.1.3.3.cmml" xref="S2.Ex11.m1.3.3.1.1.1.3.3">′</ci></apply><apply id="S2.Ex11.m1.3.3.1.1.1.1.1.1.cmml" xref="S2.Ex11.m1.3.3.1.1.1.1.1"><times id="S2.Ex11.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.1"></times><ci id="S2.Ex11.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S2.Ex11.m1.3.3.1.1.1.1.1.1.2">𝜎</ci><ci id="S2.Ex11.m1.1.1.cmml" xref="S2.Ex11.m1.1.1">𝑋</ci></apply></apply><apply id="S2.Ex11.m1.3.3.1.1.3.cmml" xref="S2.Ex11.m1.3.3.1.1.3"><times id="S2.Ex11.m1.3.3.1.1.3.1.cmml" xref="S2.Ex11.m1.3.3.1.1.3.1"></times><apply id="S2.Ex11.m1.3.3.1.1.3.2.cmml" xref="S2.Ex11.m1.3.3.1.1.3.2"><compose id="S2.Ex11.m1.3.3.1.1.3.2.1.cmml" xref="S2.Ex11.m1.3.3.1.1.3.2.1"></compose><apply id="S2.Ex11.m1.3.3.1.1.3.2.2.cmml" xref="S2.Ex11.m1.3.3.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.Ex11.m1.3.3.1.1.3.2.2.1.cmml" xref="S2.Ex11.m1.3.3.1.1.3.2.2">superscript</csymbol><ci id="S2.Ex11.m1.3.3.1.1.3.2.2.2.cmml" xref="S2.Ex11.m1.3.3.1.1.3.2.2.2">𝜎</ci><ci id="S2.Ex11.m1.3.3.1.1.3.2.2.3.cmml" xref="S2.Ex11.m1.3.3.1.1.3.2.2.3">′</ci></apply><ci id="S2.Ex11.m1.3.3.1.1.3.2.3.cmml" xref="S2.Ex11.m1.3.3.1.1.3.2.3">𝜎</ci></apply><ci id="S2.Ex11.m1.2.2.cmml" xref="S2.Ex11.m1.2.2">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex11.m1.3c">\sigma^{\prime}(\sigma(X))=\sigma^{\prime}\circ\sigma(X)\,.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex11.m1.3d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_σ ( italic_X ) ) = italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∘ italic_σ ( italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S2.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S2.I4.i2.p1"> <p class="ltx_p" id="S2.I4.i2.p1.8">The map <math alttext="\sigma^{\Sigma}" class="ltx_Math" display="inline" id="S2.I4.i2.p1.1.m1.1"><semantics id="S2.I4.i2.p1.1.m1.1a"><msup id="S2.I4.i2.p1.1.m1.1.1" xref="S2.I4.i2.p1.1.m1.1.1.cmml"><mi id="S2.I4.i2.p1.1.m1.1.1.2" xref="S2.I4.i2.p1.1.m1.1.1.2.cmml">σ</mi><mi id="S2.I4.i2.p1.1.m1.1.1.3" mathvariant="normal" xref="S2.I4.i2.p1.1.m1.1.1.3.cmml">Σ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.I4.i2.p1.1.m1.1b"><apply id="S2.I4.i2.p1.1.m1.1.1.cmml" xref="S2.I4.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.I4.i2.p1.1.m1.1.1.1.cmml" xref="S2.I4.i2.p1.1.m1.1.1">superscript</csymbol><ci id="S2.I4.i2.p1.1.m1.1.1.2.cmml" xref="S2.I4.i2.p1.1.m1.1.1.2">𝜎</ci><ci id="S2.I4.i2.p1.1.m1.1.1.3.cmml" xref="S2.I4.i2.p1.1.m1.1.1.3">Σ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i2.p1.1.m1.1c">\sigma^{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i2.p1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT</annotation></semantics></math> respects the inclusion: For any two subshifts <math alttext="X^{\prime}\subseteq X" class="ltx_Math" display="inline" id="S2.I4.i2.p1.2.m2.1"><semantics id="S2.I4.i2.p1.2.m2.1a"><mrow id="S2.I4.i2.p1.2.m2.1.1" xref="S2.I4.i2.p1.2.m2.1.1.cmml"><msup id="S2.I4.i2.p1.2.m2.1.1.2" xref="S2.I4.i2.p1.2.m2.1.1.2.cmml"><mi id="S2.I4.i2.p1.2.m2.1.1.2.2" xref="S2.I4.i2.p1.2.m2.1.1.2.2.cmml">X</mi><mo id="S2.I4.i2.p1.2.m2.1.1.2.3" xref="S2.I4.i2.p1.2.m2.1.1.2.3.cmml">′</mo></msup><mo id="S2.I4.i2.p1.2.m2.1.1.1" xref="S2.I4.i2.p1.2.m2.1.1.1.cmml">⊆</mo><mi id="S2.I4.i2.p1.2.m2.1.1.3" xref="S2.I4.i2.p1.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i2.p1.2.m2.1b"><apply id="S2.I4.i2.p1.2.m2.1.1.cmml" xref="S2.I4.i2.p1.2.m2.1.1"><subset id="S2.I4.i2.p1.2.m2.1.1.1.cmml" xref="S2.I4.i2.p1.2.m2.1.1.1"></subset><apply id="S2.I4.i2.p1.2.m2.1.1.2.cmml" xref="S2.I4.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I4.i2.p1.2.m2.1.1.2.1.cmml" xref="S2.I4.i2.p1.2.m2.1.1.2">superscript</csymbol><ci id="S2.I4.i2.p1.2.m2.1.1.2.2.cmml" xref="S2.I4.i2.p1.2.m2.1.1.2.2">𝑋</ci><ci id="S2.I4.i2.p1.2.m2.1.1.2.3.cmml" xref="S2.I4.i2.p1.2.m2.1.1.2.3">′</ci></apply><ci id="S2.I4.i2.p1.2.m2.1.1.3.cmml" xref="S2.I4.i2.p1.2.m2.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i2.p1.2.m2.1c">X^{\prime}\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i2.p1.2.m2.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_X</annotation></semantics></math> in <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.I4.i2.p1.3.m3.1"><semantics id="S2.I4.i2.p1.3.m3.1a"><mrow id="S2.I4.i2.p1.3.m3.1.2" xref="S2.I4.i2.p1.3.m3.1.2.cmml"><mi id="S2.I4.i2.p1.3.m3.1.2.2" mathvariant="normal" xref="S2.I4.i2.p1.3.m3.1.2.2.cmml">Σ</mi><mo id="S2.I4.i2.p1.3.m3.1.2.1" xref="S2.I4.i2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.I4.i2.p1.3.m3.1.2.3.2" xref="S2.I4.i2.p1.3.m3.1.2.cmml"><mo id="S2.I4.i2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S2.I4.i2.p1.3.m3.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I4.i2.p1.3.m3.1.1" xref="S2.I4.i2.p1.3.m3.1.1.cmml">𝒜</mi><mo id="S2.I4.i2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S2.I4.i2.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i2.p1.3.m3.1b"><apply id="S2.I4.i2.p1.3.m3.1.2.cmml" xref="S2.I4.i2.p1.3.m3.1.2"><times id="S2.I4.i2.p1.3.m3.1.2.1.cmml" xref="S2.I4.i2.p1.3.m3.1.2.1"></times><ci id="S2.I4.i2.p1.3.m3.1.2.2.cmml" xref="S2.I4.i2.p1.3.m3.1.2.2">Σ</ci><ci id="S2.I4.i2.p1.3.m3.1.1.cmml" xref="S2.I4.i2.p1.3.m3.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i2.p1.3.m3.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i2.p1.3.m3.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math> one has <math alttext="\sigma(X^{\prime})\subseteq\sigma(X)" class="ltx_Math" display="inline" id="S2.I4.i2.p1.4.m4.2"><semantics id="S2.I4.i2.p1.4.m4.2a"><mrow id="S2.I4.i2.p1.4.m4.2.2" xref="S2.I4.i2.p1.4.m4.2.2.cmml"><mrow id="S2.I4.i2.p1.4.m4.2.2.1" xref="S2.I4.i2.p1.4.m4.2.2.1.cmml"><mi id="S2.I4.i2.p1.4.m4.2.2.1.3" xref="S2.I4.i2.p1.4.m4.2.2.1.3.cmml">σ</mi><mo id="S2.I4.i2.p1.4.m4.2.2.1.2" xref="S2.I4.i2.p1.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S2.I4.i2.p1.4.m4.2.2.1.1.1" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.cmml"><mo id="S2.I4.i2.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.cmml">(</mo><msup id="S2.I4.i2.p1.4.m4.2.2.1.1.1.1" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.cmml"><mi id="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.2" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.2.cmml">X</mi><mo id="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.3" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.I4.i2.p1.4.m4.2.2.1.1.1.3" stretchy="false" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I4.i2.p1.4.m4.2.2.2" xref="S2.I4.i2.p1.4.m4.2.2.2.cmml">⊆</mo><mrow id="S2.I4.i2.p1.4.m4.2.2.3" xref="S2.I4.i2.p1.4.m4.2.2.3.cmml"><mi id="S2.I4.i2.p1.4.m4.2.2.3.2" xref="S2.I4.i2.p1.4.m4.2.2.3.2.cmml">σ</mi><mo id="S2.I4.i2.p1.4.m4.2.2.3.1" xref="S2.I4.i2.p1.4.m4.2.2.3.1.cmml">⁢</mo><mrow id="S2.I4.i2.p1.4.m4.2.2.3.3.2" xref="S2.I4.i2.p1.4.m4.2.2.3.cmml"><mo id="S2.I4.i2.p1.4.m4.2.2.3.3.2.1" stretchy="false" xref="S2.I4.i2.p1.4.m4.2.2.3.cmml">(</mo><mi id="S2.I4.i2.p1.4.m4.1.1" xref="S2.I4.i2.p1.4.m4.1.1.cmml">X</mi><mo id="S2.I4.i2.p1.4.m4.2.2.3.3.2.2" stretchy="false" xref="S2.I4.i2.p1.4.m4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i2.p1.4.m4.2b"><apply id="S2.I4.i2.p1.4.m4.2.2.cmml" xref="S2.I4.i2.p1.4.m4.2.2"><subset id="S2.I4.i2.p1.4.m4.2.2.2.cmml" xref="S2.I4.i2.p1.4.m4.2.2.2"></subset><apply id="S2.I4.i2.p1.4.m4.2.2.1.cmml" xref="S2.I4.i2.p1.4.m4.2.2.1"><times id="S2.I4.i2.p1.4.m4.2.2.1.2.cmml" xref="S2.I4.i2.p1.4.m4.2.2.1.2"></times><ci id="S2.I4.i2.p1.4.m4.2.2.1.3.cmml" xref="S2.I4.i2.p1.4.m4.2.2.1.3">𝜎</ci><apply id="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.cmml" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.2">𝑋</ci><ci id="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="S2.I4.i2.p1.4.m4.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S2.I4.i2.p1.4.m4.2.2.3.cmml" xref="S2.I4.i2.p1.4.m4.2.2.3"><times id="S2.I4.i2.p1.4.m4.2.2.3.1.cmml" xref="S2.I4.i2.p1.4.m4.2.2.3.1"></times><ci id="S2.I4.i2.p1.4.m4.2.2.3.2.cmml" xref="S2.I4.i2.p1.4.m4.2.2.3.2">𝜎</ci><ci id="S2.I4.i2.p1.4.m4.1.1.cmml" xref="S2.I4.i2.p1.4.m4.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i2.p1.4.m4.2c">\sigma(X^{\prime})\subseteq\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i2.p1.4.m4.2d">italic_σ ( italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⊆ italic_σ ( italic_X )</annotation></semantics></math>. For any third subshift <math alttext="X^{\prime\prime}" class="ltx_Math" display="inline" id="S2.I4.i2.p1.5.m5.1"><semantics id="S2.I4.i2.p1.5.m5.1a"><msup id="S2.I4.i2.p1.5.m5.1.1" xref="S2.I4.i2.p1.5.m5.1.1.cmml"><mi id="S2.I4.i2.p1.5.m5.1.1.2" xref="S2.I4.i2.p1.5.m5.1.1.2.cmml">X</mi><mo id="S2.I4.i2.p1.5.m5.1.1.3" xref="S2.I4.i2.p1.5.m5.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.I4.i2.p1.5.m5.1b"><apply id="S2.I4.i2.p1.5.m5.1.1.cmml" xref="S2.I4.i2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.I4.i2.p1.5.m5.1.1.1.cmml" xref="S2.I4.i2.p1.5.m5.1.1">superscript</csymbol><ci id="S2.I4.i2.p1.5.m5.1.1.2.cmml" xref="S2.I4.i2.p1.5.m5.1.1.2">𝑋</ci><ci id="S2.I4.i2.p1.5.m5.1.1.3.cmml" xref="S2.I4.i2.p1.5.m5.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i2.p1.5.m5.1c">X^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i2.p1.5.m5.1d">italic_X start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.I4.i2.p1.6.m6.1"><semantics id="S2.I4.i2.p1.6.m6.1a"><mrow id="S2.I4.i2.p1.6.m6.1.2" xref="S2.I4.i2.p1.6.m6.1.2.cmml"><mi id="S2.I4.i2.p1.6.m6.1.2.2" mathvariant="normal" xref="S2.I4.i2.p1.6.m6.1.2.2.cmml">Σ</mi><mo id="S2.I4.i2.p1.6.m6.1.2.1" xref="S2.I4.i2.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.I4.i2.p1.6.m6.1.2.3.2" xref="S2.I4.i2.p1.6.m6.1.2.cmml"><mo id="S2.I4.i2.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S2.I4.i2.p1.6.m6.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I4.i2.p1.6.m6.1.1" xref="S2.I4.i2.p1.6.m6.1.1.cmml">𝒜</mi><mo id="S2.I4.i2.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S2.I4.i2.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i2.p1.6.m6.1b"><apply id="S2.I4.i2.p1.6.m6.1.2.cmml" xref="S2.I4.i2.p1.6.m6.1.2"><times id="S2.I4.i2.p1.6.m6.1.2.1.cmml" xref="S2.I4.i2.p1.6.m6.1.2.1"></times><ci id="S2.I4.i2.p1.6.m6.1.2.2.cmml" xref="S2.I4.i2.p1.6.m6.1.2.2">Σ</ci><ci id="S2.I4.i2.p1.6.m6.1.1.cmml" xref="S2.I4.i2.p1.6.m6.1.1">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i2.p1.6.m6.1c">\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i2.p1.6.m6.1d">roman_Σ ( caligraphic_A )</annotation></semantics></math> one has <math alttext="\sigma(X\cap X^{\prime\prime})\subseteq\sigma(X)\cap\sigma(X^{\prime\prime})" class="ltx_Math" display="inline" id="S2.I4.i2.p1.7.m7.3"><semantics id="S2.I4.i2.p1.7.m7.3a"><mrow id="S2.I4.i2.p1.7.m7.3.3" xref="S2.I4.i2.p1.7.m7.3.3.cmml"><mrow id="S2.I4.i2.p1.7.m7.2.2.1" xref="S2.I4.i2.p1.7.m7.2.2.1.cmml"><mi id="S2.I4.i2.p1.7.m7.2.2.1.3" xref="S2.I4.i2.p1.7.m7.2.2.1.3.cmml">σ</mi><mo id="S2.I4.i2.p1.7.m7.2.2.1.2" xref="S2.I4.i2.p1.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S2.I4.i2.p1.7.m7.2.2.1.1.1" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.cmml"><mo id="S2.I4.i2.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.2" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.2.cmml">X</mi><mo id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.1" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.1.cmml">∩</mo><msup id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.cmml"><mi id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.2" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.2.cmml">X</mi><mo id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.3" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.3.cmml">′′</mo></msup></mrow><mo id="S2.I4.i2.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I4.i2.p1.7.m7.3.3.3" xref="S2.I4.i2.p1.7.m7.3.3.3.cmml">⊆</mo><mrow id="S2.I4.i2.p1.7.m7.3.3.2" xref="S2.I4.i2.p1.7.m7.3.3.2.cmml"><mrow id="S2.I4.i2.p1.7.m7.3.3.2.3" xref="S2.I4.i2.p1.7.m7.3.3.2.3.cmml"><mi id="S2.I4.i2.p1.7.m7.3.3.2.3.2" xref="S2.I4.i2.p1.7.m7.3.3.2.3.2.cmml">σ</mi><mo id="S2.I4.i2.p1.7.m7.3.3.2.3.1" xref="S2.I4.i2.p1.7.m7.3.3.2.3.1.cmml">⁢</mo><mrow id="S2.I4.i2.p1.7.m7.3.3.2.3.3.2" xref="S2.I4.i2.p1.7.m7.3.3.2.3.cmml"><mo id="S2.I4.i2.p1.7.m7.3.3.2.3.3.2.1" stretchy="false" xref="S2.I4.i2.p1.7.m7.3.3.2.3.cmml">(</mo><mi id="S2.I4.i2.p1.7.m7.1.1" xref="S2.I4.i2.p1.7.m7.1.1.cmml">X</mi><mo id="S2.I4.i2.p1.7.m7.3.3.2.3.3.2.2" stretchy="false" xref="S2.I4.i2.p1.7.m7.3.3.2.3.cmml">)</mo></mrow></mrow><mo id="S2.I4.i2.p1.7.m7.3.3.2.2" xref="S2.I4.i2.p1.7.m7.3.3.2.2.cmml">∩</mo><mrow id="S2.I4.i2.p1.7.m7.3.3.2.1" xref="S2.I4.i2.p1.7.m7.3.3.2.1.cmml"><mi id="S2.I4.i2.p1.7.m7.3.3.2.1.3" xref="S2.I4.i2.p1.7.m7.3.3.2.1.3.cmml">σ</mi><mo id="S2.I4.i2.p1.7.m7.3.3.2.1.2" xref="S2.I4.i2.p1.7.m7.3.3.2.1.2.cmml">⁢</mo><mrow id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.cmml"><mo id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.2" stretchy="false" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.cmml">(</mo><msup id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.cmml"><mi id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.2" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.2.cmml">X</mi><mo id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.3" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.3.cmml">′′</mo></msup><mo id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.3" stretchy="false" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i2.p1.7.m7.3b"><apply id="S2.I4.i2.p1.7.m7.3.3.cmml" xref="S2.I4.i2.p1.7.m7.3.3"><subset id="S2.I4.i2.p1.7.m7.3.3.3.cmml" xref="S2.I4.i2.p1.7.m7.3.3.3"></subset><apply id="S2.I4.i2.p1.7.m7.2.2.1.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1"><times id="S2.I4.i2.p1.7.m7.2.2.1.2.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.2"></times><ci id="S2.I4.i2.p1.7.m7.2.2.1.3.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.3">𝜎</ci><apply id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1"><intersect id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.1"></intersect><ci id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.2">𝑋</ci><apply id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.1.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3">superscript</csymbol><ci id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.2.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.2">𝑋</ci><ci id="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.3.cmml" xref="S2.I4.i2.p1.7.m7.2.2.1.1.1.1.3.3">′′</ci></apply></apply></apply><apply id="S2.I4.i2.p1.7.m7.3.3.2.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2"><intersect id="S2.I4.i2.p1.7.m7.3.3.2.2.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.2"></intersect><apply id="S2.I4.i2.p1.7.m7.3.3.2.3.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.3"><times id="S2.I4.i2.p1.7.m7.3.3.2.3.1.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.3.1"></times><ci id="S2.I4.i2.p1.7.m7.3.3.2.3.2.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.3.2">𝜎</ci><ci id="S2.I4.i2.p1.7.m7.1.1.cmml" xref="S2.I4.i2.p1.7.m7.1.1">𝑋</ci></apply><apply id="S2.I4.i2.p1.7.m7.3.3.2.1.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.1"><times id="S2.I4.i2.p1.7.m7.3.3.2.1.2.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.1.2"></times><ci id="S2.I4.i2.p1.7.m7.3.3.2.1.3.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.1.3">𝜎</ci><apply id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1"><csymbol cd="ambiguous" id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.1.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1">superscript</csymbol><ci id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.2.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.2">𝑋</ci><ci id="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.3.cmml" xref="S2.I4.i2.p1.7.m7.3.3.2.1.1.1.1.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i2.p1.7.m7.3c">\sigma(X\cap X^{\prime\prime})\subseteq\sigma(X)\cap\sigma(X^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i2.p1.7.m7.3d">italic_σ ( italic_X ∩ italic_X start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ⊆ italic_σ ( italic_X ) ∩ italic_σ ( italic_X start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\sigma(X\cup X^{\prime\prime})=\sigma(X)\cup\sigma(X^{\prime\prime})" class="ltx_Math" display="inline" id="S2.I4.i2.p1.8.m8.3"><semantics id="S2.I4.i2.p1.8.m8.3a"><mrow id="S2.I4.i2.p1.8.m8.3.3" xref="S2.I4.i2.p1.8.m8.3.3.cmml"><mrow id="S2.I4.i2.p1.8.m8.2.2.1" xref="S2.I4.i2.p1.8.m8.2.2.1.cmml"><mi id="S2.I4.i2.p1.8.m8.2.2.1.3" xref="S2.I4.i2.p1.8.m8.2.2.1.3.cmml">σ</mi><mo id="S2.I4.i2.p1.8.m8.2.2.1.2" xref="S2.I4.i2.p1.8.m8.2.2.1.2.cmml">⁢</mo><mrow id="S2.I4.i2.p1.8.m8.2.2.1.1.1" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.cmml"><mo id="S2.I4.i2.p1.8.m8.2.2.1.1.1.2" stretchy="false" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.cmml"><mi id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.2" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.2.cmml">X</mi><mo id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.1" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.1.cmml">∪</mo><msup id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.cmml"><mi id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.2" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.2.cmml">X</mi><mo id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.3" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.3.cmml">′′</mo></msup></mrow><mo id="S2.I4.i2.p1.8.m8.2.2.1.1.1.3" stretchy="false" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I4.i2.p1.8.m8.3.3.3" xref="S2.I4.i2.p1.8.m8.3.3.3.cmml">=</mo><mrow id="S2.I4.i2.p1.8.m8.3.3.2" xref="S2.I4.i2.p1.8.m8.3.3.2.cmml"><mrow id="S2.I4.i2.p1.8.m8.3.3.2.3" xref="S2.I4.i2.p1.8.m8.3.3.2.3.cmml"><mi id="S2.I4.i2.p1.8.m8.3.3.2.3.2" xref="S2.I4.i2.p1.8.m8.3.3.2.3.2.cmml">σ</mi><mo id="S2.I4.i2.p1.8.m8.3.3.2.3.1" xref="S2.I4.i2.p1.8.m8.3.3.2.3.1.cmml">⁢</mo><mrow id="S2.I4.i2.p1.8.m8.3.3.2.3.3.2" xref="S2.I4.i2.p1.8.m8.3.3.2.3.cmml"><mo id="S2.I4.i2.p1.8.m8.3.3.2.3.3.2.1" stretchy="false" xref="S2.I4.i2.p1.8.m8.3.3.2.3.cmml">(</mo><mi id="S2.I4.i2.p1.8.m8.1.1" xref="S2.I4.i2.p1.8.m8.1.1.cmml">X</mi><mo id="S2.I4.i2.p1.8.m8.3.3.2.3.3.2.2" stretchy="false" xref="S2.I4.i2.p1.8.m8.3.3.2.3.cmml">)</mo></mrow></mrow><mo id="S2.I4.i2.p1.8.m8.3.3.2.2" xref="S2.I4.i2.p1.8.m8.3.3.2.2.cmml">∪</mo><mrow id="S2.I4.i2.p1.8.m8.3.3.2.1" xref="S2.I4.i2.p1.8.m8.3.3.2.1.cmml"><mi id="S2.I4.i2.p1.8.m8.3.3.2.1.3" xref="S2.I4.i2.p1.8.m8.3.3.2.1.3.cmml">σ</mi><mo id="S2.I4.i2.p1.8.m8.3.3.2.1.2" xref="S2.I4.i2.p1.8.m8.3.3.2.1.2.cmml">⁢</mo><mrow id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.cmml"><mo id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.2" stretchy="false" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.cmml">(</mo><msup id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.cmml"><mi id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.2" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.2.cmml">X</mi><mo id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.3" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.3.cmml">′′</mo></msup><mo id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.3" stretchy="false" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i2.p1.8.m8.3b"><apply id="S2.I4.i2.p1.8.m8.3.3.cmml" xref="S2.I4.i2.p1.8.m8.3.3"><eq id="S2.I4.i2.p1.8.m8.3.3.3.cmml" xref="S2.I4.i2.p1.8.m8.3.3.3"></eq><apply id="S2.I4.i2.p1.8.m8.2.2.1.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1"><times id="S2.I4.i2.p1.8.m8.2.2.1.2.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.2"></times><ci id="S2.I4.i2.p1.8.m8.2.2.1.3.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.3">𝜎</ci><apply id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1"><union id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.1.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.1"></union><ci id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.2.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.2">𝑋</ci><apply id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.1.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3">superscript</csymbol><ci id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.2.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.2">𝑋</ci><ci id="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.3.cmml" xref="S2.I4.i2.p1.8.m8.2.2.1.1.1.1.3.3">′′</ci></apply></apply></apply><apply id="S2.I4.i2.p1.8.m8.3.3.2.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2"><union id="S2.I4.i2.p1.8.m8.3.3.2.2.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.2"></union><apply id="S2.I4.i2.p1.8.m8.3.3.2.3.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.3"><times id="S2.I4.i2.p1.8.m8.3.3.2.3.1.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.3.1"></times><ci id="S2.I4.i2.p1.8.m8.3.3.2.3.2.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.3.2">𝜎</ci><ci id="S2.I4.i2.p1.8.m8.1.1.cmml" xref="S2.I4.i2.p1.8.m8.1.1">𝑋</ci></apply><apply id="S2.I4.i2.p1.8.m8.3.3.2.1.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.1"><times id="S2.I4.i2.p1.8.m8.3.3.2.1.2.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.1.2"></times><ci id="S2.I4.i2.p1.8.m8.3.3.2.1.3.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.1.3">𝜎</ci><apply id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1"><csymbol cd="ambiguous" id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.1.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1">superscript</csymbol><ci id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.2.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.2">𝑋</ci><ci id="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.3.cmml" xref="S2.I4.i2.p1.8.m8.3.3.2.1.1.1.1.3">′′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i2.p1.8.m8.3c">\sigma(X\cup X^{\prime\prime})=\sigma(X)\cup\sigma(X^{\prime\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i2.p1.8.m8.3d">italic_σ ( italic_X ∪ italic_X start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_σ ( italic_X ) ∪ italic_σ ( italic_X start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>; the analogous statements hold for infinite intersections and for the closure of infinite unions.</p> </div> </li> <li class="ltx_item" id="S2.I4.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S2.I4.i3.p1"> <p class="ltx_p" id="S2.I4.i3.p1.5">The map <math alttext="\sigma^{\Sigma}" class="ltx_Math" display="inline" id="S2.I4.i3.p1.1.m1.1"><semantics id="S2.I4.i3.p1.1.m1.1a"><msup id="S2.I4.i3.p1.1.m1.1.1" xref="S2.I4.i3.p1.1.m1.1.1.cmml"><mi id="S2.I4.i3.p1.1.m1.1.1.2" xref="S2.I4.i3.p1.1.m1.1.1.2.cmml">σ</mi><mi id="S2.I4.i3.p1.1.m1.1.1.3" mathvariant="normal" xref="S2.I4.i3.p1.1.m1.1.1.3.cmml">Σ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.I4.i3.p1.1.m1.1b"><apply id="S2.I4.i3.p1.1.m1.1.1.cmml" xref="S2.I4.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.I4.i3.p1.1.m1.1.1.1.cmml" xref="S2.I4.i3.p1.1.m1.1.1">superscript</csymbol><ci id="S2.I4.i3.p1.1.m1.1.1.2.cmml" xref="S2.I4.i3.p1.1.m1.1.1.2">𝜎</ci><ci id="S2.I4.i3.p1.1.m1.1.1.3.cmml" xref="S2.I4.i3.p1.1.m1.1.1.3">Σ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i3.p1.1.m1.1c">\sigma^{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i3.p1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT</annotation></semantics></math> respects the subshift closure: If <math alttext="\cal L\subseteq\cal A^{*}" class="ltx_Math" display="inline" id="S2.I4.i3.p1.2.m2.1"><semantics id="S2.I4.i3.p1.2.m2.1a"><mrow id="S2.I4.i3.p1.2.m2.1.1" xref="S2.I4.i3.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i3.p1.2.m2.1.1.2" xref="S2.I4.i3.p1.2.m2.1.1.2.cmml">ℒ</mi><mo id="S2.I4.i3.p1.2.m2.1.1.1" xref="S2.I4.i3.p1.2.m2.1.1.1.cmml">⊆</mo><msup id="S2.I4.i3.p1.2.m2.1.1.3" xref="S2.I4.i3.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i3.p1.2.m2.1.1.3.2" xref="S2.I4.i3.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mo id="S2.I4.i3.p1.2.m2.1.1.3.3" xref="S2.I4.i3.p1.2.m2.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i3.p1.2.m2.1b"><apply id="S2.I4.i3.p1.2.m2.1.1.cmml" xref="S2.I4.i3.p1.2.m2.1.1"><subset id="S2.I4.i3.p1.2.m2.1.1.1.cmml" xref="S2.I4.i3.p1.2.m2.1.1.1"></subset><ci id="S2.I4.i3.p1.2.m2.1.1.2.cmml" xref="S2.I4.i3.p1.2.m2.1.1.2">ℒ</ci><apply id="S2.I4.i3.p1.2.m2.1.1.3.cmml" xref="S2.I4.i3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.I4.i3.p1.2.m2.1.1.3.1.cmml" xref="S2.I4.i3.p1.2.m2.1.1.3">superscript</csymbol><ci id="S2.I4.i3.p1.2.m2.1.1.3.2.cmml" xref="S2.I4.i3.p1.2.m2.1.1.3.2">𝒜</ci><times id="S2.I4.i3.p1.2.m2.1.1.3.3.cmml" xref="S2.I4.i3.p1.2.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i3.p1.2.m2.1c">\cal L\subseteq\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i3.p1.2.m2.1d">caligraphic_L ⊆ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is an infinite set, then the subshift <math alttext="X(\cal L)\in\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.I4.i3.p1.3.m3.2"><semantics id="S2.I4.i3.p1.3.m3.2a"><mrow id="S2.I4.i3.p1.3.m3.2.3" xref="S2.I4.i3.p1.3.m3.2.3.cmml"><mrow id="S2.I4.i3.p1.3.m3.2.3.2" xref="S2.I4.i3.p1.3.m3.2.3.2.cmml"><mi id="S2.I4.i3.p1.3.m3.2.3.2.2" xref="S2.I4.i3.p1.3.m3.2.3.2.2.cmml">X</mi><mo id="S2.I4.i3.p1.3.m3.2.3.2.1" xref="S2.I4.i3.p1.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S2.I4.i3.p1.3.m3.2.3.2.3.2" xref="S2.I4.i3.p1.3.m3.2.3.2.cmml"><mo id="S2.I4.i3.p1.3.m3.2.3.2.3.2.1" stretchy="false" xref="S2.I4.i3.p1.3.m3.2.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I4.i3.p1.3.m3.1.1" xref="S2.I4.i3.p1.3.m3.1.1.cmml">ℒ</mi><mo id="S2.I4.i3.p1.3.m3.2.3.2.3.2.2" stretchy="false" xref="S2.I4.i3.p1.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.I4.i3.p1.3.m3.2.3.1" xref="S2.I4.i3.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="S2.I4.i3.p1.3.m3.2.3.3" xref="S2.I4.i3.p1.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i3.p1.3.m3.2.3.3.2" mathvariant="script" xref="S2.I4.i3.p1.3.m3.2.3.3.2.cmml">Σ</mi><mo id="S2.I4.i3.p1.3.m3.2.3.3.1" xref="S2.I4.i3.p1.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S2.I4.i3.p1.3.m3.2.3.3.3.2" xref="S2.I4.i3.p1.3.m3.2.3.3.cmml"><mo id="S2.I4.i3.p1.3.m3.2.3.3.3.2.1" stretchy="false" xref="S2.I4.i3.p1.3.m3.2.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I4.i3.p1.3.m3.2.2" xref="S2.I4.i3.p1.3.m3.2.2.cmml">𝒜</mi><mo id="S2.I4.i3.p1.3.m3.2.3.3.3.2.2" stretchy="false" xref="S2.I4.i3.p1.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i3.p1.3.m3.2b"><apply id="S2.I4.i3.p1.3.m3.2.3.cmml" xref="S2.I4.i3.p1.3.m3.2.3"><in id="S2.I4.i3.p1.3.m3.2.3.1.cmml" xref="S2.I4.i3.p1.3.m3.2.3.1"></in><apply id="S2.I4.i3.p1.3.m3.2.3.2.cmml" xref="S2.I4.i3.p1.3.m3.2.3.2"><times id="S2.I4.i3.p1.3.m3.2.3.2.1.cmml" xref="S2.I4.i3.p1.3.m3.2.3.2.1"></times><ci id="S2.I4.i3.p1.3.m3.2.3.2.2.cmml" xref="S2.I4.i3.p1.3.m3.2.3.2.2">𝑋</ci><ci id="S2.I4.i3.p1.3.m3.1.1.cmml" xref="S2.I4.i3.p1.3.m3.1.1">ℒ</ci></apply><apply id="S2.I4.i3.p1.3.m3.2.3.3.cmml" xref="S2.I4.i3.p1.3.m3.2.3.3"><times id="S2.I4.i3.p1.3.m3.2.3.3.1.cmml" xref="S2.I4.i3.p1.3.m3.2.3.3.1"></times><ci id="S2.I4.i3.p1.3.m3.2.3.3.2.cmml" xref="S2.I4.i3.p1.3.m3.2.3.3.2">script-Σ</ci><ci id="S2.I4.i3.p1.3.m3.2.2.cmml" xref="S2.I4.i3.p1.3.m3.2.2">𝒜</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i3.p1.3.m3.2c">X(\cal L)\in\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i3.p1.3.m3.2d">italic_X ( caligraphic_L ) ∈ caligraphic_Σ ( caligraphic_A )</annotation></semantics></math> generated by <math alttext="\cal L" class="ltx_Math" display="inline" id="S2.I4.i3.p1.4.m4.1"><semantics id="S2.I4.i3.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i3.p1.4.m4.1.1" xref="S2.I4.i3.p1.4.m4.1.1.cmml">ℒ</mi><annotation-xml encoding="MathML-Content" id="S2.I4.i3.p1.4.m4.1b"><ci id="S2.I4.i3.p1.4.m4.1.1.cmml" xref="S2.I4.i3.p1.4.m4.1.1">ℒ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i3.p1.4.m4.1c">\cal L</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i3.p1.4.m4.1d">caligraphic_L</annotation></semantics></math> has as image the subshift generated by <math alttext="\sigma(\cal L)" class="ltx_Math" display="inline" id="S2.I4.i3.p1.5.m5.1"><semantics id="S2.I4.i3.p1.5.m5.1a"><mrow id="S2.I4.i3.p1.5.m5.1.2" xref="S2.I4.i3.p1.5.m5.1.2.cmml"><mi id="S2.I4.i3.p1.5.m5.1.2.2" xref="S2.I4.i3.p1.5.m5.1.2.2.cmml">σ</mi><mo id="S2.I4.i3.p1.5.m5.1.2.1" xref="S2.I4.i3.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.I4.i3.p1.5.m5.1.2.3.2" xref="S2.I4.i3.p1.5.m5.1.2.cmml"><mo id="S2.I4.i3.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S2.I4.i3.p1.5.m5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I4.i3.p1.5.m5.1.1" xref="S2.I4.i3.p1.5.m5.1.1.cmml">ℒ</mi><mo id="S2.I4.i3.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S2.I4.i3.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i3.p1.5.m5.1b"><apply id="S2.I4.i3.p1.5.m5.1.2.cmml" xref="S2.I4.i3.p1.5.m5.1.2"><times id="S2.I4.i3.p1.5.m5.1.2.1.cmml" xref="S2.I4.i3.p1.5.m5.1.2.1"></times><ci id="S2.I4.i3.p1.5.m5.1.2.2.cmml" xref="S2.I4.i3.p1.5.m5.1.2.2">𝜎</ci><ci id="S2.I4.i3.p1.5.m5.1.1.cmml" xref="S2.I4.i3.p1.5.m5.1.1">ℒ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i3.p1.5.m5.1c">\sigma(\cal L)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i3.p1.5.m5.1d">italic_σ ( caligraphic_L )</annotation></semantics></math>:</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma(X(\cal L))=X(\sigma(\cal L))" class="ltx_Math" display="block" id="S2.Ex12.m1.4"><semantics id="S2.Ex12.m1.4a"><mrow id="S2.Ex12.m1.4.4" xref="S2.Ex12.m1.4.4.cmml"><mrow id="S2.Ex12.m1.3.3.1" xref="S2.Ex12.m1.3.3.1.cmml"><mi id="S2.Ex12.m1.3.3.1.3" xref="S2.Ex12.m1.3.3.1.3.cmml">σ</mi><mo id="S2.Ex12.m1.3.3.1.2" xref="S2.Ex12.m1.3.3.1.2.cmml">⁢</mo><mrow id="S2.Ex12.m1.3.3.1.1.1" xref="S2.Ex12.m1.3.3.1.1.1.1.cmml"><mo id="S2.Ex12.m1.3.3.1.1.1.2" stretchy="false" xref="S2.Ex12.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.Ex12.m1.3.3.1.1.1.1" xref="S2.Ex12.m1.3.3.1.1.1.1.cmml"><mi id="S2.Ex12.m1.3.3.1.1.1.1.2" xref="S2.Ex12.m1.3.3.1.1.1.1.2.cmml">X</mi><mo id="S2.Ex12.m1.3.3.1.1.1.1.1" xref="S2.Ex12.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex12.m1.3.3.1.1.1.1.3.2" xref="S2.Ex12.m1.3.3.1.1.1.1.cmml"><mo id="S2.Ex12.m1.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex12.m1.3.3.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex12.m1.1.1" xref="S2.Ex12.m1.1.1.cmml">ℒ</mi><mo id="S2.Ex12.m1.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex12.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex12.m1.3.3.1.1.1.3" stretchy="false" xref="S2.Ex12.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex12.m1.4.4.3" xref="S2.Ex12.m1.4.4.3.cmml">=</mo><mrow id="S2.Ex12.m1.4.4.2" xref="S2.Ex12.m1.4.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex12.m1.4.4.2.3" xref="S2.Ex12.m1.4.4.2.3.cmml">𝒳</mi><mo id="S2.Ex12.m1.4.4.2.2" xref="S2.Ex12.m1.4.4.2.2.cmml">⁢</mo><mrow id="S2.Ex12.m1.4.4.2.1.1" xref="S2.Ex12.m1.4.4.2.1.1.1.cmml"><mo id="S2.Ex12.m1.4.4.2.1.1.2" stretchy="false" xref="S2.Ex12.m1.4.4.2.1.1.1.cmml">(</mo><mrow id="S2.Ex12.m1.4.4.2.1.1.1" xref="S2.Ex12.m1.4.4.2.1.1.1.cmml"><mi id="S2.Ex12.m1.4.4.2.1.1.1.2" xref="S2.Ex12.m1.4.4.2.1.1.1.2.cmml">σ</mi><mo id="S2.Ex12.m1.4.4.2.1.1.1.1" xref="S2.Ex12.m1.4.4.2.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex12.m1.4.4.2.1.1.1.3.2" xref="S2.Ex12.m1.4.4.2.1.1.1.cmml"><mo id="S2.Ex12.m1.4.4.2.1.1.1.3.2.1" stretchy="false" xref="S2.Ex12.m1.4.4.2.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex12.m1.2.2" xref="S2.Ex12.m1.2.2.cmml">ℒ</mi><mo id="S2.Ex12.m1.4.4.2.1.1.1.3.2.2" stretchy="false" xref="S2.Ex12.m1.4.4.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex12.m1.4.4.2.1.1.3" stretchy="false" xref="S2.Ex12.m1.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex12.m1.4b"><apply id="S2.Ex12.m1.4.4.cmml" xref="S2.Ex12.m1.4.4"><eq id="S2.Ex12.m1.4.4.3.cmml" xref="S2.Ex12.m1.4.4.3"></eq><apply id="S2.Ex12.m1.3.3.1.cmml" xref="S2.Ex12.m1.3.3.1"><times id="S2.Ex12.m1.3.3.1.2.cmml" xref="S2.Ex12.m1.3.3.1.2"></times><ci id="S2.Ex12.m1.3.3.1.3.cmml" xref="S2.Ex12.m1.3.3.1.3">𝜎</ci><apply id="S2.Ex12.m1.3.3.1.1.1.1.cmml" xref="S2.Ex12.m1.3.3.1.1.1"><times id="S2.Ex12.m1.3.3.1.1.1.1.1.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.1"></times><ci id="S2.Ex12.m1.3.3.1.1.1.1.2.cmml" xref="S2.Ex12.m1.3.3.1.1.1.1.2">𝑋</ci><ci id="S2.Ex12.m1.1.1.cmml" xref="S2.Ex12.m1.1.1">ℒ</ci></apply></apply><apply id="S2.Ex12.m1.4.4.2.cmml" xref="S2.Ex12.m1.4.4.2"><times id="S2.Ex12.m1.4.4.2.2.cmml" xref="S2.Ex12.m1.4.4.2.2"></times><ci id="S2.Ex12.m1.4.4.2.3.cmml" xref="S2.Ex12.m1.4.4.2.3">𝒳</ci><apply id="S2.Ex12.m1.4.4.2.1.1.1.cmml" xref="S2.Ex12.m1.4.4.2.1.1"><times id="S2.Ex12.m1.4.4.2.1.1.1.1.cmml" xref="S2.Ex12.m1.4.4.2.1.1.1.1"></times><ci id="S2.Ex12.m1.4.4.2.1.1.1.2.cmml" xref="S2.Ex12.m1.4.4.2.1.1.1.2">𝜎</ci><ci id="S2.Ex12.m1.2.2.cmml" xref="S2.Ex12.m1.2.2">ℒ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex12.m1.4c">\sigma(X(\cal L))=X(\sigma(\cal L))</annotation><annotation encoding="application/x-llamapun" id="S2.Ex12.m1.4d">italic_σ ( italic_X ( caligraphic_L ) ) = caligraphic_X ( italic_σ ( caligraphic_L ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S2.I4.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(4)</span> <div class="ltx_para" id="S2.I4.i4.p1"> <p class="ltx_p" id="S2.I4.i4.p1.4">For any subshift <math alttext="X\in\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.I4.i4.p1.1.m1.1"><semantics id="S2.I4.i4.p1.1.m1.1a"><mrow id="S2.I4.i4.p1.1.m1.1.2" xref="S2.I4.i4.p1.1.m1.1.2.cmml"><mi id="S2.I4.i4.p1.1.m1.1.2.2" xref="S2.I4.i4.p1.1.m1.1.2.2.cmml">X</mi><mo id="S2.I4.i4.p1.1.m1.1.2.1" xref="S2.I4.i4.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S2.I4.i4.p1.1.m1.1.2.3" xref="S2.I4.i4.p1.1.m1.1.2.3.cmml"><mi id="S2.I4.i4.p1.1.m1.1.2.3.2" mathvariant="normal" xref="S2.I4.i4.p1.1.m1.1.2.3.2.cmml">Σ</mi><mo id="S2.I4.i4.p1.1.m1.1.2.3.1" xref="S2.I4.i4.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S2.I4.i4.p1.1.m1.1.2.3.3.2" xref="S2.I4.i4.p1.1.m1.1.2.3.cmml"><mo id="S2.I4.i4.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S2.I4.i4.p1.1.m1.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I4.i4.p1.1.m1.1.1" xref="S2.I4.i4.p1.1.m1.1.1.cmml">𝒜</mi><mo id="S2.I4.i4.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S2.I4.i4.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i4.p1.1.m1.1b"><apply id="S2.I4.i4.p1.1.m1.1.2.cmml" xref="S2.I4.i4.p1.1.m1.1.2"><in id="S2.I4.i4.p1.1.m1.1.2.1.cmml" xref="S2.I4.i4.p1.1.m1.1.2.1"></in><ci id="S2.I4.i4.p1.1.m1.1.2.2.cmml" xref="S2.I4.i4.p1.1.m1.1.2.2">𝑋</ci><apply id="S2.I4.i4.p1.1.m1.1.2.3.cmml" xref="S2.I4.i4.p1.1.m1.1.2.3"><times id="S2.I4.i4.p1.1.m1.1.2.3.1.cmml" xref="S2.I4.i4.p1.1.m1.1.2.3.1"></times><ci id="S2.I4.i4.p1.1.m1.1.2.3.2.cmml" xref="S2.I4.i4.p1.1.m1.1.2.3.2">Σ</ci><ci id="S2.I4.i4.p1.1.m1.1.1.cmml" xref="S2.I4.i4.p1.1.m1.1.1">𝒜</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i4.p1.1.m1.1c">X\in\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i4.p1.1.m1.1d">italic_X ∈ roman_Σ ( caligraphic_A )</annotation></semantics></math> the map <math alttext="\sigma^{\Sigma}" class="ltx_Math" display="inline" id="S2.I4.i4.p1.2.m2.1"><semantics id="S2.I4.i4.p1.2.m2.1a"><msup id="S2.I4.i4.p1.2.m2.1.1" xref="S2.I4.i4.p1.2.m2.1.1.cmml"><mi id="S2.I4.i4.p1.2.m2.1.1.2" xref="S2.I4.i4.p1.2.m2.1.1.2.cmml">σ</mi><mi id="S2.I4.i4.p1.2.m2.1.1.3" mathvariant="normal" xref="S2.I4.i4.p1.2.m2.1.1.3.cmml">Σ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.I4.i4.p1.2.m2.1b"><apply id="S2.I4.i4.p1.2.m2.1.1.cmml" xref="S2.I4.i4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.I4.i4.p1.2.m2.1.1.1.cmml" xref="S2.I4.i4.p1.2.m2.1.1">superscript</csymbol><ci id="S2.I4.i4.p1.2.m2.1.1.2.cmml" xref="S2.I4.i4.p1.2.m2.1.1.2">𝜎</ci><ci id="S2.I4.i4.p1.2.m2.1.1.3.cmml" xref="S2.I4.i4.p1.2.m2.1.1.3">Σ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i4.p1.2.m2.1c">\sigma^{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i4.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT</annotation></semantics></math> induces on the subset <math alttext="\Sigma(X)\subseteq\Sigma(\cal A)" class="ltx_Math" display="inline" id="S2.I4.i4.p1.3.m3.2"><semantics id="S2.I4.i4.p1.3.m3.2a"><mrow id="S2.I4.i4.p1.3.m3.2.3" xref="S2.I4.i4.p1.3.m3.2.3.cmml"><mrow id="S2.I4.i4.p1.3.m3.2.3.2" xref="S2.I4.i4.p1.3.m3.2.3.2.cmml"><mi id="S2.I4.i4.p1.3.m3.2.3.2.2" mathvariant="normal" xref="S2.I4.i4.p1.3.m3.2.3.2.2.cmml">Σ</mi><mo id="S2.I4.i4.p1.3.m3.2.3.2.1" xref="S2.I4.i4.p1.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S2.I4.i4.p1.3.m3.2.3.2.3.2" xref="S2.I4.i4.p1.3.m3.2.3.2.cmml"><mo id="S2.I4.i4.p1.3.m3.2.3.2.3.2.1" stretchy="false" xref="S2.I4.i4.p1.3.m3.2.3.2.cmml">(</mo><mi id="S2.I4.i4.p1.3.m3.1.1" xref="S2.I4.i4.p1.3.m3.1.1.cmml">X</mi><mo id="S2.I4.i4.p1.3.m3.2.3.2.3.2.2" stretchy="false" xref="S2.I4.i4.p1.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.I4.i4.p1.3.m3.2.3.1" xref="S2.I4.i4.p1.3.m3.2.3.1.cmml">⊆</mo><mrow id="S2.I4.i4.p1.3.m3.2.3.3" xref="S2.I4.i4.p1.3.m3.2.3.3.cmml"><mi id="S2.I4.i4.p1.3.m3.2.3.3.2" mathvariant="normal" xref="S2.I4.i4.p1.3.m3.2.3.3.2.cmml">Σ</mi><mo id="S2.I4.i4.p1.3.m3.2.3.3.1" xref="S2.I4.i4.p1.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S2.I4.i4.p1.3.m3.2.3.3.3.2" xref="S2.I4.i4.p1.3.m3.2.3.3.cmml"><mo id="S2.I4.i4.p1.3.m3.2.3.3.3.2.1" stretchy="false" xref="S2.I4.i4.p1.3.m3.2.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.I4.i4.p1.3.m3.2.2" xref="S2.I4.i4.p1.3.m3.2.2.cmml">𝒜</mi><mo id="S2.I4.i4.p1.3.m3.2.3.3.3.2.2" stretchy="false" xref="S2.I4.i4.p1.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i4.p1.3.m3.2b"><apply id="S2.I4.i4.p1.3.m3.2.3.cmml" xref="S2.I4.i4.p1.3.m3.2.3"><subset id="S2.I4.i4.p1.3.m3.2.3.1.cmml" xref="S2.I4.i4.p1.3.m3.2.3.1"></subset><apply id="S2.I4.i4.p1.3.m3.2.3.2.cmml" xref="S2.I4.i4.p1.3.m3.2.3.2"><times id="S2.I4.i4.p1.3.m3.2.3.2.1.cmml" xref="S2.I4.i4.p1.3.m3.2.3.2.1"></times><ci id="S2.I4.i4.p1.3.m3.2.3.2.2.cmml" xref="S2.I4.i4.p1.3.m3.2.3.2.2">Σ</ci><ci id="S2.I4.i4.p1.3.m3.1.1.cmml" xref="S2.I4.i4.p1.3.m3.1.1">𝑋</ci></apply><apply id="S2.I4.i4.p1.3.m3.2.3.3.cmml" xref="S2.I4.i4.p1.3.m3.2.3.3"><times id="S2.I4.i4.p1.3.m3.2.3.3.1.cmml" xref="S2.I4.i4.p1.3.m3.2.3.3.1"></times><ci id="S2.I4.i4.p1.3.m3.2.3.3.2.cmml" xref="S2.I4.i4.p1.3.m3.2.3.3.2">Σ</ci><ci id="S2.I4.i4.p1.3.m3.2.2.cmml" xref="S2.I4.i4.p1.3.m3.2.2">𝒜</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i4.p1.3.m3.2c">\Sigma(X)\subseteq\Sigma(\cal A)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i4.p1.3.m3.2d">roman_Σ ( italic_X ) ⊆ roman_Σ ( caligraphic_A )</annotation></semantics></math> of subshifts contained in <math alttext="X" class="ltx_Math" display="inline" id="S2.I4.i4.p1.4.m4.1"><semantics id="S2.I4.i4.p1.4.m4.1a"><mi id="S2.I4.i4.p1.4.m4.1.1" xref="S2.I4.i4.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.I4.i4.p1.4.m4.1b"><ci id="S2.I4.i4.p1.4.m4.1.1.cmml" xref="S2.I4.i4.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i4.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i4.p1.4.m4.1d">italic_X</annotation></semantics></math> a map</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{\Sigma}_{X}:\Sigma(X)\to\Sigma(\sigma(X))\,." class="ltx_Math" display="block" id="S2.Ex13.m1.3"><semantics id="S2.Ex13.m1.3a"><mrow id="S2.Ex13.m1.3.3.1" xref="S2.Ex13.m1.3.3.1.1.cmml"><mrow id="S2.Ex13.m1.3.3.1.1" xref="S2.Ex13.m1.3.3.1.1.cmml"><msubsup id="S2.Ex13.m1.3.3.1.1.3" xref="S2.Ex13.m1.3.3.1.1.3.cmml"><mi id="S2.Ex13.m1.3.3.1.1.3.2.2" xref="S2.Ex13.m1.3.3.1.1.3.2.2.cmml">σ</mi><mi id="S2.Ex13.m1.3.3.1.1.3.3" xref="S2.Ex13.m1.3.3.1.1.3.3.cmml">X</mi><mi id="S2.Ex13.m1.3.3.1.1.3.2.3" mathvariant="normal" xref="S2.Ex13.m1.3.3.1.1.3.2.3.cmml">Σ</mi></msubsup><mo id="S2.Ex13.m1.3.3.1.1.2" lspace="0.278em" rspace="0.278em" xref="S2.Ex13.m1.3.3.1.1.2.cmml">:</mo><mrow id="S2.Ex13.m1.3.3.1.1.1" xref="S2.Ex13.m1.3.3.1.1.1.cmml"><mrow id="S2.Ex13.m1.3.3.1.1.1.3" xref="S2.Ex13.m1.3.3.1.1.1.3.cmml"><mi id="S2.Ex13.m1.3.3.1.1.1.3.2" mathvariant="normal" xref="S2.Ex13.m1.3.3.1.1.1.3.2.cmml">Σ</mi><mo id="S2.Ex13.m1.3.3.1.1.1.3.1" xref="S2.Ex13.m1.3.3.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex13.m1.3.3.1.1.1.3.3.2" xref="S2.Ex13.m1.3.3.1.1.1.3.cmml"><mo id="S2.Ex13.m1.3.3.1.1.1.3.3.2.1" stretchy="false" xref="S2.Ex13.m1.3.3.1.1.1.3.cmml">(</mo><mi id="S2.Ex13.m1.1.1" xref="S2.Ex13.m1.1.1.cmml">X</mi><mo id="S2.Ex13.m1.3.3.1.1.1.3.3.2.2" stretchy="false" xref="S2.Ex13.m1.3.3.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex13.m1.3.3.1.1.1.2" stretchy="false" xref="S2.Ex13.m1.3.3.1.1.1.2.cmml">→</mo><mrow id="S2.Ex13.m1.3.3.1.1.1.1" xref="S2.Ex13.m1.3.3.1.1.1.1.cmml"><mi id="S2.Ex13.m1.3.3.1.1.1.1.3" mathvariant="normal" xref="S2.Ex13.m1.3.3.1.1.1.1.3.cmml">Σ</mi><mo id="S2.Ex13.m1.3.3.1.1.1.1.2" xref="S2.Ex13.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex13.m1.3.3.1.1.1.1.1.1" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex13.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.2" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.1" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.3.2" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mi id="S2.Ex13.m1.2.2" xref="S2.Ex13.m1.2.2.cmml">X</mi><mo id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex13.m1.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.Ex13.m1.3.3.1.2" lspace="0.170em" xref="S2.Ex13.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex13.m1.3b"><apply id="S2.Ex13.m1.3.3.1.1.cmml" xref="S2.Ex13.m1.3.3.1"><ci id="S2.Ex13.m1.3.3.1.1.2.cmml" xref="S2.Ex13.m1.3.3.1.1.2">:</ci><apply id="S2.Ex13.m1.3.3.1.1.3.cmml" xref="S2.Ex13.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S2.Ex13.m1.3.3.1.1.3.1.cmml" xref="S2.Ex13.m1.3.3.1.1.3">subscript</csymbol><apply id="S2.Ex13.m1.3.3.1.1.3.2.cmml" xref="S2.Ex13.m1.3.3.1.1.3"><csymbol cd="ambiguous" id="S2.Ex13.m1.3.3.1.1.3.2.1.cmml" xref="S2.Ex13.m1.3.3.1.1.3">superscript</csymbol><ci id="S2.Ex13.m1.3.3.1.1.3.2.2.cmml" xref="S2.Ex13.m1.3.3.1.1.3.2.2">𝜎</ci><ci id="S2.Ex13.m1.3.3.1.1.3.2.3.cmml" xref="S2.Ex13.m1.3.3.1.1.3.2.3">Σ</ci></apply><ci id="S2.Ex13.m1.3.3.1.1.3.3.cmml" xref="S2.Ex13.m1.3.3.1.1.3.3">𝑋</ci></apply><apply id="S2.Ex13.m1.3.3.1.1.1.cmml" xref="S2.Ex13.m1.3.3.1.1.1"><ci id="S2.Ex13.m1.3.3.1.1.1.2.cmml" xref="S2.Ex13.m1.3.3.1.1.1.2">→</ci><apply id="S2.Ex13.m1.3.3.1.1.1.3.cmml" xref="S2.Ex13.m1.3.3.1.1.1.3"><times id="S2.Ex13.m1.3.3.1.1.1.3.1.cmml" xref="S2.Ex13.m1.3.3.1.1.1.3.1"></times><ci id="S2.Ex13.m1.3.3.1.1.1.3.2.cmml" xref="S2.Ex13.m1.3.3.1.1.1.3.2">Σ</ci><ci id="S2.Ex13.m1.1.1.cmml" xref="S2.Ex13.m1.1.1">𝑋</ci></apply><apply id="S2.Ex13.m1.3.3.1.1.1.1.cmml" xref="S2.Ex13.m1.3.3.1.1.1.1"><times id="S2.Ex13.m1.3.3.1.1.1.1.2.cmml" xref="S2.Ex13.m1.3.3.1.1.1.1.2"></times><ci id="S2.Ex13.m1.3.3.1.1.1.1.3.cmml" xref="S2.Ex13.m1.3.3.1.1.1.1.3">Σ</ci><apply id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1"><times id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.1"></times><ci id="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S2.Ex13.m1.3.3.1.1.1.1.1.1.1.2">𝜎</ci><ci id="S2.Ex13.m1.2.2.cmml" xref="S2.Ex13.m1.2.2">𝑋</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex13.m1.3c">\sigma^{\Sigma}_{X}:\Sigma(X)\to\Sigma(\sigma(X))\,.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex13.m1.3d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT : roman_Σ ( italic_X ) → roman_Σ ( italic_σ ( italic_X ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S2.I4.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(5)</span> <div class="ltx_para" id="S2.I4.i5.p1"> <p class="ltx_p" id="S2.I4.i5.p1.6">For any subshift <math alttext="Y\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.I4.i5.p1.1.m1.1"><semantics id="S2.I4.i5.p1.1.m1.1a"><mrow id="S2.I4.i5.p1.1.m1.1.1" xref="S2.I4.i5.p1.1.m1.1.1.cmml"><mi id="S2.I4.i5.p1.1.m1.1.1.2" xref="S2.I4.i5.p1.1.m1.1.1.2.cmml">Y</mi><mo id="S2.I4.i5.p1.1.m1.1.1.1" xref="S2.I4.i5.p1.1.m1.1.1.1.cmml">⊆</mo><msup id="S2.I4.i5.p1.1.m1.1.1.3" xref="S2.I4.i5.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i5.p1.1.m1.1.1.3.2" xref="S2.I4.i5.p1.1.m1.1.1.3.2.cmml">ℬ</mi><mi id="S2.I4.i5.p1.1.m1.1.1.3.3" xref="S2.I4.i5.p1.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i5.p1.1.m1.1b"><apply id="S2.I4.i5.p1.1.m1.1.1.cmml" xref="S2.I4.i5.p1.1.m1.1.1"><subset id="S2.I4.i5.p1.1.m1.1.1.1.cmml" xref="S2.I4.i5.p1.1.m1.1.1.1"></subset><ci id="S2.I4.i5.p1.1.m1.1.1.2.cmml" xref="S2.I4.i5.p1.1.m1.1.1.2">𝑌</ci><apply id="S2.I4.i5.p1.1.m1.1.1.3.cmml" xref="S2.I4.i5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.I4.i5.p1.1.m1.1.1.3.1.cmml" xref="S2.I4.i5.p1.1.m1.1.1.3">superscript</csymbol><ci id="S2.I4.i5.p1.1.m1.1.1.3.2.cmml" xref="S2.I4.i5.p1.1.m1.1.1.3.2">ℬ</ci><ci id="S2.I4.i5.p1.1.m1.1.1.3.3.cmml" xref="S2.I4.i5.p1.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i5.p1.1.m1.1c">Y\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i5.p1.1.m1.1d">italic_Y ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> the preimage <math alttext="(\sigma^{\mathbb{Z}})^{-1}(Y)\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.I4.i5.p1.2.m2.2"><semantics id="S2.I4.i5.p1.2.m2.2a"><mrow id="S2.I4.i5.p1.2.m2.2.2" xref="S2.I4.i5.p1.2.m2.2.2.cmml"><mrow id="S2.I4.i5.p1.2.m2.2.2.1" xref="S2.I4.i5.p1.2.m2.2.2.1.cmml"><msup id="S2.I4.i5.p1.2.m2.2.2.1.1" xref="S2.I4.i5.p1.2.m2.2.2.1.1.cmml"><mrow id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.cmml"><mo id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.2" stretchy="false" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.cmml">(</mo><msup id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.cmml"><mi id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.2" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.2.cmml">σ</mi><mi id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.3" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.3" stretchy="false" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.I4.i5.p1.2.m2.2.2.1.1.3" xref="S2.I4.i5.p1.2.m2.2.2.1.1.3.cmml"><mo id="S2.I4.i5.p1.2.m2.2.2.1.1.3a" xref="S2.I4.i5.p1.2.m2.2.2.1.1.3.cmml">−</mo><mn id="S2.I4.i5.p1.2.m2.2.2.1.1.3.2" xref="S2.I4.i5.p1.2.m2.2.2.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S2.I4.i5.p1.2.m2.2.2.1.2" xref="S2.I4.i5.p1.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.I4.i5.p1.2.m2.2.2.1.3.2" xref="S2.I4.i5.p1.2.m2.2.2.1.cmml"><mo id="S2.I4.i5.p1.2.m2.2.2.1.3.2.1" stretchy="false" xref="S2.I4.i5.p1.2.m2.2.2.1.cmml">(</mo><mi id="S2.I4.i5.p1.2.m2.1.1" xref="S2.I4.i5.p1.2.m2.1.1.cmml">Y</mi><mo id="S2.I4.i5.p1.2.m2.2.2.1.3.2.2" stretchy="false" xref="S2.I4.i5.p1.2.m2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.I4.i5.p1.2.m2.2.2.2" xref="S2.I4.i5.p1.2.m2.2.2.2.cmml">⊆</mo><msup id="S2.I4.i5.p1.2.m2.2.2.3" xref="S2.I4.i5.p1.2.m2.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i5.p1.2.m2.2.2.3.2" xref="S2.I4.i5.p1.2.m2.2.2.3.2.cmml">𝒜</mi><mi id="S2.I4.i5.p1.2.m2.2.2.3.3" xref="S2.I4.i5.p1.2.m2.2.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i5.p1.2.m2.2b"><apply id="S2.I4.i5.p1.2.m2.2.2.cmml" xref="S2.I4.i5.p1.2.m2.2.2"><subset id="S2.I4.i5.p1.2.m2.2.2.2.cmml" xref="S2.I4.i5.p1.2.m2.2.2.2"></subset><apply id="S2.I4.i5.p1.2.m2.2.2.1.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1"><times id="S2.I4.i5.p1.2.m2.2.2.1.2.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.2"></times><apply id="S2.I4.i5.p1.2.m2.2.2.1.1.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S2.I4.i5.p1.2.m2.2.2.1.1.2.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.1">superscript</csymbol><apply id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1">superscript</csymbol><ci id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.2">𝜎</ci><ci id="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.3.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.1.1.1.1.3">ℤ</ci></apply><apply id="S2.I4.i5.p1.2.m2.2.2.1.1.3.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.1.3"><minus id="S2.I4.i5.p1.2.m2.2.2.1.1.3.1.cmml" xref="S2.I4.i5.p1.2.m2.2.2.1.1.3"></minus><cn id="S2.I4.i5.p1.2.m2.2.2.1.1.3.2.cmml" type="integer" xref="S2.I4.i5.p1.2.m2.2.2.1.1.3.2">1</cn></apply></apply><ci id="S2.I4.i5.p1.2.m2.1.1.cmml" xref="S2.I4.i5.p1.2.m2.1.1">𝑌</ci></apply><apply id="S2.I4.i5.p1.2.m2.2.2.3.cmml" xref="S2.I4.i5.p1.2.m2.2.2.3"><csymbol cd="ambiguous" id="S2.I4.i5.p1.2.m2.2.2.3.1.cmml" xref="S2.I4.i5.p1.2.m2.2.2.3">superscript</csymbol><ci id="S2.I4.i5.p1.2.m2.2.2.3.2.cmml" xref="S2.I4.i5.p1.2.m2.2.2.3.2">𝒜</ci><ci id="S2.I4.i5.p1.2.m2.2.2.3.3.cmml" xref="S2.I4.i5.p1.2.m2.2.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i5.p1.2.m2.2c">(\sigma^{\mathbb{Z}})^{-1}(Y)\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i5.p1.2.m2.2d">( italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_Y ) ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is either the empty set, or else it is a subshift over <math alttext="\cal A" class="ltx_Math" display="inline" id="S2.I4.i5.p1.3.m3.1"><semantics id="S2.I4.i5.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i5.p1.3.m3.1.1" xref="S2.I4.i5.p1.3.m3.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S2.I4.i5.p1.3.m3.1b"><ci id="S2.I4.i5.p1.3.m3.1.1.cmml" xref="S2.I4.i5.p1.3.m3.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i5.p1.3.m3.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i5.p1.3.m3.1d">caligraphic_A</annotation></semantics></math>. Alternatively this subshift is obtained as union of all subshifts <math alttext="X_{i}" class="ltx_Math" display="inline" id="S2.I4.i5.p1.4.m4.1"><semantics id="S2.I4.i5.p1.4.m4.1a"><msub id="S2.I4.i5.p1.4.m4.1.1" xref="S2.I4.i5.p1.4.m4.1.1.cmml"><mi id="S2.I4.i5.p1.4.m4.1.1.2" xref="S2.I4.i5.p1.4.m4.1.1.2.cmml">X</mi><mi id="S2.I4.i5.p1.4.m4.1.1.3" xref="S2.I4.i5.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.I4.i5.p1.4.m4.1b"><apply id="S2.I4.i5.p1.4.m4.1.1.cmml" xref="S2.I4.i5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.I4.i5.p1.4.m4.1.1.1.cmml" xref="S2.I4.i5.p1.4.m4.1.1">subscript</csymbol><ci id="S2.I4.i5.p1.4.m4.1.1.2.cmml" xref="S2.I4.i5.p1.4.m4.1.1.2">𝑋</ci><ci id="S2.I4.i5.p1.4.m4.1.1.3.cmml" xref="S2.I4.i5.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i5.p1.4.m4.1c">X_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i5.p1.4.m4.1d">italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\sigma^{\Sigma}(X_{i})\subseteq Y" class="ltx_Math" display="inline" id="S2.I4.i5.p1.5.m5.1"><semantics id="S2.I4.i5.p1.5.m5.1a"><mrow id="S2.I4.i5.p1.5.m5.1.1" xref="S2.I4.i5.p1.5.m5.1.1.cmml"><mrow id="S2.I4.i5.p1.5.m5.1.1.1" xref="S2.I4.i5.p1.5.m5.1.1.1.cmml"><msup id="S2.I4.i5.p1.5.m5.1.1.1.3" xref="S2.I4.i5.p1.5.m5.1.1.1.3.cmml"><mi id="S2.I4.i5.p1.5.m5.1.1.1.3.2" xref="S2.I4.i5.p1.5.m5.1.1.1.3.2.cmml">σ</mi><mi id="S2.I4.i5.p1.5.m5.1.1.1.3.3" mathvariant="normal" xref="S2.I4.i5.p1.5.m5.1.1.1.3.3.cmml">Σ</mi></msup><mo id="S2.I4.i5.p1.5.m5.1.1.1.2" xref="S2.I4.i5.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S2.I4.i5.p1.5.m5.1.1.1.1.1" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S2.I4.i5.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S2.I4.i5.p1.5.m5.1.1.1.1.1.1" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.2" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.2.cmml">X</mi><mi id="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.3" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.I4.i5.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I4.i5.p1.5.m5.1.1.2" xref="S2.I4.i5.p1.5.m5.1.1.2.cmml">⊆</mo><mi id="S2.I4.i5.p1.5.m5.1.1.3" xref="S2.I4.i5.p1.5.m5.1.1.3.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i5.p1.5.m5.1b"><apply id="S2.I4.i5.p1.5.m5.1.1.cmml" xref="S2.I4.i5.p1.5.m5.1.1"><subset id="S2.I4.i5.p1.5.m5.1.1.2.cmml" xref="S2.I4.i5.p1.5.m5.1.1.2"></subset><apply id="S2.I4.i5.p1.5.m5.1.1.1.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1"><times id="S2.I4.i5.p1.5.m5.1.1.1.2.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.2"></times><apply id="S2.I4.i5.p1.5.m5.1.1.1.3.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="S2.I4.i5.p1.5.m5.1.1.1.3.1.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.3">superscript</csymbol><ci id="S2.I4.i5.p1.5.m5.1.1.1.3.2.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.3.2">𝜎</ci><ci id="S2.I4.i5.p1.5.m5.1.1.1.3.3.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.3.3">Σ</ci></apply><apply id="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.2">𝑋</ci><ci id="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S2.I4.i5.p1.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply></apply><ci id="S2.I4.i5.p1.5.m5.1.1.3.cmml" xref="S2.I4.i5.p1.5.m5.1.1.3">𝑌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i5.p1.5.m5.1c">\sigma^{\Sigma}(X_{i})\subseteq Y</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i5.p1.5.m5.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT ( italic_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ⊆ italic_Y</annotation></semantics></math>. For simplicity we denote this subshift by <math alttext="\sigma^{-1}(Y)\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.I4.i5.p1.6.m6.1"><semantics id="S2.I4.i5.p1.6.m6.1a"><mrow id="S2.I4.i5.p1.6.m6.1.2" xref="S2.I4.i5.p1.6.m6.1.2.cmml"><mrow id="S2.I4.i5.p1.6.m6.1.2.2" xref="S2.I4.i5.p1.6.m6.1.2.2.cmml"><msup id="S2.I4.i5.p1.6.m6.1.2.2.2" xref="S2.I4.i5.p1.6.m6.1.2.2.2.cmml"><mi id="S2.I4.i5.p1.6.m6.1.2.2.2.2" xref="S2.I4.i5.p1.6.m6.1.2.2.2.2.cmml">σ</mi><mrow id="S2.I4.i5.p1.6.m6.1.2.2.2.3" xref="S2.I4.i5.p1.6.m6.1.2.2.2.3.cmml"><mo id="S2.I4.i5.p1.6.m6.1.2.2.2.3a" xref="S2.I4.i5.p1.6.m6.1.2.2.2.3.cmml">−</mo><mn id="S2.I4.i5.p1.6.m6.1.2.2.2.3.2" xref="S2.I4.i5.p1.6.m6.1.2.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.I4.i5.p1.6.m6.1.2.2.1" xref="S2.I4.i5.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S2.I4.i5.p1.6.m6.1.2.2.3.2" xref="S2.I4.i5.p1.6.m6.1.2.2.cmml"><mo id="S2.I4.i5.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S2.I4.i5.p1.6.m6.1.2.2.cmml">(</mo><mi id="S2.I4.i5.p1.6.m6.1.1" xref="S2.I4.i5.p1.6.m6.1.1.cmml">Y</mi><mo id="S2.I4.i5.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S2.I4.i5.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.I4.i5.p1.6.m6.1.2.1" xref="S2.I4.i5.p1.6.m6.1.2.1.cmml">⊆</mo><msup id="S2.I4.i5.p1.6.m6.1.2.3" xref="S2.I4.i5.p1.6.m6.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i5.p1.6.m6.1.2.3.2" xref="S2.I4.i5.p1.6.m6.1.2.3.2.cmml">𝒜</mi><mi id="S2.I4.i5.p1.6.m6.1.2.3.3" xref="S2.I4.i5.p1.6.m6.1.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i5.p1.6.m6.1b"><apply id="S2.I4.i5.p1.6.m6.1.2.cmml" xref="S2.I4.i5.p1.6.m6.1.2"><subset id="S2.I4.i5.p1.6.m6.1.2.1.cmml" xref="S2.I4.i5.p1.6.m6.1.2.1"></subset><apply id="S2.I4.i5.p1.6.m6.1.2.2.cmml" xref="S2.I4.i5.p1.6.m6.1.2.2"><times id="S2.I4.i5.p1.6.m6.1.2.2.1.cmml" xref="S2.I4.i5.p1.6.m6.1.2.2.1"></times><apply id="S2.I4.i5.p1.6.m6.1.2.2.2.cmml" xref="S2.I4.i5.p1.6.m6.1.2.2.2"><csymbol cd="ambiguous" id="S2.I4.i5.p1.6.m6.1.2.2.2.1.cmml" xref="S2.I4.i5.p1.6.m6.1.2.2.2">superscript</csymbol><ci id="S2.I4.i5.p1.6.m6.1.2.2.2.2.cmml" xref="S2.I4.i5.p1.6.m6.1.2.2.2.2">𝜎</ci><apply id="S2.I4.i5.p1.6.m6.1.2.2.2.3.cmml" xref="S2.I4.i5.p1.6.m6.1.2.2.2.3"><minus id="S2.I4.i5.p1.6.m6.1.2.2.2.3.1.cmml" xref="S2.I4.i5.p1.6.m6.1.2.2.2.3"></minus><cn id="S2.I4.i5.p1.6.m6.1.2.2.2.3.2.cmml" type="integer" xref="S2.I4.i5.p1.6.m6.1.2.2.2.3.2">1</cn></apply></apply><ci id="S2.I4.i5.p1.6.m6.1.1.cmml" xref="S2.I4.i5.p1.6.m6.1.1">𝑌</ci></apply><apply id="S2.I4.i5.p1.6.m6.1.2.3.cmml" xref="S2.I4.i5.p1.6.m6.1.2.3"><csymbol cd="ambiguous" id="S2.I4.i5.p1.6.m6.1.2.3.1.cmml" xref="S2.I4.i5.p1.6.m6.1.2.3">superscript</csymbol><ci id="S2.I4.i5.p1.6.m6.1.2.3.2.cmml" xref="S2.I4.i5.p1.6.m6.1.2.3.2">𝒜</ci><ci id="S2.I4.i5.p1.6.m6.1.2.3.3.cmml" xref="S2.I4.i5.p1.6.m6.1.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i5.p1.6.m6.1c">\sigma^{-1}(Y)\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i5.p1.6.m6.1d">italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_Y ) ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I4.i6" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(6)</span> <div class="ltx_para" id="S2.I4.i6.p1"> <p class="ltx_p" id="S2.I4.i6.p1.4">By considering for any subshifts <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.I4.i6.p1.1.m1.1"><semantics id="S2.I4.i6.p1.1.m1.1a"><mrow id="S2.I4.i6.p1.1.m1.1.1" xref="S2.I4.i6.p1.1.m1.1.1.cmml"><mi id="S2.I4.i6.p1.1.m1.1.1.2" xref="S2.I4.i6.p1.1.m1.1.1.2.cmml">X</mi><mo id="S2.I4.i6.p1.1.m1.1.1.1" xref="S2.I4.i6.p1.1.m1.1.1.1.cmml">⊆</mo><msup id="S2.I4.i6.p1.1.m1.1.1.3" xref="S2.I4.i6.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.I4.i6.p1.1.m1.1.1.3.2" xref="S2.I4.i6.p1.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S2.I4.i6.p1.1.m1.1.1.3.3" xref="S2.I4.i6.p1.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i6.p1.1.m1.1b"><apply id="S2.I4.i6.p1.1.m1.1.1.cmml" xref="S2.I4.i6.p1.1.m1.1.1"><subset id="S2.I4.i6.p1.1.m1.1.1.1.cmml" xref="S2.I4.i6.p1.1.m1.1.1.1"></subset><ci id="S2.I4.i6.p1.1.m1.1.1.2.cmml" xref="S2.I4.i6.p1.1.m1.1.1.2">𝑋</ci><apply id="S2.I4.i6.p1.1.m1.1.1.3.cmml" xref="S2.I4.i6.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.I4.i6.p1.1.m1.1.1.3.1.cmml" xref="S2.I4.i6.p1.1.m1.1.1.3">superscript</csymbol><ci id="S2.I4.i6.p1.1.m1.1.1.3.2.cmml" xref="S2.I4.i6.p1.1.m1.1.1.3.2">𝒜</ci><ci id="S2.I4.i6.p1.1.m1.1.1.3.3.cmml" xref="S2.I4.i6.p1.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i6.p1.1.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i6.p1.1.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="Y\subseteq\sigma(X)" class="ltx_Math" display="inline" id="S2.I4.i6.p1.2.m2.1"><semantics id="S2.I4.i6.p1.2.m2.1a"><mrow id="S2.I4.i6.p1.2.m2.1.2" xref="S2.I4.i6.p1.2.m2.1.2.cmml"><mi id="S2.I4.i6.p1.2.m2.1.2.2" xref="S2.I4.i6.p1.2.m2.1.2.2.cmml">Y</mi><mo id="S2.I4.i6.p1.2.m2.1.2.1" xref="S2.I4.i6.p1.2.m2.1.2.1.cmml">⊆</mo><mrow id="S2.I4.i6.p1.2.m2.1.2.3" xref="S2.I4.i6.p1.2.m2.1.2.3.cmml"><mi id="S2.I4.i6.p1.2.m2.1.2.3.2" xref="S2.I4.i6.p1.2.m2.1.2.3.2.cmml">σ</mi><mo id="S2.I4.i6.p1.2.m2.1.2.3.1" xref="S2.I4.i6.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S2.I4.i6.p1.2.m2.1.2.3.3.2" xref="S2.I4.i6.p1.2.m2.1.2.3.cmml"><mo id="S2.I4.i6.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S2.I4.i6.p1.2.m2.1.2.3.cmml">(</mo><mi id="S2.I4.i6.p1.2.m2.1.1" xref="S2.I4.i6.p1.2.m2.1.1.cmml">X</mi><mo id="S2.I4.i6.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S2.I4.i6.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i6.p1.2.m2.1b"><apply id="S2.I4.i6.p1.2.m2.1.2.cmml" xref="S2.I4.i6.p1.2.m2.1.2"><subset id="S2.I4.i6.p1.2.m2.1.2.1.cmml" xref="S2.I4.i6.p1.2.m2.1.2.1"></subset><ci id="S2.I4.i6.p1.2.m2.1.2.2.cmml" xref="S2.I4.i6.p1.2.m2.1.2.2">𝑌</ci><apply id="S2.I4.i6.p1.2.m2.1.2.3.cmml" xref="S2.I4.i6.p1.2.m2.1.2.3"><times id="S2.I4.i6.p1.2.m2.1.2.3.1.cmml" xref="S2.I4.i6.p1.2.m2.1.2.3.1"></times><ci id="S2.I4.i6.p1.2.m2.1.2.3.2.cmml" xref="S2.I4.i6.p1.2.m2.1.2.3.2">𝜎</ci><ci id="S2.I4.i6.p1.2.m2.1.1.cmml" xref="S2.I4.i6.p1.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i6.p1.2.m2.1c">Y\subseteq\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i6.p1.2.m2.1d">italic_Y ⊆ italic_σ ( italic_X )</annotation></semantics></math> the subshift <math alttext="X\cap\sigma^{-1}(Y)" class="ltx_Math" display="inline" id="S2.I4.i6.p1.3.m3.1"><semantics id="S2.I4.i6.p1.3.m3.1a"><mrow id="S2.I4.i6.p1.3.m3.1.2" xref="S2.I4.i6.p1.3.m3.1.2.cmml"><mi id="S2.I4.i6.p1.3.m3.1.2.2" xref="S2.I4.i6.p1.3.m3.1.2.2.cmml">X</mi><mo id="S2.I4.i6.p1.3.m3.1.2.1" xref="S2.I4.i6.p1.3.m3.1.2.1.cmml">∩</mo><mrow id="S2.I4.i6.p1.3.m3.1.2.3" xref="S2.I4.i6.p1.3.m3.1.2.3.cmml"><msup id="S2.I4.i6.p1.3.m3.1.2.3.2" xref="S2.I4.i6.p1.3.m3.1.2.3.2.cmml"><mi id="S2.I4.i6.p1.3.m3.1.2.3.2.2" xref="S2.I4.i6.p1.3.m3.1.2.3.2.2.cmml">σ</mi><mrow id="S2.I4.i6.p1.3.m3.1.2.3.2.3" xref="S2.I4.i6.p1.3.m3.1.2.3.2.3.cmml"><mo id="S2.I4.i6.p1.3.m3.1.2.3.2.3a" xref="S2.I4.i6.p1.3.m3.1.2.3.2.3.cmml">−</mo><mn id="S2.I4.i6.p1.3.m3.1.2.3.2.3.2" xref="S2.I4.i6.p1.3.m3.1.2.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.I4.i6.p1.3.m3.1.2.3.1" xref="S2.I4.i6.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S2.I4.i6.p1.3.m3.1.2.3.3.2" xref="S2.I4.i6.p1.3.m3.1.2.3.cmml"><mo id="S2.I4.i6.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="S2.I4.i6.p1.3.m3.1.2.3.cmml">(</mo><mi id="S2.I4.i6.p1.3.m3.1.1" xref="S2.I4.i6.p1.3.m3.1.1.cmml">Y</mi><mo id="S2.I4.i6.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="S2.I4.i6.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i6.p1.3.m3.1b"><apply id="S2.I4.i6.p1.3.m3.1.2.cmml" xref="S2.I4.i6.p1.3.m3.1.2"><intersect id="S2.I4.i6.p1.3.m3.1.2.1.cmml" xref="S2.I4.i6.p1.3.m3.1.2.1"></intersect><ci id="S2.I4.i6.p1.3.m3.1.2.2.cmml" xref="S2.I4.i6.p1.3.m3.1.2.2">𝑋</ci><apply id="S2.I4.i6.p1.3.m3.1.2.3.cmml" xref="S2.I4.i6.p1.3.m3.1.2.3"><times id="S2.I4.i6.p1.3.m3.1.2.3.1.cmml" xref="S2.I4.i6.p1.3.m3.1.2.3.1"></times><apply id="S2.I4.i6.p1.3.m3.1.2.3.2.cmml" xref="S2.I4.i6.p1.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S2.I4.i6.p1.3.m3.1.2.3.2.1.cmml" xref="S2.I4.i6.p1.3.m3.1.2.3.2">superscript</csymbol><ci id="S2.I4.i6.p1.3.m3.1.2.3.2.2.cmml" xref="S2.I4.i6.p1.3.m3.1.2.3.2.2">𝜎</ci><apply id="S2.I4.i6.p1.3.m3.1.2.3.2.3.cmml" xref="S2.I4.i6.p1.3.m3.1.2.3.2.3"><minus id="S2.I4.i6.p1.3.m3.1.2.3.2.3.1.cmml" xref="S2.I4.i6.p1.3.m3.1.2.3.2.3"></minus><cn id="S2.I4.i6.p1.3.m3.1.2.3.2.3.2.cmml" type="integer" xref="S2.I4.i6.p1.3.m3.1.2.3.2.3.2">1</cn></apply></apply><ci id="S2.I4.i6.p1.3.m3.1.1.cmml" xref="S2.I4.i6.p1.3.m3.1.1">𝑌</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i6.p1.3.m3.1c">X\cap\sigma^{-1}(Y)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i6.p1.3.m3.1d">italic_X ∩ italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_Y )</annotation></semantics></math> we observe that the above map <math alttext="\sigma^{\Sigma}_{X}:\Sigma(X)\to\Sigma(\sigma(X))" class="ltx_Math" display="inline" id="S2.I4.i6.p1.4.m4.3"><semantics id="S2.I4.i6.p1.4.m4.3a"><mrow id="S2.I4.i6.p1.4.m4.3.3" xref="S2.I4.i6.p1.4.m4.3.3.cmml"><msubsup id="S2.I4.i6.p1.4.m4.3.3.3" xref="S2.I4.i6.p1.4.m4.3.3.3.cmml"><mi id="S2.I4.i6.p1.4.m4.3.3.3.2.2" xref="S2.I4.i6.p1.4.m4.3.3.3.2.2.cmml">σ</mi><mi id="S2.I4.i6.p1.4.m4.3.3.3.3" xref="S2.I4.i6.p1.4.m4.3.3.3.3.cmml">X</mi><mi id="S2.I4.i6.p1.4.m4.3.3.3.2.3" mathvariant="normal" xref="S2.I4.i6.p1.4.m4.3.3.3.2.3.cmml">Σ</mi></msubsup><mo id="S2.I4.i6.p1.4.m4.3.3.2" lspace="0.278em" rspace="0.278em" xref="S2.I4.i6.p1.4.m4.3.3.2.cmml">:</mo><mrow id="S2.I4.i6.p1.4.m4.3.3.1" xref="S2.I4.i6.p1.4.m4.3.3.1.cmml"><mrow id="S2.I4.i6.p1.4.m4.3.3.1.3" xref="S2.I4.i6.p1.4.m4.3.3.1.3.cmml"><mi id="S2.I4.i6.p1.4.m4.3.3.1.3.2" mathvariant="normal" xref="S2.I4.i6.p1.4.m4.3.3.1.3.2.cmml">Σ</mi><mo id="S2.I4.i6.p1.4.m4.3.3.1.3.1" xref="S2.I4.i6.p1.4.m4.3.3.1.3.1.cmml">⁢</mo><mrow id="S2.I4.i6.p1.4.m4.3.3.1.3.3.2" xref="S2.I4.i6.p1.4.m4.3.3.1.3.cmml"><mo id="S2.I4.i6.p1.4.m4.3.3.1.3.3.2.1" stretchy="false" xref="S2.I4.i6.p1.4.m4.3.3.1.3.cmml">(</mo><mi id="S2.I4.i6.p1.4.m4.1.1" xref="S2.I4.i6.p1.4.m4.1.1.cmml">X</mi><mo id="S2.I4.i6.p1.4.m4.3.3.1.3.3.2.2" stretchy="false" xref="S2.I4.i6.p1.4.m4.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="S2.I4.i6.p1.4.m4.3.3.1.2" stretchy="false" xref="S2.I4.i6.p1.4.m4.3.3.1.2.cmml">→</mo><mrow id="S2.I4.i6.p1.4.m4.3.3.1.1" xref="S2.I4.i6.p1.4.m4.3.3.1.1.cmml"><mi id="S2.I4.i6.p1.4.m4.3.3.1.1.3" mathvariant="normal" xref="S2.I4.i6.p1.4.m4.3.3.1.1.3.cmml">Σ</mi><mo id="S2.I4.i6.p1.4.m4.3.3.1.1.2" xref="S2.I4.i6.p1.4.m4.3.3.1.1.2.cmml">⁢</mo><mrow id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.cmml"><mo id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.2" stretchy="false" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.cmml"><mi id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.2" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.2.cmml">σ</mi><mo id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.1" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.3.2" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.cmml"><mo id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.cmml">(</mo><mi id="S2.I4.i6.p1.4.m4.2.2" xref="S2.I4.i6.p1.4.m4.2.2.cmml">X</mi><mo id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.3" stretchy="false" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i6.p1.4.m4.3b"><apply id="S2.I4.i6.p1.4.m4.3.3.cmml" xref="S2.I4.i6.p1.4.m4.3.3"><ci id="S2.I4.i6.p1.4.m4.3.3.2.cmml" xref="S2.I4.i6.p1.4.m4.3.3.2">:</ci><apply id="S2.I4.i6.p1.4.m4.3.3.3.cmml" xref="S2.I4.i6.p1.4.m4.3.3.3"><csymbol cd="ambiguous" id="S2.I4.i6.p1.4.m4.3.3.3.1.cmml" xref="S2.I4.i6.p1.4.m4.3.3.3">subscript</csymbol><apply id="S2.I4.i6.p1.4.m4.3.3.3.2.cmml" xref="S2.I4.i6.p1.4.m4.3.3.3"><csymbol cd="ambiguous" id="S2.I4.i6.p1.4.m4.3.3.3.2.1.cmml" 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xref="S2.I4.i6.p1.4.m4.3.3.1.1.3">Σ</ci><apply id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1"><times id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.1.cmml" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.1"></times><ci id="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.2.cmml" xref="S2.I4.i6.p1.4.m4.3.3.1.1.1.1.1.2">𝜎</ci><ci id="S2.I4.i6.p1.4.m4.2.2.cmml" xref="S2.I4.i6.p1.4.m4.2.2">𝑋</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i6.p1.4.m4.3c">\sigma^{\Sigma}_{X}:\Sigma(X)\to\Sigma(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i6.p1.4.m4.3d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT : roman_Σ ( italic_X ) → roman_Σ ( italic_σ ( italic_X ) )</annotation></semantics></math> is surjective. It also preserves the partial order given by the inclusion of subshifts.</p> </div> </li> <li class="ltx_item" id="S2.I4.i7" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(7)</span> <div class="ltx_para" id="S2.I4.i7.p1"> <p class="ltx_p" id="S2.I4.i7.p1.2">In particular, if <math alttext="X" class="ltx_Math" display="inline" id="S2.I4.i7.p1.1.m1.1"><semantics id="S2.I4.i7.p1.1.m1.1a"><mi id="S2.I4.i7.p1.1.m1.1.1" xref="S2.I4.i7.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.I4.i7.p1.1.m1.1b"><ci id="S2.I4.i7.p1.1.m1.1.1.cmml" xref="S2.I4.i7.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i7.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i7.p1.1.m1.1d">italic_X</annotation></semantics></math> is minimal, then so is <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S2.I4.i7.p1.2.m2.1"><semantics id="S2.I4.i7.p1.2.m2.1a"><mrow id="S2.I4.i7.p1.2.m2.1.2" xref="S2.I4.i7.p1.2.m2.1.2.cmml"><mi id="S2.I4.i7.p1.2.m2.1.2.2" xref="S2.I4.i7.p1.2.m2.1.2.2.cmml">σ</mi><mo id="S2.I4.i7.p1.2.m2.1.2.1" xref="S2.I4.i7.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.I4.i7.p1.2.m2.1.2.3.2" xref="S2.I4.i7.p1.2.m2.1.2.cmml"><mo id="S2.I4.i7.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.I4.i7.p1.2.m2.1.2.cmml">(</mo><mi id="S2.I4.i7.p1.2.m2.1.1" xref="S2.I4.i7.p1.2.m2.1.1.cmml">X</mi><mo id="S2.I4.i7.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.I4.i7.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I4.i7.p1.2.m2.1b"><apply id="S2.I4.i7.p1.2.m2.1.2.cmml" xref="S2.I4.i7.p1.2.m2.1.2"><times id="S2.I4.i7.p1.2.m2.1.2.1.cmml" xref="S2.I4.i7.p1.2.m2.1.2.1"></times><ci id="S2.I4.i7.p1.2.m2.1.2.2.cmml" xref="S2.I4.i7.p1.2.m2.1.2.2">𝜎</ci><ci id="S2.I4.i7.p1.2.m2.1.1.cmml" xref="S2.I4.i7.p1.2.m2.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i7.p1.2.m2.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i7.p1.2.m2.1d">italic_σ ( italic_X )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I4.i8" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(8)</span> <div class="ltx_para" id="S2.I4.i8.p1"> <p class="ltx_p" id="S2.I4.i8.p1.1">The map <math alttext="\sigma^{\Sigma}" class="ltx_Math" display="inline" id="S2.I4.i8.p1.1.m1.1"><semantics id="S2.I4.i8.p1.1.m1.1a"><msup id="S2.I4.i8.p1.1.m1.1.1" xref="S2.I4.i8.p1.1.m1.1.1.cmml"><mi id="S2.I4.i8.p1.1.m1.1.1.2" xref="S2.I4.i8.p1.1.m1.1.1.2.cmml">σ</mi><mi id="S2.I4.i8.p1.1.m1.1.1.3" mathvariant="normal" xref="S2.I4.i8.p1.1.m1.1.1.3.cmml">Σ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.I4.i8.p1.1.m1.1b"><apply id="S2.I4.i8.p1.1.m1.1.1.cmml" xref="S2.I4.i8.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.I4.i8.p1.1.m1.1.1.1.cmml" xref="S2.I4.i8.p1.1.m1.1.1">superscript</csymbol><ci id="S2.I4.i8.p1.1.m1.1.1.2.cmml" xref="S2.I4.i8.p1.1.m1.1.1.2">𝜎</ci><ci id="S2.I4.i8.p1.1.m1.1.1.3.cmml" xref="S2.I4.i8.p1.1.m1.1.1.3">Σ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I4.i8.p1.1.m1.1c">\sigma^{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.I4.i8.p1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT</annotation></semantics></math> is continuous with respect to the canonical topology on the subshift spaces.</p> </div> </li> </ol> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S2.Thmthm6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm6.1.1.1">Remark 2.6</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm6.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm6.p1"> <p class="ltx_p" id="S2.Thmthm6.p1.7">Although not central to the topics evoked in this paper, for completeness we would like to state here how a non-erasing morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.Thmthm6.p1.1.m1.1"><semantics id="S2.Thmthm6.p1.1.m1.1a"><mrow id="S2.Thmthm6.p1.1.m1.1.1" xref="S2.Thmthm6.p1.1.m1.1.1.cmml"><mi id="S2.Thmthm6.p1.1.m1.1.1.2" xref="S2.Thmthm6.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S2.Thmthm6.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm6.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.Thmthm6.p1.1.m1.1.1.3" xref="S2.Thmthm6.p1.1.m1.1.1.3.cmml"><msup id="S2.Thmthm6.p1.1.m1.1.1.3.2" xref="S2.Thmthm6.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm6.p1.1.m1.1.1.3.2.2" xref="S2.Thmthm6.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.Thmthm6.p1.1.m1.1.1.3.2.3" xref="S2.Thmthm6.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm6.p1.1.m1.1.1.3.1" stretchy="false" xref="S2.Thmthm6.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.Thmthm6.p1.1.m1.1.1.3.3" xref="S2.Thmthm6.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm6.p1.1.m1.1.1.3.3.2" xref="S2.Thmthm6.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.Thmthm6.p1.1.m1.1.1.3.3.3" xref="S2.Thmthm6.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm6.p1.1.m1.1b"><apply id="S2.Thmthm6.p1.1.m1.1.1.cmml" xref="S2.Thmthm6.p1.1.m1.1.1"><ci id="S2.Thmthm6.p1.1.m1.1.1.1.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.1">:</ci><ci id="S2.Thmthm6.p1.1.m1.1.1.2.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.2">𝜎</ci><apply id="S2.Thmthm6.p1.1.m1.1.1.3.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3"><ci id="S2.Thmthm6.p1.1.m1.1.1.3.1.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.1">→</ci><apply id="S2.Thmthm6.p1.1.m1.1.1.3.2.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmthm6.p1.1.m1.1.1.3.2.1.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.Thmthm6.p1.1.m1.1.1.3.2.2.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S2.Thmthm6.p1.1.m1.1.1.3.2.3.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S2.Thmthm6.p1.1.m1.1.1.3.3.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmthm6.p1.1.m1.1.1.3.3.1.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.Thmthm6.p1.1.m1.1.1.3.3.2.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S2.Thmthm6.p1.1.m1.1.1.3.3.3.cmml" xref="S2.Thmthm6.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm6.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm6.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> acts on the complexity and the topological entropy of a subshift. Recall that for any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm6.p1.2.m2.1"><semantics id="S2.Thmthm6.p1.2.m2.1a"><mrow id="S2.Thmthm6.p1.2.m2.1.1" xref="S2.Thmthm6.p1.2.m2.1.1.cmml"><mi id="S2.Thmthm6.p1.2.m2.1.1.2" xref="S2.Thmthm6.p1.2.m2.1.1.2.cmml">X</mi><mo id="S2.Thmthm6.p1.2.m2.1.1.1" xref="S2.Thmthm6.p1.2.m2.1.1.1.cmml">⊆</mo><msup id="S2.Thmthm6.p1.2.m2.1.1.3" xref="S2.Thmthm6.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm6.p1.2.m2.1.1.3.2" xref="S2.Thmthm6.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S2.Thmthm6.p1.2.m2.1.1.3.3" xref="S2.Thmthm6.p1.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm6.p1.2.m2.1b"><apply id="S2.Thmthm6.p1.2.m2.1.1.cmml" xref="S2.Thmthm6.p1.2.m2.1.1"><subset id="S2.Thmthm6.p1.2.m2.1.1.1.cmml" xref="S2.Thmthm6.p1.2.m2.1.1.1"></subset><ci id="S2.Thmthm6.p1.2.m2.1.1.2.cmml" xref="S2.Thmthm6.p1.2.m2.1.1.2">𝑋</ci><apply id="S2.Thmthm6.p1.2.m2.1.1.3.cmml" xref="S2.Thmthm6.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm6.p1.2.m2.1.1.3.1.cmml" xref="S2.Thmthm6.p1.2.m2.1.1.3">superscript</csymbol><ci id="S2.Thmthm6.p1.2.m2.1.1.3.2.cmml" xref="S2.Thmthm6.p1.2.m2.1.1.3.2">𝒜</ci><ci id="S2.Thmthm6.p1.2.m2.1.1.3.3.cmml" xref="S2.Thmthm6.p1.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm6.p1.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm6.p1.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> the <span class="ltx_text ltx_font_italic" id="S2.Thmthm6.p1.7.1">complexity</span> (also called <span class="ltx_text ltx_font_italic" id="S2.Thmthm6.p1.7.2">word complexity</span>) is given by the function <math alttext="p_{X}:\mathbb{N}\to\mathbb{N}" class="ltx_Math" display="inline" id="S2.Thmthm6.p1.3.m3.1"><semantics id="S2.Thmthm6.p1.3.m3.1a"><mrow id="S2.Thmthm6.p1.3.m3.1.1" xref="S2.Thmthm6.p1.3.m3.1.1.cmml"><msub id="S2.Thmthm6.p1.3.m3.1.1.2" xref="S2.Thmthm6.p1.3.m3.1.1.2.cmml"><mi id="S2.Thmthm6.p1.3.m3.1.1.2.2" xref="S2.Thmthm6.p1.3.m3.1.1.2.2.cmml">p</mi><mi id="S2.Thmthm6.p1.3.m3.1.1.2.3" xref="S2.Thmthm6.p1.3.m3.1.1.2.3.cmml">X</mi></msub><mo id="S2.Thmthm6.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm6.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S2.Thmthm6.p1.3.m3.1.1.3" xref="S2.Thmthm6.p1.3.m3.1.1.3.cmml"><mi id="S2.Thmthm6.p1.3.m3.1.1.3.2" xref="S2.Thmthm6.p1.3.m3.1.1.3.2.cmml">ℕ</mi><mo id="S2.Thmthm6.p1.3.m3.1.1.3.1" stretchy="false" xref="S2.Thmthm6.p1.3.m3.1.1.3.1.cmml">→</mo><mi id="S2.Thmthm6.p1.3.m3.1.1.3.3" xref="S2.Thmthm6.p1.3.m3.1.1.3.3.cmml">ℕ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm6.p1.3.m3.1b"><apply id="S2.Thmthm6.p1.3.m3.1.1.cmml" xref="S2.Thmthm6.p1.3.m3.1.1"><ci id="S2.Thmthm6.p1.3.m3.1.1.1.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.1">:</ci><apply id="S2.Thmthm6.p1.3.m3.1.1.2.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.Thmthm6.p1.3.m3.1.1.2.1.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.2">subscript</csymbol><ci id="S2.Thmthm6.p1.3.m3.1.1.2.2.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.2.2">𝑝</ci><ci id="S2.Thmthm6.p1.3.m3.1.1.2.3.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.2.3">𝑋</ci></apply><apply id="S2.Thmthm6.p1.3.m3.1.1.3.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.3"><ci id="S2.Thmthm6.p1.3.m3.1.1.3.1.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.3.1">→</ci><ci id="S2.Thmthm6.p1.3.m3.1.1.3.2.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.3.2">ℕ</ci><ci id="S2.Thmthm6.p1.3.m3.1.1.3.3.cmml" xref="S2.Thmthm6.p1.3.m3.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm6.p1.3.m3.1c">p_{X}:\mathbb{N}\to\mathbb{N}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm6.p1.3.m3.1d">italic_p start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT : blackboard_N → blackboard_N</annotation></semantics></math>, defined via <math alttext="p_{X}(n)=\mbox{card}\{w\in\cal L(X)\mid|w|=n\}" class="ltx_Math" display="inline" id="S2.Thmthm6.p1.4.m4.5"><semantics id="S2.Thmthm6.p1.4.m4.5a"><mrow id="S2.Thmthm6.p1.4.m4.5.5" xref="S2.Thmthm6.p1.4.m4.5.5.cmml"><mrow id="S2.Thmthm6.p1.4.m4.5.5.4" xref="S2.Thmthm6.p1.4.m4.5.5.4.cmml"><msub id="S2.Thmthm6.p1.4.m4.5.5.4.2" xref="S2.Thmthm6.p1.4.m4.5.5.4.2.cmml"><mi id="S2.Thmthm6.p1.4.m4.5.5.4.2.2" xref="S2.Thmthm6.p1.4.m4.5.5.4.2.2.cmml">p</mi><mi id="S2.Thmthm6.p1.4.m4.5.5.4.2.3" xref="S2.Thmthm6.p1.4.m4.5.5.4.2.3.cmml">X</mi></msub><mo id="S2.Thmthm6.p1.4.m4.5.5.4.1" xref="S2.Thmthm6.p1.4.m4.5.5.4.1.cmml">⁢</mo><mrow id="S2.Thmthm6.p1.4.m4.5.5.4.3.2" xref="S2.Thmthm6.p1.4.m4.5.5.4.cmml"><mo id="S2.Thmthm6.p1.4.m4.5.5.4.3.2.1" stretchy="false" xref="S2.Thmthm6.p1.4.m4.5.5.4.cmml">(</mo><mi id="S2.Thmthm6.p1.4.m4.1.1" xref="S2.Thmthm6.p1.4.m4.1.1.cmml">n</mi><mo id="S2.Thmthm6.p1.4.m4.5.5.4.3.2.2" stretchy="false" xref="S2.Thmthm6.p1.4.m4.5.5.4.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm6.p1.4.m4.5.5.3" xref="S2.Thmthm6.p1.4.m4.5.5.3.cmml">=</mo><mrow id="S2.Thmthm6.p1.4.m4.5.5.2" xref="S2.Thmthm6.p1.4.m4.5.5.2.cmml"><mtext id="S2.Thmthm6.p1.4.m4.5.5.2.4" xref="S2.Thmthm6.p1.4.m4.5.5.2.4a.cmml">card</mtext><mo id="S2.Thmthm6.p1.4.m4.5.5.2.3" xref="S2.Thmthm6.p1.4.m4.5.5.2.3.cmml">⁢</mo><mrow id="S2.Thmthm6.p1.4.m4.5.5.2.2.2" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.3.cmml"><mo id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.3" stretchy="false" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.3.1.cmml">{</mo><mrow id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.cmml"><mi id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.2" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.2.cmml">w</mi><mo id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.1" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.1.cmml">∈</mo><mrow id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.2" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.2.cmml">ℒ</mi><mo id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.1" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.3.2" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.cmml"><mo id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.3.2.1" stretchy="false" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm6.p1.4.m4.2.2" xref="S2.Thmthm6.p1.4.m4.2.2.cmml">𝒳</mi><mo id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.3.2.2" stretchy="false" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.4" lspace="0em" rspace="0em" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.3.1.cmml">∣</mo><mrow id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.cmml"><mrow id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.2" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.1.cmml"><mo id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.2.1" stretchy="false" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm6.p1.4.m4.3.3" xref="S2.Thmthm6.p1.4.m4.3.3.cmml">𝓌</mi><mo id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.2.2" stretchy="false" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.1.1.cmml">|</mo></mrow><mo id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.1" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.1.cmml">=</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.3" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.3.cmml">𝓃</mi></mrow><mo id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.5" stretchy="false" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm6.p1.4.m4.5b"><apply id="S2.Thmthm6.p1.4.m4.5.5.cmml" xref="S2.Thmthm6.p1.4.m4.5.5"><eq id="S2.Thmthm6.p1.4.m4.5.5.3.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.3"></eq><apply id="S2.Thmthm6.p1.4.m4.5.5.4.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.4"><times id="S2.Thmthm6.p1.4.m4.5.5.4.1.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.4.1"></times><apply id="S2.Thmthm6.p1.4.m4.5.5.4.2.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.4.2"><csymbol cd="ambiguous" id="S2.Thmthm6.p1.4.m4.5.5.4.2.1.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.4.2">subscript</csymbol><ci id="S2.Thmthm6.p1.4.m4.5.5.4.2.2.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.4.2.2">𝑝</ci><ci id="S2.Thmthm6.p1.4.m4.5.5.4.2.3.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.4.2.3">𝑋</ci></apply><ci id="S2.Thmthm6.p1.4.m4.1.1.cmml" xref="S2.Thmthm6.p1.4.m4.1.1">𝑛</ci></apply><apply id="S2.Thmthm6.p1.4.m4.5.5.2.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2"><times id="S2.Thmthm6.p1.4.m4.5.5.2.3.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.3"></times><ci id="S2.Thmthm6.p1.4.m4.5.5.2.4a.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.4"><mtext id="S2.Thmthm6.p1.4.m4.5.5.2.4.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.4">card</mtext></ci><apply id="S2.Thmthm6.p1.4.m4.5.5.2.2.3.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2"><csymbol cd="latexml" id="S2.Thmthm6.p1.4.m4.5.5.2.2.3.1.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.3">conditional-set</csymbol><apply id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.cmml" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1"><in id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.1.cmml" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.1"></in><ci id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.2.cmml" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.2">𝑤</ci><apply id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.cmml" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3"><times id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.1.cmml" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.1"></times><ci id="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.2.cmml" xref="S2.Thmthm6.p1.4.m4.4.4.1.1.1.1.3.2">ℒ</ci><ci id="S2.Thmthm6.p1.4.m4.2.2.cmml" xref="S2.Thmthm6.p1.4.m4.2.2">𝒳</ci></apply></apply><apply id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2"><eq id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.1.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.1"></eq><apply id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.1.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.2"><abs id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.1.1.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.2.2.1"></abs><ci id="S2.Thmthm6.p1.4.m4.3.3.cmml" xref="S2.Thmthm6.p1.4.m4.3.3">𝓌</ci></apply><ci id="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.3.cmml" xref="S2.Thmthm6.p1.4.m4.5.5.2.2.2.2.3">𝓃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm6.p1.4.m4.5c">p_{X}(n)=\mbox{card}\{w\in\cal L(X)\mid|w|=n\}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm6.p1.4.m4.5d">italic_p start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_n ) = card { italic_w ∈ caligraphic_L ( caligraphic_X ) ∣ | caligraphic_w | = caligraphic_n }</annotation></semantics></math>. The <span class="ltx_text ltx_font_italic" id="S2.Thmthm6.p1.7.3">topological entropy</span> of <math alttext="X" class="ltx_Math" display="inline" id="S2.Thmthm6.p1.5.m5.1"><semantics id="S2.Thmthm6.p1.5.m5.1a"><mi id="S2.Thmthm6.p1.5.m5.1.1" xref="S2.Thmthm6.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm6.p1.5.m5.1b"><ci id="S2.Thmthm6.p1.5.m5.1.1.cmml" xref="S2.Thmthm6.p1.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm6.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm6.p1.5.m5.1d">italic_X</annotation></semantics></math> is defined as <math alttext="h_{X}=\underset{n\to\infty}{\lim}\frac{\log\,p_{X}(n)}{n}" class="ltx_Math" display="inline" id="S2.Thmthm6.p1.6.m6.1"><semantics id="S2.Thmthm6.p1.6.m6.1a"><mrow id="S2.Thmthm6.p1.6.m6.1.2" xref="S2.Thmthm6.p1.6.m6.1.2.cmml"><msub id="S2.Thmthm6.p1.6.m6.1.2.2" xref="S2.Thmthm6.p1.6.m6.1.2.2.cmml"><mi id="S2.Thmthm6.p1.6.m6.1.2.2.2" xref="S2.Thmthm6.p1.6.m6.1.2.2.2.cmml">h</mi><mi id="S2.Thmthm6.p1.6.m6.1.2.2.3" xref="S2.Thmthm6.p1.6.m6.1.2.2.3.cmml">X</mi></msub><mo id="S2.Thmthm6.p1.6.m6.1.2.1" rspace="0.1389em" xref="S2.Thmthm6.p1.6.m6.1.2.1.cmml">=</mo><mrow id="S2.Thmthm6.p1.6.m6.1.2.3" xref="S2.Thmthm6.p1.6.m6.1.2.3.cmml"><munder accentunder="true" id="S2.Thmthm6.p1.6.m6.1.2.3.2" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.cmml"><mo id="S2.Thmthm6.p1.6.m6.1.2.3.2.2" lspace="0.1389em" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.2.cmml">lim</mo><mrow id="S2.Thmthm6.p1.6.m6.1.2.3.2.1" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.1.cmml"><mi id="S2.Thmthm6.p1.6.m6.1.2.3.2.1.2" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.1.2.cmml">n</mi><mo id="S2.Thmthm6.p1.6.m6.1.2.3.2.1.1" stretchy="false" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.1.1.cmml">→</mo><mi id="S2.Thmthm6.p1.6.m6.1.2.3.2.1.3" mathvariant="normal" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.1.3.cmml">∞</mi></mrow></munder><mo id="S2.Thmthm6.p1.6.m6.1.2.3.1" lspace="0.167em" xref="S2.Thmthm6.p1.6.m6.1.2.3.1.cmml">⁢</mo><mfrac id="S2.Thmthm6.p1.6.m6.1.1" xref="S2.Thmthm6.p1.6.m6.1.1.cmml"><mrow id="S2.Thmthm6.p1.6.m6.1.1.1" xref="S2.Thmthm6.p1.6.m6.1.1.1.cmml"><mrow id="S2.Thmthm6.p1.6.m6.1.1.1.3" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.cmml"><mi id="S2.Thmthm6.p1.6.m6.1.1.1.3.1" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.1.cmml">log</mi><mo id="S2.Thmthm6.p1.6.m6.1.1.1.3a" lspace="0.337em" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.cmml">⁡</mo><msub id="S2.Thmthm6.p1.6.m6.1.1.1.3.2" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.2.cmml"><mi id="S2.Thmthm6.p1.6.m6.1.1.1.3.2.2" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.2.2.cmml">p</mi><mi id="S2.Thmthm6.p1.6.m6.1.1.1.3.2.3" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.2.3.cmml">X</mi></msub></mrow><mo id="S2.Thmthm6.p1.6.m6.1.1.1.2" xref="S2.Thmthm6.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm6.p1.6.m6.1.1.1.4.2" xref="S2.Thmthm6.p1.6.m6.1.1.1.cmml"><mo id="S2.Thmthm6.p1.6.m6.1.1.1.4.2.1" stretchy="false" xref="S2.Thmthm6.p1.6.m6.1.1.1.cmml">(</mo><mi id="S2.Thmthm6.p1.6.m6.1.1.1.1" xref="S2.Thmthm6.p1.6.m6.1.1.1.1.cmml">n</mi><mo id="S2.Thmthm6.p1.6.m6.1.1.1.4.2.2" stretchy="false" xref="S2.Thmthm6.p1.6.m6.1.1.1.cmml">)</mo></mrow></mrow><mi id="S2.Thmthm6.p1.6.m6.1.1.3" xref="S2.Thmthm6.p1.6.m6.1.1.3.cmml">n</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm6.p1.6.m6.1b"><apply id="S2.Thmthm6.p1.6.m6.1.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.2"><eq id="S2.Thmthm6.p1.6.m6.1.2.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.1"></eq><apply id="S2.Thmthm6.p1.6.m6.1.2.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S2.Thmthm6.p1.6.m6.1.2.2.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.2">subscript</csymbol><ci id="S2.Thmthm6.p1.6.m6.1.2.2.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.2.2">ℎ</ci><ci id="S2.Thmthm6.p1.6.m6.1.2.2.3.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.2.3">𝑋</ci></apply><apply id="S2.Thmthm6.p1.6.m6.1.2.3.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.3"><times id="S2.Thmthm6.p1.6.m6.1.2.3.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.3.1"></times><apply id="S2.Thmthm6.p1.6.m6.1.2.3.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.3.2"><apply id="S2.Thmthm6.p1.6.m6.1.2.3.2.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.1"><ci id="S2.Thmthm6.p1.6.m6.1.2.3.2.1.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.1.1">→</ci><ci id="S2.Thmthm6.p1.6.m6.1.2.3.2.1.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.1.2">𝑛</ci><infinity id="S2.Thmthm6.p1.6.m6.1.2.3.2.1.3.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.1.3"></infinity></apply><limit id="S2.Thmthm6.p1.6.m6.1.2.3.2.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.2.3.2.2"></limit></apply><apply id="S2.Thmthm6.p1.6.m6.1.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.1"><divide id="S2.Thmthm6.p1.6.m6.1.1.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.1"></divide><apply id="S2.Thmthm6.p1.6.m6.1.1.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1"><times id="S2.Thmthm6.p1.6.m6.1.1.1.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1.2"></times><apply id="S2.Thmthm6.p1.6.m6.1.1.1.3.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1.3"><log id="S2.Thmthm6.p1.6.m6.1.1.1.3.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.1"></log><apply id="S2.Thmthm6.p1.6.m6.1.1.1.3.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmthm6.p1.6.m6.1.1.1.3.2.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.2">subscript</csymbol><ci id="S2.Thmthm6.p1.6.m6.1.1.1.3.2.2.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.2.2">𝑝</ci><ci id="S2.Thmthm6.p1.6.m6.1.1.1.3.2.3.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1.3.2.3">𝑋</ci></apply></apply><ci id="S2.Thmthm6.p1.6.m6.1.1.1.1.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.1.1">𝑛</ci></apply><ci id="S2.Thmthm6.p1.6.m6.1.1.3.cmml" xref="S2.Thmthm6.p1.6.m6.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm6.p1.6.m6.1c">h_{X}=\underset{n\to\infty}{\lim}\frac{\log\,p_{X}(n)}{n}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm6.p1.6.m6.1d">italic_h start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = start_UNDERACCENT italic_n → ∞ end_UNDERACCENT start_ARG roman_lim end_ARG divide start_ARG roman_log italic_p start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_n ) end_ARG start_ARG italic_n end_ARG</annotation></semantics></math>. A fairly standard exercise (see for instance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib13" title="">13</a>]</cite>, Lemma 2.1, or <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib9" title="">9</a>]</cite>, Lemma 2.9 and Proposition 2.11) shows, for <math alttext="Y:=\sigma(X)\," class="ltx_Math" display="inline" id="S2.Thmthm6.p1.7.m7.1"><semantics id="S2.Thmthm6.p1.7.m7.1a"><mrow id="S2.Thmthm6.p1.7.m7.1.2" xref="S2.Thmthm6.p1.7.m7.1.2.cmml"><mi id="S2.Thmthm6.p1.7.m7.1.2.2" xref="S2.Thmthm6.p1.7.m7.1.2.2.cmml">Y</mi><mo id="S2.Thmthm6.p1.7.m7.1.2.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm6.p1.7.m7.1.2.1.cmml">:=</mo><mrow id="S2.Thmthm6.p1.7.m7.1.2.3" xref="S2.Thmthm6.p1.7.m7.1.2.3.cmml"><mi id="S2.Thmthm6.p1.7.m7.1.2.3.2" xref="S2.Thmthm6.p1.7.m7.1.2.3.2.cmml">σ</mi><mo id="S2.Thmthm6.p1.7.m7.1.2.3.1" xref="S2.Thmthm6.p1.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S2.Thmthm6.p1.7.m7.1.2.3.3.2" xref="S2.Thmthm6.p1.7.m7.1.2.3.cmml"><mo id="S2.Thmthm6.p1.7.m7.1.2.3.3.2.1" stretchy="false" xref="S2.Thmthm6.p1.7.m7.1.2.3.cmml">(</mo><mi id="S2.Thmthm6.p1.7.m7.1.1" xref="S2.Thmthm6.p1.7.m7.1.1.cmml">X</mi><mo id="S2.Thmthm6.p1.7.m7.1.2.3.3.2.2" stretchy="false" xref="S2.Thmthm6.p1.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm6.p1.7.m7.1b"><apply id="S2.Thmthm6.p1.7.m7.1.2.cmml" xref="S2.Thmthm6.p1.7.m7.1.2"><csymbol cd="latexml" id="S2.Thmthm6.p1.7.m7.1.2.1.cmml" xref="S2.Thmthm6.p1.7.m7.1.2.1">assign</csymbol><ci id="S2.Thmthm6.p1.7.m7.1.2.2.cmml" xref="S2.Thmthm6.p1.7.m7.1.2.2">𝑌</ci><apply id="S2.Thmthm6.p1.7.m7.1.2.3.cmml" xref="S2.Thmthm6.p1.7.m7.1.2.3"><times id="S2.Thmthm6.p1.7.m7.1.2.3.1.cmml" xref="S2.Thmthm6.p1.7.m7.1.2.3.1"></times><ci id="S2.Thmthm6.p1.7.m7.1.2.3.2.cmml" xref="S2.Thmthm6.p1.7.m7.1.2.3.2">𝜎</ci><ci id="S2.Thmthm6.p1.7.m7.1.1.cmml" xref="S2.Thmthm6.p1.7.m7.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm6.p1.7.m7.1c">Y:=\sigma(X)\,</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm6.p1.7.m7.1d">italic_Y := italic_σ ( italic_X )</annotation></semantics></math>:</p> <ol class="ltx_enumerate" id="S2.I5"> <li class="ltx_item" id="S2.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S2.I5.i1.p1"> <p class="ltx_p" id="S2.I5.i1.p1.1">Setting <math alttext="||\sigma||:=\max\{|\sigma(a_{k})\mid a_{k}\in\cal A\}" class="ltx_Math" display="inline" id="S2.I5.i1.p1.1.m1.3"><semantics id="S2.I5.i1.p1.1.m1.3a"><mrow id="S2.I5.i1.p1.1.m1.3.3" xref="S2.I5.i1.p1.1.m1.3.3.cmml"><mrow id="S2.I5.i1.p1.1.m1.3.3.3.2" xref="S2.I5.i1.p1.1.m1.3.3.3.1.cmml"><mo id="S2.I5.i1.p1.1.m1.3.3.3.2.1" stretchy="false" xref="S2.I5.i1.p1.1.m1.3.3.3.1.1.cmml">‖</mo><mi id="S2.I5.i1.p1.1.m1.1.1" xref="S2.I5.i1.p1.1.m1.1.1.cmml">σ</mi><mo id="S2.I5.i1.p1.1.m1.3.3.3.2.2" rspace="0.278em" stretchy="false" xref="S2.I5.i1.p1.1.m1.3.3.3.1.1.cmml">‖</mo></mrow><mo id="S2.I5.i1.p1.1.m1.3.3.2" rspace="0.278em" xref="S2.I5.i1.p1.1.m1.3.3.2.cmml">:=</mo><mrow id="S2.I5.i1.p1.1.m1.3.3.1.1" xref="S2.I5.i1.p1.1.m1.3.3.1.2.cmml"><mi id="S2.I5.i1.p1.1.m1.2.2" xref="S2.I5.i1.p1.1.m1.2.2.cmml">max</mi><mo id="S2.I5.i1.p1.1.m1.3.3.1.1a" xref="S2.I5.i1.p1.1.m1.3.3.1.2.cmml">⁡</mo><mrow id="S2.I5.i1.p1.1.m1.3.3.1.1.1" xref="S2.I5.i1.p1.1.m1.3.3.1.2.cmml"><mo id="S2.I5.i1.p1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S2.I5.i1.p1.1.m1.3.3.1.2.cmml">{</mo><mrow id="S2.I5.i1.p1.1.m1.3.3.1.1.1.1" xref="S2.I5.i1.p1.1.m1.3.3.1.1.1.1.cmml"><mrow id="S2.I5.i1.p1.1.m1.3.3.1.1.1.1.1" xref="S2.I5.i1.p1.1.m1.3.3.1.1.1.1.1.cmml"><mrow id="S2.I5.i1.p1.1.m1.3.3.1.1.1.1.1.1.1" xref="S2.I5.i1.p1.1.m1.3.3.1.1.1.1.1.1.2.cmml"><mo id="S2.I5.i1.p1.1.m1.3.3.1.1.1.1.1.1.1.2" stretchy="false" 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end_POSTSUBSCRIPT ∈ caligraphic_A }</annotation></semantics></math> one has</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="p_{Y}(n)\,\,\leq\,\,||\sigma||\cdot p_{X}(n)\,." class="ltx_Math" display="block" id="S2.Ex14.m1.4"><semantics id="S2.Ex14.m1.4a"><mrow id="S2.Ex14.m1.4.4.1" xref="S2.Ex14.m1.4.4.1.1.cmml"><mrow id="S2.Ex14.m1.4.4.1.1" xref="S2.Ex14.m1.4.4.1.1.cmml"><mrow id="S2.Ex14.m1.4.4.1.1.2" xref="S2.Ex14.m1.4.4.1.1.2.cmml"><msub id="S2.Ex14.m1.4.4.1.1.2.2" xref="S2.Ex14.m1.4.4.1.1.2.2.cmml"><mi id="S2.Ex14.m1.4.4.1.1.2.2.2" xref="S2.Ex14.m1.4.4.1.1.2.2.2.cmml">p</mi><mi id="S2.Ex14.m1.4.4.1.1.2.2.3" xref="S2.Ex14.m1.4.4.1.1.2.2.3.cmml">Y</mi></msub><mo id="S2.Ex14.m1.4.4.1.1.2.1" xref="S2.Ex14.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex14.m1.4.4.1.1.2.3.2" 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id="S2.Ex14.m1.4.4.1.1.2.1.cmml" xref="S2.Ex14.m1.4.4.1.1.2.1"></times><apply id="S2.Ex14.m1.4.4.1.1.2.2.cmml" xref="S2.Ex14.m1.4.4.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex14.m1.4.4.1.1.2.2.1.cmml" xref="S2.Ex14.m1.4.4.1.1.2.2">subscript</csymbol><ci id="S2.Ex14.m1.4.4.1.1.2.2.2.cmml" xref="S2.Ex14.m1.4.4.1.1.2.2.2">𝑝</ci><ci id="S2.Ex14.m1.4.4.1.1.2.2.3.cmml" xref="S2.Ex14.m1.4.4.1.1.2.2.3">𝑌</ci></apply><ci id="S2.Ex14.m1.1.1.cmml" xref="S2.Ex14.m1.1.1">𝑛</ci></apply><apply id="S2.Ex14.m1.4.4.1.1.3.cmml" xref="S2.Ex14.m1.4.4.1.1.3"><times id="S2.Ex14.m1.4.4.1.1.3.1.cmml" xref="S2.Ex14.m1.4.4.1.1.3.1"></times><apply id="S2.Ex14.m1.4.4.1.1.3.2.cmml" xref="S2.Ex14.m1.4.4.1.1.3.2"><ci id="S2.Ex14.m1.4.4.1.1.3.2.1.cmml" xref="S2.Ex14.m1.4.4.1.1.3.2.1">⋅</ci><apply id="S2.Ex14.m1.4.4.1.1.3.2.2.1.cmml" xref="S2.Ex14.m1.4.4.1.1.3.2.2.2"><csymbol cd="latexml" id="S2.Ex14.m1.4.4.1.1.3.2.2.1.1.cmml" xref="S2.Ex14.m1.4.4.1.1.3.2.2.2.1">norm</csymbol><ci id="S2.Ex14.m1.2.2.cmml" xref="S2.Ex14.m1.2.2">𝜎</ci></apply><apply id="S2.Ex14.m1.4.4.1.1.3.2.3.cmml" xref="S2.Ex14.m1.4.4.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.Ex14.m1.4.4.1.1.3.2.3.1.cmml" xref="S2.Ex14.m1.4.4.1.1.3.2.3">subscript</csymbol><ci id="S2.Ex14.m1.4.4.1.1.3.2.3.2.cmml" xref="S2.Ex14.m1.4.4.1.1.3.2.3.2">𝑝</ci><ci id="S2.Ex14.m1.4.4.1.1.3.2.3.3.cmml" xref="S2.Ex14.m1.4.4.1.1.3.2.3.3">𝑋</ci></apply></apply><ci id="S2.Ex14.m1.3.3.cmml" xref="S2.Ex14.m1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex14.m1.4c">p_{Y}(n)\,\,\leq\,\,||\sigma||\cdot p_{X}(n)\,.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex14.m1.4d">italic_p start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ( italic_n ) ≤ | | italic_σ | | ⋅ italic_p start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_n ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S2.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S2.I5.i2.p1"> <p class="ltx_p" id="S2.I5.i2.p1.1">As direct consequence one derives</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="h_{Y}\,\,\leq\,\,h_{X}\,." class="ltx_Math" display="block" id="S2.Ex15.m1.1"><semantics id="S2.Ex15.m1.1a"><mrow id="S2.Ex15.m1.1.1.1" xref="S2.Ex15.m1.1.1.1.1.cmml"><mrow id="S2.Ex15.m1.1.1.1.1" xref="S2.Ex15.m1.1.1.1.1.cmml"><msub id="S2.Ex15.m1.1.1.1.1.2" xref="S2.Ex15.m1.1.1.1.1.2.cmml"><mi id="S2.Ex15.m1.1.1.1.1.2.2" xref="S2.Ex15.m1.1.1.1.1.2.2.cmml">h</mi><mi id="S2.Ex15.m1.1.1.1.1.2.3" xref="S2.Ex15.m1.1.1.1.1.2.3.cmml">Y</mi></msub><mo id="S2.Ex15.m1.1.1.1.1.1" lspace="0.608em" rspace="0.608em" xref="S2.Ex15.m1.1.1.1.1.1.cmml">≤</mo><msub id="S2.Ex15.m1.1.1.1.1.3" xref="S2.Ex15.m1.1.1.1.1.3.cmml"><mi id="S2.Ex15.m1.1.1.1.1.3.2" xref="S2.Ex15.m1.1.1.1.1.3.2.cmml">h</mi><mi id="S2.Ex15.m1.1.1.1.1.3.3" xref="S2.Ex15.m1.1.1.1.1.3.3.cmml">X</mi></msub></mrow><mo id="S2.Ex15.m1.1.1.1.2" lspace="0em" xref="S2.Ex15.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex15.m1.1b"><apply id="S2.Ex15.m1.1.1.1.1.cmml" xref="S2.Ex15.m1.1.1.1"><leq id="S2.Ex15.m1.1.1.1.1.1.cmml" xref="S2.Ex15.m1.1.1.1.1.1"></leq><apply id="S2.Ex15.m1.1.1.1.1.2.cmml" xref="S2.Ex15.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex15.m1.1.1.1.1.2.1.cmml" xref="S2.Ex15.m1.1.1.1.1.2">subscript</csymbol><ci id="S2.Ex15.m1.1.1.1.1.2.2.cmml" xref="S2.Ex15.m1.1.1.1.1.2.2">ℎ</ci><ci id="S2.Ex15.m1.1.1.1.1.2.3.cmml" xref="S2.Ex15.m1.1.1.1.1.2.3">𝑌</ci></apply><apply id="S2.Ex15.m1.1.1.1.1.3.cmml" xref="S2.Ex15.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex15.m1.1.1.1.1.3.1.cmml" xref="S2.Ex15.m1.1.1.1.1.3">subscript</csymbol><ci id="S2.Ex15.m1.1.1.1.1.3.2.cmml" xref="S2.Ex15.m1.1.1.1.1.3.2">ℎ</ci><ci id="S2.Ex15.m1.1.1.1.1.3.3.cmml" xref="S2.Ex15.m1.1.1.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex15.m1.1c">h_{Y}\,\,\leq\,\,h_{X}\,.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex15.m1.1d">italic_h start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ≤ italic_h start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> </ol> </div> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3. </span>About injectivity</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1"></p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.3">Injectivity of monoid morphisms <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mi id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">italic_σ</annotation></semantics></math> and of their induced maps <math alttext="\sigma^{\mathbb{Z}},\sigma^{T}" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.2"><semantics id="S2.SS3.p2.2.m2.2a"><mrow id="S2.SS3.p2.2.m2.2.2.2" xref="S2.SS3.p2.2.m2.2.2.3.cmml"><msup id="S2.SS3.p2.2.m2.1.1.1.1" xref="S2.SS3.p2.2.m2.1.1.1.1.cmml"><mi id="S2.SS3.p2.2.m2.1.1.1.1.2" xref="S2.SS3.p2.2.m2.1.1.1.1.2.cmml">σ</mi><mi id="S2.SS3.p2.2.m2.1.1.1.1.3" xref="S2.SS3.p2.2.m2.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S2.SS3.p2.2.m2.2.2.2.3" xref="S2.SS3.p2.2.m2.2.2.3.cmml">,</mo><msup id="S2.SS3.p2.2.m2.2.2.2.2" xref="S2.SS3.p2.2.m2.2.2.2.2.cmml"><mi id="S2.SS3.p2.2.m2.2.2.2.2.2" xref="S2.SS3.p2.2.m2.2.2.2.2.2.cmml">σ</mi><mi id="S2.SS3.p2.2.m2.2.2.2.2.3" xref="S2.SS3.p2.2.m2.2.2.2.2.3.cmml">T</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.2b"><list id="S2.SS3.p2.2.m2.2.2.3.cmml" xref="S2.SS3.p2.2.m2.2.2.2"><apply id="S2.SS3.p2.2.m2.1.1.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.2.m2.1.1.1.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1.1.1">superscript</csymbol><ci id="S2.SS3.p2.2.m2.1.1.1.1.2.cmml" xref="S2.SS3.p2.2.m2.1.1.1.1.2">𝜎</ci><ci id="S2.SS3.p2.2.m2.1.1.1.1.3.cmml" xref="S2.SS3.p2.2.m2.1.1.1.1.3">ℤ</ci></apply><apply id="S2.SS3.p2.2.m2.2.2.2.2.cmml" xref="S2.SS3.p2.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS3.p2.2.m2.2.2.2.2.1.cmml" xref="S2.SS3.p2.2.m2.2.2.2.2">superscript</csymbol><ci id="S2.SS3.p2.2.m2.2.2.2.2.2.cmml" xref="S2.SS3.p2.2.m2.2.2.2.2.2">𝜎</ci><ci id="S2.SS3.p2.2.m2.2.2.2.2.3.cmml" xref="S2.SS3.p2.2.m2.2.2.2.2.3">𝑇</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.2c">\sigma^{\mathbb{Z}},\sigma^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.2d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT , italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\sigma^{\Sigma}" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.1"><semantics id="S2.SS3.p2.3.m3.1a"><msup id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml"><mi id="S2.SS3.p2.3.m3.1.1.2" xref="S2.SS3.p2.3.m3.1.1.2.cmml">σ</mi><mi id="S2.SS3.p2.3.m3.1.1.3" mathvariant="normal" xref="S2.SS3.p2.3.m3.1.1.3.cmml">Σ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.1b"><apply id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS3.p2.3.m3.1.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1">superscript</csymbol><ci id="S2.SS3.p2.3.m3.1.1.2.cmml" xref="S2.SS3.p2.3.m3.1.1.2">𝜎</ci><ci id="S2.SS3.p2.3.m3.1.1.3.cmml" xref="S2.SS3.p2.3.m3.1.1.3">Σ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.1c">\sigma^{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.1d">italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT</annotation></semantics></math> is an important and often tricky issue. We start out in sub-Subsection <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS3.SSS1" title="2.3.1. Typical injectivity problems ‣ 2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.3.1</span></a> to list several problems and give examples where these often undesired phenomena do occur. In sub-Subsection <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS3.SSS2" title="2.3.2. Shift-period preservation ‣ 2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.3.2</span></a>, which can be read independently, we will then define the subtle notion of “shift-period preservation”, which will be used in Sections <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5" title="5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6" title="6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6</span></a> below as well as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>.</p> </div> <section class="ltx_subsubsection" id="S2.SS3.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.3.1. </span>Typical injectivity problems</h4> <div class="ltx_para" id="S2.SS3.SSS1.p1"> <p class="ltx_p" id="S2.SS3.SSS1.p1.1"></p> </div> <div class="ltx_para" id="S2.SS3.SSS1.p2"> <p class="ltx_p" id="S2.SS3.SSS1.p2.1">For starters, we have the following well known phenomenon:</p> </div> <div class="ltx_theorem ltx_theorem_rem" id="S2.Thmthm7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm7.1.1.1">Remark 2.7</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm7.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm7.p1"> <p class="ltx_p" id="S2.Thmthm7.p1.2">There exist non-erasing morphisms <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.Thmthm7.p1.1.m1.1"><semantics id="S2.Thmthm7.p1.1.m1.1a"><mrow id="S2.Thmthm7.p1.1.m1.1.1" xref="S2.Thmthm7.p1.1.m1.1.1.cmml"><mi id="S2.Thmthm7.p1.1.m1.1.1.2" xref="S2.Thmthm7.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S2.Thmthm7.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm7.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.Thmthm7.p1.1.m1.1.1.3" xref="S2.Thmthm7.p1.1.m1.1.1.3.cmml"><msup id="S2.Thmthm7.p1.1.m1.1.1.3.2" xref="S2.Thmthm7.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm7.p1.1.m1.1.1.3.2.2" xref="S2.Thmthm7.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.Thmthm7.p1.1.m1.1.1.3.2.3" xref="S2.Thmthm7.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm7.p1.1.m1.1.1.3.1" stretchy="false" xref="S2.Thmthm7.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.Thmthm7.p1.1.m1.1.1.3.3" xref="S2.Thmthm7.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm7.p1.1.m1.1.1.3.3.2" xref="S2.Thmthm7.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.Thmthm7.p1.1.m1.1.1.3.3.3" xref="S2.Thmthm7.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm7.p1.1.m1.1b"><apply id="S2.Thmthm7.p1.1.m1.1.1.cmml" xref="S2.Thmthm7.p1.1.m1.1.1"><ci id="S2.Thmthm7.p1.1.m1.1.1.1.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.1">:</ci><ci id="S2.Thmthm7.p1.1.m1.1.1.2.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.2">𝜎</ci><apply id="S2.Thmthm7.p1.1.m1.1.1.3.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3"><ci id="S2.Thmthm7.p1.1.m1.1.1.3.1.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.1">→</ci><apply id="S2.Thmthm7.p1.1.m1.1.1.3.2.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmthm7.p1.1.m1.1.1.3.2.1.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.Thmthm7.p1.1.m1.1.1.3.2.2.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S2.Thmthm7.p1.1.m1.1.1.3.2.3.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S2.Thmthm7.p1.1.m1.1.1.3.3.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmthm7.p1.1.m1.1.1.3.3.1.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.Thmthm7.p1.1.m1.1.1.3.3.2.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S2.Thmthm7.p1.1.m1.1.1.3.3.3.cmml" xref="S2.Thmthm7.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm7.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm7.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> which are injective, while the associated free group homomorphism <math alttext="F(\cal A)\to F(\cal B)" class="ltx_Math" display="inline" id="S2.Thmthm7.p1.2.m2.2"><semantics id="S2.Thmthm7.p1.2.m2.2a"><mrow id="S2.Thmthm7.p1.2.m2.2.3" xref="S2.Thmthm7.p1.2.m2.2.3.cmml"><mrow id="S2.Thmthm7.p1.2.m2.2.3.2" xref="S2.Thmthm7.p1.2.m2.2.3.2.cmml"><mi id="S2.Thmthm7.p1.2.m2.2.3.2.2" xref="S2.Thmthm7.p1.2.m2.2.3.2.2.cmml">F</mi><mo id="S2.Thmthm7.p1.2.m2.2.3.2.1" xref="S2.Thmthm7.p1.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S2.Thmthm7.p1.2.m2.2.3.2.3.2" xref="S2.Thmthm7.p1.2.m2.2.3.2.cmml"><mo id="S2.Thmthm7.p1.2.m2.2.3.2.3.2.1" stretchy="false" xref="S2.Thmthm7.p1.2.m2.2.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm7.p1.2.m2.1.1" xref="S2.Thmthm7.p1.2.m2.1.1.cmml">𝒜</mi><mo id="S2.Thmthm7.p1.2.m2.2.3.2.3.2.2" stretchy="false" xref="S2.Thmthm7.p1.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm7.p1.2.m2.2.3.1" stretchy="false" xref="S2.Thmthm7.p1.2.m2.2.3.1.cmml">→</mo><mrow id="S2.Thmthm7.p1.2.m2.2.3.3" xref="S2.Thmthm7.p1.2.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm7.p1.2.m2.2.3.3.2" xref="S2.Thmthm7.p1.2.m2.2.3.3.2.cmml">ℱ</mi><mo id="S2.Thmthm7.p1.2.m2.2.3.3.1" xref="S2.Thmthm7.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S2.Thmthm7.p1.2.m2.2.3.3.3.2" xref="S2.Thmthm7.p1.2.m2.2.3.3.cmml"><mo id="S2.Thmthm7.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="S2.Thmthm7.p1.2.m2.2.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm7.p1.2.m2.2.2" xref="S2.Thmthm7.p1.2.m2.2.2.cmml">ℬ</mi><mo id="S2.Thmthm7.p1.2.m2.2.3.3.3.2.2" stretchy="false" xref="S2.Thmthm7.p1.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm7.p1.2.m2.2b"><apply id="S2.Thmthm7.p1.2.m2.2.3.cmml" xref="S2.Thmthm7.p1.2.m2.2.3"><ci id="S2.Thmthm7.p1.2.m2.2.3.1.cmml" xref="S2.Thmthm7.p1.2.m2.2.3.1">→</ci><apply id="S2.Thmthm7.p1.2.m2.2.3.2.cmml" xref="S2.Thmthm7.p1.2.m2.2.3.2"><times id="S2.Thmthm7.p1.2.m2.2.3.2.1.cmml" xref="S2.Thmthm7.p1.2.m2.2.3.2.1"></times><ci id="S2.Thmthm7.p1.2.m2.2.3.2.2.cmml" xref="S2.Thmthm7.p1.2.m2.2.3.2.2">𝐹</ci><ci id="S2.Thmthm7.p1.2.m2.1.1.cmml" xref="S2.Thmthm7.p1.2.m2.1.1">𝒜</ci></apply><apply id="S2.Thmthm7.p1.2.m2.2.3.3.cmml" xref="S2.Thmthm7.p1.2.m2.2.3.3"><times id="S2.Thmthm7.p1.2.m2.2.3.3.1.cmml" xref="S2.Thmthm7.p1.2.m2.2.3.3.1"></times><ci id="S2.Thmthm7.p1.2.m2.2.3.3.2.cmml" xref="S2.Thmthm7.p1.2.m2.2.3.3.2">ℱ</ci><ci id="S2.Thmthm7.p1.2.m2.2.2.cmml" xref="S2.Thmthm7.p1.2.m2.2.2">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm7.p1.2.m2.2c">F(\cal A)\to F(\cal B)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm7.p1.2.m2.2d">italic_F ( caligraphic_A ) → caligraphic_F ( caligraphic_B )</annotation></semantics></math> is not injective. An example is given by</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma:\{a,b,c\}^{*}\to\{d,e\}^{*}\,,\,\,a\mapsto dde,\,\,b\mapsto dee,\,\,c% \mapsto dede\," class="ltx_Math" display="block" id="S2.Ex16.m1.7"><semantics id="S2.Ex16.m1.7a"><mrow id="S2.Ex16.m1.7.7" xref="S2.Ex16.m1.7.7.cmml"><mi id="S2.Ex16.m1.7.7.4" xref="S2.Ex16.m1.7.7.4.cmml">σ</mi><mo id="S2.Ex16.m1.7.7.3" lspace="0.278em" rspace="0.278em" xref="S2.Ex16.m1.7.7.3.cmml">:</mo><mrow id="S2.Ex16.m1.7.7.2.2" xref="S2.Ex16.m1.7.7.2.3.cmml"><mrow id="S2.Ex16.m1.6.6.1.1.1" xref="S2.Ex16.m1.6.6.1.1.1.cmml"><msup id="S2.Ex16.m1.6.6.1.1.1.2" xref="S2.Ex16.m1.6.6.1.1.1.2.cmml"><mrow id="S2.Ex16.m1.6.6.1.1.1.2.2.2" xref="S2.Ex16.m1.6.6.1.1.1.2.2.1.cmml"><mo id="S2.Ex16.m1.6.6.1.1.1.2.2.2.1" stretchy="false" xref="S2.Ex16.m1.6.6.1.1.1.2.2.1.cmml">{</mo><mi id="S2.Ex16.m1.1.1" xref="S2.Ex16.m1.1.1.cmml">a</mi><mo id="S2.Ex16.m1.6.6.1.1.1.2.2.2.2" xref="S2.Ex16.m1.6.6.1.1.1.2.2.1.cmml">,</mo><mi id="S2.Ex16.m1.2.2" xref="S2.Ex16.m1.2.2.cmml">b</mi><mo id="S2.Ex16.m1.6.6.1.1.1.2.2.2.3" xref="S2.Ex16.m1.6.6.1.1.1.2.2.1.cmml">,</mo><mi id="S2.Ex16.m1.3.3" xref="S2.Ex16.m1.3.3.cmml">c</mi><mo id="S2.Ex16.m1.6.6.1.1.1.2.2.2.4" stretchy="false" xref="S2.Ex16.m1.6.6.1.1.1.2.2.1.cmml">}</mo></mrow><mo id="S2.Ex16.m1.6.6.1.1.1.2.3" xref="S2.Ex16.m1.6.6.1.1.1.2.3.cmml">∗</mo></msup><mo id="S2.Ex16.m1.6.6.1.1.1.1" stretchy="false" xref="S2.Ex16.m1.6.6.1.1.1.1.cmml">→</mo><msup id="S2.Ex16.m1.6.6.1.1.1.3" xref="S2.Ex16.m1.6.6.1.1.1.3.cmml"><mrow id="S2.Ex16.m1.6.6.1.1.1.3.2.2" xref="S2.Ex16.m1.6.6.1.1.1.3.2.1.cmml"><mo id="S2.Ex16.m1.6.6.1.1.1.3.2.2.1" stretchy="false" xref="S2.Ex16.m1.6.6.1.1.1.3.2.1.cmml">{</mo><mi id="S2.Ex16.m1.4.4" xref="S2.Ex16.m1.4.4.cmml">d</mi><mo id="S2.Ex16.m1.6.6.1.1.1.3.2.2.2" xref="S2.Ex16.m1.6.6.1.1.1.3.2.1.cmml">,</mo><mi id="S2.Ex16.m1.5.5" xref="S2.Ex16.m1.5.5.cmml">e</mi><mo id="S2.Ex16.m1.6.6.1.1.1.3.2.2.3" stretchy="false" xref="S2.Ex16.m1.6.6.1.1.1.3.2.1.cmml">}</mo></mrow><mo id="S2.Ex16.m1.6.6.1.1.1.3.3" xref="S2.Ex16.m1.6.6.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S2.Ex16.m1.7.7.2.2.3" rspace="0.497em" xref="S2.Ex16.m1.7.7.2.3a.cmml">,</mo><mrow id="S2.Ex16.m1.7.7.2.2.2.2" xref="S2.Ex16.m1.7.7.2.2.2.3.cmml"><mrow id="S2.Ex16.m1.7.7.2.2.2.1.1" xref="S2.Ex16.m1.7.7.2.2.2.1.1.cmml"><mi id="S2.Ex16.m1.7.7.2.2.2.1.1.2" xref="S2.Ex16.m1.7.7.2.2.2.1.1.2.cmml">a</mi><mo id="S2.Ex16.m1.7.7.2.2.2.1.1.1" stretchy="false" xref="S2.Ex16.m1.7.7.2.2.2.1.1.1.cmml">↦</mo><mrow id="S2.Ex16.m1.7.7.2.2.2.1.1.3" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.cmml"><mi id="S2.Ex16.m1.7.7.2.2.2.1.1.3.2" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.2.cmml">d</mi><mo id="S2.Ex16.m1.7.7.2.2.2.1.1.3.1" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex16.m1.7.7.2.2.2.1.1.3.3" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.3.cmml">d</mi><mo id="S2.Ex16.m1.7.7.2.2.2.1.1.3.1a" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex16.m1.7.7.2.2.2.1.1.3.4" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.4.cmml">e</mi></mrow></mrow><mo id="S2.Ex16.m1.7.7.2.2.2.2.3" rspace="0.497em" xref="S2.Ex16.m1.7.7.2.2.2.3a.cmml">,</mo><mrow id="S2.Ex16.m1.7.7.2.2.2.2.2.2" xref="S2.Ex16.m1.7.7.2.2.2.2.2.3.cmml"><mrow id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.cmml"><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.2" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.2.cmml">b</mi><mo id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.1" stretchy="false" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.1.cmml">↦</mo><mrow id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.cmml"><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.2" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.2.cmml">d</mi><mo id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.1" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.3" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.3.cmml">e</mi><mo id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.1a" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.4" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.4.cmml">e</mi></mrow></mrow><mo id="S2.Ex16.m1.7.7.2.2.2.2.2.2.3" rspace="0.497em" xref="S2.Ex16.m1.7.7.2.2.2.2.2.3a.cmml">,</mo><mrow id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.cmml"><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.2" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.2.cmml">c</mi><mo id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.1" stretchy="false" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.1.cmml">↦</mo><mrow id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.cmml"><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.2" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.2.cmml">d</mi><mo id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.1" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.3" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.3.cmml">e</mi><mo id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.1a" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.4" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.4.cmml">d</mi><mo id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.1b" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.1.cmml">⁢</mo><mi id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.5" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.5.cmml">e</mi></mrow></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex16.m1.7b"><apply id="S2.Ex16.m1.7.7.cmml" xref="S2.Ex16.m1.7.7"><ci id="S2.Ex16.m1.7.7.3.cmml" xref="S2.Ex16.m1.7.7.3">:</ci><ci id="S2.Ex16.m1.7.7.4.cmml" xref="S2.Ex16.m1.7.7.4">𝜎</ci><apply id="S2.Ex16.m1.7.7.2.3.cmml" xref="S2.Ex16.m1.7.7.2.2"><csymbol cd="ambiguous" id="S2.Ex16.m1.7.7.2.3a.cmml" xref="S2.Ex16.m1.7.7.2.2.3">formulae-sequence</csymbol><apply id="S2.Ex16.m1.6.6.1.1.1.cmml" xref="S2.Ex16.m1.6.6.1.1.1"><ci id="S2.Ex16.m1.6.6.1.1.1.1.cmml" xref="S2.Ex16.m1.6.6.1.1.1.1">→</ci><apply id="S2.Ex16.m1.6.6.1.1.1.2.cmml" xref="S2.Ex16.m1.6.6.1.1.1.2"><csymbol cd="ambiguous" id="S2.Ex16.m1.6.6.1.1.1.2.1.cmml" xref="S2.Ex16.m1.6.6.1.1.1.2">superscript</csymbol><set id="S2.Ex16.m1.6.6.1.1.1.2.2.1.cmml" xref="S2.Ex16.m1.6.6.1.1.1.2.2.2"><ci id="S2.Ex16.m1.1.1.cmml" xref="S2.Ex16.m1.1.1">𝑎</ci><ci id="S2.Ex16.m1.2.2.cmml" xref="S2.Ex16.m1.2.2">𝑏</ci><ci id="S2.Ex16.m1.3.3.cmml" xref="S2.Ex16.m1.3.3">𝑐</ci></set><times id="S2.Ex16.m1.6.6.1.1.1.2.3.cmml" xref="S2.Ex16.m1.6.6.1.1.1.2.3"></times></apply><apply id="S2.Ex16.m1.6.6.1.1.1.3.cmml" xref="S2.Ex16.m1.6.6.1.1.1.3"><csymbol cd="ambiguous" id="S2.Ex16.m1.6.6.1.1.1.3.1.cmml" xref="S2.Ex16.m1.6.6.1.1.1.3">superscript</csymbol><set id="S2.Ex16.m1.6.6.1.1.1.3.2.1.cmml" xref="S2.Ex16.m1.6.6.1.1.1.3.2.2"><ci id="S2.Ex16.m1.4.4.cmml" xref="S2.Ex16.m1.4.4">𝑑</ci><ci id="S2.Ex16.m1.5.5.cmml" xref="S2.Ex16.m1.5.5">𝑒</ci></set><times id="S2.Ex16.m1.6.6.1.1.1.3.3.cmml" xref="S2.Ex16.m1.6.6.1.1.1.3.3"></times></apply></apply><apply id="S2.Ex16.m1.7.7.2.2.2.3.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2"><csymbol cd="ambiguous" id="S2.Ex16.m1.7.7.2.2.2.3a.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.3">formulae-sequence</csymbol><apply id="S2.Ex16.m1.7.7.2.2.2.1.1.cmml" xref="S2.Ex16.m1.7.7.2.2.2.1.1"><csymbol cd="latexml" id="S2.Ex16.m1.7.7.2.2.2.1.1.1.cmml" xref="S2.Ex16.m1.7.7.2.2.2.1.1.1">maps-to</csymbol><ci id="S2.Ex16.m1.7.7.2.2.2.1.1.2.cmml" xref="S2.Ex16.m1.7.7.2.2.2.1.1.2">𝑎</ci><apply id="S2.Ex16.m1.7.7.2.2.2.1.1.3.cmml" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3"><times id="S2.Ex16.m1.7.7.2.2.2.1.1.3.1.cmml" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.1"></times><ci id="S2.Ex16.m1.7.7.2.2.2.1.1.3.2.cmml" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.2">𝑑</ci><ci id="S2.Ex16.m1.7.7.2.2.2.1.1.3.3.cmml" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.3">𝑑</ci><ci id="S2.Ex16.m1.7.7.2.2.2.1.1.3.4.cmml" xref="S2.Ex16.m1.7.7.2.2.2.1.1.3.4">𝑒</ci></apply></apply><apply id="S2.Ex16.m1.7.7.2.2.2.2.2.3.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Ex16.m1.7.7.2.2.2.2.2.3a.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.3">formulae-sequence</csymbol><apply id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1"><csymbol cd="latexml" id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.1.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.1">maps-to</csymbol><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.2.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.2">𝑏</ci><apply id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3"><times id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.1.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.1"></times><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.2.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.2">𝑑</ci><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.3.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.3">𝑒</ci><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.4.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.1.1.3.4">𝑒</ci></apply></apply><apply id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2"><csymbol cd="latexml" id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.1.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.1">maps-to</csymbol><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.2.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.2">𝑐</ci><apply id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3"><times id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.1.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.1"></times><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.2.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.2">𝑑</ci><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.3.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.3">𝑒</ci><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.4.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.4">𝑑</ci><ci id="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.5.cmml" xref="S2.Ex16.m1.7.7.2.2.2.2.2.2.2.3.5">𝑒</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex16.m1.7c">\sigma:\{a,b,c\}^{*}\to\{d,e\}^{*}\,,\,\,a\mapsto dde,\,\,b\mapsto dee,\,\,c% \mapsto dede\,</annotation><annotation encoding="application/x-llamapun" id="S2.Ex16.m1.7d">italic_σ : { italic_a , italic_b , italic_c } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → { italic_d , italic_e } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_a ↦ italic_d italic_d italic_e , italic_b ↦ italic_d italic_e italic_e , italic_c ↦ italic_d italic_e italic_d italic_e</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmthm7.p1.4">which maps <math alttext="c(ac^{-1}b)^{-2}" class="ltx_Math" display="inline" id="S2.Thmthm7.p1.3.m1.1"><semantics id="S2.Thmthm7.p1.3.m1.1a"><mrow id="S2.Thmthm7.p1.3.m1.1.1" xref="S2.Thmthm7.p1.3.m1.1.1.cmml"><mi id="S2.Thmthm7.p1.3.m1.1.1.3" xref="S2.Thmthm7.p1.3.m1.1.1.3.cmml">c</mi><mo id="S2.Thmthm7.p1.3.m1.1.1.2" xref="S2.Thmthm7.p1.3.m1.1.1.2.cmml">⁢</mo><msup id="S2.Thmthm7.p1.3.m1.1.1.1" xref="S2.Thmthm7.p1.3.m1.1.1.1.cmml"><mrow id="S2.Thmthm7.p1.3.m1.1.1.1.1.1" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.cmml"><mo id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.cmml"><mi id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.2" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.2.cmml">a</mi><mo id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.1" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.cmml"><mi id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.2" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.2.cmml">c</mi><mrow id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3.cmml"><mo id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3a" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3.cmml">−</mo><mn id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3.2" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.1a" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.4" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.4.cmml">b</mi></mrow><mo id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.Thmthm7.p1.3.m1.1.1.1.3" xref="S2.Thmthm7.p1.3.m1.1.1.1.3.cmml"><mo id="S2.Thmthm7.p1.3.m1.1.1.1.3a" xref="S2.Thmthm7.p1.3.m1.1.1.1.3.cmml">−</mo><mn id="S2.Thmthm7.p1.3.m1.1.1.1.3.2" xref="S2.Thmthm7.p1.3.m1.1.1.1.3.2.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm7.p1.3.m1.1b"><apply id="S2.Thmthm7.p1.3.m1.1.1.cmml" xref="S2.Thmthm7.p1.3.m1.1.1"><times id="S2.Thmthm7.p1.3.m1.1.1.2.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.2"></times><ci id="S2.Thmthm7.p1.3.m1.1.1.3.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.3">𝑐</ci><apply id="S2.Thmthm7.p1.3.m1.1.1.1.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm7.p1.3.m1.1.1.1.2.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1">superscript</csymbol><apply id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1"><times id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.1.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.1"></times><ci id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.2.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.2">𝑎</ci><apply id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3">superscript</csymbol><ci id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.2">𝑐</ci><apply id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3"><minus id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3.1.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3"></minus><cn id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3.2.cmml" type="integer" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.3.3.2">1</cn></apply></apply><ci id="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.4.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.1.1.1.4">𝑏</ci></apply><apply id="S2.Thmthm7.p1.3.m1.1.1.1.3.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.3"><minus id="S2.Thmthm7.p1.3.m1.1.1.1.3.1.cmml" xref="S2.Thmthm7.p1.3.m1.1.1.1.3"></minus><cn id="S2.Thmthm7.p1.3.m1.1.1.1.3.2.cmml" type="integer" xref="S2.Thmthm7.p1.3.m1.1.1.1.3.2">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm7.p1.3.m1.1c">c(ac^{-1}b)^{-2}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm7.p1.3.m1.1d">italic_c ( italic_a italic_c start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_b ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> to the neutral element <math alttext="1\in F(\{d,e\})" class="ltx_Math" display="inline" id="S2.Thmthm7.p1.4.m2.3"><semantics id="S2.Thmthm7.p1.4.m2.3a"><mrow id="S2.Thmthm7.p1.4.m2.3.3" xref="S2.Thmthm7.p1.4.m2.3.3.cmml"><mn id="S2.Thmthm7.p1.4.m2.3.3.3" xref="S2.Thmthm7.p1.4.m2.3.3.3.cmml">1</mn><mo id="S2.Thmthm7.p1.4.m2.3.3.2" xref="S2.Thmthm7.p1.4.m2.3.3.2.cmml">∈</mo><mrow id="S2.Thmthm7.p1.4.m2.3.3.1" xref="S2.Thmthm7.p1.4.m2.3.3.1.cmml"><mi id="S2.Thmthm7.p1.4.m2.3.3.1.3" xref="S2.Thmthm7.p1.4.m2.3.3.1.3.cmml">F</mi><mo id="S2.Thmthm7.p1.4.m2.3.3.1.2" xref="S2.Thmthm7.p1.4.m2.3.3.1.2.cmml">⁢</mo><mrow id="S2.Thmthm7.p1.4.m2.3.3.1.1.1" xref="S2.Thmthm7.p1.4.m2.3.3.1.cmml"><mo id="S2.Thmthm7.p1.4.m2.3.3.1.1.1.2" stretchy="false" xref="S2.Thmthm7.p1.4.m2.3.3.1.cmml">(</mo><mrow id="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.2" xref="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.1.cmml"><mo id="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.2.1" stretchy="false" xref="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.1.cmml">{</mo><mi id="S2.Thmthm7.p1.4.m2.1.1" xref="S2.Thmthm7.p1.4.m2.1.1.cmml">d</mi><mo id="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.2.2" xref="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.1.cmml">,</mo><mi id="S2.Thmthm7.p1.4.m2.2.2" xref="S2.Thmthm7.p1.4.m2.2.2.cmml">e</mi><mo id="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.2.3" stretchy="false" xref="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.1.cmml">}</mo></mrow><mo id="S2.Thmthm7.p1.4.m2.3.3.1.1.1.3" stretchy="false" xref="S2.Thmthm7.p1.4.m2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm7.p1.4.m2.3b"><apply id="S2.Thmthm7.p1.4.m2.3.3.cmml" xref="S2.Thmthm7.p1.4.m2.3.3"><in id="S2.Thmthm7.p1.4.m2.3.3.2.cmml" xref="S2.Thmthm7.p1.4.m2.3.3.2"></in><cn id="S2.Thmthm7.p1.4.m2.3.3.3.cmml" type="integer" xref="S2.Thmthm7.p1.4.m2.3.3.3">1</cn><apply id="S2.Thmthm7.p1.4.m2.3.3.1.cmml" xref="S2.Thmthm7.p1.4.m2.3.3.1"><times id="S2.Thmthm7.p1.4.m2.3.3.1.2.cmml" xref="S2.Thmthm7.p1.4.m2.3.3.1.2"></times><ci id="S2.Thmthm7.p1.4.m2.3.3.1.3.cmml" xref="S2.Thmthm7.p1.4.m2.3.3.1.3">𝐹</ci><set id="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.1.cmml" xref="S2.Thmthm7.p1.4.m2.3.3.1.1.1.1.2"><ci id="S2.Thmthm7.p1.4.m2.1.1.cmml" xref="S2.Thmthm7.p1.4.m2.1.1">𝑑</ci><ci id="S2.Thmthm7.p1.4.m2.2.2.cmml" xref="S2.Thmthm7.p1.4.m2.2.2">𝑒</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm7.p1.4.m2.3c">1\in F(\{d,e\})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm7.p1.4.m2.3d">1 ∈ italic_F ( { italic_d , italic_e } )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.SS3.SSS1.p3"> <p class="ltx_p" id="S2.SS3.SSS1.p3.6">A second, also well known disturbance comes from the fact that injective free group morphisms need not induce injective maps on the set of conjugacy classes. This happens for example for the morphism induced by the quotient map, if two boundary curves of a surface (with at least one more boundary component to ensure free fundamental groups) are glued together. In a classical monoid morphism situation we observe that for the “Thue-Morse morphism”</p> <table class="ltx_equation ltx_eqn_table" id="S2.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(2.11)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma_{\text{TM}}:\{a,b\}^{*}\to\{c,d\}^{*}\,,\,\,\,a\mapsto cd,\,\,b\mapsto dc" class="ltx_Math" display="block" id="S2.E11.m1.6"><semantics id="S2.E11.m1.6a"><mrow id="S2.E11.m1.6.6" xref="S2.E11.m1.6.6.cmml"><msub id="S2.E11.m1.6.6.4" xref="S2.E11.m1.6.6.4.cmml"><mi id="S2.E11.m1.6.6.4.2" xref="S2.E11.m1.6.6.4.2.cmml">σ</mi><mtext id="S2.E11.m1.6.6.4.3" xref="S2.E11.m1.6.6.4.3a.cmml">TM</mtext></msub><mo id="S2.E11.m1.6.6.3" lspace="0.278em" rspace="0.278em" xref="S2.E11.m1.6.6.3.cmml">:</mo><mrow 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stretchy="false" xref="S2.E11.m1.5.5.1.1.1.3.2.1.cmml">{</mo><mi id="S2.E11.m1.3.3" xref="S2.E11.m1.3.3.cmml">c</mi><mo id="S2.E11.m1.5.5.1.1.1.3.2.2.2" xref="S2.E11.m1.5.5.1.1.1.3.2.1.cmml">,</mo><mi id="S2.E11.m1.4.4" xref="S2.E11.m1.4.4.cmml">d</mi><mo id="S2.E11.m1.5.5.1.1.1.3.2.2.3" stretchy="false" xref="S2.E11.m1.5.5.1.1.1.3.2.1.cmml">}</mo></mrow><mo id="S2.E11.m1.5.5.1.1.1.3.3" xref="S2.E11.m1.5.5.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S2.E11.m1.6.6.2.2.3" rspace="0.667em" xref="S2.E11.m1.6.6.2.3a.cmml">,</mo><mrow id="S2.E11.m1.6.6.2.2.2.2" xref="S2.E11.m1.6.6.2.2.2.3.cmml"><mrow id="S2.E11.m1.6.6.2.2.2.1.1" xref="S2.E11.m1.6.6.2.2.2.1.1.cmml"><mi id="S2.E11.m1.6.6.2.2.2.1.1.2" xref="S2.E11.m1.6.6.2.2.2.1.1.2.cmml">a</mi><mo id="S2.E11.m1.6.6.2.2.2.1.1.1" stretchy="false" xref="S2.E11.m1.6.6.2.2.2.1.1.1.cmml">↦</mo><mrow id="S2.E11.m1.6.6.2.2.2.1.1.3" xref="S2.E11.m1.6.6.2.2.2.1.1.3.cmml"><mi id="S2.E11.m1.6.6.2.2.2.1.1.3.2" 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xref="S2.E11.m1.6.6.2.2.2.2.2.3.2">𝑑</ci><ci id="S2.E11.m1.6.6.2.2.2.2.2.3.3.cmml" xref="S2.E11.m1.6.6.2.2.2.2.2.3.3">𝑐</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E11.m1.6c">\sigma_{\text{TM}}:\{a,b\}^{*}\to\{c,d\}^{*}\,,\,\,\,a\mapsto cd,\,\,b\mapsto dc</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m1.6d">italic_σ start_POSTSUBSCRIPT TM end_POSTSUBSCRIPT : { italic_a , italic_b } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → { italic_c , italic_d } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_a ↦ italic_c italic_d , italic_b ↦ italic_d italic_c</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS1.p3.5">the images <math alttext="\sigma_{\text{TM}}(a)" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p3.1.m1.1"><semantics id="S2.SS3.SSS1.p3.1.m1.1a"><mrow id="S2.SS3.SSS1.p3.1.m1.1.2" xref="S2.SS3.SSS1.p3.1.m1.1.2.cmml"><msub id="S2.SS3.SSS1.p3.1.m1.1.2.2" xref="S2.SS3.SSS1.p3.1.m1.1.2.2.cmml"><mi id="S2.SS3.SSS1.p3.1.m1.1.2.2.2" xref="S2.SS3.SSS1.p3.1.m1.1.2.2.2.cmml">σ</mi><mtext id="S2.SS3.SSS1.p3.1.m1.1.2.2.3" xref="S2.SS3.SSS1.p3.1.m1.1.2.2.3a.cmml">TM</mtext></msub><mo id="S2.SS3.SSS1.p3.1.m1.1.2.1" xref="S2.SS3.SSS1.p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS1.p3.1.m1.1.2.3.2" xref="S2.SS3.SSS1.p3.1.m1.1.2.cmml"><mo id="S2.SS3.SSS1.p3.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS3.SSS1.p3.1.m1.1.2.cmml">(</mo><mi id="S2.SS3.SSS1.p3.1.m1.1.1" xref="S2.SS3.SSS1.p3.1.m1.1.1.cmml">a</mi><mo id="S2.SS3.SSS1.p3.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS3.SSS1.p3.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p3.1.m1.1b"><apply id="S2.SS3.SSS1.p3.1.m1.1.2.cmml" xref="S2.SS3.SSS1.p3.1.m1.1.2"><times id="S2.SS3.SSS1.p3.1.m1.1.2.1.cmml" xref="S2.SS3.SSS1.p3.1.m1.1.2.1"></times><apply id="S2.SS3.SSS1.p3.1.m1.1.2.2.cmml" xref="S2.SS3.SSS1.p3.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS1.p3.1.m1.1.2.2.1.cmml" xref="S2.SS3.SSS1.p3.1.m1.1.2.2">subscript</csymbol><ci id="S2.SS3.SSS1.p3.1.m1.1.2.2.2.cmml" xref="S2.SS3.SSS1.p3.1.m1.1.2.2.2">𝜎</ci><ci id="S2.SS3.SSS1.p3.1.m1.1.2.2.3a.cmml" xref="S2.SS3.SSS1.p3.1.m1.1.2.2.3"><mtext id="S2.SS3.SSS1.p3.1.m1.1.2.2.3.cmml" mathsize="70%" xref="S2.SS3.SSS1.p3.1.m1.1.2.2.3">TM</mtext></ci></apply><ci id="S2.SS3.SSS1.p3.1.m1.1.1.cmml" xref="S2.SS3.SSS1.p3.1.m1.1.1">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p3.1.m1.1c">\sigma_{\text{TM}}(a)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p3.1.m1.1d">italic_σ start_POSTSUBSCRIPT TM end_POSTSUBSCRIPT ( italic_a )</annotation></semantics></math> and <math alttext="\sigma_{\text{TM}}(b)" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p3.2.m2.1"><semantics id="S2.SS3.SSS1.p3.2.m2.1a"><mrow id="S2.SS3.SSS1.p3.2.m2.1.2" xref="S2.SS3.SSS1.p3.2.m2.1.2.cmml"><msub id="S2.SS3.SSS1.p3.2.m2.1.2.2" xref="S2.SS3.SSS1.p3.2.m2.1.2.2.cmml"><mi id="S2.SS3.SSS1.p3.2.m2.1.2.2.2" xref="S2.SS3.SSS1.p3.2.m2.1.2.2.2.cmml">σ</mi><mtext id="S2.SS3.SSS1.p3.2.m2.1.2.2.3" xref="S2.SS3.SSS1.p3.2.m2.1.2.2.3a.cmml">TM</mtext></msub><mo id="S2.SS3.SSS1.p3.2.m2.1.2.1" xref="S2.SS3.SSS1.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS1.p3.2.m2.1.2.3.2" xref="S2.SS3.SSS1.p3.2.m2.1.2.cmml"><mo id="S2.SS3.SSS1.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS3.SSS1.p3.2.m2.1.2.cmml">(</mo><mi id="S2.SS3.SSS1.p3.2.m2.1.1" xref="S2.SS3.SSS1.p3.2.m2.1.1.cmml">b</mi><mo id="S2.SS3.SSS1.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS3.SSS1.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p3.2.m2.1b"><apply id="S2.SS3.SSS1.p3.2.m2.1.2.cmml" xref="S2.SS3.SSS1.p3.2.m2.1.2"><times id="S2.SS3.SSS1.p3.2.m2.1.2.1.cmml" xref="S2.SS3.SSS1.p3.2.m2.1.2.1"></times><apply id="S2.SS3.SSS1.p3.2.m2.1.2.2.cmml" xref="S2.SS3.SSS1.p3.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS1.p3.2.m2.1.2.2.1.cmml" xref="S2.SS3.SSS1.p3.2.m2.1.2.2">subscript</csymbol><ci id="S2.SS3.SSS1.p3.2.m2.1.2.2.2.cmml" xref="S2.SS3.SSS1.p3.2.m2.1.2.2.2">𝜎</ci><ci id="S2.SS3.SSS1.p3.2.m2.1.2.2.3a.cmml" xref="S2.SS3.SSS1.p3.2.m2.1.2.2.3"><mtext id="S2.SS3.SSS1.p3.2.m2.1.2.2.3.cmml" mathsize="70%" xref="S2.SS3.SSS1.p3.2.m2.1.2.2.3">TM</mtext></ci></apply><ci id="S2.SS3.SSS1.p3.2.m2.1.1.cmml" xref="S2.SS3.SSS1.p3.2.m2.1.1">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p3.2.m2.1c">\sigma_{\text{TM}}(b)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p3.2.m2.1d">italic_σ start_POSTSUBSCRIPT TM end_POSTSUBSCRIPT ( italic_b )</annotation></semantics></math> are conjugate in <math alttext="F(\{c,d\})" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p3.3.m3.3"><semantics id="S2.SS3.SSS1.p3.3.m3.3a"><mrow id="S2.SS3.SSS1.p3.3.m3.3.3" xref="S2.SS3.SSS1.p3.3.m3.3.3.cmml"><mi id="S2.SS3.SSS1.p3.3.m3.3.3.3" xref="S2.SS3.SSS1.p3.3.m3.3.3.3.cmml">F</mi><mo id="S2.SS3.SSS1.p3.3.m3.3.3.2" xref="S2.SS3.SSS1.p3.3.m3.3.3.2.cmml">⁢</mo><mrow id="S2.SS3.SSS1.p3.3.m3.3.3.1.1" xref="S2.SS3.SSS1.p3.3.m3.3.3.cmml"><mo id="S2.SS3.SSS1.p3.3.m3.3.3.1.1.2" stretchy="false" xref="S2.SS3.SSS1.p3.3.m3.3.3.cmml">(</mo><mrow id="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.2" xref="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.1.cmml"><mo id="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.2.1" stretchy="false" xref="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.1.cmml">{</mo><mi id="S2.SS3.SSS1.p3.3.m3.1.1" xref="S2.SS3.SSS1.p3.3.m3.1.1.cmml">c</mi><mo id="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.2.2" xref="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.1.cmml">,</mo><mi id="S2.SS3.SSS1.p3.3.m3.2.2" xref="S2.SS3.SSS1.p3.3.m3.2.2.cmml">d</mi><mo id="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.2.3" stretchy="false" xref="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.1.cmml">}</mo></mrow><mo id="S2.SS3.SSS1.p3.3.m3.3.3.1.1.3" stretchy="false" xref="S2.SS3.SSS1.p3.3.m3.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p3.3.m3.3b"><apply id="S2.SS3.SSS1.p3.3.m3.3.3.cmml" xref="S2.SS3.SSS1.p3.3.m3.3.3"><times id="S2.SS3.SSS1.p3.3.m3.3.3.2.cmml" xref="S2.SS3.SSS1.p3.3.m3.3.3.2"></times><ci id="S2.SS3.SSS1.p3.3.m3.3.3.3.cmml" xref="S2.SS3.SSS1.p3.3.m3.3.3.3">𝐹</ci><set id="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.1.cmml" xref="S2.SS3.SSS1.p3.3.m3.3.3.1.1.1.2"><ci id="S2.SS3.SSS1.p3.3.m3.1.1.cmml" xref="S2.SS3.SSS1.p3.3.m3.1.1">𝑐</ci><ci id="S2.SS3.SSS1.p3.3.m3.2.2.cmml" xref="S2.SS3.SSS1.p3.3.m3.2.2">𝑑</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p3.3.m3.3c">F(\{c,d\})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p3.3.m3.3d">italic_F ( { italic_c , italic_d } )</annotation></semantics></math>, while <math alttext="\sigma_{\text{TM}}" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p3.4.m4.1"><semantics id="S2.SS3.SSS1.p3.4.m4.1a"><msub id="S2.SS3.SSS1.p3.4.m4.1.1" xref="S2.SS3.SSS1.p3.4.m4.1.1.cmml"><mi id="S2.SS3.SSS1.p3.4.m4.1.1.2" xref="S2.SS3.SSS1.p3.4.m4.1.1.2.cmml">σ</mi><mtext id="S2.SS3.SSS1.p3.4.m4.1.1.3" xref="S2.SS3.SSS1.p3.4.m4.1.1.3a.cmml">TM</mtext></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p3.4.m4.1b"><apply id="S2.SS3.SSS1.p3.4.m4.1.1.cmml" xref="S2.SS3.SSS1.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS1.p3.4.m4.1.1.1.cmml" xref="S2.SS3.SSS1.p3.4.m4.1.1">subscript</csymbol><ci id="S2.SS3.SSS1.p3.4.m4.1.1.2.cmml" xref="S2.SS3.SSS1.p3.4.m4.1.1.2">𝜎</ci><ci id="S2.SS3.SSS1.p3.4.m4.1.1.3a.cmml" xref="S2.SS3.SSS1.p3.4.m4.1.1.3"><mtext id="S2.SS3.SSS1.p3.4.m4.1.1.3.cmml" mathsize="70%" xref="S2.SS3.SSS1.p3.4.m4.1.1.3">TM</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p3.4.m4.1c">\sigma_{\text{TM}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p3.4.m4.1d">italic_σ start_POSTSUBSCRIPT TM end_POSTSUBSCRIPT</annotation></semantics></math> is injective on all of <math alttext="F(\{a,b\})" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p3.5.m5.3"><semantics id="S2.SS3.SSS1.p3.5.m5.3a"><mrow id="S2.SS3.SSS1.p3.5.m5.3.3" xref="S2.SS3.SSS1.p3.5.m5.3.3.cmml"><mi id="S2.SS3.SSS1.p3.5.m5.3.3.3" xref="S2.SS3.SSS1.p3.5.m5.3.3.3.cmml">F</mi><mo id="S2.SS3.SSS1.p3.5.m5.3.3.2" xref="S2.SS3.SSS1.p3.5.m5.3.3.2.cmml">⁢</mo><mrow id="S2.SS3.SSS1.p3.5.m5.3.3.1.1" xref="S2.SS3.SSS1.p3.5.m5.3.3.cmml"><mo id="S2.SS3.SSS1.p3.5.m5.3.3.1.1.2" stretchy="false" xref="S2.SS3.SSS1.p3.5.m5.3.3.cmml">(</mo><mrow id="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.2" xref="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.1.cmml"><mo id="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.2.1" stretchy="false" xref="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.1.cmml">{</mo><mi id="S2.SS3.SSS1.p3.5.m5.1.1" xref="S2.SS3.SSS1.p3.5.m5.1.1.cmml">a</mi><mo id="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.2.2" xref="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.1.cmml">,</mo><mi id="S2.SS3.SSS1.p3.5.m5.2.2" xref="S2.SS3.SSS1.p3.5.m5.2.2.cmml">b</mi><mo id="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.2.3" stretchy="false" xref="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.1.cmml">}</mo></mrow><mo id="S2.SS3.SSS1.p3.5.m5.3.3.1.1.3" stretchy="false" xref="S2.SS3.SSS1.p3.5.m5.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p3.5.m5.3b"><apply id="S2.SS3.SSS1.p3.5.m5.3.3.cmml" xref="S2.SS3.SSS1.p3.5.m5.3.3"><times id="S2.SS3.SSS1.p3.5.m5.3.3.2.cmml" xref="S2.SS3.SSS1.p3.5.m5.3.3.2"></times><ci id="S2.SS3.SSS1.p3.5.m5.3.3.3.cmml" xref="S2.SS3.SSS1.p3.5.m5.3.3.3">𝐹</ci><set id="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.1.cmml" xref="S2.SS3.SSS1.p3.5.m5.3.3.1.1.1.2"><ci id="S2.SS3.SSS1.p3.5.m5.1.1.cmml" xref="S2.SS3.SSS1.p3.5.m5.1.1">𝑎</ci><ci id="S2.SS3.SSS1.p3.5.m5.2.2.cmml" xref="S2.SS3.SSS1.p3.5.m5.2.2">𝑏</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p3.5.m5.3c">F(\{a,b\})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p3.5.m5.3d">italic_F ( { italic_a , italic_b } )</annotation></semantics></math>. This yields:</p> </div> <div class="ltx_theorem ltx_theorem_rem" id="S2.Thmthm8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm8.1.1.1">Remark 2.8</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm8.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm8.p1"> <p class="ltx_p" id="S2.Thmthm8.p1.3">There exist non-erasing monoid morphisms <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.Thmthm8.p1.1.m1.1"><semantics id="S2.Thmthm8.p1.1.m1.1a"><mi id="S2.Thmthm8.p1.1.m1.1.1" xref="S2.Thmthm8.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm8.p1.1.m1.1b"><ci id="S2.Thmthm8.p1.1.m1.1.1.cmml" xref="S2.Thmthm8.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm8.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm8.p1.1.m1.1d">italic_σ</annotation></semantics></math> which are injective, but for which the induced map <math alttext="\sigma^{T}" class="ltx_Math" display="inline" id="S2.Thmthm8.p1.2.m2.1"><semantics id="S2.Thmthm8.p1.2.m2.1a"><msup id="S2.Thmthm8.p1.2.m2.1.1" xref="S2.Thmthm8.p1.2.m2.1.1.cmml"><mi id="S2.Thmthm8.p1.2.m2.1.1.2" xref="S2.Thmthm8.p1.2.m2.1.1.2.cmml">σ</mi><mi id="S2.Thmthm8.p1.2.m2.1.1.3" xref="S2.Thmthm8.p1.2.m2.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm8.p1.2.m2.1b"><apply id="S2.Thmthm8.p1.2.m2.1.1.cmml" xref="S2.Thmthm8.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.Thmthm8.p1.2.m2.1.1.1.cmml" xref="S2.Thmthm8.p1.2.m2.1.1">superscript</csymbol><ci id="S2.Thmthm8.p1.2.m2.1.1.2.cmml" xref="S2.Thmthm8.p1.2.m2.1.1.2">𝜎</ci><ci id="S2.Thmthm8.p1.2.m2.1.1.3.cmml" xref="S2.Thmthm8.p1.2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm8.p1.2.m2.1c">\sigma^{T}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm8.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> is not injective. For example, for the Thue-Morse morphism one has <math alttext="\cal O(\sigma_{\text{TM}}(a^{\pm\infty}))=\cal O(\sigma_{\text{TM}}(b^{\pm% \infty}))" class="ltx_Math" display="inline" id="S2.Thmthm8.p1.3.m3.2"><semantics id="S2.Thmthm8.p1.3.m3.2a"><mrow id="S2.Thmthm8.p1.3.m3.2.2" xref="S2.Thmthm8.p1.3.m3.2.2.cmml"><mrow id="S2.Thmthm8.p1.3.m3.1.1.1" xref="S2.Thmthm8.p1.3.m3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm8.p1.3.m3.1.1.1.3" xref="S2.Thmthm8.p1.3.m3.1.1.1.3.cmml">𝒪</mi><mo id="S2.Thmthm8.p1.3.m3.1.1.1.2" xref="S2.Thmthm8.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm8.p1.3.m3.1.1.1.1.1" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.cmml"><msub id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.3" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.3.cmml"><mi id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.3.2" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.3.2.cmml">σ</mi><mtext id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.3.3" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.3.3a.cmml">TM</mtext></msub><mo id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.2" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.2" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.2.cmml">𝒶</mi><mrow id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3a" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3.cmml">±</mo><mi id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm8.p1.3.m3.2.2.3" xref="S2.Thmthm8.p1.3.m3.2.2.3.cmml">=</mo><mrow id="S2.Thmthm8.p1.3.m3.2.2.2" xref="S2.Thmthm8.p1.3.m3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm8.p1.3.m3.2.2.2.3" xref="S2.Thmthm8.p1.3.m3.2.2.2.3.cmml">𝒪</mi><mo id="S2.Thmthm8.p1.3.m3.2.2.2.2" xref="S2.Thmthm8.p1.3.m3.2.2.2.2.cmml">⁢</mo><mrow id="S2.Thmthm8.p1.3.m3.2.2.2.1.1" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.cmml"><mo id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.2" stretchy="false" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.cmml"><msub id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.cmml"><mi id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.2" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.2.cmml">σ</mi><mtext id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.3" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.3a.cmml">TM</mtext></msub><mo id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.2" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.cmml"><mo id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.2" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.2.cmml">𝒷</mi><mrow id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3.cmml"><mo id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3a" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3.cmml">±</mo><mi id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.3" stretchy="false" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm8.p1.3.m3.2b"><apply id="S2.Thmthm8.p1.3.m3.2.2.cmml" xref="S2.Thmthm8.p1.3.m3.2.2"><eq id="S2.Thmthm8.p1.3.m3.2.2.3.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.3"></eq><apply id="S2.Thmthm8.p1.3.m3.1.1.1.cmml" xref="S2.Thmthm8.p1.3.m3.1.1.1"><times id="S2.Thmthm8.p1.3.m3.1.1.1.2.cmml" xref="S2.Thmthm8.p1.3.m3.1.1.1.2"></times><ci id="S2.Thmthm8.p1.3.m3.1.1.1.3.cmml" xref="S2.Thmthm8.p1.3.m3.1.1.1.3">𝒪</ci><apply id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.cmml" 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xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.2">𝒶</ci><apply id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm8.p1.3.m3.1.1.1.1.1.1.1.1.1.3.2"></infinity></apply></apply></apply></apply><apply id="S2.Thmthm8.p1.3.m3.2.2.2.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2"><times id="S2.Thmthm8.p1.3.m3.2.2.2.2.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.2"></times><ci id="S2.Thmthm8.p1.3.m3.2.2.2.3.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.3">𝒪</ci><apply id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1"><times id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.2.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.2"></times><apply id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.1.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3">subscript</csymbol><ci id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.2.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.2">𝜎</ci><ci id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.3a.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.3"><mtext id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.3.cmml" mathsize="70%" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.3.3">TM</mtext></ci></apply><apply id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.1.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1">superscript</csymbol><ci id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.2.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.2">𝒷</ci><apply id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm8.p1.3.m3.2.2.2.1.1.1.1.1.1.3.2"></infinity></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm8.p1.3.m3.2c">\cal O(\sigma_{\text{TM}}(a^{\pm\infty}))=\cal O(\sigma_{\text{TM}}(b^{\pm% \infty}))</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm8.p1.3.m3.2d">caligraphic_O ( italic_σ start_POSTSUBSCRIPT TM end_POSTSUBSCRIPT ( caligraphic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) ) = caligraphic_O ( italic_σ start_POSTSUBSCRIPT TM end_POSTSUBSCRIPT ( caligraphic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.SS3.SSS1.p4"> <p class="ltx_p" id="S2.SS3.SSS1.p4.1">Two further, more subtle “injectivity disturbances” can be observed by the following two examples, despite the fact that in both cases the induced map on the shift-orbits is injective. Indeed, in both cases we consider a subshift which consists of a single periodic shift-orbit: in the first case the map <math alttext="\sigma^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS3.SSS1.p4.1.m1.1"><semantics id="S2.SS3.SSS1.p4.1.m1.1a"><msup id="S2.SS3.SSS1.p4.1.m1.1.1" xref="S2.SS3.SSS1.p4.1.m1.1.1.cmml"><mi id="S2.SS3.SSS1.p4.1.m1.1.1.2" xref="S2.SS3.SSS1.p4.1.m1.1.1.2.cmml">σ</mi><mi id="S2.SS3.SSS1.p4.1.m1.1.1.3" xref="S2.SS3.SSS1.p4.1.m1.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS1.p4.1.m1.1b"><apply id="S2.SS3.SSS1.p4.1.m1.1.1.cmml" xref="S2.SS3.SSS1.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS1.p4.1.m1.1.1.1.cmml" xref="S2.SS3.SSS1.p4.1.m1.1.1">superscript</csymbol><ci id="S2.SS3.SSS1.p4.1.m1.1.1.2.cmml" xref="S2.SS3.SSS1.p4.1.m1.1.1.2">𝜎</ci><ci id="S2.SS3.SSS1.p4.1.m1.1.1.3.cmml" xref="S2.SS3.SSS1.p4.1.m1.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS1.p4.1.m1.1c">\sigma^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS1.p4.1.m1.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> restricted to this orbit is not injective, while in the second case it is, but a minimal period in the preimage orbit is not mapped to a minimal period in the image.</p> </div> <div class="ltx_theorem ltx_theorem_rem" id="S2.Thmthm9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm9.1.1.1">Remark 2.9</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm9.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm9.p1"> <p class="ltx_p" id="S2.Thmthm9.p1.5">(1) The morphism <math alttext="\sigma_{1}:\{a,b\}^{*}\to\{c\}^{*}\,,\,\,\sigma_{1}(a)=\sigma_{1}(b)=c" class="ltx_Math" display="inline" id="S2.Thmthm9.p1.1.m1.7"><semantics id="S2.Thmthm9.p1.1.m1.7a"><mrow id="S2.Thmthm9.p1.1.m1.7.7" xref="S2.Thmthm9.p1.1.m1.7.7.cmml"><msub id="S2.Thmthm9.p1.1.m1.7.7.4" xref="S2.Thmthm9.p1.1.m1.7.7.4.cmml"><mi id="S2.Thmthm9.p1.1.m1.7.7.4.2" xref="S2.Thmthm9.p1.1.m1.7.7.4.2.cmml">σ</mi><mn id="S2.Thmthm9.p1.1.m1.7.7.4.3" xref="S2.Thmthm9.p1.1.m1.7.7.4.3.cmml">1</mn></msub><mo id="S2.Thmthm9.p1.1.m1.7.7.3" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm9.p1.1.m1.7.7.3.cmml">:</mo><mrow id="S2.Thmthm9.p1.1.m1.7.7.2.2" xref="S2.Thmthm9.p1.1.m1.7.7.2.3.cmml"><mrow id="S2.Thmthm9.p1.1.m1.6.6.1.1.1" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.cmml"><msup id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.cmml"><mrow id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.2.2" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.2.1.cmml"><mo id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.2.2.1" stretchy="false" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.2.1.cmml">{</mo><mi id="S2.Thmthm9.p1.1.m1.1.1" xref="S2.Thmthm9.p1.1.m1.1.1.cmml">a</mi><mo id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.2.2.2" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.2.1.cmml">,</mo><mi id="S2.Thmthm9.p1.1.m1.2.2" xref="S2.Thmthm9.p1.1.m1.2.2.cmml">b</mi><mo id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.2.2.3" stretchy="false" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.2.1.cmml">}</mo></mrow><mo id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.3" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.1" stretchy="false" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.1.cmml">→</mo><msup id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3.cmml"><mrow id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3.2.2" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3.2.1.cmml"><mo id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3.2.2.1" stretchy="false" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3.2.1.cmml">{</mo><mi id="S2.Thmthm9.p1.1.m1.3.3" xref="S2.Thmthm9.p1.1.m1.3.3.cmml">c</mi><mo id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3.2.2.2" stretchy="false" xref="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3.2.1.cmml">}</mo></mrow><mo id="S2.Thmthm9.p1.1.m1.6.6.1.1.1.3.3" 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xref="S2.Thmthm9.p1.1.m1.7.7.2.2.2.4.2.3">1</cn></apply><ci id="S2.Thmthm9.p1.1.m1.5.5.cmml" xref="S2.Thmthm9.p1.1.m1.5.5">𝑏</ci></apply></apply><apply id="S2.Thmthm9.p1.1.m1.7.7.2.2.2c.cmml" xref="S2.Thmthm9.p1.1.m1.7.7.2.2.2"><eq id="S2.Thmthm9.p1.1.m1.7.7.2.2.2.5.cmml" xref="S2.Thmthm9.p1.1.m1.7.7.2.2.2.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S2.Thmthm9.p1.1.m1.7.7.2.2.2.4.cmml" id="S2.Thmthm9.p1.1.m1.7.7.2.2.2d.cmml" xref="S2.Thmthm9.p1.1.m1.7.7.2.2.2"></share><ci id="S2.Thmthm9.p1.1.m1.7.7.2.2.2.6.cmml" xref="S2.Thmthm9.p1.1.m1.7.7.2.2.2.6">𝑐</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p1.1.m1.7c">\sigma_{1}:\{a,b\}^{*}\to\{c\}^{*}\,,\,\,\sigma_{1}(a)=\sigma_{1}(b)=c</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p1.1.m1.7d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : { italic_a , italic_b } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → { italic_c } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a ) = italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_b ) = italic_c</annotation></semantics></math> maps the orbit <math alttext="\cal O((ab)^{\pm\infty})" class="ltx_Math" display="inline" id="S2.Thmthm9.p1.2.m2.1"><semantics id="S2.Thmthm9.p1.2.m2.1a"><mrow id="S2.Thmthm9.p1.2.m2.1.1" xref="S2.Thmthm9.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p1.2.m2.1.1.3" xref="S2.Thmthm9.p1.2.m2.1.1.3.cmml">𝒪</mi><mo id="S2.Thmthm9.p1.2.m2.1.1.2" xref="S2.Thmthm9.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm9.p1.2.m2.1.1.1.1" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.cmml"><mo id="S2.Thmthm9.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmthm9.p1.2.m2.1.1.1.1.1" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.cmml"><mrow id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.2" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.2.cmml">𝒶</mi><mo id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.1" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.3" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.3.cmml">𝒷</mi></mrow><mo id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3.cmml"><mo id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3a" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm9.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p1.2.m2.1b"><apply id="S2.Thmthm9.p1.2.m2.1.1.cmml" xref="S2.Thmthm9.p1.2.m2.1.1"><times id="S2.Thmthm9.p1.2.m2.1.1.2.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.2"></times><ci id="S2.Thmthm9.p1.2.m2.1.1.3.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.3">𝒪</ci><apply id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.2.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1">superscript</csymbol><apply id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1"><times id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.1"></times><ci id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.2">𝒶</ci><ci id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.1.1.1.3">𝒷</ci></apply><apply id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm9.p1.2.m2.1.1.1.1.1.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p1.2.m2.1c">\cal O((ab)^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p1.2.m2.1d">caligraphic_O ( ( caligraphic_a caligraphic_b ) start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math> to <math alttext="\cal O(c^{\pm\infty})" class="ltx_Math" display="inline" id="S2.Thmthm9.p1.3.m3.1"><semantics id="S2.Thmthm9.p1.3.m3.1a"><mrow id="S2.Thmthm9.p1.3.m3.1.1" xref="S2.Thmthm9.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p1.3.m3.1.1.3" xref="S2.Thmthm9.p1.3.m3.1.1.3.cmml">𝒪</mi><mo id="S2.Thmthm9.p1.3.m3.1.1.2" xref="S2.Thmthm9.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm9.p1.3.m3.1.1.1.1" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.cmml"><mo id="S2.Thmthm9.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmthm9.p1.3.m3.1.1.1.1.1" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.2" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.2.cmml">𝒸</mi><mrow id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3.cmml"><mo id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3a" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm9.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p1.3.m3.1b"><apply id="S2.Thmthm9.p1.3.m3.1.1.cmml" xref="S2.Thmthm9.p1.3.m3.1.1"><times id="S2.Thmthm9.p1.3.m3.1.1.2.cmml" xref="S2.Thmthm9.p1.3.m3.1.1.2"></times><ci id="S2.Thmthm9.p1.3.m3.1.1.3.cmml" xref="S2.Thmthm9.p1.3.m3.1.1.3">𝒪</ci><apply id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.cmml" xref="S2.Thmthm9.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.1.cmml" xref="S2.Thmthm9.p1.3.m3.1.1.1.1">superscript</csymbol><ci id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.2.cmml" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.2">𝒸</ci><apply id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3.cmml" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm9.p1.3.m3.1.1.1.1.1.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p1.3.m3.1c">\cal O(c^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p1.3.m3.1d">caligraphic_O ( caligraphic_c start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math>, but the restriction of <math alttext="\sigma_{1}^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm9.p1.4.m4.1"><semantics id="S2.Thmthm9.p1.4.m4.1a"><msubsup id="S2.Thmthm9.p1.4.m4.1.1" xref="S2.Thmthm9.p1.4.m4.1.1.cmml"><mi id="S2.Thmthm9.p1.4.m4.1.1.2.2" xref="S2.Thmthm9.p1.4.m4.1.1.2.2.cmml">σ</mi><mn id="S2.Thmthm9.p1.4.m4.1.1.2.3" xref="S2.Thmthm9.p1.4.m4.1.1.2.3.cmml">1</mn><mi id="S2.Thmthm9.p1.4.m4.1.1.3" xref="S2.Thmthm9.p1.4.m4.1.1.3.cmml">ℤ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p1.4.m4.1b"><apply id="S2.Thmthm9.p1.4.m4.1.1.cmml" xref="S2.Thmthm9.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p1.4.m4.1.1.1.cmml" xref="S2.Thmthm9.p1.4.m4.1.1">superscript</csymbol><apply id="S2.Thmthm9.p1.4.m4.1.1.2.cmml" xref="S2.Thmthm9.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p1.4.m4.1.1.2.1.cmml" xref="S2.Thmthm9.p1.4.m4.1.1">subscript</csymbol><ci id="S2.Thmthm9.p1.4.m4.1.1.2.2.cmml" xref="S2.Thmthm9.p1.4.m4.1.1.2.2">𝜎</ci><cn id="S2.Thmthm9.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S2.Thmthm9.p1.4.m4.1.1.2.3">1</cn></apply><ci id="S2.Thmthm9.p1.4.m4.1.1.3.cmml" xref="S2.Thmthm9.p1.4.m4.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p1.4.m4.1c">\sigma_{1}^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p1.4.m4.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\cal O((ab)^{\pm\infty})" class="ltx_Math" display="inline" id="S2.Thmthm9.p1.5.m5.1"><semantics id="S2.Thmthm9.p1.5.m5.1a"><mrow id="S2.Thmthm9.p1.5.m5.1.1" xref="S2.Thmthm9.p1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p1.5.m5.1.1.3" xref="S2.Thmthm9.p1.5.m5.1.1.3.cmml">𝒪</mi><mo id="S2.Thmthm9.p1.5.m5.1.1.2" xref="S2.Thmthm9.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm9.p1.5.m5.1.1.1.1" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.cmml"><mo id="S2.Thmthm9.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmthm9.p1.5.m5.1.1.1.1.1" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.cmml"><mrow id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.2" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.2.cmml">𝒶</mi><mo id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.1" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.3" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.3.cmml">𝒷</mi></mrow><mo id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3.cmml"><mo id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3a" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm9.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p1.5.m5.1b"><apply id="S2.Thmthm9.p1.5.m5.1.1.cmml" xref="S2.Thmthm9.p1.5.m5.1.1"><times id="S2.Thmthm9.p1.5.m5.1.1.2.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.2"></times><ci id="S2.Thmthm9.p1.5.m5.1.1.3.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.3">𝒪</ci><apply id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.2.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1">superscript</csymbol><apply id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1"><times id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.1"></times><ci id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.2.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.2">𝒶</ci><ci id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.3.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.1.1.1.3">𝒷</ci></apply><apply id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm9.p1.5.m5.1.1.1.1.1.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p1.5.m5.1c">\cal O((ab)^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p1.5.m5.1d">caligraphic_O ( ( caligraphic_a caligraphic_b ) start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math> is not injective.</p> </div> <div class="ltx_para ltx_noindent" id="S2.Thmthm9.p2"> <p class="ltx_p" id="S2.Thmthm9.p2.9">(2) The morphism <math alttext="\sigma_{2}:\{a\}^{*}\to\{b\}^{*}\,,\,\,\sigma_{2}(a)=b^{2}" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.1.m1.5"><semantics id="S2.Thmthm9.p2.1.m1.5a"><mrow id="S2.Thmthm9.p2.1.m1.5.5" xref="S2.Thmthm9.p2.1.m1.5.5.cmml"><msub id="S2.Thmthm9.p2.1.m1.5.5.4" xref="S2.Thmthm9.p2.1.m1.5.5.4.cmml"><mi id="S2.Thmthm9.p2.1.m1.5.5.4.2" xref="S2.Thmthm9.p2.1.m1.5.5.4.2.cmml">σ</mi><mn id="S2.Thmthm9.p2.1.m1.5.5.4.3" xref="S2.Thmthm9.p2.1.m1.5.5.4.3.cmml">2</mn></msub><mo id="S2.Thmthm9.p2.1.m1.5.5.3" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm9.p2.1.m1.5.5.3.cmml">:</mo><mrow id="S2.Thmthm9.p2.1.m1.5.5.2.2" xref="S2.Thmthm9.p2.1.m1.5.5.2.3.cmml"><mrow id="S2.Thmthm9.p2.1.m1.4.4.1.1.1" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.cmml"><msup id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.cmml"><mrow id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.2.2" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.2.1.cmml"><mo id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.2.2.1" stretchy="false" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.2.1.cmml">{</mo><mi id="S2.Thmthm9.p2.1.m1.1.1" xref="S2.Thmthm9.p2.1.m1.1.1.cmml">a</mi><mo id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.2.2.2" stretchy="false" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.2.1.cmml">}</mo></mrow><mo id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.3" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.1" stretchy="false" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.1.cmml">→</mo><msup id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.cmml"><mrow id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.2.2" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.2.1.cmml"><mo id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.2.2.1" stretchy="false" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.2.1.cmml">{</mo><mi id="S2.Thmthm9.p2.1.m1.2.2" xref="S2.Thmthm9.p2.1.m1.2.2.cmml">b</mi><mo id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.2.2.2" stretchy="false" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.2.1.cmml">}</mo></mrow><mo id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.3" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S2.Thmthm9.p2.1.m1.5.5.2.2.3" rspace="0.497em" xref="S2.Thmthm9.p2.1.m1.5.5.2.3a.cmml">,</mo><mrow id="S2.Thmthm9.p2.1.m1.5.5.2.2.2" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.cmml"><mrow id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.cmml"><msub id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.cmml"><mi id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.2" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.2.cmml">σ</mi><mn id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.3" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.1" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.3.2" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.cmml"><mo id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.3.2.1" stretchy="false" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.cmml">(</mo><mi id="S2.Thmthm9.p2.1.m1.3.3" xref="S2.Thmthm9.p2.1.m1.3.3.cmml">a</mi><mo id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.3.2.2" stretchy="false" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.1" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.1.cmml">=</mo><msup id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.cmml"><mi id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.2" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.2.cmml">b</mi><mn id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.3" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.3.cmml">2</mn></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.1.m1.5b"><apply id="S2.Thmthm9.p2.1.m1.5.5.cmml" xref="S2.Thmthm9.p2.1.m1.5.5"><ci id="S2.Thmthm9.p2.1.m1.5.5.3.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.3">:</ci><apply id="S2.Thmthm9.p2.1.m1.5.5.4.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.4"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.1.m1.5.5.4.1.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.4">subscript</csymbol><ci id="S2.Thmthm9.p2.1.m1.5.5.4.2.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.4.2">𝜎</ci><cn id="S2.Thmthm9.p2.1.m1.5.5.4.3.cmml" type="integer" xref="S2.Thmthm9.p2.1.m1.5.5.4.3">2</cn></apply><apply id="S2.Thmthm9.p2.1.m1.5.5.2.3.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.1.m1.5.5.2.3a.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.3">formulae-sequence</csymbol><apply id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1"><ci id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.1.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.1">→</ci><apply id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.1.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2">superscript</csymbol><set id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.2.1.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.2.2"><ci id="S2.Thmthm9.p2.1.m1.1.1.cmml" xref="S2.Thmthm9.p2.1.m1.1.1">𝑎</ci></set><times id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.3.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.2.3"></times></apply><apply id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.1.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3">superscript</csymbol><set id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.2.1.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.2.2"><ci id="S2.Thmthm9.p2.1.m1.2.2.cmml" xref="S2.Thmthm9.p2.1.m1.2.2">𝑏</ci></set><times id="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.3.cmml" xref="S2.Thmthm9.p2.1.m1.4.4.1.1.1.3.3"></times></apply></apply><apply id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2"><eq id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.1.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.1"></eq><apply id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2"><times id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.1.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.1"></times><apply id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.1.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2">subscript</csymbol><ci id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.2.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.2">𝜎</ci><cn id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.3.cmml" type="integer" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.2.2.3">2</cn></apply><ci id="S2.Thmthm9.p2.1.m1.3.3.cmml" xref="S2.Thmthm9.p2.1.m1.3.3">𝑎</ci></apply><apply id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.1.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3">superscript</csymbol><ci id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.2.cmml" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.2">𝑏</ci><cn id="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.3.cmml" type="integer" xref="S2.Thmthm9.p2.1.m1.5.5.2.2.2.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.1.m1.5c">\sigma_{2}:\{a\}^{*}\to\{b\}^{*}\,,\,\,\sigma_{2}(a)=b^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.1.m1.5d">italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : { italic_a } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → { italic_b } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_a ) = italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> maps the orbit <math alttext="\cal O(a^{\pm\infty})" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.2.m2.1"><semantics id="S2.Thmthm9.p2.2.m2.1a"><mrow id="S2.Thmthm9.p2.2.m2.1.1" xref="S2.Thmthm9.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p2.2.m2.1.1.3" xref="S2.Thmthm9.p2.2.m2.1.1.3.cmml">𝒪</mi><mo id="S2.Thmthm9.p2.2.m2.1.1.2" xref="S2.Thmthm9.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm9.p2.2.m2.1.1.1.1" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.cmml"><mo id="S2.Thmthm9.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmthm9.p2.2.m2.1.1.1.1.1" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.2" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.2.cmml">𝒶</mi><mrow id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3.cmml"><mo id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3a" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm9.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.2.m2.1b"><apply id="S2.Thmthm9.p2.2.m2.1.1.cmml" xref="S2.Thmthm9.p2.2.m2.1.1"><times id="S2.Thmthm9.p2.2.m2.1.1.2.cmml" xref="S2.Thmthm9.p2.2.m2.1.1.2"></times><ci id="S2.Thmthm9.p2.2.m2.1.1.3.cmml" xref="S2.Thmthm9.p2.2.m2.1.1.3">𝒪</ci><apply id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.cmml" xref="S2.Thmthm9.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.1.cmml" xref="S2.Thmthm9.p2.2.m2.1.1.1.1">superscript</csymbol><ci id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.2.cmml" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.2">𝒶</ci><apply id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3.cmml" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm9.p2.2.m2.1.1.1.1.1.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.2.m2.1c">\cal O(a^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.2.m2.1d">caligraphic_O ( caligraphic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math> injectively to <math alttext="\cal O(b^{\pm\infty})" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.3.m3.1"><semantics id="S2.Thmthm9.p2.3.m3.1a"><mrow id="S2.Thmthm9.p2.3.m3.1.1" xref="S2.Thmthm9.p2.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p2.3.m3.1.1.3" xref="S2.Thmthm9.p2.3.m3.1.1.3.cmml">𝒪</mi><mo id="S2.Thmthm9.p2.3.m3.1.1.2" xref="S2.Thmthm9.p2.3.m3.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm9.p2.3.m3.1.1.1.1" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.cmml"><mo id="S2.Thmthm9.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmthm9.p2.3.m3.1.1.1.1.1" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.2" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.2.cmml">𝒷</mi><mrow id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3.cmml"><mo id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3a" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S2.Thmthm9.p2.3.m3.1.1.1.1.3" stretchy="false" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.3.m3.1b"><apply id="S2.Thmthm9.p2.3.m3.1.1.cmml" xref="S2.Thmthm9.p2.3.m3.1.1"><times id="S2.Thmthm9.p2.3.m3.1.1.2.cmml" xref="S2.Thmthm9.p2.3.m3.1.1.2"></times><ci id="S2.Thmthm9.p2.3.m3.1.1.3.cmml" xref="S2.Thmthm9.p2.3.m3.1.1.3">𝒪</ci><apply id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.cmml" xref="S2.Thmthm9.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.1.cmml" xref="S2.Thmthm9.p2.3.m3.1.1.1.1">superscript</csymbol><ci id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.2.cmml" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.2">𝒷</ci><apply id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3.cmml" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3.1.cmml" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3.2.cmml" xref="S2.Thmthm9.p2.3.m3.1.1.1.1.1.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.3.m3.1c">\cal O(b^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.3.m3.1d">caligraphic_O ( caligraphic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math>, but the “shift-period” is not preserved: For both, <math alttext="a^{\pm\infty}" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.4.m4.1"><semantics id="S2.Thmthm9.p2.4.m4.1a"><msup id="S2.Thmthm9.p2.4.m4.1.1" xref="S2.Thmthm9.p2.4.m4.1.1.cmml"><mi id="S2.Thmthm9.p2.4.m4.1.1.2" xref="S2.Thmthm9.p2.4.m4.1.1.2.cmml">a</mi><mrow id="S2.Thmthm9.p2.4.m4.1.1.3" xref="S2.Thmthm9.p2.4.m4.1.1.3.cmml"><mo id="S2.Thmthm9.p2.4.m4.1.1.3a" xref="S2.Thmthm9.p2.4.m4.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p2.4.m4.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p2.4.m4.1.1.3.2.cmml">∞</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.4.m4.1b"><apply id="S2.Thmthm9.p2.4.m4.1.1.cmml" xref="S2.Thmthm9.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.4.m4.1.1.1.cmml" xref="S2.Thmthm9.p2.4.m4.1.1">superscript</csymbol><ci id="S2.Thmthm9.p2.4.m4.1.1.2.cmml" xref="S2.Thmthm9.p2.4.m4.1.1.2">𝑎</ci><apply id="S2.Thmthm9.p2.4.m4.1.1.3.cmml" xref="S2.Thmthm9.p2.4.m4.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p2.4.m4.1.1.3.1.cmml" xref="S2.Thmthm9.p2.4.m4.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p2.4.m4.1.1.3.2.cmml" xref="S2.Thmthm9.p2.4.m4.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.4.m4.1c">a^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.4.m4.1d">italic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="b^{\pm\infty}" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.5.m5.1"><semantics id="S2.Thmthm9.p2.5.m5.1a"><msup id="S2.Thmthm9.p2.5.m5.1.1" xref="S2.Thmthm9.p2.5.m5.1.1.cmml"><mi id="S2.Thmthm9.p2.5.m5.1.1.2" xref="S2.Thmthm9.p2.5.m5.1.1.2.cmml">b</mi><mrow id="S2.Thmthm9.p2.5.m5.1.1.3" xref="S2.Thmthm9.p2.5.m5.1.1.3.cmml"><mo id="S2.Thmthm9.p2.5.m5.1.1.3a" xref="S2.Thmthm9.p2.5.m5.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p2.5.m5.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p2.5.m5.1.1.3.2.cmml">∞</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.5.m5.1b"><apply id="S2.Thmthm9.p2.5.m5.1.1.cmml" xref="S2.Thmthm9.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.5.m5.1.1.1.cmml" xref="S2.Thmthm9.p2.5.m5.1.1">superscript</csymbol><ci id="S2.Thmthm9.p2.5.m5.1.1.2.cmml" xref="S2.Thmthm9.p2.5.m5.1.1.2">𝑏</ci><apply id="S2.Thmthm9.p2.5.m5.1.1.3.cmml" xref="S2.Thmthm9.p2.5.m5.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p2.5.m5.1.1.3.1.cmml" xref="S2.Thmthm9.p2.5.m5.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p2.5.m5.1.1.3.2.cmml" xref="S2.Thmthm9.p2.5.m5.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.5.m5.1c">b^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.5.m5.1d">italic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> the minimal shift-period has length 1, but <math alttext="\sigma_{2}" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.6.m6.1"><semantics id="S2.Thmthm9.p2.6.m6.1a"><msub id="S2.Thmthm9.p2.6.m6.1.1" xref="S2.Thmthm9.p2.6.m6.1.1.cmml"><mi id="S2.Thmthm9.p2.6.m6.1.1.2" xref="S2.Thmthm9.p2.6.m6.1.1.2.cmml">σ</mi><mn id="S2.Thmthm9.p2.6.m6.1.1.3" xref="S2.Thmthm9.p2.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.6.m6.1b"><apply id="S2.Thmthm9.p2.6.m6.1.1.cmml" xref="S2.Thmthm9.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.6.m6.1.1.1.cmml" xref="S2.Thmthm9.p2.6.m6.1.1">subscript</csymbol><ci id="S2.Thmthm9.p2.6.m6.1.1.2.cmml" xref="S2.Thmthm9.p2.6.m6.1.1.2">𝜎</ci><cn id="S2.Thmthm9.p2.6.m6.1.1.3.cmml" type="integer" xref="S2.Thmthm9.p2.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.6.m6.1c">\sigma_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.6.m6.1d">italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> maps the shift-period <math alttext="a" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.7.m7.1"><semantics id="S2.Thmthm9.p2.7.m7.1a"><mi id="S2.Thmthm9.p2.7.m7.1.1" xref="S2.Thmthm9.p2.7.m7.1.1.cmml">a</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.7.m7.1b"><ci id="S2.Thmthm9.p2.7.m7.1.1.cmml" xref="S2.Thmthm9.p2.7.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.7.m7.1c">a</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.7.m7.1d">italic_a</annotation></semantics></math> of <math alttext="a^{\pm\infty}" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.8.m8.1"><semantics id="S2.Thmthm9.p2.8.m8.1a"><msup id="S2.Thmthm9.p2.8.m8.1.1" xref="S2.Thmthm9.p2.8.m8.1.1.cmml"><mi id="S2.Thmthm9.p2.8.m8.1.1.2" xref="S2.Thmthm9.p2.8.m8.1.1.2.cmml">a</mi><mrow id="S2.Thmthm9.p2.8.m8.1.1.3" xref="S2.Thmthm9.p2.8.m8.1.1.3.cmml"><mo id="S2.Thmthm9.p2.8.m8.1.1.3a" xref="S2.Thmthm9.p2.8.m8.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p2.8.m8.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p2.8.m8.1.1.3.2.cmml">∞</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.8.m8.1b"><apply id="S2.Thmthm9.p2.8.m8.1.1.cmml" xref="S2.Thmthm9.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.8.m8.1.1.1.cmml" xref="S2.Thmthm9.p2.8.m8.1.1">superscript</csymbol><ci id="S2.Thmthm9.p2.8.m8.1.1.2.cmml" xref="S2.Thmthm9.p2.8.m8.1.1.2">𝑎</ci><apply id="S2.Thmthm9.p2.8.m8.1.1.3.cmml" xref="S2.Thmthm9.p2.8.m8.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p2.8.m8.1.1.3.1.cmml" xref="S2.Thmthm9.p2.8.m8.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p2.8.m8.1.1.3.2.cmml" xref="S2.Thmthm9.p2.8.m8.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.8.m8.1c">a^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.8.m8.1d">italic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> to a word of length 2, which is hence not the minimal shift-period of <math alttext="b^{\pm\infty}" class="ltx_Math" display="inline" id="S2.Thmthm9.p2.9.m9.1"><semantics id="S2.Thmthm9.p2.9.m9.1a"><msup id="S2.Thmthm9.p2.9.m9.1.1" xref="S2.Thmthm9.p2.9.m9.1.1.cmml"><mi id="S2.Thmthm9.p2.9.m9.1.1.2" xref="S2.Thmthm9.p2.9.m9.1.1.2.cmml">b</mi><mrow id="S2.Thmthm9.p2.9.m9.1.1.3" xref="S2.Thmthm9.p2.9.m9.1.1.3.cmml"><mo id="S2.Thmthm9.p2.9.m9.1.1.3a" xref="S2.Thmthm9.p2.9.m9.1.1.3.cmml">±</mo><mi id="S2.Thmthm9.p2.9.m9.1.1.3.2" mathvariant="normal" xref="S2.Thmthm9.p2.9.m9.1.1.3.2.cmml">∞</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.Thmthm9.p2.9.m9.1b"><apply id="S2.Thmthm9.p2.9.m9.1.1.cmml" xref="S2.Thmthm9.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S2.Thmthm9.p2.9.m9.1.1.1.cmml" xref="S2.Thmthm9.p2.9.m9.1.1">superscript</csymbol><ci id="S2.Thmthm9.p2.9.m9.1.1.2.cmml" xref="S2.Thmthm9.p2.9.m9.1.1.2">𝑏</ci><apply id="S2.Thmthm9.p2.9.m9.1.1.3.cmml" xref="S2.Thmthm9.p2.9.m9.1.1.3"><csymbol cd="latexml" id="S2.Thmthm9.p2.9.m9.1.1.3.1.cmml" xref="S2.Thmthm9.p2.9.m9.1.1.3">plus-or-minus</csymbol><infinity id="S2.Thmthm9.p2.9.m9.1.1.3.2.cmml" xref="S2.Thmthm9.p2.9.m9.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm9.p2.9.m9.1c">b^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm9.p2.9.m9.1d">italic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </div> </section> <section class="ltx_subsubsection" id="S2.SS3.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">2.3.2. </span>Shift-period preservation</h4> <div class="ltx_para" id="S2.SS3.SSS2.p1"> <p class="ltx_p" id="S2.SS3.SSS2.p1.1"></p> </div> <div class="ltx_para" id="S2.SS3.SSS2.p2"> <p class="ltx_p" id="S2.SS3.SSS2.p2.13">For any periodic word <math alttext="{\bf x}\in\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.1.m1.1"><semantics id="S2.SS3.SSS2.p2.1.m1.1a"><mrow id="S2.SS3.SSS2.p2.1.m1.1.1" xref="S2.SS3.SSS2.p2.1.m1.1.1.cmml"><mi id="S2.SS3.SSS2.p2.1.m1.1.1.2" xref="S2.SS3.SSS2.p2.1.m1.1.1.2.cmml">𝐱</mi><mo id="S2.SS3.SSS2.p2.1.m1.1.1.1" xref="S2.SS3.SSS2.p2.1.m1.1.1.1.cmml">∈</mo><msup id="S2.SS3.SSS2.p2.1.m1.1.1.3" xref="S2.SS3.SSS2.p2.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.SSS2.p2.1.m1.1.1.3.2" xref="S2.SS3.SSS2.p2.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S2.SS3.SSS2.p2.1.m1.1.1.3.3" xref="S2.SS3.SSS2.p2.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.1.m1.1b"><apply id="S2.SS3.SSS2.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1"><in id="S2.SS3.SSS2.p2.1.m1.1.1.1.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1.1"></in><ci id="S2.SS3.SSS2.p2.1.m1.1.1.2.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1.2">𝐱</ci><apply id="S2.SS3.SSS2.p2.1.m1.1.1.3.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p2.1.m1.1.1.3.1.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS3.SSS2.p2.1.m1.1.1.3.2.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1.3.2">𝒜</ci><ci id="S2.SS3.SSS2.p2.1.m1.1.1.3.3.cmml" xref="S2.SS3.SSS2.p2.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.1.m1.1c">{\bf x}\in\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.1.m1.1d">bold_x ∈ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> we define the <span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p2.13.1">shift-period exponent</span> of <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.2.m2.1"><semantics id="S2.SS3.SSS2.p2.2.m2.1a"><mi id="S2.SS3.SSS2.p2.2.m2.1.1" xref="S2.SS3.SSS2.p2.2.m2.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.2.m2.1b"><ci id="S2.SS3.SSS2.p2.2.m2.1.1.cmml" xref="S2.SS3.SSS2.p2.2.m2.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.2.m2.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.2.m2.1d">bold_x</annotation></semantics></math> to be the smallest integer <math alttext="k\geq 1" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.3.m3.1"><semantics id="S2.SS3.SSS2.p2.3.m3.1a"><mrow id="S2.SS3.SSS2.p2.3.m3.1.1" xref="S2.SS3.SSS2.p2.3.m3.1.1.cmml"><mi id="S2.SS3.SSS2.p2.3.m3.1.1.2" xref="S2.SS3.SSS2.p2.3.m3.1.1.2.cmml">k</mi><mo id="S2.SS3.SSS2.p2.3.m3.1.1.1" xref="S2.SS3.SSS2.p2.3.m3.1.1.1.cmml">≥</mo><mn id="S2.SS3.SSS2.p2.3.m3.1.1.3" xref="S2.SS3.SSS2.p2.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.3.m3.1b"><apply id="S2.SS3.SSS2.p2.3.m3.1.1.cmml" xref="S2.SS3.SSS2.p2.3.m3.1.1"><geq id="S2.SS3.SSS2.p2.3.m3.1.1.1.cmml" xref="S2.SS3.SSS2.p2.3.m3.1.1.1"></geq><ci id="S2.SS3.SSS2.p2.3.m3.1.1.2.cmml" xref="S2.SS3.SSS2.p2.3.m3.1.1.2">𝑘</ci><cn id="S2.SS3.SSS2.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS3.SSS2.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.3.m3.1c">k\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.3.m3.1d">italic_k ≥ 1</annotation></semantics></math> such that <math alttext="T^{k}({\bf x})={\bf x}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.4.m4.1"><semantics id="S2.SS3.SSS2.p2.4.m4.1a"><mrow id="S2.SS3.SSS2.p2.4.m4.1.2" xref="S2.SS3.SSS2.p2.4.m4.1.2.cmml"><mrow id="S2.SS3.SSS2.p2.4.m4.1.2.2" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.cmml"><msup id="S2.SS3.SSS2.p2.4.m4.1.2.2.2" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.2.cmml"><mi id="S2.SS3.SSS2.p2.4.m4.1.2.2.2.2" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.2.2.cmml">T</mi><mi id="S2.SS3.SSS2.p2.4.m4.1.2.2.2.3" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.2.3.cmml">k</mi></msup><mo id="S2.SS3.SSS2.p2.4.m4.1.2.2.1" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS2.p2.4.m4.1.2.2.3.2" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.cmml"><mo id="S2.SS3.SSS2.p2.4.m4.1.2.2.3.2.1" stretchy="false" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.cmml">(</mo><mi id="S2.SS3.SSS2.p2.4.m4.1.1" xref="S2.SS3.SSS2.p2.4.m4.1.1.cmml">𝐱</mi><mo id="S2.SS3.SSS2.p2.4.m4.1.2.2.3.2.2" stretchy="false" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS3.SSS2.p2.4.m4.1.2.1" xref="S2.SS3.SSS2.p2.4.m4.1.2.1.cmml">=</mo><mi id="S2.SS3.SSS2.p2.4.m4.1.2.3" xref="S2.SS3.SSS2.p2.4.m4.1.2.3.cmml">𝐱</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.4.m4.1b"><apply id="S2.SS3.SSS2.p2.4.m4.1.2.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2"><eq id="S2.SS3.SSS2.p2.4.m4.1.2.1.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2.1"></eq><apply id="S2.SS3.SSS2.p2.4.m4.1.2.2.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2.2"><times id="S2.SS3.SSS2.p2.4.m4.1.2.2.1.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.1"></times><apply id="S2.SS3.SSS2.p2.4.m4.1.2.2.2.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p2.4.m4.1.2.2.2.1.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.2">superscript</csymbol><ci id="S2.SS3.SSS2.p2.4.m4.1.2.2.2.2.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.2.2">𝑇</ci><ci id="S2.SS3.SSS2.p2.4.m4.1.2.2.2.3.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2.2.2.3">𝑘</ci></apply><ci id="S2.SS3.SSS2.p2.4.m4.1.1.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.1">𝐱</ci></apply><ci id="S2.SS3.SSS2.p2.4.m4.1.2.3.cmml" xref="S2.SS3.SSS2.p2.4.m4.1.2.3">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.4.m4.1c">T^{k}({\bf x})={\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.4.m4.1d">italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( bold_x ) = bold_x</annotation></semantics></math>. If <math alttext="{\bf x}=w^{\pm\infty}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.5.m5.1"><semantics id="S2.SS3.SSS2.p2.5.m5.1a"><mrow id="S2.SS3.SSS2.p2.5.m5.1.1" xref="S2.SS3.SSS2.p2.5.m5.1.1.cmml"><mi id="S2.SS3.SSS2.p2.5.m5.1.1.2" xref="S2.SS3.SSS2.p2.5.m5.1.1.2.cmml">𝐱</mi><mo id="S2.SS3.SSS2.p2.5.m5.1.1.1" xref="S2.SS3.SSS2.p2.5.m5.1.1.1.cmml">=</mo><msup id="S2.SS3.SSS2.p2.5.m5.1.1.3" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.cmml"><mi id="S2.SS3.SSS2.p2.5.m5.1.1.3.2" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.2.cmml">w</mi><mrow id="S2.SS3.SSS2.p2.5.m5.1.1.3.3" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.3.cmml"><mo id="S2.SS3.SSS2.p2.5.m5.1.1.3.3a" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.3.cmml">±</mo><mi id="S2.SS3.SSS2.p2.5.m5.1.1.3.3.2" mathvariant="normal" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.3.2.cmml">∞</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.5.m5.1b"><apply id="S2.SS3.SSS2.p2.5.m5.1.1.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1"><eq id="S2.SS3.SSS2.p2.5.m5.1.1.1.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1.1"></eq><ci id="S2.SS3.SSS2.p2.5.m5.1.1.2.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1.2">𝐱</ci><apply id="S2.SS3.SSS2.p2.5.m5.1.1.3.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p2.5.m5.1.1.3.1.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1.3">superscript</csymbol><ci id="S2.SS3.SSS2.p2.5.m5.1.1.3.2.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.2">𝑤</ci><apply id="S2.SS3.SSS2.p2.5.m5.1.1.3.3.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.3"><csymbol cd="latexml" id="S2.SS3.SSS2.p2.5.m5.1.1.3.3.1.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.3">plus-or-minus</csymbol><infinity id="S2.SS3.SSS2.p2.5.m5.1.1.3.3.2.cmml" xref="S2.SS3.SSS2.p2.5.m5.1.1.3.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.5.m5.1c">{\bf x}=w^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.5.m5.1d">bold_x = italic_w start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> for some <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.6.m6.1"><semantics id="S2.SS3.SSS2.p2.6.m6.1a"><mrow id="S2.SS3.SSS2.p2.6.m6.1.1" xref="S2.SS3.SSS2.p2.6.m6.1.1.cmml"><mi id="S2.SS3.SSS2.p2.6.m6.1.1.2" xref="S2.SS3.SSS2.p2.6.m6.1.1.2.cmml">w</mi><mo id="S2.SS3.SSS2.p2.6.m6.1.1.1" xref="S2.SS3.SSS2.p2.6.m6.1.1.1.cmml">∈</mo><msup id="S2.SS3.SSS2.p2.6.m6.1.1.3" xref="S2.SS3.SSS2.p2.6.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.SSS2.p2.6.m6.1.1.3.2" xref="S2.SS3.SSS2.p2.6.m6.1.1.3.2.cmml">𝒜</mi><mo id="S2.SS3.SSS2.p2.6.m6.1.1.3.3" xref="S2.SS3.SSS2.p2.6.m6.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.6.m6.1b"><apply id="S2.SS3.SSS2.p2.6.m6.1.1.cmml" xref="S2.SS3.SSS2.p2.6.m6.1.1"><in id="S2.SS3.SSS2.p2.6.m6.1.1.1.cmml" xref="S2.SS3.SSS2.p2.6.m6.1.1.1"></in><ci id="S2.SS3.SSS2.p2.6.m6.1.1.2.cmml" xref="S2.SS3.SSS2.p2.6.m6.1.1.2">𝑤</ci><apply id="S2.SS3.SSS2.p2.6.m6.1.1.3.cmml" xref="S2.SS3.SSS2.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS2.p2.6.m6.1.1.3.1.cmml" xref="S2.SS3.SSS2.p2.6.m6.1.1.3">superscript</csymbol><ci id="S2.SS3.SSS2.p2.6.m6.1.1.3.2.cmml" xref="S2.SS3.SSS2.p2.6.m6.1.1.3.2">𝒜</ci><times id="S2.SS3.SSS2.p2.6.m6.1.1.3.3.cmml" xref="S2.SS3.SSS2.p2.6.m6.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.6.m6.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.6.m6.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="k" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.7.m7.1"><semantics id="S2.SS3.SSS2.p2.7.m7.1a"><mi id="S2.SS3.SSS2.p2.7.m7.1.1" xref="S2.SS3.SSS2.p2.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.7.m7.1b"><ci id="S2.SS3.SSS2.p2.7.m7.1.1.cmml" xref="S2.SS3.SSS2.p2.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.7.m7.1d">italic_k</annotation></semantics></math> divides <math alttext="|w|" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.8.m8.1"><semantics id="S2.SS3.SSS2.p2.8.m8.1a"><mrow id="S2.SS3.SSS2.p2.8.m8.1.2.2" xref="S2.SS3.SSS2.p2.8.m8.1.2.1.cmml"><mo id="S2.SS3.SSS2.p2.8.m8.1.2.2.1" stretchy="false" xref="S2.SS3.SSS2.p2.8.m8.1.2.1.1.cmml">|</mo><mi id="S2.SS3.SSS2.p2.8.m8.1.1" xref="S2.SS3.SSS2.p2.8.m8.1.1.cmml">w</mi><mo id="S2.SS3.SSS2.p2.8.m8.1.2.2.2" stretchy="false" xref="S2.SS3.SSS2.p2.8.m8.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.8.m8.1b"><apply id="S2.SS3.SSS2.p2.8.m8.1.2.1.cmml" xref="S2.SS3.SSS2.p2.8.m8.1.2.2"><abs id="S2.SS3.SSS2.p2.8.m8.1.2.1.1.cmml" xref="S2.SS3.SSS2.p2.8.m8.1.2.2.1"></abs><ci id="S2.SS3.SSS2.p2.8.m8.1.1.cmml" xref="S2.SS3.SSS2.p2.8.m8.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.8.m8.1c">|w|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.8.m8.1d">| italic_w |</annotation></semantics></math>. If <math alttext="w" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.9.m9.1"><semantics id="S2.SS3.SSS2.p2.9.m9.1a"><mi id="S2.SS3.SSS2.p2.9.m9.1.1" xref="S2.SS3.SSS2.p2.9.m9.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.9.m9.1b"><ci id="S2.SS3.SSS2.p2.9.m9.1.1.cmml" xref="S2.SS3.SSS2.p2.9.m9.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.9.m9.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.9.m9.1d">italic_w</annotation></semantics></math> can not be written as proper power (see (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E4" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.4</span></a>)), then the shift-period exponent of <math alttext="\bf x" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.10.m10.1"><semantics id="S2.SS3.SSS2.p2.10.m10.1a"><mi id="S2.SS3.SSS2.p2.10.m10.1.1" xref="S2.SS3.SSS2.p2.10.m10.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.10.m10.1b"><ci id="S2.SS3.SSS2.p2.10.m10.1.1.cmml" xref="S2.SS3.SSS2.p2.10.m10.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.10.m10.1c">\bf x</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.10.m10.1d">bold_x</annotation></semantics></math> is given by <math alttext="k=|w|" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.11.m11.1"><semantics id="S2.SS3.SSS2.p2.11.m11.1a"><mrow id="S2.SS3.SSS2.p2.11.m11.1.2" xref="S2.SS3.SSS2.p2.11.m11.1.2.cmml"><mi id="S2.SS3.SSS2.p2.11.m11.1.2.2" xref="S2.SS3.SSS2.p2.11.m11.1.2.2.cmml">k</mi><mo id="S2.SS3.SSS2.p2.11.m11.1.2.1" xref="S2.SS3.SSS2.p2.11.m11.1.2.1.cmml">=</mo><mrow id="S2.SS3.SSS2.p2.11.m11.1.2.3.2" xref="S2.SS3.SSS2.p2.11.m11.1.2.3.1.cmml"><mo id="S2.SS3.SSS2.p2.11.m11.1.2.3.2.1" stretchy="false" xref="S2.SS3.SSS2.p2.11.m11.1.2.3.1.1.cmml">|</mo><mi id="S2.SS3.SSS2.p2.11.m11.1.1" xref="S2.SS3.SSS2.p2.11.m11.1.1.cmml">w</mi><mo id="S2.SS3.SSS2.p2.11.m11.1.2.3.2.2" stretchy="false" xref="S2.SS3.SSS2.p2.11.m11.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.11.m11.1b"><apply id="S2.SS3.SSS2.p2.11.m11.1.2.cmml" xref="S2.SS3.SSS2.p2.11.m11.1.2"><eq id="S2.SS3.SSS2.p2.11.m11.1.2.1.cmml" xref="S2.SS3.SSS2.p2.11.m11.1.2.1"></eq><ci id="S2.SS3.SSS2.p2.11.m11.1.2.2.cmml" xref="S2.SS3.SSS2.p2.11.m11.1.2.2">𝑘</ci><apply id="S2.SS3.SSS2.p2.11.m11.1.2.3.1.cmml" xref="S2.SS3.SSS2.p2.11.m11.1.2.3.2"><abs id="S2.SS3.SSS2.p2.11.m11.1.2.3.1.1.cmml" xref="S2.SS3.SSS2.p2.11.m11.1.2.3.2.1"></abs><ci id="S2.SS3.SSS2.p2.11.m11.1.1.cmml" xref="S2.SS3.SSS2.p2.11.m11.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.11.m11.1c">k=|w|</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.11.m11.1d">italic_k = | italic_w |</annotation></semantics></math>. In this case any cyclic permutation of <math alttext="w" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.12.m12.1"><semantics id="S2.SS3.SSS2.p2.12.m12.1a"><mi id="S2.SS3.SSS2.p2.12.m12.1.1" xref="S2.SS3.SSS2.p2.12.m12.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.12.m12.1b"><ci id="S2.SS3.SSS2.p2.12.m12.1.1.cmml" xref="S2.SS3.SSS2.p2.12.m12.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.12.m12.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.12.m12.1d">italic_w</annotation></semantics></math> will be called a <span class="ltx_text ltx_font_italic" id="S2.SS3.SSS2.p2.13.2">shift-period</span> of the periodic word <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S2.SS3.SSS2.p2.13.m13.1"><semantics id="S2.SS3.SSS2.p2.13.m13.1a"><mi id="S2.SS3.SSS2.p2.13.m13.1.1" xref="S2.SS3.SSS2.p2.13.m13.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS2.p2.13.m13.1b"><ci id="S2.SS3.SSS2.p2.13.m13.1.1.cmml" xref="S2.SS3.SSS2.p2.13.m13.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS2.p2.13.m13.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS2.p2.13.m13.1d">bold_x</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_defn" id="S2.Thmthm10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm10.1.1.1">Definition 2.10</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm10.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm10.p1"> <p class="ltx_p" id="S2.Thmthm10.p1.6">A morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S2.Thmthm10.p1.1.m1.1"><semantics id="S2.Thmthm10.p1.1.m1.1a"><mrow id="S2.Thmthm10.p1.1.m1.1.1" xref="S2.Thmthm10.p1.1.m1.1.1.cmml"><mi id="S2.Thmthm10.p1.1.m1.1.1.2" xref="S2.Thmthm10.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S2.Thmthm10.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.Thmthm10.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S2.Thmthm10.p1.1.m1.1.1.3" xref="S2.Thmthm10.p1.1.m1.1.1.3.cmml"><msup id="S2.Thmthm10.p1.1.m1.1.1.3.2" xref="S2.Thmthm10.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm10.p1.1.m1.1.1.3.2.2" xref="S2.Thmthm10.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.Thmthm10.p1.1.m1.1.1.3.2.3" xref="S2.Thmthm10.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S2.Thmthm10.p1.1.m1.1.1.3.1" stretchy="false" xref="S2.Thmthm10.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S2.Thmthm10.p1.1.m1.1.1.3.3" xref="S2.Thmthm10.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm10.p1.1.m1.1.1.3.3.2" xref="S2.Thmthm10.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S2.Thmthm10.p1.1.m1.1.1.3.3.3" xref="S2.Thmthm10.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p1.1.m1.1b"><apply id="S2.Thmthm10.p1.1.m1.1.1.cmml" xref="S2.Thmthm10.p1.1.m1.1.1"><ci id="S2.Thmthm10.p1.1.m1.1.1.1.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.1">:</ci><ci id="S2.Thmthm10.p1.1.m1.1.1.2.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.2">𝜎</ci><apply id="S2.Thmthm10.p1.1.m1.1.1.3.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3"><ci id="S2.Thmthm10.p1.1.m1.1.1.3.1.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.1">→</ci><apply id="S2.Thmthm10.p1.1.m1.1.1.3.2.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmthm10.p1.1.m1.1.1.3.2.1.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.Thmthm10.p1.1.m1.1.1.3.2.2.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S2.Thmthm10.p1.1.m1.1.1.3.2.3.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S2.Thmthm10.p1.1.m1.1.1.3.3.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmthm10.p1.1.m1.1.1.3.3.1.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.Thmthm10.p1.1.m1.1.1.3.3.2.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S2.Thmthm10.p1.1.m1.1.1.3.3.3.cmml" xref="S2.Thmthm10.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is said to <span class="ltx_text ltx_font_italic" id="S2.Thmthm10.p1.6.1">preserve the shift-period</span> of some biinfinite periodic word <math alttext="{\bf x}\in\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm10.p1.2.m2.1"><semantics id="S2.Thmthm10.p1.2.m2.1a"><mrow id="S2.Thmthm10.p1.2.m2.1.1" xref="S2.Thmthm10.p1.2.m2.1.1.cmml"><mi id="S2.Thmthm10.p1.2.m2.1.1.2" xref="S2.Thmthm10.p1.2.m2.1.1.2.cmml">𝐱</mi><mo id="S2.Thmthm10.p1.2.m2.1.1.1" xref="S2.Thmthm10.p1.2.m2.1.1.1.cmml">∈</mo><msup id="S2.Thmthm10.p1.2.m2.1.1.3" xref="S2.Thmthm10.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm10.p1.2.m2.1.1.3.2" xref="S2.Thmthm10.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S2.Thmthm10.p1.2.m2.1.1.3.3" xref="S2.Thmthm10.p1.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p1.2.m2.1b"><apply id="S2.Thmthm10.p1.2.m2.1.1.cmml" xref="S2.Thmthm10.p1.2.m2.1.1"><in id="S2.Thmthm10.p1.2.m2.1.1.1.cmml" xref="S2.Thmthm10.p1.2.m2.1.1.1"></in><ci id="S2.Thmthm10.p1.2.m2.1.1.2.cmml" xref="S2.Thmthm10.p1.2.m2.1.1.2">𝐱</ci><apply id="S2.Thmthm10.p1.2.m2.1.1.3.cmml" xref="S2.Thmthm10.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm10.p1.2.m2.1.1.3.1.cmml" xref="S2.Thmthm10.p1.2.m2.1.1.3">superscript</csymbol><ci id="S2.Thmthm10.p1.2.m2.1.1.3.2.cmml" xref="S2.Thmthm10.p1.2.m2.1.1.3.2">𝒜</ci><ci id="S2.Thmthm10.p1.2.m2.1.1.3.3.cmml" xref="S2.Thmthm10.p1.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p1.2.m2.1c">{\bf x}\in\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p1.2.m2.1d">bold_x ∈ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> if any shift-period <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S2.Thmthm10.p1.3.m3.1"><semantics id="S2.Thmthm10.p1.3.m3.1a"><mrow id="S2.Thmthm10.p1.3.m3.1.1" xref="S2.Thmthm10.p1.3.m3.1.1.cmml"><mi id="S2.Thmthm10.p1.3.m3.1.1.2" xref="S2.Thmthm10.p1.3.m3.1.1.2.cmml">w</mi><mo id="S2.Thmthm10.p1.3.m3.1.1.1" xref="S2.Thmthm10.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S2.Thmthm10.p1.3.m3.1.1.3" xref="S2.Thmthm10.p1.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm10.p1.3.m3.1.1.3.2" xref="S2.Thmthm10.p1.3.m3.1.1.3.2.cmml">𝒜</mi><mo id="S2.Thmthm10.p1.3.m3.1.1.3.3" xref="S2.Thmthm10.p1.3.m3.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p1.3.m3.1b"><apply id="S2.Thmthm10.p1.3.m3.1.1.cmml" xref="S2.Thmthm10.p1.3.m3.1.1"><in id="S2.Thmthm10.p1.3.m3.1.1.1.cmml" xref="S2.Thmthm10.p1.3.m3.1.1.1"></in><ci id="S2.Thmthm10.p1.3.m3.1.1.2.cmml" xref="S2.Thmthm10.p1.3.m3.1.1.2">𝑤</ci><apply id="S2.Thmthm10.p1.3.m3.1.1.3.cmml" xref="S2.Thmthm10.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm10.p1.3.m3.1.1.3.1.cmml" xref="S2.Thmthm10.p1.3.m3.1.1.3">superscript</csymbol><ci id="S2.Thmthm10.p1.3.m3.1.1.3.2.cmml" xref="S2.Thmthm10.p1.3.m3.1.1.3.2">𝒜</ci><times id="S2.Thmthm10.p1.3.m3.1.1.3.3.cmml" xref="S2.Thmthm10.p1.3.m3.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p1.3.m3.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p1.3.m3.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S2.Thmthm10.p1.4.m4.1"><semantics id="S2.Thmthm10.p1.4.m4.1a"><mi id="S2.Thmthm10.p1.4.m4.1.1" xref="S2.Thmthm10.p1.4.m4.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p1.4.m4.1b"><ci id="S2.Thmthm10.p1.4.m4.1.1.cmml" xref="S2.Thmthm10.p1.4.m4.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p1.4.m4.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p1.4.m4.1d">bold_x</annotation></semantics></math> is mapped by <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.Thmthm10.p1.5.m5.1"><semantics id="S2.Thmthm10.p1.5.m5.1a"><mi id="S2.Thmthm10.p1.5.m5.1.1" xref="S2.Thmthm10.p1.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p1.5.m5.1b"><ci id="S2.Thmthm10.p1.5.m5.1.1.cmml" xref="S2.Thmthm10.p1.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p1.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p1.5.m5.1d">italic_σ</annotation></semantics></math> to a shift-period of the image word <math alttext="\sigma^{\mathbb{Z}}({\bf x})\in\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S2.Thmthm10.p1.6.m6.1"><semantics id="S2.Thmthm10.p1.6.m6.1a"><mrow id="S2.Thmthm10.p1.6.m6.1.2" xref="S2.Thmthm10.p1.6.m6.1.2.cmml"><mrow id="S2.Thmthm10.p1.6.m6.1.2.2" xref="S2.Thmthm10.p1.6.m6.1.2.2.cmml"><msup id="S2.Thmthm10.p1.6.m6.1.2.2.2" xref="S2.Thmthm10.p1.6.m6.1.2.2.2.cmml"><mi id="S2.Thmthm10.p1.6.m6.1.2.2.2.2" xref="S2.Thmthm10.p1.6.m6.1.2.2.2.2.cmml">σ</mi><mi id="S2.Thmthm10.p1.6.m6.1.2.2.2.3" xref="S2.Thmthm10.p1.6.m6.1.2.2.2.3.cmml">ℤ</mi></msup><mo id="S2.Thmthm10.p1.6.m6.1.2.2.1" xref="S2.Thmthm10.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S2.Thmthm10.p1.6.m6.1.2.2.3.2" xref="S2.Thmthm10.p1.6.m6.1.2.2.cmml"><mo id="S2.Thmthm10.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S2.Thmthm10.p1.6.m6.1.2.2.cmml">(</mo><mi id="S2.Thmthm10.p1.6.m6.1.1" xref="S2.Thmthm10.p1.6.m6.1.1.cmml">𝐱</mi><mo id="S2.Thmthm10.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S2.Thmthm10.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm10.p1.6.m6.1.2.1" xref="S2.Thmthm10.p1.6.m6.1.2.1.cmml">∈</mo><msup id="S2.Thmthm10.p1.6.m6.1.2.3" xref="S2.Thmthm10.p1.6.m6.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm10.p1.6.m6.1.2.3.2" xref="S2.Thmthm10.p1.6.m6.1.2.3.2.cmml">ℬ</mi><mi id="S2.Thmthm10.p1.6.m6.1.2.3.3" xref="S2.Thmthm10.p1.6.m6.1.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p1.6.m6.1b"><apply id="S2.Thmthm10.p1.6.m6.1.2.cmml" xref="S2.Thmthm10.p1.6.m6.1.2"><in id="S2.Thmthm10.p1.6.m6.1.2.1.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.1"></in><apply id="S2.Thmthm10.p1.6.m6.1.2.2.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.2"><times id="S2.Thmthm10.p1.6.m6.1.2.2.1.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.2.1"></times><apply id="S2.Thmthm10.p1.6.m6.1.2.2.2.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.2.2"><csymbol cd="ambiguous" id="S2.Thmthm10.p1.6.m6.1.2.2.2.1.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.2.2">superscript</csymbol><ci id="S2.Thmthm10.p1.6.m6.1.2.2.2.2.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.2.2.2">𝜎</ci><ci id="S2.Thmthm10.p1.6.m6.1.2.2.2.3.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.2.2.3">ℤ</ci></apply><ci id="S2.Thmthm10.p1.6.m6.1.1.cmml" xref="S2.Thmthm10.p1.6.m6.1.1">𝐱</ci></apply><apply id="S2.Thmthm10.p1.6.m6.1.2.3.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.3"><csymbol cd="ambiguous" id="S2.Thmthm10.p1.6.m6.1.2.3.1.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.3">superscript</csymbol><ci id="S2.Thmthm10.p1.6.m6.1.2.3.2.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.3.2">ℬ</ci><ci id="S2.Thmthm10.p1.6.m6.1.2.3.3.cmml" xref="S2.Thmthm10.p1.6.m6.1.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p1.6.m6.1c">\sigma^{\mathbb{Z}}({\bf x})\in\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p1.6.m6.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x ) ∈ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.Thmthm10.p2"> <p class="ltx_p" id="S2.Thmthm10.p2.5">In other words, if <math alttext="{\bf x}=w^{\pm\infty}" class="ltx_Math" display="inline" id="S2.Thmthm10.p2.1.m1.1"><semantics id="S2.Thmthm10.p2.1.m1.1a"><mrow id="S2.Thmthm10.p2.1.m1.1.1" xref="S2.Thmthm10.p2.1.m1.1.1.cmml"><mi id="S2.Thmthm10.p2.1.m1.1.1.2" xref="S2.Thmthm10.p2.1.m1.1.1.2.cmml">𝐱</mi><mo id="S2.Thmthm10.p2.1.m1.1.1.1" xref="S2.Thmthm10.p2.1.m1.1.1.1.cmml">=</mo><msup id="S2.Thmthm10.p2.1.m1.1.1.3" xref="S2.Thmthm10.p2.1.m1.1.1.3.cmml"><mi id="S2.Thmthm10.p2.1.m1.1.1.3.2" xref="S2.Thmthm10.p2.1.m1.1.1.3.2.cmml">w</mi><mrow id="S2.Thmthm10.p2.1.m1.1.1.3.3" xref="S2.Thmthm10.p2.1.m1.1.1.3.3.cmml"><mo id="S2.Thmthm10.p2.1.m1.1.1.3.3a" xref="S2.Thmthm10.p2.1.m1.1.1.3.3.cmml">±</mo><mi id="S2.Thmthm10.p2.1.m1.1.1.3.3.2" mathvariant="normal" xref="S2.Thmthm10.p2.1.m1.1.1.3.3.2.cmml">∞</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p2.1.m1.1b"><apply id="S2.Thmthm10.p2.1.m1.1.1.cmml" xref="S2.Thmthm10.p2.1.m1.1.1"><eq id="S2.Thmthm10.p2.1.m1.1.1.1.cmml" xref="S2.Thmthm10.p2.1.m1.1.1.1"></eq><ci id="S2.Thmthm10.p2.1.m1.1.1.2.cmml" xref="S2.Thmthm10.p2.1.m1.1.1.2">𝐱</ci><apply id="S2.Thmthm10.p2.1.m1.1.1.3.cmml" xref="S2.Thmthm10.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmthm10.p2.1.m1.1.1.3.1.cmml" xref="S2.Thmthm10.p2.1.m1.1.1.3">superscript</csymbol><ci id="S2.Thmthm10.p2.1.m1.1.1.3.2.cmml" xref="S2.Thmthm10.p2.1.m1.1.1.3.2">𝑤</ci><apply id="S2.Thmthm10.p2.1.m1.1.1.3.3.cmml" xref="S2.Thmthm10.p2.1.m1.1.1.3.3"><csymbol cd="latexml" id="S2.Thmthm10.p2.1.m1.1.1.3.3.1.cmml" xref="S2.Thmthm10.p2.1.m1.1.1.3.3">plus-or-minus</csymbol><infinity id="S2.Thmthm10.p2.1.m1.1.1.3.3.2.cmml" xref="S2.Thmthm10.p2.1.m1.1.1.3.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p2.1.m1.1c">{\bf x}=w^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p2.1.m1.1d">bold_x = italic_w start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S2.Thmthm10.p2.2.m2.1"><semantics id="S2.Thmthm10.p2.2.m2.1a"><mrow id="S2.Thmthm10.p2.2.m2.1.2" xref="S2.Thmthm10.p2.2.m2.1.2.cmml"><mi id="S2.Thmthm10.p2.2.m2.1.2.2" xref="S2.Thmthm10.p2.2.m2.1.2.2.cmml">σ</mi><mo id="S2.Thmthm10.p2.2.m2.1.2.1" xref="S2.Thmthm10.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.Thmthm10.p2.2.m2.1.2.3.2" xref="S2.Thmthm10.p2.2.m2.1.2.cmml"><mo id="S2.Thmthm10.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S2.Thmthm10.p2.2.m2.1.2.cmml">(</mo><mi id="S2.Thmthm10.p2.2.m2.1.1" xref="S2.Thmthm10.p2.2.m2.1.1.cmml">w</mi><mo id="S2.Thmthm10.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S2.Thmthm10.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p2.2.m2.1b"><apply id="S2.Thmthm10.p2.2.m2.1.2.cmml" xref="S2.Thmthm10.p2.2.m2.1.2"><times id="S2.Thmthm10.p2.2.m2.1.2.1.cmml" xref="S2.Thmthm10.p2.2.m2.1.2.1"></times><ci id="S2.Thmthm10.p2.2.m2.1.2.2.cmml" xref="S2.Thmthm10.p2.2.m2.1.2.2">𝜎</ci><ci id="S2.Thmthm10.p2.2.m2.1.1.cmml" xref="S2.Thmthm10.p2.2.m2.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p2.2.m2.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p2.2.m2.1d">italic_σ ( italic_w )</annotation></semantics></math> is a proper power, then so is <math alttext="w" class="ltx_Math" display="inline" id="S2.Thmthm10.p2.3.m3.1"><semantics id="S2.Thmthm10.p2.3.m3.1a"><mi id="S2.Thmthm10.p2.3.m3.1.1" xref="S2.Thmthm10.p2.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p2.3.m3.1b"><ci id="S2.Thmthm10.p2.3.m3.1.1.cmml" xref="S2.Thmthm10.p2.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p2.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p2.3.m3.1d">italic_w</annotation></semantics></math> (or else <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.Thmthm10.p2.4.m4.1"><semantics id="S2.Thmthm10.p2.4.m4.1a"><mi id="S2.Thmthm10.p2.4.m4.1.1" xref="S2.Thmthm10.p2.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p2.4.m4.1b"><ci id="S2.Thmthm10.p2.4.m4.1.1.cmml" xref="S2.Thmthm10.p2.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p2.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p2.4.m4.1d">italic_σ</annotation></semantics></math> doesn’t preserve the shift-period of <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S2.Thmthm10.p2.5.m5.1"><semantics id="S2.Thmthm10.p2.5.m5.1a"><mi id="S2.Thmthm10.p2.5.m5.1.1" xref="S2.Thmthm10.p2.5.m5.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm10.p2.5.m5.1b"><ci id="S2.Thmthm10.p2.5.m5.1.1.cmml" xref="S2.Thmthm10.p2.5.m5.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm10.p2.5.m5.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm10.p2.5.m5.1d">bold_x</annotation></semantics></math>).</p> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S2.Thmthm11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmthm11.1.1.1">Remark 2.11</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmthm11.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmthm11.p1"> <p class="ltx_p" id="S2.Thmthm11.p1.6">In the special case where <math alttext="|\sigma(a_{i})|=1" class="ltx_Math" display="inline" id="S2.Thmthm11.p1.1.m1.1"><semantics id="S2.Thmthm11.p1.1.m1.1a"><mrow id="S2.Thmthm11.p1.1.m1.1.1" xref="S2.Thmthm11.p1.1.m1.1.1.cmml"><mrow id="S2.Thmthm11.p1.1.m1.1.1.1.1" xref="S2.Thmthm11.p1.1.m1.1.1.1.2.cmml"><mo id="S2.Thmthm11.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S2.Thmthm11.p1.1.m1.1.1.1.2.1.cmml">|</mo><mrow id="S2.Thmthm11.p1.1.m1.1.1.1.1.1" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.cmml"><mi id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.3" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.3.cmml">σ</mi><mo id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.2" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.2" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.3" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Thmthm11.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S2.Thmthm11.p1.1.m1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S2.Thmthm11.p1.1.m1.1.1.2" xref="S2.Thmthm11.p1.1.m1.1.1.2.cmml">=</mo><mn id="S2.Thmthm11.p1.1.m1.1.1.3" xref="S2.Thmthm11.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm11.p1.1.m1.1b"><apply id="S2.Thmthm11.p1.1.m1.1.1.cmml" xref="S2.Thmthm11.p1.1.m1.1.1"><eq id="S2.Thmthm11.p1.1.m1.1.1.2.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.2"></eq><apply id="S2.Thmthm11.p1.1.m1.1.1.1.2.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1"><abs id="S2.Thmthm11.p1.1.m1.1.1.1.2.1.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.2"></abs><apply id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1"><times id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.2.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.2"></times><ci id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.3.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.3">𝜎</ci><apply id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.Thmthm11.p1.1.m1.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply><cn id="S2.Thmthm11.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.Thmthm11.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm11.p1.1.m1.1c">|\sigma(a_{i})|=1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm11.p1.1.m1.1d">| italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) | = 1</annotation></semantics></math> for all <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S2.Thmthm11.p1.2.m2.1"><semantics id="S2.Thmthm11.p1.2.m2.1a"><mrow id="S2.Thmthm11.p1.2.m2.1.1" xref="S2.Thmthm11.p1.2.m2.1.1.cmml"><msub id="S2.Thmthm11.p1.2.m2.1.1.2" xref="S2.Thmthm11.p1.2.m2.1.1.2.cmml"><mi id="S2.Thmthm11.p1.2.m2.1.1.2.2" xref="S2.Thmthm11.p1.2.m2.1.1.2.2.cmml">a</mi><mi id="S2.Thmthm11.p1.2.m2.1.1.2.3" xref="S2.Thmthm11.p1.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S2.Thmthm11.p1.2.m2.1.1.1" xref="S2.Thmthm11.p1.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S2.Thmthm11.p1.2.m2.1.1.3" xref="S2.Thmthm11.p1.2.m2.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm11.p1.2.m2.1b"><apply id="S2.Thmthm11.p1.2.m2.1.1.cmml" xref="S2.Thmthm11.p1.2.m2.1.1"><in id="S2.Thmthm11.p1.2.m2.1.1.1.cmml" xref="S2.Thmthm11.p1.2.m2.1.1.1"></in><apply id="S2.Thmthm11.p1.2.m2.1.1.2.cmml" xref="S2.Thmthm11.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.Thmthm11.p1.2.m2.1.1.2.1.cmml" xref="S2.Thmthm11.p1.2.m2.1.1.2">subscript</csymbol><ci id="S2.Thmthm11.p1.2.m2.1.1.2.2.cmml" xref="S2.Thmthm11.p1.2.m2.1.1.2.2">𝑎</ci><ci id="S2.Thmthm11.p1.2.m2.1.1.2.3.cmml" xref="S2.Thmthm11.p1.2.m2.1.1.2.3">𝑖</ci></apply><ci id="S2.Thmthm11.p1.2.m2.1.1.3.cmml" xref="S2.Thmthm11.p1.2.m2.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm11.p1.2.m2.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm11.p1.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> we observe that <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.Thmthm11.p1.3.m3.1"><semantics id="S2.Thmthm11.p1.3.m3.1a"><mi id="S2.Thmthm11.p1.3.m3.1.1" xref="S2.Thmthm11.p1.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm11.p1.3.m3.1b"><ci id="S2.Thmthm11.p1.3.m3.1.1.cmml" xref="S2.Thmthm11.p1.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm11.p1.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm11.p1.3.m3.1d">italic_σ</annotation></semantics></math> preserves the shift-period of some periodic biinfinite word <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S2.Thmthm11.p1.4.m4.1"><semantics id="S2.Thmthm11.p1.4.m4.1a"><mi id="S2.Thmthm11.p1.4.m4.1.1" xref="S2.Thmthm11.p1.4.m4.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm11.p1.4.m4.1b"><ci id="S2.Thmthm11.p1.4.m4.1.1.cmml" xref="S2.Thmthm11.p1.4.m4.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm11.p1.4.m4.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm11.p1.4.m4.1d">bold_x</annotation></semantics></math> if and only if the shift-period exponents of <math alttext="\bf x" class="ltx_Math" display="inline" id="S2.Thmthm11.p1.5.m5.1"><semantics id="S2.Thmthm11.p1.5.m5.1a"><mi id="S2.Thmthm11.p1.5.m5.1.1" xref="S2.Thmthm11.p1.5.m5.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.Thmthm11.p1.5.m5.1b"><ci id="S2.Thmthm11.p1.5.m5.1.1.cmml" xref="S2.Thmthm11.p1.5.m5.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm11.p1.5.m5.1c">\bf x</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm11.p1.5.m5.1d">bold_x</annotation></semantics></math> and of its image <math alttext="\sigma^{\mathbb{Z}}({\bf x})" class="ltx_Math" display="inline" id="S2.Thmthm11.p1.6.m6.1"><semantics id="S2.Thmthm11.p1.6.m6.1a"><mrow id="S2.Thmthm11.p1.6.m6.1.2" xref="S2.Thmthm11.p1.6.m6.1.2.cmml"><msup id="S2.Thmthm11.p1.6.m6.1.2.2" xref="S2.Thmthm11.p1.6.m6.1.2.2.cmml"><mi id="S2.Thmthm11.p1.6.m6.1.2.2.2" xref="S2.Thmthm11.p1.6.m6.1.2.2.2.cmml">σ</mi><mi id="S2.Thmthm11.p1.6.m6.1.2.2.3" xref="S2.Thmthm11.p1.6.m6.1.2.2.3.cmml">ℤ</mi></msup><mo id="S2.Thmthm11.p1.6.m6.1.2.1" xref="S2.Thmthm11.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.Thmthm11.p1.6.m6.1.2.3.2" xref="S2.Thmthm11.p1.6.m6.1.2.cmml"><mo id="S2.Thmthm11.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S2.Thmthm11.p1.6.m6.1.2.cmml">(</mo><mi id="S2.Thmthm11.p1.6.m6.1.1" xref="S2.Thmthm11.p1.6.m6.1.1.cmml">𝐱</mi><mo id="S2.Thmthm11.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S2.Thmthm11.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmthm11.p1.6.m6.1b"><apply id="S2.Thmthm11.p1.6.m6.1.2.cmml" xref="S2.Thmthm11.p1.6.m6.1.2"><times id="S2.Thmthm11.p1.6.m6.1.2.1.cmml" xref="S2.Thmthm11.p1.6.m6.1.2.1"></times><apply id="S2.Thmthm11.p1.6.m6.1.2.2.cmml" xref="S2.Thmthm11.p1.6.m6.1.2.2"><csymbol cd="ambiguous" id="S2.Thmthm11.p1.6.m6.1.2.2.1.cmml" xref="S2.Thmthm11.p1.6.m6.1.2.2">superscript</csymbol><ci id="S2.Thmthm11.p1.6.m6.1.2.2.2.cmml" xref="S2.Thmthm11.p1.6.m6.1.2.2.2">𝜎</ci><ci id="S2.Thmthm11.p1.6.m6.1.2.2.3.cmml" xref="S2.Thmthm11.p1.6.m6.1.2.2.3">ℤ</ci></apply><ci id="S2.Thmthm11.p1.6.m6.1.1.cmml" xref="S2.Thmthm11.p1.6.m6.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmthm11.p1.6.m6.1c">\sigma^{\mathbb{Z}}({\bf x})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmthm11.p1.6.m6.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x )</annotation></semantics></math> agree.</p> </div> </div> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3. </span>The measure transfer</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.6">In this section we will carefully define for any non-erasing monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mrow id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mi id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S3.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml"><msup id="S3.p1.1.m1.1.1.3.2" xref="S3.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.1.m1.1.1.3.2.2" xref="S3.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.p1.1.m1.1.1.3.2.3" xref="S3.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.p1.1.m1.1.1.3.1" stretchy="false" xref="S3.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.p1.1.m1.1.1.3.3" xref="S3.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.1.m1.1.1.3.3.2" xref="S3.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.p1.1.m1.1.1.3.3.3" xref="S3.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><ci id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1.1">:</ci><ci id="S3.p1.1.m1.1.1.2.cmml" xref="S3.p1.1.m1.1.1.2">𝜎</ci><apply id="S3.p1.1.m1.1.1.3.cmml" xref="S3.p1.1.m1.1.1.3"><ci id="S3.p1.1.m1.1.1.3.1.cmml" xref="S3.p1.1.m1.1.1.3.1">→</ci><apply id="S3.p1.1.m1.1.1.3.2.cmml" xref="S3.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.3.2.1.cmml" xref="S3.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.p1.1.m1.1.1.3.2.2.cmml" xref="S3.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.p1.1.m1.1.1.3.2.3.cmml" xref="S3.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S3.p1.1.m1.1.1.3.3.cmml" xref="S3.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.3.3.1.cmml" xref="S3.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.p1.1.m1.1.1.3.3.2.cmml" xref="S3.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.p1.1.m1.1.1.3.3.3.cmml" xref="S3.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mi id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><ci id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><msup id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.3.m3.1.1.2" xref="S3.p1.3.m3.1.1.2.cmml">𝒜</mi><mi id="S3.p1.3.m3.1.1.3" xref="S3.p1.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><apply id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.1.1.cmml" xref="S3.p1.3.m3.1.1">superscript</csymbol><ci id="S3.p1.3.m3.1.1.2.cmml" xref="S3.p1.3.m3.1.1.2">𝒜</ci><ci id="S3.p1.3.m3.1.1.3.cmml" xref="S3.p1.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> a shift-invariant “image measure” <math alttext="\mu^{\sigma}" class="ltx_Math" display="inline" id="S3.p1.4.m4.1"><semantics id="S3.p1.4.m4.1a"><msup id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><mi id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml">μ</mi><mi id="S3.p1.4.m4.1.1.3" xref="S3.p1.4.m4.1.1.3.cmml">σ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><apply id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.p1.4.m4.1.1.1.cmml" xref="S3.p1.4.m4.1.1">superscript</csymbol><ci id="S3.p1.4.m4.1.1.2.cmml" xref="S3.p1.4.m4.1.1.2">𝜇</ci><ci id="S3.p1.4.m4.1.1.3.cmml" xref="S3.p1.4.m4.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">\mu^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math> on <math alttext="\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><msup id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.5.m5.1.1.2" xref="S3.p1.5.m5.1.1.2.cmml">ℬ</mi><mi id="S3.p1.5.m5.1.1.3" xref="S3.p1.5.m5.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><apply id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.1.cmml" xref="S3.p1.5.m5.1.1">superscript</csymbol><ci id="S3.p1.5.m5.1.1.2.cmml" xref="S3.p1.5.m5.1.1.2">ℬ</ci><ci id="S3.p1.5.m5.1.1.3.cmml" xref="S3.p1.5.m5.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. The simplest and most natural way to understand this measure transfer is achieved by decomposing the given morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.p1.6.m6.1"><semantics id="S3.p1.6.m6.1a"><mi id="S3.p1.6.m6.1.1" xref="S3.p1.6.m6.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><ci id="S3.p1.6.m6.1.1.cmml" xref="S3.p1.6.m6.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.1d">italic_σ</annotation></semantics></math> in a canonical way into two morphisms of very elementary type. We start our detailed presentation by considering first each of these two elementary morphism types separately.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1. </span>Subdivision morphisms</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1"></p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.6">Let <math alttext="\cal A" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">caligraphic_A</annotation></semantics></math> be a finite alphabet, and let <math alttext="\ell:\cal A\to\mathbb{Z}_{\geq 1}" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2" mathvariant="normal" xref="S3.SS1.p2.2.m2.1.1.2.cmml">ℓ</mi><mo id="S3.SS1.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.2.m2.1.1.3.2" xref="S3.SS1.p2.2.m2.1.1.3.2.cmml">𝒜</mi><mo id="S3.SS1.p2.2.m2.1.1.3.1" stretchy="false" xref="S3.SS1.p2.2.m2.1.1.3.1.cmml">→</mo><msub id="S3.SS1.p2.2.m2.1.1.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.3.2" xref="S3.SS1.p2.2.m2.1.1.3.3.2.cmml">ℤ</mi><mrow id="S3.SS1.p2.2.m2.1.1.3.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.3.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.3.3.2" xref="S3.SS1.p2.2.m2.1.1.3.3.3.2.cmml"></mi><mo id="S3.SS1.p2.2.m2.1.1.3.3.3.1" xref="S3.SS1.p2.2.m2.1.1.3.3.3.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S3.SS1.p2.2.m2.1.1.3.3.3.3" mathvariant="script" xref="S3.SS1.p2.2.m2.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><ci id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1">:</ci><ci id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2">ℓ</ci><apply id="S3.SS1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3"><ci id="S3.SS1.p2.2.m2.1.1.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3.1">→</ci><ci id="S3.SS1.p2.2.m2.1.1.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2">𝒜</ci><apply id="S3.SS1.p2.2.m2.1.1.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.3.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3">subscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.3.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.2">ℤ</ci><apply id="S3.SS1.p2.2.m2.1.1.3.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.3"><geq id="S3.SS1.p2.2.m2.1.1.3.3.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S3.SS1.p2.2.m2.1.1.3.3.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3.3.2">absent</csymbol><cn id="S3.SS1.p2.2.m2.1.1.3.3.3.3.cmml" type="integer" xref="S3.SS1.p2.2.m2.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">\ell:\cal A\to\mathbb{Z}_{\geq 1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">roman_ℓ : caligraphic_A → blackboard_Z start_POSTSUBSCRIPT ≥ caligraphic_1 end_POSTSUBSCRIPT</annotation></semantics></math> be any map, called <span class="ltx_text ltx_font_italic" id="S3.SS1.p2.6.1">subdivision length function</span>. We now define a new <span class="ltx_text ltx_font_italic" id="S3.SS1.p2.6.2">subdivision alphabet</span> <math alttext="\cal A_{\ell}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><msub id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml">𝒜</mi><mi id="S3.SS1.p2.3.m3.1.1.3" mathvariant="normal" xref="S3.SS1.p2.3.m3.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2">𝒜</ci><ci id="S3.SS1.p2.3.m3.1.1.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">\cal A_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> which consists of letters <math alttext="a_{i}(k)" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.1"><semantics id="S3.SS1.p2.4.m4.1a"><mrow id="S3.SS1.p2.4.m4.1.2" xref="S3.SS1.p2.4.m4.1.2.cmml"><msub id="S3.SS1.p2.4.m4.1.2.2" xref="S3.SS1.p2.4.m4.1.2.2.cmml"><mi id="S3.SS1.p2.4.m4.1.2.2.2" xref="S3.SS1.p2.4.m4.1.2.2.2.cmml">a</mi><mi id="S3.SS1.p2.4.m4.1.2.2.3" xref="S3.SS1.p2.4.m4.1.2.2.3.cmml">i</mi></msub><mo id="S3.SS1.p2.4.m4.1.2.1" xref="S3.SS1.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p2.4.m4.1.2.3.2" xref="S3.SS1.p2.4.m4.1.2.cmml"><mo id="S3.SS1.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S3.SS1.p2.4.m4.1.2.cmml">(</mo><mi id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml">k</mi><mo id="S3.SS1.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S3.SS1.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.1b"><apply id="S3.SS1.p2.4.m4.1.2.cmml" xref="S3.SS1.p2.4.m4.1.2"><times id="S3.SS1.p2.4.m4.1.2.1.cmml" xref="S3.SS1.p2.4.m4.1.2.1"></times><apply id="S3.SS1.p2.4.m4.1.2.2.cmml" xref="S3.SS1.p2.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.4.m4.1.2.2.1.cmml" xref="S3.SS1.p2.4.m4.1.2.2">subscript</csymbol><ci id="S3.SS1.p2.4.m4.1.2.2.2.cmml" xref="S3.SS1.p2.4.m4.1.2.2.2">𝑎</ci><ci id="S3.SS1.p2.4.m4.1.2.2.3.cmml" xref="S3.SS1.p2.4.m4.1.2.2.3">𝑖</ci></apply><ci id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.1c">a_{i}(k)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k )</annotation></semantics></math> for any <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m5.1"><semantics id="S3.SS1.p2.5.m5.1a"><mrow id="S3.SS1.p2.5.m5.1.1" xref="S3.SS1.p2.5.m5.1.1.cmml"><msub id="S3.SS1.p2.5.m5.1.1.2" xref="S3.SS1.p2.5.m5.1.1.2.cmml"><mi id="S3.SS1.p2.5.m5.1.1.2.2" xref="S3.SS1.p2.5.m5.1.1.2.2.cmml">a</mi><mi id="S3.SS1.p2.5.m5.1.1.2.3" xref="S3.SS1.p2.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.p2.5.m5.1.1.1" xref="S3.SS1.p2.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.5.m5.1.1.3" xref="S3.SS1.p2.5.m5.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m5.1b"><apply id="S3.SS1.p2.5.m5.1.1.cmml" xref="S3.SS1.p2.5.m5.1.1"><in id="S3.SS1.p2.5.m5.1.1.1.cmml" xref="S3.SS1.p2.5.m5.1.1.1"></in><apply id="S3.SS1.p2.5.m5.1.1.2.cmml" xref="S3.SS1.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p2.5.m5.1.1.2.1.cmml" xref="S3.SS1.p2.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS1.p2.5.m5.1.1.2.2.cmml" xref="S3.SS1.p2.5.m5.1.1.2.2">𝑎</ci><ci id="S3.SS1.p2.5.m5.1.1.2.3.cmml" xref="S3.SS1.p2.5.m5.1.1.2.3">𝑖</ci></apply><ci id="S3.SS1.p2.5.m5.1.1.3.cmml" xref="S3.SS1.p2.5.m5.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.5.m5.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> and any <math alttext="k\in\{1,\ldots,\ell(a_{i})\}" class="ltx_Math" display="inline" id="S3.SS1.p2.6.m6.3"><semantics id="S3.SS1.p2.6.m6.3a"><mrow id="S3.SS1.p2.6.m6.3.3" xref="S3.SS1.p2.6.m6.3.3.cmml"><mi id="S3.SS1.p2.6.m6.3.3.3" xref="S3.SS1.p2.6.m6.3.3.3.cmml">k</mi><mo id="S3.SS1.p2.6.m6.3.3.2" xref="S3.SS1.p2.6.m6.3.3.2.cmml">∈</mo><mrow id="S3.SS1.p2.6.m6.3.3.1.1" xref="S3.SS1.p2.6.m6.3.3.1.2.cmml"><mo id="S3.SS1.p2.6.m6.3.3.1.1.2" stretchy="false" xref="S3.SS1.p2.6.m6.3.3.1.2.cmml">{</mo><mn id="S3.SS1.p2.6.m6.1.1" xref="S3.SS1.p2.6.m6.1.1.cmml">1</mn><mo id="S3.SS1.p2.6.m6.3.3.1.1.3" xref="S3.SS1.p2.6.m6.3.3.1.2.cmml">,</mo><mi id="S3.SS1.p2.6.m6.2.2" mathvariant="normal" xref="S3.SS1.p2.6.m6.2.2.cmml">…</mi><mo id="S3.SS1.p2.6.m6.3.3.1.1.4" xref="S3.SS1.p2.6.m6.3.3.1.2.cmml">,</mo><mrow id="S3.SS1.p2.6.m6.3.3.1.1.1" xref="S3.SS1.p2.6.m6.3.3.1.1.1.cmml"><mi id="S3.SS1.p2.6.m6.3.3.1.1.1.3" mathvariant="normal" xref="S3.SS1.p2.6.m6.3.3.1.1.1.3.cmml">ℓ</mi><mo id="S3.SS1.p2.6.m6.3.3.1.1.1.2" xref="S3.SS1.p2.6.m6.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.cmml"><mo id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.2" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.3" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p2.6.m6.3.3.1.1.5" stretchy="false" xref="S3.SS1.p2.6.m6.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.6.m6.3b"><apply id="S3.SS1.p2.6.m6.3.3.cmml" xref="S3.SS1.p2.6.m6.3.3"><in id="S3.SS1.p2.6.m6.3.3.2.cmml" xref="S3.SS1.p2.6.m6.3.3.2"></in><ci id="S3.SS1.p2.6.m6.3.3.3.cmml" xref="S3.SS1.p2.6.m6.3.3.3">𝑘</ci><set id="S3.SS1.p2.6.m6.3.3.1.2.cmml" xref="S3.SS1.p2.6.m6.3.3.1.1"><cn id="S3.SS1.p2.6.m6.1.1.cmml" type="integer" xref="S3.SS1.p2.6.m6.1.1">1</cn><ci id="S3.SS1.p2.6.m6.2.2.cmml" xref="S3.SS1.p2.6.m6.2.2">…</ci><apply id="S3.SS1.p2.6.m6.3.3.1.1.1.cmml" xref="S3.SS1.p2.6.m6.3.3.1.1.1"><times id="S3.SS1.p2.6.m6.3.3.1.1.1.2.cmml" xref="S3.SS1.p2.6.m6.3.3.1.1.1.2"></times><ci id="S3.SS1.p2.6.m6.3.3.1.1.1.3.cmml" xref="S3.SS1.p2.6.m6.3.3.1.1.1.3">ℓ</ci><apply id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p2.6.m6.3.3.1.1.1.1.1.1.3">𝑖</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.6.m6.3c">k\in\{1,\ldots,\ell(a_{i})\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.6.m6.3d">italic_k ∈ { 1 , … , roman_ℓ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) }</annotation></semantics></math>. We then define the associated <span class="ltx_text ltx_font_italic" id="S3.SS1.p2.6.3">subdivision morphism</span> given by</p> <table class="ltx_equation ltx_eqn_table" id="S3.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(3.1)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}\,,\,\,a_{i}\mapsto a_{i}(1)\ldots a_% {i}({\ell(a_{i}))}\,." class="ltx_Math" display="block" id="S3.E1.m1.2"><semantics id="S3.E1.m1.2a"><mrow id="S3.E1.m1.2.2.1" xref="S3.E1.m1.2.2.1.1.cmml"><mrow id="S3.E1.m1.2.2.1.1" xref="S3.E1.m1.2.2.1.1.cmml"><msub id="S3.E1.m1.2.2.1.1.4" xref="S3.E1.m1.2.2.1.1.4.cmml"><mi id="S3.E1.m1.2.2.1.1.4.2" xref="S3.E1.m1.2.2.1.1.4.2.cmml">π</mi><mi id="S3.E1.m1.2.2.1.1.4.3" mathvariant="normal" xref="S3.E1.m1.2.2.1.1.4.3.cmml">ℓ</mi></msub><mo id="S3.E1.m1.2.2.1.1.3" lspace="0.278em" rspace="0.278em" xref="S3.E1.m1.2.2.1.1.3.cmml">:</mo><mrow id="S3.E1.m1.2.2.1.1.2.2" xref="S3.E1.m1.2.2.1.1.2.3.cmml"><mrow id="S3.E1.m1.2.2.1.1.1.1.1" xref="S3.E1.m1.2.2.1.1.1.1.1.cmml"><msup id="S3.E1.m1.2.2.1.1.1.1.1.2" xref="S3.E1.m1.2.2.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.2.2.1.1.1.1.1.2.2" xref="S3.E1.m1.2.2.1.1.1.1.1.2.2.cmml">𝒜</mi><mo id="S3.E1.m1.2.2.1.1.1.1.1.2.3" xref="S3.E1.m1.2.2.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S3.E1.m1.2.2.1.1.1.1.1.1" stretchy="false" xref="S3.E1.m1.2.2.1.1.1.1.1.1.cmml">→</mo><msubsup id="S3.E1.m1.2.2.1.1.1.1.1.3" xref="S3.E1.m1.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.2.2.1.1.1.1.1.3.2.2" xref="S3.E1.m1.2.2.1.1.1.1.1.3.2.2.cmml">𝒜</mi><mi id="S3.E1.m1.2.2.1.1.1.1.1.3.2.3" mathvariant="normal" xref="S3.E1.m1.2.2.1.1.1.1.1.3.2.3.cmml">ℓ</mi><mo id="S3.E1.m1.2.2.1.1.1.1.1.3.3" xref="S3.E1.m1.2.2.1.1.1.1.1.3.3.cmml">∗</mo></msubsup></mrow><mo id="S3.E1.m1.2.2.1.1.2.2.3" rspace="0.497em" xref="S3.E1.m1.2.2.1.1.2.3a.cmml">,</mo><mrow id="S3.E1.m1.2.2.1.1.2.2.2" xref="S3.E1.m1.2.2.1.1.2.2.2.cmml"><msub id="S3.E1.m1.2.2.1.1.2.2.2.3" xref="S3.E1.m1.2.2.1.1.2.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.2.2.1.1.2.2.2.3.2" xref="S3.E1.m1.2.2.1.1.2.2.2.3.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.2.2.1.1.2.2.2.3.3" xref="S3.E1.m1.2.2.1.1.2.2.2.3.3.cmml">𝒾</mi></msub><mo id="S3.E1.m1.2.2.1.1.2.2.2.2" stretchy="false" xref="S3.E1.m1.2.2.1.1.2.2.2.2.cmml">↦</mo><mrow id="S3.E1.m1.2.2.1.1.2.2.2.1" xref="S3.E1.m1.2.2.1.1.2.2.2.1.cmml"><msub id="S3.E1.m1.2.2.1.1.2.2.2.1.3" xref="S3.E1.m1.2.2.1.1.2.2.2.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.2.2.1.1.2.2.2.1.3.2" xref="S3.E1.m1.2.2.1.1.2.2.2.1.3.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.2.2.1.1.2.2.2.1.3.3" xref="S3.E1.m1.2.2.1.1.2.2.2.1.3.3.cmml">𝒾</mi></msub><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.2" xref="S3.E1.m1.2.2.1.1.2.2.2.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.2.2.1.1.2.2.2.1.4.2" xref="S3.E1.m1.2.2.1.1.2.2.2.1.cmml"><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.4.2.1" stretchy="false" xref="S3.E1.m1.2.2.1.1.2.2.2.1.cmml">(</mo><mn class="ltx_font_mathcaligraphic" id="S3.E1.m1.1.1" mathvariant="script" xref="S3.E1.m1.1.1.cmml">1</mn><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.4.2.2" stretchy="false" xref="S3.E1.m1.2.2.1.1.2.2.2.1.cmml">)</mo></mrow><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.2a" xref="S3.E1.m1.2.2.1.1.2.2.2.1.2.cmml">⁢</mo><mi id="S3.E1.m1.2.2.1.1.2.2.2.1.5" mathvariant="normal" xref="S3.E1.m1.2.2.1.1.2.2.2.1.5.cmml">…</mi><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.2b" xref="S3.E1.m1.2.2.1.1.2.2.2.1.2.cmml">⁢</mo><msub id="S3.E1.m1.2.2.1.1.2.2.2.1.6" xref="S3.E1.m1.2.2.1.1.2.2.2.1.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.2.2.1.1.2.2.2.1.6.2" xref="S3.E1.m1.2.2.1.1.2.2.2.1.6.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.2.2.1.1.2.2.2.1.6.3" xref="S3.E1.m1.2.2.1.1.2.2.2.1.6.3.cmml">𝒾</mi></msub><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.2c" xref="S3.E1.m1.2.2.1.1.2.2.2.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.cmml"><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.2" stretchy="false" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.cmml"><mi id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.3" mathvariant="normal" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.3.cmml">ℓ</mi><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.2" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.cmml"><mo id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.cmml"><mi 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id="S3.E1.m1.2.2.1.1.4.2.cmml" xref="S3.E1.m1.2.2.1.1.4.2">𝜋</ci><ci id="S3.E1.m1.2.2.1.1.4.3.cmml" xref="S3.E1.m1.2.2.1.1.4.3">ℓ</ci></apply><apply id="S3.E1.m1.2.2.1.1.2.3.cmml" xref="S3.E1.m1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.2.3a.cmml" xref="S3.E1.m1.2.2.1.1.2.2.3">formulae-sequence</csymbol><apply id="S3.E1.m1.2.2.1.1.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1"><ci id="S3.E1.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.1">→</ci><apply id="S3.E1.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.1.1.1.2.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.2">superscript</csymbol><ci id="S3.E1.m1.2.2.1.1.1.1.1.2.2.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.2.2">𝒜</ci><times id="S3.E1.m1.2.2.1.1.1.1.1.2.3.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.2.3"></times></apply><apply id="S3.E1.m1.2.2.1.1.1.1.1.3.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.1.1.1.3.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.3">superscript</csymbol><apply id="S3.E1.m1.2.2.1.1.1.1.1.3.2.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.1.1.1.3.2.1.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.3">subscript</csymbol><ci id="S3.E1.m1.2.2.1.1.1.1.1.3.2.2.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.3.2.2">𝒜</ci><ci id="S3.E1.m1.2.2.1.1.1.1.1.3.2.3.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.3.2.3">ℓ</ci></apply><times id="S3.E1.m1.2.2.1.1.1.1.1.3.3.cmml" xref="S3.E1.m1.2.2.1.1.1.1.1.3.3"></times></apply></apply><apply id="S3.E1.m1.2.2.1.1.2.2.2.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2"><csymbol cd="latexml" id="S3.E1.m1.2.2.1.1.2.2.2.2.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.2">maps-to</csymbol><apply id="S3.E1.m1.2.2.1.1.2.2.2.3.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.3"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.2.2.2.3.1.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.3">subscript</csymbol><ci id="S3.E1.m1.2.2.1.1.2.2.2.3.2.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.3.2">𝒶</ci><ci id="S3.E1.m1.2.2.1.1.2.2.2.3.3.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.3.3">𝒾</ci></apply><apply id="S3.E1.m1.2.2.1.1.2.2.2.1.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1"><times id="S3.E1.m1.2.2.1.1.2.2.2.1.2.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.2"></times><apply id="S3.E1.m1.2.2.1.1.2.2.2.1.3.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.3"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.2.2.2.1.3.1.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.3">subscript</csymbol><ci id="S3.E1.m1.2.2.1.1.2.2.2.1.3.2.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.3.2">𝒶</ci><ci id="S3.E1.m1.2.2.1.1.2.2.2.1.3.3.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.3.3">𝒾</ci></apply><cn id="S3.E1.m1.1.1.cmml" type="integer" xref="S3.E1.m1.1.1">1</cn><ci id="S3.E1.m1.2.2.1.1.2.2.2.1.5.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.5">…</ci><apply id="S3.E1.m1.2.2.1.1.2.2.2.1.6.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.6"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.2.2.2.1.6.1.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.6">subscript</csymbol><ci id="S3.E1.m1.2.2.1.1.2.2.2.1.6.2.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.6.2">𝒶</ci><ci id="S3.E1.m1.2.2.1.1.2.2.2.1.6.3.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.6.3">𝒾</ci></apply><apply id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1"><times id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.2.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.2"></times><ci id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.3.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.3">ℓ</ci><apply id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.2.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.2">𝒶</ci><ci id="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.3.cmml" xref="S3.E1.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.1.3">𝒾</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1.m1.2c">\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}\,,\,\,a_{i}\mapsto a_{i}(1)\ldots a_% {i}({\ell(a_{i}))}\,.</annotation><annotation encoding="application/x-llamapun" id="S3.E1.m1.2d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ↦ caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ( caligraphic_1 ) … caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ( roman_ℓ ( caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_theorem ltx_theorem_rem" id="S3.Thmthm1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm1.1.1.1">Remark 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm1.p1"> <p class="ltx_p" id="S3.Thmthm1.p1.10">(1) The name and the intuition here comes from picturing <math alttext="\cal A" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.1.m1.1"><semantics id="S3.Thmthm1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm1.p1.1.m1.1.1" xref="S3.Thmthm1.p1.1.m1.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.1.m1.1b"><ci id="S3.Thmthm1.p1.1.m1.1.1.cmml" xref="S3.Thmthm1.p1.1.m1.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.1.m1.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.1.m1.1d">caligraphic_A</annotation></semantics></math> as edge labels of an oriented “rose” <math alttext="RA" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.2.m2.1"><semantics id="S3.Thmthm1.p1.2.m2.1a"><mrow id="S3.Thmthm1.p1.2.m2.1.1" xref="S3.Thmthm1.p1.2.m2.1.1.cmml"><mi id="S3.Thmthm1.p1.2.m2.1.1.2" xref="S3.Thmthm1.p1.2.m2.1.1.2.cmml">R</mi><mo id="S3.Thmthm1.p1.2.m2.1.1.1" xref="S3.Thmthm1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm1.p1.2.m2.1.1.3" xref="S3.Thmthm1.p1.2.m2.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.2.m2.1b"><apply id="S3.Thmthm1.p1.2.m2.1.1.cmml" xref="S3.Thmthm1.p1.2.m2.1.1"><times id="S3.Thmthm1.p1.2.m2.1.1.1.cmml" xref="S3.Thmthm1.p1.2.m2.1.1.1"></times><ci id="S3.Thmthm1.p1.2.m2.1.1.2.cmml" xref="S3.Thmthm1.p1.2.m2.1.1.2">𝑅</ci><ci id="S3.Thmthm1.p1.2.m2.1.1.3.cmml" xref="S3.Thmthm1.p1.2.m2.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.2.m2.1c">RA</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.2.m2.1d">italic_R italic_A</annotation></semantics></math>, i.e. a 1-vertex graph with <math alttext="\mbox{card}(A)" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.3.m3.1"><semantics id="S3.Thmthm1.p1.3.m3.1a"><mrow id="S3.Thmthm1.p1.3.m3.1.2" xref="S3.Thmthm1.p1.3.m3.1.2.cmml"><mtext id="S3.Thmthm1.p1.3.m3.1.2.2" xref="S3.Thmthm1.p1.3.m3.1.2.2a.cmml">card</mtext><mo id="S3.Thmthm1.p1.3.m3.1.2.1" xref="S3.Thmthm1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm1.p1.3.m3.1.2.3.2" xref="S3.Thmthm1.p1.3.m3.1.2.cmml"><mo id="S3.Thmthm1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S3.Thmthm1.p1.3.m3.1.2.cmml">(</mo><mi id="S3.Thmthm1.p1.3.m3.1.1" xref="S3.Thmthm1.p1.3.m3.1.1.cmml">A</mi><mo id="S3.Thmthm1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S3.Thmthm1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.3.m3.1b"><apply id="S3.Thmthm1.p1.3.m3.1.2.cmml" xref="S3.Thmthm1.p1.3.m3.1.2"><times id="S3.Thmthm1.p1.3.m3.1.2.1.cmml" xref="S3.Thmthm1.p1.3.m3.1.2.1"></times><ci id="S3.Thmthm1.p1.3.m3.1.2.2a.cmml" xref="S3.Thmthm1.p1.3.m3.1.2.2"><mtext id="S3.Thmthm1.p1.3.m3.1.2.2.cmml" xref="S3.Thmthm1.p1.3.m3.1.2.2">card</mtext></ci><ci id="S3.Thmthm1.p1.3.m3.1.1.cmml" xref="S3.Thmthm1.p1.3.m3.1.1">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.3.m3.1c">\mbox{card}(A)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.3.m3.1d">card ( italic_A )</annotation></semantics></math> oriented edges. Then any edge with label <math alttext="a_{i}" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.4.m4.1"><semantics id="S3.Thmthm1.p1.4.m4.1a"><msub id="S3.Thmthm1.p1.4.m4.1.1" xref="S3.Thmthm1.p1.4.m4.1.1.cmml"><mi id="S3.Thmthm1.p1.4.m4.1.1.2" xref="S3.Thmthm1.p1.4.m4.1.1.2.cmml">a</mi><mi id="S3.Thmthm1.p1.4.m4.1.1.3" xref="S3.Thmthm1.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.4.m4.1b"><apply id="S3.Thmthm1.p1.4.m4.1.1.cmml" xref="S3.Thmthm1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p1.4.m4.1.1.1.cmml" xref="S3.Thmthm1.p1.4.m4.1.1">subscript</csymbol><ci id="S3.Thmthm1.p1.4.m4.1.1.2.cmml" xref="S3.Thmthm1.p1.4.m4.1.1.2">𝑎</ci><ci id="S3.Thmthm1.p1.4.m4.1.1.3.cmml" xref="S3.Thmthm1.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.4.m4.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is subdivided by introducing <math alttext="\ell(a_{i})-1" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.5.m5.1"><semantics id="S3.Thmthm1.p1.5.m5.1a"><mrow id="S3.Thmthm1.p1.5.m5.1.1" xref="S3.Thmthm1.p1.5.m5.1.1.cmml"><mrow id="S3.Thmthm1.p1.5.m5.1.1.1" xref="S3.Thmthm1.p1.5.m5.1.1.1.cmml"><mi id="S3.Thmthm1.p1.5.m5.1.1.1.3" mathvariant="normal" xref="S3.Thmthm1.p1.5.m5.1.1.1.3.cmml">ℓ</mi><mo id="S3.Thmthm1.p1.5.m5.1.1.1.2" xref="S3.Thmthm1.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmthm1.p1.5.m5.1.1.1.1.1" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.2" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.3" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm1.p1.5.m5.1.1.2" xref="S3.Thmthm1.p1.5.m5.1.1.2.cmml">−</mo><mn id="S3.Thmthm1.p1.5.m5.1.1.3" xref="S3.Thmthm1.p1.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.5.m5.1b"><apply id="S3.Thmthm1.p1.5.m5.1.1.cmml" xref="S3.Thmthm1.p1.5.m5.1.1"><minus id="S3.Thmthm1.p1.5.m5.1.1.2.cmml" xref="S3.Thmthm1.p1.5.m5.1.1.2"></minus><apply id="S3.Thmthm1.p1.5.m5.1.1.1.cmml" xref="S3.Thmthm1.p1.5.m5.1.1.1"><times id="S3.Thmthm1.p1.5.m5.1.1.1.2.cmml" xref="S3.Thmthm1.p1.5.m5.1.1.1.2"></times><ci id="S3.Thmthm1.p1.5.m5.1.1.1.3.cmml" xref="S3.Thmthm1.p1.5.m5.1.1.1.3">ℓ</ci><apply id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm1.p1.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S3.Thmthm1.p1.5.m5.1.1.3.cmml" type="integer" xref="S3.Thmthm1.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.5.m5.1c">\ell(a_{i})-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.5.m5.1d">roman_ℓ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - 1</annotation></semantics></math> new vertices in its interior, and by labeling the obtained new edges (in the order given by the orientation) by <math alttext="a_{i}(1),a_{i}(2),\ldots,a_{i}(\ell(a_{i}))" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.6.m6.6"><semantics id="S3.Thmthm1.p1.6.m6.6a"><mrow id="S3.Thmthm1.p1.6.m6.6.6.3" xref="S3.Thmthm1.p1.6.m6.6.6.4.cmml"><mrow id="S3.Thmthm1.p1.6.m6.4.4.1.1" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.cmml"><msub id="S3.Thmthm1.p1.6.m6.4.4.1.1.2" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.2.cmml"><mi id="S3.Thmthm1.p1.6.m6.4.4.1.1.2.2" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.2.2.cmml">a</mi><mi id="S3.Thmthm1.p1.6.m6.4.4.1.1.2.3" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.2.3.cmml">i</mi></msub><mo id="S3.Thmthm1.p1.6.m6.4.4.1.1.1" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.1.cmml">⁢</mo><mrow id="S3.Thmthm1.p1.6.m6.4.4.1.1.3.2" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.cmml"><mo id="S3.Thmthm1.p1.6.m6.4.4.1.1.3.2.1" stretchy="false" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.cmml">(</mo><mn id="S3.Thmthm1.p1.6.m6.1.1" xref="S3.Thmthm1.p1.6.m6.1.1.cmml">1</mn><mo id="S3.Thmthm1.p1.6.m6.4.4.1.1.3.2.2" stretchy="false" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm1.p1.6.m6.6.6.3.4" xref="S3.Thmthm1.p1.6.m6.6.6.4.cmml">,</mo><mrow id="S3.Thmthm1.p1.6.m6.5.5.2.2" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.cmml"><msub id="S3.Thmthm1.p1.6.m6.5.5.2.2.2" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.2.cmml"><mi id="S3.Thmthm1.p1.6.m6.5.5.2.2.2.2" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.2.2.cmml">a</mi><mi id="S3.Thmthm1.p1.6.m6.5.5.2.2.2.3" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.2.3.cmml">i</mi></msub><mo id="S3.Thmthm1.p1.6.m6.5.5.2.2.1" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm1.p1.6.m6.5.5.2.2.3.2" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.cmml"><mo id="S3.Thmthm1.p1.6.m6.5.5.2.2.3.2.1" stretchy="false" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.cmml">(</mo><mn id="S3.Thmthm1.p1.6.m6.2.2" xref="S3.Thmthm1.p1.6.m6.2.2.cmml">2</mn><mo id="S3.Thmthm1.p1.6.m6.5.5.2.2.3.2.2" stretchy="false" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm1.p1.6.m6.6.6.3.5" xref="S3.Thmthm1.p1.6.m6.6.6.4.cmml">,</mo><mi id="S3.Thmthm1.p1.6.m6.3.3" mathvariant="normal" xref="S3.Thmthm1.p1.6.m6.3.3.cmml">…</mi><mo id="S3.Thmthm1.p1.6.m6.6.6.3.6" xref="S3.Thmthm1.p1.6.m6.6.6.4.cmml">,</mo><mrow id="S3.Thmthm1.p1.6.m6.6.6.3.3" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.cmml"><msub id="S3.Thmthm1.p1.6.m6.6.6.3.3.3" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.3.cmml"><mi id="S3.Thmthm1.p1.6.m6.6.6.3.3.3.2" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.3.2.cmml">a</mi><mi id="S3.Thmthm1.p1.6.m6.6.6.3.3.3.3" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.3.3.cmml">i</mi></msub><mo id="S3.Thmthm1.p1.6.m6.6.6.3.3.2" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.2.cmml">⁢</mo><mrow id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.cmml"><mo id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.2" stretchy="false" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.cmml">(</mo><mrow id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.cmml"><mi id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.3" mathvariant="normal" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.3.cmml">ℓ</mi><mo id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.2" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.cmml"><mo id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.2" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.3" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.3" stretchy="false" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.6.m6.6b"><list id="S3.Thmthm1.p1.6.m6.6.6.4.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3"><apply id="S3.Thmthm1.p1.6.m6.4.4.1.1.cmml" xref="S3.Thmthm1.p1.6.m6.4.4.1.1"><times id="S3.Thmthm1.p1.6.m6.4.4.1.1.1.cmml" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.1"></times><apply id="S3.Thmthm1.p1.6.m6.4.4.1.1.2.cmml" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm1.p1.6.m6.4.4.1.1.2.1.cmml" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.2">subscript</csymbol><ci id="S3.Thmthm1.p1.6.m6.4.4.1.1.2.2.cmml" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.2.2">𝑎</ci><ci id="S3.Thmthm1.p1.6.m6.4.4.1.1.2.3.cmml" xref="S3.Thmthm1.p1.6.m6.4.4.1.1.2.3">𝑖</ci></apply><cn id="S3.Thmthm1.p1.6.m6.1.1.cmml" type="integer" xref="S3.Thmthm1.p1.6.m6.1.1">1</cn></apply><apply id="S3.Thmthm1.p1.6.m6.5.5.2.2.cmml" xref="S3.Thmthm1.p1.6.m6.5.5.2.2"><times id="S3.Thmthm1.p1.6.m6.5.5.2.2.1.cmml" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.1"></times><apply id="S3.Thmthm1.p1.6.m6.5.5.2.2.2.cmml" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.2"><csymbol cd="ambiguous" id="S3.Thmthm1.p1.6.m6.5.5.2.2.2.1.cmml" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.2">subscript</csymbol><ci id="S3.Thmthm1.p1.6.m6.5.5.2.2.2.2.cmml" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.2.2">𝑎</ci><ci id="S3.Thmthm1.p1.6.m6.5.5.2.2.2.3.cmml" xref="S3.Thmthm1.p1.6.m6.5.5.2.2.2.3">𝑖</ci></apply><cn id="S3.Thmthm1.p1.6.m6.2.2.cmml" type="integer" xref="S3.Thmthm1.p1.6.m6.2.2">2</cn></apply><ci id="S3.Thmthm1.p1.6.m6.3.3.cmml" xref="S3.Thmthm1.p1.6.m6.3.3">…</ci><apply id="S3.Thmthm1.p1.6.m6.6.6.3.3.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3"><times id="S3.Thmthm1.p1.6.m6.6.6.3.3.2.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.2"></times><apply id="S3.Thmthm1.p1.6.m6.6.6.3.3.3.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.3"><csymbol cd="ambiguous" id="S3.Thmthm1.p1.6.m6.6.6.3.3.3.1.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.3">subscript</csymbol><ci id="S3.Thmthm1.p1.6.m6.6.6.3.3.3.2.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.3.2">𝑎</ci><ci id="S3.Thmthm1.p1.6.m6.6.6.3.3.3.3.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.3.3">𝑖</ci></apply><apply id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1"><times id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.2.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.2"></times><ci id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.3.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.3">ℓ</ci><apply id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm1.p1.6.m6.6.6.3.3.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.6.m6.6c">a_{i}(1),a_{i}(2),\ldots,a_{i}(\ell(a_{i}))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.6.m6.6d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 ) , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 2 ) , … , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( roman_ℓ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) )</annotation></semantics></math>. Then any edge path <math alttext="\gamma(w)" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.7.m7.1"><semantics id="S3.Thmthm1.p1.7.m7.1a"><mrow id="S3.Thmthm1.p1.7.m7.1.2" xref="S3.Thmthm1.p1.7.m7.1.2.cmml"><mi id="S3.Thmthm1.p1.7.m7.1.2.2" xref="S3.Thmthm1.p1.7.m7.1.2.2.cmml">γ</mi><mo id="S3.Thmthm1.p1.7.m7.1.2.1" xref="S3.Thmthm1.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm1.p1.7.m7.1.2.3.2" xref="S3.Thmthm1.p1.7.m7.1.2.cmml"><mo id="S3.Thmthm1.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S3.Thmthm1.p1.7.m7.1.2.cmml">(</mo><mi id="S3.Thmthm1.p1.7.m7.1.1" xref="S3.Thmthm1.p1.7.m7.1.1.cmml">w</mi><mo id="S3.Thmthm1.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S3.Thmthm1.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.7.m7.1b"><apply id="S3.Thmthm1.p1.7.m7.1.2.cmml" xref="S3.Thmthm1.p1.7.m7.1.2"><times id="S3.Thmthm1.p1.7.m7.1.2.1.cmml" xref="S3.Thmthm1.p1.7.m7.1.2.1"></times><ci id="S3.Thmthm1.p1.7.m7.1.2.2.cmml" xref="S3.Thmthm1.p1.7.m7.1.2.2">𝛾</ci><ci id="S3.Thmthm1.p1.7.m7.1.1.cmml" xref="S3.Thmthm1.p1.7.m7.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.7.m7.1c">\gamma(w)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.7.m7.1d">italic_γ ( italic_w )</annotation></semantics></math> in <math alttext="RA" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.8.m8.1"><semantics id="S3.Thmthm1.p1.8.m8.1a"><mrow id="S3.Thmthm1.p1.8.m8.1.1" xref="S3.Thmthm1.p1.8.m8.1.1.cmml"><mi id="S3.Thmthm1.p1.8.m8.1.1.2" xref="S3.Thmthm1.p1.8.m8.1.1.2.cmml">R</mi><mo id="S3.Thmthm1.p1.8.m8.1.1.1" xref="S3.Thmthm1.p1.8.m8.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm1.p1.8.m8.1.1.3" xref="S3.Thmthm1.p1.8.m8.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.8.m8.1b"><apply id="S3.Thmthm1.p1.8.m8.1.1.cmml" xref="S3.Thmthm1.p1.8.m8.1.1"><times id="S3.Thmthm1.p1.8.m8.1.1.1.cmml" xref="S3.Thmthm1.p1.8.m8.1.1.1"></times><ci id="S3.Thmthm1.p1.8.m8.1.1.2.cmml" xref="S3.Thmthm1.p1.8.m8.1.1.2">𝑅</ci><ci id="S3.Thmthm1.p1.8.m8.1.1.3.cmml" xref="S3.Thmthm1.p1.8.m8.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.8.m8.1c">RA</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.8.m8.1d">italic_R italic_A</annotation></semantics></math>, which reads off a word <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.9.m9.1"><semantics id="S3.Thmthm1.p1.9.m9.1a"><mrow id="S3.Thmthm1.p1.9.m9.1.1" xref="S3.Thmthm1.p1.9.m9.1.1.cmml"><mi id="S3.Thmthm1.p1.9.m9.1.1.2" xref="S3.Thmthm1.p1.9.m9.1.1.2.cmml">w</mi><mo id="S3.Thmthm1.p1.9.m9.1.1.1" xref="S3.Thmthm1.p1.9.m9.1.1.1.cmml">∈</mo><msup id="S3.Thmthm1.p1.9.m9.1.1.3" xref="S3.Thmthm1.p1.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm1.p1.9.m9.1.1.3.2" xref="S3.Thmthm1.p1.9.m9.1.1.3.2.cmml">𝒜</mi><mo id="S3.Thmthm1.p1.9.m9.1.1.3.3" xref="S3.Thmthm1.p1.9.m9.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.9.m9.1b"><apply id="S3.Thmthm1.p1.9.m9.1.1.cmml" xref="S3.Thmthm1.p1.9.m9.1.1"><in id="S3.Thmthm1.p1.9.m9.1.1.1.cmml" xref="S3.Thmthm1.p1.9.m9.1.1.1"></in><ci id="S3.Thmthm1.p1.9.m9.1.1.2.cmml" xref="S3.Thmthm1.p1.9.m9.1.1.2">𝑤</ci><apply id="S3.Thmthm1.p1.9.m9.1.1.3.cmml" xref="S3.Thmthm1.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm1.p1.9.m9.1.1.3.1.cmml" xref="S3.Thmthm1.p1.9.m9.1.1.3">superscript</csymbol><ci id="S3.Thmthm1.p1.9.m9.1.1.3.2.cmml" xref="S3.Thmthm1.p1.9.m9.1.1.3.2">𝒜</ci><times id="S3.Thmthm1.p1.9.m9.1.1.3.3.cmml" xref="S3.Thmthm1.p1.9.m9.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.9.m9.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.9.m9.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, will after the subdivision read off the word <math alttext="\pi_{\ell}(w)" class="ltx_Math" display="inline" id="S3.Thmthm1.p1.10.m10.1"><semantics id="S3.Thmthm1.p1.10.m10.1a"><mrow id="S3.Thmthm1.p1.10.m10.1.2" xref="S3.Thmthm1.p1.10.m10.1.2.cmml"><msub id="S3.Thmthm1.p1.10.m10.1.2.2" xref="S3.Thmthm1.p1.10.m10.1.2.2.cmml"><mi id="S3.Thmthm1.p1.10.m10.1.2.2.2" xref="S3.Thmthm1.p1.10.m10.1.2.2.2.cmml">π</mi><mi id="S3.Thmthm1.p1.10.m10.1.2.2.3" mathvariant="normal" xref="S3.Thmthm1.p1.10.m10.1.2.2.3.cmml">ℓ</mi></msub><mo id="S3.Thmthm1.p1.10.m10.1.2.1" xref="S3.Thmthm1.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm1.p1.10.m10.1.2.3.2" xref="S3.Thmthm1.p1.10.m10.1.2.cmml"><mo id="S3.Thmthm1.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S3.Thmthm1.p1.10.m10.1.2.cmml">(</mo><mi id="S3.Thmthm1.p1.10.m10.1.1" xref="S3.Thmthm1.p1.10.m10.1.1.cmml">w</mi><mo id="S3.Thmthm1.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S3.Thmthm1.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p1.10.m10.1b"><apply id="S3.Thmthm1.p1.10.m10.1.2.cmml" xref="S3.Thmthm1.p1.10.m10.1.2"><times id="S3.Thmthm1.p1.10.m10.1.2.1.cmml" xref="S3.Thmthm1.p1.10.m10.1.2.1"></times><apply id="S3.Thmthm1.p1.10.m10.1.2.2.cmml" xref="S3.Thmthm1.p1.10.m10.1.2.2"><csymbol cd="ambiguous" id="S3.Thmthm1.p1.10.m10.1.2.2.1.cmml" xref="S3.Thmthm1.p1.10.m10.1.2.2">subscript</csymbol><ci id="S3.Thmthm1.p1.10.m10.1.2.2.2.cmml" xref="S3.Thmthm1.p1.10.m10.1.2.2.2">𝜋</ci><ci id="S3.Thmthm1.p1.10.m10.1.2.2.3.cmml" xref="S3.Thmthm1.p1.10.m10.1.2.2.3">ℓ</ci></apply><ci id="S3.Thmthm1.p1.10.m10.1.1.cmml" xref="S3.Thmthm1.p1.10.m10.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p1.10.m10.1c">\pi_{\ell}(w)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p1.10.m10.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S3.Thmthm1.p2"> <p class="ltx_p" id="S3.Thmthm1.p2.7">(2) The natural “geometrization” of the <span class="ltx_text ltx_font_italic" id="S3.Thmthm1.p2.7.1">subdivision monoid</span> <math alttext="\cal A_{\ell}^{*}" class="ltx_Math" display="inline" id="S3.Thmthm1.p2.1.m1.1"><semantics id="S3.Thmthm1.p2.1.m1.1a"><msubsup id="S3.Thmthm1.p2.1.m1.1.1" xref="S3.Thmthm1.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm1.p2.1.m1.1.1.2.2" xref="S3.Thmthm1.p2.1.m1.1.1.2.2.cmml">𝒜</mi><mi id="S3.Thmthm1.p2.1.m1.1.1.2.3" mathvariant="normal" xref="S3.Thmthm1.p2.1.m1.1.1.2.3.cmml">ℓ</mi><mo id="S3.Thmthm1.p2.1.m1.1.1.3" xref="S3.Thmthm1.p2.1.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p2.1.m1.1b"><apply id="S3.Thmthm1.p2.1.m1.1.1.cmml" xref="S3.Thmthm1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.1.m1.1.1.1.cmml" xref="S3.Thmthm1.p2.1.m1.1.1">superscript</csymbol><apply id="S3.Thmthm1.p2.1.m1.1.1.2.cmml" xref="S3.Thmthm1.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.1.m1.1.1.2.1.cmml" xref="S3.Thmthm1.p2.1.m1.1.1">subscript</csymbol><ci id="S3.Thmthm1.p2.1.m1.1.1.2.2.cmml" xref="S3.Thmthm1.p2.1.m1.1.1.2.2">𝒜</ci><ci id="S3.Thmthm1.p2.1.m1.1.1.2.3.cmml" xref="S3.Thmthm1.p2.1.m1.1.1.2.3">ℓ</ci></apply><times id="S3.Thmthm1.p2.1.m1.1.1.3.cmml" xref="S3.Thmthm1.p2.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p2.1.m1.1c">\cal A_{\ell}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p2.1.m1.1d">caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> as rose <math alttext="R_{\cal A_{\ell}}" class="ltx_Math" display="inline" id="S3.Thmthm1.p2.2.m2.1"><semantics id="S3.Thmthm1.p2.2.m2.1a"><msub id="S3.Thmthm1.p2.2.m2.1.1" xref="S3.Thmthm1.p2.2.m2.1.1.cmml"><mi id="S3.Thmthm1.p2.2.m2.1.1.2" xref="S3.Thmthm1.p2.2.m2.1.1.2.cmml">R</mi><msub id="S3.Thmthm1.p2.2.m2.1.1.3" xref="S3.Thmthm1.p2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm1.p2.2.m2.1.1.3.2" xref="S3.Thmthm1.p2.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S3.Thmthm1.p2.2.m2.1.1.3.3" mathvariant="normal" xref="S3.Thmthm1.p2.2.m2.1.1.3.3.cmml">ℓ</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p2.2.m2.1b"><apply id="S3.Thmthm1.p2.2.m2.1.1.cmml" xref="S3.Thmthm1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.2.m2.1.1.1.cmml" xref="S3.Thmthm1.p2.2.m2.1.1">subscript</csymbol><ci id="S3.Thmthm1.p2.2.m2.1.1.2.cmml" xref="S3.Thmthm1.p2.2.m2.1.1.2">𝑅</ci><apply id="S3.Thmthm1.p2.2.m2.1.1.3.cmml" xref="S3.Thmthm1.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.2.m2.1.1.3.1.cmml" xref="S3.Thmthm1.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.Thmthm1.p2.2.m2.1.1.3.2.cmml" xref="S3.Thmthm1.p2.2.m2.1.1.3.2">𝒜</ci><ci id="S3.Thmthm1.p2.2.m2.1.1.3.3.cmml" xref="S3.Thmthm1.p2.2.m2.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p2.2.m2.1c">R_{\cal A_{\ell}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p2.2.m2.1d">italic_R start_POSTSUBSCRIPT caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is however not quite the above subdivision of the rose <math alttext="RA" class="ltx_Math" display="inline" id="S3.Thmthm1.p2.3.m3.1"><semantics id="S3.Thmthm1.p2.3.m3.1a"><mrow id="S3.Thmthm1.p2.3.m3.1.1" xref="S3.Thmthm1.p2.3.m3.1.1.cmml"><mi id="S3.Thmthm1.p2.3.m3.1.1.2" xref="S3.Thmthm1.p2.3.m3.1.1.2.cmml">R</mi><mo id="S3.Thmthm1.p2.3.m3.1.1.1" xref="S3.Thmthm1.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm1.p2.3.m3.1.1.3" xref="S3.Thmthm1.p2.3.m3.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p2.3.m3.1b"><apply id="S3.Thmthm1.p2.3.m3.1.1.cmml" xref="S3.Thmthm1.p2.3.m3.1.1"><times id="S3.Thmthm1.p2.3.m3.1.1.1.cmml" xref="S3.Thmthm1.p2.3.m3.1.1.1"></times><ci id="S3.Thmthm1.p2.3.m3.1.1.2.cmml" xref="S3.Thmthm1.p2.3.m3.1.1.2">𝑅</ci><ci id="S3.Thmthm1.p2.3.m3.1.1.3.cmml" xref="S3.Thmthm1.p2.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p2.3.m3.1c">RA</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p2.3.m3.1d">italic_R italic_A</annotation></semantics></math>, but is obtained from the latter by identifying all subdivision vertices into a single vertex. This is reflected by the fact that the subdivision morphism <math alttext="\pi_{\ell}" class="ltx_Math" display="inline" id="S3.Thmthm1.p2.4.m4.1"><semantics id="S3.Thmthm1.p2.4.m4.1a"><msub id="S3.Thmthm1.p2.4.m4.1.1" xref="S3.Thmthm1.p2.4.m4.1.1.cmml"><mi id="S3.Thmthm1.p2.4.m4.1.1.2" xref="S3.Thmthm1.p2.4.m4.1.1.2.cmml">π</mi><mi id="S3.Thmthm1.p2.4.m4.1.1.3" mathvariant="normal" xref="S3.Thmthm1.p2.4.m4.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p2.4.m4.1b"><apply id="S3.Thmthm1.p2.4.m4.1.1.cmml" xref="S3.Thmthm1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.4.m4.1.1.1.cmml" xref="S3.Thmthm1.p2.4.m4.1.1">subscript</csymbol><ci id="S3.Thmthm1.p2.4.m4.1.1.2.cmml" xref="S3.Thmthm1.p2.4.m4.1.1.2">𝜋</ci><ci id="S3.Thmthm1.p2.4.m4.1.1.3.cmml" xref="S3.Thmthm1.p2.4.m4.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p2.4.m4.1c">\pi_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p2.4.m4.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> defined above is not surjective (except in the trivial case where all <math alttext="\ell(a_{i})=1" class="ltx_Math" display="inline" id="S3.Thmthm1.p2.5.m5.1"><semantics id="S3.Thmthm1.p2.5.m5.1a"><mrow id="S3.Thmthm1.p2.5.m5.1.1" xref="S3.Thmthm1.p2.5.m5.1.1.cmml"><mrow id="S3.Thmthm1.p2.5.m5.1.1.1" xref="S3.Thmthm1.p2.5.m5.1.1.1.cmml"><mi id="S3.Thmthm1.p2.5.m5.1.1.1.3" mathvariant="normal" xref="S3.Thmthm1.p2.5.m5.1.1.1.3.cmml">ℓ</mi><mo id="S3.Thmthm1.p2.5.m5.1.1.1.2" xref="S3.Thmthm1.p2.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmthm1.p2.5.m5.1.1.1.1.1" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.2" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.3" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm1.p2.5.m5.1.1.2" xref="S3.Thmthm1.p2.5.m5.1.1.2.cmml">=</mo><mn id="S3.Thmthm1.p2.5.m5.1.1.3" xref="S3.Thmthm1.p2.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p2.5.m5.1b"><apply id="S3.Thmthm1.p2.5.m5.1.1.cmml" xref="S3.Thmthm1.p2.5.m5.1.1"><eq id="S3.Thmthm1.p2.5.m5.1.1.2.cmml" xref="S3.Thmthm1.p2.5.m5.1.1.2"></eq><apply id="S3.Thmthm1.p2.5.m5.1.1.1.cmml" xref="S3.Thmthm1.p2.5.m5.1.1.1"><times id="S3.Thmthm1.p2.5.m5.1.1.1.2.cmml" xref="S3.Thmthm1.p2.5.m5.1.1.1.2"></times><ci id="S3.Thmthm1.p2.5.m5.1.1.1.3.cmml" xref="S3.Thmthm1.p2.5.m5.1.1.1.3">ℓ</ci><apply id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.cmml" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm1.p2.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply></apply><cn id="S3.Thmthm1.p2.5.m5.1.1.3.cmml" type="integer" xref="S3.Thmthm1.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p2.5.m5.1c">\ell(a_{i})=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p2.5.m5.1d">roman_ℓ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = 1</annotation></semantics></math> so that <math alttext="\pi_{\ell}" class="ltx_Math" display="inline" id="S3.Thmthm1.p2.6.m6.1"><semantics id="S3.Thmthm1.p2.6.m6.1a"><msub id="S3.Thmthm1.p2.6.m6.1.1" xref="S3.Thmthm1.p2.6.m6.1.1.cmml"><mi id="S3.Thmthm1.p2.6.m6.1.1.2" xref="S3.Thmthm1.p2.6.m6.1.1.2.cmml">π</mi><mi id="S3.Thmthm1.p2.6.m6.1.1.3" mathvariant="normal" xref="S3.Thmthm1.p2.6.m6.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p2.6.m6.1b"><apply id="S3.Thmthm1.p2.6.m6.1.1.cmml" xref="S3.Thmthm1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.6.m6.1.1.1.cmml" xref="S3.Thmthm1.p2.6.m6.1.1">subscript</csymbol><ci id="S3.Thmthm1.p2.6.m6.1.1.2.cmml" xref="S3.Thmthm1.p2.6.m6.1.1.2">𝜋</ci><ci id="S3.Thmthm1.p2.6.m6.1.1.3.cmml" xref="S3.Thmthm1.p2.6.m6.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p2.6.m6.1c">\pi_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p2.6.m6.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> is a bijection). Its image generates a subshift of finite type in <math alttext="\cal A_{\ell}^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm1.p2.7.m7.1"><semantics id="S3.Thmthm1.p2.7.m7.1a"><msubsup id="S3.Thmthm1.p2.7.m7.1.1" xref="S3.Thmthm1.p2.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm1.p2.7.m7.1.1.2.2" xref="S3.Thmthm1.p2.7.m7.1.1.2.2.cmml">𝒜</mi><mi id="S3.Thmthm1.p2.7.m7.1.1.2.3" mathvariant="normal" xref="S3.Thmthm1.p2.7.m7.1.1.2.3.cmml">ℓ</mi><mi id="S3.Thmthm1.p2.7.m7.1.1.3" xref="S3.Thmthm1.p2.7.m7.1.1.3.cmml">ℤ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmthm1.p2.7.m7.1b"><apply id="S3.Thmthm1.p2.7.m7.1.1.cmml" xref="S3.Thmthm1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.7.m7.1.1.1.cmml" xref="S3.Thmthm1.p2.7.m7.1.1">superscript</csymbol><apply id="S3.Thmthm1.p2.7.m7.1.1.2.cmml" xref="S3.Thmthm1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S3.Thmthm1.p2.7.m7.1.1.2.1.cmml" xref="S3.Thmthm1.p2.7.m7.1.1">subscript</csymbol><ci id="S3.Thmthm1.p2.7.m7.1.1.2.2.cmml" xref="S3.Thmthm1.p2.7.m7.1.1.2.2">𝒜</ci><ci id="S3.Thmthm1.p2.7.m7.1.1.2.3.cmml" xref="S3.Thmthm1.p2.7.m7.1.1.2.3">ℓ</ci></apply><ci id="S3.Thmthm1.p2.7.m7.1.1.3.cmml" xref="S3.Thmthm1.p2.7.m7.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm1.p2.7.m7.1c">\cal A_{\ell}^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm1.p2.7.m7.1d">caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.1">From the above definitions we deduce directly the following:</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="S3.Thmthm2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm2.1.1.1">Lemma 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm2.p1"> <p class="ltx_p" id="S3.Thmthm2.p1.1"><span class="ltx_text ltx_font_italic" id="S3.Thmthm2.p1.1.1">Let <math alttext="\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}" class="ltx_Math" display="inline" id="S3.Thmthm2.p1.1.1.m1.1"><semantics id="S3.Thmthm2.p1.1.1.m1.1a"><mrow id="S3.Thmthm2.p1.1.1.m1.1.1" xref="S3.Thmthm2.p1.1.1.m1.1.1.cmml"><msub id="S3.Thmthm2.p1.1.1.m1.1.1.2" xref="S3.Thmthm2.p1.1.1.m1.1.1.2.cmml"><mi id="S3.Thmthm2.p1.1.1.m1.1.1.2.2" xref="S3.Thmthm2.p1.1.1.m1.1.1.2.2.cmml">π</mi><mi id="S3.Thmthm2.p1.1.1.m1.1.1.2.3" mathvariant="normal" xref="S3.Thmthm2.p1.1.1.m1.1.1.2.3.cmml">ℓ</mi></msub><mo id="S3.Thmthm2.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm2.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm2.p1.1.1.m1.1.1.3" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.cmml"><msup id="S3.Thmthm2.p1.1.1.m1.1.1.3.2" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm2.p1.1.1.m1.1.1.3.2.2" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm2.p1.1.1.m1.1.1.3.2.3" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm2.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.1.cmml">→</mo><msubsup id="S3.Thmthm2.p1.1.1.m1.1.1.3.3" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.2" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.3" mathvariant="normal" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.3.cmml">ℓ</mi><mo id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.3" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm2.p1.1.1.m1.1b"><apply id="S3.Thmthm2.p1.1.1.m1.1.1.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1"><ci id="S3.Thmthm2.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.1">:</ci><apply id="S3.Thmthm2.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm2.p1.1.1.m1.1.1.2.1.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.2">subscript</csymbol><ci id="S3.Thmthm2.p1.1.1.m1.1.1.2.2.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.2.2">𝜋</ci><ci id="S3.Thmthm2.p1.1.1.m1.1.1.2.3.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.2.3">ℓ</ci></apply><apply id="S3.Thmthm2.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3"><ci id="S3.Thmthm2.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm2.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm2.p1.1.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm2.p1.1.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm2.p1.1.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3">superscript</csymbol><apply id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.1.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.2.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.2">𝒜</ci><ci id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.3.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3.2.3">ℓ</ci></apply><times id="S3.Thmthm2.p1.1.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm2.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm2.p1.1.1.m1.1c">\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm2.p1.1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a subdivision morphism as in (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E1" title="In 3.1. Subdivision morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.1</span></a>). Then the following holds:</span></p> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.1"><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.1.1">The monoid morphism </span><math alttext="\pi_{\ell}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><msub id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml">π</mi><mi id="S3.I1.i1.p1.1.m1.1.1.3" mathvariant="normal" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2">𝜋</ci><ci id="S3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">\pi_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i1.p1.1.2"> is injective.</span></p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.1"><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.1.1">The induced map </span><math alttext="\pi_{\ell}^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal A_{\ell}^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><mrow id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><msubsup id="S3.I1.i2.p1.1.m1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.2.2.2" xref="S3.I1.i2.p1.1.m1.1.1.2.2.2.cmml">π</mi><mi id="S3.I1.i2.p1.1.m1.1.1.2.2.3" mathvariant="normal" xref="S3.I1.i2.p1.1.m1.1.1.2.2.3.cmml">ℓ</mi><mi id="S3.I1.i2.p1.1.m1.1.1.2.3" xref="S3.I1.i2.p1.1.m1.1.1.2.3.cmml">ℤ</mi></msubsup><mo id="S3.I1.i2.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.I1.i2.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml"><msup id="S3.I1.i2.p1.1.m1.1.1.3.2" xref="S3.I1.i2.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i2.p1.1.m1.1.1.3.2.2" xref="S3.I1.i2.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mi id="S3.I1.i2.p1.1.m1.1.1.3.2.3" xref="S3.I1.i2.p1.1.m1.1.1.3.2.3.cmml">ℤ</mi></msup><mo id="S3.I1.i2.p1.1.m1.1.1.3.1" stretchy="false" xref="S3.I1.i2.p1.1.m1.1.1.3.1.cmml">→</mo><msubsup id="S3.I1.i2.p1.1.m1.1.1.3.3" xref="S3.I1.i2.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i2.p1.1.m1.1.1.3.3.2.2" xref="S3.I1.i2.p1.1.m1.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S3.I1.i2.p1.1.m1.1.1.3.3.2.3" mathvariant="normal" xref="S3.I1.i2.p1.1.m1.1.1.3.3.2.3.cmml">ℓ</mi><mi id="S3.I1.i2.p1.1.m1.1.1.3.3.3" xref="S3.I1.i2.p1.1.m1.1.1.3.3.3.cmml">ℤ</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><ci id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1">:</ci><apply id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2">superscript</csymbol><apply id="S3.I1.i2.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.2.2.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.2.2.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2.2.2">𝜋</ci><ci id="S3.I1.i2.p1.1.m1.1.1.2.2.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2.2.3">ℓ</ci></apply><ci id="S3.I1.i2.p1.1.m1.1.1.2.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2.3">ℤ</ci></apply><apply id="S3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3"><ci id="S3.I1.i2.p1.1.m1.1.1.3.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.1">→</ci><apply id="S3.I1.i2.p1.1.m1.1.1.3.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.3.2.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.3.2.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.2.2">𝒜</ci><ci id="S3.I1.i2.p1.1.m1.1.1.3.2.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.2.3">ℤ</ci></apply><apply id="S3.I1.i2.p1.1.m1.1.1.3.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.3.3.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.3">superscript</csymbol><apply id="S3.I1.i2.p1.1.m1.1.1.3.3.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.3.3.2.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.3.3.2.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.3.2.2">𝒜</ci><ci id="S3.I1.i2.p1.1.m1.1.1.3.3.2.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.3.2.3">ℓ</ci></apply><ci id="S3.I1.i2.p1.1.m1.1.1.3.3.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.3.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">\pi_{\ell}^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal A_{\ell}^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT : caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i2.p1.1.2"> is injective.</span></p> </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.1"><span class="ltx_text ltx_font_italic" id="S3.I1.i3.p1.1.1">The map </span><math alttext="\pi_{\ell}^{T}" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.m1.1"><semantics id="S3.I1.i3.p1.1.m1.1a"><msubsup id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml"><mi id="S3.I1.i3.p1.1.m1.1.1.2.2" xref="S3.I1.i3.p1.1.m1.1.1.2.2.cmml">π</mi><mi id="S3.I1.i3.p1.1.m1.1.1.2.3" mathvariant="normal" xref="S3.I1.i3.p1.1.m1.1.1.2.3.cmml">ℓ</mi><mi id="S3.I1.i3.p1.1.m1.1.1.3" xref="S3.I1.i3.p1.1.m1.1.1.3.cmml">T</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.m1.1b"><apply id="S3.I1.i3.p1.1.m1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.1.m1.1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1">superscript</csymbol><apply id="S3.I1.i3.p1.1.m1.1.1.2.cmml" xref="S3.I1.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I1.i3.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i3.p1.1.m1.1.1.2.2">𝜋</ci><ci id="S3.I1.i3.p1.1.m1.1.1.2.3.cmml" xref="S3.I1.i3.p1.1.m1.1.1.2.3">ℓ</ci></apply><ci id="S3.I1.i3.p1.1.m1.1.1.3.cmml" xref="S3.I1.i3.p1.1.m1.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.m1.1c">\pi_{\ell}^{T}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i3.p1.1.2"> induced on shift-orbits is injective.</span></p> </div> </li> <li class="ltx_item" id="S3.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(4)</span> <div class="ltx_para" id="S3.I1.i4.p1"> <p class="ltx_p" id="S3.I1.i4.p1.2"><span class="ltx_text ltx_font_italic" id="S3.I1.i4.p1.2.1">The morphism </span><math alttext="\pi_{\ell}" class="ltx_Math" display="inline" id="S3.I1.i4.p1.1.m1.1"><semantics id="S3.I1.i4.p1.1.m1.1a"><msub id="S3.I1.i4.p1.1.m1.1.1" xref="S3.I1.i4.p1.1.m1.1.1.cmml"><mi id="S3.I1.i4.p1.1.m1.1.1.2" xref="S3.I1.i4.p1.1.m1.1.1.2.cmml">π</mi><mi id="S3.I1.i4.p1.1.m1.1.1.3" mathvariant="normal" xref="S3.I1.i4.p1.1.m1.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.1.m1.1b"><apply id="S3.I1.i4.p1.1.m1.1.1.cmml" xref="S3.I1.i4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i4.p1.1.m1.1.1.1.cmml" xref="S3.I1.i4.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I1.i4.p1.1.m1.1.1.2.cmml" xref="S3.I1.i4.p1.1.m1.1.1.2">𝜋</ci><ci id="S3.I1.i4.p1.1.m1.1.1.3.cmml" xref="S3.I1.i4.p1.1.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.1.m1.1c">\pi_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i4.p1.2.2"> preserves the shift-period of any biinfinite periodic word </span><math alttext="{\bf x}\in\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.I1.i4.p1.2.m2.1"><semantics id="S3.I1.i4.p1.2.m2.1a"><mrow id="S3.I1.i4.p1.2.m2.1.1" xref="S3.I1.i4.p1.2.m2.1.1.cmml"><mi id="S3.I1.i4.p1.2.m2.1.1.2" xref="S3.I1.i4.p1.2.m2.1.1.2.cmml">𝐱</mi><mo id="S3.I1.i4.p1.2.m2.1.1.1" xref="S3.I1.i4.p1.2.m2.1.1.1.cmml">∈</mo><msup id="S3.I1.i4.p1.2.m2.1.1.3" xref="S3.I1.i4.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i4.p1.2.m2.1.1.3.2" xref="S3.I1.i4.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S3.I1.i4.p1.2.m2.1.1.3.3" xref="S3.I1.i4.p1.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.2.m2.1b"><apply id="S3.I1.i4.p1.2.m2.1.1.cmml" xref="S3.I1.i4.p1.2.m2.1.1"><in id="S3.I1.i4.p1.2.m2.1.1.1.cmml" xref="S3.I1.i4.p1.2.m2.1.1.1"></in><ci id="S3.I1.i4.p1.2.m2.1.1.2.cmml" xref="S3.I1.i4.p1.2.m2.1.1.2">𝐱</ci><apply id="S3.I1.i4.p1.2.m2.1.1.3.cmml" xref="S3.I1.i4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i4.p1.2.m2.1.1.3.1.cmml" xref="S3.I1.i4.p1.2.m2.1.1.3">superscript</csymbol><ci id="S3.I1.i4.p1.2.m2.1.1.3.2.cmml" xref="S3.I1.i4.p1.2.m2.1.1.3.2">𝒜</ci><ci id="S3.I1.i4.p1.2.m2.1.1.3.3.cmml" xref="S3.I1.i4.p1.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.2.m2.1c">{\bf x}\in\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.2.m2.1d">bold_x ∈ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i4.p1.2.3">.</span></p> </div> </li> <li class="ltx_item" id="S3.I1.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(5)</span> <div class="ltx_para" id="S3.I1.i5.p1"> <p class="ltx_p" id="S3.I1.i5.p1.2"><span class="ltx_text ltx_font_italic" id="S3.I1.i5.p1.2.1">The map </span><math alttext="\pi_{\ell}^{\Sigma}" class="ltx_Math" display="inline" id="S3.I1.i5.p1.1.m1.1"><semantics id="S3.I1.i5.p1.1.m1.1a"><msubsup id="S3.I1.i5.p1.1.m1.1.1" xref="S3.I1.i5.p1.1.m1.1.1.cmml"><mi id="S3.I1.i5.p1.1.m1.1.1.2.2" xref="S3.I1.i5.p1.1.m1.1.1.2.2.cmml">π</mi><mi id="S3.I1.i5.p1.1.m1.1.1.2.3" mathvariant="normal" xref="S3.I1.i5.p1.1.m1.1.1.2.3.cmml">ℓ</mi><mi id="S3.I1.i5.p1.1.m1.1.1.3" mathvariant="normal" xref="S3.I1.i5.p1.1.m1.1.1.3.cmml">Σ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.I1.i5.p1.1.m1.1b"><apply id="S3.I1.i5.p1.1.m1.1.1.cmml" xref="S3.I1.i5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i5.p1.1.m1.1.1.1.cmml" xref="S3.I1.i5.p1.1.m1.1.1">superscript</csymbol><apply id="S3.I1.i5.p1.1.m1.1.1.2.cmml" xref="S3.I1.i5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i5.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i5.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I1.i5.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i5.p1.1.m1.1.1.2.2">𝜋</ci><ci id="S3.I1.i5.p1.1.m1.1.1.2.3.cmml" xref="S3.I1.i5.p1.1.m1.1.1.2.3">ℓ</ci></apply><ci id="S3.I1.i5.p1.1.m1.1.1.3.cmml" xref="S3.I1.i5.p1.1.m1.1.1.3">Σ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i5.p1.1.m1.1c">\pi_{\ell}^{\Sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i5.p1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i5.p1.2.2"> induced on subshifts over </span><math alttext="\cal A" class="ltx_Math" display="inline" id="S3.I1.i5.p1.2.m2.1"><semantics id="S3.I1.i5.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.I1.i5.p1.2.m2.1.1" xref="S3.I1.i5.p1.2.m2.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i5.p1.2.m2.1b"><ci id="S3.I1.i5.p1.2.m2.1.1.cmml" xref="S3.I1.i5.p1.2.m2.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i5.p1.2.m2.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i5.p1.2.m2.1d">caligraphic_A</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.i5.p1.2.3"> is injective.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S3.SS1.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.1.p1"> <p class="ltx_p" id="S3.SS1.1.p1.7">All the listed properties follow directly from the definition of the map <math alttext="\pi_{\ell}" class="ltx_Math" display="inline" id="S3.SS1.1.p1.1.m1.1"><semantics id="S3.SS1.1.p1.1.m1.1a"><msub id="S3.SS1.1.p1.1.m1.1.1" xref="S3.SS1.1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.1.p1.1.m1.1.1.2" xref="S3.SS1.1.p1.1.m1.1.1.2.cmml">π</mi><mi id="S3.SS1.1.p1.1.m1.1.1.3" mathvariant="normal" xref="S3.SS1.1.p1.1.m1.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.1.p1.1.m1.1b"><apply id="S3.SS1.1.p1.1.m1.1.1.cmml" xref="S3.SS1.1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.1.p1.1.m1.1.1.2">𝜋</ci><ci id="S3.SS1.1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.1.p1.1.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.p1.1.m1.1c">\pi_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.p1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>, since for any element <math alttext="w^{\prime}\in\pi_{\ell}(\cal A^{*})" class="ltx_Math" display="inline" id="S3.SS1.1.p1.2.m2.1"><semantics id="S3.SS1.1.p1.2.m2.1a"><mrow id="S3.SS1.1.p1.2.m2.1.1" xref="S3.SS1.1.p1.2.m2.1.1.cmml"><msup id="S3.SS1.1.p1.2.m2.1.1.3" xref="S3.SS1.1.p1.2.m2.1.1.3.cmml"><mi id="S3.SS1.1.p1.2.m2.1.1.3.2" xref="S3.SS1.1.p1.2.m2.1.1.3.2.cmml">w</mi><mo id="S3.SS1.1.p1.2.m2.1.1.3.3" xref="S3.SS1.1.p1.2.m2.1.1.3.3.cmml">′</mo></msup><mo id="S3.SS1.1.p1.2.m2.1.1.2" xref="S3.SS1.1.p1.2.m2.1.1.2.cmml">∈</mo><mrow id="S3.SS1.1.p1.2.m2.1.1.1" xref="S3.SS1.1.p1.2.m2.1.1.1.cmml"><msub id="S3.SS1.1.p1.2.m2.1.1.1.3" xref="S3.SS1.1.p1.2.m2.1.1.1.3.cmml"><mi id="S3.SS1.1.p1.2.m2.1.1.1.3.2" xref="S3.SS1.1.p1.2.m2.1.1.1.3.2.cmml">π</mi><mi id="S3.SS1.1.p1.2.m2.1.1.1.3.3" mathvariant="normal" xref="S3.SS1.1.p1.2.m2.1.1.1.3.3.cmml">ℓ</mi></msub><mo id="S3.SS1.1.p1.2.m2.1.1.1.2" xref="S3.SS1.1.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.1.p1.2.m2.1.1.1.1.1" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS1.1.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.1.p1.2.m2.1.1.1.1.1.1" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.2" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.2.cmml">𝒜</mi><mo id="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.3" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.1.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.1.p1.2.m2.1b"><apply id="S3.SS1.1.p1.2.m2.1.1.cmml" xref="S3.SS1.1.p1.2.m2.1.1"><in id="S3.SS1.1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.1.p1.2.m2.1.1.2"></in><apply id="S3.SS1.1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.1.p1.2.m2.1.1.3.1.cmml" xref="S3.SS1.1.p1.2.m2.1.1.3">superscript</csymbol><ci id="S3.SS1.1.p1.2.m2.1.1.3.2.cmml" xref="S3.SS1.1.p1.2.m2.1.1.3.2">𝑤</ci><ci id="S3.SS1.1.p1.2.m2.1.1.3.3.cmml" xref="S3.SS1.1.p1.2.m2.1.1.3.3">′</ci></apply><apply id="S3.SS1.1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1"><times id="S3.SS1.1.p1.2.m2.1.1.1.2.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.2"></times><apply id="S3.SS1.1.p1.2.m2.1.1.1.3.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.1.p1.2.m2.1.1.1.3.1.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.3">subscript</csymbol><ci id="S3.SS1.1.p1.2.m2.1.1.1.3.2.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.3.2">𝜋</ci><ci id="S3.SS1.1.p1.2.m2.1.1.1.3.3.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.3.3">ℓ</ci></apply><apply id="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.2">𝒜</ci><times id="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS1.1.p1.2.m2.1.1.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.p1.2.m2.1c">w^{\prime}\in\pi_{\ell}(\cal A^{*})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.p1.2.m2.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math> the (uniquely determined) preimage <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S3.SS1.1.p1.3.m3.1"><semantics id="S3.SS1.1.p1.3.m3.1a"><mrow id="S3.SS1.1.p1.3.m3.1.1" xref="S3.SS1.1.p1.3.m3.1.1.cmml"><mi id="S3.SS1.1.p1.3.m3.1.1.2" xref="S3.SS1.1.p1.3.m3.1.1.2.cmml">w</mi><mo id="S3.SS1.1.p1.3.m3.1.1.1" xref="S3.SS1.1.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S3.SS1.1.p1.3.m3.1.1.3" xref="S3.SS1.1.p1.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.1.p1.3.m3.1.1.3.2" xref="S3.SS1.1.p1.3.m3.1.1.3.2.cmml">𝒜</mi><mo id="S3.SS1.1.p1.3.m3.1.1.3.3" xref="S3.SS1.1.p1.3.m3.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.1.p1.3.m3.1b"><apply id="S3.SS1.1.p1.3.m3.1.1.cmml" xref="S3.SS1.1.p1.3.m3.1.1"><in id="S3.SS1.1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.1.p1.3.m3.1.1.1"></in><ci id="S3.SS1.1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.1.p1.3.m3.1.1.2">𝑤</ci><apply id="S3.SS1.1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.1.p1.3.m3.1.1.3.1.cmml" xref="S3.SS1.1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S3.SS1.1.p1.3.m3.1.1.3.2.cmml" xref="S3.SS1.1.p1.3.m3.1.1.3.2">𝒜</ci><times id="S3.SS1.1.p1.3.m3.1.1.3.3.cmml" xref="S3.SS1.1.p1.3.m3.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.p1.3.m3.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.p1.3.m3.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is directly visible through replacing every factor <math alttext="a_{i}(1)\ldots a_{i}(\ell(a_{i}))" class="ltx_Math" display="inline" id="S3.SS1.1.p1.4.m4.2"><semantics id="S3.SS1.1.p1.4.m4.2a"><mrow id="S3.SS1.1.p1.4.m4.2.2" xref="S3.SS1.1.p1.4.m4.2.2.cmml"><msub id="S3.SS1.1.p1.4.m4.2.2.3" xref="S3.SS1.1.p1.4.m4.2.2.3.cmml"><mi id="S3.SS1.1.p1.4.m4.2.2.3.2" xref="S3.SS1.1.p1.4.m4.2.2.3.2.cmml">a</mi><mi id="S3.SS1.1.p1.4.m4.2.2.3.3" xref="S3.SS1.1.p1.4.m4.2.2.3.3.cmml">i</mi></msub><mo id="S3.SS1.1.p1.4.m4.2.2.2" xref="S3.SS1.1.p1.4.m4.2.2.2.cmml">⁢</mo><mrow id="S3.SS1.1.p1.4.m4.2.2.4.2" xref="S3.SS1.1.p1.4.m4.2.2.cmml"><mo id="S3.SS1.1.p1.4.m4.2.2.4.2.1" stretchy="false" xref="S3.SS1.1.p1.4.m4.2.2.cmml">(</mo><mn id="S3.SS1.1.p1.4.m4.1.1" xref="S3.SS1.1.p1.4.m4.1.1.cmml">1</mn><mo id="S3.SS1.1.p1.4.m4.2.2.4.2.2" stretchy="false" xref="S3.SS1.1.p1.4.m4.2.2.cmml">)</mo></mrow><mo id="S3.SS1.1.p1.4.m4.2.2.2a" xref="S3.SS1.1.p1.4.m4.2.2.2.cmml">⁢</mo><mi id="S3.SS1.1.p1.4.m4.2.2.5" mathvariant="normal" xref="S3.SS1.1.p1.4.m4.2.2.5.cmml">…</mi><mo id="S3.SS1.1.p1.4.m4.2.2.2b" xref="S3.SS1.1.p1.4.m4.2.2.2.cmml">⁢</mo><msub id="S3.SS1.1.p1.4.m4.2.2.6" xref="S3.SS1.1.p1.4.m4.2.2.6.cmml"><mi id="S3.SS1.1.p1.4.m4.2.2.6.2" xref="S3.SS1.1.p1.4.m4.2.2.6.2.cmml">a</mi><mi id="S3.SS1.1.p1.4.m4.2.2.6.3" xref="S3.SS1.1.p1.4.m4.2.2.6.3.cmml">i</mi></msub><mo id="S3.SS1.1.p1.4.m4.2.2.2c" xref="S3.SS1.1.p1.4.m4.2.2.2.cmml">⁢</mo><mrow id="S3.SS1.1.p1.4.m4.2.2.1.1" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.cmml"><mo id="S3.SS1.1.p1.4.m4.2.2.1.1.2" stretchy="false" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS1.1.p1.4.m4.2.2.1.1.1" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.cmml"><mi id="S3.SS1.1.p1.4.m4.2.2.1.1.1.3" mathvariant="normal" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.3.cmml">ℓ</mi><mo id="S3.SS1.1.p1.4.m4.2.2.1.1.1.2" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.cmml"><mo id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.cmml"><mi id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.2" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.3" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.1.p1.4.m4.2.2.1.1.3" stretchy="false" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.1.p1.4.m4.2b"><apply id="S3.SS1.1.p1.4.m4.2.2.cmml" xref="S3.SS1.1.p1.4.m4.2.2"><times id="S3.SS1.1.p1.4.m4.2.2.2.cmml" xref="S3.SS1.1.p1.4.m4.2.2.2"></times><apply id="S3.SS1.1.p1.4.m4.2.2.3.cmml" xref="S3.SS1.1.p1.4.m4.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.1.p1.4.m4.2.2.3.1.cmml" xref="S3.SS1.1.p1.4.m4.2.2.3">subscript</csymbol><ci id="S3.SS1.1.p1.4.m4.2.2.3.2.cmml" xref="S3.SS1.1.p1.4.m4.2.2.3.2">𝑎</ci><ci id="S3.SS1.1.p1.4.m4.2.2.3.3.cmml" xref="S3.SS1.1.p1.4.m4.2.2.3.3">𝑖</ci></apply><cn id="S3.SS1.1.p1.4.m4.1.1.cmml" type="integer" xref="S3.SS1.1.p1.4.m4.1.1">1</cn><ci id="S3.SS1.1.p1.4.m4.2.2.5.cmml" xref="S3.SS1.1.p1.4.m4.2.2.5">…</ci><apply id="S3.SS1.1.p1.4.m4.2.2.6.cmml" xref="S3.SS1.1.p1.4.m4.2.2.6"><csymbol cd="ambiguous" id="S3.SS1.1.p1.4.m4.2.2.6.1.cmml" xref="S3.SS1.1.p1.4.m4.2.2.6">subscript</csymbol><ci id="S3.SS1.1.p1.4.m4.2.2.6.2.cmml" xref="S3.SS1.1.p1.4.m4.2.2.6.2">𝑎</ci><ci id="S3.SS1.1.p1.4.m4.2.2.6.3.cmml" xref="S3.SS1.1.p1.4.m4.2.2.6.3">𝑖</ci></apply><apply id="S3.SS1.1.p1.4.m4.2.2.1.1.1.cmml" xref="S3.SS1.1.p1.4.m4.2.2.1.1"><times id="S3.SS1.1.p1.4.m4.2.2.1.1.1.2.cmml" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.2"></times><ci id="S3.SS1.1.p1.4.m4.2.2.1.1.1.3.cmml" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.3">ℓ</ci><apply id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.cmml" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.1.cmml" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.2.cmml" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.3.cmml" xref="S3.SS1.1.p1.4.m4.2.2.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.p1.4.m4.2c">a_{i}(1)\ldots a_{i}(\ell(a_{i}))</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.p1.4.m4.2d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 ) … italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( roman_ℓ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) )</annotation></semantics></math> by the letter <math alttext="a_{i}\," class="ltx_Math" display="inline" id="S3.SS1.1.p1.5.m5.1"><semantics id="S3.SS1.1.p1.5.m5.1a"><msub id="S3.SS1.1.p1.5.m5.1.1" xref="S3.SS1.1.p1.5.m5.1.1.cmml"><mi id="S3.SS1.1.p1.5.m5.1.1.2" xref="S3.SS1.1.p1.5.m5.1.1.2.cmml">a</mi><mi id="S3.SS1.1.p1.5.m5.1.1.3" xref="S3.SS1.1.p1.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.1.p1.5.m5.1b"><apply id="S3.SS1.1.p1.5.m5.1.1.cmml" xref="S3.SS1.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS1.1.p1.5.m5.1.1.1.cmml" xref="S3.SS1.1.p1.5.m5.1.1">subscript</csymbol><ci id="S3.SS1.1.p1.5.m5.1.1.2.cmml" xref="S3.SS1.1.p1.5.m5.1.1.2">𝑎</ci><ci id="S3.SS1.1.p1.5.m5.1.1.3.cmml" xref="S3.SS1.1.p1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.p1.5.m5.1c">a_{i}\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.p1.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S3.SS1.1.p1.6.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S3.SS1.1.p1.6.1.m1.1"><semantics id="S3.SS1.1.p1.6.1.m1.1a"><mo id="S3.SS1.1.p1.6.1.m1.1.1" xref="S3.SS1.1.p1.6.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S3.SS1.1.p1.6.1.m1.1b"><csymbol cd="latexml" id="S3.SS1.1.p1.6.1.m1.1.1.cmml" xref="S3.SS1.1.p1.6.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.p1.6.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.p1.6.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S3.SS1.1.p1.7.m6.1"><semantics id="S3.SS1.1.p1.7.m6.1a"><mo id="S3.SS1.1.p1.7.m6.1.1" xref="S3.SS1.1.p1.7.m6.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S3.SS1.1.p1.7.m6.1b"><csymbol cd="latexml" id="S3.SS1.1.p1.7.m6.1.1.cmml" xref="S3.SS1.1.p1.7.m6.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.p1.7.m6.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.p1.7.m6.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_defn" id="S3.Thmthm3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm3.1.1.1">Definition 3.3</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm3.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm3.p1"> <p class="ltx_p" id="S3.Thmthm3.p1.6">Let <math alttext="\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}" class="ltx_Math" display="inline" id="S3.Thmthm3.p1.1.m1.1"><semantics id="S3.Thmthm3.p1.1.m1.1a"><mrow id="S3.Thmthm3.p1.1.m1.1.1" xref="S3.Thmthm3.p1.1.m1.1.1.cmml"><msub id="S3.Thmthm3.p1.1.m1.1.1.2" xref="S3.Thmthm3.p1.1.m1.1.1.2.cmml"><mi id="S3.Thmthm3.p1.1.m1.1.1.2.2" xref="S3.Thmthm3.p1.1.m1.1.1.2.2.cmml">π</mi><mi id="S3.Thmthm3.p1.1.m1.1.1.2.3" mathvariant="normal" xref="S3.Thmthm3.p1.1.m1.1.1.2.3.cmml">ℓ</mi></msub><mo id="S3.Thmthm3.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm3.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm3.p1.1.m1.1.1.3" xref="S3.Thmthm3.p1.1.m1.1.1.3.cmml"><msup id="S3.Thmthm3.p1.1.m1.1.1.3.2" xref="S3.Thmthm3.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p1.1.m1.1.1.3.2.2" xref="S3.Thmthm3.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm3.p1.1.m1.1.1.3.2.3" xref="S3.Thmthm3.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm3.p1.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm3.p1.1.m1.1.1.3.1.cmml">→</mo><msubsup id="S3.Thmthm3.p1.1.m1.1.1.3.3" xref="S3.Thmthm3.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p1.1.m1.1.1.3.3.2.2" xref="S3.Thmthm3.p1.1.m1.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S3.Thmthm3.p1.1.m1.1.1.3.3.2.3" mathvariant="normal" xref="S3.Thmthm3.p1.1.m1.1.1.3.3.2.3.cmml">ℓ</mi><mo id="S3.Thmthm3.p1.1.m1.1.1.3.3.3" xref="S3.Thmthm3.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p1.1.m1.1b"><apply id="S3.Thmthm3.p1.1.m1.1.1.cmml" xref="S3.Thmthm3.p1.1.m1.1.1"><ci id="S3.Thmthm3.p1.1.m1.1.1.1.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.1">:</ci><apply id="S3.Thmthm3.p1.1.m1.1.1.2.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p1.1.m1.1.1.2.1.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.Thmthm3.p1.1.m1.1.1.2.2.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.2.2">𝜋</ci><ci id="S3.Thmthm3.p1.1.m1.1.1.2.3.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.2.3">ℓ</ci></apply><apply id="S3.Thmthm3.p1.1.m1.1.1.3.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3"><ci id="S3.Thmthm3.p1.1.m1.1.1.3.1.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm3.p1.1.m1.1.1.3.2.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p1.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm3.p1.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm3.p1.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm3.p1.1.m1.1.1.3.3.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm3.p1.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.3">superscript</csymbol><apply id="S3.Thmthm3.p1.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm3.p1.1.m1.1.1.3.3.2.1.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.Thmthm3.p1.1.m1.1.1.3.3.2.2.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.3.2.2">𝒜</ci><ci id="S3.Thmthm3.p1.1.m1.1.1.3.3.2.3.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.3.2.3">ℓ</ci></apply><times id="S3.Thmthm3.p1.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm3.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p1.1.m1.1c">\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a subdivision morphism as in (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E1" title="In 3.1. Subdivision morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.1</span></a>), and let <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm3.p1.2.m2.1"><semantics id="S3.Thmthm3.p1.2.m2.1a"><mi id="S3.Thmthm3.p1.2.m2.1.1" xref="S3.Thmthm3.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p1.2.m2.1b"><ci id="S3.Thmthm3.p1.2.m2.1.1.cmml" xref="S3.Thmthm3.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p1.2.m2.1d">italic_μ</annotation></semantics></math> be an invariant measure on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm3.p1.3.m3.1"><semantics id="S3.Thmthm3.p1.3.m3.1a"><msup id="S3.Thmthm3.p1.3.m3.1.1" xref="S3.Thmthm3.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p1.3.m3.1.1.2" xref="S3.Thmthm3.p1.3.m3.1.1.2.cmml">𝒜</mi><mi id="S3.Thmthm3.p1.3.m3.1.1.3" xref="S3.Thmthm3.p1.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p1.3.m3.1b"><apply id="S3.Thmthm3.p1.3.m3.1.1.cmml" xref="S3.Thmthm3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmthm3.p1.3.m3.1.1.1.cmml" xref="S3.Thmthm3.p1.3.m3.1.1">superscript</csymbol><ci id="S3.Thmthm3.p1.3.m3.1.1.2.cmml" xref="S3.Thmthm3.p1.3.m3.1.1.2">𝒜</ci><ci id="S3.Thmthm3.p1.3.m3.1.1.3.cmml" xref="S3.Thmthm3.p1.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p1.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p1.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Consider the weight function <math alttext="\cal A^{*}\to\mathbb{R}_{\geq 0}\,,\,\,w\mapsto\mu([w])" class="ltx_Math" display="inline" id="S3.Thmthm3.p1.4.m4.3"><semantics id="S3.Thmthm3.p1.4.m4.3a"><mrow id="S3.Thmthm3.p1.4.m4.3.3.2" xref="S3.Thmthm3.p1.4.m4.3.3.3.cmml"><mrow id="S3.Thmthm3.p1.4.m4.2.2.1.1" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.cmml"><msup id="S3.Thmthm3.p1.4.m4.2.2.1.1.2" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p1.4.m4.2.2.1.1.2.2" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.2.2.cmml">𝒜</mi><mo id="S3.Thmthm3.p1.4.m4.2.2.1.1.2.3" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm3.p1.4.m4.2.2.1.1.1" stretchy="false" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.1.cmml">→</mo><msub id="S3.Thmthm3.p1.4.m4.2.2.1.1.3" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.cmml"><mi id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.2" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.2.cmml">ℝ</mi><mrow id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.cmml"><mi id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.2" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.2.cmml"></mi><mo id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.1" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.3" mathvariant="script" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.3.cmml">0</mn></mrow></msub></mrow><mo id="S3.Thmthm3.p1.4.m4.3.3.2.3" rspace="0.497em" xref="S3.Thmthm3.p1.4.m4.3.3.3a.cmml">,</mo><mrow id="S3.Thmthm3.p1.4.m4.3.3.2.2" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p1.4.m4.3.3.2.2.3" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.3.cmml">𝓌</mi><mo id="S3.Thmthm3.p1.4.m4.3.3.2.2.2" stretchy="false" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.2.cmml">↦</mo><mrow id="S3.Thmthm3.p1.4.m4.3.3.2.2.1" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.cmml"><mi id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.3" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.3.cmml">μ</mi><mo id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.2" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.cmml"><mo id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.2" stretchy="false" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.cmml">(</mo><mrow id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.2" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.1.cmml"><mo id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.2.1" stretchy="false" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.1.1.cmml">[</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p1.4.m4.1.1" xref="S3.Thmthm3.p1.4.m4.1.1.cmml">𝓌</mi><mo id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.2.2" stretchy="false" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.3" stretchy="false" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p1.4.m4.3b"><apply id="S3.Thmthm3.p1.4.m4.3.3.3.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p1.4.m4.3.3.3a.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.3">formulae-sequence</csymbol><apply id="S3.Thmthm3.p1.4.m4.2.2.1.1.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1"><ci id="S3.Thmthm3.p1.4.m4.2.2.1.1.1.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.1">→</ci><apply id="S3.Thmthm3.p1.4.m4.2.2.1.1.2.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p1.4.m4.2.2.1.1.2.1.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.2">superscript</csymbol><ci id="S3.Thmthm3.p1.4.m4.2.2.1.1.2.2.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.2.2">𝒜</ci><times id="S3.Thmthm3.p1.4.m4.2.2.1.1.2.3.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.2.3"></times></apply><apply id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.1.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3">subscript</csymbol><ci id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.2.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.2">ℝ</ci><apply id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3"><geq id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.1.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.1"></geq><csymbol cd="latexml" id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.2.cmml" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.2">absent</csymbol><cn id="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.3.cmml" type="integer" xref="S3.Thmthm3.p1.4.m4.2.2.1.1.3.3.3">0</cn></apply></apply></apply><apply id="S3.Thmthm3.p1.4.m4.3.3.2.2.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.2"><csymbol cd="latexml" id="S3.Thmthm3.p1.4.m4.3.3.2.2.2.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.2">maps-to</csymbol><ci id="S3.Thmthm3.p1.4.m4.3.3.2.2.3.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.3">𝓌</ci><apply id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1"><times id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.2.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.2"></times><ci id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.3.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.3">𝜇</ci><apply id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.1.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.2"><csymbol cd="latexml" id="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.1.1.cmml" xref="S3.Thmthm3.p1.4.m4.3.3.2.2.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.Thmthm3.p1.4.m4.1.1.cmml" xref="S3.Thmthm3.p1.4.m4.1.1">𝓌</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p1.4.m4.3c">\cal A^{*}\to\mathbb{R}_{\geq 0}\,,\,\,w\mapsto\mu([w])</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p1.4.m4.3d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → blackboard_R start_POSTSUBSCRIPT ≥ caligraphic_0 end_POSTSUBSCRIPT , caligraphic_w ↦ italic_μ ( [ caligraphic_w ] )</annotation></semantics></math> associated to <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm3.p1.5.m5.1"><semantics id="S3.Thmthm3.p1.5.m5.1a"><mi id="S3.Thmthm3.p1.5.m5.1.1" xref="S3.Thmthm3.p1.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p1.5.m5.1b"><ci id="S3.Thmthm3.p1.5.m5.1.1.cmml" xref="S3.Thmthm3.p1.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p1.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p1.5.m5.1d">italic_μ</annotation></semantics></math>, which for simplicity is also denoted by <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm3.p1.6.m6.1"><semantics id="S3.Thmthm3.p1.6.m6.1a"><mi id="S3.Thmthm3.p1.6.m6.1.1" xref="S3.Thmthm3.p1.6.m6.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p1.6.m6.1b"><ci id="S3.Thmthm3.p1.6.m6.1.1.cmml" xref="S3.Thmthm3.p1.6.m6.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p1.6.m6.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p1.6.m6.1d">italic_μ</annotation></semantics></math> (see (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E3" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.3</span></a>)).</p> </div> <div class="ltx_para" id="S3.Thmthm3.p2"> <p class="ltx_p" id="S3.Thmthm3.p2.1">Define a function <math alttext="\mu_{\ell}:\cal A_{\ell}^{*}\to\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.1.m1.1"><semantics id="S3.Thmthm3.p2.1.m1.1a"><mrow id="S3.Thmthm3.p2.1.m1.1.1" xref="S3.Thmthm3.p2.1.m1.1.1.cmml"><msub id="S3.Thmthm3.p2.1.m1.1.1.2" xref="S3.Thmthm3.p2.1.m1.1.1.2.cmml"><mi id="S3.Thmthm3.p2.1.m1.1.1.2.2" xref="S3.Thmthm3.p2.1.m1.1.1.2.2.cmml">μ</mi><mi id="S3.Thmthm3.p2.1.m1.1.1.2.3" mathvariant="normal" xref="S3.Thmthm3.p2.1.m1.1.1.2.3.cmml">ℓ</mi></msub><mo id="S3.Thmthm3.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm3.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm3.p2.1.m1.1.1.3" xref="S3.Thmthm3.p2.1.m1.1.1.3.cmml"><msubsup id="S3.Thmthm3.p2.1.m1.1.1.3.2" xref="S3.Thmthm3.p2.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p2.1.m1.1.1.3.2.2.2" xref="S3.Thmthm3.p2.1.m1.1.1.3.2.2.2.cmml">𝒜</mi><mi id="S3.Thmthm3.p2.1.m1.1.1.3.2.2.3" mathvariant="normal" xref="S3.Thmthm3.p2.1.m1.1.1.3.2.2.3.cmml">ℓ</mi><mo id="S3.Thmthm3.p2.1.m1.1.1.3.2.3" xref="S3.Thmthm3.p2.1.m1.1.1.3.2.3.cmml">∗</mo></msubsup><mo id="S3.Thmthm3.p2.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm3.p2.1.m1.1.1.3.1.cmml">→</mo><msub id="S3.Thmthm3.p2.1.m1.1.1.3.3" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.cmml"><mi id="S3.Thmthm3.p2.1.m1.1.1.3.3.2" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.2.cmml">ℝ</mi><mrow id="S3.Thmthm3.p2.1.m1.1.1.3.3.3" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.3.cmml"><mi id="S3.Thmthm3.p2.1.m1.1.1.3.3.3.2" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.3.2.cmml"></mi><mo id="S3.Thmthm3.p2.1.m1.1.1.3.3.3.1" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.3.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p2.1.m1.1.1.3.3.3.3" mathvariant="script" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.3.3.cmml">0</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.1.m1.1b"><apply id="S3.Thmthm3.p2.1.m1.1.1.cmml" xref="S3.Thmthm3.p2.1.m1.1.1"><ci id="S3.Thmthm3.p2.1.m1.1.1.1.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.1">:</ci><apply id="S3.Thmthm3.p2.1.m1.1.1.2.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p2.1.m1.1.1.2.1.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.2">subscript</csymbol><ci id="S3.Thmthm3.p2.1.m1.1.1.2.2.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.2.2">𝜇</ci><ci id="S3.Thmthm3.p2.1.m1.1.1.2.3.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.2.3">ℓ</ci></apply><apply id="S3.Thmthm3.p2.1.m1.1.1.3.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3"><ci id="S3.Thmthm3.p2.1.m1.1.1.3.1.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm3.p2.1.m1.1.1.3.2.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p2.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.2">superscript</csymbol><apply id="S3.Thmthm3.p2.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p2.1.m1.1.1.3.2.2.1.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.2">subscript</csymbol><ci id="S3.Thmthm3.p2.1.m1.1.1.3.2.2.2.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.2.2.2">𝒜</ci><ci id="S3.Thmthm3.p2.1.m1.1.1.3.2.2.3.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.2.2.3">ℓ</ci></apply><times id="S3.Thmthm3.p2.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm3.p2.1.m1.1.1.3.3.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm3.p2.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.Thmthm3.p2.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.2">ℝ</ci><apply id="S3.Thmthm3.p2.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.3"><geq id="S3.Thmthm3.p2.1.m1.1.1.3.3.3.1.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S3.Thmthm3.p2.1.m1.1.1.3.3.3.2.cmml" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.3.2">absent</csymbol><cn id="S3.Thmthm3.p2.1.m1.1.1.3.3.3.3.cmml" type="integer" xref="S3.Thmthm3.p2.1.m1.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.1.m1.1c">\mu_{\ell}:\cal A_{\ell}^{*}\to\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.1.m1.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → blackboard_R start_POSTSUBSCRIPT ≥ caligraphic_0 end_POSTSUBSCRIPT</annotation></semantics></math> by</p> <table class="ltx_equation ltx_eqn_table" id="S3.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(3.2)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu_{\ell}(w)=\mu(\widehat{w})\,," class="ltx_Math" display="block" id="S3.E2.m1.3"><semantics id="S3.E2.m1.3a"><mrow id="S3.E2.m1.3.3.1" xref="S3.E2.m1.3.3.1.1.cmml"><mrow id="S3.E2.m1.3.3.1.1" xref="S3.E2.m1.3.3.1.1.cmml"><mrow id="S3.E2.m1.3.3.1.1.2" xref="S3.E2.m1.3.3.1.1.2.cmml"><msub id="S3.E2.m1.3.3.1.1.2.2" xref="S3.E2.m1.3.3.1.1.2.2.cmml"><mi id="S3.E2.m1.3.3.1.1.2.2.2" xref="S3.E2.m1.3.3.1.1.2.2.2.cmml">μ</mi><mi id="S3.E2.m1.3.3.1.1.2.2.3" mathvariant="normal" xref="S3.E2.m1.3.3.1.1.2.2.3.cmml">ℓ</mi></msub><mo id="S3.E2.m1.3.3.1.1.2.1" xref="S3.E2.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S3.E2.m1.3.3.1.1.2.3.2" xref="S3.E2.m1.3.3.1.1.2.cmml"><mo id="S3.E2.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S3.E2.m1.3.3.1.1.2.cmml">(</mo><mi id="S3.E2.m1.1.1" xref="S3.E2.m1.1.1.cmml">w</mi><mo id="S3.E2.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="S3.E2.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E2.m1.3.3.1.1.1" xref="S3.E2.m1.3.3.1.1.1.cmml">=</mo><mrow id="S3.E2.m1.3.3.1.1.3" xref="S3.E2.m1.3.3.1.1.3.cmml"><mi id="S3.E2.m1.3.3.1.1.3.2" xref="S3.E2.m1.3.3.1.1.3.2.cmml">μ</mi><mo id="S3.E2.m1.3.3.1.1.3.1" xref="S3.E2.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S3.E2.m1.3.3.1.1.3.3.2" xref="S3.E2.m1.2.2.cmml"><mo id="S3.E2.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S3.E2.m1.2.2.cmml">(</mo><mover accent="true" id="S3.E2.m1.2.2" xref="S3.E2.m1.2.2.cmml"><mi id="S3.E2.m1.2.2.2" xref="S3.E2.m1.2.2.2.cmml">w</mi><mo id="S3.E2.m1.2.2.1" xref="S3.E2.m1.2.2.1.cmml">^</mo></mover><mo id="S3.E2.m1.3.3.1.1.3.3.2.2" rspace="0.170em" stretchy="false" xref="S3.E2.m1.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E2.m1.3.3.1.2" xref="S3.E2.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E2.m1.3b"><apply id="S3.E2.m1.3.3.1.1.cmml" xref="S3.E2.m1.3.3.1"><eq id="S3.E2.m1.3.3.1.1.1.cmml" xref="S3.E2.m1.3.3.1.1.1"></eq><apply id="S3.E2.m1.3.3.1.1.2.cmml" xref="S3.E2.m1.3.3.1.1.2"><times id="S3.E2.m1.3.3.1.1.2.1.cmml" xref="S3.E2.m1.3.3.1.1.2.1"></times><apply id="S3.E2.m1.3.3.1.1.2.2.cmml" xref="S3.E2.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.E2.m1.3.3.1.1.2.2.1.cmml" xref="S3.E2.m1.3.3.1.1.2.2">subscript</csymbol><ci id="S3.E2.m1.3.3.1.1.2.2.2.cmml" xref="S3.E2.m1.3.3.1.1.2.2.2">𝜇</ci><ci id="S3.E2.m1.3.3.1.1.2.2.3.cmml" xref="S3.E2.m1.3.3.1.1.2.2.3">ℓ</ci></apply><ci id="S3.E2.m1.1.1.cmml" xref="S3.E2.m1.1.1">𝑤</ci></apply><apply id="S3.E2.m1.3.3.1.1.3.cmml" xref="S3.E2.m1.3.3.1.1.3"><times id="S3.E2.m1.3.3.1.1.3.1.cmml" xref="S3.E2.m1.3.3.1.1.3.1"></times><ci id="S3.E2.m1.3.3.1.1.3.2.cmml" xref="S3.E2.m1.3.3.1.1.3.2">𝜇</ci><apply id="S3.E2.m1.2.2.cmml" xref="S3.E2.m1.3.3.1.1.3.3.2"><ci id="S3.E2.m1.2.2.1.cmml" xref="S3.E2.m1.2.2.1">^</ci><ci id="S3.E2.m1.2.2.2.cmml" xref="S3.E2.m1.2.2.2">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.3c">\mu_{\ell}(w)=\mu(\widehat{w})\,,</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.3d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_w ) = italic_μ ( over^ start_ARG italic_w end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmthm3.p2.10">where <math alttext="\widehat{w}\in\cal A^{*}" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.2.m1.1"><semantics id="S3.Thmthm3.p2.2.m1.1a"><mrow id="S3.Thmthm3.p2.2.m1.1.1" xref="S3.Thmthm3.p2.2.m1.1.1.cmml"><mover accent="true" id="S3.Thmthm3.p2.2.m1.1.1.2" xref="S3.Thmthm3.p2.2.m1.1.1.2.cmml"><mi id="S3.Thmthm3.p2.2.m1.1.1.2.2" xref="S3.Thmthm3.p2.2.m1.1.1.2.2.cmml">w</mi><mo id="S3.Thmthm3.p2.2.m1.1.1.2.1" xref="S3.Thmthm3.p2.2.m1.1.1.2.1.cmml">^</mo></mover><mo id="S3.Thmthm3.p2.2.m1.1.1.1" xref="S3.Thmthm3.p2.2.m1.1.1.1.cmml">∈</mo><msup id="S3.Thmthm3.p2.2.m1.1.1.3" xref="S3.Thmthm3.p2.2.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p2.2.m1.1.1.3.2" xref="S3.Thmthm3.p2.2.m1.1.1.3.2.cmml">𝒜</mi><mo id="S3.Thmthm3.p2.2.m1.1.1.3.3" xref="S3.Thmthm3.p2.2.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.2.m1.1b"><apply id="S3.Thmthm3.p2.2.m1.1.1.cmml" xref="S3.Thmthm3.p2.2.m1.1.1"><in id="S3.Thmthm3.p2.2.m1.1.1.1.cmml" xref="S3.Thmthm3.p2.2.m1.1.1.1"></in><apply id="S3.Thmthm3.p2.2.m1.1.1.2.cmml" xref="S3.Thmthm3.p2.2.m1.1.1.2"><ci id="S3.Thmthm3.p2.2.m1.1.1.2.1.cmml" xref="S3.Thmthm3.p2.2.m1.1.1.2.1">^</ci><ci id="S3.Thmthm3.p2.2.m1.1.1.2.2.cmml" xref="S3.Thmthm3.p2.2.m1.1.1.2.2">𝑤</ci></apply><apply id="S3.Thmthm3.p2.2.m1.1.1.3.cmml" xref="S3.Thmthm3.p2.2.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm3.p2.2.m1.1.1.3.1.cmml" xref="S3.Thmthm3.p2.2.m1.1.1.3">superscript</csymbol><ci id="S3.Thmthm3.p2.2.m1.1.1.3.2.cmml" xref="S3.Thmthm3.p2.2.m1.1.1.3.2">𝒜</ci><times id="S3.Thmthm3.p2.2.m1.1.1.3.3.cmml" xref="S3.Thmthm3.p2.2.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.2.m1.1c">\widehat{w}\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.2.m1.1d">over^ start_ARG italic_w end_ARG ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is the shortest word in <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.3.m2.1"><semantics id="S3.Thmthm3.p2.3.m2.1a"><msup id="S3.Thmthm3.p2.3.m2.1.1" xref="S3.Thmthm3.p2.3.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p2.3.m2.1.1.2" xref="S3.Thmthm3.p2.3.m2.1.1.2.cmml">𝒜</mi><mo id="S3.Thmthm3.p2.3.m2.1.1.3" xref="S3.Thmthm3.p2.3.m2.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.3.m2.1b"><apply id="S3.Thmthm3.p2.3.m2.1.1.cmml" xref="S3.Thmthm3.p2.3.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmthm3.p2.3.m2.1.1.1.cmml" xref="S3.Thmthm3.p2.3.m2.1.1">superscript</csymbol><ci id="S3.Thmthm3.p2.3.m2.1.1.2.cmml" xref="S3.Thmthm3.p2.3.m2.1.1.2">𝒜</ci><times id="S3.Thmthm3.p2.3.m2.1.1.3.cmml" xref="S3.Thmthm3.p2.3.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.3.m2.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.3.m2.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\pi_{\ell}(\widehat{w})" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.4.m3.1"><semantics id="S3.Thmthm3.p2.4.m3.1a"><mrow id="S3.Thmthm3.p2.4.m3.1.2" xref="S3.Thmthm3.p2.4.m3.1.2.cmml"><msub id="S3.Thmthm3.p2.4.m3.1.2.2" xref="S3.Thmthm3.p2.4.m3.1.2.2.cmml"><mi id="S3.Thmthm3.p2.4.m3.1.2.2.2" xref="S3.Thmthm3.p2.4.m3.1.2.2.2.cmml">π</mi><mi id="S3.Thmthm3.p2.4.m3.1.2.2.3" mathvariant="normal" xref="S3.Thmthm3.p2.4.m3.1.2.2.3.cmml">ℓ</mi></msub><mo id="S3.Thmthm3.p2.4.m3.1.2.1" xref="S3.Thmthm3.p2.4.m3.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm3.p2.4.m3.1.2.3.2" xref="S3.Thmthm3.p2.4.m3.1.1.cmml"><mo id="S3.Thmthm3.p2.4.m3.1.2.3.2.1" stretchy="false" xref="S3.Thmthm3.p2.4.m3.1.1.cmml">(</mo><mover accent="true" id="S3.Thmthm3.p2.4.m3.1.1" xref="S3.Thmthm3.p2.4.m3.1.1.cmml"><mi id="S3.Thmthm3.p2.4.m3.1.1.2" xref="S3.Thmthm3.p2.4.m3.1.1.2.cmml">w</mi><mo id="S3.Thmthm3.p2.4.m3.1.1.1" xref="S3.Thmthm3.p2.4.m3.1.1.1.cmml">^</mo></mover><mo id="S3.Thmthm3.p2.4.m3.1.2.3.2.2" stretchy="false" xref="S3.Thmthm3.p2.4.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.4.m3.1b"><apply id="S3.Thmthm3.p2.4.m3.1.2.cmml" xref="S3.Thmthm3.p2.4.m3.1.2"><times id="S3.Thmthm3.p2.4.m3.1.2.1.cmml" xref="S3.Thmthm3.p2.4.m3.1.2.1"></times><apply id="S3.Thmthm3.p2.4.m3.1.2.2.cmml" xref="S3.Thmthm3.p2.4.m3.1.2.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p2.4.m3.1.2.2.1.cmml" xref="S3.Thmthm3.p2.4.m3.1.2.2">subscript</csymbol><ci id="S3.Thmthm3.p2.4.m3.1.2.2.2.cmml" xref="S3.Thmthm3.p2.4.m3.1.2.2.2">𝜋</ci><ci id="S3.Thmthm3.p2.4.m3.1.2.2.3.cmml" xref="S3.Thmthm3.p2.4.m3.1.2.2.3">ℓ</ci></apply><apply id="S3.Thmthm3.p2.4.m3.1.1.cmml" xref="S3.Thmthm3.p2.4.m3.1.2.3.2"><ci id="S3.Thmthm3.p2.4.m3.1.1.1.cmml" xref="S3.Thmthm3.p2.4.m3.1.1.1">^</ci><ci id="S3.Thmthm3.p2.4.m3.1.1.2.cmml" xref="S3.Thmthm3.p2.4.m3.1.1.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.4.m3.1c">\pi_{\ell}(\widehat{w})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.4.m3.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over^ start_ARG italic_w end_ARG )</annotation></semantics></math> contains <math alttext="w" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.5.m4.1"><semantics id="S3.Thmthm3.p2.5.m4.1a"><mi id="S3.Thmthm3.p2.5.m4.1.1" xref="S3.Thmthm3.p2.5.m4.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.5.m4.1b"><ci id="S3.Thmthm3.p2.5.m4.1.1.cmml" xref="S3.Thmthm3.p2.5.m4.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.5.m4.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.5.m4.1d">italic_w</annotation></semantics></math> as factor. If such <math alttext="\widehat{w}" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.6.m5.1"><semantics id="S3.Thmthm3.p2.6.m5.1a"><mover accent="true" id="S3.Thmthm3.p2.6.m5.1.1" xref="S3.Thmthm3.p2.6.m5.1.1.cmml"><mi id="S3.Thmthm3.p2.6.m5.1.1.2" xref="S3.Thmthm3.p2.6.m5.1.1.2.cmml">w</mi><mo id="S3.Thmthm3.p2.6.m5.1.1.1" xref="S3.Thmthm3.p2.6.m5.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.6.m5.1b"><apply id="S3.Thmthm3.p2.6.m5.1.1.cmml" xref="S3.Thmthm3.p2.6.m5.1.1"><ci id="S3.Thmthm3.p2.6.m5.1.1.1.cmml" xref="S3.Thmthm3.p2.6.m5.1.1.1">^</ci><ci id="S3.Thmthm3.p2.6.m5.1.1.2.cmml" xref="S3.Thmthm3.p2.6.m5.1.1.2">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.6.m5.1c">\widehat{w}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.6.m5.1d">over^ start_ARG italic_w end_ARG</annotation></semantics></math> exists, then it is uniquely defined by <math alttext="w" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.7.m6.1"><semantics id="S3.Thmthm3.p2.7.m6.1a"><mi id="S3.Thmthm3.p2.7.m6.1.1" xref="S3.Thmthm3.p2.7.m6.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.7.m6.1b"><ci id="S3.Thmthm3.p2.7.m6.1.1.cmml" xref="S3.Thmthm3.p2.7.m6.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.7.m6.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.7.m6.1d">italic_w</annotation></semantics></math>. If there is no such word <math alttext="\widehat{w}" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.8.m7.1"><semantics id="S3.Thmthm3.p2.8.m7.1a"><mover accent="true" id="S3.Thmthm3.p2.8.m7.1.1" xref="S3.Thmthm3.p2.8.m7.1.1.cmml"><mi id="S3.Thmthm3.p2.8.m7.1.1.2" xref="S3.Thmthm3.p2.8.m7.1.1.2.cmml">w</mi><mo id="S3.Thmthm3.p2.8.m7.1.1.1" xref="S3.Thmthm3.p2.8.m7.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.8.m7.1b"><apply id="S3.Thmthm3.p2.8.m7.1.1.cmml" xref="S3.Thmthm3.p2.8.m7.1.1"><ci id="S3.Thmthm3.p2.8.m7.1.1.1.cmml" xref="S3.Thmthm3.p2.8.m7.1.1.1">^</ci><ci id="S3.Thmthm3.p2.8.m7.1.1.2.cmml" xref="S3.Thmthm3.p2.8.m7.1.1.2">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.8.m7.1c">\widehat{w}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.8.m7.1d">over^ start_ARG italic_w end_ARG</annotation></semantics></math>, we set formally <math alttext="\mu(\widehat{w})=0" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.9.m8.1"><semantics id="S3.Thmthm3.p2.9.m8.1a"><mrow id="S3.Thmthm3.p2.9.m8.1.2" xref="S3.Thmthm3.p2.9.m8.1.2.cmml"><mrow id="S3.Thmthm3.p2.9.m8.1.2.2" xref="S3.Thmthm3.p2.9.m8.1.2.2.cmml"><mi id="S3.Thmthm3.p2.9.m8.1.2.2.2" xref="S3.Thmthm3.p2.9.m8.1.2.2.2.cmml">μ</mi><mo id="S3.Thmthm3.p2.9.m8.1.2.2.1" xref="S3.Thmthm3.p2.9.m8.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm3.p2.9.m8.1.2.2.3.2" xref="S3.Thmthm3.p2.9.m8.1.1.cmml"><mo id="S3.Thmthm3.p2.9.m8.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm3.p2.9.m8.1.1.cmml">(</mo><mover accent="true" id="S3.Thmthm3.p2.9.m8.1.1" xref="S3.Thmthm3.p2.9.m8.1.1.cmml"><mi id="S3.Thmthm3.p2.9.m8.1.1.2" xref="S3.Thmthm3.p2.9.m8.1.1.2.cmml">w</mi><mo id="S3.Thmthm3.p2.9.m8.1.1.1" xref="S3.Thmthm3.p2.9.m8.1.1.1.cmml">^</mo></mover><mo id="S3.Thmthm3.p2.9.m8.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm3.p2.9.m8.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm3.p2.9.m8.1.2.1" xref="S3.Thmthm3.p2.9.m8.1.2.1.cmml">=</mo><mn id="S3.Thmthm3.p2.9.m8.1.2.3" xref="S3.Thmthm3.p2.9.m8.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.9.m8.1b"><apply id="S3.Thmthm3.p2.9.m8.1.2.cmml" xref="S3.Thmthm3.p2.9.m8.1.2"><eq id="S3.Thmthm3.p2.9.m8.1.2.1.cmml" xref="S3.Thmthm3.p2.9.m8.1.2.1"></eq><apply id="S3.Thmthm3.p2.9.m8.1.2.2.cmml" xref="S3.Thmthm3.p2.9.m8.1.2.2"><times id="S3.Thmthm3.p2.9.m8.1.2.2.1.cmml" xref="S3.Thmthm3.p2.9.m8.1.2.2.1"></times><ci id="S3.Thmthm3.p2.9.m8.1.2.2.2.cmml" xref="S3.Thmthm3.p2.9.m8.1.2.2.2">𝜇</ci><apply id="S3.Thmthm3.p2.9.m8.1.1.cmml" xref="S3.Thmthm3.p2.9.m8.1.2.2.3.2"><ci id="S3.Thmthm3.p2.9.m8.1.1.1.cmml" xref="S3.Thmthm3.p2.9.m8.1.1.1">^</ci><ci id="S3.Thmthm3.p2.9.m8.1.1.2.cmml" xref="S3.Thmthm3.p2.9.m8.1.1.2">𝑤</ci></apply></apply><cn id="S3.Thmthm3.p2.9.m8.1.2.3.cmml" type="integer" xref="S3.Thmthm3.p2.9.m8.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.9.m8.1c">\mu(\widehat{w})=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.9.m8.1d">italic_μ ( over^ start_ARG italic_w end_ARG ) = 0</annotation></semantics></math> and thus <math alttext="\mu_{\ell}(w)=0" class="ltx_Math" display="inline" id="S3.Thmthm3.p2.10.m9.1"><semantics id="S3.Thmthm3.p2.10.m9.1a"><mrow id="S3.Thmthm3.p2.10.m9.1.2" xref="S3.Thmthm3.p2.10.m9.1.2.cmml"><mrow id="S3.Thmthm3.p2.10.m9.1.2.2" xref="S3.Thmthm3.p2.10.m9.1.2.2.cmml"><msub id="S3.Thmthm3.p2.10.m9.1.2.2.2" xref="S3.Thmthm3.p2.10.m9.1.2.2.2.cmml"><mi id="S3.Thmthm3.p2.10.m9.1.2.2.2.2" xref="S3.Thmthm3.p2.10.m9.1.2.2.2.2.cmml">μ</mi><mi id="S3.Thmthm3.p2.10.m9.1.2.2.2.3" mathvariant="normal" xref="S3.Thmthm3.p2.10.m9.1.2.2.2.3.cmml">ℓ</mi></msub><mo id="S3.Thmthm3.p2.10.m9.1.2.2.1" xref="S3.Thmthm3.p2.10.m9.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm3.p2.10.m9.1.2.2.3.2" xref="S3.Thmthm3.p2.10.m9.1.2.2.cmml"><mo id="S3.Thmthm3.p2.10.m9.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm3.p2.10.m9.1.2.2.cmml">(</mo><mi id="S3.Thmthm3.p2.10.m9.1.1" xref="S3.Thmthm3.p2.10.m9.1.1.cmml">w</mi><mo id="S3.Thmthm3.p2.10.m9.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm3.p2.10.m9.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm3.p2.10.m9.1.2.1" xref="S3.Thmthm3.p2.10.m9.1.2.1.cmml">=</mo><mn id="S3.Thmthm3.p2.10.m9.1.2.3" xref="S3.Thmthm3.p2.10.m9.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p2.10.m9.1b"><apply id="S3.Thmthm3.p2.10.m9.1.2.cmml" xref="S3.Thmthm3.p2.10.m9.1.2"><eq id="S3.Thmthm3.p2.10.m9.1.2.1.cmml" xref="S3.Thmthm3.p2.10.m9.1.2.1"></eq><apply id="S3.Thmthm3.p2.10.m9.1.2.2.cmml" xref="S3.Thmthm3.p2.10.m9.1.2.2"><times id="S3.Thmthm3.p2.10.m9.1.2.2.1.cmml" xref="S3.Thmthm3.p2.10.m9.1.2.2.1"></times><apply id="S3.Thmthm3.p2.10.m9.1.2.2.2.cmml" xref="S3.Thmthm3.p2.10.m9.1.2.2.2"><csymbol cd="ambiguous" id="S3.Thmthm3.p2.10.m9.1.2.2.2.1.cmml" xref="S3.Thmthm3.p2.10.m9.1.2.2.2">subscript</csymbol><ci id="S3.Thmthm3.p2.10.m9.1.2.2.2.2.cmml" xref="S3.Thmthm3.p2.10.m9.1.2.2.2.2">𝜇</ci><ci id="S3.Thmthm3.p2.10.m9.1.2.2.2.3.cmml" xref="S3.Thmthm3.p2.10.m9.1.2.2.2.3">ℓ</ci></apply><ci id="S3.Thmthm3.p2.10.m9.1.1.cmml" xref="S3.Thmthm3.p2.10.m9.1.1">𝑤</ci></apply><cn id="S3.Thmthm3.p2.10.m9.1.2.3.cmml" type="integer" xref="S3.Thmthm3.p2.10.m9.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p2.10.m9.1c">\mu_{\ell}(w)=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p2.10.m9.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_w ) = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.Thmthm3.p3"> <p class="ltx_p" id="S3.Thmthm3.p3.5">It is shown in Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm4" title="Lemma 3.4. ‣ 3.1. Subdivision morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.4</span></a> just below that the function <math alttext="\mu_{\ell}" class="ltx_Math" display="inline" id="S3.Thmthm3.p3.1.m1.1"><semantics id="S3.Thmthm3.p3.1.m1.1a"><msub id="S3.Thmthm3.p3.1.m1.1.1" xref="S3.Thmthm3.p3.1.m1.1.1.cmml"><mi id="S3.Thmthm3.p3.1.m1.1.1.2" xref="S3.Thmthm3.p3.1.m1.1.1.2.cmml">μ</mi><mi id="S3.Thmthm3.p3.1.m1.1.1.3" mathvariant="normal" xref="S3.Thmthm3.p3.1.m1.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p3.1.m1.1b"><apply id="S3.Thmthm3.p3.1.m1.1.1.cmml" xref="S3.Thmthm3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm3.p3.1.m1.1.1.1.cmml" xref="S3.Thmthm3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.Thmthm3.p3.1.m1.1.1.2.cmml" xref="S3.Thmthm3.p3.1.m1.1.1.2">𝜇</ci><ci id="S3.Thmthm3.p3.1.m1.1.1.3.cmml" xref="S3.Thmthm3.p3.1.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p3.1.m1.1c">\mu_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p3.1.m1.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> is a weight function, so that (see Section <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS1" title="2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.1</span></a>) it defines an invariant measure on <math alttext="\cal A_{\ell}^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm3.p3.2.m2.1"><semantics id="S3.Thmthm3.p3.2.m2.1a"><msubsup id="S3.Thmthm3.p3.2.m2.1.1" xref="S3.Thmthm3.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm3.p3.2.m2.1.1.2.2" xref="S3.Thmthm3.p3.2.m2.1.1.2.2.cmml">𝒜</mi><mi id="S3.Thmthm3.p3.2.m2.1.1.2.3" mathvariant="normal" xref="S3.Thmthm3.p3.2.m2.1.1.2.3.cmml">ℓ</mi><mi id="S3.Thmthm3.p3.2.m2.1.1.3" xref="S3.Thmthm3.p3.2.m2.1.1.3.cmml">ℤ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p3.2.m2.1b"><apply id="S3.Thmthm3.p3.2.m2.1.1.cmml" xref="S3.Thmthm3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmthm3.p3.2.m2.1.1.1.cmml" xref="S3.Thmthm3.p3.2.m2.1.1">superscript</csymbol><apply id="S3.Thmthm3.p3.2.m2.1.1.2.cmml" xref="S3.Thmthm3.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmthm3.p3.2.m2.1.1.2.1.cmml" xref="S3.Thmthm3.p3.2.m2.1.1">subscript</csymbol><ci id="S3.Thmthm3.p3.2.m2.1.1.2.2.cmml" xref="S3.Thmthm3.p3.2.m2.1.1.2.2">𝒜</ci><ci id="S3.Thmthm3.p3.2.m2.1.1.2.3.cmml" xref="S3.Thmthm3.p3.2.m2.1.1.2.3">ℓ</ci></apply><ci id="S3.Thmthm3.p3.2.m2.1.1.3.cmml" xref="S3.Thmthm3.p3.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p3.2.m2.1c">\cal A_{\ell}^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p3.2.m2.1d">caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> which will also be denoted by <math alttext="\mu_{\ell}" class="ltx_Math" display="inline" id="S3.Thmthm3.p3.3.m3.1"><semantics id="S3.Thmthm3.p3.3.m3.1a"><msub id="S3.Thmthm3.p3.3.m3.1.1" xref="S3.Thmthm3.p3.3.m3.1.1.cmml"><mi id="S3.Thmthm3.p3.3.m3.1.1.2" xref="S3.Thmthm3.p3.3.m3.1.1.2.cmml">μ</mi><mi id="S3.Thmthm3.p3.3.m3.1.1.3" mathvariant="normal" xref="S3.Thmthm3.p3.3.m3.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p3.3.m3.1b"><apply id="S3.Thmthm3.p3.3.m3.1.1.cmml" xref="S3.Thmthm3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmthm3.p3.3.m3.1.1.1.cmml" xref="S3.Thmthm3.p3.3.m3.1.1">subscript</csymbol><ci id="S3.Thmthm3.p3.3.m3.1.1.2.cmml" xref="S3.Thmthm3.p3.3.m3.1.1.2">𝜇</ci><ci id="S3.Thmthm3.p3.3.m3.1.1.3.cmml" xref="S3.Thmthm3.p3.3.m3.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p3.3.m3.1c">\mu_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p3.3.m3.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. We call <math alttext="\mu_{\ell}" class="ltx_Math" display="inline" id="S3.Thmthm3.p3.4.m4.1"><semantics id="S3.Thmthm3.p3.4.m4.1a"><msub id="S3.Thmthm3.p3.4.m4.1.1" xref="S3.Thmthm3.p3.4.m4.1.1.cmml"><mi id="S3.Thmthm3.p3.4.m4.1.1.2" xref="S3.Thmthm3.p3.4.m4.1.1.2.cmml">μ</mi><mi id="S3.Thmthm3.p3.4.m4.1.1.3" mathvariant="normal" xref="S3.Thmthm3.p3.4.m4.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p3.4.m4.1b"><apply id="S3.Thmthm3.p3.4.m4.1.1.cmml" xref="S3.Thmthm3.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmthm3.p3.4.m4.1.1.1.cmml" xref="S3.Thmthm3.p3.4.m4.1.1">subscript</csymbol><ci id="S3.Thmthm3.p3.4.m4.1.1.2.cmml" xref="S3.Thmthm3.p3.4.m4.1.1.2">𝜇</ci><ci id="S3.Thmthm3.p3.4.m4.1.1.3.cmml" xref="S3.Thmthm3.p3.4.m4.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p3.4.m4.1c">\mu_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p3.4.m4.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> the <span class="ltx_text ltx_font_italic" id="S3.Thmthm3.p3.5.1">subdivision measure</span> defined by <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm3.p3.5.m5.1"><semantics id="S3.Thmthm3.p3.5.m5.1a"><mi id="S3.Thmthm3.p3.5.m5.1.1" xref="S3.Thmthm3.p3.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm3.p3.5.m5.1b"><ci id="S3.Thmthm3.p3.5.m5.1.1.cmml" xref="S3.Thmthm3.p3.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm3.p3.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm3.p3.5.m5.1d">italic_μ</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_lem" id="S3.Thmthm4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm4.1.1.1">Lemma 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm4.p1"> <p class="ltx_p" id="S3.Thmthm4.p1.3"><span class="ltx_text ltx_font_italic" id="S3.Thmthm4.p1.3.3">The function <math alttext="\mu_{\ell}" class="ltx_Math" display="inline" id="S3.Thmthm4.p1.1.1.m1.1"><semantics id="S3.Thmthm4.p1.1.1.m1.1a"><msub id="S3.Thmthm4.p1.1.1.m1.1.1" xref="S3.Thmthm4.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmthm4.p1.1.1.m1.1.1.2" xref="S3.Thmthm4.p1.1.1.m1.1.1.2.cmml">μ</mi><mi id="S3.Thmthm4.p1.1.1.m1.1.1.3" mathvariant="normal" xref="S3.Thmthm4.p1.1.1.m1.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm4.p1.1.1.m1.1b"><apply id="S3.Thmthm4.p1.1.1.m1.1.1.cmml" xref="S3.Thmthm4.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm4.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmthm4.p1.1.1.m1.1.1">subscript</csymbol><ci id="S3.Thmthm4.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmthm4.p1.1.1.m1.1.1.2">𝜇</ci><ci id="S3.Thmthm4.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmthm4.p1.1.1.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm4.p1.1.1.m1.1c">\mu_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm4.p1.1.1.m1.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> inherits from <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm4.p1.2.2.m2.1"><semantics id="S3.Thmthm4.p1.2.2.m2.1a"><mi id="S3.Thmthm4.p1.2.2.m2.1.1" xref="S3.Thmthm4.p1.2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm4.p1.2.2.m2.1b"><ci id="S3.Thmthm4.p1.2.2.m2.1.1.cmml" xref="S3.Thmthm4.p1.2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm4.p1.2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm4.p1.2.2.m2.1d">italic_μ</annotation></semantics></math> the Kirchhoff equalities (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E2" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.2</span></a>), so that <math alttext="\mu_{\ell}" class="ltx_Math" display="inline" id="S3.Thmthm4.p1.3.3.m3.1"><semantics id="S3.Thmthm4.p1.3.3.m3.1a"><msub id="S3.Thmthm4.p1.3.3.m3.1.1" xref="S3.Thmthm4.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmthm4.p1.3.3.m3.1.1.2" xref="S3.Thmthm4.p1.3.3.m3.1.1.2.cmml">μ</mi><mi id="S3.Thmthm4.p1.3.3.m3.1.1.3" mathvariant="normal" xref="S3.Thmthm4.p1.3.3.m3.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm4.p1.3.3.m3.1b"><apply id="S3.Thmthm4.p1.3.3.m3.1.1.cmml" xref="S3.Thmthm4.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmthm4.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmthm4.p1.3.3.m3.1.1">subscript</csymbol><ci id="S3.Thmthm4.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmthm4.p1.3.3.m3.1.1.2">𝜇</ci><ci id="S3.Thmthm4.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmthm4.p1.3.3.m3.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm4.p1.3.3.m3.1c">\mu_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm4.p1.3.3.m3.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> is itself a weight function.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS1.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.2.p1"> <p class="ltx_p" id="S3.SS1.2.p1.1">By symmetry it suffices to prove the first equality of (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E2" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.2</span></a>) for the function <math alttext="\mu_{\ell}" class="ltx_Math" display="inline" id="S3.SS1.2.p1.1.m1.1"><semantics id="S3.SS1.2.p1.1.m1.1a"><msub id="S3.SS1.2.p1.1.m1.1.1" xref="S3.SS1.2.p1.1.m1.1.1.cmml"><mi id="S3.SS1.2.p1.1.m1.1.1.2" xref="S3.SS1.2.p1.1.m1.1.1.2.cmml">μ</mi><mi id="S3.SS1.2.p1.1.m1.1.1.3" mathvariant="normal" xref="S3.SS1.2.p1.1.m1.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p1.1.m1.1b"><apply id="S3.SS1.2.p1.1.m1.1.1.cmml" xref="S3.SS1.2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.2.p1.1.m1.1.1.1.cmml" xref="S3.SS1.2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS1.2.p1.1.m1.1.1.2.cmml" xref="S3.SS1.2.p1.1.m1.1.1.2">𝜇</ci><ci id="S3.SS1.2.p1.1.m1.1.1.3.cmml" xref="S3.SS1.2.p1.1.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p1.1.m1.1c">\mu_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p1.1.m1.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. We have to distinguish three cases:</p> </div> <div class="ltx_para" id="S3.SS1.3.p2"> <p class="ltx_p" id="S3.SS1.3.p2.4">If <math alttext="w\in\cal A_{\ell}^{*}" class="ltx_Math" display="inline" id="S3.SS1.3.p2.1.m1.1"><semantics id="S3.SS1.3.p2.1.m1.1a"><mrow id="S3.SS1.3.p2.1.m1.1.1" xref="S3.SS1.3.p2.1.m1.1.1.cmml"><mi id="S3.SS1.3.p2.1.m1.1.1.2" xref="S3.SS1.3.p2.1.m1.1.1.2.cmml">w</mi><mo id="S3.SS1.3.p2.1.m1.1.1.1" xref="S3.SS1.3.p2.1.m1.1.1.1.cmml">∈</mo><msubsup id="S3.SS1.3.p2.1.m1.1.1.3" xref="S3.SS1.3.p2.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.3.p2.1.m1.1.1.3.2.2" xref="S3.SS1.3.p2.1.m1.1.1.3.2.2.cmml">𝒜</mi><mi id="S3.SS1.3.p2.1.m1.1.1.3.2.3" mathvariant="normal" xref="S3.SS1.3.p2.1.m1.1.1.3.2.3.cmml">ℓ</mi><mo id="S3.SS1.3.p2.1.m1.1.1.3.3" xref="S3.SS1.3.p2.1.m1.1.1.3.3.cmml">∗</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p2.1.m1.1b"><apply id="S3.SS1.3.p2.1.m1.1.1.cmml" xref="S3.SS1.3.p2.1.m1.1.1"><in id="S3.SS1.3.p2.1.m1.1.1.1.cmml" xref="S3.SS1.3.p2.1.m1.1.1.1"></in><ci id="S3.SS1.3.p2.1.m1.1.1.2.cmml" xref="S3.SS1.3.p2.1.m1.1.1.2">𝑤</ci><apply id="S3.SS1.3.p2.1.m1.1.1.3.cmml" xref="S3.SS1.3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.3.p2.1.m1.1.1.3.1.cmml" xref="S3.SS1.3.p2.1.m1.1.1.3">superscript</csymbol><apply id="S3.SS1.3.p2.1.m1.1.1.3.2.cmml" xref="S3.SS1.3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.3.p2.1.m1.1.1.3.2.1.cmml" xref="S3.SS1.3.p2.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS1.3.p2.1.m1.1.1.3.2.2.cmml" xref="S3.SS1.3.p2.1.m1.1.1.3.2.2">𝒜</ci><ci id="S3.SS1.3.p2.1.m1.1.1.3.2.3.cmml" xref="S3.SS1.3.p2.1.m1.1.1.3.2.3">ℓ</ci></apply><times id="S3.SS1.3.p2.1.m1.1.1.3.3.cmml" xref="S3.SS1.3.p2.1.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p2.1.m1.1c">w\in\cal A_{\ell}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p2.1.m1.1d">italic_w ∈ caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is not a factor of any element from <math alttext="\pi_{\ell}(\cal A^{*})" class="ltx_Math" display="inline" id="S3.SS1.3.p2.2.m2.1"><semantics id="S3.SS1.3.p2.2.m2.1a"><mrow id="S3.SS1.3.p2.2.m2.1.1" xref="S3.SS1.3.p2.2.m2.1.1.cmml"><msub id="S3.SS1.3.p2.2.m2.1.1.3" xref="S3.SS1.3.p2.2.m2.1.1.3.cmml"><mi id="S3.SS1.3.p2.2.m2.1.1.3.2" xref="S3.SS1.3.p2.2.m2.1.1.3.2.cmml">π</mi><mi id="S3.SS1.3.p2.2.m2.1.1.3.3" mathvariant="normal" xref="S3.SS1.3.p2.2.m2.1.1.3.3.cmml">ℓ</mi></msub><mo id="S3.SS1.3.p2.2.m2.1.1.2" xref="S3.SS1.3.p2.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.3.p2.2.m2.1.1.1.1" xref="S3.SS1.3.p2.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS1.3.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS1.3.p2.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.3.p2.2.m2.1.1.1.1.1" xref="S3.SS1.3.p2.2.m2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.3.p2.2.m2.1.1.1.1.1.2" xref="S3.SS1.3.p2.2.m2.1.1.1.1.1.2.cmml">𝒜</mi><mo id="S3.SS1.3.p2.2.m2.1.1.1.1.1.3" xref="S3.SS1.3.p2.2.m2.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.3.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS1.3.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p2.2.m2.1b"><apply id="S3.SS1.3.p2.2.m2.1.1.cmml" xref="S3.SS1.3.p2.2.m2.1.1"><times id="S3.SS1.3.p2.2.m2.1.1.2.cmml" xref="S3.SS1.3.p2.2.m2.1.1.2"></times><apply id="S3.SS1.3.p2.2.m2.1.1.3.cmml" xref="S3.SS1.3.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.3.p2.2.m2.1.1.3.1.cmml" xref="S3.SS1.3.p2.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.3.p2.2.m2.1.1.3.2.cmml" xref="S3.SS1.3.p2.2.m2.1.1.3.2">𝜋</ci><ci id="S3.SS1.3.p2.2.m2.1.1.3.3.cmml" xref="S3.SS1.3.p2.2.m2.1.1.3.3">ℓ</ci></apply><apply id="S3.SS1.3.p2.2.m2.1.1.1.1.1.cmml" xref="S3.SS1.3.p2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.3.p2.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS1.3.p2.2.m2.1.1.1.1">superscript</csymbol><ci id="S3.SS1.3.p2.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS1.3.p2.2.m2.1.1.1.1.1.2">𝒜</ci><times id="S3.SS1.3.p2.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS1.3.p2.2.m2.1.1.1.1.1.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p2.2.m2.1c">\pi_{\ell}(\cal A^{*})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p2.2.m2.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>, then the same is true for <math alttext="xw" class="ltx_Math" display="inline" id="S3.SS1.3.p2.3.m3.1"><semantics id="S3.SS1.3.p2.3.m3.1a"><mrow id="S3.SS1.3.p2.3.m3.1.1" xref="S3.SS1.3.p2.3.m3.1.1.cmml"><mi id="S3.SS1.3.p2.3.m3.1.1.2" xref="S3.SS1.3.p2.3.m3.1.1.2.cmml">x</mi><mo id="S3.SS1.3.p2.3.m3.1.1.1" xref="S3.SS1.3.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS1.3.p2.3.m3.1.1.3" xref="S3.SS1.3.p2.3.m3.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p2.3.m3.1b"><apply id="S3.SS1.3.p2.3.m3.1.1.cmml" xref="S3.SS1.3.p2.3.m3.1.1"><times id="S3.SS1.3.p2.3.m3.1.1.1.cmml" xref="S3.SS1.3.p2.3.m3.1.1.1"></times><ci id="S3.SS1.3.p2.3.m3.1.1.2.cmml" xref="S3.SS1.3.p2.3.m3.1.1.2">𝑥</ci><ci id="S3.SS1.3.p2.3.m3.1.1.3.cmml" xref="S3.SS1.3.p2.3.m3.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p2.3.m3.1c">xw</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p2.3.m3.1d">italic_x italic_w</annotation></semantics></math> for any <math alttext="x\in\cal A_{\ell}" class="ltx_Math" display="inline" id="S3.SS1.3.p2.4.m4.1"><semantics id="S3.SS1.3.p2.4.m4.1a"><mrow id="S3.SS1.3.p2.4.m4.1.1" xref="S3.SS1.3.p2.4.m4.1.1.cmml"><mi id="S3.SS1.3.p2.4.m4.1.1.2" xref="S3.SS1.3.p2.4.m4.1.1.2.cmml">x</mi><mo id="S3.SS1.3.p2.4.m4.1.1.1" xref="S3.SS1.3.p2.4.m4.1.1.1.cmml">∈</mo><msub id="S3.SS1.3.p2.4.m4.1.1.3" xref="S3.SS1.3.p2.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.3.p2.4.m4.1.1.3.2" xref="S3.SS1.3.p2.4.m4.1.1.3.2.cmml">𝒜</mi><mi id="S3.SS1.3.p2.4.m4.1.1.3.3" mathvariant="normal" xref="S3.SS1.3.p2.4.m4.1.1.3.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p2.4.m4.1b"><apply id="S3.SS1.3.p2.4.m4.1.1.cmml" xref="S3.SS1.3.p2.4.m4.1.1"><in id="S3.SS1.3.p2.4.m4.1.1.1.cmml" xref="S3.SS1.3.p2.4.m4.1.1.1"></in><ci id="S3.SS1.3.p2.4.m4.1.1.2.cmml" xref="S3.SS1.3.p2.4.m4.1.1.2">𝑥</ci><apply id="S3.SS1.3.p2.4.m4.1.1.3.cmml" xref="S3.SS1.3.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.3.p2.4.m4.1.1.3.1.cmml" xref="S3.SS1.3.p2.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS1.3.p2.4.m4.1.1.3.2.cmml" xref="S3.SS1.3.p2.4.m4.1.1.3.2">𝒜</ci><ci id="S3.SS1.3.p2.4.m4.1.1.3.3.cmml" xref="S3.SS1.3.p2.4.m4.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p2.4.m4.1c">x\in\cal A_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p2.4.m4.1d">italic_x ∈ caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. In this case both, the left and the right hand side of the desired equality, are equal to 0.</p> </div> <div class="ltx_para" id="S3.SS1.4.p3"> <p class="ltx_p" id="S3.SS1.4.p3.10">If <math alttext="w" class="ltx_Math" display="inline" id="S3.SS1.4.p3.1.m1.1"><semantics id="S3.SS1.4.p3.1.m1.1a"><mi id="S3.SS1.4.p3.1.m1.1.1" xref="S3.SS1.4.p3.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.1.m1.1b"><ci id="S3.SS1.4.p3.1.m1.1.1.cmml" xref="S3.SS1.4.p3.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.1.m1.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.1.m1.1d">italic_w</annotation></semantics></math> is a factor of some element in <math alttext="\pi_{\ell}(\cal A^{*})" class="ltx_Math" display="inline" id="S3.SS1.4.p3.2.m2.1"><semantics id="S3.SS1.4.p3.2.m2.1a"><mrow id="S3.SS1.4.p3.2.m2.1.1" xref="S3.SS1.4.p3.2.m2.1.1.cmml"><msub id="S3.SS1.4.p3.2.m2.1.1.3" xref="S3.SS1.4.p3.2.m2.1.1.3.cmml"><mi id="S3.SS1.4.p3.2.m2.1.1.3.2" xref="S3.SS1.4.p3.2.m2.1.1.3.2.cmml">π</mi><mi id="S3.SS1.4.p3.2.m2.1.1.3.3" mathvariant="normal" xref="S3.SS1.4.p3.2.m2.1.1.3.3.cmml">ℓ</mi></msub><mo id="S3.SS1.4.p3.2.m2.1.1.2" xref="S3.SS1.4.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.4.p3.2.m2.1.1.1.1" xref="S3.SS1.4.p3.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS1.4.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS1.4.p3.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.4.p3.2.m2.1.1.1.1.1" xref="S3.SS1.4.p3.2.m2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.4.p3.2.m2.1.1.1.1.1.2" xref="S3.SS1.4.p3.2.m2.1.1.1.1.1.2.cmml">𝒜</mi><mo id="S3.SS1.4.p3.2.m2.1.1.1.1.1.3" xref="S3.SS1.4.p3.2.m2.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.4.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS1.4.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.2.m2.1b"><apply id="S3.SS1.4.p3.2.m2.1.1.cmml" xref="S3.SS1.4.p3.2.m2.1.1"><times id="S3.SS1.4.p3.2.m2.1.1.2.cmml" xref="S3.SS1.4.p3.2.m2.1.1.2"></times><apply id="S3.SS1.4.p3.2.m2.1.1.3.cmml" xref="S3.SS1.4.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.4.p3.2.m2.1.1.3.1.cmml" xref="S3.SS1.4.p3.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS1.4.p3.2.m2.1.1.3.2.cmml" xref="S3.SS1.4.p3.2.m2.1.1.3.2">𝜋</ci><ci id="S3.SS1.4.p3.2.m2.1.1.3.3.cmml" xref="S3.SS1.4.p3.2.m2.1.1.3.3">ℓ</ci></apply><apply id="S3.SS1.4.p3.2.m2.1.1.1.1.1.cmml" xref="S3.SS1.4.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p3.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS1.4.p3.2.m2.1.1.1.1">superscript</csymbol><ci id="S3.SS1.4.p3.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS1.4.p3.2.m2.1.1.1.1.1.2">𝒜</ci><times id="S3.SS1.4.p3.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS1.4.p3.2.m2.1.1.1.1.1.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.2.m2.1c">\pi_{\ell}(\cal A^{*})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.2.m2.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>, then we consider the first letter <math alttext="a_{i}(k)" class="ltx_Math" display="inline" id="S3.SS1.4.p3.3.m3.1"><semantics id="S3.SS1.4.p3.3.m3.1a"><mrow id="S3.SS1.4.p3.3.m3.1.2" xref="S3.SS1.4.p3.3.m3.1.2.cmml"><msub id="S3.SS1.4.p3.3.m3.1.2.2" xref="S3.SS1.4.p3.3.m3.1.2.2.cmml"><mi id="S3.SS1.4.p3.3.m3.1.2.2.2" xref="S3.SS1.4.p3.3.m3.1.2.2.2.cmml">a</mi><mi id="S3.SS1.4.p3.3.m3.1.2.2.3" xref="S3.SS1.4.p3.3.m3.1.2.2.3.cmml">i</mi></msub><mo id="S3.SS1.4.p3.3.m3.1.2.1" xref="S3.SS1.4.p3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.4.p3.3.m3.1.2.3.2" xref="S3.SS1.4.p3.3.m3.1.2.cmml"><mo id="S3.SS1.4.p3.3.m3.1.2.3.2.1" stretchy="false" xref="S3.SS1.4.p3.3.m3.1.2.cmml">(</mo><mi id="S3.SS1.4.p3.3.m3.1.1" xref="S3.SS1.4.p3.3.m3.1.1.cmml">k</mi><mo id="S3.SS1.4.p3.3.m3.1.2.3.2.2" stretchy="false" xref="S3.SS1.4.p3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.3.m3.1b"><apply id="S3.SS1.4.p3.3.m3.1.2.cmml" xref="S3.SS1.4.p3.3.m3.1.2"><times id="S3.SS1.4.p3.3.m3.1.2.1.cmml" xref="S3.SS1.4.p3.3.m3.1.2.1"></times><apply id="S3.SS1.4.p3.3.m3.1.2.2.cmml" xref="S3.SS1.4.p3.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.4.p3.3.m3.1.2.2.1.cmml" xref="S3.SS1.4.p3.3.m3.1.2.2">subscript</csymbol><ci id="S3.SS1.4.p3.3.m3.1.2.2.2.cmml" xref="S3.SS1.4.p3.3.m3.1.2.2.2">𝑎</ci><ci id="S3.SS1.4.p3.3.m3.1.2.2.3.cmml" xref="S3.SS1.4.p3.3.m3.1.2.2.3">𝑖</ci></apply><ci id="S3.SS1.4.p3.3.m3.1.1.cmml" xref="S3.SS1.4.p3.3.m3.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.3.m3.1c">a_{i}(k)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.3.m3.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k )</annotation></semantics></math> of <math alttext="w" class="ltx_Math" display="inline" id="S3.SS1.4.p3.4.m4.1"><semantics id="S3.SS1.4.p3.4.m4.1a"><mi id="S3.SS1.4.p3.4.m4.1.1" xref="S3.SS1.4.p3.4.m4.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.4.m4.1b"><ci id="S3.SS1.4.p3.4.m4.1.1.cmml" xref="S3.SS1.4.p3.4.m4.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.4.m4.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.4.m4.1d">italic_w</annotation></semantics></math>. If we have <math alttext="k\geq 2" class="ltx_Math" display="inline" id="S3.SS1.4.p3.5.m5.1"><semantics id="S3.SS1.4.p3.5.m5.1a"><mrow id="S3.SS1.4.p3.5.m5.1.1" xref="S3.SS1.4.p3.5.m5.1.1.cmml"><mi id="S3.SS1.4.p3.5.m5.1.1.2" xref="S3.SS1.4.p3.5.m5.1.1.2.cmml">k</mi><mo id="S3.SS1.4.p3.5.m5.1.1.1" xref="S3.SS1.4.p3.5.m5.1.1.1.cmml">≥</mo><mn id="S3.SS1.4.p3.5.m5.1.1.3" xref="S3.SS1.4.p3.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.5.m5.1b"><apply id="S3.SS1.4.p3.5.m5.1.1.cmml" xref="S3.SS1.4.p3.5.m5.1.1"><geq id="S3.SS1.4.p3.5.m5.1.1.1.cmml" xref="S3.SS1.4.p3.5.m5.1.1.1"></geq><ci id="S3.SS1.4.p3.5.m5.1.1.2.cmml" xref="S3.SS1.4.p3.5.m5.1.1.2">𝑘</ci><cn id="S3.SS1.4.p3.5.m5.1.1.3.cmml" type="integer" xref="S3.SS1.4.p3.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.5.m5.1c">k\geq 2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.5.m5.1d">italic_k ≥ 2</annotation></semantics></math>, then there is only one letter <math alttext="x\in\cal A_{\ell}" class="ltx_Math" display="inline" id="S3.SS1.4.p3.6.m6.1"><semantics id="S3.SS1.4.p3.6.m6.1a"><mrow id="S3.SS1.4.p3.6.m6.1.1" xref="S3.SS1.4.p3.6.m6.1.1.cmml"><mi id="S3.SS1.4.p3.6.m6.1.1.2" xref="S3.SS1.4.p3.6.m6.1.1.2.cmml">x</mi><mo id="S3.SS1.4.p3.6.m6.1.1.1" xref="S3.SS1.4.p3.6.m6.1.1.1.cmml">∈</mo><msub id="S3.SS1.4.p3.6.m6.1.1.3" xref="S3.SS1.4.p3.6.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.4.p3.6.m6.1.1.3.2" xref="S3.SS1.4.p3.6.m6.1.1.3.2.cmml">𝒜</mi><mi id="S3.SS1.4.p3.6.m6.1.1.3.3" mathvariant="normal" xref="S3.SS1.4.p3.6.m6.1.1.3.3.cmml">ℓ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.6.m6.1b"><apply id="S3.SS1.4.p3.6.m6.1.1.cmml" xref="S3.SS1.4.p3.6.m6.1.1"><in id="S3.SS1.4.p3.6.m6.1.1.1.cmml" xref="S3.SS1.4.p3.6.m6.1.1.1"></in><ci id="S3.SS1.4.p3.6.m6.1.1.2.cmml" xref="S3.SS1.4.p3.6.m6.1.1.2">𝑥</ci><apply id="S3.SS1.4.p3.6.m6.1.1.3.cmml" xref="S3.SS1.4.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.4.p3.6.m6.1.1.3.1.cmml" xref="S3.SS1.4.p3.6.m6.1.1.3">subscript</csymbol><ci id="S3.SS1.4.p3.6.m6.1.1.3.2.cmml" xref="S3.SS1.4.p3.6.m6.1.1.3.2">𝒜</ci><ci id="S3.SS1.4.p3.6.m6.1.1.3.3.cmml" xref="S3.SS1.4.p3.6.m6.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.6.m6.1c">x\in\cal A_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.6.m6.1d">italic_x ∈ caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="xw" class="ltx_Math" display="inline" id="S3.SS1.4.p3.7.m7.1"><semantics id="S3.SS1.4.p3.7.m7.1a"><mrow id="S3.SS1.4.p3.7.m7.1.1" xref="S3.SS1.4.p3.7.m7.1.1.cmml"><mi id="S3.SS1.4.p3.7.m7.1.1.2" xref="S3.SS1.4.p3.7.m7.1.1.2.cmml">x</mi><mo id="S3.SS1.4.p3.7.m7.1.1.1" xref="S3.SS1.4.p3.7.m7.1.1.1.cmml">⁢</mo><mi id="S3.SS1.4.p3.7.m7.1.1.3" xref="S3.SS1.4.p3.7.m7.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.7.m7.1b"><apply id="S3.SS1.4.p3.7.m7.1.1.cmml" xref="S3.SS1.4.p3.7.m7.1.1"><times id="S3.SS1.4.p3.7.m7.1.1.1.cmml" xref="S3.SS1.4.p3.7.m7.1.1.1"></times><ci id="S3.SS1.4.p3.7.m7.1.1.2.cmml" xref="S3.SS1.4.p3.7.m7.1.1.2">𝑥</ci><ci id="S3.SS1.4.p3.7.m7.1.1.3.cmml" xref="S3.SS1.4.p3.7.m7.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.7.m7.1c">xw</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.7.m7.1d">italic_x italic_w</annotation></semantics></math> is a factor of some element from <math alttext="\pi_{\ell}(\cal A^{*})" class="ltx_Math" display="inline" id="S3.SS1.4.p3.8.m8.1"><semantics id="S3.SS1.4.p3.8.m8.1a"><mrow id="S3.SS1.4.p3.8.m8.1.1" xref="S3.SS1.4.p3.8.m8.1.1.cmml"><msub id="S3.SS1.4.p3.8.m8.1.1.3" xref="S3.SS1.4.p3.8.m8.1.1.3.cmml"><mi id="S3.SS1.4.p3.8.m8.1.1.3.2" xref="S3.SS1.4.p3.8.m8.1.1.3.2.cmml">π</mi><mi id="S3.SS1.4.p3.8.m8.1.1.3.3" mathvariant="normal" xref="S3.SS1.4.p3.8.m8.1.1.3.3.cmml">ℓ</mi></msub><mo id="S3.SS1.4.p3.8.m8.1.1.2" xref="S3.SS1.4.p3.8.m8.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.4.p3.8.m8.1.1.1.1" xref="S3.SS1.4.p3.8.m8.1.1.1.1.1.cmml"><mo id="S3.SS1.4.p3.8.m8.1.1.1.1.2" stretchy="false" xref="S3.SS1.4.p3.8.m8.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.4.p3.8.m8.1.1.1.1.1" xref="S3.SS1.4.p3.8.m8.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.4.p3.8.m8.1.1.1.1.1.2" xref="S3.SS1.4.p3.8.m8.1.1.1.1.1.2.cmml">𝒜</mi><mo id="S3.SS1.4.p3.8.m8.1.1.1.1.1.3" xref="S3.SS1.4.p3.8.m8.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.4.p3.8.m8.1.1.1.1.3" stretchy="false" xref="S3.SS1.4.p3.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.8.m8.1b"><apply id="S3.SS1.4.p3.8.m8.1.1.cmml" xref="S3.SS1.4.p3.8.m8.1.1"><times id="S3.SS1.4.p3.8.m8.1.1.2.cmml" xref="S3.SS1.4.p3.8.m8.1.1.2"></times><apply id="S3.SS1.4.p3.8.m8.1.1.3.cmml" xref="S3.SS1.4.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.4.p3.8.m8.1.1.3.1.cmml" xref="S3.SS1.4.p3.8.m8.1.1.3">subscript</csymbol><ci id="S3.SS1.4.p3.8.m8.1.1.3.2.cmml" xref="S3.SS1.4.p3.8.m8.1.1.3.2">𝜋</ci><ci id="S3.SS1.4.p3.8.m8.1.1.3.3.cmml" xref="S3.SS1.4.p3.8.m8.1.1.3.3">ℓ</ci></apply><apply id="S3.SS1.4.p3.8.m8.1.1.1.1.1.cmml" xref="S3.SS1.4.p3.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p3.8.m8.1.1.1.1.1.1.cmml" xref="S3.SS1.4.p3.8.m8.1.1.1.1">superscript</csymbol><ci id="S3.SS1.4.p3.8.m8.1.1.1.1.1.2.cmml" xref="S3.SS1.4.p3.8.m8.1.1.1.1.1.2">𝒜</ci><times id="S3.SS1.4.p3.8.m8.1.1.1.1.1.3.cmml" xref="S3.SS1.4.p3.8.m8.1.1.1.1.1.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.8.m8.1c">\pi_{\ell}(\cal A^{*})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.8.m8.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>, namely <math alttext="x=a_{i}(k-1)" class="ltx_Math" display="inline" id="S3.SS1.4.p3.9.m9.1"><semantics id="S3.SS1.4.p3.9.m9.1a"><mrow id="S3.SS1.4.p3.9.m9.1.1" xref="S3.SS1.4.p3.9.m9.1.1.cmml"><mi id="S3.SS1.4.p3.9.m9.1.1.3" xref="S3.SS1.4.p3.9.m9.1.1.3.cmml">x</mi><mo id="S3.SS1.4.p3.9.m9.1.1.2" xref="S3.SS1.4.p3.9.m9.1.1.2.cmml">=</mo><mrow id="S3.SS1.4.p3.9.m9.1.1.1" xref="S3.SS1.4.p3.9.m9.1.1.1.cmml"><msub id="S3.SS1.4.p3.9.m9.1.1.1.3" xref="S3.SS1.4.p3.9.m9.1.1.1.3.cmml"><mi id="S3.SS1.4.p3.9.m9.1.1.1.3.2" xref="S3.SS1.4.p3.9.m9.1.1.1.3.2.cmml">a</mi><mi id="S3.SS1.4.p3.9.m9.1.1.1.3.3" xref="S3.SS1.4.p3.9.m9.1.1.1.3.3.cmml">i</mi></msub><mo id="S3.SS1.4.p3.9.m9.1.1.1.2" xref="S3.SS1.4.p3.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.4.p3.9.m9.1.1.1.1.1" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.cmml"><mo id="S3.SS1.4.p3.9.m9.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.4.p3.9.m9.1.1.1.1.1.1" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.cmml"><mi id="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.2" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.2.cmml">k</mi><mo id="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.1" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.3" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS1.4.p3.9.m9.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.9.m9.1b"><apply id="S3.SS1.4.p3.9.m9.1.1.cmml" xref="S3.SS1.4.p3.9.m9.1.1"><eq id="S3.SS1.4.p3.9.m9.1.1.2.cmml" xref="S3.SS1.4.p3.9.m9.1.1.2"></eq><ci id="S3.SS1.4.p3.9.m9.1.1.3.cmml" xref="S3.SS1.4.p3.9.m9.1.1.3">𝑥</ci><apply id="S3.SS1.4.p3.9.m9.1.1.1.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1"><times id="S3.SS1.4.p3.9.m9.1.1.1.2.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1.2"></times><apply id="S3.SS1.4.p3.9.m9.1.1.1.3.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.4.p3.9.m9.1.1.1.3.1.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1.3">subscript</csymbol><ci id="S3.SS1.4.p3.9.m9.1.1.1.3.2.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1.3.2">𝑎</ci><ci id="S3.SS1.4.p3.9.m9.1.1.1.3.3.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1.3.3">𝑖</ci></apply><apply id="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1"><minus id="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.1"></minus><ci id="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.2.cmml" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.2">𝑘</ci><cn id="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.4.p3.9.m9.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.9.m9.1c">x=a_{i}(k-1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.9.m9.1d">italic_x = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k - 1 )</annotation></semantics></math>. In this case we have <math alttext="\widehat{xw}=\widehat{w}" class="ltx_Math" display="inline" id="S3.SS1.4.p3.10.m10.1"><semantics id="S3.SS1.4.p3.10.m10.1a"><mrow id="S3.SS1.4.p3.10.m10.1.1" xref="S3.SS1.4.p3.10.m10.1.1.cmml"><mover accent="true" id="S3.SS1.4.p3.10.m10.1.1.2" xref="S3.SS1.4.p3.10.m10.1.1.2.cmml"><mrow id="S3.SS1.4.p3.10.m10.1.1.2.2" xref="S3.SS1.4.p3.10.m10.1.1.2.2.cmml"><mi id="S3.SS1.4.p3.10.m10.1.1.2.2.2" xref="S3.SS1.4.p3.10.m10.1.1.2.2.2.cmml">x</mi><mo id="S3.SS1.4.p3.10.m10.1.1.2.2.1" xref="S3.SS1.4.p3.10.m10.1.1.2.2.1.cmml">⁢</mo><mi id="S3.SS1.4.p3.10.m10.1.1.2.2.3" xref="S3.SS1.4.p3.10.m10.1.1.2.2.3.cmml">w</mi></mrow><mo id="S3.SS1.4.p3.10.m10.1.1.2.1" xref="S3.SS1.4.p3.10.m10.1.1.2.1.cmml">^</mo></mover><mo id="S3.SS1.4.p3.10.m10.1.1.1" xref="S3.SS1.4.p3.10.m10.1.1.1.cmml">=</mo><mover accent="true" id="S3.SS1.4.p3.10.m10.1.1.3" xref="S3.SS1.4.p3.10.m10.1.1.3.cmml"><mi id="S3.SS1.4.p3.10.m10.1.1.3.2" xref="S3.SS1.4.p3.10.m10.1.1.3.2.cmml">w</mi><mo id="S3.SS1.4.p3.10.m10.1.1.3.1" xref="S3.SS1.4.p3.10.m10.1.1.3.1.cmml">^</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p3.10.m10.1b"><apply id="S3.SS1.4.p3.10.m10.1.1.cmml" xref="S3.SS1.4.p3.10.m10.1.1"><eq id="S3.SS1.4.p3.10.m10.1.1.1.cmml" xref="S3.SS1.4.p3.10.m10.1.1.1"></eq><apply id="S3.SS1.4.p3.10.m10.1.1.2.cmml" xref="S3.SS1.4.p3.10.m10.1.1.2"><ci id="S3.SS1.4.p3.10.m10.1.1.2.1.cmml" xref="S3.SS1.4.p3.10.m10.1.1.2.1">^</ci><apply id="S3.SS1.4.p3.10.m10.1.1.2.2.cmml" xref="S3.SS1.4.p3.10.m10.1.1.2.2"><times id="S3.SS1.4.p3.10.m10.1.1.2.2.1.cmml" xref="S3.SS1.4.p3.10.m10.1.1.2.2.1"></times><ci id="S3.SS1.4.p3.10.m10.1.1.2.2.2.cmml" xref="S3.SS1.4.p3.10.m10.1.1.2.2.2">𝑥</ci><ci id="S3.SS1.4.p3.10.m10.1.1.2.2.3.cmml" xref="S3.SS1.4.p3.10.m10.1.1.2.2.3">𝑤</ci></apply></apply><apply id="S3.SS1.4.p3.10.m10.1.1.3.cmml" xref="S3.SS1.4.p3.10.m10.1.1.3"><ci id="S3.SS1.4.p3.10.m10.1.1.3.1.cmml" xref="S3.SS1.4.p3.10.m10.1.1.3.1">^</ci><ci id="S3.SS1.4.p3.10.m10.1.1.3.2.cmml" xref="S3.SS1.4.p3.10.m10.1.1.3.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p3.10.m10.1c">\widehat{xw}=\widehat{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p3.10.m10.1d">over^ start_ARG italic_x italic_w end_ARG = over^ start_ARG italic_w end_ARG</annotation></semantics></math>, so that again the desired equality holds.</p> </div> <div class="ltx_para" id="S3.SS1.5.p4"> <p class="ltx_p" id="S3.SS1.5.p4.11">Finally, if the first letter of <math alttext="w" class="ltx_Math" display="inline" id="S3.SS1.5.p4.1.m1.1"><semantics id="S3.SS1.5.p4.1.m1.1a"><mi id="S3.SS1.5.p4.1.m1.1.1" xref="S3.SS1.5.p4.1.m1.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.1.m1.1b"><ci id="S3.SS1.5.p4.1.m1.1.1.cmml" xref="S3.SS1.5.p4.1.m1.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.1.m1.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.1.m1.1d">italic_w</annotation></semantics></math> is equal to some <math alttext="a_{i}(1)" class="ltx_Math" display="inline" id="S3.SS1.5.p4.2.m2.1"><semantics id="S3.SS1.5.p4.2.m2.1a"><mrow id="S3.SS1.5.p4.2.m2.1.2" xref="S3.SS1.5.p4.2.m2.1.2.cmml"><msub id="S3.SS1.5.p4.2.m2.1.2.2" xref="S3.SS1.5.p4.2.m2.1.2.2.cmml"><mi id="S3.SS1.5.p4.2.m2.1.2.2.2" xref="S3.SS1.5.p4.2.m2.1.2.2.2.cmml">a</mi><mi id="S3.SS1.5.p4.2.m2.1.2.2.3" xref="S3.SS1.5.p4.2.m2.1.2.2.3.cmml">i</mi></msub><mo id="S3.SS1.5.p4.2.m2.1.2.1" xref="S3.SS1.5.p4.2.m2.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.5.p4.2.m2.1.2.3.2" xref="S3.SS1.5.p4.2.m2.1.2.cmml"><mo id="S3.SS1.5.p4.2.m2.1.2.3.2.1" stretchy="false" xref="S3.SS1.5.p4.2.m2.1.2.cmml">(</mo><mn id="S3.SS1.5.p4.2.m2.1.1" xref="S3.SS1.5.p4.2.m2.1.1.cmml">1</mn><mo id="S3.SS1.5.p4.2.m2.1.2.3.2.2" stretchy="false" xref="S3.SS1.5.p4.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.2.m2.1b"><apply id="S3.SS1.5.p4.2.m2.1.2.cmml" xref="S3.SS1.5.p4.2.m2.1.2"><times id="S3.SS1.5.p4.2.m2.1.2.1.cmml" xref="S3.SS1.5.p4.2.m2.1.2.1"></times><apply id="S3.SS1.5.p4.2.m2.1.2.2.cmml" xref="S3.SS1.5.p4.2.m2.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.5.p4.2.m2.1.2.2.1.cmml" xref="S3.SS1.5.p4.2.m2.1.2.2">subscript</csymbol><ci id="S3.SS1.5.p4.2.m2.1.2.2.2.cmml" xref="S3.SS1.5.p4.2.m2.1.2.2.2">𝑎</ci><ci id="S3.SS1.5.p4.2.m2.1.2.2.3.cmml" xref="S3.SS1.5.p4.2.m2.1.2.2.3">𝑖</ci></apply><cn id="S3.SS1.5.p4.2.m2.1.1.cmml" type="integer" xref="S3.SS1.5.p4.2.m2.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.2.m2.1c">a_{i}(1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 )</annotation></semantics></math>, then <math alttext="xw" class="ltx_Math" display="inline" id="S3.SS1.5.p4.3.m3.1"><semantics id="S3.SS1.5.p4.3.m3.1a"><mrow id="S3.SS1.5.p4.3.m3.1.1" xref="S3.SS1.5.p4.3.m3.1.1.cmml"><mi id="S3.SS1.5.p4.3.m3.1.1.2" xref="S3.SS1.5.p4.3.m3.1.1.2.cmml">x</mi><mo id="S3.SS1.5.p4.3.m3.1.1.1" xref="S3.SS1.5.p4.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS1.5.p4.3.m3.1.1.3" xref="S3.SS1.5.p4.3.m3.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.3.m3.1b"><apply id="S3.SS1.5.p4.3.m3.1.1.cmml" xref="S3.SS1.5.p4.3.m3.1.1"><times id="S3.SS1.5.p4.3.m3.1.1.1.cmml" xref="S3.SS1.5.p4.3.m3.1.1.1"></times><ci id="S3.SS1.5.p4.3.m3.1.1.2.cmml" xref="S3.SS1.5.p4.3.m3.1.1.2">𝑥</ci><ci id="S3.SS1.5.p4.3.m3.1.1.3.cmml" xref="S3.SS1.5.p4.3.m3.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.3.m3.1c">xw</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.3.m3.1d">italic_x italic_w</annotation></semantics></math> is a factor of some element from <math alttext="\pi_{\ell}(\cal A^{*})" class="ltx_Math" display="inline" id="S3.SS1.5.p4.4.m4.1"><semantics id="S3.SS1.5.p4.4.m4.1a"><mrow id="S3.SS1.5.p4.4.m4.1.1" xref="S3.SS1.5.p4.4.m4.1.1.cmml"><msub id="S3.SS1.5.p4.4.m4.1.1.3" xref="S3.SS1.5.p4.4.m4.1.1.3.cmml"><mi id="S3.SS1.5.p4.4.m4.1.1.3.2" xref="S3.SS1.5.p4.4.m4.1.1.3.2.cmml">π</mi><mi id="S3.SS1.5.p4.4.m4.1.1.3.3" mathvariant="normal" xref="S3.SS1.5.p4.4.m4.1.1.3.3.cmml">ℓ</mi></msub><mo id="S3.SS1.5.p4.4.m4.1.1.2" xref="S3.SS1.5.p4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.5.p4.4.m4.1.1.1.1" xref="S3.SS1.5.p4.4.m4.1.1.1.1.1.cmml"><mo id="S3.SS1.5.p4.4.m4.1.1.1.1.2" stretchy="false" xref="S3.SS1.5.p4.4.m4.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.5.p4.4.m4.1.1.1.1.1" xref="S3.SS1.5.p4.4.m4.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.5.p4.4.m4.1.1.1.1.1.2" xref="S3.SS1.5.p4.4.m4.1.1.1.1.1.2.cmml">𝒜</mi><mo id="S3.SS1.5.p4.4.m4.1.1.1.1.1.3" xref="S3.SS1.5.p4.4.m4.1.1.1.1.1.3.cmml">∗</mo></msup><mo id="S3.SS1.5.p4.4.m4.1.1.1.1.3" stretchy="false" xref="S3.SS1.5.p4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.4.m4.1b"><apply id="S3.SS1.5.p4.4.m4.1.1.cmml" xref="S3.SS1.5.p4.4.m4.1.1"><times id="S3.SS1.5.p4.4.m4.1.1.2.cmml" xref="S3.SS1.5.p4.4.m4.1.1.2"></times><apply id="S3.SS1.5.p4.4.m4.1.1.3.cmml" xref="S3.SS1.5.p4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.5.p4.4.m4.1.1.3.1.cmml" xref="S3.SS1.5.p4.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS1.5.p4.4.m4.1.1.3.2.cmml" xref="S3.SS1.5.p4.4.m4.1.1.3.2">𝜋</ci><ci id="S3.SS1.5.p4.4.m4.1.1.3.3.cmml" xref="S3.SS1.5.p4.4.m4.1.1.3.3">ℓ</ci></apply><apply id="S3.SS1.5.p4.4.m4.1.1.1.1.1.cmml" xref="S3.SS1.5.p4.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.5.p4.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS1.5.p4.4.m4.1.1.1.1">superscript</csymbol><ci id="S3.SS1.5.p4.4.m4.1.1.1.1.1.2.cmml" xref="S3.SS1.5.p4.4.m4.1.1.1.1.1.2">𝒜</ci><times id="S3.SS1.5.p4.4.m4.1.1.1.1.1.3.cmml" xref="S3.SS1.5.p4.4.m4.1.1.1.1.1.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.4.m4.1c">\pi_{\ell}(\cal A^{*})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.4.m4.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math> precisely if <math alttext="x=a_{j}(\ell(j))" class="ltx_Math" display="inline" id="S3.SS1.5.p4.5.m5.2"><semantics id="S3.SS1.5.p4.5.m5.2a"><mrow id="S3.SS1.5.p4.5.m5.2.2" xref="S3.SS1.5.p4.5.m5.2.2.cmml"><mi id="S3.SS1.5.p4.5.m5.2.2.3" xref="S3.SS1.5.p4.5.m5.2.2.3.cmml">x</mi><mo id="S3.SS1.5.p4.5.m5.2.2.2" xref="S3.SS1.5.p4.5.m5.2.2.2.cmml">=</mo><mrow id="S3.SS1.5.p4.5.m5.2.2.1" xref="S3.SS1.5.p4.5.m5.2.2.1.cmml"><msub id="S3.SS1.5.p4.5.m5.2.2.1.3" xref="S3.SS1.5.p4.5.m5.2.2.1.3.cmml"><mi id="S3.SS1.5.p4.5.m5.2.2.1.3.2" xref="S3.SS1.5.p4.5.m5.2.2.1.3.2.cmml">a</mi><mi id="S3.SS1.5.p4.5.m5.2.2.1.3.3" xref="S3.SS1.5.p4.5.m5.2.2.1.3.3.cmml">j</mi></msub><mo id="S3.SS1.5.p4.5.m5.2.2.1.2" xref="S3.SS1.5.p4.5.m5.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.5.p4.5.m5.2.2.1.1.1" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.cmml"><mo id="S3.SS1.5.p4.5.m5.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.cmml"><mi id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.2" mathvariant="normal" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.2.cmml">ℓ</mi><mo id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.1" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.3.2" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.cmml"><mo id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.cmml">(</mo><mi id="S3.SS1.5.p4.5.m5.1.1" xref="S3.SS1.5.p4.5.m5.1.1.cmml">j</mi><mo id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.5.p4.5.m5.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.5.m5.2b"><apply id="S3.SS1.5.p4.5.m5.2.2.cmml" xref="S3.SS1.5.p4.5.m5.2.2"><eq id="S3.SS1.5.p4.5.m5.2.2.2.cmml" xref="S3.SS1.5.p4.5.m5.2.2.2"></eq><ci id="S3.SS1.5.p4.5.m5.2.2.3.cmml" xref="S3.SS1.5.p4.5.m5.2.2.3">𝑥</ci><apply id="S3.SS1.5.p4.5.m5.2.2.1.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1"><times id="S3.SS1.5.p4.5.m5.2.2.1.2.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1.2"></times><apply id="S3.SS1.5.p4.5.m5.2.2.1.3.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1.3"><csymbol cd="ambiguous" id="S3.SS1.5.p4.5.m5.2.2.1.3.1.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1.3">subscript</csymbol><ci id="S3.SS1.5.p4.5.m5.2.2.1.3.2.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1.3.2">𝑎</ci><ci id="S3.SS1.5.p4.5.m5.2.2.1.3.3.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1.3.3">𝑗</ci></apply><apply id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1"><times id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.1.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.1"></times><ci id="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.2.cmml" xref="S3.SS1.5.p4.5.m5.2.2.1.1.1.1.2">ℓ</ci><ci id="S3.SS1.5.p4.5.m5.1.1.cmml" xref="S3.SS1.5.p4.5.m5.1.1">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.5.m5.2c">x=a_{j}(\ell(j))</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.5.m5.2d">italic_x = italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( roman_ℓ ( italic_j ) )</annotation></semantics></math> for any of the <math alttext="a_{j}\in\cal A" class="ltx_Math" display="inline" id="S3.SS1.5.p4.6.m6.1"><semantics id="S3.SS1.5.p4.6.m6.1a"><mrow id="S3.SS1.5.p4.6.m6.1.1" xref="S3.SS1.5.p4.6.m6.1.1.cmml"><msub id="S3.SS1.5.p4.6.m6.1.1.2" xref="S3.SS1.5.p4.6.m6.1.1.2.cmml"><mi id="S3.SS1.5.p4.6.m6.1.1.2.2" xref="S3.SS1.5.p4.6.m6.1.1.2.2.cmml">a</mi><mi id="S3.SS1.5.p4.6.m6.1.1.2.3" xref="S3.SS1.5.p4.6.m6.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS1.5.p4.6.m6.1.1.1" xref="S3.SS1.5.p4.6.m6.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.5.p4.6.m6.1.1.3" xref="S3.SS1.5.p4.6.m6.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.6.m6.1b"><apply id="S3.SS1.5.p4.6.m6.1.1.cmml" xref="S3.SS1.5.p4.6.m6.1.1"><in id="S3.SS1.5.p4.6.m6.1.1.1.cmml" xref="S3.SS1.5.p4.6.m6.1.1.1"></in><apply id="S3.SS1.5.p4.6.m6.1.1.2.cmml" xref="S3.SS1.5.p4.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.5.p4.6.m6.1.1.2.1.cmml" xref="S3.SS1.5.p4.6.m6.1.1.2">subscript</csymbol><ci id="S3.SS1.5.p4.6.m6.1.1.2.2.cmml" xref="S3.SS1.5.p4.6.m6.1.1.2.2">𝑎</ci><ci id="S3.SS1.5.p4.6.m6.1.1.2.3.cmml" xref="S3.SS1.5.p4.6.m6.1.1.2.3">𝑗</ci></apply><ci id="S3.SS1.5.p4.6.m6.1.1.3.cmml" xref="S3.SS1.5.p4.6.m6.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.6.m6.1c">a_{j}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.6.m6.1d">italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math>. In this case we have <math alttext="\widehat{xw}=a_{j}\widehat{w}" class="ltx_Math" display="inline" id="S3.SS1.5.p4.7.m7.1"><semantics id="S3.SS1.5.p4.7.m7.1a"><mrow id="S3.SS1.5.p4.7.m7.1.1" xref="S3.SS1.5.p4.7.m7.1.1.cmml"><mover accent="true" id="S3.SS1.5.p4.7.m7.1.1.2" xref="S3.SS1.5.p4.7.m7.1.1.2.cmml"><mrow id="S3.SS1.5.p4.7.m7.1.1.2.2" xref="S3.SS1.5.p4.7.m7.1.1.2.2.cmml"><mi id="S3.SS1.5.p4.7.m7.1.1.2.2.2" xref="S3.SS1.5.p4.7.m7.1.1.2.2.2.cmml">x</mi><mo id="S3.SS1.5.p4.7.m7.1.1.2.2.1" xref="S3.SS1.5.p4.7.m7.1.1.2.2.1.cmml">⁢</mo><mi id="S3.SS1.5.p4.7.m7.1.1.2.2.3" xref="S3.SS1.5.p4.7.m7.1.1.2.2.3.cmml">w</mi></mrow><mo id="S3.SS1.5.p4.7.m7.1.1.2.1" xref="S3.SS1.5.p4.7.m7.1.1.2.1.cmml">^</mo></mover><mo id="S3.SS1.5.p4.7.m7.1.1.1" xref="S3.SS1.5.p4.7.m7.1.1.1.cmml">=</mo><mrow id="S3.SS1.5.p4.7.m7.1.1.3" xref="S3.SS1.5.p4.7.m7.1.1.3.cmml"><msub id="S3.SS1.5.p4.7.m7.1.1.3.2" xref="S3.SS1.5.p4.7.m7.1.1.3.2.cmml"><mi id="S3.SS1.5.p4.7.m7.1.1.3.2.2" xref="S3.SS1.5.p4.7.m7.1.1.3.2.2.cmml">a</mi><mi id="S3.SS1.5.p4.7.m7.1.1.3.2.3" xref="S3.SS1.5.p4.7.m7.1.1.3.2.3.cmml">j</mi></msub><mo id="S3.SS1.5.p4.7.m7.1.1.3.1" xref="S3.SS1.5.p4.7.m7.1.1.3.1.cmml">⁢</mo><mover accent="true" id="S3.SS1.5.p4.7.m7.1.1.3.3" xref="S3.SS1.5.p4.7.m7.1.1.3.3.cmml"><mi id="S3.SS1.5.p4.7.m7.1.1.3.3.2" xref="S3.SS1.5.p4.7.m7.1.1.3.3.2.cmml">w</mi><mo id="S3.SS1.5.p4.7.m7.1.1.3.3.1" xref="S3.SS1.5.p4.7.m7.1.1.3.3.1.cmml">^</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.7.m7.1b"><apply id="S3.SS1.5.p4.7.m7.1.1.cmml" xref="S3.SS1.5.p4.7.m7.1.1"><eq id="S3.SS1.5.p4.7.m7.1.1.1.cmml" xref="S3.SS1.5.p4.7.m7.1.1.1"></eq><apply id="S3.SS1.5.p4.7.m7.1.1.2.cmml" xref="S3.SS1.5.p4.7.m7.1.1.2"><ci id="S3.SS1.5.p4.7.m7.1.1.2.1.cmml" xref="S3.SS1.5.p4.7.m7.1.1.2.1">^</ci><apply id="S3.SS1.5.p4.7.m7.1.1.2.2.cmml" xref="S3.SS1.5.p4.7.m7.1.1.2.2"><times id="S3.SS1.5.p4.7.m7.1.1.2.2.1.cmml" xref="S3.SS1.5.p4.7.m7.1.1.2.2.1"></times><ci id="S3.SS1.5.p4.7.m7.1.1.2.2.2.cmml" xref="S3.SS1.5.p4.7.m7.1.1.2.2.2">𝑥</ci><ci id="S3.SS1.5.p4.7.m7.1.1.2.2.3.cmml" xref="S3.SS1.5.p4.7.m7.1.1.2.2.3">𝑤</ci></apply></apply><apply id="S3.SS1.5.p4.7.m7.1.1.3.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3"><times id="S3.SS1.5.p4.7.m7.1.1.3.1.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3.1"></times><apply id="S3.SS1.5.p4.7.m7.1.1.3.2.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.5.p4.7.m7.1.1.3.2.1.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3.2">subscript</csymbol><ci id="S3.SS1.5.p4.7.m7.1.1.3.2.2.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3.2.2">𝑎</ci><ci id="S3.SS1.5.p4.7.m7.1.1.3.2.3.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3.2.3">𝑗</ci></apply><apply id="S3.SS1.5.p4.7.m7.1.1.3.3.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3.3"><ci id="S3.SS1.5.p4.7.m7.1.1.3.3.1.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3.3.1">^</ci><ci id="S3.SS1.5.p4.7.m7.1.1.3.3.2.cmml" xref="S3.SS1.5.p4.7.m7.1.1.3.3.2">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.7.m7.1c">\widehat{xw}=a_{j}\widehat{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.7.m7.1d">over^ start_ARG italic_x italic_w end_ARG = italic_a start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT over^ start_ARG italic_w end_ARG</annotation></semantics></math>, so that the first Kirchhoff equality for <math alttext="\mu(\widehat{w})" class="ltx_Math" display="inline" id="S3.SS1.5.p4.8.m8.1"><semantics id="S3.SS1.5.p4.8.m8.1a"><mrow id="S3.SS1.5.p4.8.m8.1.2" xref="S3.SS1.5.p4.8.m8.1.2.cmml"><mi id="S3.SS1.5.p4.8.m8.1.2.2" xref="S3.SS1.5.p4.8.m8.1.2.2.cmml">μ</mi><mo id="S3.SS1.5.p4.8.m8.1.2.1" xref="S3.SS1.5.p4.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.5.p4.8.m8.1.2.3.2" xref="S3.SS1.5.p4.8.m8.1.1.cmml"><mo id="S3.SS1.5.p4.8.m8.1.2.3.2.1" stretchy="false" xref="S3.SS1.5.p4.8.m8.1.1.cmml">(</mo><mover accent="true" id="S3.SS1.5.p4.8.m8.1.1" xref="S3.SS1.5.p4.8.m8.1.1.cmml"><mi id="S3.SS1.5.p4.8.m8.1.1.2" xref="S3.SS1.5.p4.8.m8.1.1.2.cmml">w</mi><mo id="S3.SS1.5.p4.8.m8.1.1.1" xref="S3.SS1.5.p4.8.m8.1.1.1.cmml">^</mo></mover><mo id="S3.SS1.5.p4.8.m8.1.2.3.2.2" stretchy="false" xref="S3.SS1.5.p4.8.m8.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.8.m8.1b"><apply id="S3.SS1.5.p4.8.m8.1.2.cmml" xref="S3.SS1.5.p4.8.m8.1.2"><times id="S3.SS1.5.p4.8.m8.1.2.1.cmml" xref="S3.SS1.5.p4.8.m8.1.2.1"></times><ci id="S3.SS1.5.p4.8.m8.1.2.2.cmml" xref="S3.SS1.5.p4.8.m8.1.2.2">𝜇</ci><apply id="S3.SS1.5.p4.8.m8.1.1.cmml" xref="S3.SS1.5.p4.8.m8.1.2.3.2"><ci id="S3.SS1.5.p4.8.m8.1.1.1.cmml" xref="S3.SS1.5.p4.8.m8.1.1.1">^</ci><ci id="S3.SS1.5.p4.8.m8.1.1.2.cmml" xref="S3.SS1.5.p4.8.m8.1.1.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.8.m8.1c">\mu(\widehat{w})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.8.m8.1d">italic_μ ( over^ start_ARG italic_w end_ARG )</annotation></semantics></math> gives directly the desired equality for <math alttext="\mu_{\ell}(w)" class="ltx_Math" display="inline" id="S3.SS1.5.p4.9.m9.1"><semantics id="S3.SS1.5.p4.9.m9.1a"><mrow id="S3.SS1.5.p4.9.m9.1.2" xref="S3.SS1.5.p4.9.m9.1.2.cmml"><msub id="S3.SS1.5.p4.9.m9.1.2.2" xref="S3.SS1.5.p4.9.m9.1.2.2.cmml"><mi id="S3.SS1.5.p4.9.m9.1.2.2.2" xref="S3.SS1.5.p4.9.m9.1.2.2.2.cmml">μ</mi><mi id="S3.SS1.5.p4.9.m9.1.2.2.3" mathvariant="normal" xref="S3.SS1.5.p4.9.m9.1.2.2.3.cmml">ℓ</mi></msub><mo id="S3.SS1.5.p4.9.m9.1.2.1" xref="S3.SS1.5.p4.9.m9.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.5.p4.9.m9.1.2.3.2" xref="S3.SS1.5.p4.9.m9.1.2.cmml"><mo id="S3.SS1.5.p4.9.m9.1.2.3.2.1" stretchy="false" xref="S3.SS1.5.p4.9.m9.1.2.cmml">(</mo><mi id="S3.SS1.5.p4.9.m9.1.1" xref="S3.SS1.5.p4.9.m9.1.1.cmml">w</mi><mo id="S3.SS1.5.p4.9.m9.1.2.3.2.2" stretchy="false" xref="S3.SS1.5.p4.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.9.m9.1b"><apply id="S3.SS1.5.p4.9.m9.1.2.cmml" xref="S3.SS1.5.p4.9.m9.1.2"><times id="S3.SS1.5.p4.9.m9.1.2.1.cmml" xref="S3.SS1.5.p4.9.m9.1.2.1"></times><apply id="S3.SS1.5.p4.9.m9.1.2.2.cmml" xref="S3.SS1.5.p4.9.m9.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.5.p4.9.m9.1.2.2.1.cmml" xref="S3.SS1.5.p4.9.m9.1.2.2">subscript</csymbol><ci id="S3.SS1.5.p4.9.m9.1.2.2.2.cmml" xref="S3.SS1.5.p4.9.m9.1.2.2.2">𝜇</ci><ci id="S3.SS1.5.p4.9.m9.1.2.2.3.cmml" xref="S3.SS1.5.p4.9.m9.1.2.2.3">ℓ</ci></apply><ci id="S3.SS1.5.p4.9.m9.1.1.cmml" xref="S3.SS1.5.p4.9.m9.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.9.m9.1c">\mu_{\ell}(w)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.9.m9.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S3.SS1.5.p4.10.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S3.SS1.5.p4.10.1.m1.1"><semantics id="S3.SS1.5.p4.10.1.m1.1a"><mo id="S3.SS1.5.p4.10.1.m1.1.1" xref="S3.SS1.5.p4.10.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.10.1.m1.1b"><csymbol cd="latexml" id="S3.SS1.5.p4.10.1.m1.1.1.cmml" xref="S3.SS1.5.p4.10.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.10.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.10.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S3.SS1.5.p4.11.m10.1"><semantics id="S3.SS1.5.p4.11.m10.1a"><mo id="S3.SS1.5.p4.11.m10.1.1" xref="S3.SS1.5.p4.11.m10.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p4.11.m10.1b"><csymbol cd="latexml" id="S3.SS1.5.p4.11.m10.1.1.cmml" xref="S3.SS1.5.p4.11.m10.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p4.11.m10.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p4.11.m10.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S3.Thmthm5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm5.1.1.1">Remark 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm5.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm5.p1"> <p class="ltx_p" id="S3.Thmthm5.p1.1">The subdivision measure defined by a probability measure will in general not be probability: Unless <math alttext="\ell" class="ltx_Math" display="inline" id="S3.Thmthm5.p1.1.m1.1"><semantics id="S3.Thmthm5.p1.1.m1.1a"><mi id="S3.Thmthm5.p1.1.m1.1.1" mathvariant="normal" xref="S3.Thmthm5.p1.1.m1.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm5.p1.1.m1.1b"><ci id="S3.Thmthm5.p1.1.m1.1.1.cmml" xref="S3.Thmthm5.p1.1.m1.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm5.p1.1.m1.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm5.p1.1.m1.1d">roman_ℓ</annotation></semantics></math> is the constant function with value 1, for the total measure we will have</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu_{\ell}(\cal A_{\ell}^{\mathbb{Z}})>1\,." class="ltx_Math" display="block" id="S3.Ex1.m1.1"><semantics id="S3.Ex1.m1.1a"><mrow id="S3.Ex1.m1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.cmml"><mrow id="S3.Ex1.m1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.cmml"><mrow id="S3.Ex1.m1.1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.cmml"><msub id="S3.Ex1.m1.1.1.1.1.1.3" xref="S3.Ex1.m1.1.1.1.1.1.3.cmml"><mi id="S3.Ex1.m1.1.1.1.1.1.3.2" xref="S3.Ex1.m1.1.1.1.1.1.3.2.cmml">μ</mi><mi id="S3.Ex1.m1.1.1.1.1.1.3.3" mathvariant="normal" xref="S3.Ex1.m1.1.1.1.1.1.3.3.cmml">ℓ</mi></msub><mo id="S3.Ex1.m1.1.1.1.1.1.2" xref="S3.Ex1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex1.m1.1.1.1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.Ex1.m1.1.1.1.1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.2" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.2.cmml">𝒜</mi><mi id="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.3" mathvariant="normal" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.3.cmml">ℓ</mi><mi id="S3.Ex1.m1.1.1.1.1.1.1.1.1.3" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msubsup><mo id="S3.Ex1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex1.m1.1.1.1.1.2" xref="S3.Ex1.m1.1.1.1.1.2.cmml">></mo><mn class="ltx_font_mathcaligraphic" id="S3.Ex1.m1.1.1.1.1.3" mathvariant="script" xref="S3.Ex1.m1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.Ex1.m1.1.1.1.2" lspace="0.170em" xref="S3.Ex1.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex1.m1.1b"><apply id="S3.Ex1.m1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1"><gt id="S3.Ex1.m1.1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.1.2"></gt><apply id="S3.Ex1.m1.1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1"><times id="S3.Ex1.m1.1.1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.1.1.2"></times><apply id="S3.Ex1.m1.1.1.1.1.1.3.cmml" xref="S3.Ex1.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex1.m1.1.1.1.1.1.3.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.Ex1.m1.1.1.1.1.1.3.2.cmml" xref="S3.Ex1.m1.1.1.1.1.1.3.2">𝜇</ci><ci id="S3.Ex1.m1.1.1.1.1.1.3.3.cmml" xref="S3.Ex1.m1.1.1.1.1.1.3.3">ℓ</ci></apply><apply id="S3.Ex1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.2">𝒜</ci><ci id="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.2.3">ℓ</ci></apply><ci id="S3.Ex1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.3">ℤ</ci></apply></apply><cn id="S3.Ex1.m1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex1.m1.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.1c">\mu_{\ell}(\cal A_{\ell}^{\mathbb{Z}})>1\,.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) > caligraphic_1 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmthm5.p1.2">In fact, one easily derives from (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E2" title="In Definition 3.3. ‣ 3.1. Subdivision morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.2</span></a>) that <math alttext="\mu_{\ell}(a_{i}(k))=\mu(a_{i})" class="ltx_Math" display="inline" id="S3.Thmthm5.p1.2.m1.3"><semantics id="S3.Thmthm5.p1.2.m1.3a"><mrow id="S3.Thmthm5.p1.2.m1.3.3" xref="S3.Thmthm5.p1.2.m1.3.3.cmml"><mrow id="S3.Thmthm5.p1.2.m1.2.2.1" xref="S3.Thmthm5.p1.2.m1.2.2.1.cmml"><msub id="S3.Thmthm5.p1.2.m1.2.2.1.3" xref="S3.Thmthm5.p1.2.m1.2.2.1.3.cmml"><mi id="S3.Thmthm5.p1.2.m1.2.2.1.3.2" xref="S3.Thmthm5.p1.2.m1.2.2.1.3.2.cmml">μ</mi><mi id="S3.Thmthm5.p1.2.m1.2.2.1.3.3" mathvariant="normal" xref="S3.Thmthm5.p1.2.m1.2.2.1.3.3.cmml">ℓ</mi></msub><mo id="S3.Thmthm5.p1.2.m1.2.2.1.2" xref="S3.Thmthm5.p1.2.m1.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm5.p1.2.m1.2.2.1.1.1" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.cmml"><mo id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.2" stretchy="false" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.cmml"><msub id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.cmml"><mi id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.2" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.2.cmml">a</mi><mi id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.3" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.1" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.3.2" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.cmml"><mo id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.cmml">(</mo><mi id="S3.Thmthm5.p1.2.m1.1.1" xref="S3.Thmthm5.p1.2.m1.1.1.cmml">k</mi><mo id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.3" stretchy="false" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm5.p1.2.m1.3.3.3" xref="S3.Thmthm5.p1.2.m1.3.3.3.cmml">=</mo><mrow id="S3.Thmthm5.p1.2.m1.3.3.2" xref="S3.Thmthm5.p1.2.m1.3.3.2.cmml"><mi id="S3.Thmthm5.p1.2.m1.3.3.2.3" xref="S3.Thmthm5.p1.2.m1.3.3.2.3.cmml">μ</mi><mo id="S3.Thmthm5.p1.2.m1.3.3.2.2" xref="S3.Thmthm5.p1.2.m1.3.3.2.2.cmml">⁢</mo><mrow id="S3.Thmthm5.p1.2.m1.3.3.2.1.1" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.cmml"><mo id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.2" stretchy="false" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.cmml">(</mo><msub id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.cmml"><mi id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.2" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.2.cmml">a</mi><mi id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.3" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.3.cmml">i</mi></msub><mo id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.3" stretchy="false" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm5.p1.2.m1.3b"><apply 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id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.1.cmml" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2">subscript</csymbol><ci id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.2.cmml" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.2">𝑎</ci><ci id="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmthm5.p1.2.m1.2.2.1.1.1.1.2.3">𝑖</ci></apply><ci id="S3.Thmthm5.p1.2.m1.1.1.cmml" xref="S3.Thmthm5.p1.2.m1.1.1">𝑘</ci></apply></apply><apply id="S3.Thmthm5.p1.2.m1.3.3.2.cmml" xref="S3.Thmthm5.p1.2.m1.3.3.2"><times id="S3.Thmthm5.p1.2.m1.3.3.2.2.cmml" xref="S3.Thmthm5.p1.2.m1.3.3.2.2"></times><ci id="S3.Thmthm5.p1.2.m1.3.3.2.3.cmml" xref="S3.Thmthm5.p1.2.m1.3.3.2.3">𝜇</ci><apply id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.cmml" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.1.cmml" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1">subscript</csymbol><ci id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.2.cmml" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.2">𝑎</ci><ci id="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.3.cmml" xref="S3.Thmthm5.p1.2.m1.3.3.2.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm5.p1.2.m1.3c">\mu_{\ell}(a_{i}(k))=\mu(a_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm5.p1.2.m1.3d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_k ) ) = italic_μ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>, which yields</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu_{\ell}(\cal A_{\ell}^{\mathbb{Z}})=\sum_{a_{i}(k)\,\in\,\cal A_{\ell}}\mu_% {\ell}(a_{i}(k))=\sum_{a_{i}\in\cal A}\ell(a_{i})\cdot\mu(a_{i})\,." class="ltx_Math" display="block" id="S3.Ex2.m1.3"><semantics id="S3.Ex2.m1.3a"><mrow 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xref="S3.Ex2.m1.3.3.1.1.4.2.2.1.1.3">𝒾</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex2.m1.3c">\mu_{\ell}(\cal A_{\ell}^{\mathbb{Z}})=\sum_{a_{i}(k)\,\in\,\cal A_{\ell}}\mu_% {\ell}(a_{i}(k))=\sum_{a_{i}\in\cal A}\ell(a_{i})\cdot\mu(a_{i})\,.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex2.m1.3d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ( caligraphic_k ) ∈ caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ( caligraphic_k ) ) = ∑ start_POSTSUBSCRIPT caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT roman_ℓ ( caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ) ⋅ italic_μ ( caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2. </span>Letter-to-letter morphisms</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1"></p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.5">Recall that a monoid morphism <math alttext="\alpha:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mrow id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mi id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">α</mi><mo id="S3.SS2.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS2.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S3.SS2.p2.1.m1.1.1.3" xref="S3.SS2.p2.1.m1.1.1.3.cmml"><msup id="S3.SS2.p2.1.m1.1.1.3.2" xref="S3.SS2.p2.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.1.m1.1.1.3.2.2" xref="S3.SS2.p2.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.SS2.p2.1.m1.1.1.3.2.3" xref="S3.SS2.p2.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.SS2.p2.1.m1.1.1.3.1" stretchy="false" xref="S3.SS2.p2.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.SS2.p2.1.m1.1.1.3.3" xref="S3.SS2.p2.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.1.m1.1.1.3.3.2" xref="S3.SS2.p2.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.SS2.p2.1.m1.1.1.3.3.3" xref="S3.SS2.p2.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><ci id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1">:</ci><ci id="S3.SS2.p2.1.m1.1.1.2.cmml" xref="S3.SS2.p2.1.m1.1.1.2">𝛼</ci><apply id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3"><ci id="S3.SS2.p2.1.m1.1.1.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.3.1">→</ci><apply id="S3.SS2.p2.1.m1.1.1.3.2.cmml" xref="S3.SS2.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.3.2.1.cmml" xref="S3.SS2.p2.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.3.2.2.cmml" xref="S3.SS2.p2.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.SS2.p2.1.m1.1.1.3.2.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3.2.3"></times></apply><apply id="S3.SS2.p2.1.m1.1.1.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.p2.1.m1.1.1.3.3.1.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.SS2.p2.1.m1.1.1.3.3.2.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.SS2.p2.1.m1.1.1.3.3.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">\alpha:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_α : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S3.SS2.p2.5.1">letter-to-letter</span> if for any letter <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mrow id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml"><msub id="S3.SS2.p2.2.m2.1.1.2" xref="S3.SS2.p2.2.m2.1.1.2.cmml"><mi id="S3.SS2.p2.2.m2.1.1.2.2" xref="S3.SS2.p2.2.m2.1.1.2.2.cmml">a</mi><mi id="S3.SS2.p2.2.m2.1.1.2.3" xref="S3.SS2.p2.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS2.p2.2.m2.1.1.1" xref="S3.SS2.p2.2.m2.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.2.m2.1.1.3" xref="S3.SS2.p2.2.m2.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><apply id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1"><in id="S3.SS2.p2.2.m2.1.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1.1"></in><apply id="S3.SS2.p2.2.m2.1.1.2.cmml" xref="S3.SS2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p2.2.m2.1.1.2.1.cmml" xref="S3.SS2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS2.p2.2.m2.1.1.2.2.cmml" xref="S3.SS2.p2.2.m2.1.1.2.2">𝑎</ci><ci id="S3.SS2.p2.2.m2.1.1.2.3.cmml" xref="S3.SS2.p2.2.m2.1.1.2.3">𝑖</ci></apply><ci id="S3.SS2.p2.2.m2.1.1.3.cmml" xref="S3.SS2.p2.2.m2.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> the length of its image is equal to <math alttext="|\alpha(a_{i})|=1" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.1"><semantics id="S3.SS2.p2.3.m3.1a"><mrow id="S3.SS2.p2.3.m3.1.1" xref="S3.SS2.p2.3.m3.1.1.cmml"><mrow id="S3.SS2.p2.3.m3.1.1.1.1" xref="S3.SS2.p2.3.m3.1.1.1.2.cmml"><mo id="S3.SS2.p2.3.m3.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.3.m3.1.1.1.2.1.cmml">|</mo><mrow id="S3.SS2.p2.3.m3.1.1.1.1.1" xref="S3.SS2.p2.3.m3.1.1.1.1.1.cmml"><mi id="S3.SS2.p2.3.m3.1.1.1.1.1.3" xref="S3.SS2.p2.3.m3.1.1.1.1.1.3.cmml">α</mi><mo id="S3.SS2.p2.3.m3.1.1.1.1.1.2" xref="S3.SS2.p2.3.m3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.2" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.3" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p2.3.m3.1.1.1.1.3" stretchy="false" xref="S3.SS2.p2.3.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.SS2.p2.3.m3.1.1.2" xref="S3.SS2.p2.3.m3.1.1.2.cmml">=</mo><mn id="S3.SS2.p2.3.m3.1.1.3" xref="S3.SS2.p2.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.1b"><apply id="S3.SS2.p2.3.m3.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1"><eq id="S3.SS2.p2.3.m3.1.1.2.cmml" xref="S3.SS2.p2.3.m3.1.1.2"></eq><apply id="S3.SS2.p2.3.m3.1.1.1.2.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1"><abs id="S3.SS2.p2.3.m3.1.1.1.2.1.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.2"></abs><apply id="S3.SS2.p2.3.m3.1.1.1.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.1"><times id="S3.SS2.p2.3.m3.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.1.2"></times><ci id="S3.SS2.p2.3.m3.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.1.3">𝛼</ci><apply id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p2.3.m3.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply><cn id="S3.SS2.p2.3.m3.1.1.3.cmml" type="integer" xref="S3.SS2.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m3.1c">|\alpha(a_{i})|=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m3.1d">| italic_α ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) | = 1</annotation></semantics></math>. In other words: <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m4.1"><semantics id="S3.SS2.p2.4.m4.1a"><mi id="S3.SS2.p2.4.m4.1.1" xref="S3.SS2.p2.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.1b"><ci id="S3.SS2.p2.4.m4.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.1d">italic_α</annotation></semantics></math> is induced by a map <math alttext="\alpha A:\cal A\to\cal B" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m5.1"><semantics id="S3.SS2.p2.5.m5.1a"><mrow id="S3.SS2.p2.5.m5.1.1" xref="S3.SS2.p2.5.m5.1.1.cmml"><mrow id="S3.SS2.p2.5.m5.1.1.2" xref="S3.SS2.p2.5.m5.1.1.2.cmml"><mi id="S3.SS2.p2.5.m5.1.1.2.2" xref="S3.SS2.p2.5.m5.1.1.2.2.cmml">α</mi><mo id="S3.SS2.p2.5.m5.1.1.2.1" xref="S3.SS2.p2.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S3.SS2.p2.5.m5.1.1.2.3" xref="S3.SS2.p2.5.m5.1.1.2.3.cmml">A</mi></mrow><mo id="S3.SS2.p2.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS2.p2.5.m5.1.1.1.cmml">:</mo><mrow id="S3.SS2.p2.5.m5.1.1.3" xref="S3.SS2.p2.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.5.m5.1.1.3.2" xref="S3.SS2.p2.5.m5.1.1.3.2.cmml">𝒜</mi><mo id="S3.SS2.p2.5.m5.1.1.3.1" stretchy="false" xref="S3.SS2.p2.5.m5.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p2.5.m5.1.1.3.3" xref="S3.SS2.p2.5.m5.1.1.3.3.cmml">ℬ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m5.1b"><apply id="S3.SS2.p2.5.m5.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1"><ci id="S3.SS2.p2.5.m5.1.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1.1">:</ci><apply id="S3.SS2.p2.5.m5.1.1.2.cmml" xref="S3.SS2.p2.5.m5.1.1.2"><times id="S3.SS2.p2.5.m5.1.1.2.1.cmml" xref="S3.SS2.p2.5.m5.1.1.2.1"></times><ci id="S3.SS2.p2.5.m5.1.1.2.2.cmml" xref="S3.SS2.p2.5.m5.1.1.2.2">𝛼</ci><ci id="S3.SS2.p2.5.m5.1.1.2.3.cmml" xref="S3.SS2.p2.5.m5.1.1.2.3">𝐴</ci></apply><apply id="S3.SS2.p2.5.m5.1.1.3.cmml" xref="S3.SS2.p2.5.m5.1.1.3"><ci id="S3.SS2.p2.5.m5.1.1.3.1.cmml" xref="S3.SS2.p2.5.m5.1.1.3.1">→</ci><ci id="S3.SS2.p2.5.m5.1.1.3.2.cmml" xref="S3.SS2.p2.5.m5.1.1.3.2">𝒜</ci><ci id="S3.SS2.p2.5.m5.1.1.3.3.cmml" xref="S3.SS2.p2.5.m5.1.1.3.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m5.1c">\alpha A:\cal A\to\cal B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m5.1d">italic_α italic_A : caligraphic_A → caligraphic_B</annotation></semantics></math> on the alphabets <span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span> Some authors (see for instance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib7" title="">7</a>]</cite>) require in addition that a letter-to-letter morphism must be surjective. All letter-to-letter morphisms occurring in this paper are indeed surjective, but formally we do not need this condition anywhere.</span></span></span>.</p> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.7">It follows directly that both, the letter-to-letter morphism <math alttext="\alpha:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mrow id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml"><mi id="S3.SS2.p3.1.m1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.2.cmml">α</mi><mo id="S3.SS2.p3.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS2.p3.1.m1.1.1.1.cmml">:</mo><mrow id="S3.SS2.p3.1.m1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.3.cmml"><msup id="S3.SS2.p3.1.m1.1.1.3.2" xref="S3.SS2.p3.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.1.m1.1.1.3.2.2" xref="S3.SS2.p3.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.SS2.p3.1.m1.1.1.3.2.3" xref="S3.SS2.p3.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.SS2.p3.1.m1.1.1.3.1" stretchy="false" xref="S3.SS2.p3.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.SS2.p3.1.m1.1.1.3.3" xref="S3.SS2.p3.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.1.m1.1.1.3.3.2" xref="S3.SS2.p3.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.SS2.p3.1.m1.1.1.3.3.3" xref="S3.SS2.p3.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><apply id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1"><ci id="S3.SS2.p3.1.m1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1.1">:</ci><ci id="S3.SS2.p3.1.m1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.1.2">𝛼</ci><apply id="S3.SS2.p3.1.m1.1.1.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3"><ci id="S3.SS2.p3.1.m1.1.1.3.1.cmml" xref="S3.SS2.p3.1.m1.1.1.3.1">→</ci><apply id="S3.SS2.p3.1.m1.1.1.3.2.cmml" xref="S3.SS2.p3.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.p3.1.m1.1.1.3.2.1.cmml" xref="S3.SS2.p3.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.SS2.p3.1.m1.1.1.3.2.2.cmml" xref="S3.SS2.p3.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.SS2.p3.1.m1.1.1.3.2.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3.2.3"></times></apply><apply id="S3.SS2.p3.1.m1.1.1.3.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.p3.1.m1.1.1.3.3.1.cmml" xref="S3.SS2.p3.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.SS2.p3.1.m1.1.1.3.3.2.cmml" xref="S3.SS2.p3.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.SS2.p3.1.m1.1.1.3.3.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">\alpha:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_α : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and the induced map <math alttext="\alpha^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m2.1"><semantics id="S3.SS2.p3.2.m2.1a"><mrow id="S3.SS2.p3.2.m2.1.1" xref="S3.SS2.p3.2.m2.1.1.cmml"><msup id="S3.SS2.p3.2.m2.1.1.2" xref="S3.SS2.p3.2.m2.1.1.2.cmml"><mi id="S3.SS2.p3.2.m2.1.1.2.2" xref="S3.SS2.p3.2.m2.1.1.2.2.cmml">α</mi><mi id="S3.SS2.p3.2.m2.1.1.2.3" xref="S3.SS2.p3.2.m2.1.1.2.3.cmml">ℤ</mi></msup><mo id="S3.SS2.p3.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS2.p3.2.m2.1.1.1.cmml">:</mo><mrow id="S3.SS2.p3.2.m2.1.1.3" xref="S3.SS2.p3.2.m2.1.1.3.cmml"><msup id="S3.SS2.p3.2.m2.1.1.3.2" xref="S3.SS2.p3.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.2.m2.1.1.3.2.2" xref="S3.SS2.p3.2.m2.1.1.3.2.2.cmml">𝒜</mi><mi id="S3.SS2.p3.2.m2.1.1.3.2.3" xref="S3.SS2.p3.2.m2.1.1.3.2.3.cmml">ℤ</mi></msup><mo id="S3.SS2.p3.2.m2.1.1.3.1" stretchy="false" xref="S3.SS2.p3.2.m2.1.1.3.1.cmml">→</mo><msup id="S3.SS2.p3.2.m2.1.1.3.3" xref="S3.SS2.p3.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.2.m2.1.1.3.3.2" xref="S3.SS2.p3.2.m2.1.1.3.3.2.cmml">ℬ</mi><mi id="S3.SS2.p3.2.m2.1.1.3.3.3" xref="S3.SS2.p3.2.m2.1.1.3.3.3.cmml">ℤ</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m2.1b"><apply id="S3.SS2.p3.2.m2.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1"><ci id="S3.SS2.p3.2.m2.1.1.1.cmml" xref="S3.SS2.p3.2.m2.1.1.1">:</ci><apply id="S3.SS2.p3.2.m2.1.1.2.cmml" xref="S3.SS2.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p3.2.m2.1.1.2.1.cmml" xref="S3.SS2.p3.2.m2.1.1.2">superscript</csymbol><ci id="S3.SS2.p3.2.m2.1.1.2.2.cmml" xref="S3.SS2.p3.2.m2.1.1.2.2">𝛼</ci><ci id="S3.SS2.p3.2.m2.1.1.2.3.cmml" xref="S3.SS2.p3.2.m2.1.1.2.3">ℤ</ci></apply><apply id="S3.SS2.p3.2.m2.1.1.3.cmml" xref="S3.SS2.p3.2.m2.1.1.3"><ci id="S3.SS2.p3.2.m2.1.1.3.1.cmml" xref="S3.SS2.p3.2.m2.1.1.3.1">→</ci><apply id="S3.SS2.p3.2.m2.1.1.3.2.cmml" xref="S3.SS2.p3.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.p3.2.m2.1.1.3.2.1.cmml" xref="S3.SS2.p3.2.m2.1.1.3.2">superscript</csymbol><ci id="S3.SS2.p3.2.m2.1.1.3.2.2.cmml" xref="S3.SS2.p3.2.m2.1.1.3.2.2">𝒜</ci><ci id="S3.SS2.p3.2.m2.1.1.3.2.3.cmml" xref="S3.SS2.p3.2.m2.1.1.3.2.3">ℤ</ci></apply><apply id="S3.SS2.p3.2.m2.1.1.3.3.cmml" xref="S3.SS2.p3.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.p3.2.m2.1.1.3.3.1.cmml" xref="S3.SS2.p3.2.m2.1.1.3.3">superscript</csymbol><ci id="S3.SS2.p3.2.m2.1.1.3.3.2.cmml" xref="S3.SS2.p3.2.m2.1.1.3.3.2">ℬ</ci><ci id="S3.SS2.p3.2.m2.1.1.3.3.3.cmml" xref="S3.SS2.p3.2.m2.1.1.3.3.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m2.1c">\alpha^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m2.1d">italic_α start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT : caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, are injective and/or surjective if and only if <math alttext="\alpha A" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m3.1"><semantics id="S3.SS2.p3.3.m3.1a"><mrow id="S3.SS2.p3.3.m3.1.1" xref="S3.SS2.p3.3.m3.1.1.cmml"><mi id="S3.SS2.p3.3.m3.1.1.2" xref="S3.SS2.p3.3.m3.1.1.2.cmml">α</mi><mo id="S3.SS2.p3.3.m3.1.1.1" xref="S3.SS2.p3.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS2.p3.3.m3.1.1.3" xref="S3.SS2.p3.3.m3.1.1.3.cmml">A</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m3.1b"><apply id="S3.SS2.p3.3.m3.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1"><times id="S3.SS2.p3.3.m3.1.1.1.cmml" xref="S3.SS2.p3.3.m3.1.1.1"></times><ci id="S3.SS2.p3.3.m3.1.1.2.cmml" xref="S3.SS2.p3.3.m3.1.1.2">𝛼</ci><ci id="S3.SS2.p3.3.m3.1.1.3.cmml" xref="S3.SS2.p3.3.m3.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m3.1c">\alpha A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m3.1d">italic_α italic_A</annotation></semantics></math> is injective and/or surjective. Furthermore, <math alttext="\alpha^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m4.1"><semantics id="S3.SS2.p3.4.m4.1a"><msup id="S3.SS2.p3.4.m4.1.1" xref="S3.SS2.p3.4.m4.1.1.cmml"><mi id="S3.SS2.p3.4.m4.1.1.2" xref="S3.SS2.p3.4.m4.1.1.2.cmml">α</mi><mi id="S3.SS2.p3.4.m4.1.1.3" xref="S3.SS2.p3.4.m4.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m4.1b"><apply id="S3.SS2.p3.4.m4.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.4.m4.1.1.1.cmml" xref="S3.SS2.p3.4.m4.1.1">superscript</csymbol><ci id="S3.SS2.p3.4.m4.1.1.2.cmml" xref="S3.SS2.p3.4.m4.1.1.2">𝛼</ci><ci id="S3.SS2.p3.4.m4.1.1.3.cmml" xref="S3.SS2.p3.4.m4.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m4.1c">\alpha^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m4.1d">italic_α start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> commutes with the shift maps (both denoted by <math alttext="T" class="ltx_Math" display="inline" id="S3.SS2.p3.5.m5.1"><semantics id="S3.SS2.p3.5.m5.1a"><mi id="S3.SS2.p3.5.m5.1.1" xref="S3.SS2.p3.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.5.m5.1b"><ci id="S3.SS2.p3.5.m5.1.1.cmml" xref="S3.SS2.p3.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.5.m5.1d">italic_T</annotation></semantics></math>) on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.SS2.p3.6.m6.1"><semantics id="S3.SS2.p3.6.m6.1a"><msup id="S3.SS2.p3.6.m6.1.1" xref="S3.SS2.p3.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.6.m6.1.1.2" xref="S3.SS2.p3.6.m6.1.1.2.cmml">𝒜</mi><mi id="S3.SS2.p3.6.m6.1.1.3" xref="S3.SS2.p3.6.m6.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.6.m6.1b"><apply id="S3.SS2.p3.6.m6.1.1.cmml" xref="S3.SS2.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.6.m6.1.1.1.cmml" xref="S3.SS2.p3.6.m6.1.1">superscript</csymbol><ci id="S3.SS2.p3.6.m6.1.1.2.cmml" xref="S3.SS2.p3.6.m6.1.1.2">𝒜</ci><ci id="S3.SS2.p3.6.m6.1.1.3.cmml" xref="S3.SS2.p3.6.m6.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.6.m6.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.6.m6.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\cal B^{\mathbb{Z}}\," class="ltx_Math" display="inline" id="S3.SS2.p3.7.m7.1"><semantics id="S3.SS2.p3.7.m7.1a"><msup id="S3.SS2.p3.7.m7.1.1" xref="S3.SS2.p3.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.7.m7.1.1.2" xref="S3.SS2.p3.7.m7.1.1.2.cmml">ℬ</mi><mi id="S3.SS2.p3.7.m7.1.1.3" xref="S3.SS2.p3.7.m7.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.7.m7.1b"><apply id="S3.SS2.p3.7.m7.1.1.cmml" xref="S3.SS2.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.7.m7.1.1.1.cmml" xref="S3.SS2.p3.7.m7.1.1">superscript</csymbol><ci id="S3.SS2.p3.7.m7.1.1.2.cmml" xref="S3.SS2.p3.7.m7.1.1.2">ℬ</ci><ci id="S3.SS2.p3.7.m7.1.1.3.cmml" xref="S3.SS2.p3.7.m7.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.7.m7.1c">\cal B^{\mathbb{Z}}\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.7.m7.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, thus giving</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="T\circ\alpha=\alpha\circ T\,." class="ltx_Math" display="block" id="S3.Ex3.m1.1"><semantics id="S3.Ex3.m1.1a"><mrow id="S3.Ex3.m1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.cmml"><mrow id="S3.Ex3.m1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.cmml"><mrow id="S3.Ex3.m1.1.1.1.1.2" xref="S3.Ex3.m1.1.1.1.1.2.cmml"><mi id="S3.Ex3.m1.1.1.1.1.2.2" xref="S3.Ex3.m1.1.1.1.1.2.2.cmml">T</mi><mo id="S3.Ex3.m1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Ex3.m1.1.1.1.1.2.1.cmml">∘</mo><mi id="S3.Ex3.m1.1.1.1.1.2.3" xref="S3.Ex3.m1.1.1.1.1.2.3.cmml">α</mi></mrow><mo id="S3.Ex3.m1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.cmml">=</mo><mrow id="S3.Ex3.m1.1.1.1.1.3" xref="S3.Ex3.m1.1.1.1.1.3.cmml"><mi id="S3.Ex3.m1.1.1.1.1.3.2" xref="S3.Ex3.m1.1.1.1.1.3.2.cmml">α</mi><mo id="S3.Ex3.m1.1.1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.Ex3.m1.1.1.1.1.3.1.cmml">∘</mo><mi id="S3.Ex3.m1.1.1.1.1.3.3" xref="S3.Ex3.m1.1.1.1.1.3.3.cmml">T</mi></mrow></mrow><mo id="S3.Ex3.m1.1.1.1.2" lspace="0.170em" xref="S3.Ex3.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex3.m1.1b"><apply id="S3.Ex3.m1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1"><eq id="S3.Ex3.m1.1.1.1.1.1.cmml" xref="S3.Ex3.m1.1.1.1.1.1"></eq><apply id="S3.Ex3.m1.1.1.1.1.2.cmml" xref="S3.Ex3.m1.1.1.1.1.2"><compose id="S3.Ex3.m1.1.1.1.1.2.1.cmml" xref="S3.Ex3.m1.1.1.1.1.2.1"></compose><ci id="S3.Ex3.m1.1.1.1.1.2.2.cmml" xref="S3.Ex3.m1.1.1.1.1.2.2">𝑇</ci><ci id="S3.Ex3.m1.1.1.1.1.2.3.cmml" xref="S3.Ex3.m1.1.1.1.1.2.3">𝛼</ci></apply><apply id="S3.Ex3.m1.1.1.1.1.3.cmml" xref="S3.Ex3.m1.1.1.1.1.3"><compose id="S3.Ex3.m1.1.1.1.1.3.1.cmml" xref="S3.Ex3.m1.1.1.1.1.3.1"></compose><ci id="S3.Ex3.m1.1.1.1.1.3.2.cmml" xref="S3.Ex3.m1.1.1.1.1.3.2">𝛼</ci><ci id="S3.Ex3.m1.1.1.1.1.3.3.cmml" xref="S3.Ex3.m1.1.1.1.1.3.3">𝑇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex3.m1.1c">T\circ\alpha=\alpha\circ T\,.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex3.m1.1d">italic_T ∘ italic_α = italic_α ∘ italic_T .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p3.18">As a consequence, the image <math alttext="\alpha^{\mathbb{Z}}(X)" class="ltx_Math" display="inline" id="S3.SS2.p3.8.m1.1"><semantics id="S3.SS2.p3.8.m1.1a"><mrow id="S3.SS2.p3.8.m1.1.2" xref="S3.SS2.p3.8.m1.1.2.cmml"><msup id="S3.SS2.p3.8.m1.1.2.2" xref="S3.SS2.p3.8.m1.1.2.2.cmml"><mi id="S3.SS2.p3.8.m1.1.2.2.2" xref="S3.SS2.p3.8.m1.1.2.2.2.cmml">α</mi><mi id="S3.SS2.p3.8.m1.1.2.2.3" xref="S3.SS2.p3.8.m1.1.2.2.3.cmml">ℤ</mi></msup><mo id="S3.SS2.p3.8.m1.1.2.1" xref="S3.SS2.p3.8.m1.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p3.8.m1.1.2.3.2" xref="S3.SS2.p3.8.m1.1.2.cmml"><mo id="S3.SS2.p3.8.m1.1.2.3.2.1" stretchy="false" xref="S3.SS2.p3.8.m1.1.2.cmml">(</mo><mi id="S3.SS2.p3.8.m1.1.1" xref="S3.SS2.p3.8.m1.1.1.cmml">X</mi><mo id="S3.SS2.p3.8.m1.1.2.3.2.2" stretchy="false" xref="S3.SS2.p3.8.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.8.m1.1b"><apply id="S3.SS2.p3.8.m1.1.2.cmml" xref="S3.SS2.p3.8.m1.1.2"><times id="S3.SS2.p3.8.m1.1.2.1.cmml" xref="S3.SS2.p3.8.m1.1.2.1"></times><apply id="S3.SS2.p3.8.m1.1.2.2.cmml" xref="S3.SS2.p3.8.m1.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.p3.8.m1.1.2.2.1.cmml" xref="S3.SS2.p3.8.m1.1.2.2">superscript</csymbol><ci id="S3.SS2.p3.8.m1.1.2.2.2.cmml" xref="S3.SS2.p3.8.m1.1.2.2.2">𝛼</ci><ci id="S3.SS2.p3.8.m1.1.2.2.3.cmml" xref="S3.SS2.p3.8.m1.1.2.2.3">ℤ</ci></apply><ci id="S3.SS2.p3.8.m1.1.1.cmml" xref="S3.SS2.p3.8.m1.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.8.m1.1c">\alpha^{\mathbb{Z}}(X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.8.m1.1d">italic_α start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> of any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.SS2.p3.9.m2.1"><semantics id="S3.SS2.p3.9.m2.1a"><mrow id="S3.SS2.p3.9.m2.1.1" xref="S3.SS2.p3.9.m2.1.1.cmml"><mi id="S3.SS2.p3.9.m2.1.1.2" xref="S3.SS2.p3.9.m2.1.1.2.cmml">X</mi><mo id="S3.SS2.p3.9.m2.1.1.1" xref="S3.SS2.p3.9.m2.1.1.1.cmml">⊆</mo><msup id="S3.SS2.p3.9.m2.1.1.3" xref="S3.SS2.p3.9.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.9.m2.1.1.3.2" xref="S3.SS2.p3.9.m2.1.1.3.2.cmml">𝒜</mi><mi id="S3.SS2.p3.9.m2.1.1.3.3" xref="S3.SS2.p3.9.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.9.m2.1b"><apply id="S3.SS2.p3.9.m2.1.1.cmml" xref="S3.SS2.p3.9.m2.1.1"><subset id="S3.SS2.p3.9.m2.1.1.1.cmml" xref="S3.SS2.p3.9.m2.1.1.1"></subset><ci id="S3.SS2.p3.9.m2.1.1.2.cmml" xref="S3.SS2.p3.9.m2.1.1.2">𝑋</ci><apply id="S3.SS2.p3.9.m2.1.1.3.cmml" xref="S3.SS2.p3.9.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p3.9.m2.1.1.3.1.cmml" xref="S3.SS2.p3.9.m2.1.1.3">superscript</csymbol><ci id="S3.SS2.p3.9.m2.1.1.3.2.cmml" xref="S3.SS2.p3.9.m2.1.1.3.2">𝒜</ci><ci id="S3.SS2.p3.9.m2.1.1.3.3.cmml" xref="S3.SS2.p3.9.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.9.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.9.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is equal to the image subshift <math alttext="\alpha^{\Sigma}(X)" class="ltx_Math" display="inline" id="S3.SS2.p3.10.m3.1"><semantics id="S3.SS2.p3.10.m3.1a"><mrow id="S3.SS2.p3.10.m3.1.2" xref="S3.SS2.p3.10.m3.1.2.cmml"><msup id="S3.SS2.p3.10.m3.1.2.2" xref="S3.SS2.p3.10.m3.1.2.2.cmml"><mi id="S3.SS2.p3.10.m3.1.2.2.2" xref="S3.SS2.p3.10.m3.1.2.2.2.cmml">α</mi><mi id="S3.SS2.p3.10.m3.1.2.2.3" mathvariant="normal" xref="S3.SS2.p3.10.m3.1.2.2.3.cmml">Σ</mi></msup><mo id="S3.SS2.p3.10.m3.1.2.1" xref="S3.SS2.p3.10.m3.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p3.10.m3.1.2.3.2" xref="S3.SS2.p3.10.m3.1.2.cmml"><mo id="S3.SS2.p3.10.m3.1.2.3.2.1" stretchy="false" xref="S3.SS2.p3.10.m3.1.2.cmml">(</mo><mi id="S3.SS2.p3.10.m3.1.1" xref="S3.SS2.p3.10.m3.1.1.cmml">X</mi><mo id="S3.SS2.p3.10.m3.1.2.3.2.2" stretchy="false" xref="S3.SS2.p3.10.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.10.m3.1b"><apply id="S3.SS2.p3.10.m3.1.2.cmml" xref="S3.SS2.p3.10.m3.1.2"><times id="S3.SS2.p3.10.m3.1.2.1.cmml" xref="S3.SS2.p3.10.m3.1.2.1"></times><apply id="S3.SS2.p3.10.m3.1.2.2.cmml" xref="S3.SS2.p3.10.m3.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.p3.10.m3.1.2.2.1.cmml" xref="S3.SS2.p3.10.m3.1.2.2">superscript</csymbol><ci id="S3.SS2.p3.10.m3.1.2.2.2.cmml" xref="S3.SS2.p3.10.m3.1.2.2.2">𝛼</ci><ci id="S3.SS2.p3.10.m3.1.2.2.3.cmml" xref="S3.SS2.p3.10.m3.1.2.2.3">Σ</ci></apply><ci id="S3.SS2.p3.10.m3.1.1.cmml" xref="S3.SS2.p3.10.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.10.m3.1c">\alpha^{\Sigma}(X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.10.m3.1d">italic_α start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> over <math alttext="\cal B" class="ltx_Math" display="inline" id="S3.SS2.p3.11.m4.1"><semantics id="S3.SS2.p3.11.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.11.m4.1.1" xref="S3.SS2.p3.11.m4.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.11.m4.1b"><ci id="S3.SS2.p3.11.m4.1.1.cmml" xref="S3.SS2.p3.11.m4.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.11.m4.1c">\cal B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.11.m4.1d">caligraphic_B</annotation></semantics></math>. Furthermore, for any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS2.p3.12.m5.1"><semantics id="S3.SS2.p3.12.m5.1a"><mi id="S3.SS2.p3.12.m5.1.1" xref="S3.SS2.p3.12.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.12.m5.1b"><ci id="S3.SS2.p3.12.m5.1.1.cmml" xref="S3.SS2.p3.12.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.12.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.12.m5.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.SS2.p3.13.m6.1"><semantics id="S3.SS2.p3.13.m6.1a"><msup id="S3.SS2.p3.13.m6.1.1" xref="S3.SS2.p3.13.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.13.m6.1.1.2" xref="S3.SS2.p3.13.m6.1.1.2.cmml">𝒜</mi><mi id="S3.SS2.p3.13.m6.1.1.3" xref="S3.SS2.p3.13.m6.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.13.m6.1b"><apply id="S3.SS2.p3.13.m6.1.1.cmml" xref="S3.SS2.p3.13.m6.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.13.m6.1.1.1.cmml" xref="S3.SS2.p3.13.m6.1.1">superscript</csymbol><ci id="S3.SS2.p3.13.m6.1.1.2.cmml" xref="S3.SS2.p3.13.m6.1.1.2">𝒜</ci><ci id="S3.SS2.p3.13.m6.1.1.3.cmml" xref="S3.SS2.p3.13.m6.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.13.m6.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.13.m6.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> the classical push-forward measure <math alttext="\alpha_{*}(\mu)" class="ltx_Math" display="inline" id="S3.SS2.p3.14.m7.1"><semantics id="S3.SS2.p3.14.m7.1a"><mrow id="S3.SS2.p3.14.m7.1.2" xref="S3.SS2.p3.14.m7.1.2.cmml"><msub id="S3.SS2.p3.14.m7.1.2.2" xref="S3.SS2.p3.14.m7.1.2.2.cmml"><mi id="S3.SS2.p3.14.m7.1.2.2.2" xref="S3.SS2.p3.14.m7.1.2.2.2.cmml">α</mi><mo id="S3.SS2.p3.14.m7.1.2.2.3" xref="S3.SS2.p3.14.m7.1.2.2.3.cmml">∗</mo></msub><mo id="S3.SS2.p3.14.m7.1.2.1" xref="S3.SS2.p3.14.m7.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p3.14.m7.1.2.3.2" xref="S3.SS2.p3.14.m7.1.2.cmml"><mo id="S3.SS2.p3.14.m7.1.2.3.2.1" stretchy="false" xref="S3.SS2.p3.14.m7.1.2.cmml">(</mo><mi id="S3.SS2.p3.14.m7.1.1" xref="S3.SS2.p3.14.m7.1.1.cmml">μ</mi><mo id="S3.SS2.p3.14.m7.1.2.3.2.2" stretchy="false" xref="S3.SS2.p3.14.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.14.m7.1b"><apply id="S3.SS2.p3.14.m7.1.2.cmml" xref="S3.SS2.p3.14.m7.1.2"><times id="S3.SS2.p3.14.m7.1.2.1.cmml" xref="S3.SS2.p3.14.m7.1.2.1"></times><apply id="S3.SS2.p3.14.m7.1.2.2.cmml" xref="S3.SS2.p3.14.m7.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.p3.14.m7.1.2.2.1.cmml" xref="S3.SS2.p3.14.m7.1.2.2">subscript</csymbol><ci id="S3.SS2.p3.14.m7.1.2.2.2.cmml" xref="S3.SS2.p3.14.m7.1.2.2.2">𝛼</ci><times id="S3.SS2.p3.14.m7.1.2.2.3.cmml" xref="S3.SS2.p3.14.m7.1.2.2.3"></times></apply><ci id="S3.SS2.p3.14.m7.1.1.cmml" xref="S3.SS2.p3.14.m7.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.14.m7.1c">\alpha_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.14.m7.1d">italic_α start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math> is an invariant measure on <math alttext="\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.SS2.p3.15.m8.1"><semantics id="S3.SS2.p3.15.m8.1a"><msup id="S3.SS2.p3.15.m8.1.1" xref="S3.SS2.p3.15.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p3.15.m8.1.1.2" xref="S3.SS2.p3.15.m8.1.1.2.cmml">ℬ</mi><mi id="S3.SS2.p3.15.m8.1.1.3" xref="S3.SS2.p3.15.m8.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.15.m8.1b"><apply id="S3.SS2.p3.15.m8.1.1.cmml" xref="S3.SS2.p3.15.m8.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.15.m8.1.1.1.cmml" xref="S3.SS2.p3.15.m8.1.1">superscript</csymbol><ci id="S3.SS2.p3.15.m8.1.1.2.cmml" xref="S3.SS2.p3.15.m8.1.1.2">ℬ</ci><ci id="S3.SS2.p3.15.m8.1.1.3.cmml" xref="S3.SS2.p3.15.m8.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.15.m8.1c">\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.15.m8.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, with support equal to <math alttext="\alpha^{\mathbb{Z}}(\mbox{Supp}(\mu))" class="ltx_Math" display="inline" id="S3.SS2.p3.16.m9.2"><semantics id="S3.SS2.p3.16.m9.2a"><mrow id="S3.SS2.p3.16.m9.2.2" xref="S3.SS2.p3.16.m9.2.2.cmml"><msup id="S3.SS2.p3.16.m9.2.2.3" xref="S3.SS2.p3.16.m9.2.2.3.cmml"><mi id="S3.SS2.p3.16.m9.2.2.3.2" xref="S3.SS2.p3.16.m9.2.2.3.2.cmml">α</mi><mi id="S3.SS2.p3.16.m9.2.2.3.3" xref="S3.SS2.p3.16.m9.2.2.3.3.cmml">ℤ</mi></msup><mo id="S3.SS2.p3.16.m9.2.2.2" xref="S3.SS2.p3.16.m9.2.2.2.cmml">⁢</mo><mrow id="S3.SS2.p3.16.m9.2.2.1.1" xref="S3.SS2.p3.16.m9.2.2.1.1.1.cmml"><mo id="S3.SS2.p3.16.m9.2.2.1.1.2" stretchy="false" xref="S3.SS2.p3.16.m9.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS2.p3.16.m9.2.2.1.1.1" xref="S3.SS2.p3.16.m9.2.2.1.1.1.cmml"><mtext id="S3.SS2.p3.16.m9.2.2.1.1.1.2" xref="S3.SS2.p3.16.m9.2.2.1.1.1.2a.cmml">Supp</mtext><mo id="S3.SS2.p3.16.m9.2.2.1.1.1.1" xref="S3.SS2.p3.16.m9.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS2.p3.16.m9.2.2.1.1.1.3.2" xref="S3.SS2.p3.16.m9.2.2.1.1.1.cmml"><mo id="S3.SS2.p3.16.m9.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.SS2.p3.16.m9.2.2.1.1.1.cmml">(</mo><mi id="S3.SS2.p3.16.m9.1.1" xref="S3.SS2.p3.16.m9.1.1.cmml">μ</mi><mo id="S3.SS2.p3.16.m9.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.SS2.p3.16.m9.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p3.16.m9.2.2.1.1.3" stretchy="false" xref="S3.SS2.p3.16.m9.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.16.m9.2b"><apply id="S3.SS2.p3.16.m9.2.2.cmml" xref="S3.SS2.p3.16.m9.2.2"><times id="S3.SS2.p3.16.m9.2.2.2.cmml" xref="S3.SS2.p3.16.m9.2.2.2"></times><apply id="S3.SS2.p3.16.m9.2.2.3.cmml" xref="S3.SS2.p3.16.m9.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p3.16.m9.2.2.3.1.cmml" xref="S3.SS2.p3.16.m9.2.2.3">superscript</csymbol><ci id="S3.SS2.p3.16.m9.2.2.3.2.cmml" xref="S3.SS2.p3.16.m9.2.2.3.2">𝛼</ci><ci id="S3.SS2.p3.16.m9.2.2.3.3.cmml" xref="S3.SS2.p3.16.m9.2.2.3.3">ℤ</ci></apply><apply id="S3.SS2.p3.16.m9.2.2.1.1.1.cmml" xref="S3.SS2.p3.16.m9.2.2.1.1"><times id="S3.SS2.p3.16.m9.2.2.1.1.1.1.cmml" xref="S3.SS2.p3.16.m9.2.2.1.1.1.1"></times><ci id="S3.SS2.p3.16.m9.2.2.1.1.1.2a.cmml" xref="S3.SS2.p3.16.m9.2.2.1.1.1.2"><mtext id="S3.SS2.p3.16.m9.2.2.1.1.1.2.cmml" xref="S3.SS2.p3.16.m9.2.2.1.1.1.2">Supp</mtext></ci><ci id="S3.SS2.p3.16.m9.1.1.cmml" xref="S3.SS2.p3.16.m9.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.16.m9.2c">\alpha^{\mathbb{Z}}(\mbox{Supp}(\mu))</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.16.m9.2d">italic_α start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( Supp ( italic_μ ) )</annotation></semantics></math>. In particular, if <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS2.p3.17.m10.1"><semantics id="S3.SS2.p3.17.m10.1a"><mi id="S3.SS2.p3.17.m10.1.1" xref="S3.SS2.p3.17.m10.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.17.m10.1b"><ci id="S3.SS2.p3.17.m10.1.1.cmml" xref="S3.SS2.p3.17.m10.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.17.m10.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.17.m10.1d">italic_μ</annotation></semantics></math> is a probability measure, then so is <math alttext="\alpha_{*}(\mu)" class="ltx_Math" display="inline" id="S3.SS2.p3.18.m11.1"><semantics id="S3.SS2.p3.18.m11.1a"><mrow id="S3.SS2.p3.18.m11.1.2" xref="S3.SS2.p3.18.m11.1.2.cmml"><msub id="S3.SS2.p3.18.m11.1.2.2" xref="S3.SS2.p3.18.m11.1.2.2.cmml"><mi id="S3.SS2.p3.18.m11.1.2.2.2" xref="S3.SS2.p3.18.m11.1.2.2.2.cmml">α</mi><mo id="S3.SS2.p3.18.m11.1.2.2.3" xref="S3.SS2.p3.18.m11.1.2.2.3.cmml">∗</mo></msub><mo id="S3.SS2.p3.18.m11.1.2.1" xref="S3.SS2.p3.18.m11.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p3.18.m11.1.2.3.2" xref="S3.SS2.p3.18.m11.1.2.cmml"><mo id="S3.SS2.p3.18.m11.1.2.3.2.1" stretchy="false" xref="S3.SS2.p3.18.m11.1.2.cmml">(</mo><mi id="S3.SS2.p3.18.m11.1.1" xref="S3.SS2.p3.18.m11.1.1.cmml">μ</mi><mo id="S3.SS2.p3.18.m11.1.2.3.2.2" stretchy="false" xref="S3.SS2.p3.18.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.18.m11.1b"><apply id="S3.SS2.p3.18.m11.1.2.cmml" xref="S3.SS2.p3.18.m11.1.2"><times id="S3.SS2.p3.18.m11.1.2.1.cmml" xref="S3.SS2.p3.18.m11.1.2.1"></times><apply id="S3.SS2.p3.18.m11.1.2.2.cmml" xref="S3.SS2.p3.18.m11.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.p3.18.m11.1.2.2.1.cmml" xref="S3.SS2.p3.18.m11.1.2.2">subscript</csymbol><ci id="S3.SS2.p3.18.m11.1.2.2.2.cmml" xref="S3.SS2.p3.18.m11.1.2.2.2">𝛼</ci><times id="S3.SS2.p3.18.m11.1.2.2.3.cmml" xref="S3.SS2.p3.18.m11.1.2.2.3"></times></apply><ci id="S3.SS2.p3.18.m11.1.1.cmml" xref="S3.SS2.p3.18.m11.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.18.m11.1c">\alpha_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.18.m11.1d">italic_α start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.5">According to the defining equation for the push-forward measure, <math alttext="\mu_{*}(S):=\mu(f^{-1}(S))" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.3"><semantics id="S3.SS2.p4.1.m1.3a"><mrow id="S3.SS2.p4.1.m1.3.3" xref="S3.SS2.p4.1.m1.3.3.cmml"><mrow id="S3.SS2.p4.1.m1.3.3.3" xref="S3.SS2.p4.1.m1.3.3.3.cmml"><msub id="S3.SS2.p4.1.m1.3.3.3.2" xref="S3.SS2.p4.1.m1.3.3.3.2.cmml"><mi id="S3.SS2.p4.1.m1.3.3.3.2.2" xref="S3.SS2.p4.1.m1.3.3.3.2.2.cmml">μ</mi><mo id="S3.SS2.p4.1.m1.3.3.3.2.3" xref="S3.SS2.p4.1.m1.3.3.3.2.3.cmml">∗</mo></msub><mo id="S3.SS2.p4.1.m1.3.3.3.1" xref="S3.SS2.p4.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS2.p4.1.m1.3.3.3.3.2" xref="S3.SS2.p4.1.m1.3.3.3.cmml"><mo id="S3.SS2.p4.1.m1.3.3.3.3.2.1" stretchy="false" xref="S3.SS2.p4.1.m1.3.3.3.cmml">(</mo><mi id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml">S</mi><mo id="S3.SS2.p4.1.m1.3.3.3.3.2.2" rspace="0.278em" stretchy="false" xref="S3.SS2.p4.1.m1.3.3.3.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p4.1.m1.3.3.2" rspace="0.278em" xref="S3.SS2.p4.1.m1.3.3.2.cmml">:=</mo><mrow id="S3.SS2.p4.1.m1.3.3.1" xref="S3.SS2.p4.1.m1.3.3.1.cmml"><mi id="S3.SS2.p4.1.m1.3.3.1.3" xref="S3.SS2.p4.1.m1.3.3.1.3.cmml">μ</mi><mo id="S3.SS2.p4.1.m1.3.3.1.2" xref="S3.SS2.p4.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS2.p4.1.m1.3.3.1.1.1" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.cmml"><mo id="S3.SS2.p4.1.m1.3.3.1.1.1.2" stretchy="false" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p4.1.m1.3.3.1.1.1.1" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.cmml"><msup id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.cmml"><mi id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.2" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.2.cmml">f</mi><mrow id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3.cmml"><mo id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3a" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3.cmml">−</mo><mn id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3.2" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.SS2.p4.1.m1.3.3.1.1.1.1.1" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS2.p4.1.m1.3.3.1.1.1.1.3.2" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.cmml"><mo id="S3.SS2.p4.1.m1.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.cmml">(</mo><mi id="S3.SS2.p4.1.m1.2.2" xref="S3.SS2.p4.1.m1.2.2.cmml">S</mi><mo id="S3.SS2.p4.1.m1.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS2.p4.1.m1.3.3.1.1.1.3" stretchy="false" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.3b"><apply id="S3.SS2.p4.1.m1.3.3.cmml" xref="S3.SS2.p4.1.m1.3.3"><csymbol cd="latexml" id="S3.SS2.p4.1.m1.3.3.2.cmml" xref="S3.SS2.p4.1.m1.3.3.2">assign</csymbol><apply id="S3.SS2.p4.1.m1.3.3.3.cmml" xref="S3.SS2.p4.1.m1.3.3.3"><times id="S3.SS2.p4.1.m1.3.3.3.1.cmml" xref="S3.SS2.p4.1.m1.3.3.3.1"></times><apply id="S3.SS2.p4.1.m1.3.3.3.2.cmml" xref="S3.SS2.p4.1.m1.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.p4.1.m1.3.3.3.2.1.cmml" xref="S3.SS2.p4.1.m1.3.3.3.2">subscript</csymbol><ci id="S3.SS2.p4.1.m1.3.3.3.2.2.cmml" xref="S3.SS2.p4.1.m1.3.3.3.2.2">𝜇</ci><times id="S3.SS2.p4.1.m1.3.3.3.2.3.cmml" xref="S3.SS2.p4.1.m1.3.3.3.2.3"></times></apply><ci id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1">𝑆</ci></apply><apply id="S3.SS2.p4.1.m1.3.3.1.cmml" xref="S3.SS2.p4.1.m1.3.3.1"><times id="S3.SS2.p4.1.m1.3.3.1.2.cmml" xref="S3.SS2.p4.1.m1.3.3.1.2"></times><ci id="S3.SS2.p4.1.m1.3.3.1.3.cmml" xref="S3.SS2.p4.1.m1.3.3.1.3">𝜇</ci><apply id="S3.SS2.p4.1.m1.3.3.1.1.1.1.cmml" xref="S3.SS2.p4.1.m1.3.3.1.1.1"><times id="S3.SS2.p4.1.m1.3.3.1.1.1.1.1.cmml" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.1"></times><apply id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.cmml" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.1.cmml" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2">superscript</csymbol><ci id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.2.cmml" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.2">𝑓</ci><apply id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3.cmml" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3"><minus id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3.1.cmml" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3"></minus><cn id="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3.2.cmml" type="integer" xref="S3.SS2.p4.1.m1.3.3.1.1.1.1.2.3.2">1</cn></apply></apply><ci id="S3.SS2.p4.1.m1.2.2.cmml" xref="S3.SS2.p4.1.m1.2.2">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.3c">\mu_{*}(S):=\mu(f^{-1}(S))</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.3d">italic_μ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_S ) := italic_μ ( italic_f start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_S ) )</annotation></semantics></math> for any measurable set <math alttext="S" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m2.1"><semantics id="S3.SS2.p4.2.m2.1a"><mi id="S3.SS2.p4.2.m2.1.1" xref="S3.SS2.p4.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m2.1b"><ci id="S3.SS2.p4.2.m2.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.1d">italic_S</annotation></semantics></math> in the range of any measurable map <math alttext="f" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m3.1"><semantics id="S3.SS2.p4.3.m3.1a"><mi id="S3.SS2.p4.3.m3.1.1" xref="S3.SS2.p4.3.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m3.1b"><ci id="S3.SS2.p4.3.m3.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m3.1d">italic_f</annotation></semantics></math>, we obtain for the weight function associated to <math alttext="\alpha_{*}(\mu)" class="ltx_Math" display="inline" id="S3.SS2.p4.4.m4.1"><semantics id="S3.SS2.p4.4.m4.1a"><mrow id="S3.SS2.p4.4.m4.1.2" xref="S3.SS2.p4.4.m4.1.2.cmml"><msub id="S3.SS2.p4.4.m4.1.2.2" xref="S3.SS2.p4.4.m4.1.2.2.cmml"><mi id="S3.SS2.p4.4.m4.1.2.2.2" xref="S3.SS2.p4.4.m4.1.2.2.2.cmml">α</mi><mo id="S3.SS2.p4.4.m4.1.2.2.3" xref="S3.SS2.p4.4.m4.1.2.2.3.cmml">∗</mo></msub><mo id="S3.SS2.p4.4.m4.1.2.1" xref="S3.SS2.p4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.SS2.p4.4.m4.1.2.3.2" xref="S3.SS2.p4.4.m4.1.2.cmml"><mo id="S3.SS2.p4.4.m4.1.2.3.2.1" stretchy="false" xref="S3.SS2.p4.4.m4.1.2.cmml">(</mo><mi id="S3.SS2.p4.4.m4.1.1" xref="S3.SS2.p4.4.m4.1.1.cmml">μ</mi><mo id="S3.SS2.p4.4.m4.1.2.3.2.2" stretchy="false" xref="S3.SS2.p4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.4.m4.1b"><apply id="S3.SS2.p4.4.m4.1.2.cmml" xref="S3.SS2.p4.4.m4.1.2"><times id="S3.SS2.p4.4.m4.1.2.1.cmml" xref="S3.SS2.p4.4.m4.1.2.1"></times><apply id="S3.SS2.p4.4.m4.1.2.2.cmml" xref="S3.SS2.p4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.p4.4.m4.1.2.2.1.cmml" xref="S3.SS2.p4.4.m4.1.2.2">subscript</csymbol><ci id="S3.SS2.p4.4.m4.1.2.2.2.cmml" xref="S3.SS2.p4.4.m4.1.2.2.2">𝛼</ci><times id="S3.SS2.p4.4.m4.1.2.2.3.cmml" xref="S3.SS2.p4.4.m4.1.2.2.3"></times></apply><ci id="S3.SS2.p4.4.m4.1.1.cmml" xref="S3.SS2.p4.4.m4.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.4.m4.1c">\alpha_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.4.m4.1d">italic_α start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math> and for any <math alttext="w\in\cal B^{*}" class="ltx_Math" display="inline" id="S3.SS2.p4.5.m5.1"><semantics id="S3.SS2.p4.5.m5.1a"><mrow id="S3.SS2.p4.5.m5.1.1" xref="S3.SS2.p4.5.m5.1.1.cmml"><mi id="S3.SS2.p4.5.m5.1.1.2" xref="S3.SS2.p4.5.m5.1.1.2.cmml">w</mi><mo id="S3.SS2.p4.5.m5.1.1.1" xref="S3.SS2.p4.5.m5.1.1.1.cmml">∈</mo><msup id="S3.SS2.p4.5.m5.1.1.3" xref="S3.SS2.p4.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p4.5.m5.1.1.3.2" xref="S3.SS2.p4.5.m5.1.1.3.2.cmml">ℬ</mi><mo id="S3.SS2.p4.5.m5.1.1.3.3" xref="S3.SS2.p4.5.m5.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.5.m5.1b"><apply id="S3.SS2.p4.5.m5.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1"><in id="S3.SS2.p4.5.m5.1.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1.1"></in><ci id="S3.SS2.p4.5.m5.1.1.2.cmml" xref="S3.SS2.p4.5.m5.1.1.2">𝑤</ci><apply id="S3.SS2.p4.5.m5.1.1.3.cmml" xref="S3.SS2.p4.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p4.5.m5.1.1.3.1.cmml" xref="S3.SS2.p4.5.m5.1.1.3">superscript</csymbol><ci id="S3.SS2.p4.5.m5.1.1.3.2.cmml" xref="S3.SS2.p4.5.m5.1.1.3.2">ℬ</ci><times id="S3.SS2.p4.5.m5.1.1.3.3.cmml" xref="S3.SS2.p4.5.m5.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.5.m5.1c">w\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.5.m5.1d">italic_w ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the finite sum decomposition</p> <table class="ltx_equation ltx_eqn_table" id="S3.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(3.3)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\alpha_{*}(\mu)(w)=\sum_{u\,\in\,\alpha^{-1}(w)}\mu(u)\,." class="ltx_Math" display="block" id="S3.E3.m1.5"><semantics id="S3.E3.m1.5a"><mrow id="S3.E3.m1.5.5.1" xref="S3.E3.m1.5.5.1.1.cmml"><mrow id="S3.E3.m1.5.5.1.1" xref="S3.E3.m1.5.5.1.1.cmml"><mrow id="S3.E3.m1.5.5.1.1.2" xref="S3.E3.m1.5.5.1.1.2.cmml"><msub id="S3.E3.m1.5.5.1.1.2.2" xref="S3.E3.m1.5.5.1.1.2.2.cmml"><mi id="S3.E3.m1.5.5.1.1.2.2.2" xref="S3.E3.m1.5.5.1.1.2.2.2.cmml">α</mi><mo id="S3.E3.m1.5.5.1.1.2.2.3" xref="S3.E3.m1.5.5.1.1.2.2.3.cmml">∗</mo></msub><mo id="S3.E3.m1.5.5.1.1.2.1" xref="S3.E3.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S3.E3.m1.5.5.1.1.2.3.2" xref="S3.E3.m1.5.5.1.1.2.cmml"><mo id="S3.E3.m1.5.5.1.1.2.3.2.1" stretchy="false" xref="S3.E3.m1.5.5.1.1.2.cmml">(</mo><mi id="S3.E3.m1.2.2" xref="S3.E3.m1.2.2.cmml">μ</mi><mo id="S3.E3.m1.5.5.1.1.2.3.2.2" stretchy="false" xref="S3.E3.m1.5.5.1.1.2.cmml">)</mo></mrow><mo id="S3.E3.m1.5.5.1.1.2.1a" xref="S3.E3.m1.5.5.1.1.2.1.cmml">⁢</mo><mrow id="S3.E3.m1.5.5.1.1.2.4.2" xref="S3.E3.m1.5.5.1.1.2.cmml"><mo id="S3.E3.m1.5.5.1.1.2.4.2.1" stretchy="false" xref="S3.E3.m1.5.5.1.1.2.cmml">(</mo><mi id="S3.E3.m1.3.3" xref="S3.E3.m1.3.3.cmml">w</mi><mo id="S3.E3.m1.5.5.1.1.2.4.2.2" stretchy="false" xref="S3.E3.m1.5.5.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E3.m1.5.5.1.1.1" rspace="0.111em" xref="S3.E3.m1.5.5.1.1.1.cmml">=</mo><mrow id="S3.E3.m1.5.5.1.1.3" xref="S3.E3.m1.5.5.1.1.3.cmml"><munder id="S3.E3.m1.5.5.1.1.3.1" xref="S3.E3.m1.5.5.1.1.3.1.cmml"><mo id="S3.E3.m1.5.5.1.1.3.1.2" movablelimits="false" xref="S3.E3.m1.5.5.1.1.3.1.2.cmml">∑</mo><mrow id="S3.E3.m1.1.1.1" xref="S3.E3.m1.1.1.1.cmml"><mi id="S3.E3.m1.1.1.1.3" xref="S3.E3.m1.1.1.1.3.cmml">u</mi><mo id="S3.E3.m1.1.1.1.2" lspace="0.448em" rspace="0.448em" xref="S3.E3.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.E3.m1.1.1.1.4" xref="S3.E3.m1.1.1.1.4.cmml"><msup id="S3.E3.m1.1.1.1.4.2" 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xref="S3.E3.m1.1.1.1.4.2.3"><minus id="S3.E3.m1.1.1.1.4.2.3.1.cmml" xref="S3.E3.m1.1.1.1.4.2.3"></minus><cn id="S3.E3.m1.1.1.1.4.2.3.2.cmml" type="integer" xref="S3.E3.m1.1.1.1.4.2.3.2">1</cn></apply></apply><ci id="S3.E3.m1.1.1.1.1.cmml" xref="S3.E3.m1.1.1.1.1">𝑤</ci></apply></apply></apply><apply id="S3.E3.m1.5.5.1.1.3.2.cmml" xref="S3.E3.m1.5.5.1.1.3.2"><times id="S3.E3.m1.5.5.1.1.3.2.1.cmml" xref="S3.E3.m1.5.5.1.1.3.2.1"></times><ci id="S3.E3.m1.5.5.1.1.3.2.2.cmml" xref="S3.E3.m1.5.5.1.1.3.2.2">𝜇</ci><ci id="S3.E3.m1.4.4.cmml" xref="S3.E3.m1.4.4">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m1.5c">\alpha_{*}(\mu)(w)=\sum_{u\,\in\,\alpha^{-1}(w)}\mu(u)\,.</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.5d">italic_α start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( italic_w ) = ∑ start_POSTSUBSCRIPT italic_u ∈ italic_α start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w ) end_POSTSUBSCRIPT italic_μ ( italic_u ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p4.7">Note here that any <math alttext="u\in\alpha^{-1}(w)" class="ltx_Math" display="inline" id="S3.SS2.p4.6.m1.1"><semantics id="S3.SS2.p4.6.m1.1a"><mrow id="S3.SS2.p4.6.m1.1.2" xref="S3.SS2.p4.6.m1.1.2.cmml"><mi id="S3.SS2.p4.6.m1.1.2.2" xref="S3.SS2.p4.6.m1.1.2.2.cmml">u</mi><mo id="S3.SS2.p4.6.m1.1.2.1" xref="S3.SS2.p4.6.m1.1.2.1.cmml">∈</mo><mrow id="S3.SS2.p4.6.m1.1.2.3" xref="S3.SS2.p4.6.m1.1.2.3.cmml"><msup id="S3.SS2.p4.6.m1.1.2.3.2" xref="S3.SS2.p4.6.m1.1.2.3.2.cmml"><mi id="S3.SS2.p4.6.m1.1.2.3.2.2" xref="S3.SS2.p4.6.m1.1.2.3.2.2.cmml">α</mi><mrow id="S3.SS2.p4.6.m1.1.2.3.2.3" xref="S3.SS2.p4.6.m1.1.2.3.2.3.cmml"><mo id="S3.SS2.p4.6.m1.1.2.3.2.3a" xref="S3.SS2.p4.6.m1.1.2.3.2.3.cmml">−</mo><mn id="S3.SS2.p4.6.m1.1.2.3.2.3.2" xref="S3.SS2.p4.6.m1.1.2.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.SS2.p4.6.m1.1.2.3.1" xref="S3.SS2.p4.6.m1.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS2.p4.6.m1.1.2.3.3.2" xref="S3.SS2.p4.6.m1.1.2.3.cmml"><mo id="S3.SS2.p4.6.m1.1.2.3.3.2.1" stretchy="false" xref="S3.SS2.p4.6.m1.1.2.3.cmml">(</mo><mi id="S3.SS2.p4.6.m1.1.1" xref="S3.SS2.p4.6.m1.1.1.cmml">w</mi><mo id="S3.SS2.p4.6.m1.1.2.3.3.2.2" stretchy="false" xref="S3.SS2.p4.6.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.6.m1.1b"><apply id="S3.SS2.p4.6.m1.1.2.cmml" xref="S3.SS2.p4.6.m1.1.2"><in id="S3.SS2.p4.6.m1.1.2.1.cmml" xref="S3.SS2.p4.6.m1.1.2.1"></in><ci id="S3.SS2.p4.6.m1.1.2.2.cmml" xref="S3.SS2.p4.6.m1.1.2.2">𝑢</ci><apply id="S3.SS2.p4.6.m1.1.2.3.cmml" xref="S3.SS2.p4.6.m1.1.2.3"><times id="S3.SS2.p4.6.m1.1.2.3.1.cmml" xref="S3.SS2.p4.6.m1.1.2.3.1"></times><apply id="S3.SS2.p4.6.m1.1.2.3.2.cmml" xref="S3.SS2.p4.6.m1.1.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.p4.6.m1.1.2.3.2.1.cmml" xref="S3.SS2.p4.6.m1.1.2.3.2">superscript</csymbol><ci id="S3.SS2.p4.6.m1.1.2.3.2.2.cmml" xref="S3.SS2.p4.6.m1.1.2.3.2.2">𝛼</ci><apply id="S3.SS2.p4.6.m1.1.2.3.2.3.cmml" xref="S3.SS2.p4.6.m1.1.2.3.2.3"><minus id="S3.SS2.p4.6.m1.1.2.3.2.3.1.cmml" xref="S3.SS2.p4.6.m1.1.2.3.2.3"></minus><cn id="S3.SS2.p4.6.m1.1.2.3.2.3.2.cmml" type="integer" xref="S3.SS2.p4.6.m1.1.2.3.2.3.2">1</cn></apply></apply><ci id="S3.SS2.p4.6.m1.1.1.cmml" xref="S3.SS2.p4.6.m1.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.6.m1.1c">u\in\alpha^{-1}(w)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.6.m1.1d">italic_u ∈ italic_α start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w )</annotation></semantics></math> has length <math alttext="|u|=|w|" class="ltx_Math" display="inline" id="S3.SS2.p4.7.m2.2"><semantics id="S3.SS2.p4.7.m2.2a"><mrow id="S3.SS2.p4.7.m2.2.3" xref="S3.SS2.p4.7.m2.2.3.cmml"><mrow id="S3.SS2.p4.7.m2.2.3.2.2" xref="S3.SS2.p4.7.m2.2.3.2.1.cmml"><mo id="S3.SS2.p4.7.m2.2.3.2.2.1" stretchy="false" xref="S3.SS2.p4.7.m2.2.3.2.1.1.cmml">|</mo><mi id="S3.SS2.p4.7.m2.1.1" xref="S3.SS2.p4.7.m2.1.1.cmml">u</mi><mo id="S3.SS2.p4.7.m2.2.3.2.2.2" stretchy="false" xref="S3.SS2.p4.7.m2.2.3.2.1.1.cmml">|</mo></mrow><mo id="S3.SS2.p4.7.m2.2.3.1" xref="S3.SS2.p4.7.m2.2.3.1.cmml">=</mo><mrow id="S3.SS2.p4.7.m2.2.3.3.2" xref="S3.SS2.p4.7.m2.2.3.3.1.cmml"><mo id="S3.SS2.p4.7.m2.2.3.3.2.1" stretchy="false" xref="S3.SS2.p4.7.m2.2.3.3.1.1.cmml">|</mo><mi id="S3.SS2.p4.7.m2.2.2" xref="S3.SS2.p4.7.m2.2.2.cmml">w</mi><mo id="S3.SS2.p4.7.m2.2.3.3.2.2" stretchy="false" xref="S3.SS2.p4.7.m2.2.3.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.7.m2.2b"><apply id="S3.SS2.p4.7.m2.2.3.cmml" xref="S3.SS2.p4.7.m2.2.3"><eq id="S3.SS2.p4.7.m2.2.3.1.cmml" xref="S3.SS2.p4.7.m2.2.3.1"></eq><apply id="S3.SS2.p4.7.m2.2.3.2.1.cmml" xref="S3.SS2.p4.7.m2.2.3.2.2"><abs id="S3.SS2.p4.7.m2.2.3.2.1.1.cmml" xref="S3.SS2.p4.7.m2.2.3.2.2.1"></abs><ci id="S3.SS2.p4.7.m2.1.1.cmml" xref="S3.SS2.p4.7.m2.1.1">𝑢</ci></apply><apply id="S3.SS2.p4.7.m2.2.3.3.1.cmml" xref="S3.SS2.p4.7.m2.2.3.3.2"><abs id="S3.SS2.p4.7.m2.2.3.3.1.1.cmml" xref="S3.SS2.p4.7.m2.2.3.3.2.1"></abs><ci id="S3.SS2.p4.7.m2.2.2.cmml" xref="S3.SS2.p4.7.m2.2.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.7.m2.2c">|u|=|w|</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.7.m2.2d">| italic_u | = | italic_w |</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3. </span>The induced measure morphisms</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1"></p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.5">We now consider an arbitrary non-erasing monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><mrow id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml"><mi id="S3.SS3.p2.1.m1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.2.cmml">σ</mi><mo id="S3.SS3.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S3.SS3.p2.1.m1.1.1.3" xref="S3.SS3.p2.1.m1.1.1.3.cmml"><msup id="S3.SS3.p2.1.m1.1.1.3.2" xref="S3.SS3.p2.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.1.m1.1.1.3.2.2" xref="S3.SS3.p2.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.SS3.p2.1.m1.1.1.3.2.3" xref="S3.SS3.p2.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.SS3.p2.1.m1.1.1.3.1" stretchy="false" xref="S3.SS3.p2.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.SS3.p2.1.m1.1.1.3.3" xref="S3.SS3.p2.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.1.m1.1.1.3.3.2" xref="S3.SS3.p2.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.SS3.p2.1.m1.1.1.3.3.3" xref="S3.SS3.p2.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><apply id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1"><ci id="S3.SS3.p2.1.m1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1.1">:</ci><ci id="S3.SS3.p2.1.m1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.1.1.2">𝜎</ci><apply id="S3.SS3.p2.1.m1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3"><ci id="S3.SS3.p2.1.m1.1.1.3.1.cmml" xref="S3.SS3.p2.1.m1.1.1.3.1">→</ci><apply id="S3.SS3.p2.1.m1.1.1.3.2.cmml" xref="S3.SS3.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.1.1.3.2.1.cmml" xref="S3.SS3.p2.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.SS3.p2.1.m1.1.1.3.2.2.cmml" xref="S3.SS3.p2.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.SS3.p2.1.m1.1.1.3.2.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3.2.3"></times></apply><apply id="S3.SS3.p2.1.m1.1.1.3.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.1.1.3.3.1.cmml" xref="S3.SS3.p2.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.SS3.p2.1.m1.1.1.3.3.2.cmml" xref="S3.SS3.p2.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.SS3.p2.1.m1.1.1.3.3.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> (as usual for finite alphabets <math alttext="\cal A" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.1"><semantics id="S3.SS3.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.2.m2.1.1" xref="S3.SS3.p2.2.m2.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.1b"><ci id="S3.SS3.p2.2.m2.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.1d">caligraphic_A</annotation></semantics></math> and <math alttext="\cal B" class="ltx_Math" display="inline" id="S3.SS3.p2.3.m3.1"><semantics id="S3.SS3.p2.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.3.m3.1.1" xref="S3.SS3.p2.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.3.m3.1b"><ci id="S3.SS3.p2.3.m3.1.1.cmml" xref="S3.SS3.p2.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.3.m3.1c">\cal B</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.3.m3.1d">caligraphic_B</annotation></semantics></math>). Then <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS3.p2.4.m4.1"><semantics id="S3.SS3.p2.4.m4.1a"><mi id="S3.SS3.p2.4.m4.1.1" xref="S3.SS3.p2.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.4.m4.1b"><ci id="S3.SS3.p2.4.m4.1.1.cmml" xref="S3.SS3.p2.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.4.m4.1d">italic_σ</annotation></semantics></math> defines a subdivision length function <math alttext="\ell_{\sigma}:\cal A\to\mathbb{Z}_{\geq 1}\," class="ltx_Math" display="inline" id="S3.SS3.p2.5.m5.1"><semantics id="S3.SS3.p2.5.m5.1a"><mrow id="S3.SS3.p2.5.m5.1.1" xref="S3.SS3.p2.5.m5.1.1.cmml"><msub id="S3.SS3.p2.5.m5.1.1.2" xref="S3.SS3.p2.5.m5.1.1.2.cmml"><mi id="S3.SS3.p2.5.m5.1.1.2.2" mathvariant="normal" xref="S3.SS3.p2.5.m5.1.1.2.2.cmml">ℓ</mi><mi id="S3.SS3.p2.5.m5.1.1.2.3" xref="S3.SS3.p2.5.m5.1.1.2.3.cmml">σ</mi></msub><mo id="S3.SS3.p2.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.p2.5.m5.1.1.1.cmml">:</mo><mrow id="S3.SS3.p2.5.m5.1.1.3" xref="S3.SS3.p2.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.5.m5.1.1.3.2" xref="S3.SS3.p2.5.m5.1.1.3.2.cmml">𝒜</mi><mo id="S3.SS3.p2.5.m5.1.1.3.1" stretchy="false" xref="S3.SS3.p2.5.m5.1.1.3.1.cmml">→</mo><msub id="S3.SS3.p2.5.m5.1.1.3.3" xref="S3.SS3.p2.5.m5.1.1.3.3.cmml"><mi id="S3.SS3.p2.5.m5.1.1.3.3.2" xref="S3.SS3.p2.5.m5.1.1.3.3.2.cmml">ℤ</mi><mrow id="S3.SS3.p2.5.m5.1.1.3.3.3" xref="S3.SS3.p2.5.m5.1.1.3.3.3.cmml"><mi id="S3.SS3.p2.5.m5.1.1.3.3.3.2" xref="S3.SS3.p2.5.m5.1.1.3.3.3.2.cmml"></mi><mo id="S3.SS3.p2.5.m5.1.1.3.3.3.1" xref="S3.SS3.p2.5.m5.1.1.3.3.3.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S3.SS3.p2.5.m5.1.1.3.3.3.3" mathvariant="script" xref="S3.SS3.p2.5.m5.1.1.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.5.m5.1b"><apply id="S3.SS3.p2.5.m5.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1"><ci id="S3.SS3.p2.5.m5.1.1.1.cmml" xref="S3.SS3.p2.5.m5.1.1.1">:</ci><apply id="S3.SS3.p2.5.m5.1.1.2.cmml" xref="S3.SS3.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p2.5.m5.1.1.2.1.cmml" xref="S3.SS3.p2.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS3.p2.5.m5.1.1.2.2.cmml" xref="S3.SS3.p2.5.m5.1.1.2.2">ℓ</ci><ci id="S3.SS3.p2.5.m5.1.1.2.3.cmml" xref="S3.SS3.p2.5.m5.1.1.2.3">𝜎</ci></apply><apply id="S3.SS3.p2.5.m5.1.1.3.cmml" xref="S3.SS3.p2.5.m5.1.1.3"><ci id="S3.SS3.p2.5.m5.1.1.3.1.cmml" xref="S3.SS3.p2.5.m5.1.1.3.1">→</ci><ci id="S3.SS3.p2.5.m5.1.1.3.2.cmml" xref="S3.SS3.p2.5.m5.1.1.3.2">𝒜</ci><apply id="S3.SS3.p2.5.m5.1.1.3.3.cmml" xref="S3.SS3.p2.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS3.p2.5.m5.1.1.3.3.1.cmml" xref="S3.SS3.p2.5.m5.1.1.3.3">subscript</csymbol><ci id="S3.SS3.p2.5.m5.1.1.3.3.2.cmml" xref="S3.SS3.p2.5.m5.1.1.3.3.2">ℤ</ci><apply id="S3.SS3.p2.5.m5.1.1.3.3.3.cmml" xref="S3.SS3.p2.5.m5.1.1.3.3.3"><geq id="S3.SS3.p2.5.m5.1.1.3.3.3.1.cmml" xref="S3.SS3.p2.5.m5.1.1.3.3.3.1"></geq><csymbol cd="latexml" id="S3.SS3.p2.5.m5.1.1.3.3.3.2.cmml" xref="S3.SS3.p2.5.m5.1.1.3.3.3.2">absent</csymbol><cn id="S3.SS3.p2.5.m5.1.1.3.3.3.3.cmml" type="integer" xref="S3.SS3.p2.5.m5.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.5.m5.1c">\ell_{\sigma}:\cal A\to\mathbb{Z}_{\geq 1}\,</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.5.m5.1d">roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT : caligraphic_A → blackboard_Z start_POSTSUBSCRIPT ≥ caligraphic_1 end_POSTSUBSCRIPT</annotation></semantics></math>, given by</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ell_{\sigma}(a_{i}):=|\sigma(a_{i})|" class="ltx_Math" display="block" id="S3.Ex4.m1.2"><semantics id="S3.Ex4.m1.2a"><mrow id="S3.Ex4.m1.2.2" xref="S3.Ex4.m1.2.2.cmml"><mrow id="S3.Ex4.m1.1.1.1" xref="S3.Ex4.m1.1.1.1.cmml"><msub id="S3.Ex4.m1.1.1.1.3" xref="S3.Ex4.m1.1.1.1.3.cmml"><mi id="S3.Ex4.m1.1.1.1.3.2" mathvariant="normal" xref="S3.Ex4.m1.1.1.1.3.2.cmml">ℓ</mi><mi id="S3.Ex4.m1.1.1.1.3.3" xref="S3.Ex4.m1.1.1.1.3.3.cmml">σ</mi></msub><mo id="S3.Ex4.m1.1.1.1.2" xref="S3.Ex4.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex4.m1.1.1.1.1.1" xref="S3.Ex4.m1.1.1.1.1.1.1.cmml"><mo id="S3.Ex4.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex4.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex4.m1.1.1.1.1.1.1" xref="S3.Ex4.m1.1.1.1.1.1.1.cmml"><mi id="S3.Ex4.m1.1.1.1.1.1.1.2" xref="S3.Ex4.m1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.Ex4.m1.1.1.1.1.1.1.3" xref="S3.Ex4.m1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.Ex4.m1.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S3.Ex4.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex4.m1.2.2.3" rspace="0.278em" xref="S3.Ex4.m1.2.2.3.cmml">:=</mo><mrow id="S3.Ex4.m1.2.2.2.1" xref="S3.Ex4.m1.2.2.2.2.cmml"><mo id="S3.Ex4.m1.2.2.2.1.2" stretchy="false" xref="S3.Ex4.m1.2.2.2.2.1.cmml">|</mo><mrow id="S3.Ex4.m1.2.2.2.1.1" xref="S3.Ex4.m1.2.2.2.1.1.cmml"><mi id="S3.Ex4.m1.2.2.2.1.1.3" xref="S3.Ex4.m1.2.2.2.1.1.3.cmml">σ</mi><mo id="S3.Ex4.m1.2.2.2.1.1.2" xref="S3.Ex4.m1.2.2.2.1.1.2.cmml">⁢</mo><mrow id="S3.Ex4.m1.2.2.2.1.1.1.1" xref="S3.Ex4.m1.2.2.2.1.1.1.1.1.cmml"><mo id="S3.Ex4.m1.2.2.2.1.1.1.1.2" stretchy="false" xref="S3.Ex4.m1.2.2.2.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex4.m1.2.2.2.1.1.1.1.1" xref="S3.Ex4.m1.2.2.2.1.1.1.1.1.cmml"><mi id="S3.Ex4.m1.2.2.2.1.1.1.1.1.2" xref="S3.Ex4.m1.2.2.2.1.1.1.1.1.2.cmml">a</mi><mi id="S3.Ex4.m1.2.2.2.1.1.1.1.1.3" xref="S3.Ex4.m1.2.2.2.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.Ex4.m1.2.2.2.1.1.1.1.3" stretchy="false" xref="S3.Ex4.m1.2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex4.m1.2.2.2.1.3" stretchy="false" xref="S3.Ex4.m1.2.2.2.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex4.m1.2b"><apply id="S3.Ex4.m1.2.2.cmml" xref="S3.Ex4.m1.2.2"><csymbol cd="latexml" id="S3.Ex4.m1.2.2.3.cmml" xref="S3.Ex4.m1.2.2.3">assign</csymbol><apply id="S3.Ex4.m1.1.1.1.cmml" xref="S3.Ex4.m1.1.1.1"><times id="S3.Ex4.m1.1.1.1.2.cmml" xref="S3.Ex4.m1.1.1.1.2"></times><apply id="S3.Ex4.m1.1.1.1.3.cmml" xref="S3.Ex4.m1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex4.m1.1.1.1.3.1.cmml" xref="S3.Ex4.m1.1.1.1.3">subscript</csymbol><ci id="S3.Ex4.m1.1.1.1.3.2.cmml" xref="S3.Ex4.m1.1.1.1.3.2">ℓ</ci><ci id="S3.Ex4.m1.1.1.1.3.3.cmml" xref="S3.Ex4.m1.1.1.1.3.3">𝜎</ci></apply><apply id="S3.Ex4.m1.1.1.1.1.1.1.cmml" xref="S3.Ex4.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex4.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex4.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex4.m1.1.1.1.1.1.1.2.cmml" xref="S3.Ex4.m1.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.Ex4.m1.1.1.1.1.1.1.3.cmml" xref="S3.Ex4.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S3.Ex4.m1.2.2.2.2.cmml" xref="S3.Ex4.m1.2.2.2.1"><abs id="S3.Ex4.m1.2.2.2.2.1.cmml" xref="S3.Ex4.m1.2.2.2.1.2"></abs><apply id="S3.Ex4.m1.2.2.2.1.1.cmml" xref="S3.Ex4.m1.2.2.2.1.1"><times id="S3.Ex4.m1.2.2.2.1.1.2.cmml" xref="S3.Ex4.m1.2.2.2.1.1.2"></times><ci id="S3.Ex4.m1.2.2.2.1.1.3.cmml" xref="S3.Ex4.m1.2.2.2.1.1.3">𝜎</ci><apply id="S3.Ex4.m1.2.2.2.1.1.1.1.1.cmml" xref="S3.Ex4.m1.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex4.m1.2.2.2.1.1.1.1.1.1.cmml" xref="S3.Ex4.m1.2.2.2.1.1.1.1">subscript</csymbol><ci id="S3.Ex4.m1.2.2.2.1.1.1.1.1.2.cmml" xref="S3.Ex4.m1.2.2.2.1.1.1.1.1.2">𝑎</ci><ci id="S3.Ex4.m1.2.2.2.1.1.1.1.1.3.cmml" xref="S3.Ex4.m1.2.2.2.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex4.m1.2c">\ell_{\sigma}(a_{i}):=|\sigma(a_{i})|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex4.m1.2d">roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) := | italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p2.7">for any <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S3.SS3.p2.6.m1.1"><semantics id="S3.SS3.p2.6.m1.1a"><mrow id="S3.SS3.p2.6.m1.1.1" xref="S3.SS3.p2.6.m1.1.1.cmml"><msub id="S3.SS3.p2.6.m1.1.1.2" xref="S3.SS3.p2.6.m1.1.1.2.cmml"><mi id="S3.SS3.p2.6.m1.1.1.2.2" xref="S3.SS3.p2.6.m1.1.1.2.2.cmml">a</mi><mi id="S3.SS3.p2.6.m1.1.1.2.3" xref="S3.SS3.p2.6.m1.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS3.p2.6.m1.1.1.1" xref="S3.SS3.p2.6.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.6.m1.1.1.3" xref="S3.SS3.p2.6.m1.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.6.m1.1b"><apply id="S3.SS3.p2.6.m1.1.1.cmml" xref="S3.SS3.p2.6.m1.1.1"><in id="S3.SS3.p2.6.m1.1.1.1.cmml" xref="S3.SS3.p2.6.m1.1.1.1"></in><apply id="S3.SS3.p2.6.m1.1.1.2.cmml" xref="S3.SS3.p2.6.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p2.6.m1.1.1.2.1.cmml" xref="S3.SS3.p2.6.m1.1.1.2">subscript</csymbol><ci id="S3.SS3.p2.6.m1.1.1.2.2.cmml" xref="S3.SS3.p2.6.m1.1.1.2.2">𝑎</ci><ci id="S3.SS3.p2.6.m1.1.1.2.3.cmml" xref="S3.SS3.p2.6.m1.1.1.2.3">𝑖</ci></apply><ci id="S3.SS3.p2.6.m1.1.1.3.cmml" xref="S3.SS3.p2.6.m1.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.6.m1.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.6.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math>. This gives a subdivision alphabet <math alttext="\cal A_{\sigma}:=\cal A_{\ell_{\sigma}}" class="ltx_Math" display="inline" id="S3.SS3.p2.7.m2.1"><semantics id="S3.SS3.p2.7.m2.1a"><mrow id="S3.SS3.p2.7.m2.1.1" xref="S3.SS3.p2.7.m2.1.1.cmml"><msub id="S3.SS3.p2.7.m2.1.1.2" xref="S3.SS3.p2.7.m2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.7.m2.1.1.2.2" xref="S3.SS3.p2.7.m2.1.1.2.2.cmml">𝒜</mi><mi id="S3.SS3.p2.7.m2.1.1.2.3" xref="S3.SS3.p2.7.m2.1.1.2.3.cmml">σ</mi></msub><mo id="S3.SS3.p2.7.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS3.p2.7.m2.1.1.1.cmml">:=</mo><msub id="S3.SS3.p2.7.m2.1.1.3" xref="S3.SS3.p2.7.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.7.m2.1.1.3.2" xref="S3.SS3.p2.7.m2.1.1.3.2.cmml">𝒜</mi><msub id="S3.SS3.p2.7.m2.1.1.3.3" xref="S3.SS3.p2.7.m2.1.1.3.3.cmml"><mi id="S3.SS3.p2.7.m2.1.1.3.3.2" mathvariant="normal" xref="S3.SS3.p2.7.m2.1.1.3.3.2.cmml">ℓ</mi><mi id="S3.SS3.p2.7.m2.1.1.3.3.3" xref="S3.SS3.p2.7.m2.1.1.3.3.3.cmml">σ</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.7.m2.1b"><apply id="S3.SS3.p2.7.m2.1.1.cmml" xref="S3.SS3.p2.7.m2.1.1"><csymbol cd="latexml" id="S3.SS3.p2.7.m2.1.1.1.cmml" xref="S3.SS3.p2.7.m2.1.1.1">assign</csymbol><apply id="S3.SS3.p2.7.m2.1.1.2.cmml" xref="S3.SS3.p2.7.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p2.7.m2.1.1.2.1.cmml" xref="S3.SS3.p2.7.m2.1.1.2">subscript</csymbol><ci id="S3.SS3.p2.7.m2.1.1.2.2.cmml" xref="S3.SS3.p2.7.m2.1.1.2.2">𝒜</ci><ci id="S3.SS3.p2.7.m2.1.1.2.3.cmml" xref="S3.SS3.p2.7.m2.1.1.2.3">𝜎</ci></apply><apply id="S3.SS3.p2.7.m2.1.1.3.cmml" xref="S3.SS3.p2.7.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p2.7.m2.1.1.3.1.cmml" xref="S3.SS3.p2.7.m2.1.1.3">subscript</csymbol><ci id="S3.SS3.p2.7.m2.1.1.3.2.cmml" xref="S3.SS3.p2.7.m2.1.1.3.2">𝒜</ci><apply id="S3.SS3.p2.7.m2.1.1.3.3.cmml" xref="S3.SS3.p2.7.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS3.p2.7.m2.1.1.3.3.1.cmml" xref="S3.SS3.p2.7.m2.1.1.3.3">subscript</csymbol><ci id="S3.SS3.p2.7.m2.1.1.3.3.2.cmml" xref="S3.SS3.p2.7.m2.1.1.3.3.2">ℓ</ci><ci id="S3.SS3.p2.7.m2.1.1.3.3.3.cmml" xref="S3.SS3.p2.7.m2.1.1.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.7.m2.1c">\cal A_{\sigma}:=\cal A_{\ell_{\sigma}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.7.m2.1d">caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT := caligraphic_A start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> as well as a subdivision morphism</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\pi_{\sigma}:=\pi_{\ell_{\sigma}}:\cal A^{*}\to\cal A_{\sigma}^{*}\,." class="ltx_Math" display="block" id="S3.Ex5.m1.1"><semantics id="S3.Ex5.m1.1a"><mrow id="S3.Ex5.m1.1.1.1" xref="S3.Ex5.m1.1.1.1.1.cmml"><mrow id="S3.Ex5.m1.1.1.1.1" xref="S3.Ex5.m1.1.1.1.1.cmml"><mrow id="S3.Ex5.m1.1.1.1.1.2" xref="S3.Ex5.m1.1.1.1.1.2.cmml"><msub id="S3.Ex5.m1.1.1.1.1.2.2" xref="S3.Ex5.m1.1.1.1.1.2.2.cmml"><mi id="S3.Ex5.m1.1.1.1.1.2.2.2" xref="S3.Ex5.m1.1.1.1.1.2.2.2.cmml">π</mi><mi id="S3.Ex5.m1.1.1.1.1.2.2.3" xref="S3.Ex5.m1.1.1.1.1.2.2.3.cmml">σ</mi></msub><mo id="S3.Ex5.m1.1.1.1.1.2.1" lspace="0.278em" rspace="0.278em" xref="S3.Ex5.m1.1.1.1.1.2.1.cmml">:=</mo><msub id="S3.Ex5.m1.1.1.1.1.2.3" xref="S3.Ex5.m1.1.1.1.1.2.3.cmml"><mi id="S3.Ex5.m1.1.1.1.1.2.3.2" xref="S3.Ex5.m1.1.1.1.1.2.3.2.cmml">π</mi><msub id="S3.Ex5.m1.1.1.1.1.2.3.3" xref="S3.Ex5.m1.1.1.1.1.2.3.3.cmml"><mi id="S3.Ex5.m1.1.1.1.1.2.3.3.2" mathvariant="normal" xref="S3.Ex5.m1.1.1.1.1.2.3.3.2.cmml">ℓ</mi><mi id="S3.Ex5.m1.1.1.1.1.2.3.3.3" xref="S3.Ex5.m1.1.1.1.1.2.3.3.3.cmml">σ</mi></msub></msub></mrow><mo id="S3.Ex5.m1.1.1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Ex5.m1.1.1.1.1.1.cmml">:</mo><mrow id="S3.Ex5.m1.1.1.1.1.3" xref="S3.Ex5.m1.1.1.1.1.3.cmml"><msup id="S3.Ex5.m1.1.1.1.1.3.2" xref="S3.Ex5.m1.1.1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex5.m1.1.1.1.1.3.2.2" xref="S3.Ex5.m1.1.1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Ex5.m1.1.1.1.1.3.2.3" xref="S3.Ex5.m1.1.1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Ex5.m1.1.1.1.1.3.1" stretchy="false" xref="S3.Ex5.m1.1.1.1.1.3.1.cmml">→</mo><msubsup id="S3.Ex5.m1.1.1.1.1.3.3" xref="S3.Ex5.m1.1.1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex5.m1.1.1.1.1.3.3.2.2" xref="S3.Ex5.m1.1.1.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S3.Ex5.m1.1.1.1.1.3.3.2.3" xref="S3.Ex5.m1.1.1.1.1.3.3.2.3.cmml">σ</mi><mo id="S3.Ex5.m1.1.1.1.1.3.3.3" xref="S3.Ex5.m1.1.1.1.1.3.3.3.cmml">∗</mo></msubsup></mrow></mrow><mo id="S3.Ex5.m1.1.1.1.2" lspace="0em" xref="S3.Ex5.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" 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id="S3.Ex5.m1.1.1.1.1.2.3.3.1.cmml" xref="S3.Ex5.m1.1.1.1.1.2.3.3">subscript</csymbol><ci id="S3.Ex5.m1.1.1.1.1.2.3.3.2.cmml" xref="S3.Ex5.m1.1.1.1.1.2.3.3.2">ℓ</ci><ci id="S3.Ex5.m1.1.1.1.1.2.3.3.3.cmml" xref="S3.Ex5.m1.1.1.1.1.2.3.3.3">𝜎</ci></apply></apply></apply><apply id="S3.Ex5.m1.1.1.1.1.3.cmml" xref="S3.Ex5.m1.1.1.1.1.3"><ci id="S3.Ex5.m1.1.1.1.1.3.1.cmml" xref="S3.Ex5.m1.1.1.1.1.3.1">→</ci><apply id="S3.Ex5.m1.1.1.1.1.3.2.cmml" xref="S3.Ex5.m1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.1.1.1.3.2.1.cmml" xref="S3.Ex5.m1.1.1.1.1.3.2">superscript</csymbol><ci id="S3.Ex5.m1.1.1.1.1.3.2.2.cmml" xref="S3.Ex5.m1.1.1.1.1.3.2.2">𝒜</ci><times id="S3.Ex5.m1.1.1.1.1.3.2.3.cmml" xref="S3.Ex5.m1.1.1.1.1.3.2.3"></times></apply><apply id="S3.Ex5.m1.1.1.1.1.3.3.cmml" xref="S3.Ex5.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.1.1.1.3.3.1.cmml" xref="S3.Ex5.m1.1.1.1.1.3.3">superscript</csymbol><apply id="S3.Ex5.m1.1.1.1.1.3.3.2.cmml" xref="S3.Ex5.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Ex5.m1.1.1.1.1.3.3.2.1.cmml" xref="S3.Ex5.m1.1.1.1.1.3.3">subscript</csymbol><ci id="S3.Ex5.m1.1.1.1.1.3.3.2.2.cmml" xref="S3.Ex5.m1.1.1.1.1.3.3.2.2">𝒜</ci><ci id="S3.Ex5.m1.1.1.1.1.3.3.2.3.cmml" xref="S3.Ex5.m1.1.1.1.1.3.3.2.3">𝜎</ci></apply><times id="S3.Ex5.m1.1.1.1.1.3.3.3.cmml" xref="S3.Ex5.m1.1.1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex5.m1.1c">\pi_{\sigma}:=\pi_{\ell_{\sigma}}:\cal A^{*}\to\cal A_{\sigma}^{*}\,.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5.m1.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT := italic_π start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p2.8">Furthermore, <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS3.p2.8.m1.1"><semantics id="S3.SS3.p2.8.m1.1a"><mi id="S3.SS3.p2.8.m1.1.1" xref="S3.SS3.p2.8.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.8.m1.1b"><ci id="S3.SS3.p2.8.m1.1.1.cmml" xref="S3.SS3.p2.8.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.8.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.8.m1.1d">italic_σ</annotation></semantics></math> defines a letter-to-letter morphism given by</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\alpha_{\sigma}:\cal A_{\sigma}^{*}\to\cal B^{*}\,,\,\,a_{i}(k)\mapsto\sigma(a% _{i})_{k}\,," class="ltx_Math" display="block" id="S3.Ex6.m1.2"><semantics id="S3.Ex6.m1.2a"><mrow id="S3.Ex6.m1.2.2.1" xref="S3.Ex6.m1.2.2.1.1.cmml"><mrow id="S3.Ex6.m1.2.2.1.1" xref="S3.Ex6.m1.2.2.1.1.cmml"><msub id="S3.Ex6.m1.2.2.1.1.4" xref="S3.Ex6.m1.2.2.1.1.4.cmml"><mi id="S3.Ex6.m1.2.2.1.1.4.2" xref="S3.Ex6.m1.2.2.1.1.4.2.cmml">α</mi><mi id="S3.Ex6.m1.2.2.1.1.4.3" xref="S3.Ex6.m1.2.2.1.1.4.3.cmml">σ</mi></msub><mo id="S3.Ex6.m1.2.2.1.1.3" lspace="0.278em" rspace="0.278em" xref="S3.Ex6.m1.2.2.1.1.3.cmml">:</mo><mrow id="S3.Ex6.m1.2.2.1.1.2.2" xref="S3.Ex6.m1.2.2.1.1.2.3.cmml"><mrow id="S3.Ex6.m1.2.2.1.1.1.1.1" xref="S3.Ex6.m1.2.2.1.1.1.1.1.cmml"><msubsup id="S3.Ex6.m1.2.2.1.1.1.1.1.2" xref="S3.Ex6.m1.2.2.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex6.m1.2.2.1.1.1.1.1.2.2.2" xref="S3.Ex6.m1.2.2.1.1.1.1.1.2.2.2.cmml">𝒜</mi><mi id="S3.Ex6.m1.2.2.1.1.1.1.1.2.2.3" xref="S3.Ex6.m1.2.2.1.1.1.1.1.2.2.3.cmml">σ</mi><mo id="S3.Ex6.m1.2.2.1.1.1.1.1.2.3" xref="S3.Ex6.m1.2.2.1.1.1.1.1.2.3.cmml">∗</mo></msubsup><mo id="S3.Ex6.m1.2.2.1.1.1.1.1.1" stretchy="false" xref="S3.Ex6.m1.2.2.1.1.1.1.1.1.cmml">→</mo><msup id="S3.Ex6.m1.2.2.1.1.1.1.1.3" xref="S3.Ex6.m1.2.2.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex6.m1.2.2.1.1.1.1.1.3.2" xref="S3.Ex6.m1.2.2.1.1.1.1.1.3.2.cmml">ℬ</mi><mo id="S3.Ex6.m1.2.2.1.1.1.1.1.3.3" xref="S3.Ex6.m1.2.2.1.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S3.Ex6.m1.2.2.1.1.2.2.3" rspace="0.497em" xref="S3.Ex6.m1.2.2.1.1.2.3a.cmml">,</mo><mrow id="S3.Ex6.m1.2.2.1.1.2.2.2" xref="S3.Ex6.m1.2.2.1.1.2.2.2.cmml"><mrow id="S3.Ex6.m1.2.2.1.1.2.2.2.3" xref="S3.Ex6.m1.2.2.1.1.2.2.2.3.cmml"><msub id="S3.Ex6.m1.2.2.1.1.2.2.2.3.2" xref="S3.Ex6.m1.2.2.1.1.2.2.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex6.m1.2.2.1.1.2.2.2.3.2.2" xref="S3.Ex6.m1.2.2.1.1.2.2.2.3.2.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S3.Ex6.m1.2.2.1.1.2.2.2.3.2.3" xref="S3.Ex6.m1.2.2.1.1.2.2.2.3.2.3.cmml">𝒾</mi></msub><mo id="S3.Ex6.m1.2.2.1.1.2.2.2.3.1" xref="S3.Ex6.m1.2.2.1.1.2.2.2.3.1.cmml">⁢</mo><mrow id="S3.Ex6.m1.2.2.1.1.2.2.2.3.3.2" xref="S3.Ex6.m1.2.2.1.1.2.2.2.3.cmml"><mo id="S3.Ex6.m1.2.2.1.1.2.2.2.3.3.2.1" stretchy="false" xref="S3.Ex6.m1.2.2.1.1.2.2.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex6.m1.1.1" xref="S3.Ex6.m1.1.1.cmml">𝓀</mi><mo id="S3.Ex6.m1.2.2.1.1.2.2.2.3.3.2.2" stretchy="false" xref="S3.Ex6.m1.2.2.1.1.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex6.m1.2.2.1.1.2.2.2.2" stretchy="false" xref="S3.Ex6.m1.2.2.1.1.2.2.2.2.cmml">↦</mo><mrow id="S3.Ex6.m1.2.2.1.1.2.2.2.1" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.cmml"><mi id="S3.Ex6.m1.2.2.1.1.2.2.2.1.3" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.3.cmml">σ</mi><mo id="S3.Ex6.m1.2.2.1.1.2.2.2.1.2" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.2.cmml">⁢</mo><msub id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.cmml"><mrow id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.cmml"><mo id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.2" stretchy="false" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.2" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.2.cmml">𝒶</mi><mi class="ltx_font_mathcaligraphic" id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.3" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.3.cmml">𝒾</mi></msub><mo id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.3" stretchy="false" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mi class="ltx_font_mathcaligraphic" id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.3" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.3.cmml">𝓀</mi></msub></mrow></mrow></mrow></mrow><mo id="S3.Ex6.m1.2.2.1.2" xref="S3.Ex6.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex6.m1.2b"><apply id="S3.Ex6.m1.2.2.1.1.cmml" xref="S3.Ex6.m1.2.2.1"><ci id="S3.Ex6.m1.2.2.1.1.3.cmml" xref="S3.Ex6.m1.2.2.1.1.3">:</ci><apply id="S3.Ex6.m1.2.2.1.1.4.cmml" xref="S3.Ex6.m1.2.2.1.1.4"><csymbol cd="ambiguous" id="S3.Ex6.m1.2.2.1.1.4.1.cmml" xref="S3.Ex6.m1.2.2.1.1.4">subscript</csymbol><ci id="S3.Ex6.m1.2.2.1.1.4.2.cmml" xref="S3.Ex6.m1.2.2.1.1.4.2">𝛼</ci><ci id="S3.Ex6.m1.2.2.1.1.4.3.cmml" xref="S3.Ex6.m1.2.2.1.1.4.3">𝜎</ci></apply><apply id="S3.Ex6.m1.2.2.1.1.2.3.cmml" xref="S3.Ex6.m1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex6.m1.2.2.1.1.2.3a.cmml" xref="S3.Ex6.m1.2.2.1.1.2.2.3">formulae-sequence</csymbol><apply id="S3.Ex6.m1.2.2.1.1.1.1.1.cmml" xref="S3.Ex6.m1.2.2.1.1.1.1.1"><ci id="S3.Ex6.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.Ex6.m1.2.2.1.1.1.1.1.1">→</ci><apply id="S3.Ex6.m1.2.2.1.1.1.1.1.2.cmml" xref="S3.Ex6.m1.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex6.m1.2.2.1.1.1.1.1.2.1.cmml" xref="S3.Ex6.m1.2.2.1.1.1.1.1.2">superscript</csymbol><apply id="S3.Ex6.m1.2.2.1.1.1.1.1.2.2.cmml" xref="S3.Ex6.m1.2.2.1.1.1.1.1.2"><csymbol 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xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1">subscript</csymbol><apply id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.cmml" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.1.cmml" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1">subscript</csymbol><ci id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.2.cmml" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.2">𝒶</ci><ci id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.3.cmml" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.1.1.1.3">𝒾</ci></apply><ci id="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.3.cmml" xref="S3.Ex6.m1.2.2.1.1.2.2.2.1.1.3">𝓀</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex6.m1.2c">\alpha_{\sigma}:\cal A_{\sigma}^{*}\to\cal B^{*}\,,\,\,a_{i}(k)\mapsto\sigma(a% _{i})_{k}\,,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex6.m1.2d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ( caligraphic_k ) ↦ italic_σ ( caligraphic_a start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT caligraphic_k end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p2.13">for any <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S3.SS3.p2.9.m1.1"><semantics id="S3.SS3.p2.9.m1.1a"><mrow id="S3.SS3.p2.9.m1.1.1" xref="S3.SS3.p2.9.m1.1.1.cmml"><msub id="S3.SS3.p2.9.m1.1.1.2" xref="S3.SS3.p2.9.m1.1.1.2.cmml"><mi id="S3.SS3.p2.9.m1.1.1.2.2" xref="S3.SS3.p2.9.m1.1.1.2.2.cmml">a</mi><mi id="S3.SS3.p2.9.m1.1.1.2.3" xref="S3.SS3.p2.9.m1.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS3.p2.9.m1.1.1.1" xref="S3.SS3.p2.9.m1.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.9.m1.1.1.3" xref="S3.SS3.p2.9.m1.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.9.m1.1b"><apply id="S3.SS3.p2.9.m1.1.1.cmml" xref="S3.SS3.p2.9.m1.1.1"><in id="S3.SS3.p2.9.m1.1.1.1.cmml" xref="S3.SS3.p2.9.m1.1.1.1"></in><apply id="S3.SS3.p2.9.m1.1.1.2.cmml" xref="S3.SS3.p2.9.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p2.9.m1.1.1.2.1.cmml" xref="S3.SS3.p2.9.m1.1.1.2">subscript</csymbol><ci id="S3.SS3.p2.9.m1.1.1.2.2.cmml" xref="S3.SS3.p2.9.m1.1.1.2.2">𝑎</ci><ci id="S3.SS3.p2.9.m1.1.1.2.3.cmml" xref="S3.SS3.p2.9.m1.1.1.2.3">𝑖</ci></apply><ci id="S3.SS3.p2.9.m1.1.1.3.cmml" xref="S3.SS3.p2.9.m1.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.9.m1.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.9.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> and <math alttext="1\leq k\leq|\sigma(a_{i})|" class="ltx_Math" display="inline" id="S3.SS3.p2.10.m2.1"><semantics id="S3.SS3.p2.10.m2.1a"><mrow id="S3.SS3.p2.10.m2.1.1" xref="S3.SS3.p2.10.m2.1.1.cmml"><mn id="S3.SS3.p2.10.m2.1.1.3" xref="S3.SS3.p2.10.m2.1.1.3.cmml">1</mn><mo id="S3.SS3.p2.10.m2.1.1.4" xref="S3.SS3.p2.10.m2.1.1.4.cmml">≤</mo><mi id="S3.SS3.p2.10.m2.1.1.5" xref="S3.SS3.p2.10.m2.1.1.5.cmml">k</mi><mo id="S3.SS3.p2.10.m2.1.1.6" xref="S3.SS3.p2.10.m2.1.1.6.cmml">≤</mo><mrow id="S3.SS3.p2.10.m2.1.1.1.1" xref="S3.SS3.p2.10.m2.1.1.1.2.cmml"><mo id="S3.SS3.p2.10.m2.1.1.1.1.2" stretchy="false" xref="S3.SS3.p2.10.m2.1.1.1.2.1.cmml">|</mo><mrow id="S3.SS3.p2.10.m2.1.1.1.1.1" xref="S3.SS3.p2.10.m2.1.1.1.1.1.cmml"><mi id="S3.SS3.p2.10.m2.1.1.1.1.1.3" xref="S3.SS3.p2.10.m2.1.1.1.1.1.3.cmml">σ</mi><mo id="S3.SS3.p2.10.m2.1.1.1.1.1.2" xref="S3.SS3.p2.10.m2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.2" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.3" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p2.10.m2.1.1.1.1.3" stretchy="false" xref="S3.SS3.p2.10.m2.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.10.m2.1b"><apply id="S3.SS3.p2.10.m2.1.1.cmml" xref="S3.SS3.p2.10.m2.1.1"><and id="S3.SS3.p2.10.m2.1.1a.cmml" xref="S3.SS3.p2.10.m2.1.1"></and><apply id="S3.SS3.p2.10.m2.1.1b.cmml" xref="S3.SS3.p2.10.m2.1.1"><leq id="S3.SS3.p2.10.m2.1.1.4.cmml" xref="S3.SS3.p2.10.m2.1.1.4"></leq><cn id="S3.SS3.p2.10.m2.1.1.3.cmml" type="integer" xref="S3.SS3.p2.10.m2.1.1.3">1</cn><ci id="S3.SS3.p2.10.m2.1.1.5.cmml" xref="S3.SS3.p2.10.m2.1.1.5">𝑘</ci></apply><apply id="S3.SS3.p2.10.m2.1.1c.cmml" xref="S3.SS3.p2.10.m2.1.1"><leq id="S3.SS3.p2.10.m2.1.1.6.cmml" xref="S3.SS3.p2.10.m2.1.1.6"></leq><share href="https://arxiv.org/html/2211.11234v4#S3.SS3.p2.10.m2.1.1.5.cmml" id="S3.SS3.p2.10.m2.1.1d.cmml" xref="S3.SS3.p2.10.m2.1.1"></share><apply id="S3.SS3.p2.10.m2.1.1.1.2.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1"><abs id="S3.SS3.p2.10.m2.1.1.1.2.1.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1.2"></abs><apply id="S3.SS3.p2.10.m2.1.1.1.1.1.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1.1"><times id="S3.SS3.p2.10.m2.1.1.1.1.1.2.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1.1.2"></times><ci id="S3.SS3.p2.10.m2.1.1.1.1.1.3.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1.1.3">𝜎</ci><apply id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p2.10.m2.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.10.m2.1c">1\leq k\leq|\sigma(a_{i})|</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.10.m2.1d">1 ≤ italic_k ≤ | italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) |</annotation></semantics></math>, where <math alttext="\sigma(a_{i})_{k}\in\cal B" class="ltx_Math" display="inline" id="S3.SS3.p2.11.m3.1"><semantics id="S3.SS3.p2.11.m3.1a"><mrow id="S3.SS3.p2.11.m3.1.1" xref="S3.SS3.p2.11.m3.1.1.cmml"><mrow id="S3.SS3.p2.11.m3.1.1.1" xref="S3.SS3.p2.11.m3.1.1.1.cmml"><mi id="S3.SS3.p2.11.m3.1.1.1.3" xref="S3.SS3.p2.11.m3.1.1.1.3.cmml">σ</mi><mo id="S3.SS3.p2.11.m3.1.1.1.2" xref="S3.SS3.p2.11.m3.1.1.1.2.cmml">⁢</mo><msub id="S3.SS3.p2.11.m3.1.1.1.1" xref="S3.SS3.p2.11.m3.1.1.1.1.cmml"><mrow id="S3.SS3.p2.11.m3.1.1.1.1.1.1" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p2.11.m3.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p2.11.m3.1.1.1.1.1.1.1" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.2" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.3" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS3.p2.11.m3.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.SS3.p2.11.m3.1.1.1.1.3" xref="S3.SS3.p2.11.m3.1.1.1.1.3.cmml">k</mi></msub></mrow><mo id="S3.SS3.p2.11.m3.1.1.2" xref="S3.SS3.p2.11.m3.1.1.2.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.11.m3.1.1.3" xref="S3.SS3.p2.11.m3.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.11.m3.1b"><apply id="S3.SS3.p2.11.m3.1.1.cmml" xref="S3.SS3.p2.11.m3.1.1"><in id="S3.SS3.p2.11.m3.1.1.2.cmml" xref="S3.SS3.p2.11.m3.1.1.2"></in><apply id="S3.SS3.p2.11.m3.1.1.1.cmml" xref="S3.SS3.p2.11.m3.1.1.1"><times id="S3.SS3.p2.11.m3.1.1.1.2.cmml" xref="S3.SS3.p2.11.m3.1.1.1.2"></times><ci id="S3.SS3.p2.11.m3.1.1.1.3.cmml" xref="S3.SS3.p2.11.m3.1.1.1.3">𝜎</ci><apply id="S3.SS3.p2.11.m3.1.1.1.1.cmml" xref="S3.SS3.p2.11.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.11.m3.1.1.1.1.2.cmml" xref="S3.SS3.p2.11.m3.1.1.1.1">subscript</csymbol><apply id="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p2.11.m3.1.1.1.1.1.1.1.3">𝑖</ci></apply><ci id="S3.SS3.p2.11.m3.1.1.1.1.3.cmml" xref="S3.SS3.p2.11.m3.1.1.1.1.3">𝑘</ci></apply></apply><ci id="S3.SS3.p2.11.m3.1.1.3.cmml" xref="S3.SS3.p2.11.m3.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.11.m3.1c">\sigma(a_{i})_{k}\in\cal B</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.11.m3.1d">italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_B</annotation></semantics></math> denotes the <math alttext="k" class="ltx_Math" display="inline" id="S3.SS3.p2.12.m4.1"><semantics id="S3.SS3.p2.12.m4.1a"><mi id="S3.SS3.p2.12.m4.1.1" xref="S3.SS3.p2.12.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.12.m4.1b"><ci id="S3.SS3.p2.12.m4.1.1.cmml" xref="S3.SS3.p2.12.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.12.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.12.m4.1d">italic_k</annotation></semantics></math>-th letter of the word <math alttext="\sigma(a_{i})\in\cal B^{*}" class="ltx_Math" display="inline" id="S3.SS3.p2.13.m5.1"><semantics id="S3.SS3.p2.13.m5.1a"><mrow id="S3.SS3.p2.13.m5.1.1" xref="S3.SS3.p2.13.m5.1.1.cmml"><mrow id="S3.SS3.p2.13.m5.1.1.1" xref="S3.SS3.p2.13.m5.1.1.1.cmml"><mi id="S3.SS3.p2.13.m5.1.1.1.3" xref="S3.SS3.p2.13.m5.1.1.1.3.cmml">σ</mi><mo id="S3.SS3.p2.13.m5.1.1.1.2" xref="S3.SS3.p2.13.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p2.13.m5.1.1.1.1.1" xref="S3.SS3.p2.13.m5.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p2.13.m5.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p2.13.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p2.13.m5.1.1.1.1.1.1" xref="S3.SS3.p2.13.m5.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p2.13.m5.1.1.1.1.1.1.2" xref="S3.SS3.p2.13.m5.1.1.1.1.1.1.2.cmml">a</mi><mi id="S3.SS3.p2.13.m5.1.1.1.1.1.1.3" xref="S3.SS3.p2.13.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS3.p2.13.m5.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p2.13.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p2.13.m5.1.1.2" xref="S3.SS3.p2.13.m5.1.1.2.cmml">∈</mo><msup id="S3.SS3.p2.13.m5.1.1.3" xref="S3.SS3.p2.13.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.13.m5.1.1.3.2" xref="S3.SS3.p2.13.m5.1.1.3.2.cmml">ℬ</mi><mo id="S3.SS3.p2.13.m5.1.1.3.3" xref="S3.SS3.p2.13.m5.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.13.m5.1b"><apply id="S3.SS3.p2.13.m5.1.1.cmml" xref="S3.SS3.p2.13.m5.1.1"><in id="S3.SS3.p2.13.m5.1.1.2.cmml" xref="S3.SS3.p2.13.m5.1.1.2"></in><apply id="S3.SS3.p2.13.m5.1.1.1.cmml" xref="S3.SS3.p2.13.m5.1.1.1"><times id="S3.SS3.p2.13.m5.1.1.1.2.cmml" xref="S3.SS3.p2.13.m5.1.1.1.2"></times><ci id="S3.SS3.p2.13.m5.1.1.1.3.cmml" xref="S3.SS3.p2.13.m5.1.1.1.3">𝜎</ci><apply id="S3.SS3.p2.13.m5.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.13.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.13.m5.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p2.13.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p2.13.m5.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p2.13.m5.1.1.1.1.1.1.2">𝑎</ci><ci id="S3.SS3.p2.13.m5.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p2.13.m5.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S3.SS3.p2.13.m5.1.1.3.cmml" xref="S3.SS3.p2.13.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p2.13.m5.1.1.3.1.cmml" xref="S3.SS3.p2.13.m5.1.1.3">superscript</csymbol><ci id="S3.SS3.p2.13.m5.1.1.3.2.cmml" xref="S3.SS3.p2.13.m5.1.1.3.2">ℬ</ci><times id="S3.SS3.p2.13.m5.1.1.3.3.cmml" xref="S3.SS3.p2.13.m5.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.13.m5.1c">\sigma(a_{i})\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.13.m5.1d">italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_defnrem" id="S3.Thmthm6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm6.1.1.1">Definition-Remark 3.6</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm6.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm6.p1"> <p class="ltx_p" id="S3.Thmthm6.p1.1">(1) From the above definitions we observe that any non-erasing monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm6.p1.1.m1.1"><semantics id="S3.Thmthm6.p1.1.m1.1a"><mrow id="S3.Thmthm6.p1.1.m1.1.1" xref="S3.Thmthm6.p1.1.m1.1.1.cmml"><mi id="S3.Thmthm6.p1.1.m1.1.1.2" xref="S3.Thmthm6.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S3.Thmthm6.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm6.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm6.p1.1.m1.1.1.3" xref="S3.Thmthm6.p1.1.m1.1.1.3.cmml"><msup id="S3.Thmthm6.p1.1.m1.1.1.3.2" xref="S3.Thmthm6.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p1.1.m1.1.1.3.2.2" xref="S3.Thmthm6.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm6.p1.1.m1.1.1.3.2.3" xref="S3.Thmthm6.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm6.p1.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm6.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.Thmthm6.p1.1.m1.1.1.3.3" xref="S3.Thmthm6.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p1.1.m1.1.1.3.3.2" xref="S3.Thmthm6.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.Thmthm6.p1.1.m1.1.1.3.3.3" xref="S3.Thmthm6.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p1.1.m1.1b"><apply id="S3.Thmthm6.p1.1.m1.1.1.cmml" xref="S3.Thmthm6.p1.1.m1.1.1"><ci id="S3.Thmthm6.p1.1.m1.1.1.1.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.1">:</ci><ci id="S3.Thmthm6.p1.1.m1.1.1.2.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.2">𝜎</ci><apply id="S3.Thmthm6.p1.1.m1.1.1.3.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3"><ci id="S3.Thmthm6.p1.1.m1.1.1.3.1.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm6.p1.1.m1.1.1.3.2.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm6.p1.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm6.p1.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm6.p1.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm6.p1.1.m1.1.1.3.3.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm6.p1.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.Thmthm6.p1.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.Thmthm6.p1.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm6.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> admits a <span class="ltx_text ltx_font_italic" id="S3.Thmthm6.p1.1.1">canonical decomposition</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(3.4)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma=\alpha_{\sigma}\circ\pi_{\sigma}" class="ltx_Math" display="block" id="S3.E4.m1.1"><semantics id="S3.E4.m1.1a"><mrow id="S3.E4.m1.1.1" xref="S3.E4.m1.1.1.cmml"><mi id="S3.E4.m1.1.1.2" xref="S3.E4.m1.1.1.2.cmml">σ</mi><mo id="S3.E4.m1.1.1.1" xref="S3.E4.m1.1.1.1.cmml">=</mo><mrow id="S3.E4.m1.1.1.3" xref="S3.E4.m1.1.1.3.cmml"><msub id="S3.E4.m1.1.1.3.2" xref="S3.E4.m1.1.1.3.2.cmml"><mi id="S3.E4.m1.1.1.3.2.2" xref="S3.E4.m1.1.1.3.2.2.cmml">α</mi><mi id="S3.E4.m1.1.1.3.2.3" xref="S3.E4.m1.1.1.3.2.3.cmml">σ</mi></msub><mo id="S3.E4.m1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.E4.m1.1.1.3.1.cmml">∘</mo><msub id="S3.E4.m1.1.1.3.3" xref="S3.E4.m1.1.1.3.3.cmml"><mi id="S3.E4.m1.1.1.3.3.2" xref="S3.E4.m1.1.1.3.3.2.cmml">π</mi><mi id="S3.E4.m1.1.1.3.3.3" xref="S3.E4.m1.1.1.3.3.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E4.m1.1b"><apply id="S3.E4.m1.1.1.cmml" xref="S3.E4.m1.1.1"><eq id="S3.E4.m1.1.1.1.cmml" xref="S3.E4.m1.1.1.1"></eq><ci id="S3.E4.m1.1.1.2.cmml" xref="S3.E4.m1.1.1.2">𝜎</ci><apply id="S3.E4.m1.1.1.3.cmml" xref="S3.E4.m1.1.1.3"><compose id="S3.E4.m1.1.1.3.1.cmml" xref="S3.E4.m1.1.1.3.1"></compose><apply id="S3.E4.m1.1.1.3.2.cmml" xref="S3.E4.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.E4.m1.1.1.3.2.1.cmml" xref="S3.E4.m1.1.1.3.2">subscript</csymbol><ci id="S3.E4.m1.1.1.3.2.2.cmml" xref="S3.E4.m1.1.1.3.2.2">𝛼</ci><ci id="S3.E4.m1.1.1.3.2.3.cmml" xref="S3.E4.m1.1.1.3.2.3">𝜎</ci></apply><apply id="S3.E4.m1.1.1.3.3.cmml" xref="S3.E4.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.E4.m1.1.1.3.3.1.cmml" xref="S3.E4.m1.1.1.3.3">subscript</csymbol><ci id="S3.E4.m1.1.1.3.3.2.cmml" xref="S3.E4.m1.1.1.3.3.2">𝜋</ci><ci id="S3.E4.m1.1.1.3.3.3.cmml" xref="S3.E4.m1.1.1.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E4.m1.1c">\sigma=\alpha_{\sigma}\circ\pi_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.1d">italic_σ = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∘ italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmthm6.p1.3">as product of a subdivision morphism <math alttext="\pi_{\sigma}" class="ltx_Math" display="inline" id="S3.Thmthm6.p1.2.m1.1"><semantics id="S3.Thmthm6.p1.2.m1.1a"><msub id="S3.Thmthm6.p1.2.m1.1.1" xref="S3.Thmthm6.p1.2.m1.1.1.cmml"><mi id="S3.Thmthm6.p1.2.m1.1.1.2" xref="S3.Thmthm6.p1.2.m1.1.1.2.cmml">π</mi><mi id="S3.Thmthm6.p1.2.m1.1.1.3" xref="S3.Thmthm6.p1.2.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p1.2.m1.1b"><apply id="S3.Thmthm6.p1.2.m1.1.1.cmml" xref="S3.Thmthm6.p1.2.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p1.2.m1.1.1.1.cmml" xref="S3.Thmthm6.p1.2.m1.1.1">subscript</csymbol><ci id="S3.Thmthm6.p1.2.m1.1.1.2.cmml" xref="S3.Thmthm6.p1.2.m1.1.1.2">𝜋</ci><ci id="S3.Thmthm6.p1.2.m1.1.1.3.cmml" xref="S3.Thmthm6.p1.2.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p1.2.m1.1c">\pi_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p1.2.m1.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> with a subsequent letter-to-letter morphism <math alttext="\alpha_{\sigma}" class="ltx_Math" display="inline" id="S3.Thmthm6.p1.3.m2.1"><semantics id="S3.Thmthm6.p1.3.m2.1a"><msub id="S3.Thmthm6.p1.3.m2.1.1" xref="S3.Thmthm6.p1.3.m2.1.1.cmml"><mi id="S3.Thmthm6.p1.3.m2.1.1.2" xref="S3.Thmthm6.p1.3.m2.1.1.2.cmml">α</mi><mi id="S3.Thmthm6.p1.3.m2.1.1.3" xref="S3.Thmthm6.p1.3.m2.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p1.3.m2.1b"><apply id="S3.Thmthm6.p1.3.m2.1.1.cmml" xref="S3.Thmthm6.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p1.3.m2.1.1.1.cmml" xref="S3.Thmthm6.p1.3.m2.1.1">subscript</csymbol><ci id="S3.Thmthm6.p1.3.m2.1.1.2.cmml" xref="S3.Thmthm6.p1.3.m2.1.1.2">𝛼</ci><ci id="S3.Thmthm6.p1.3.m2.1.1.3.cmml" xref="S3.Thmthm6.p1.3.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p1.3.m2.1c">\alpha_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p1.3.m2.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S3.Thmthm6.p2"> <p class="ltx_p" id="S3.Thmthm6.p2.9">(2) As a consequence, the morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.1.m1.1"><semantics id="S3.Thmthm6.p2.1.m1.1a"><mi id="S3.Thmthm6.p2.1.m1.1.1" xref="S3.Thmthm6.p2.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.1.m1.1b"><ci id="S3.Thmthm6.p2.1.m1.1.1.cmml" xref="S3.Thmthm6.p2.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.1.m1.1d">italic_σ</annotation></semantics></math> induces a canonical <span class="ltx_text ltx_font_italic" id="S3.Thmthm6.p2.9.1">measure transfer</span> <math alttext="\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.2.m2.2"><semantics id="S3.Thmthm6.p2.2.m2.2a"><mrow id="S3.Thmthm6.p2.2.m2.2.2" xref="S3.Thmthm6.p2.2.m2.2.2.cmml"><mrow id="S3.Thmthm6.p2.2.m2.2.2.4" xref="S3.Thmthm6.p2.2.m2.2.2.4.cmml"><mi id="S3.Thmthm6.p2.2.m2.2.2.4.2" xref="S3.Thmthm6.p2.2.m2.2.2.4.2.cmml">σ</mi><mo id="S3.Thmthm6.p2.2.m2.2.2.4.1" xref="S3.Thmthm6.p2.2.m2.2.2.4.1.cmml">⁢</mo><mi id="S3.Thmthm6.p2.2.m2.2.2.4.3" xref="S3.Thmthm6.p2.2.m2.2.2.4.3.cmml">M</mi></mrow><mo id="S3.Thmthm6.p2.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm6.p2.2.m2.2.2.3.cmml">:</mo><mrow id="S3.Thmthm6.p2.2.m2.2.2.2" xref="S3.Thmthm6.p2.2.m2.2.2.2.cmml"><mrow id="S3.Thmthm6.p2.2.m2.1.1.1.1" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p2.2.m2.1.1.1.1.3" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.3.cmml">ℳ</mi><mo id="S3.Thmthm6.p2.2.m2.1.1.1.1.2" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.cmml"><mo id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.2" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.3" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm6.p2.2.m2.2.2.2.3" stretchy="false" xref="S3.Thmthm6.p2.2.m2.2.2.2.3.cmml">→</mo><mrow id="S3.Thmthm6.p2.2.m2.2.2.2.2" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p2.2.m2.2.2.2.2.3" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.3.cmml">ℳ</mi><mo id="S3.Thmthm6.p2.2.m2.2.2.2.2.2" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.2.cmml">⁢</mo><mrow id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.cmml"><mo id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.2" stretchy="false" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.cmml">(</mo><msup id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.2" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.2.cmml">ℬ</mi><mi id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.3" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.3" stretchy="false" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.2.m2.2b"><apply id="S3.Thmthm6.p2.2.m2.2.2.cmml" xref="S3.Thmthm6.p2.2.m2.2.2"><ci id="S3.Thmthm6.p2.2.m2.2.2.3.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.3">:</ci><apply id="S3.Thmthm6.p2.2.m2.2.2.4.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.4"><times id="S3.Thmthm6.p2.2.m2.2.2.4.1.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.4.1"></times><ci id="S3.Thmthm6.p2.2.m2.2.2.4.2.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.4.2">𝜎</ci><ci id="S3.Thmthm6.p2.2.m2.2.2.4.3.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.4.3">𝑀</ci></apply><apply id="S3.Thmthm6.p2.2.m2.2.2.2.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2"><ci id="S3.Thmthm6.p2.2.m2.2.2.2.3.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2.3">→</ci><apply id="S3.Thmthm6.p2.2.m2.1.1.1.1.cmml" xref="S3.Thmthm6.p2.2.m2.1.1.1.1"><times id="S3.Thmthm6.p2.2.m2.1.1.1.1.2.cmml" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.2"></times><ci id="S3.Thmthm6.p2.2.m2.1.1.1.1.3.cmml" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.3">ℳ</ci><apply id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.2">𝒜</ci><ci id="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm6.p2.2.m2.1.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S3.Thmthm6.p2.2.m2.2.2.2.2.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2.2"><times id="S3.Thmthm6.p2.2.m2.2.2.2.2.2.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.2"></times><ci id="S3.Thmthm6.p2.2.m2.2.2.2.2.3.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.3">ℳ</ci><apply id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.1.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1">superscript</csymbol><ci id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.2.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.2">ℬ</ci><ci id="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.3.cmml" xref="S3.Thmthm6.p2.2.m2.2.2.2.2.1.1.1.3">ℤ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.2.m2.2c">\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.2.m2.2d">italic_σ italic_M : caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> which maps any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.3.m3.1"><semantics id="S3.Thmthm6.p2.3.m3.1a"><mi id="S3.Thmthm6.p2.3.m3.1.1" xref="S3.Thmthm6.p2.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.3.m3.1b"><ci id="S3.Thmthm6.p2.3.m3.1.1.cmml" xref="S3.Thmthm6.p2.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.3.m3.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.4.m4.1"><semantics id="S3.Thmthm6.p2.4.m4.1a"><msup id="S3.Thmthm6.p2.4.m4.1.1" xref="S3.Thmthm6.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p2.4.m4.1.1.2" xref="S3.Thmthm6.p2.4.m4.1.1.2.cmml">𝒜</mi><mi id="S3.Thmthm6.p2.4.m4.1.1.3" xref="S3.Thmthm6.p2.4.m4.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.4.m4.1b"><apply id="S3.Thmthm6.p2.4.m4.1.1.cmml" xref="S3.Thmthm6.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.4.m4.1.1.1.cmml" xref="S3.Thmthm6.p2.4.m4.1.1">superscript</csymbol><ci id="S3.Thmthm6.p2.4.m4.1.1.2.cmml" xref="S3.Thmthm6.p2.4.m4.1.1.2">𝒜</ci><ci id="S3.Thmthm6.p2.4.m4.1.1.3.cmml" xref="S3.Thmthm6.p2.4.m4.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.4.m4.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.4.m4.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> to the measure <math alttext="(\alpha_{\sigma})_{*}(\mu_{\ell_{\sigma}})" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.5.m5.2"><semantics id="S3.Thmthm6.p2.5.m5.2a"><mrow id="S3.Thmthm6.p2.5.m5.2.2" xref="S3.Thmthm6.p2.5.m5.2.2.cmml"><msub id="S3.Thmthm6.p2.5.m5.1.1.1" xref="S3.Thmthm6.p2.5.m5.1.1.1.cmml"><mrow id="S3.Thmthm6.p2.5.m5.1.1.1.1.1" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.2" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.2.cmml">α</mi><mi id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.3" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.3.cmml">σ</mi></msub><mo id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmthm6.p2.5.m5.1.1.1.3" xref="S3.Thmthm6.p2.5.m5.1.1.1.3.cmml">∗</mo></msub><mo id="S3.Thmthm6.p2.5.m5.2.2.3" xref="S3.Thmthm6.p2.5.m5.2.2.3.cmml">⁢</mo><mrow id="S3.Thmthm6.p2.5.m5.2.2.2.1" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.cmml"><mo id="S3.Thmthm6.p2.5.m5.2.2.2.1.2" stretchy="false" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.cmml">(</mo><msub id="S3.Thmthm6.p2.5.m5.2.2.2.1.1" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.cmml"><mi id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.2" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.2.cmml">μ</mi><msub id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.cmml"><mi id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.2" mathvariant="normal" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.2.cmml">ℓ</mi><mi id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.3" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.3.cmml">σ</mi></msub></msub><mo id="S3.Thmthm6.p2.5.m5.2.2.2.1.3" stretchy="false" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.5.m5.2b"><apply id="S3.Thmthm6.p2.5.m5.2.2.cmml" xref="S3.Thmthm6.p2.5.m5.2.2"><times id="S3.Thmthm6.p2.5.m5.2.2.3.cmml" xref="S3.Thmthm6.p2.5.m5.2.2.3"></times><apply id="S3.Thmthm6.p2.5.m5.1.1.1.cmml" xref="S3.Thmthm6.p2.5.m5.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.5.m5.1.1.1.2.cmml" xref="S3.Thmthm6.p2.5.m5.1.1.1">subscript</csymbol><apply id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.cmml" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.2">𝛼</ci><ci id="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm6.p2.5.m5.1.1.1.1.1.1.3">𝜎</ci></apply><times id="S3.Thmthm6.p2.5.m5.1.1.1.3.cmml" xref="S3.Thmthm6.p2.5.m5.1.1.1.3"></times></apply><apply id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.cmml" xref="S3.Thmthm6.p2.5.m5.2.2.2.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.1.cmml" xref="S3.Thmthm6.p2.5.m5.2.2.2.1">subscript</csymbol><ci id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.2.cmml" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.2">𝜇</ci><apply id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.cmml" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.1.cmml" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3">subscript</csymbol><ci id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.2.cmml" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.2">ℓ</ci><ci id="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.3.cmml" xref="S3.Thmthm6.p2.5.m5.2.2.2.1.1.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.5.m5.2c">(\alpha_{\sigma})_{*}(\mu_{\ell_{\sigma}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.5.m5.2d">( italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>, which for simplicity will sometimes be denoted by <math alttext="\mu^{\sigma}" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.6.m6.1"><semantics id="S3.Thmthm6.p2.6.m6.1a"><msup id="S3.Thmthm6.p2.6.m6.1.1" xref="S3.Thmthm6.p2.6.m6.1.1.cmml"><mi id="S3.Thmthm6.p2.6.m6.1.1.2" xref="S3.Thmthm6.p2.6.m6.1.1.2.cmml">μ</mi><mi id="S3.Thmthm6.p2.6.m6.1.1.3" xref="S3.Thmthm6.p2.6.m6.1.1.3.cmml">σ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.6.m6.1b"><apply id="S3.Thmthm6.p2.6.m6.1.1.cmml" xref="S3.Thmthm6.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.6.m6.1.1.1.cmml" xref="S3.Thmthm6.p2.6.m6.1.1">superscript</csymbol><ci id="S3.Thmthm6.p2.6.m6.1.1.2.cmml" xref="S3.Thmthm6.p2.6.m6.1.1.2">𝜇</ci><ci id="S3.Thmthm6.p2.6.m6.1.1.3.cmml" xref="S3.Thmthm6.p2.6.m6.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.6.m6.1c">\mu^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.6.m6.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math>. For the associated weight function <math alttext="\sigma M(\mu)" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.7.m7.1"><semantics id="S3.Thmthm6.p2.7.m7.1a"><mrow id="S3.Thmthm6.p2.7.m7.1.2" xref="S3.Thmthm6.p2.7.m7.1.2.cmml"><mi id="S3.Thmthm6.p2.7.m7.1.2.2" xref="S3.Thmthm6.p2.7.m7.1.2.2.cmml">σ</mi><mo id="S3.Thmthm6.p2.7.m7.1.2.1" xref="S3.Thmthm6.p2.7.m7.1.2.1.cmml">⁢</mo><mi id="S3.Thmthm6.p2.7.m7.1.2.3" xref="S3.Thmthm6.p2.7.m7.1.2.3.cmml">M</mi><mo id="S3.Thmthm6.p2.7.m7.1.2.1a" xref="S3.Thmthm6.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm6.p2.7.m7.1.2.4.2" xref="S3.Thmthm6.p2.7.m7.1.2.cmml"><mo id="S3.Thmthm6.p2.7.m7.1.2.4.2.1" stretchy="false" xref="S3.Thmthm6.p2.7.m7.1.2.cmml">(</mo><mi id="S3.Thmthm6.p2.7.m7.1.1" xref="S3.Thmthm6.p2.7.m7.1.1.cmml">μ</mi><mo id="S3.Thmthm6.p2.7.m7.1.2.4.2.2" stretchy="false" xref="S3.Thmthm6.p2.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.7.m7.1b"><apply id="S3.Thmthm6.p2.7.m7.1.2.cmml" xref="S3.Thmthm6.p2.7.m7.1.2"><times id="S3.Thmthm6.p2.7.m7.1.2.1.cmml" xref="S3.Thmthm6.p2.7.m7.1.2.1"></times><ci id="S3.Thmthm6.p2.7.m7.1.2.2.cmml" xref="S3.Thmthm6.p2.7.m7.1.2.2">𝜎</ci><ci id="S3.Thmthm6.p2.7.m7.1.2.3.cmml" xref="S3.Thmthm6.p2.7.m7.1.2.3">𝑀</ci><ci id="S3.Thmthm6.p2.7.m7.1.1.cmml" xref="S3.Thmthm6.p2.7.m7.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.7.m7.1c">\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.7.m7.1d">italic_σ italic_M ( italic_μ )</annotation></semantics></math> on <math alttext="\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.8.m8.1"><semantics id="S3.Thmthm6.p2.8.m8.1a"><msup id="S3.Thmthm6.p2.8.m8.1.1" xref="S3.Thmthm6.p2.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p2.8.m8.1.1.2" xref="S3.Thmthm6.p2.8.m8.1.1.2.cmml">ℬ</mi><mo id="S3.Thmthm6.p2.8.m8.1.1.3" xref="S3.Thmthm6.p2.8.m8.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.8.m8.1b"><apply id="S3.Thmthm6.p2.8.m8.1.1.cmml" xref="S3.Thmthm6.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.8.m8.1.1.1.cmml" xref="S3.Thmthm6.p2.8.m8.1.1">superscript</csymbol><ci id="S3.Thmthm6.p2.8.m8.1.1.2.cmml" xref="S3.Thmthm6.p2.8.m8.1.1.2">ℬ</ci><times id="S3.Thmthm6.p2.8.m8.1.1.3.cmml" xref="S3.Thmthm6.p2.8.m8.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.8.m8.1c">\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.8.m8.1d">caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> we obtain, for any <math alttext="w\in\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.9.m9.1"><semantics id="S3.Thmthm6.p2.9.m9.1a"><mrow id="S3.Thmthm6.p2.9.m9.1.1" xref="S3.Thmthm6.p2.9.m9.1.1.cmml"><mi id="S3.Thmthm6.p2.9.m9.1.1.2" xref="S3.Thmthm6.p2.9.m9.1.1.2.cmml">w</mi><mo id="S3.Thmthm6.p2.9.m9.1.1.1" xref="S3.Thmthm6.p2.9.m9.1.1.1.cmml">∈</mo><msup id="S3.Thmthm6.p2.9.m9.1.1.3" xref="S3.Thmthm6.p2.9.m9.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p2.9.m9.1.1.3.2" xref="S3.Thmthm6.p2.9.m9.1.1.3.2.cmml">ℬ</mi><mo id="S3.Thmthm6.p2.9.m9.1.1.3.3" xref="S3.Thmthm6.p2.9.m9.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.9.m9.1b"><apply id="S3.Thmthm6.p2.9.m9.1.1.cmml" xref="S3.Thmthm6.p2.9.m9.1.1"><in id="S3.Thmthm6.p2.9.m9.1.1.1.cmml" xref="S3.Thmthm6.p2.9.m9.1.1.1"></in><ci id="S3.Thmthm6.p2.9.m9.1.1.2.cmml" xref="S3.Thmthm6.p2.9.m9.1.1.2">𝑤</ci><apply id="S3.Thmthm6.p2.9.m9.1.1.3.cmml" xref="S3.Thmthm6.p2.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.9.m9.1.1.3.1.cmml" xref="S3.Thmthm6.p2.9.m9.1.1.3">superscript</csymbol><ci id="S3.Thmthm6.p2.9.m9.1.1.3.2.cmml" xref="S3.Thmthm6.p2.9.m9.1.1.3.2">ℬ</ci><times id="S3.Thmthm6.p2.9.m9.1.1.3.3.cmml" xref="S3.Thmthm6.p2.9.m9.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.9.m9.1c">w\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.9.m9.1d">italic_w ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, the formula</p> <table class="ltx_equation ltx_eqn_table" id="S3.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(3.5)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma M(\mu)(w)\,\,\,[=:\mu^{\sigma}(w)]=\sum_{u\,\in\,{\alpha_{\sigma}^{-1}(% w)}}\mu_{\ell_{\sigma}}(u)\,=\sum_{u\,\in\,{\alpha_{\sigma}^{-1}(w)}}\mu(% \widehat{u})\,," class="ltx_math_unparsed" display="block" id="S3.E5.m1.4"><semantics id="S3.E5.m1.4a"><mrow id="S3.E5.m1.4b"><mi id="S3.E5.m1.4.5">σ</mi><mi id="S3.E5.m1.4.6">M</mi><mrow id="S3.E5.m1.4.7"><mo id="S3.E5.m1.4.7.1" stretchy="false">(</mo><mi id="S3.E5.m1.3.3">μ</mi><mo id="S3.E5.m1.4.7.2" stretchy="false">)</mo></mrow><mrow id="S3.E5.m1.4.8"><mo id="S3.E5.m1.4.8.1" stretchy="false">(</mo><mi id="S3.E5.m1.4.4">w</mi><mo id="S3.E5.m1.4.8.2" rspace="0.500em" stretchy="false">)</mo></mrow><mrow id="S3.E5.m1.4.9"><mo id="S3.E5.m1.4.9.1" stretchy="false">[</mo><mo id="S3.E5.m1.4.9.2" lspace="0em" rspace="0em">=</mo><mo id="S3.E5.m1.4.9.3" rspace="0.278em">:</mo><msup id="S3.E5.m1.4.9.4"><mi id="S3.E5.m1.4.9.4.2">μ</mi><mi id="S3.E5.m1.4.9.4.3">σ</mi></msup><mrow id="S3.E5.m1.4.9.5"><mo id="S3.E5.m1.4.9.5.1" stretchy="false">(</mo><mi id="S3.E5.m1.4.9.5.2">w</mi><mo id="S3.E5.m1.4.9.5.3" stretchy="false">)</mo></mrow><mo id="S3.E5.m1.4.9.6" stretchy="false">]</mo></mrow><mo id="S3.E5.m1.4.10" rspace="0.111em">=</mo><munder id="S3.E5.m1.4.11"><mo id="S3.E5.m1.4.11.2" movablelimits="false">∑</mo><mrow id="S3.E5.m1.1.1.1"><mi id="S3.E5.m1.1.1.1.3">u</mi><mo id="S3.E5.m1.1.1.1.2" lspace="0.448em" rspace="0.448em">∈</mo><mrow id="S3.E5.m1.1.1.1.4"><msubsup id="S3.E5.m1.1.1.1.4.2"><mi id="S3.E5.m1.1.1.1.4.2.2.2">α</mi><mi id="S3.E5.m1.1.1.1.4.2.2.3">σ</mi><mrow id="S3.E5.m1.1.1.1.4.2.3"><mo id="S3.E5.m1.1.1.1.4.2.3a">−</mo><mn id="S3.E5.m1.1.1.1.4.2.3.2">1</mn></mrow></msubsup><mo id="S3.E5.m1.1.1.1.4.1">⁢</mo><mrow id="S3.E5.m1.1.1.1.4.3.2"><mo id="S3.E5.m1.1.1.1.4.3.2.1" stretchy="false">(</mo><mi id="S3.E5.m1.1.1.1.1">w</mi><mo id="S3.E5.m1.1.1.1.4.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow></munder><msub id="S3.E5.m1.4.12"><mi id="S3.E5.m1.4.12.2">μ</mi><msub id="S3.E5.m1.4.12.3"><mi id="S3.E5.m1.4.12.3.2" mathvariant="normal">ℓ</mi><mi id="S3.E5.m1.4.12.3.3">σ</mi></msub></msub><mrow id="S3.E5.m1.4.13"><mo id="S3.E5.m1.4.13.1" stretchy="false">(</mo><mi id="S3.E5.m1.4.13.2">u</mi><mo id="S3.E5.m1.4.13.3" rspace="0.170em" stretchy="false">)</mo></mrow><mo id="S3.E5.m1.4.14" rspace="0.111em">=</mo><munder id="S3.E5.m1.4.15"><mo id="S3.E5.m1.4.15.2" movablelimits="false">∑</mo><mrow id="S3.E5.m1.2.2.1"><mi id="S3.E5.m1.2.2.1.3">u</mi><mo id="S3.E5.m1.2.2.1.2" lspace="0.448em" rspace="0.448em">∈</mo><mrow id="S3.E5.m1.2.2.1.4"><msubsup id="S3.E5.m1.2.2.1.4.2"><mi id="S3.E5.m1.2.2.1.4.2.2.2">α</mi><mi id="S3.E5.m1.2.2.1.4.2.2.3">σ</mi><mrow id="S3.E5.m1.2.2.1.4.2.3"><mo id="S3.E5.m1.2.2.1.4.2.3a">−</mo><mn id="S3.E5.m1.2.2.1.4.2.3.2">1</mn></mrow></msubsup><mo id="S3.E5.m1.2.2.1.4.1">⁢</mo><mrow id="S3.E5.m1.2.2.1.4.3.2"><mo id="S3.E5.m1.2.2.1.4.3.2.1" stretchy="false">(</mo><mi id="S3.E5.m1.2.2.1.1">w</mi><mo id="S3.E5.m1.2.2.1.4.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow></munder><mi id="S3.E5.m1.4.16">μ</mi><mrow id="S3.E5.m1.4.17"><mo id="S3.E5.m1.4.17.1" stretchy="false">(</mo><mover accent="true" id="S3.E5.m1.4.17.2"><mi id="S3.E5.m1.4.17.2.2">u</mi><mo id="S3.E5.m1.4.17.2.1">^</mo></mover><mo id="S3.E5.m1.4.17.3" rspace="0.170em" stretchy="false">)</mo></mrow><mo id="S3.E5.m1.4.18">,</mo></mrow><annotation encoding="application/x-tex" id="S3.E5.m1.4c">\sigma M(\mu)(w)\,\,\,[=:\mu^{\sigma}(w)]=\sum_{u\,\in\,{\alpha_{\sigma}^{-1}(% w)}}\mu_{\ell_{\sigma}}(u)\,=\sum_{u\,\in\,{\alpha_{\sigma}^{-1}(w)}}\mu(% \widehat{u})\,,</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.4d">italic_σ italic_M ( italic_μ ) ( italic_w ) [ = : italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_w ) ] = ∑ start_POSTSUBSCRIPT italic_u ∈ italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w ) end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ) = ∑ start_POSTSUBSCRIPT italic_u ∈ italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w ) end_POSTSUBSCRIPT italic_μ ( over^ start_ARG italic_u end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmthm6.p2.13">where <math alttext="\widehat{u}\in\cal A^{*}" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.10.m1.1"><semantics id="S3.Thmthm6.p2.10.m1.1a"><mrow id="S3.Thmthm6.p2.10.m1.1.1" xref="S3.Thmthm6.p2.10.m1.1.1.cmml"><mover accent="true" id="S3.Thmthm6.p2.10.m1.1.1.2" xref="S3.Thmthm6.p2.10.m1.1.1.2.cmml"><mi id="S3.Thmthm6.p2.10.m1.1.1.2.2" xref="S3.Thmthm6.p2.10.m1.1.1.2.2.cmml">u</mi><mo id="S3.Thmthm6.p2.10.m1.1.1.2.1" xref="S3.Thmthm6.p2.10.m1.1.1.2.1.cmml">^</mo></mover><mo id="S3.Thmthm6.p2.10.m1.1.1.1" xref="S3.Thmthm6.p2.10.m1.1.1.1.cmml">∈</mo><msup id="S3.Thmthm6.p2.10.m1.1.1.3" xref="S3.Thmthm6.p2.10.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm6.p2.10.m1.1.1.3.2" xref="S3.Thmthm6.p2.10.m1.1.1.3.2.cmml">𝒜</mi><mo id="S3.Thmthm6.p2.10.m1.1.1.3.3" xref="S3.Thmthm6.p2.10.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.10.m1.1b"><apply id="S3.Thmthm6.p2.10.m1.1.1.cmml" xref="S3.Thmthm6.p2.10.m1.1.1"><in id="S3.Thmthm6.p2.10.m1.1.1.1.cmml" xref="S3.Thmthm6.p2.10.m1.1.1.1"></in><apply id="S3.Thmthm6.p2.10.m1.1.1.2.cmml" xref="S3.Thmthm6.p2.10.m1.1.1.2"><ci id="S3.Thmthm6.p2.10.m1.1.1.2.1.cmml" xref="S3.Thmthm6.p2.10.m1.1.1.2.1">^</ci><ci id="S3.Thmthm6.p2.10.m1.1.1.2.2.cmml" xref="S3.Thmthm6.p2.10.m1.1.1.2.2">𝑢</ci></apply><apply id="S3.Thmthm6.p2.10.m1.1.1.3.cmml" xref="S3.Thmthm6.p2.10.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.10.m1.1.1.3.1.cmml" xref="S3.Thmthm6.p2.10.m1.1.1.3">superscript</csymbol><ci id="S3.Thmthm6.p2.10.m1.1.1.3.2.cmml" xref="S3.Thmthm6.p2.10.m1.1.1.3.2">𝒜</ci><times id="S3.Thmthm6.p2.10.m1.1.1.3.3.cmml" xref="S3.Thmthm6.p2.10.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.10.m1.1c">\widehat{u}\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.10.m1.1d">over^ start_ARG italic_u end_ARG ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is defined above in Definition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm3" title="Definition 3.3. ‣ 3.1. Subdivision morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.3</span></a>. Recall from Definition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm3" title="Definition 3.3. ‣ 3.1. Subdivision morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.3</span></a> that if <math alttext="\widehat{u}" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.11.m2.1"><semantics id="S3.Thmthm6.p2.11.m2.1a"><mover accent="true" id="S3.Thmthm6.p2.11.m2.1.1" xref="S3.Thmthm6.p2.11.m2.1.1.cmml"><mi id="S3.Thmthm6.p2.11.m2.1.1.2" xref="S3.Thmthm6.p2.11.m2.1.1.2.cmml">u</mi><mo id="S3.Thmthm6.p2.11.m2.1.1.1" xref="S3.Thmthm6.p2.11.m2.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.11.m2.1b"><apply id="S3.Thmthm6.p2.11.m2.1.1.cmml" xref="S3.Thmthm6.p2.11.m2.1.1"><ci id="S3.Thmthm6.p2.11.m2.1.1.1.cmml" xref="S3.Thmthm6.p2.11.m2.1.1.1">^</ci><ci id="S3.Thmthm6.p2.11.m2.1.1.2.cmml" xref="S3.Thmthm6.p2.11.m2.1.1.2">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.11.m2.1c">\widehat{u}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.11.m2.1d">over^ start_ARG italic_u end_ARG</annotation></semantics></math> doesn’t exist for some <math alttext="u\in{\alpha_{\sigma}^{-1}(w)}" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.12.m3.1"><semantics id="S3.Thmthm6.p2.12.m3.1a"><mrow id="S3.Thmthm6.p2.12.m3.1.2" xref="S3.Thmthm6.p2.12.m3.1.2.cmml"><mi id="S3.Thmthm6.p2.12.m3.1.2.2" xref="S3.Thmthm6.p2.12.m3.1.2.2.cmml">u</mi><mo id="S3.Thmthm6.p2.12.m3.1.2.1" xref="S3.Thmthm6.p2.12.m3.1.2.1.cmml">∈</mo><mrow id="S3.Thmthm6.p2.12.m3.1.2.3" xref="S3.Thmthm6.p2.12.m3.1.2.3.cmml"><msubsup id="S3.Thmthm6.p2.12.m3.1.2.3.2" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.cmml"><mi id="S3.Thmthm6.p2.12.m3.1.2.3.2.2.2" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.2.2.cmml">α</mi><mi id="S3.Thmthm6.p2.12.m3.1.2.3.2.2.3" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.2.3.cmml">σ</mi><mrow id="S3.Thmthm6.p2.12.m3.1.2.3.2.3" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.3.cmml"><mo id="S3.Thmthm6.p2.12.m3.1.2.3.2.3a" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.3.cmml">−</mo><mn id="S3.Thmthm6.p2.12.m3.1.2.3.2.3.2" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S3.Thmthm6.p2.12.m3.1.2.3.1" xref="S3.Thmthm6.p2.12.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm6.p2.12.m3.1.2.3.3.2" xref="S3.Thmthm6.p2.12.m3.1.2.3.cmml"><mo id="S3.Thmthm6.p2.12.m3.1.2.3.3.2.1" stretchy="false" xref="S3.Thmthm6.p2.12.m3.1.2.3.cmml">(</mo><mi id="S3.Thmthm6.p2.12.m3.1.1" xref="S3.Thmthm6.p2.12.m3.1.1.cmml">w</mi><mo id="S3.Thmthm6.p2.12.m3.1.2.3.3.2.2" stretchy="false" xref="S3.Thmthm6.p2.12.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.12.m3.1b"><apply id="S3.Thmthm6.p2.12.m3.1.2.cmml" xref="S3.Thmthm6.p2.12.m3.1.2"><in id="S3.Thmthm6.p2.12.m3.1.2.1.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.1"></in><ci id="S3.Thmthm6.p2.12.m3.1.2.2.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.2">𝑢</ci><apply id="S3.Thmthm6.p2.12.m3.1.2.3.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3"><times id="S3.Thmthm6.p2.12.m3.1.2.3.1.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.1"></times><apply id="S3.Thmthm6.p2.12.m3.1.2.3.2.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.12.m3.1.2.3.2.1.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.2">superscript</csymbol><apply id="S3.Thmthm6.p2.12.m3.1.2.3.2.2.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm6.p2.12.m3.1.2.3.2.2.1.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.2">subscript</csymbol><ci id="S3.Thmthm6.p2.12.m3.1.2.3.2.2.2.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.2.2">𝛼</ci><ci id="S3.Thmthm6.p2.12.m3.1.2.3.2.2.3.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.2.3">𝜎</ci></apply><apply id="S3.Thmthm6.p2.12.m3.1.2.3.2.3.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.3"><minus id="S3.Thmthm6.p2.12.m3.1.2.3.2.3.1.cmml" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.3"></minus><cn id="S3.Thmthm6.p2.12.m3.1.2.3.2.3.2.cmml" type="integer" xref="S3.Thmthm6.p2.12.m3.1.2.3.2.3.2">1</cn></apply></apply><ci id="S3.Thmthm6.p2.12.m3.1.1.cmml" xref="S3.Thmthm6.p2.12.m3.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.12.m3.1c">u\in{\alpha_{\sigma}^{-1}(w)}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.12.m3.1d">italic_u ∈ italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w )</annotation></semantics></math>, then one has formally set <math alttext="\mu(\widehat{u})=0" class="ltx_Math" display="inline" id="S3.Thmthm6.p2.13.m4.1"><semantics id="S3.Thmthm6.p2.13.m4.1a"><mrow id="S3.Thmthm6.p2.13.m4.1.2" xref="S3.Thmthm6.p2.13.m4.1.2.cmml"><mrow id="S3.Thmthm6.p2.13.m4.1.2.2" xref="S3.Thmthm6.p2.13.m4.1.2.2.cmml"><mi id="S3.Thmthm6.p2.13.m4.1.2.2.2" xref="S3.Thmthm6.p2.13.m4.1.2.2.2.cmml">μ</mi><mo id="S3.Thmthm6.p2.13.m4.1.2.2.1" xref="S3.Thmthm6.p2.13.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm6.p2.13.m4.1.2.2.3.2" xref="S3.Thmthm6.p2.13.m4.1.1.cmml"><mo id="S3.Thmthm6.p2.13.m4.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm6.p2.13.m4.1.1.cmml">(</mo><mover accent="true" id="S3.Thmthm6.p2.13.m4.1.1" xref="S3.Thmthm6.p2.13.m4.1.1.cmml"><mi id="S3.Thmthm6.p2.13.m4.1.1.2" xref="S3.Thmthm6.p2.13.m4.1.1.2.cmml">u</mi><mo id="S3.Thmthm6.p2.13.m4.1.1.1" xref="S3.Thmthm6.p2.13.m4.1.1.1.cmml">^</mo></mover><mo id="S3.Thmthm6.p2.13.m4.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm6.p2.13.m4.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm6.p2.13.m4.1.2.1" xref="S3.Thmthm6.p2.13.m4.1.2.1.cmml">=</mo><mn id="S3.Thmthm6.p2.13.m4.1.2.3" xref="S3.Thmthm6.p2.13.m4.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm6.p2.13.m4.1b"><apply id="S3.Thmthm6.p2.13.m4.1.2.cmml" xref="S3.Thmthm6.p2.13.m4.1.2"><eq id="S3.Thmthm6.p2.13.m4.1.2.1.cmml" xref="S3.Thmthm6.p2.13.m4.1.2.1"></eq><apply id="S3.Thmthm6.p2.13.m4.1.2.2.cmml" xref="S3.Thmthm6.p2.13.m4.1.2.2"><times id="S3.Thmthm6.p2.13.m4.1.2.2.1.cmml" xref="S3.Thmthm6.p2.13.m4.1.2.2.1"></times><ci id="S3.Thmthm6.p2.13.m4.1.2.2.2.cmml" xref="S3.Thmthm6.p2.13.m4.1.2.2.2">𝜇</ci><apply id="S3.Thmthm6.p2.13.m4.1.1.cmml" xref="S3.Thmthm6.p2.13.m4.1.2.2.3.2"><ci id="S3.Thmthm6.p2.13.m4.1.1.1.cmml" xref="S3.Thmthm6.p2.13.m4.1.1.1">^</ci><ci id="S3.Thmthm6.p2.13.m4.1.1.2.cmml" xref="S3.Thmthm6.p2.13.m4.1.1.2">𝑢</ci></apply></apply><cn id="S3.Thmthm6.p2.13.m4.1.2.3.cmml" type="integer" xref="S3.Thmthm6.p2.13.m4.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm6.p2.13.m4.1c">\mu(\widehat{u})=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm6.p2.13.m4.1d">italic_μ ( over^ start_ARG italic_u end_ARG ) = 0</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.1">The above canonical decomposition of an arbitrary morphism has been used previously; it can be found for instance in Lemma 3.4 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib6" title="">6</a>]</cite>.</p> </div> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.4. </span>Basic properties of the measure transfer map</h3> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.1"></p> </div> <div class="ltx_para" id="S3.SS4.p2"> <p class="ltx_p" id="S3.SS4.p2.2">In this subsection we want to show some first properties of the measure transfer map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S3.SS4.p2.1.m1.1"><semantics id="S3.SS4.p2.1.m1.1a"><mrow id="S3.SS4.p2.1.m1.1.1" xref="S3.SS4.p2.1.m1.1.1.cmml"><mi id="S3.SS4.p2.1.m1.1.1.2" xref="S3.SS4.p2.1.m1.1.1.2.cmml">σ</mi><mo id="S3.SS4.p2.1.m1.1.1.1" xref="S3.SS4.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.SS4.p2.1.m1.1.1.3" xref="S3.SS4.p2.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.1.m1.1b"><apply id="S3.SS4.p2.1.m1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1"><times id="S3.SS4.p2.1.m1.1.1.1.cmml" xref="S3.SS4.p2.1.m1.1.1.1"></times><ci id="S3.SS4.p2.1.m1.1.1.2.cmml" xref="S3.SS4.p2.1.m1.1.1.2">𝜎</ci><ci id="S3.SS4.p2.1.m1.1.1.3.cmml" xref="S3.SS4.p2.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.1.m1.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.1.m1.1d">italic_σ italic_M</annotation></semantics></math> defined in the previous subsection. We start out with some basic facts: their proof is an elementary (and not very illuminating) exercise, based on the canonical decomposition (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E4" title="In Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.4</span></a>) of any morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS4.p2.2.m2.1"><semantics id="S3.SS4.p2.2.m2.1a"><mi id="S3.SS4.p2.2.m2.1.1" xref="S3.SS4.p2.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p2.2.m2.1b"><ci id="S3.SS4.p2.2.m2.1.1.cmml" xref="S3.SS4.p2.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p2.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p2.2.m2.1d">italic_σ</annotation></semantics></math>; it is hence not carried through here.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="S3.Thmthm7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm7.1.1.1">Lemma 3.7</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm7.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm7.p1"> <p class="ltx_p" id="S3.Thmthm7.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmthm7.p1.2.2">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm7.p1.1.1.m1.1"><semantics id="S3.Thmthm7.p1.1.1.m1.1a"><mrow id="S3.Thmthm7.p1.1.1.m1.1.1" xref="S3.Thmthm7.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmthm7.p1.1.1.m1.1.1.2" xref="S3.Thmthm7.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S3.Thmthm7.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm7.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm7.p1.1.1.m1.1.1.3" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.cmml"><msup id="S3.Thmthm7.p1.1.1.m1.1.1.3.2" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm7.p1.1.1.m1.1.1.3.2.2" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm7.p1.1.1.m1.1.1.3.2.3" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm7.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.Thmthm7.p1.1.1.m1.1.1.3.3" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm7.p1.1.1.m1.1.1.3.3.2" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.Thmthm7.p1.1.1.m1.1.1.3.3.3" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm7.p1.1.1.m1.1b"><apply id="S3.Thmthm7.p1.1.1.m1.1.1.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1"><ci id="S3.Thmthm7.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.1">:</ci><ci id="S3.Thmthm7.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S3.Thmthm7.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3"><ci id="S3.Thmthm7.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm7.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm7.p1.1.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm7.p1.1.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm7.p1.1.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm7.p1.1.1.m1.1.1.3.3.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm7.p1.1.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.Thmthm7.p1.1.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.Thmthm7.p1.1.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm7.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm7.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm7.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-erasing monoid morphism. Then the induced measure transfer map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S3.Thmthm7.p1.2.2.m2.1"><semantics id="S3.Thmthm7.p1.2.2.m2.1a"><mrow id="S3.Thmthm7.p1.2.2.m2.1.1" xref="S3.Thmthm7.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmthm7.p1.2.2.m2.1.1.2" xref="S3.Thmthm7.p1.2.2.m2.1.1.2.cmml">σ</mi><mo id="S3.Thmthm7.p1.2.2.m2.1.1.1" xref="S3.Thmthm7.p1.2.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm7.p1.2.2.m2.1.1.3" xref="S3.Thmthm7.p1.2.2.m2.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm7.p1.2.2.m2.1b"><apply id="S3.Thmthm7.p1.2.2.m2.1.1.cmml" xref="S3.Thmthm7.p1.2.2.m2.1.1"><times id="S3.Thmthm7.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmthm7.p1.2.2.m2.1.1.1"></times><ci id="S3.Thmthm7.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmthm7.p1.2.2.m2.1.1.2">𝜎</ci><ci id="S3.Thmthm7.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmthm7.p1.2.2.m2.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm7.p1.2.2.m2.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm7.p1.2.2.m2.1d">italic_σ italic_M</annotation></semantics></math> has the following properties:</span></p> <ol class="ltx_enumerate" id="S3.I2"> <li class="ltx_item" id="S3.I2.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S3.I2.ix1.p1"> <p class="ltx_p" id="S3.I2.ix1.p1.2"><span class="ltx_text ltx_font_italic" id="S3.I2.ix1.p1.2.1">The map </span><math alttext="\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.I2.ix1.p1.1.m1.2"><semantics id="S3.I2.ix1.p1.1.m1.2a"><mrow id="S3.I2.ix1.p1.1.m1.2.2" xref="S3.I2.ix1.p1.1.m1.2.2.cmml"><mrow id="S3.I2.ix1.p1.1.m1.2.2.4" xref="S3.I2.ix1.p1.1.m1.2.2.4.cmml"><mi id="S3.I2.ix1.p1.1.m1.2.2.4.2" xref="S3.I2.ix1.p1.1.m1.2.2.4.2.cmml">σ</mi><mo id="S3.I2.ix1.p1.1.m1.2.2.4.1" xref="S3.I2.ix1.p1.1.m1.2.2.4.1.cmml">⁢</mo><mi id="S3.I2.ix1.p1.1.m1.2.2.4.3" xref="S3.I2.ix1.p1.1.m1.2.2.4.3.cmml">M</mi></mrow><mo id="S3.I2.ix1.p1.1.m1.2.2.3" lspace="0.278em" rspace="0.278em" xref="S3.I2.ix1.p1.1.m1.2.2.3.cmml">:</mo><mrow id="S3.I2.ix1.p1.1.m1.2.2.2" xref="S3.I2.ix1.p1.1.m1.2.2.2.cmml"><mrow id="S3.I2.ix1.p1.1.m1.1.1.1.1" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix1.p1.1.m1.1.1.1.1.3" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.3.cmml">ℳ</mi><mo id="S3.I2.ix1.p1.1.m1.1.1.1.1.2" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.2" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.3" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I2.ix1.p1.1.m1.2.2.2.3" stretchy="false" xref="S3.I2.ix1.p1.1.m1.2.2.2.3.cmml">→</mo><mrow id="S3.I2.ix1.p1.1.m1.2.2.2.2" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix1.p1.1.m1.2.2.2.2.3" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.3.cmml">ℳ</mi><mo id="S3.I2.ix1.p1.1.m1.2.2.2.2.2" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.2.cmml">⁢</mo><mrow id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.cmml"><mo id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.2" stretchy="false" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.cmml">(</mo><msup id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.2" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.2.cmml">ℬ</mi><mi id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.3" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.3" stretchy="false" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix1.p1.1.m1.2b"><apply id="S3.I2.ix1.p1.1.m1.2.2.cmml" xref="S3.I2.ix1.p1.1.m1.2.2"><ci id="S3.I2.ix1.p1.1.m1.2.2.3.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.3">:</ci><apply id="S3.I2.ix1.p1.1.m1.2.2.4.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.4"><times id="S3.I2.ix1.p1.1.m1.2.2.4.1.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.4.1"></times><ci id="S3.I2.ix1.p1.1.m1.2.2.4.2.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.4.2">𝜎</ci><ci id="S3.I2.ix1.p1.1.m1.2.2.4.3.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.4.3">𝑀</ci></apply><apply id="S3.I2.ix1.p1.1.m1.2.2.2.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2"><ci id="S3.I2.ix1.p1.1.m1.2.2.2.3.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2.3">→</ci><apply id="S3.I2.ix1.p1.1.m1.1.1.1.1.cmml" xref="S3.I2.ix1.p1.1.m1.1.1.1.1"><times id="S3.I2.ix1.p1.1.m1.1.1.1.1.2.cmml" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.2"></times><ci id="S3.I2.ix1.p1.1.m1.1.1.1.1.3.cmml" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.3">ℳ</ci><apply id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1">superscript</csymbol><ci id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.2">𝒜</ci><ci id="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.I2.ix1.p1.1.m1.1.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S3.I2.ix1.p1.1.m1.2.2.2.2.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2.2"><times id="S3.I2.ix1.p1.1.m1.2.2.2.2.2.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.2"></times><ci id="S3.I2.ix1.p1.1.m1.2.2.2.2.3.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.3">ℳ</ci><apply id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.1.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.2.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.2">ℬ</ci><ci id="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.3.cmml" xref="S3.I2.ix1.p1.1.m1.2.2.2.2.1.1.1.3">ℤ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix1.p1.1.m1.2c">\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix1.p1.1.m1.2d">italic_σ italic_M : caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix1.p1.2.2"> is </span><math alttext="\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S3.I2.ix1.p1.2.m2.1"><semantics id="S3.I2.ix1.p1.2.m2.1a"><msub id="S3.I2.ix1.p1.2.m2.1.1" xref="S3.I2.ix1.p1.2.m2.1.1.cmml"><mi id="S3.I2.ix1.p1.2.m2.1.1.2" xref="S3.I2.ix1.p1.2.m2.1.1.2.cmml">ℝ</mi><mrow id="S3.I2.ix1.p1.2.m2.1.1.3" xref="S3.I2.ix1.p1.2.m2.1.1.3.cmml"><mi id="S3.I2.ix1.p1.2.m2.1.1.3.2" xref="S3.I2.ix1.p1.2.m2.1.1.3.2.cmml"></mi><mo id="S3.I2.ix1.p1.2.m2.1.1.3.1" xref="S3.I2.ix1.p1.2.m2.1.1.3.1.cmml">≥</mo><mn id="S3.I2.ix1.p1.2.m2.1.1.3.3" xref="S3.I2.ix1.p1.2.m2.1.1.3.3.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I2.ix1.p1.2.m2.1b"><apply id="S3.I2.ix1.p1.2.m2.1.1.cmml" xref="S3.I2.ix1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I2.ix1.p1.2.m2.1.1.1.cmml" xref="S3.I2.ix1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I2.ix1.p1.2.m2.1.1.2.cmml" xref="S3.I2.ix1.p1.2.m2.1.1.2">ℝ</ci><apply id="S3.I2.ix1.p1.2.m2.1.1.3.cmml" xref="S3.I2.ix1.p1.2.m2.1.1.3"><geq id="S3.I2.ix1.p1.2.m2.1.1.3.1.cmml" xref="S3.I2.ix1.p1.2.m2.1.1.3.1"></geq><csymbol cd="latexml" id="S3.I2.ix1.p1.2.m2.1.1.3.2.cmml" xref="S3.I2.ix1.p1.2.m2.1.1.3.2">absent</csymbol><cn id="S3.I2.ix1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S3.I2.ix1.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix1.p1.2.m2.1c">\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix1.p1.2.m2.1d">blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix1.p1.2.3">-linear.</span></p> </div> </li> <li class="ltx_item" id="S3.I2.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S3.I2.ix2.p1"> <p class="ltx_p" id="S3.I2.ix2.p1.3"><span class="ltx_text ltx_font_italic" id="S3.I2.ix2.p1.3.1">The map </span><math alttext="\sigma M" class="ltx_Math" display="inline" id="S3.I2.ix2.p1.1.m1.1"><semantics id="S3.I2.ix2.p1.1.m1.1a"><mrow id="S3.I2.ix2.p1.1.m1.1.1" xref="S3.I2.ix2.p1.1.m1.1.1.cmml"><mi id="S3.I2.ix2.p1.1.m1.1.1.2" xref="S3.I2.ix2.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S3.I2.ix2.p1.1.m1.1.1.1" xref="S3.I2.ix2.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.I2.ix2.p1.1.m1.1.1.3" xref="S3.I2.ix2.p1.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix2.p1.1.m1.1b"><apply id="S3.I2.ix2.p1.1.m1.1.1.cmml" xref="S3.I2.ix2.p1.1.m1.1.1"><times id="S3.I2.ix2.p1.1.m1.1.1.1.cmml" xref="S3.I2.ix2.p1.1.m1.1.1.1"></times><ci id="S3.I2.ix2.p1.1.m1.1.1.2.cmml" xref="S3.I2.ix2.p1.1.m1.1.1.2">𝜎</ci><ci id="S3.I2.ix2.p1.1.m1.1.1.3.cmml" xref="S3.I2.ix2.p1.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix2.p1.1.m1.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix2.p1.1.m1.1d">italic_σ italic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix2.p1.3.2"> is functorial: For any second non-erasing monoid morphism </span><math alttext="\sigma^{\prime}:\cal B^{*}\to\cal C^{*}" class="ltx_Math" display="inline" id="S3.I2.ix2.p1.2.m2.1"><semantics id="S3.I2.ix2.p1.2.m2.1a"><mrow id="S3.I2.ix2.p1.2.m2.1.1" xref="S3.I2.ix2.p1.2.m2.1.1.cmml"><msup id="S3.I2.ix2.p1.2.m2.1.1.2" xref="S3.I2.ix2.p1.2.m2.1.1.2.cmml"><mi id="S3.I2.ix2.p1.2.m2.1.1.2.2" xref="S3.I2.ix2.p1.2.m2.1.1.2.2.cmml">σ</mi><mo id="S3.I2.ix2.p1.2.m2.1.1.2.3" xref="S3.I2.ix2.p1.2.m2.1.1.2.3.cmml">′</mo></msup><mo id="S3.I2.ix2.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.I2.ix2.p1.2.m2.1.1.1.cmml">:</mo><mrow id="S3.I2.ix2.p1.2.m2.1.1.3" xref="S3.I2.ix2.p1.2.m2.1.1.3.cmml"><msup id="S3.I2.ix2.p1.2.m2.1.1.3.2" xref="S3.I2.ix2.p1.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix2.p1.2.m2.1.1.3.2.2" xref="S3.I2.ix2.p1.2.m2.1.1.3.2.2.cmml">ℬ</mi><mo id="S3.I2.ix2.p1.2.m2.1.1.3.2.3" xref="S3.I2.ix2.p1.2.m2.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.I2.ix2.p1.2.m2.1.1.3.1" stretchy="false" xref="S3.I2.ix2.p1.2.m2.1.1.3.1.cmml">→</mo><msup id="S3.I2.ix2.p1.2.m2.1.1.3.3" xref="S3.I2.ix2.p1.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix2.p1.2.m2.1.1.3.3.2" xref="S3.I2.ix2.p1.2.m2.1.1.3.3.2.cmml">𝒞</mi><mo id="S3.I2.ix2.p1.2.m2.1.1.3.3.3" xref="S3.I2.ix2.p1.2.m2.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix2.p1.2.m2.1b"><apply id="S3.I2.ix2.p1.2.m2.1.1.cmml" xref="S3.I2.ix2.p1.2.m2.1.1"><ci id="S3.I2.ix2.p1.2.m2.1.1.1.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.1">:</ci><apply id="S3.I2.ix2.p1.2.m2.1.1.2.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.I2.ix2.p1.2.m2.1.1.2.1.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.2">superscript</csymbol><ci id="S3.I2.ix2.p1.2.m2.1.1.2.2.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.2.2">𝜎</ci><ci id="S3.I2.ix2.p1.2.m2.1.1.2.3.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.2.3">′</ci></apply><apply id="S3.I2.ix2.p1.2.m2.1.1.3.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3"><ci id="S3.I2.ix2.p1.2.m2.1.1.3.1.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.1">→</ci><apply id="S3.I2.ix2.p1.2.m2.1.1.3.2.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S3.I2.ix2.p1.2.m2.1.1.3.2.1.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S3.I2.ix2.p1.2.m2.1.1.3.2.2.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.2.2">ℬ</ci><times id="S3.I2.ix2.p1.2.m2.1.1.3.2.3.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.2.3"></times></apply><apply id="S3.I2.ix2.p1.2.m2.1.1.3.3.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.I2.ix2.p1.2.m2.1.1.3.3.1.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.3">superscript</csymbol><ci id="S3.I2.ix2.p1.2.m2.1.1.3.3.2.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.3.2">𝒞</ci><times id="S3.I2.ix2.p1.2.m2.1.1.3.3.3.cmml" xref="S3.I2.ix2.p1.2.m2.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix2.p1.2.m2.1c">\sigma^{\prime}:\cal B^{*}\to\cal C^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix2.p1.2.m2.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix2.p1.3.3"> one has </span><math alttext="(\sigma^{\prime}\sigma)M={\sigma^{\prime}}M{\sigma}M" class="ltx_Math" display="inline" id="S3.I2.ix2.p1.3.m3.1"><semantics id="S3.I2.ix2.p1.3.m3.1a"><mrow id="S3.I2.ix2.p1.3.m3.1.1" xref="S3.I2.ix2.p1.3.m3.1.1.cmml"><mrow id="S3.I2.ix2.p1.3.m3.1.1.1" xref="S3.I2.ix2.p1.3.m3.1.1.1.cmml"><mrow id="S3.I2.ix2.p1.3.m3.1.1.1.1.1" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.cmml"><msup id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.cmml"><mi id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.2" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.2.cmml">σ</mi><mo id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.3" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.1" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.3" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.3.cmml">σ</mi></mrow><mo id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.I2.ix2.p1.3.m3.1.1.1.2" xref="S3.I2.ix2.p1.3.m3.1.1.1.2.cmml">⁢</mo><mi id="S3.I2.ix2.p1.3.m3.1.1.1.3" xref="S3.I2.ix2.p1.3.m3.1.1.1.3.cmml">M</mi></mrow><mo id="S3.I2.ix2.p1.3.m3.1.1.2" xref="S3.I2.ix2.p1.3.m3.1.1.2.cmml">=</mo><mrow id="S3.I2.ix2.p1.3.m3.1.1.3" xref="S3.I2.ix2.p1.3.m3.1.1.3.cmml"><msup id="S3.I2.ix2.p1.3.m3.1.1.3.2" xref="S3.I2.ix2.p1.3.m3.1.1.3.2.cmml"><mi id="S3.I2.ix2.p1.3.m3.1.1.3.2.2" xref="S3.I2.ix2.p1.3.m3.1.1.3.2.2.cmml">σ</mi><mo id="S3.I2.ix2.p1.3.m3.1.1.3.2.3" xref="S3.I2.ix2.p1.3.m3.1.1.3.2.3.cmml">′</mo></msup><mo id="S3.I2.ix2.p1.3.m3.1.1.3.1" xref="S3.I2.ix2.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.I2.ix2.p1.3.m3.1.1.3.3" xref="S3.I2.ix2.p1.3.m3.1.1.3.3.cmml">M</mi><mo id="S3.I2.ix2.p1.3.m3.1.1.3.1a" xref="S3.I2.ix2.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.I2.ix2.p1.3.m3.1.1.3.4" xref="S3.I2.ix2.p1.3.m3.1.1.3.4.cmml">σ</mi><mo id="S3.I2.ix2.p1.3.m3.1.1.3.1b" xref="S3.I2.ix2.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.I2.ix2.p1.3.m3.1.1.3.5" xref="S3.I2.ix2.p1.3.m3.1.1.3.5.cmml">M</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix2.p1.3.m3.1b"><apply id="S3.I2.ix2.p1.3.m3.1.1.cmml" xref="S3.I2.ix2.p1.3.m3.1.1"><eq id="S3.I2.ix2.p1.3.m3.1.1.2.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.2"></eq><apply id="S3.I2.ix2.p1.3.m3.1.1.1.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1"><times id="S3.I2.ix2.p1.3.m3.1.1.1.2.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.2"></times><apply id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1"><times id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.1"></times><apply id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.1.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2">superscript</csymbol><ci id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.2.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.2">𝜎</ci><ci id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.3.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.2.3">′</ci></apply><ci id="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.1.1.1.3">𝜎</ci></apply><ci id="S3.I2.ix2.p1.3.m3.1.1.1.3.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.1.3">𝑀</ci></apply><apply id="S3.I2.ix2.p1.3.m3.1.1.3.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3"><times id="S3.I2.ix2.p1.3.m3.1.1.3.1.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3.1"></times><apply id="S3.I2.ix2.p1.3.m3.1.1.3.2.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S3.I2.ix2.p1.3.m3.1.1.3.2.1.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S3.I2.ix2.p1.3.m3.1.1.3.2.2.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3.2.2">𝜎</ci><ci id="S3.I2.ix2.p1.3.m3.1.1.3.2.3.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3.2.3">′</ci></apply><ci id="S3.I2.ix2.p1.3.m3.1.1.3.3.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3.3">𝑀</ci><ci id="S3.I2.ix2.p1.3.m3.1.1.3.4.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3.4">𝜎</ci><ci id="S3.I2.ix2.p1.3.m3.1.1.3.5.cmml" xref="S3.I2.ix2.p1.3.m3.1.1.3.5">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix2.p1.3.m3.1c">(\sigma^{\prime}\sigma)M={\sigma^{\prime}}M{\sigma}M</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix2.p1.3.m3.1d">( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_σ ) italic_M = italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_M italic_σ italic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix2.p1.3.4">.</span></p> </div> </li> <li class="ltx_item" id="S3.I2.ix3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(c)</span> <div class="ltx_para" id="S3.I2.ix3.p1"> <p class="ltx_p" id="S3.I2.ix3.p1.4"><span class="ltx_text ltx_font_italic" id="S3.I2.ix3.p1.4.1">The image </span><math alttext="\sigma M(\mu)" class="ltx_Math" display="inline" id="S3.I2.ix3.p1.1.m1.1"><semantics id="S3.I2.ix3.p1.1.m1.1a"><mrow id="S3.I2.ix3.p1.1.m1.1.2" xref="S3.I2.ix3.p1.1.m1.1.2.cmml"><mi id="S3.I2.ix3.p1.1.m1.1.2.2" xref="S3.I2.ix3.p1.1.m1.1.2.2.cmml">σ</mi><mo id="S3.I2.ix3.p1.1.m1.1.2.1" xref="S3.I2.ix3.p1.1.m1.1.2.1.cmml">⁢</mo><mi id="S3.I2.ix3.p1.1.m1.1.2.3" xref="S3.I2.ix3.p1.1.m1.1.2.3.cmml">M</mi><mo id="S3.I2.ix3.p1.1.m1.1.2.1a" xref="S3.I2.ix3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.I2.ix3.p1.1.m1.1.2.4.2" xref="S3.I2.ix3.p1.1.m1.1.2.cmml"><mo id="S3.I2.ix3.p1.1.m1.1.2.4.2.1" stretchy="false" xref="S3.I2.ix3.p1.1.m1.1.2.cmml">(</mo><mi id="S3.I2.ix3.p1.1.m1.1.1" xref="S3.I2.ix3.p1.1.m1.1.1.cmml">μ</mi><mo id="S3.I2.ix3.p1.1.m1.1.2.4.2.2" stretchy="false" xref="S3.I2.ix3.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix3.p1.1.m1.1b"><apply id="S3.I2.ix3.p1.1.m1.1.2.cmml" xref="S3.I2.ix3.p1.1.m1.1.2"><times id="S3.I2.ix3.p1.1.m1.1.2.1.cmml" xref="S3.I2.ix3.p1.1.m1.1.2.1"></times><ci id="S3.I2.ix3.p1.1.m1.1.2.2.cmml" xref="S3.I2.ix3.p1.1.m1.1.2.2">𝜎</ci><ci id="S3.I2.ix3.p1.1.m1.1.2.3.cmml" xref="S3.I2.ix3.p1.1.m1.1.2.3">𝑀</ci><ci id="S3.I2.ix3.p1.1.m1.1.1.cmml" xref="S3.I2.ix3.p1.1.m1.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix3.p1.1.m1.1c">\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix3.p1.1.m1.1d">italic_σ italic_M ( italic_μ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix3.p1.4.2"> of a probability measure </span><math alttext="\mu" class="ltx_Math" display="inline" id="S3.I2.ix3.p1.2.m2.1"><semantics id="S3.I2.ix3.p1.2.m2.1a"><mi id="S3.I2.ix3.p1.2.m2.1.1" xref="S3.I2.ix3.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.I2.ix3.p1.2.m2.1b"><ci id="S3.I2.ix3.p1.2.m2.1.1.cmml" xref="S3.I2.ix3.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix3.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix3.p1.2.m2.1d">italic_μ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix3.p1.4.3"> on </span><math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.I2.ix3.p1.3.m3.1"><semantics id="S3.I2.ix3.p1.3.m3.1a"><msup id="S3.I2.ix3.p1.3.m3.1.1" xref="S3.I2.ix3.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix3.p1.3.m3.1.1.2" xref="S3.I2.ix3.p1.3.m3.1.1.2.cmml">𝒜</mi><mi id="S3.I2.ix3.p1.3.m3.1.1.3" xref="S3.I2.ix3.p1.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.I2.ix3.p1.3.m3.1b"><apply id="S3.I2.ix3.p1.3.m3.1.1.cmml" xref="S3.I2.ix3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I2.ix3.p1.3.m3.1.1.1.cmml" xref="S3.I2.ix3.p1.3.m3.1.1">superscript</csymbol><ci id="S3.I2.ix3.p1.3.m3.1.1.2.cmml" xref="S3.I2.ix3.p1.3.m3.1.1.2">𝒜</ci><ci id="S3.I2.ix3.p1.3.m3.1.1.3.cmml" xref="S3.I2.ix3.p1.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix3.p1.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix3.p1.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix3.p1.4.4"> is in general not a probability measure on </span><math alttext="\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.I2.ix3.p1.4.m4.1"><semantics id="S3.I2.ix3.p1.4.m4.1a"><msup id="S3.I2.ix3.p1.4.m4.1.1" xref="S3.I2.ix3.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix3.p1.4.m4.1.1.2" xref="S3.I2.ix3.p1.4.m4.1.1.2.cmml">ℬ</mi><mi id="S3.I2.ix3.p1.4.m4.1.1.3" xref="S3.I2.ix3.p1.4.m4.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.I2.ix3.p1.4.m4.1b"><apply id="S3.I2.ix3.p1.4.m4.1.1.cmml" xref="S3.I2.ix3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.I2.ix3.p1.4.m4.1.1.1.cmml" xref="S3.I2.ix3.p1.4.m4.1.1">superscript</csymbol><ci id="S3.I2.ix3.p1.4.m4.1.1.2.cmml" xref="S3.I2.ix3.p1.4.m4.1.1.2">ℬ</ci><ci id="S3.I2.ix3.p1.4.m4.1.1.3.cmml" xref="S3.I2.ix3.p1.4.m4.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix3.p1.4.m4.1c">\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix3.p1.4.m4.1d">caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix3.p1.4.5">.</span></p> </div> </li> <li class="ltx_item" id="S3.I2.ix4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(d)</span> <div class="ltx_para" id="S3.I2.ix4.p1"> <p class="ltx_p" id="S3.I2.ix4.p1.4"><span class="ltx_text ltx_font_italic" id="S3.I2.ix4.p1.4.1">For any word </span><math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S3.I2.ix4.p1.1.m1.1"><semantics id="S3.I2.ix4.p1.1.m1.1a"><mrow id="S3.I2.ix4.p1.1.m1.1.1" xref="S3.I2.ix4.p1.1.m1.1.1.cmml"><mi id="S3.I2.ix4.p1.1.m1.1.1.2" xref="S3.I2.ix4.p1.1.m1.1.1.2.cmml">w</mi><mo id="S3.I2.ix4.p1.1.m1.1.1.1" xref="S3.I2.ix4.p1.1.m1.1.1.1.cmml">∈</mo><msup id="S3.I2.ix4.p1.1.m1.1.1.3" xref="S3.I2.ix4.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix4.p1.1.m1.1.1.3.2" xref="S3.I2.ix4.p1.1.m1.1.1.3.2.cmml">𝒜</mi><mo id="S3.I2.ix4.p1.1.m1.1.1.3.3" xref="S3.I2.ix4.p1.1.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix4.p1.1.m1.1b"><apply id="S3.I2.ix4.p1.1.m1.1.1.cmml" xref="S3.I2.ix4.p1.1.m1.1.1"><in id="S3.I2.ix4.p1.1.m1.1.1.1.cmml" xref="S3.I2.ix4.p1.1.m1.1.1.1"></in><ci id="S3.I2.ix4.p1.1.m1.1.1.2.cmml" xref="S3.I2.ix4.p1.1.m1.1.1.2">𝑤</ci><apply id="S3.I2.ix4.p1.1.m1.1.1.3.cmml" xref="S3.I2.ix4.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.I2.ix4.p1.1.m1.1.1.3.1.cmml" xref="S3.I2.ix4.p1.1.m1.1.1.3">superscript</csymbol><ci id="S3.I2.ix4.p1.1.m1.1.1.3.2.cmml" xref="S3.I2.ix4.p1.1.m1.1.1.3.2">𝒜</ci><times id="S3.I2.ix4.p1.1.m1.1.1.3.3.cmml" xref="S3.I2.ix4.p1.1.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix4.p1.1.m1.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix4.p1.1.m1.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix4.p1.4.2"> the characteristic measure </span><math alttext="\mu_{w}" class="ltx_Math" display="inline" id="S3.I2.ix4.p1.2.m2.1"><semantics id="S3.I2.ix4.p1.2.m2.1a"><msub id="S3.I2.ix4.p1.2.m2.1.1" xref="S3.I2.ix4.p1.2.m2.1.1.cmml"><mi id="S3.I2.ix4.p1.2.m2.1.1.2" xref="S3.I2.ix4.p1.2.m2.1.1.2.cmml">μ</mi><mi id="S3.I2.ix4.p1.2.m2.1.1.3" xref="S3.I2.ix4.p1.2.m2.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I2.ix4.p1.2.m2.1b"><apply id="S3.I2.ix4.p1.2.m2.1.1.cmml" xref="S3.I2.ix4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I2.ix4.p1.2.m2.1.1.1.cmml" xref="S3.I2.ix4.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I2.ix4.p1.2.m2.1.1.2.cmml" xref="S3.I2.ix4.p1.2.m2.1.1.2">𝜇</ci><ci id="S3.I2.ix4.p1.2.m2.1.1.3.cmml" xref="S3.I2.ix4.p1.2.m2.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix4.p1.2.m2.1c">\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix4.p1.2.m2.1d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix4.p1.4.3"> is mapped by </span><math alttext="\sigma M" class="ltx_Math" display="inline" id="S3.I2.ix4.p1.3.m3.1"><semantics id="S3.I2.ix4.p1.3.m3.1a"><mrow id="S3.I2.ix4.p1.3.m3.1.1" xref="S3.I2.ix4.p1.3.m3.1.1.cmml"><mi id="S3.I2.ix4.p1.3.m3.1.1.2" xref="S3.I2.ix4.p1.3.m3.1.1.2.cmml">σ</mi><mo id="S3.I2.ix4.p1.3.m3.1.1.1" xref="S3.I2.ix4.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.I2.ix4.p1.3.m3.1.1.3" xref="S3.I2.ix4.p1.3.m3.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix4.p1.3.m3.1b"><apply id="S3.I2.ix4.p1.3.m3.1.1.cmml" xref="S3.I2.ix4.p1.3.m3.1.1"><times id="S3.I2.ix4.p1.3.m3.1.1.1.cmml" xref="S3.I2.ix4.p1.3.m3.1.1.1"></times><ci id="S3.I2.ix4.p1.3.m3.1.1.2.cmml" xref="S3.I2.ix4.p1.3.m3.1.1.2">𝜎</ci><ci id="S3.I2.ix4.p1.3.m3.1.1.3.cmml" xref="S3.I2.ix4.p1.3.m3.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix4.p1.3.m3.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix4.p1.3.m3.1d">italic_σ italic_M</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix4.p1.4.4"> to the characteristic measure </span><math alttext="\mu_{\sigma(w)}" class="ltx_Math" display="inline" id="S3.I2.ix4.p1.4.m4.1"><semantics id="S3.I2.ix4.p1.4.m4.1a"><msub id="S3.I2.ix4.p1.4.m4.1.2" xref="S3.I2.ix4.p1.4.m4.1.2.cmml"><mi id="S3.I2.ix4.p1.4.m4.1.2.2" xref="S3.I2.ix4.p1.4.m4.1.2.2.cmml">μ</mi><mrow id="S3.I2.ix4.p1.4.m4.1.1.1" xref="S3.I2.ix4.p1.4.m4.1.1.1.cmml"><mi id="S3.I2.ix4.p1.4.m4.1.1.1.3" xref="S3.I2.ix4.p1.4.m4.1.1.1.3.cmml">σ</mi><mo id="S3.I2.ix4.p1.4.m4.1.1.1.2" xref="S3.I2.ix4.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.I2.ix4.p1.4.m4.1.1.1.4.2" xref="S3.I2.ix4.p1.4.m4.1.1.1.cmml"><mo id="S3.I2.ix4.p1.4.m4.1.1.1.4.2.1" stretchy="false" xref="S3.I2.ix4.p1.4.m4.1.1.1.cmml">(</mo><mi id="S3.I2.ix4.p1.4.m4.1.1.1.1" xref="S3.I2.ix4.p1.4.m4.1.1.1.1.cmml">w</mi><mo id="S3.I2.ix4.p1.4.m4.1.1.1.4.2.2" stretchy="false" xref="S3.I2.ix4.p1.4.m4.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I2.ix4.p1.4.m4.1b"><apply id="S3.I2.ix4.p1.4.m4.1.2.cmml" xref="S3.I2.ix4.p1.4.m4.1.2"><csymbol cd="ambiguous" id="S3.I2.ix4.p1.4.m4.1.2.1.cmml" xref="S3.I2.ix4.p1.4.m4.1.2">subscript</csymbol><ci id="S3.I2.ix4.p1.4.m4.1.2.2.cmml" xref="S3.I2.ix4.p1.4.m4.1.2.2">𝜇</ci><apply id="S3.I2.ix4.p1.4.m4.1.1.1.cmml" xref="S3.I2.ix4.p1.4.m4.1.1.1"><times id="S3.I2.ix4.p1.4.m4.1.1.1.2.cmml" xref="S3.I2.ix4.p1.4.m4.1.1.1.2"></times><ci id="S3.I2.ix4.p1.4.m4.1.1.1.3.cmml" xref="S3.I2.ix4.p1.4.m4.1.1.1.3">𝜎</ci><ci id="S3.I2.ix4.p1.4.m4.1.1.1.1.cmml" xref="S3.I2.ix4.p1.4.m4.1.1.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix4.p1.4.m4.1c">\mu_{\sigma(w)}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix4.p1.4.m4.1d">italic_μ start_POSTSUBSCRIPT italic_σ ( italic_w ) end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix4.p1.4.5">.</span></p> </div> </li> <li class="ltx_item" id="S3.I2.ix5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(e)</span> <div class="ltx_para" id="S3.I2.ix5.p1"> <p class="ltx_p" id="S3.I2.ix5.p1.3"><span class="ltx_text ltx_font_italic" id="S3.I2.ix5.p1.3.1">For any word </span><math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S3.I2.ix5.p1.1.m1.1"><semantics id="S3.I2.ix5.p1.1.m1.1a"><mrow id="S3.I2.ix5.p1.1.m1.1.1" xref="S3.I2.ix5.p1.1.m1.1.1.cmml"><mi id="S3.I2.ix5.p1.1.m1.1.1.2" xref="S3.I2.ix5.p1.1.m1.1.1.2.cmml">w</mi><mo id="S3.I2.ix5.p1.1.m1.1.1.1" xref="S3.I2.ix5.p1.1.m1.1.1.1.cmml">∈</mo><msup id="S3.I2.ix5.p1.1.m1.1.1.3" xref="S3.I2.ix5.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix5.p1.1.m1.1.1.3.2" xref="S3.I2.ix5.p1.1.m1.1.1.3.2.cmml">𝒜</mi><mo id="S3.I2.ix5.p1.1.m1.1.1.3.3" xref="S3.I2.ix5.p1.1.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix5.p1.1.m1.1b"><apply id="S3.I2.ix5.p1.1.m1.1.1.cmml" xref="S3.I2.ix5.p1.1.m1.1.1"><in id="S3.I2.ix5.p1.1.m1.1.1.1.cmml" xref="S3.I2.ix5.p1.1.m1.1.1.1"></in><ci id="S3.I2.ix5.p1.1.m1.1.1.2.cmml" xref="S3.I2.ix5.p1.1.m1.1.1.2">𝑤</ci><apply id="S3.I2.ix5.p1.1.m1.1.1.3.cmml" xref="S3.I2.ix5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.I2.ix5.p1.1.m1.1.1.3.1.cmml" xref="S3.I2.ix5.p1.1.m1.1.1.3">superscript</csymbol><ci id="S3.I2.ix5.p1.1.m1.1.1.3.2.cmml" xref="S3.I2.ix5.p1.1.m1.1.1.3.2">𝒜</ci><times id="S3.I2.ix5.p1.1.m1.1.1.3.3.cmml" xref="S3.I2.ix5.p1.1.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix5.p1.1.m1.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix5.p1.1.m1.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix5.p1.3.2"> the measures of corresponding cylinders </span><math alttext="[w]\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.I2.ix5.p1.2.m2.1"><semantics id="S3.I2.ix5.p1.2.m2.1a"><mrow id="S3.I2.ix5.p1.2.m2.1.2" xref="S3.I2.ix5.p1.2.m2.1.2.cmml"><mrow id="S3.I2.ix5.p1.2.m2.1.2.2.2" xref="S3.I2.ix5.p1.2.m2.1.2.2.1.cmml"><mo id="S3.I2.ix5.p1.2.m2.1.2.2.2.1" stretchy="false" xref="S3.I2.ix5.p1.2.m2.1.2.2.1.1.cmml">[</mo><mi id="S3.I2.ix5.p1.2.m2.1.1" xref="S3.I2.ix5.p1.2.m2.1.1.cmml">w</mi><mo id="S3.I2.ix5.p1.2.m2.1.2.2.2.2" stretchy="false" xref="S3.I2.ix5.p1.2.m2.1.2.2.1.1.cmml">]</mo></mrow><mo id="S3.I2.ix5.p1.2.m2.1.2.1" xref="S3.I2.ix5.p1.2.m2.1.2.1.cmml">⊆</mo><msup id="S3.I2.ix5.p1.2.m2.1.2.3" xref="S3.I2.ix5.p1.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix5.p1.2.m2.1.2.3.2" xref="S3.I2.ix5.p1.2.m2.1.2.3.2.cmml">𝒜</mi><mi id="S3.I2.ix5.p1.2.m2.1.2.3.3" xref="S3.I2.ix5.p1.2.m2.1.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix5.p1.2.m2.1b"><apply id="S3.I2.ix5.p1.2.m2.1.2.cmml" xref="S3.I2.ix5.p1.2.m2.1.2"><subset id="S3.I2.ix5.p1.2.m2.1.2.1.cmml" xref="S3.I2.ix5.p1.2.m2.1.2.1"></subset><apply id="S3.I2.ix5.p1.2.m2.1.2.2.1.cmml" xref="S3.I2.ix5.p1.2.m2.1.2.2.2"><csymbol cd="latexml" id="S3.I2.ix5.p1.2.m2.1.2.2.1.1.cmml" xref="S3.I2.ix5.p1.2.m2.1.2.2.2.1">delimited-[]</csymbol><ci id="S3.I2.ix5.p1.2.m2.1.1.cmml" xref="S3.I2.ix5.p1.2.m2.1.1">𝑤</ci></apply><apply id="S3.I2.ix5.p1.2.m2.1.2.3.cmml" xref="S3.I2.ix5.p1.2.m2.1.2.3"><csymbol cd="ambiguous" id="S3.I2.ix5.p1.2.m2.1.2.3.1.cmml" xref="S3.I2.ix5.p1.2.m2.1.2.3">superscript</csymbol><ci id="S3.I2.ix5.p1.2.m2.1.2.3.2.cmml" xref="S3.I2.ix5.p1.2.m2.1.2.3.2">𝒜</ci><ci id="S3.I2.ix5.p1.2.m2.1.2.3.3.cmml" xref="S3.I2.ix5.p1.2.m2.1.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix5.p1.2.m2.1c">[w]\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix5.p1.2.m2.1d">[ italic_w ] ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix5.p1.3.3"> and </span><math alttext="[\sigma(w)]\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.I2.ix5.p1.3.m3.2"><semantics id="S3.I2.ix5.p1.3.m3.2a"><mrow id="S3.I2.ix5.p1.3.m3.2.2" xref="S3.I2.ix5.p1.3.m3.2.2.cmml"><mrow id="S3.I2.ix5.p1.3.m3.2.2.1.1" xref="S3.I2.ix5.p1.3.m3.2.2.1.2.cmml"><mo id="S3.I2.ix5.p1.3.m3.2.2.1.1.2" stretchy="false" xref="S3.I2.ix5.p1.3.m3.2.2.1.2.1.cmml">[</mo><mrow id="S3.I2.ix5.p1.3.m3.2.2.1.1.1" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1.cmml"><mi id="S3.I2.ix5.p1.3.m3.2.2.1.1.1.2" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1.2.cmml">σ</mi><mo id="S3.I2.ix5.p1.3.m3.2.2.1.1.1.1" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.I2.ix5.p1.3.m3.2.2.1.1.1.3.2" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1.cmml"><mo id="S3.I2.ix5.p1.3.m3.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1.cmml">(</mo><mi id="S3.I2.ix5.p1.3.m3.1.1" xref="S3.I2.ix5.p1.3.m3.1.1.cmml">w</mi><mo id="S3.I2.ix5.p1.3.m3.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I2.ix5.p1.3.m3.2.2.1.1.3" stretchy="false" xref="S3.I2.ix5.p1.3.m3.2.2.1.2.1.cmml">]</mo></mrow><mo id="S3.I2.ix5.p1.3.m3.2.2.2" xref="S3.I2.ix5.p1.3.m3.2.2.2.cmml">⊆</mo><msup id="S3.I2.ix5.p1.3.m3.2.2.3" xref="S3.I2.ix5.p1.3.m3.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.ix5.p1.3.m3.2.2.3.2" xref="S3.I2.ix5.p1.3.m3.2.2.3.2.cmml">ℬ</mi><mi id="S3.I2.ix5.p1.3.m3.2.2.3.3" xref="S3.I2.ix5.p1.3.m3.2.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.ix5.p1.3.m3.2b"><apply id="S3.I2.ix5.p1.3.m3.2.2.cmml" xref="S3.I2.ix5.p1.3.m3.2.2"><subset id="S3.I2.ix5.p1.3.m3.2.2.2.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.2"></subset><apply id="S3.I2.ix5.p1.3.m3.2.2.1.2.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.1.1"><csymbol cd="latexml" id="S3.I2.ix5.p1.3.m3.2.2.1.2.1.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.2">delimited-[]</csymbol><apply id="S3.I2.ix5.p1.3.m3.2.2.1.1.1.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1"><times id="S3.I2.ix5.p1.3.m3.2.2.1.1.1.1.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1.1"></times><ci id="S3.I2.ix5.p1.3.m3.2.2.1.1.1.2.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.1.1.1.2">𝜎</ci><ci id="S3.I2.ix5.p1.3.m3.1.1.cmml" xref="S3.I2.ix5.p1.3.m3.1.1">𝑤</ci></apply></apply><apply id="S3.I2.ix5.p1.3.m3.2.2.3.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.3"><csymbol cd="ambiguous" id="S3.I2.ix5.p1.3.m3.2.2.3.1.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.3">superscript</csymbol><ci id="S3.I2.ix5.p1.3.m3.2.2.3.2.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.3.2">ℬ</ci><ci id="S3.I2.ix5.p1.3.m3.2.2.3.3.cmml" xref="S3.I2.ix5.p1.3.m3.2.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix5.p1.3.m3.2c">[\sigma(w)]\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix5.p1.3.m3.2d">[ italic_σ ( italic_w ) ] ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix5.p1.3.4"> satisfy the inequality</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(3.6)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu([w])\leq\sigma M(\mu)([\sigma(w)])\,." class="ltx_Math" display="block" id="S3.E6.m1.4"><semantics id="S3.E6.m1.4a"><mrow id="S3.E6.m1.4.4.1" xref="S3.E6.m1.4.4.1.1.cmml"><mrow id="S3.E6.m1.4.4.1.1" xref="S3.E6.m1.4.4.1.1.cmml"><mrow id="S3.E6.m1.4.4.1.1.1" xref="S3.E6.m1.4.4.1.1.1.cmml"><mi id="S3.E6.m1.4.4.1.1.1.3" xref="S3.E6.m1.4.4.1.1.1.3.cmml">μ</mi><mo id="S3.E6.m1.4.4.1.1.1.2" xref="S3.E6.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S3.E6.m1.4.4.1.1.1.1.1" xref="S3.E6.m1.4.4.1.1.1.cmml"><mo id="S3.E6.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.E6.m1.4.4.1.1.1.cmml">(</mo><mrow id="S3.E6.m1.4.4.1.1.1.1.1.1.2" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.cmml"><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.2.1" stretchy="false" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.cmml">[</mo><mi id="S3.E6.m1.1.1" xref="S3.E6.m1.1.1.cmml">w</mi><mo id="S3.E6.m1.4.4.1.1.1.1.1.1.2.2" stretchy="false" xref="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.E6.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="S3.E6.m1.4.4.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E6.m1.4.4.1.1.3" xref="S3.E6.m1.4.4.1.1.3.cmml">≤</mo><mrow id="S3.E6.m1.4.4.1.1.2" xref="S3.E6.m1.4.4.1.1.2.cmml"><mi id="S3.E6.m1.4.4.1.1.2.3" xref="S3.E6.m1.4.4.1.1.2.3.cmml">σ</mi><mo id="S3.E6.m1.4.4.1.1.2.2" xref="S3.E6.m1.4.4.1.1.2.2.cmml">⁢</mo><mi id="S3.E6.m1.4.4.1.1.2.4" xref="S3.E6.m1.4.4.1.1.2.4.cmml">M</mi><mo id="S3.E6.m1.4.4.1.1.2.2a" xref="S3.E6.m1.4.4.1.1.2.2.cmml">⁢</mo><mrow id="S3.E6.m1.4.4.1.1.2.5.2" xref="S3.E6.m1.4.4.1.1.2.cmml"><mo id="S3.E6.m1.4.4.1.1.2.5.2.1" stretchy="false" xref="S3.E6.m1.4.4.1.1.2.cmml">(</mo><mi id="S3.E6.m1.2.2" xref="S3.E6.m1.2.2.cmml">μ</mi><mo id="S3.E6.m1.4.4.1.1.2.5.2.2" stretchy="false" xref="S3.E6.m1.4.4.1.1.2.cmml">)</mo></mrow><mo id="S3.E6.m1.4.4.1.1.2.2b" xref="S3.E6.m1.4.4.1.1.2.2.cmml">⁢</mo><mrow id="S3.E6.m1.4.4.1.1.2.1.1" xref="S3.E6.m1.4.4.1.1.2.cmml"><mo id="S3.E6.m1.4.4.1.1.2.1.1.2" stretchy="false" xref="S3.E6.m1.4.4.1.1.2.cmml">(</mo><mrow id="S3.E6.m1.4.4.1.1.2.1.1.1.1" xref="S3.E6.m1.4.4.1.1.2.1.1.1.2.cmml"><mo id="S3.E6.m1.4.4.1.1.2.1.1.1.1.2" stretchy="false" xref="S3.E6.m1.4.4.1.1.2.1.1.1.2.1.cmml">[</mo><mrow id="S3.E6.m1.4.4.1.1.2.1.1.1.1.1" xref="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.cmml"><mi id="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.2" xref="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.2.cmml">σ</mi><mo id="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.1" 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xref="S3.E6.m1.4.4.1.1.1.2"></times><ci id="S3.E6.m1.4.4.1.1.1.3.cmml" xref="S3.E6.m1.4.4.1.1.1.3">𝜇</ci><apply id="S3.E6.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2"><csymbol cd="latexml" id="S3.E6.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="S3.E6.m1.4.4.1.1.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.E6.m1.1.1.cmml" xref="S3.E6.m1.1.1">𝑤</ci></apply></apply><apply id="S3.E6.m1.4.4.1.1.2.cmml" xref="S3.E6.m1.4.4.1.1.2"><times id="S3.E6.m1.4.4.1.1.2.2.cmml" xref="S3.E6.m1.4.4.1.1.2.2"></times><ci id="S3.E6.m1.4.4.1.1.2.3.cmml" xref="S3.E6.m1.4.4.1.1.2.3">𝜎</ci><ci id="S3.E6.m1.4.4.1.1.2.4.cmml" xref="S3.E6.m1.4.4.1.1.2.4">𝑀</ci><ci id="S3.E6.m1.2.2.cmml" xref="S3.E6.m1.2.2">𝜇</ci><apply id="S3.E6.m1.4.4.1.1.2.1.1.1.2.cmml" xref="S3.E6.m1.4.4.1.1.2.1.1.1.1"><csymbol cd="latexml" id="S3.E6.m1.4.4.1.1.2.1.1.1.2.1.cmml" xref="S3.E6.m1.4.4.1.1.2.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.cmml" xref="S3.E6.m1.4.4.1.1.2.1.1.1.1.1"><times id="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.1.cmml" xref="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.1"></times><ci id="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.2.cmml" xref="S3.E6.m1.4.4.1.1.2.1.1.1.1.1.2">𝜎</ci><ci id="S3.E6.m1.3.3.cmml" xref="S3.E6.m1.3.3">𝑤</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E6.m1.4c">\mu([w])\leq\sigma M(\mu)([\sigma(w)])\,.</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.4d">italic_μ ( [ italic_w ] ) ≤ italic_σ italic_M ( italic_μ ) ( [ italic_σ ( italic_w ) ] ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.I2.ix5.p1.6"><span class="ltx_text ltx_font_italic" id="S3.I2.ix5.p1.6.2">If </span><math alttext="\sigma" class="ltx_Math" display="inline" id="S3.I2.ix5.p1.4.m1.1"><semantics id="S3.I2.ix5.p1.4.m1.1a"><mi id="S3.I2.ix5.p1.4.m1.1.1" xref="S3.I2.ix5.p1.4.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.I2.ix5.p1.4.m1.1b"><ci id="S3.I2.ix5.p1.4.m1.1.1.cmml" xref="S3.I2.ix5.p1.4.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix5.p1.4.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix5.p1.4.m1.1d">italic_σ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I2.ix5.p1.6.3"> is a subdivision morphism, then (</span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2211.11234v4#S3.E6" title="In item (e) ‣ Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.6</span></a><span class="ltx_text ltx_font_italic" id="S3.I2.ix5.p1.6.4">) becomes an equality, but for a letter-to-letter morphism the inequality (</span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2211.11234v4#S3.E6" title="In item (e) ‣ Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.6</span></a><span class="ltx_text ltx_font_italic" id="S3.I2.ix5.p1.6.5">) will in general be strict. </span><span class="ltx_text ltx_font_italic ltx_inline-block" id="S3.I2.ix5.p1.5.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S3.I2.ix5.p1.5.1.m1.1"><semantics id="S3.I2.ix5.p1.5.1.m1.1a"><mo id="S3.I2.ix5.p1.5.1.m1.1.1" xref="S3.I2.ix5.p1.5.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S3.I2.ix5.p1.5.1.m1.1b"><csymbol cd="latexml" id="S3.I2.ix5.p1.5.1.m1.1.1.cmml" xref="S3.I2.ix5.p1.5.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix5.p1.5.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix5.p1.5.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S3.I2.ix5.p1.6.m2.1"><semantics id="S3.I2.ix5.p1.6.m2.1a"><mo id="S3.I2.ix5.p1.6.m2.1.1" xref="S3.I2.ix5.p1.6.m2.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S3.I2.ix5.p1.6.m2.1b"><csymbol cd="latexml" id="S3.I2.ix5.p1.6.m2.1.1.cmml" xref="S3.I2.ix5.p1.6.m2.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.ix5.p1.6.m2.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S3.I2.ix5.p1.6.m2.1d">⊓</annotation></semantics></math></p> </div> </li> </ol> </div> </div> <div class="ltx_para" id="S3.SS4.p3"> <p class="ltx_p" id="S3.SS4.p3.5">For the next observation we recall from Section <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS1" title="2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.1</span></a> that the classical weak<sup class="ltx_sup" id="S3.SS4.p3.5.1">∗</sup>-topology on the space <math alttext="\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.SS4.p3.2.m2.1"><semantics id="S3.SS4.p3.2.m2.1a"><mrow id="S3.SS4.p3.2.m2.1.1" xref="S3.SS4.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.2.m2.1.1.3" xref="S3.SS4.p3.2.m2.1.1.3.cmml">ℳ</mi><mo id="S3.SS4.p3.2.m2.1.1.2" xref="S3.SS4.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.p3.2.m2.1.1.1.1" xref="S3.SS4.p3.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS4.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS4.p3.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.p3.2.m2.1.1.1.1.1" xref="S3.SS4.p3.2.m2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.2.m2.1.1.1.1.1.2" xref="S3.SS4.p3.2.m2.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.SS4.p3.2.m2.1.1.1.1.1.3" xref="S3.SS4.p3.2.m2.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS4.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.2.m2.1b"><apply id="S3.SS4.p3.2.m2.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1"><times id="S3.SS4.p3.2.m2.1.1.2.cmml" xref="S3.SS4.p3.2.m2.1.1.2"></times><ci id="S3.SS4.p3.2.m2.1.1.3.cmml" xref="S3.SS4.p3.2.m2.1.1.3">ℳ</ci><apply id="S3.SS4.p3.2.m2.1.1.1.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1.1.1">superscript</csymbol><ci id="S3.SS4.p3.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS4.p3.2.m2.1.1.1.1.1.2">𝒜</ci><ci id="S3.SS4.p3.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS4.p3.2.m2.1.1.1.1.1.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.2.m2.1c">\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.2.m2.1d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> is equivalent to the topology induced by the product topology on the space <math alttext="\mathbb{R}_{\geq 0}^{\cal A^{*}}" class="ltx_Math" display="inline" id="S3.SS4.p3.3.m3.1"><semantics id="S3.SS4.p3.3.m3.1a"><msubsup id="S3.SS4.p3.3.m3.1.1" xref="S3.SS4.p3.3.m3.1.1.cmml"><mi id="S3.SS4.p3.3.m3.1.1.2.2" xref="S3.SS4.p3.3.m3.1.1.2.2.cmml">ℝ</mi><mrow id="S3.SS4.p3.3.m3.1.1.2.3" xref="S3.SS4.p3.3.m3.1.1.2.3.cmml"><mi id="S3.SS4.p3.3.m3.1.1.2.3.2" xref="S3.SS4.p3.3.m3.1.1.2.3.2.cmml"></mi><mo id="S3.SS4.p3.3.m3.1.1.2.3.1" xref="S3.SS4.p3.3.m3.1.1.2.3.1.cmml">≥</mo><mn id="S3.SS4.p3.3.m3.1.1.2.3.3" xref="S3.SS4.p3.3.m3.1.1.2.3.3.cmml">0</mn></mrow><msup id="S3.SS4.p3.3.m3.1.1.3" xref="S3.SS4.p3.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.3.m3.1.1.3.2" xref="S3.SS4.p3.3.m3.1.1.3.2.cmml">𝒜</mi><mo id="S3.SS4.p3.3.m3.1.1.3.3" xref="S3.SS4.p3.3.m3.1.1.3.3.cmml">∗</mo></msup></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.3.m3.1b"><apply id="S3.SS4.p3.3.m3.1.1.cmml" xref="S3.SS4.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.3.m3.1.1.1.cmml" xref="S3.SS4.p3.3.m3.1.1">superscript</csymbol><apply id="S3.SS4.p3.3.m3.1.1.2.cmml" xref="S3.SS4.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.3.m3.1.1.2.1.cmml" xref="S3.SS4.p3.3.m3.1.1">subscript</csymbol><ci id="S3.SS4.p3.3.m3.1.1.2.2.cmml" xref="S3.SS4.p3.3.m3.1.1.2.2">ℝ</ci><apply id="S3.SS4.p3.3.m3.1.1.2.3.cmml" xref="S3.SS4.p3.3.m3.1.1.2.3"><geq id="S3.SS4.p3.3.m3.1.1.2.3.1.cmml" xref="S3.SS4.p3.3.m3.1.1.2.3.1"></geq><csymbol cd="latexml" id="S3.SS4.p3.3.m3.1.1.2.3.2.cmml" xref="S3.SS4.p3.3.m3.1.1.2.3.2">absent</csymbol><cn id="S3.SS4.p3.3.m3.1.1.2.3.3.cmml" type="integer" xref="S3.SS4.p3.3.m3.1.1.2.3.3">0</cn></apply></apply><apply id="S3.SS4.p3.3.m3.1.1.3.cmml" xref="S3.SS4.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p3.3.m3.1.1.3.1.cmml" xref="S3.SS4.p3.3.m3.1.1.3">superscript</csymbol><ci id="S3.SS4.p3.3.m3.1.1.3.2.cmml" xref="S3.SS4.p3.3.m3.1.1.3.2">𝒜</ci><times id="S3.SS4.p3.3.m3.1.1.3.3.cmml" xref="S3.SS4.p3.3.m3.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.3.m3.1c">\mathbb{R}_{\geq 0}^{\cal A^{*}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.3.m3.1d">blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, via the embedding <math alttext="\cal M(\cal A^{\mathbb{Z}})\subseteq\mathbb{R}_{\geq 0}^{\cal A^{*}}" class="ltx_Math" display="inline" id="S3.SS4.p3.4.m4.1"><semantics id="S3.SS4.p3.4.m4.1a"><mrow id="S3.SS4.p3.4.m4.1.1" xref="S3.SS4.p3.4.m4.1.1.cmml"><mrow id="S3.SS4.p3.4.m4.1.1.1" xref="S3.SS4.p3.4.m4.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.4.m4.1.1.1.3" xref="S3.SS4.p3.4.m4.1.1.1.3.cmml">ℳ</mi><mo id="S3.SS4.p3.4.m4.1.1.1.2" xref="S3.SS4.p3.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.p3.4.m4.1.1.1.1.1" xref="S3.SS4.p3.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS4.p3.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.p3.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.p3.4.m4.1.1.1.1.1.1" xref="S3.SS4.p3.4.m4.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.4.m4.1.1.1.1.1.1.2" xref="S3.SS4.p3.4.m4.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.SS4.p3.4.m4.1.1.1.1.1.1.3" xref="S3.SS4.p3.4.m4.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.p3.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.p3.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.p3.4.m4.1.1.2" xref="S3.SS4.p3.4.m4.1.1.2.cmml">⊆</mo><msubsup id="S3.SS4.p3.4.m4.1.1.3" xref="S3.SS4.p3.4.m4.1.1.3.cmml"><mi id="S3.SS4.p3.4.m4.1.1.3.2.2" xref="S3.SS4.p3.4.m4.1.1.3.2.2.cmml">ℝ</mi><mrow id="S3.SS4.p3.4.m4.1.1.3.2.3" xref="S3.SS4.p3.4.m4.1.1.3.2.3.cmml"><mi id="S3.SS4.p3.4.m4.1.1.3.2.3.2" xref="S3.SS4.p3.4.m4.1.1.3.2.3.2.cmml"></mi><mo id="S3.SS4.p3.4.m4.1.1.3.2.3.1" xref="S3.SS4.p3.4.m4.1.1.3.2.3.1.cmml">≥</mo><mn class="ltx_font_mathcaligraphic" id="S3.SS4.p3.4.m4.1.1.3.2.3.3" mathvariant="script" xref="S3.SS4.p3.4.m4.1.1.3.2.3.3.cmml">0</mn></mrow><msup id="S3.SS4.p3.4.m4.1.1.3.3" xref="S3.SS4.p3.4.m4.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.4.m4.1.1.3.3.2" xref="S3.SS4.p3.4.m4.1.1.3.3.2.cmml">𝒜</mi><mo id="S3.SS4.p3.4.m4.1.1.3.3.3" xref="S3.SS4.p3.4.m4.1.1.3.3.3.cmml">∗</mo></msup></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.4.m4.1b"><apply id="S3.SS4.p3.4.m4.1.1.cmml" xref="S3.SS4.p3.4.m4.1.1"><subset id="S3.SS4.p3.4.m4.1.1.2.cmml" xref="S3.SS4.p3.4.m4.1.1.2"></subset><apply id="S3.SS4.p3.4.m4.1.1.1.cmml" xref="S3.SS4.p3.4.m4.1.1.1"><times id="S3.SS4.p3.4.m4.1.1.1.2.cmml" xref="S3.SS4.p3.4.m4.1.1.1.2"></times><ci id="S3.SS4.p3.4.m4.1.1.1.3.cmml" xref="S3.SS4.p3.4.m4.1.1.1.3">ℳ</ci><apply id="S3.SS4.p3.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS4.p3.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS4.p3.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.p3.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS4.p3.4.m4.1.1.1.1.1.1.2">𝒜</ci><ci id="S3.SS4.p3.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS4.p3.4.m4.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S3.SS4.p3.4.m4.1.1.3.cmml" xref="S3.SS4.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p3.4.m4.1.1.3.1.cmml" xref="S3.SS4.p3.4.m4.1.1.3">superscript</csymbol><apply id="S3.SS4.p3.4.m4.1.1.3.2.cmml" xref="S3.SS4.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p3.4.m4.1.1.3.2.1.cmml" xref="S3.SS4.p3.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS4.p3.4.m4.1.1.3.2.2.cmml" xref="S3.SS4.p3.4.m4.1.1.3.2.2">ℝ</ci><apply id="S3.SS4.p3.4.m4.1.1.3.2.3.cmml" xref="S3.SS4.p3.4.m4.1.1.3.2.3"><geq id="S3.SS4.p3.4.m4.1.1.3.2.3.1.cmml" xref="S3.SS4.p3.4.m4.1.1.3.2.3.1"></geq><csymbol cd="latexml" id="S3.SS4.p3.4.m4.1.1.3.2.3.2.cmml" xref="S3.SS4.p3.4.m4.1.1.3.2.3.2">absent</csymbol><cn id="S3.SS4.p3.4.m4.1.1.3.2.3.3.cmml" type="integer" xref="S3.SS4.p3.4.m4.1.1.3.2.3.3">0</cn></apply></apply><apply id="S3.SS4.p3.4.m4.1.1.3.3.cmml" xref="S3.SS4.p3.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.p3.4.m4.1.1.3.3.1.cmml" xref="S3.SS4.p3.4.m4.1.1.3.3">superscript</csymbol><ci id="S3.SS4.p3.4.m4.1.1.3.3.2.cmml" xref="S3.SS4.p3.4.m4.1.1.3.3.2">𝒜</ci><times id="S3.SS4.p3.4.m4.1.1.3.3.3.cmml" xref="S3.SS4.p3.4.m4.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.4.m4.1c">\cal M(\cal A^{\mathbb{Z}})\subseteq\mathbb{R}_{\geq 0}^{\cal A^{*}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.4.m4.1d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) ⊆ blackboard_R start_POSTSUBSCRIPT ≥ caligraphic_0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> given by <math alttext="\mu\mapsto(\mu([w]))_{w\in\cal A^{*}}" class="ltx_Math" display="inline" id="S3.SS4.p3.5.m5.2"><semantics id="S3.SS4.p3.5.m5.2a"><mrow id="S3.SS4.p3.5.m5.2.2" xref="S3.SS4.p3.5.m5.2.2.cmml"><mi id="S3.SS4.p3.5.m5.2.2.3" xref="S3.SS4.p3.5.m5.2.2.3.cmml">μ</mi><mo id="S3.SS4.p3.5.m5.2.2.2" stretchy="false" xref="S3.SS4.p3.5.m5.2.2.2.cmml">↦</mo><msub id="S3.SS4.p3.5.m5.2.2.1" xref="S3.SS4.p3.5.m5.2.2.1.cmml"><mrow id="S3.SS4.p3.5.m5.2.2.1.1.1" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.cmml"><mo id="S3.SS4.p3.5.m5.2.2.1.1.1.2" stretchy="false" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p3.5.m5.2.2.1.1.1.1" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.cmml"><mi id="S3.SS4.p3.5.m5.2.2.1.1.1.1.3" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.3.cmml">μ</mi><mo id="S3.SS4.p3.5.m5.2.2.1.1.1.1.2" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.cmml"><mo id="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.2" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.2.1" stretchy="false" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.1.1.cmml">[</mo><mi id="S3.SS4.p3.5.m5.1.1" xref="S3.SS4.p3.5.m5.1.1.cmml">w</mi><mo id="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.2.2" stretchy="false" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.1.1.cmml">]</mo></mrow><mo id="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.p3.5.m5.2.2.1.1.1.3" stretchy="false" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS4.p3.5.m5.2.2.1.3" xref="S3.SS4.p3.5.m5.2.2.1.3.cmml"><mi id="S3.SS4.p3.5.m5.2.2.1.3.2" xref="S3.SS4.p3.5.m5.2.2.1.3.2.cmml">w</mi><mo id="S3.SS4.p3.5.m5.2.2.1.3.1" xref="S3.SS4.p3.5.m5.2.2.1.3.1.cmml">∈</mo><msup id="S3.SS4.p3.5.m5.2.2.1.3.3" xref="S3.SS4.p3.5.m5.2.2.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p3.5.m5.2.2.1.3.3.2" xref="S3.SS4.p3.5.m5.2.2.1.3.3.2.cmml">𝒜</mi><mo id="S3.SS4.p3.5.m5.2.2.1.3.3.3" xref="S3.SS4.p3.5.m5.2.2.1.3.3.3.cmml">∗</mo></msup></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.5.m5.2b"><apply id="S3.SS4.p3.5.m5.2.2.cmml" xref="S3.SS4.p3.5.m5.2.2"><csymbol cd="latexml" id="S3.SS4.p3.5.m5.2.2.2.cmml" xref="S3.SS4.p3.5.m5.2.2.2">maps-to</csymbol><ci id="S3.SS4.p3.5.m5.2.2.3.cmml" xref="S3.SS4.p3.5.m5.2.2.3">𝜇</ci><apply id="S3.SS4.p3.5.m5.2.2.1.cmml" xref="S3.SS4.p3.5.m5.2.2.1"><csymbol cd="ambiguous" id="S3.SS4.p3.5.m5.2.2.1.2.cmml" xref="S3.SS4.p3.5.m5.2.2.1">subscript</csymbol><apply id="S3.SS4.p3.5.m5.2.2.1.1.1.1.cmml" xref="S3.SS4.p3.5.m5.2.2.1.1.1"><times id="S3.SS4.p3.5.m5.2.2.1.1.1.1.2.cmml" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.2"></times><ci id="S3.SS4.p3.5.m5.2.2.1.1.1.1.3.cmml" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.3">𝜇</ci><apply id="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.2"><csymbol cd="latexml" id="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.p3.5.m5.2.2.1.1.1.1.1.1.1.2.1">delimited-[]</csymbol><ci id="S3.SS4.p3.5.m5.1.1.cmml" xref="S3.SS4.p3.5.m5.1.1">𝑤</ci></apply></apply><apply id="S3.SS4.p3.5.m5.2.2.1.3.cmml" xref="S3.SS4.p3.5.m5.2.2.1.3"><in id="S3.SS4.p3.5.m5.2.2.1.3.1.cmml" xref="S3.SS4.p3.5.m5.2.2.1.3.1"></in><ci id="S3.SS4.p3.5.m5.2.2.1.3.2.cmml" xref="S3.SS4.p3.5.m5.2.2.1.3.2">𝑤</ci><apply id="S3.SS4.p3.5.m5.2.2.1.3.3.cmml" xref="S3.SS4.p3.5.m5.2.2.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.p3.5.m5.2.2.1.3.3.1.cmml" xref="S3.SS4.p3.5.m5.2.2.1.3.3">superscript</csymbol><ci id="S3.SS4.p3.5.m5.2.2.1.3.3.2.cmml" xref="S3.SS4.p3.5.m5.2.2.1.3.3.2">𝒜</ci><times id="S3.SS4.p3.5.m5.2.2.1.3.3.3.cmml" xref="S3.SS4.p3.5.m5.2.2.1.3.3.3"></times></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.5.m5.2c">\mu\mapsto(\mu([w]))_{w\in\cal A^{*}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.5.m5.2d">italic_μ ↦ ( italic_μ ( [ italic_w ] ) ) start_POSTSUBSCRIPT italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="S3.Thmthm8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm8.1.1.1">Lemma 3.8</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm8.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm8.p1"> <p class="ltx_p" id="S3.Thmthm8.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmthm8.p1.2.2">For any non-erasing monoid morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm8.p1.1.1.m1.1"><semantics id="S3.Thmthm8.p1.1.1.m1.1a"><mrow id="S3.Thmthm8.p1.1.1.m1.1.1" xref="S3.Thmthm8.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmthm8.p1.1.1.m1.1.1.2" xref="S3.Thmthm8.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S3.Thmthm8.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm8.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm8.p1.1.1.m1.1.1.3" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.cmml"><msup id="S3.Thmthm8.p1.1.1.m1.1.1.3.2" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm8.p1.1.1.m1.1.1.3.2.2" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm8.p1.1.1.m1.1.1.3.2.3" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm8.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.Thmthm8.p1.1.1.m1.1.1.3.3" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm8.p1.1.1.m1.1.1.3.3.2" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.Thmthm8.p1.1.1.m1.1.1.3.3.3" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm8.p1.1.1.m1.1b"><apply id="S3.Thmthm8.p1.1.1.m1.1.1.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1"><ci id="S3.Thmthm8.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.1">:</ci><ci id="S3.Thmthm8.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S3.Thmthm8.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3"><ci id="S3.Thmthm8.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm8.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm8.p1.1.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm8.p1.1.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm8.p1.1.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm8.p1.1.1.m1.1.1.3.3.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm8.p1.1.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.Thmthm8.p1.1.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.Thmthm8.p1.1.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm8.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm8.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm8.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the induced map <math alttext="\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.Thmthm8.p1.2.2.m2.2"><semantics id="S3.Thmthm8.p1.2.2.m2.2a"><mrow id="S3.Thmthm8.p1.2.2.m2.2.2" xref="S3.Thmthm8.p1.2.2.m2.2.2.cmml"><mrow id="S3.Thmthm8.p1.2.2.m2.2.2.4" xref="S3.Thmthm8.p1.2.2.m2.2.2.4.cmml"><mi id="S3.Thmthm8.p1.2.2.m2.2.2.4.2" xref="S3.Thmthm8.p1.2.2.m2.2.2.4.2.cmml">σ</mi><mo id="S3.Thmthm8.p1.2.2.m2.2.2.4.1" xref="S3.Thmthm8.p1.2.2.m2.2.2.4.1.cmml">⁢</mo><mi id="S3.Thmthm8.p1.2.2.m2.2.2.4.3" xref="S3.Thmthm8.p1.2.2.m2.2.2.4.3.cmml">M</mi></mrow><mo id="S3.Thmthm8.p1.2.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm8.p1.2.2.m2.2.2.3.cmml">:</mo><mrow id="S3.Thmthm8.p1.2.2.m2.2.2.2" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.cmml"><mrow id="S3.Thmthm8.p1.2.2.m2.1.1.1.1" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.3" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.3.cmml">ℳ</mi><mo id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.2" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.cmml"><mo id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.2" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.3" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm8.p1.2.2.m2.2.2.2.3" stretchy="false" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.3.cmml">→</mo><mrow id="S3.Thmthm8.p1.2.2.m2.2.2.2.2" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.3" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.3.cmml">ℳ</mi><mo id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.2" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.2.cmml">⁢</mo><mrow id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.cmml"><mo id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.2" stretchy="false" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.cmml">(</mo><msup id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.2" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.2.cmml">ℬ</mi><mi id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.3" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.3" stretchy="false" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm8.p1.2.2.m2.2b"><apply id="S3.Thmthm8.p1.2.2.m2.2.2.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2"><ci id="S3.Thmthm8.p1.2.2.m2.2.2.3.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.3">:</ci><apply id="S3.Thmthm8.p1.2.2.m2.2.2.4.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.4"><times id="S3.Thmthm8.p1.2.2.m2.2.2.4.1.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.4.1"></times><ci id="S3.Thmthm8.p1.2.2.m2.2.2.4.2.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.4.2">𝜎</ci><ci id="S3.Thmthm8.p1.2.2.m2.2.2.4.3.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.4.3">𝑀</ci></apply><apply id="S3.Thmthm8.p1.2.2.m2.2.2.2.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2"><ci id="S3.Thmthm8.p1.2.2.m2.2.2.2.3.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.3">→</ci><apply id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.cmml" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1"><times id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.2.cmml" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.2"></times><ci id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.3.cmml" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.3">ℳ</ci><apply id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.2">𝒜</ci><ci id="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm8.p1.2.2.m2.1.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2"><times id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.2.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.2"></times><ci id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.3.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.3">ℳ</ci><apply id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.1.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1">superscript</csymbol><ci id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.2.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.2">ℬ</ci><ci id="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.3.cmml" xref="S3.Thmthm8.p1.2.2.m2.2.2.2.2.1.1.1.3">ℤ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm8.p1.2.2.m2.2c">\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm8.p1.2.2.m2.2d">italic_σ italic_M : caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> is continuous.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS4.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS4.1.p1"> <p class="ltx_p" id="S3.SS4.1.p1.15">Following (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E4" title="In Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.4</span></a>) we decompose <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.SS4.1.p1.1.m1.1"><semantics id="S3.SS4.1.p1.1.m1.1a"><mi id="S3.SS4.1.p1.1.m1.1.1" xref="S3.SS4.1.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.1.m1.1b"><ci id="S3.SS4.1.p1.1.m1.1.1.cmml" xref="S3.SS4.1.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.1.m1.1d">italic_σ</annotation></semantics></math> as product <math alttext="\sigma=\alpha_{\sigma}\circ\pi_{\sigma}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.2.m2.1"><semantics id="S3.SS4.1.p1.2.m2.1a"><mrow id="S3.SS4.1.p1.2.m2.1.1" xref="S3.SS4.1.p1.2.m2.1.1.cmml"><mi id="S3.SS4.1.p1.2.m2.1.1.2" xref="S3.SS4.1.p1.2.m2.1.1.2.cmml">σ</mi><mo id="S3.SS4.1.p1.2.m2.1.1.1" xref="S3.SS4.1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS4.1.p1.2.m2.1.1.3" xref="S3.SS4.1.p1.2.m2.1.1.3.cmml"><msub id="S3.SS4.1.p1.2.m2.1.1.3.2" xref="S3.SS4.1.p1.2.m2.1.1.3.2.cmml"><mi id="S3.SS4.1.p1.2.m2.1.1.3.2.2" xref="S3.SS4.1.p1.2.m2.1.1.3.2.2.cmml">α</mi><mi id="S3.SS4.1.p1.2.m2.1.1.3.2.3" xref="S3.SS4.1.p1.2.m2.1.1.3.2.3.cmml">σ</mi></msub><mo id="S3.SS4.1.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS4.1.p1.2.m2.1.1.3.1.cmml">∘</mo><msub id="S3.SS4.1.p1.2.m2.1.1.3.3" xref="S3.SS4.1.p1.2.m2.1.1.3.3.cmml"><mi id="S3.SS4.1.p1.2.m2.1.1.3.3.2" xref="S3.SS4.1.p1.2.m2.1.1.3.3.2.cmml">π</mi><mi id="S3.SS4.1.p1.2.m2.1.1.3.3.3" xref="S3.SS4.1.p1.2.m2.1.1.3.3.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.2.m2.1b"><apply id="S3.SS4.1.p1.2.m2.1.1.cmml" xref="S3.SS4.1.p1.2.m2.1.1"><eq id="S3.SS4.1.p1.2.m2.1.1.1.cmml" xref="S3.SS4.1.p1.2.m2.1.1.1"></eq><ci id="S3.SS4.1.p1.2.m2.1.1.2.cmml" xref="S3.SS4.1.p1.2.m2.1.1.2">𝜎</ci><apply id="S3.SS4.1.p1.2.m2.1.1.3.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3"><compose id="S3.SS4.1.p1.2.m2.1.1.3.1.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.1"></compose><apply id="S3.SS4.1.p1.2.m2.1.1.3.2.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.2.m2.1.1.3.2.1.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.2">subscript</csymbol><ci id="S3.SS4.1.p1.2.m2.1.1.3.2.2.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.2.2">𝛼</ci><ci id="S3.SS4.1.p1.2.m2.1.1.3.2.3.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.2.3">𝜎</ci></apply><apply id="S3.SS4.1.p1.2.m2.1.1.3.3.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.2.m2.1.1.3.3.1.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S3.SS4.1.p1.2.m2.1.1.3.3.2.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.3.2">𝜋</ci><ci id="S3.SS4.1.p1.2.m2.1.1.3.3.3.cmml" xref="S3.SS4.1.p1.2.m2.1.1.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.2.m2.1c">\sigma=\alpha_{\sigma}\circ\pi_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.2.m2.1d">italic_σ = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∘ italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. In order to show that the first factor of this decomposition, the morphism <math alttext="\pi_{\sigma}:\cal A^{*}\to\cal A_{\sigma}^{*}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.3.m3.1"><semantics id="S3.SS4.1.p1.3.m3.1a"><mrow id="S3.SS4.1.p1.3.m3.1.1" xref="S3.SS4.1.p1.3.m3.1.1.cmml"><msub id="S3.SS4.1.p1.3.m3.1.1.2" xref="S3.SS4.1.p1.3.m3.1.1.2.cmml"><mi id="S3.SS4.1.p1.3.m3.1.1.2.2" xref="S3.SS4.1.p1.3.m3.1.1.2.2.cmml">π</mi><mi id="S3.SS4.1.p1.3.m3.1.1.2.3" xref="S3.SS4.1.p1.3.m3.1.1.2.3.cmml">σ</mi></msub><mo id="S3.SS4.1.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS4.1.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S3.SS4.1.p1.3.m3.1.1.3" xref="S3.SS4.1.p1.3.m3.1.1.3.cmml"><msup id="S3.SS4.1.p1.3.m3.1.1.3.2" xref="S3.SS4.1.p1.3.m3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.3.m3.1.1.3.2.2" xref="S3.SS4.1.p1.3.m3.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.SS4.1.p1.3.m3.1.1.3.2.3" xref="S3.SS4.1.p1.3.m3.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.SS4.1.p1.3.m3.1.1.3.1" stretchy="false" xref="S3.SS4.1.p1.3.m3.1.1.3.1.cmml">→</mo><msubsup id="S3.SS4.1.p1.3.m3.1.1.3.3" xref="S3.SS4.1.p1.3.m3.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.3.m3.1.1.3.3.2.2" xref="S3.SS4.1.p1.3.m3.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S3.SS4.1.p1.3.m3.1.1.3.3.2.3" xref="S3.SS4.1.p1.3.m3.1.1.3.3.2.3.cmml">σ</mi><mo id="S3.SS4.1.p1.3.m3.1.1.3.3.3" xref="S3.SS4.1.p1.3.m3.1.1.3.3.3.cmml">∗</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.3.m3.1b"><apply id="S3.SS4.1.p1.3.m3.1.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1"><ci id="S3.SS4.1.p1.3.m3.1.1.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1.1">:</ci><apply id="S3.SS4.1.p1.3.m3.1.1.2.cmml" xref="S3.SS4.1.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.3.m3.1.1.2.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS4.1.p1.3.m3.1.1.2.2.cmml" xref="S3.SS4.1.p1.3.m3.1.1.2.2">𝜋</ci><ci id="S3.SS4.1.p1.3.m3.1.1.2.3.cmml" xref="S3.SS4.1.p1.3.m3.1.1.2.3">𝜎</ci></apply><apply id="S3.SS4.1.p1.3.m3.1.1.3.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3"><ci id="S3.SS4.1.p1.3.m3.1.1.3.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.1">→</ci><apply id="S3.SS4.1.p1.3.m3.1.1.3.2.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.3.m3.1.1.3.2.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S3.SS4.1.p1.3.m3.1.1.3.2.2.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.2.2">𝒜</ci><times id="S3.SS4.1.p1.3.m3.1.1.3.2.3.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.2.3"></times></apply><apply id="S3.SS4.1.p1.3.m3.1.1.3.3.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.3.m3.1.1.3.3.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.3">superscript</csymbol><apply id="S3.SS4.1.p1.3.m3.1.1.3.3.2.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.3.m3.1.1.3.3.2.1.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.3">subscript</csymbol><ci id="S3.SS4.1.p1.3.m3.1.1.3.3.2.2.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.3.2.2">𝒜</ci><ci id="S3.SS4.1.p1.3.m3.1.1.3.3.2.3.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.3.2.3">𝜎</ci></apply><times id="S3.SS4.1.p1.3.m3.1.1.3.3.3.cmml" xref="S3.SS4.1.p1.3.m3.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.3.m3.1c">\pi_{\sigma}:\cal A^{*}\to\cal A_{\sigma}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.3.m3.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, induces a continuous map on <math alttext="\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.SS4.1.p1.4.m4.1"><semantics id="S3.SS4.1.p1.4.m4.1a"><mrow id="S3.SS4.1.p1.4.m4.1.1" xref="S3.SS4.1.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.4.m4.1.1.3" xref="S3.SS4.1.p1.4.m4.1.1.3.cmml">ℳ</mi><mo id="S3.SS4.1.p1.4.m4.1.1.2" xref="S3.SS4.1.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.1.p1.4.m4.1.1.1.1" xref="S3.SS4.1.p1.4.m4.1.1.1.1.1.cmml"><mo id="S3.SS4.1.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S3.SS4.1.p1.4.m4.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.1.p1.4.m4.1.1.1.1.1" xref="S3.SS4.1.p1.4.m4.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.4.m4.1.1.1.1.1.2" xref="S3.SS4.1.p1.4.m4.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.SS4.1.p1.4.m4.1.1.1.1.1.3" xref="S3.SS4.1.p1.4.m4.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.1.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S3.SS4.1.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.4.m4.1b"><apply id="S3.SS4.1.p1.4.m4.1.1.cmml" xref="S3.SS4.1.p1.4.m4.1.1"><times id="S3.SS4.1.p1.4.m4.1.1.2.cmml" xref="S3.SS4.1.p1.4.m4.1.1.2"></times><ci id="S3.SS4.1.p1.4.m4.1.1.3.cmml" xref="S3.SS4.1.p1.4.m4.1.1.3">ℳ</ci><apply id="S3.SS4.1.p1.4.m4.1.1.1.1.1.cmml" xref="S3.SS4.1.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS4.1.p1.4.m4.1.1.1.1">superscript</csymbol><ci id="S3.SS4.1.p1.4.m4.1.1.1.1.1.2.cmml" xref="S3.SS4.1.p1.4.m4.1.1.1.1.1.2">𝒜</ci><ci id="S3.SS4.1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S3.SS4.1.p1.4.m4.1.1.1.1.1.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.4.m4.1c">\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.4.m4.1d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math>, we recall that by definition this map is defined via <math alttext="\mu\mapsto\mu_{\ell_{\sigma}}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.5.m5.1"><semantics id="S3.SS4.1.p1.5.m5.1a"><mrow id="S3.SS4.1.p1.5.m5.1.1" xref="S3.SS4.1.p1.5.m5.1.1.cmml"><mi id="S3.SS4.1.p1.5.m5.1.1.2" xref="S3.SS4.1.p1.5.m5.1.1.2.cmml">μ</mi><mo id="S3.SS4.1.p1.5.m5.1.1.1" stretchy="false" xref="S3.SS4.1.p1.5.m5.1.1.1.cmml">↦</mo><msub id="S3.SS4.1.p1.5.m5.1.1.3" xref="S3.SS4.1.p1.5.m5.1.1.3.cmml"><mi id="S3.SS4.1.p1.5.m5.1.1.3.2" xref="S3.SS4.1.p1.5.m5.1.1.3.2.cmml">μ</mi><msub id="S3.SS4.1.p1.5.m5.1.1.3.3" xref="S3.SS4.1.p1.5.m5.1.1.3.3.cmml"><mi id="S3.SS4.1.p1.5.m5.1.1.3.3.2" mathvariant="normal" xref="S3.SS4.1.p1.5.m5.1.1.3.3.2.cmml">ℓ</mi><mi id="S3.SS4.1.p1.5.m5.1.1.3.3.3" xref="S3.SS4.1.p1.5.m5.1.1.3.3.3.cmml">σ</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.5.m5.1b"><apply id="S3.SS4.1.p1.5.m5.1.1.cmml" xref="S3.SS4.1.p1.5.m5.1.1"><csymbol cd="latexml" id="S3.SS4.1.p1.5.m5.1.1.1.cmml" xref="S3.SS4.1.p1.5.m5.1.1.1">maps-to</csymbol><ci id="S3.SS4.1.p1.5.m5.1.1.2.cmml" xref="S3.SS4.1.p1.5.m5.1.1.2">𝜇</ci><apply id="S3.SS4.1.p1.5.m5.1.1.3.cmml" xref="S3.SS4.1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.5.m5.1.1.3.1.cmml" xref="S3.SS4.1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.SS4.1.p1.5.m5.1.1.3.2.cmml" xref="S3.SS4.1.p1.5.m5.1.1.3.2">𝜇</ci><apply id="S3.SS4.1.p1.5.m5.1.1.3.3.cmml" xref="S3.SS4.1.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.5.m5.1.1.3.3.1.cmml" xref="S3.SS4.1.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S3.SS4.1.p1.5.m5.1.1.3.3.2.cmml" xref="S3.SS4.1.p1.5.m5.1.1.3.3.2">ℓ</ci><ci id="S3.SS4.1.p1.5.m5.1.1.3.3.3.cmml" xref="S3.SS4.1.p1.5.m5.1.1.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.5.m5.1c">\mu\mapsto\mu_{\ell_{\sigma}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.5.m5.1d">italic_μ ↦ italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, with <math alttext="\mu_{\ell_{\sigma}}(w)=\mu(\widehat{w})" class="ltx_Math" display="inline" id="S3.SS4.1.p1.6.m6.2"><semantics id="S3.SS4.1.p1.6.m6.2a"><mrow id="S3.SS4.1.p1.6.m6.2.3" xref="S3.SS4.1.p1.6.m6.2.3.cmml"><mrow id="S3.SS4.1.p1.6.m6.2.3.2" xref="S3.SS4.1.p1.6.m6.2.3.2.cmml"><msub id="S3.SS4.1.p1.6.m6.2.3.2.2" xref="S3.SS4.1.p1.6.m6.2.3.2.2.cmml"><mi id="S3.SS4.1.p1.6.m6.2.3.2.2.2" xref="S3.SS4.1.p1.6.m6.2.3.2.2.2.cmml">μ</mi><msub id="S3.SS4.1.p1.6.m6.2.3.2.2.3" xref="S3.SS4.1.p1.6.m6.2.3.2.2.3.cmml"><mi id="S3.SS4.1.p1.6.m6.2.3.2.2.3.2" mathvariant="normal" xref="S3.SS4.1.p1.6.m6.2.3.2.2.3.2.cmml">ℓ</mi><mi id="S3.SS4.1.p1.6.m6.2.3.2.2.3.3" xref="S3.SS4.1.p1.6.m6.2.3.2.2.3.3.cmml">σ</mi></msub></msub><mo id="S3.SS4.1.p1.6.m6.2.3.2.1" xref="S3.SS4.1.p1.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S3.SS4.1.p1.6.m6.2.3.2.3.2" xref="S3.SS4.1.p1.6.m6.2.3.2.cmml"><mo id="S3.SS4.1.p1.6.m6.2.3.2.3.2.1" stretchy="false" xref="S3.SS4.1.p1.6.m6.2.3.2.cmml">(</mo><mi id="S3.SS4.1.p1.6.m6.1.1" xref="S3.SS4.1.p1.6.m6.1.1.cmml">w</mi><mo id="S3.SS4.1.p1.6.m6.2.3.2.3.2.2" stretchy="false" xref="S3.SS4.1.p1.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.SS4.1.p1.6.m6.2.3.1" xref="S3.SS4.1.p1.6.m6.2.3.1.cmml">=</mo><mrow id="S3.SS4.1.p1.6.m6.2.3.3" xref="S3.SS4.1.p1.6.m6.2.3.3.cmml"><mi id="S3.SS4.1.p1.6.m6.2.3.3.2" xref="S3.SS4.1.p1.6.m6.2.3.3.2.cmml">μ</mi><mo id="S3.SS4.1.p1.6.m6.2.3.3.1" xref="S3.SS4.1.p1.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S3.SS4.1.p1.6.m6.2.3.3.3.2" xref="S3.SS4.1.p1.6.m6.2.2.cmml"><mo id="S3.SS4.1.p1.6.m6.2.3.3.3.2.1" stretchy="false" xref="S3.SS4.1.p1.6.m6.2.2.cmml">(</mo><mover accent="true" id="S3.SS4.1.p1.6.m6.2.2" xref="S3.SS4.1.p1.6.m6.2.2.cmml"><mi id="S3.SS4.1.p1.6.m6.2.2.2" xref="S3.SS4.1.p1.6.m6.2.2.2.cmml">w</mi><mo id="S3.SS4.1.p1.6.m6.2.2.1" xref="S3.SS4.1.p1.6.m6.2.2.1.cmml">^</mo></mover><mo id="S3.SS4.1.p1.6.m6.2.3.3.3.2.2" stretchy="false" xref="S3.SS4.1.p1.6.m6.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.6.m6.2b"><apply id="S3.SS4.1.p1.6.m6.2.3.cmml" xref="S3.SS4.1.p1.6.m6.2.3"><eq id="S3.SS4.1.p1.6.m6.2.3.1.cmml" xref="S3.SS4.1.p1.6.m6.2.3.1"></eq><apply id="S3.SS4.1.p1.6.m6.2.3.2.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2"><times id="S3.SS4.1.p1.6.m6.2.3.2.1.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2.1"></times><apply id="S3.SS4.1.p1.6.m6.2.3.2.2.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.6.m6.2.3.2.2.1.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2.2">subscript</csymbol><ci id="S3.SS4.1.p1.6.m6.2.3.2.2.2.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2.2.2">𝜇</ci><apply id="S3.SS4.1.p1.6.m6.2.3.2.2.3.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2.2.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.6.m6.2.3.2.2.3.1.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2.2.3">subscript</csymbol><ci id="S3.SS4.1.p1.6.m6.2.3.2.2.3.2.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2.2.3.2">ℓ</ci><ci id="S3.SS4.1.p1.6.m6.2.3.2.2.3.3.cmml" xref="S3.SS4.1.p1.6.m6.2.3.2.2.3.3">𝜎</ci></apply></apply><ci id="S3.SS4.1.p1.6.m6.1.1.cmml" xref="S3.SS4.1.p1.6.m6.1.1">𝑤</ci></apply><apply id="S3.SS4.1.p1.6.m6.2.3.3.cmml" xref="S3.SS4.1.p1.6.m6.2.3.3"><times id="S3.SS4.1.p1.6.m6.2.3.3.1.cmml" xref="S3.SS4.1.p1.6.m6.2.3.3.1"></times><ci id="S3.SS4.1.p1.6.m6.2.3.3.2.cmml" xref="S3.SS4.1.p1.6.m6.2.3.3.2">𝜇</ci><apply id="S3.SS4.1.p1.6.m6.2.2.cmml" xref="S3.SS4.1.p1.6.m6.2.3.3.3.2"><ci id="S3.SS4.1.p1.6.m6.2.2.1.cmml" xref="S3.SS4.1.p1.6.m6.2.2.1">^</ci><ci id="S3.SS4.1.p1.6.m6.2.2.2.cmml" xref="S3.SS4.1.p1.6.m6.2.2.2">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.6.m6.2c">\mu_{\ell_{\sigma}}(w)=\mu(\widehat{w})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.6.m6.2d">italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_w ) = italic_μ ( over^ start_ARG italic_w end_ARG )</annotation></semantics></math> for any <math alttext="w\in\cal A_{\sigma}^{*}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.7.m7.1"><semantics id="S3.SS4.1.p1.7.m7.1a"><mrow id="S3.SS4.1.p1.7.m7.1.1" xref="S3.SS4.1.p1.7.m7.1.1.cmml"><mi id="S3.SS4.1.p1.7.m7.1.1.2" xref="S3.SS4.1.p1.7.m7.1.1.2.cmml">w</mi><mo id="S3.SS4.1.p1.7.m7.1.1.1" xref="S3.SS4.1.p1.7.m7.1.1.1.cmml">∈</mo><msubsup id="S3.SS4.1.p1.7.m7.1.1.3" xref="S3.SS4.1.p1.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.7.m7.1.1.3.2.2" xref="S3.SS4.1.p1.7.m7.1.1.3.2.2.cmml">𝒜</mi><mi id="S3.SS4.1.p1.7.m7.1.1.3.2.3" xref="S3.SS4.1.p1.7.m7.1.1.3.2.3.cmml">σ</mi><mo id="S3.SS4.1.p1.7.m7.1.1.3.3" xref="S3.SS4.1.p1.7.m7.1.1.3.3.cmml">∗</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.7.m7.1b"><apply id="S3.SS4.1.p1.7.m7.1.1.cmml" xref="S3.SS4.1.p1.7.m7.1.1"><in id="S3.SS4.1.p1.7.m7.1.1.1.cmml" xref="S3.SS4.1.p1.7.m7.1.1.1"></in><ci id="S3.SS4.1.p1.7.m7.1.1.2.cmml" xref="S3.SS4.1.p1.7.m7.1.1.2">𝑤</ci><apply id="S3.SS4.1.p1.7.m7.1.1.3.cmml" xref="S3.SS4.1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.7.m7.1.1.3.1.cmml" xref="S3.SS4.1.p1.7.m7.1.1.3">superscript</csymbol><apply id="S3.SS4.1.p1.7.m7.1.1.3.2.cmml" xref="S3.SS4.1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.7.m7.1.1.3.2.1.cmml" xref="S3.SS4.1.p1.7.m7.1.1.3">subscript</csymbol><ci id="S3.SS4.1.p1.7.m7.1.1.3.2.2.cmml" xref="S3.SS4.1.p1.7.m7.1.1.3.2.2">𝒜</ci><ci id="S3.SS4.1.p1.7.m7.1.1.3.2.3.cmml" xref="S3.SS4.1.p1.7.m7.1.1.3.2.3">𝜎</ci></apply><times id="S3.SS4.1.p1.7.m7.1.1.3.3.cmml" xref="S3.SS4.1.p1.7.m7.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.7.m7.1c">w\in\cal A_{\sigma}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.7.m7.1d">italic_w ∈ caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\widehat{w}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.8.m8.1"><semantics id="S3.SS4.1.p1.8.m8.1a"><mover accent="true" id="S3.SS4.1.p1.8.m8.1.1" xref="S3.SS4.1.p1.8.m8.1.1.cmml"><mi id="S3.SS4.1.p1.8.m8.1.1.2" xref="S3.SS4.1.p1.8.m8.1.1.2.cmml">w</mi><mo id="S3.SS4.1.p1.8.m8.1.1.1" xref="S3.SS4.1.p1.8.m8.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.8.m8.1b"><apply id="S3.SS4.1.p1.8.m8.1.1.cmml" xref="S3.SS4.1.p1.8.m8.1.1"><ci id="S3.SS4.1.p1.8.m8.1.1.1.cmml" xref="S3.SS4.1.p1.8.m8.1.1.1">^</ci><ci id="S3.SS4.1.p1.8.m8.1.1.2.cmml" xref="S3.SS4.1.p1.8.m8.1.1.2">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.8.m8.1c">\widehat{w}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.8.m8.1d">over^ start_ARG italic_w end_ARG</annotation></semantics></math> is the shortest word in <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.9.m9.1"><semantics id="S3.SS4.1.p1.9.m9.1a"><msup id="S3.SS4.1.p1.9.m9.1.1" xref="S3.SS4.1.p1.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.9.m9.1.1.2" xref="S3.SS4.1.p1.9.m9.1.1.2.cmml">𝒜</mi><mo id="S3.SS4.1.p1.9.m9.1.1.3" xref="S3.SS4.1.p1.9.m9.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.9.m9.1b"><apply id="S3.SS4.1.p1.9.m9.1.1.cmml" xref="S3.SS4.1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.9.m9.1.1.1.cmml" xref="S3.SS4.1.p1.9.m9.1.1">superscript</csymbol><ci id="S3.SS4.1.p1.9.m9.1.1.2.cmml" xref="S3.SS4.1.p1.9.m9.1.1.2">𝒜</ci><times id="S3.SS4.1.p1.9.m9.1.1.3.cmml" xref="S3.SS4.1.p1.9.m9.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.9.m9.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.9.m9.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="w" class="ltx_Math" display="inline" id="S3.SS4.1.p1.10.m10.1"><semantics id="S3.SS4.1.p1.10.m10.1a"><mi id="S3.SS4.1.p1.10.m10.1.1" xref="S3.SS4.1.p1.10.m10.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.10.m10.1b"><ci id="S3.SS4.1.p1.10.m10.1.1.cmml" xref="S3.SS4.1.p1.10.m10.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.10.m10.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.10.m10.1d">italic_w</annotation></semantics></math> is a factor of <math alttext="\pi_{\sigma}(\widehat{w})" class="ltx_Math" display="inline" id="S3.SS4.1.p1.11.m11.1"><semantics id="S3.SS4.1.p1.11.m11.1a"><mrow id="S3.SS4.1.p1.11.m11.1.2" xref="S3.SS4.1.p1.11.m11.1.2.cmml"><msub id="S3.SS4.1.p1.11.m11.1.2.2" xref="S3.SS4.1.p1.11.m11.1.2.2.cmml"><mi id="S3.SS4.1.p1.11.m11.1.2.2.2" xref="S3.SS4.1.p1.11.m11.1.2.2.2.cmml">π</mi><mi id="S3.SS4.1.p1.11.m11.1.2.2.3" xref="S3.SS4.1.p1.11.m11.1.2.2.3.cmml">σ</mi></msub><mo id="S3.SS4.1.p1.11.m11.1.2.1" xref="S3.SS4.1.p1.11.m11.1.2.1.cmml">⁢</mo><mrow id="S3.SS4.1.p1.11.m11.1.2.3.2" xref="S3.SS4.1.p1.11.m11.1.1.cmml"><mo id="S3.SS4.1.p1.11.m11.1.2.3.2.1" stretchy="false" xref="S3.SS4.1.p1.11.m11.1.1.cmml">(</mo><mover accent="true" id="S3.SS4.1.p1.11.m11.1.1" xref="S3.SS4.1.p1.11.m11.1.1.cmml"><mi id="S3.SS4.1.p1.11.m11.1.1.2" xref="S3.SS4.1.p1.11.m11.1.1.2.cmml">w</mi><mo id="S3.SS4.1.p1.11.m11.1.1.1" xref="S3.SS4.1.p1.11.m11.1.1.1.cmml">^</mo></mover><mo id="S3.SS4.1.p1.11.m11.1.2.3.2.2" stretchy="false" xref="S3.SS4.1.p1.11.m11.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.11.m11.1b"><apply id="S3.SS4.1.p1.11.m11.1.2.cmml" xref="S3.SS4.1.p1.11.m11.1.2"><times id="S3.SS4.1.p1.11.m11.1.2.1.cmml" xref="S3.SS4.1.p1.11.m11.1.2.1"></times><apply id="S3.SS4.1.p1.11.m11.1.2.2.cmml" xref="S3.SS4.1.p1.11.m11.1.2.2"><csymbol cd="ambiguous" id="S3.SS4.1.p1.11.m11.1.2.2.1.cmml" xref="S3.SS4.1.p1.11.m11.1.2.2">subscript</csymbol><ci id="S3.SS4.1.p1.11.m11.1.2.2.2.cmml" xref="S3.SS4.1.p1.11.m11.1.2.2.2">𝜋</ci><ci id="S3.SS4.1.p1.11.m11.1.2.2.3.cmml" xref="S3.SS4.1.p1.11.m11.1.2.2.3">𝜎</ci></apply><apply id="S3.SS4.1.p1.11.m11.1.1.cmml" xref="S3.SS4.1.p1.11.m11.1.2.3.2"><ci id="S3.SS4.1.p1.11.m11.1.1.1.cmml" xref="S3.SS4.1.p1.11.m11.1.1.1">^</ci><ci id="S3.SS4.1.p1.11.m11.1.1.2.cmml" xref="S3.SS4.1.p1.11.m11.1.1.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.11.m11.1c">\pi_{\sigma}(\widehat{w})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.11.m11.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over^ start_ARG italic_w end_ARG )</annotation></semantics></math>. It follows directly that small variations of <math alttext="\mu" class="ltx_Math" display="inline" id="S3.SS4.1.p1.12.m12.1"><semantics id="S3.SS4.1.p1.12.m12.1a"><mi id="S3.SS4.1.p1.12.m12.1.1" xref="S3.SS4.1.p1.12.m12.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.12.m12.1b"><ci id="S3.SS4.1.p1.12.m12.1.1.cmml" xref="S3.SS4.1.p1.12.m12.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.12.m12.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.12.m12.1d">italic_μ</annotation></semantics></math> imply small variations of <math alttext="\mu_{\ell_{\sigma}}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.13.m13.1"><semantics id="S3.SS4.1.p1.13.m13.1a"><msub id="S3.SS4.1.p1.13.m13.1.1" xref="S3.SS4.1.p1.13.m13.1.1.cmml"><mi id="S3.SS4.1.p1.13.m13.1.1.2" xref="S3.SS4.1.p1.13.m13.1.1.2.cmml">μ</mi><msub id="S3.SS4.1.p1.13.m13.1.1.3" xref="S3.SS4.1.p1.13.m13.1.1.3.cmml"><mi id="S3.SS4.1.p1.13.m13.1.1.3.2" mathvariant="normal" xref="S3.SS4.1.p1.13.m13.1.1.3.2.cmml">ℓ</mi><mi id="S3.SS4.1.p1.13.m13.1.1.3.3" xref="S3.SS4.1.p1.13.m13.1.1.3.3.cmml">σ</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.13.m13.1b"><apply id="S3.SS4.1.p1.13.m13.1.1.cmml" xref="S3.SS4.1.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.13.m13.1.1.1.cmml" xref="S3.SS4.1.p1.13.m13.1.1">subscript</csymbol><ci id="S3.SS4.1.p1.13.m13.1.1.2.cmml" xref="S3.SS4.1.p1.13.m13.1.1.2">𝜇</ci><apply id="S3.SS4.1.p1.13.m13.1.1.3.cmml" xref="S3.SS4.1.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.1.p1.13.m13.1.1.3.1.cmml" xref="S3.SS4.1.p1.13.m13.1.1.3">subscript</csymbol><ci id="S3.SS4.1.p1.13.m13.1.1.3.2.cmml" xref="S3.SS4.1.p1.13.m13.1.1.3.2">ℓ</ci><ci id="S3.SS4.1.p1.13.m13.1.1.3.3.cmml" xref="S3.SS4.1.p1.13.m13.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.13.m13.1c">\mu_{\ell_{\sigma}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.13.m13.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, so that <math alttext="\pi_{\sigma}" class="ltx_Math" display="inline" id="S3.SS4.1.p1.14.m14.1"><semantics id="S3.SS4.1.p1.14.m14.1a"><msub id="S3.SS4.1.p1.14.m14.1.1" xref="S3.SS4.1.p1.14.m14.1.1.cmml"><mi id="S3.SS4.1.p1.14.m14.1.1.2" xref="S3.SS4.1.p1.14.m14.1.1.2.cmml">π</mi><mi id="S3.SS4.1.p1.14.m14.1.1.3" xref="S3.SS4.1.p1.14.m14.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.14.m14.1b"><apply id="S3.SS4.1.p1.14.m14.1.1.cmml" xref="S3.SS4.1.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.14.m14.1.1.1.cmml" xref="S3.SS4.1.p1.14.m14.1.1">subscript</csymbol><ci id="S3.SS4.1.p1.14.m14.1.1.2.cmml" xref="S3.SS4.1.p1.14.m14.1.1.2">𝜋</ci><ci id="S3.SS4.1.p1.14.m14.1.1.3.cmml" xref="S3.SS4.1.p1.14.m14.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.14.m14.1c">\pi_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.14.m14.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> induces a continuous map <math alttext="\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal A_{\sigma}^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.SS4.1.p1.15.m15.2"><semantics id="S3.SS4.1.p1.15.m15.2a"><mrow id="S3.SS4.1.p1.15.m15.2.2" xref="S3.SS4.1.p1.15.m15.2.2.cmml"><mrow id="S3.SS4.1.p1.15.m15.1.1.1" xref="S3.SS4.1.p1.15.m15.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.15.m15.1.1.1.3" xref="S3.SS4.1.p1.15.m15.1.1.1.3.cmml">ℳ</mi><mo id="S3.SS4.1.p1.15.m15.1.1.1.2" xref="S3.SS4.1.p1.15.m15.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.1.p1.15.m15.1.1.1.1.1" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.cmml"><mo id="S3.SS4.1.p1.15.m15.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.1.p1.15.m15.1.1.1.1.1.1" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.2" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.3" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.1.p1.15.m15.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.1.p1.15.m15.2.2.3" stretchy="false" xref="S3.SS4.1.p1.15.m15.2.2.3.cmml">→</mo><mrow id="S3.SS4.1.p1.15.m15.2.2.2" xref="S3.SS4.1.p1.15.m15.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.15.m15.2.2.2.3" xref="S3.SS4.1.p1.15.m15.2.2.2.3.cmml">ℳ</mi><mo id="S3.SS4.1.p1.15.m15.2.2.2.2" xref="S3.SS4.1.p1.15.m15.2.2.2.2.cmml">⁢</mo><mrow id="S3.SS4.1.p1.15.m15.2.2.2.1.1" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.cmml"><mo id="S3.SS4.1.p1.15.m15.2.2.2.1.1.2" stretchy="false" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.cmml">(</mo><msubsup id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.2" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.2.cmml">𝒜</mi><mi id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.3" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.3.cmml">σ</mi><mi id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.3" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.3.cmml">ℤ</mi></msubsup><mo id="S3.SS4.1.p1.15.m15.2.2.2.1.1.3" stretchy="false" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.1.p1.15.m15.2b"><apply id="S3.SS4.1.p1.15.m15.2.2.cmml" xref="S3.SS4.1.p1.15.m15.2.2"><ci id="S3.SS4.1.p1.15.m15.2.2.3.cmml" xref="S3.SS4.1.p1.15.m15.2.2.3">→</ci><apply id="S3.SS4.1.p1.15.m15.1.1.1.cmml" xref="S3.SS4.1.p1.15.m15.1.1.1"><times id="S3.SS4.1.p1.15.m15.1.1.1.2.cmml" xref="S3.SS4.1.p1.15.m15.1.1.1.2"></times><ci id="S3.SS4.1.p1.15.m15.1.1.1.3.cmml" xref="S3.SS4.1.p1.15.m15.1.1.1.3">ℳ</ci><apply id="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.cmml" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.1.cmml" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.2.cmml" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.2">𝒜</ci><ci id="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.3.cmml" xref="S3.SS4.1.p1.15.m15.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S3.SS4.1.p1.15.m15.2.2.2.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2"><times id="S3.SS4.1.p1.15.m15.2.2.2.2.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.2"></times><ci id="S3.SS4.1.p1.15.m15.2.2.2.3.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.3">ℳ</ci><apply id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.1.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1">superscript</csymbol><apply id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.1.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1">subscript</csymbol><ci id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.2.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.2">𝒜</ci><ci id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.3.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.2.3">𝜎</ci></apply><ci id="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.3.cmml" xref="S3.SS4.1.p1.15.m15.2.2.2.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.1.p1.15.m15.2c">\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal A_{\sigma}^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.1.p1.15.m15.2d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS4.2.p2"> <p class="ltx_p" id="S3.SS4.2.p2.4">The second factor <math alttext="\alpha_{\sigma}:\cal A_{\sigma}^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.SS4.2.p2.1.m1.1"><semantics id="S3.SS4.2.p2.1.m1.1a"><mrow id="S3.SS4.2.p2.1.m1.1.1" xref="S3.SS4.2.p2.1.m1.1.1.cmml"><msub id="S3.SS4.2.p2.1.m1.1.1.2" xref="S3.SS4.2.p2.1.m1.1.1.2.cmml"><mi id="S3.SS4.2.p2.1.m1.1.1.2.2" xref="S3.SS4.2.p2.1.m1.1.1.2.2.cmml">α</mi><mi id="S3.SS4.2.p2.1.m1.1.1.2.3" xref="S3.SS4.2.p2.1.m1.1.1.2.3.cmml">σ</mi></msub><mo id="S3.SS4.2.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS4.2.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S3.SS4.2.p2.1.m1.1.1.3" xref="S3.SS4.2.p2.1.m1.1.1.3.cmml"><msubsup id="S3.SS4.2.p2.1.m1.1.1.3.2" xref="S3.SS4.2.p2.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.1.m1.1.1.3.2.2.2" xref="S3.SS4.2.p2.1.m1.1.1.3.2.2.2.cmml">𝒜</mi><mi id="S3.SS4.2.p2.1.m1.1.1.3.2.2.3" xref="S3.SS4.2.p2.1.m1.1.1.3.2.2.3.cmml">σ</mi><mo id="S3.SS4.2.p2.1.m1.1.1.3.2.3" xref="S3.SS4.2.p2.1.m1.1.1.3.2.3.cmml">∗</mo></msubsup><mo id="S3.SS4.2.p2.1.m1.1.1.3.1" stretchy="false" xref="S3.SS4.2.p2.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.SS4.2.p2.1.m1.1.1.3.3" xref="S3.SS4.2.p2.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.1.m1.1.1.3.3.2" xref="S3.SS4.2.p2.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.SS4.2.p2.1.m1.1.1.3.3.3" xref="S3.SS4.2.p2.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.2.p2.1.m1.1b"><apply id="S3.SS4.2.p2.1.m1.1.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1"><ci id="S3.SS4.2.p2.1.m1.1.1.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1.1">:</ci><apply id="S3.SS4.2.p2.1.m1.1.1.2.cmml" xref="S3.SS4.2.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.2.p2.1.m1.1.1.2.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS4.2.p2.1.m1.1.1.2.2.cmml" xref="S3.SS4.2.p2.1.m1.1.1.2.2">𝛼</ci><ci id="S3.SS4.2.p2.1.m1.1.1.2.3.cmml" xref="S3.SS4.2.p2.1.m1.1.1.2.3">𝜎</ci></apply><apply id="S3.SS4.2.p2.1.m1.1.1.3.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3"><ci id="S3.SS4.2.p2.1.m1.1.1.3.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.1">→</ci><apply id="S3.SS4.2.p2.1.m1.1.1.3.2.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.2.p2.1.m1.1.1.3.2.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.2">superscript</csymbol><apply id="S3.SS4.2.p2.1.m1.1.1.3.2.2.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS4.2.p2.1.m1.1.1.3.2.2.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.2">subscript</csymbol><ci id="S3.SS4.2.p2.1.m1.1.1.3.2.2.2.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.2.2.2">𝒜</ci><ci id="S3.SS4.2.p2.1.m1.1.1.3.2.2.3.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.2.2.3">𝜎</ci></apply><times id="S3.SS4.2.p2.1.m1.1.1.3.2.3.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.2.3"></times></apply><apply id="S3.SS4.2.p2.1.m1.1.1.3.3.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS4.2.p2.1.m1.1.1.3.3.1.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.SS4.2.p2.1.m1.1.1.3.3.2.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.SS4.2.p2.1.m1.1.1.3.3.3.cmml" xref="S3.SS4.2.p2.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.2.p2.1.m1.1c">\alpha_{\sigma}:\cal A_{\sigma}^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.2.p2.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of the above decomposition induces a map <math alttext="\cal M(\cal A_{\sigma}^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.SS4.2.p2.2.m2.2"><semantics id="S3.SS4.2.p2.2.m2.2a"><mrow id="S3.SS4.2.p2.2.m2.2.2" xref="S3.SS4.2.p2.2.m2.2.2.cmml"><mrow id="S3.SS4.2.p2.2.m2.1.1.1" xref="S3.SS4.2.p2.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.2.m2.1.1.1.3" xref="S3.SS4.2.p2.2.m2.1.1.1.3.cmml">ℳ</mi><mo id="S3.SS4.2.p2.2.m2.1.1.1.2" xref="S3.SS4.2.p2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.2.p2.2.m2.1.1.1.1.1" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS4.2.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.2" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.2.cmml">𝒜</mi><mi id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.3" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.3.cmml">σ</mi><mi id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.3" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.3.cmml">ℤ</mi></msubsup><mo id="S3.SS4.2.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.2.p2.2.m2.2.2.3" stretchy="false" xref="S3.SS4.2.p2.2.m2.2.2.3.cmml">→</mo><mrow id="S3.SS4.2.p2.2.m2.2.2.2" xref="S3.SS4.2.p2.2.m2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.2.m2.2.2.2.3" xref="S3.SS4.2.p2.2.m2.2.2.2.3.cmml">ℳ</mi><mo id="S3.SS4.2.p2.2.m2.2.2.2.2" xref="S3.SS4.2.p2.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S3.SS4.2.p2.2.m2.2.2.2.1.1" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.cmml"><mo id="S3.SS4.2.p2.2.m2.2.2.2.1.1.2" stretchy="false" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.cmml">(</mo><msup id="S3.SS4.2.p2.2.m2.2.2.2.1.1.1" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.2" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.2.cmml">ℬ</mi><mi id="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.3" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.2.p2.2.m2.2.2.2.1.1.3" stretchy="false" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.2.p2.2.m2.2b"><apply id="S3.SS4.2.p2.2.m2.2.2.cmml" xref="S3.SS4.2.p2.2.m2.2.2"><ci id="S3.SS4.2.p2.2.m2.2.2.3.cmml" xref="S3.SS4.2.p2.2.m2.2.2.3">→</ci><apply id="S3.SS4.2.p2.2.m2.1.1.1.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1"><times id="S3.SS4.2.p2.2.m2.1.1.1.2.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.2"></times><ci id="S3.SS4.2.p2.2.m2.1.1.1.3.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.3">ℳ</ci><apply id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1">superscript</csymbol><apply id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.1.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.2.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.2">𝒜</ci><ci id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.3.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.2.3">𝜎</ci></apply><ci id="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS4.2.p2.2.m2.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S3.SS4.2.p2.2.m2.2.2.2.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2"><times id="S3.SS4.2.p2.2.m2.2.2.2.2.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2.2"></times><ci id="S3.SS4.2.p2.2.m2.2.2.2.3.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2.3">ℳ</ci><apply id="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.1.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1">superscript</csymbol><ci id="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.2.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.2">ℬ</ci><ci id="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.3.cmml" xref="S3.SS4.2.p2.2.m2.2.2.2.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.2.p2.2.m2.2c">\cal M(\cal A_{\sigma}^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.2.p2.2.m2.2d">caligraphic_M ( caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> that is given by the classical push-forward definition <math alttext="\mu^{\prime}\mapsto(\alpha_{\sigma})_{*}(\mu^{\prime})" class="ltx_Math" display="inline" id="S3.SS4.2.p2.3.m3.2"><semantics id="S3.SS4.2.p2.3.m3.2a"><mrow id="S3.SS4.2.p2.3.m3.2.2" xref="S3.SS4.2.p2.3.m3.2.2.cmml"><msup id="S3.SS4.2.p2.3.m3.2.2.4" xref="S3.SS4.2.p2.3.m3.2.2.4.cmml"><mi id="S3.SS4.2.p2.3.m3.2.2.4.2" xref="S3.SS4.2.p2.3.m3.2.2.4.2.cmml">μ</mi><mo id="S3.SS4.2.p2.3.m3.2.2.4.3" xref="S3.SS4.2.p2.3.m3.2.2.4.3.cmml">′</mo></msup><mo id="S3.SS4.2.p2.3.m3.2.2.3" stretchy="false" xref="S3.SS4.2.p2.3.m3.2.2.3.cmml">↦</mo><mrow id="S3.SS4.2.p2.3.m3.2.2.2" xref="S3.SS4.2.p2.3.m3.2.2.2.cmml"><msub id="S3.SS4.2.p2.3.m3.1.1.1.1" xref="S3.SS4.2.p2.3.m3.1.1.1.1.cmml"><mrow id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.cmml"><mi id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.2" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.2.cmml">α</mi><mi id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.3" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.3.cmml">σ</mi></msub><mo id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS4.2.p2.3.m3.1.1.1.1.3" xref="S3.SS4.2.p2.3.m3.1.1.1.1.3.cmml">∗</mo></msub><mo id="S3.SS4.2.p2.3.m3.2.2.2.3" xref="S3.SS4.2.p2.3.m3.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS4.2.p2.3.m3.2.2.2.2.1" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.cmml"><mo id="S3.SS4.2.p2.3.m3.2.2.2.2.1.2" stretchy="false" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.cmml">(</mo><msup id="S3.SS4.2.p2.3.m3.2.2.2.2.1.1" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.cmml"><mi id="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.2" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.2.cmml">μ</mi><mo id="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.3" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.3.cmml">′</mo></msup><mo id="S3.SS4.2.p2.3.m3.2.2.2.2.1.3" stretchy="false" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.2.p2.3.m3.2b"><apply id="S3.SS4.2.p2.3.m3.2.2.cmml" xref="S3.SS4.2.p2.3.m3.2.2"><csymbol cd="latexml" id="S3.SS4.2.p2.3.m3.2.2.3.cmml" xref="S3.SS4.2.p2.3.m3.2.2.3">maps-to</csymbol><apply id="S3.SS4.2.p2.3.m3.2.2.4.cmml" xref="S3.SS4.2.p2.3.m3.2.2.4"><csymbol cd="ambiguous" id="S3.SS4.2.p2.3.m3.2.2.4.1.cmml" xref="S3.SS4.2.p2.3.m3.2.2.4">superscript</csymbol><ci id="S3.SS4.2.p2.3.m3.2.2.4.2.cmml" xref="S3.SS4.2.p2.3.m3.2.2.4.2">𝜇</ci><ci id="S3.SS4.2.p2.3.m3.2.2.4.3.cmml" xref="S3.SS4.2.p2.3.m3.2.2.4.3">′</ci></apply><apply id="S3.SS4.2.p2.3.m3.2.2.2.cmml" xref="S3.SS4.2.p2.3.m3.2.2.2"><times id="S3.SS4.2.p2.3.m3.2.2.2.3.cmml" xref="S3.SS4.2.p2.3.m3.2.2.2.3"></times><apply id="S3.SS4.2.p2.3.m3.1.1.1.1.cmml" xref="S3.SS4.2.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.2.p2.3.m3.1.1.1.1.2.cmml" xref="S3.SS4.2.p2.3.m3.1.1.1.1">subscript</csymbol><apply id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.2">𝛼</ci><ci id="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.3.cmml" xref="S3.SS4.2.p2.3.m3.1.1.1.1.1.1.1.3">𝜎</ci></apply><times id="S3.SS4.2.p2.3.m3.1.1.1.1.3.cmml" xref="S3.SS4.2.p2.3.m3.1.1.1.1.3"></times></apply><apply id="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.cmml" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.1.cmml" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1">superscript</csymbol><ci id="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.2.cmml" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.2">𝜇</ci><ci id="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.3.cmml" xref="S3.SS4.2.p2.3.m3.2.2.2.2.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.2.p2.3.m3.2c">\mu^{\prime}\mapsto(\alpha_{\sigma})_{*}(\mu^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.2.p2.3.m3.2d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ↦ ( italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> for any <math alttext="\mu^{\prime}\in\cal M(\cal A_{\sigma}^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.SS4.2.p2.4.m4.1"><semantics id="S3.SS4.2.p2.4.m4.1a"><mrow id="S3.SS4.2.p2.4.m4.1.1" xref="S3.SS4.2.p2.4.m4.1.1.cmml"><msup id="S3.SS4.2.p2.4.m4.1.1.3" xref="S3.SS4.2.p2.4.m4.1.1.3.cmml"><mi id="S3.SS4.2.p2.4.m4.1.1.3.2" xref="S3.SS4.2.p2.4.m4.1.1.3.2.cmml">μ</mi><mo id="S3.SS4.2.p2.4.m4.1.1.3.3" xref="S3.SS4.2.p2.4.m4.1.1.3.3.cmml">′</mo></msup><mo id="S3.SS4.2.p2.4.m4.1.1.2" xref="S3.SS4.2.p2.4.m4.1.1.2.cmml">∈</mo><mrow id="S3.SS4.2.p2.4.m4.1.1.1" xref="S3.SS4.2.p2.4.m4.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.4.m4.1.1.1.3" xref="S3.SS4.2.p2.4.m4.1.1.1.3.cmml">ℳ</mi><mo id="S3.SS4.2.p2.4.m4.1.1.1.2" xref="S3.SS4.2.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.2.p2.4.m4.1.1.1.1.1" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS4.2.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.2" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.2.cmml">𝒜</mi><mi id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.3" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.3.cmml">σ</mi><mi id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.3" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.3.cmml">ℤ</mi></msubsup><mo id="S3.SS4.2.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.2.p2.4.m4.1b"><apply id="S3.SS4.2.p2.4.m4.1.1.cmml" xref="S3.SS4.2.p2.4.m4.1.1"><in id="S3.SS4.2.p2.4.m4.1.1.2.cmml" xref="S3.SS4.2.p2.4.m4.1.1.2"></in><apply id="S3.SS4.2.p2.4.m4.1.1.3.cmml" xref="S3.SS4.2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.2.p2.4.m4.1.1.3.1.cmml" xref="S3.SS4.2.p2.4.m4.1.1.3">superscript</csymbol><ci id="S3.SS4.2.p2.4.m4.1.1.3.2.cmml" xref="S3.SS4.2.p2.4.m4.1.1.3.2">𝜇</ci><ci id="S3.SS4.2.p2.4.m4.1.1.3.3.cmml" xref="S3.SS4.2.p2.4.m4.1.1.3.3">′</ci></apply><apply id="S3.SS4.2.p2.4.m4.1.1.1.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1"><times id="S3.SS4.2.p2.4.m4.1.1.1.2.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.2"></times><ci id="S3.SS4.2.p2.4.m4.1.1.1.3.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.3">ℳ</ci><apply id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1">superscript</csymbol><apply id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.1.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.2.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.2">𝒜</ci><ci id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.3.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.2.3">𝜎</ci></apply><ci id="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS4.2.p2.4.m4.1.1.1.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.2.p2.4.m4.1c">\mu^{\prime}\in\cal M(\cal A_{\sigma}^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.2.p2.4.m4.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_M ( caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math>, for which the continuity is well known (and easy to prove, by a similar approach as used in the previous paragraph).</p> </div> <div class="ltx_para" id="S3.SS4.3.p3"> <p class="ltx_p" id="S3.SS4.3.p3.4">It follows that the composition <math alttext="\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal A_{\sigma}^{\mathbb{Z}})\to\cal M(% \cal B^{\mathbb{Z}})\,,\,\,\mu\mapsto\mu_{\ell_{\sigma}}\mapsto(\alpha_{\sigma% })_{*}(\mu_{\ell_{\sigma}})" class="ltx_Math" display="inline" id="S3.SS4.3.p3.1.m1.2"><semantics id="S3.SS4.3.p3.1.m1.2a"><mrow id="S3.SS4.3.p3.1.m1.2.2.2" xref="S3.SS4.3.p3.1.m1.2.2.3.cmml"><mrow id="S3.SS4.3.p3.1.m1.1.1.1.1" xref="S3.SS4.3.p3.1.m1.1.1.1.1.cmml"><mrow id="S3.SS4.3.p3.1.m1.1.1.1.1.1" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.1.m1.1.1.1.1.1.3" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.3.cmml">ℳ</mi><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.1.2" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1.2" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1.3" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.3.p3.1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.5" stretchy="false" xref="S3.SS4.3.p3.1.m1.1.1.1.1.5.cmml">→</mo><mrow id="S3.SS4.3.p3.1.m1.1.1.1.1.2" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.1.m1.1.1.1.1.2.3" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.3.cmml">ℳ</mi><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.2.2" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.cmml"><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.2" stretchy="false" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.cmml">(</mo><msubsup id="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.2.2" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.2.2.cmml">𝒜</mi><mi id="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.2.3" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.2.3.cmml">σ</mi><mi id="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.3" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.3.cmml">ℤ</mi></msubsup><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.3" stretchy="false" xref="S3.SS4.3.p3.1.m1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.6" stretchy="false" xref="S3.SS4.3.p3.1.m1.1.1.1.1.6.cmml">→</mo><mrow id="S3.SS4.3.p3.1.m1.1.1.1.1.3" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.1.m1.1.1.1.1.3.3" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.3.cmml">ℳ</mi><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.3.2" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.2.cmml">⁢</mo><mrow id="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1.cmml"><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.2" stretchy="false" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1.cmml">(</mo><msup id="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1.2" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1.2.cmml">ℬ</mi><mi id="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1.3" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.3" rspace="0.170em" stretchy="false" xref="S3.SS4.3.p3.1.m1.1.1.1.1.3.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo 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id="S3.SS4.3.p3.1.m1.2.2.2.2.2.2.1.1.3.1.cmml" xref="S3.SS4.3.p3.1.m1.2.2.2.2.2.2.1.1.3">subscript</csymbol><ci id="S3.SS4.3.p3.1.m1.2.2.2.2.2.2.1.1.3.2.cmml" xref="S3.SS4.3.p3.1.m1.2.2.2.2.2.2.1.1.3.2">ℓ</ci><ci id="S3.SS4.3.p3.1.m1.2.2.2.2.2.2.1.1.3.3.cmml" xref="S3.SS4.3.p3.1.m1.2.2.2.2.2.2.1.1.3.3">𝜎</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.1.m1.2c">\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal A_{\sigma}^{\mathbb{Z}})\to\cal M(% \cal B^{\mathbb{Z}})\,,\,\,\mu\mapsto\mu_{\ell_{\sigma}}\mapsto(\alpha_{\sigma% })_{*}(\mu_{\ell_{\sigma}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.1.m1.2d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) , italic_μ ↦ italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ↦ ( italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math> is continuous. But this is precisely how the map <math alttext="\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.SS4.3.p3.2.m2.2"><semantics id="S3.SS4.3.p3.2.m2.2a"><mrow id="S3.SS4.3.p3.2.m2.2.2" xref="S3.SS4.3.p3.2.m2.2.2.cmml"><mrow id="S3.SS4.3.p3.2.m2.2.2.4" xref="S3.SS4.3.p3.2.m2.2.2.4.cmml"><mi id="S3.SS4.3.p3.2.m2.2.2.4.2" xref="S3.SS4.3.p3.2.m2.2.2.4.2.cmml">σ</mi><mo id="S3.SS4.3.p3.2.m2.2.2.4.1" xref="S3.SS4.3.p3.2.m2.2.2.4.1.cmml">⁢</mo><mi id="S3.SS4.3.p3.2.m2.2.2.4.3" xref="S3.SS4.3.p3.2.m2.2.2.4.3.cmml">M</mi></mrow><mo id="S3.SS4.3.p3.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S3.SS4.3.p3.2.m2.2.2.3.cmml">:</mo><mrow id="S3.SS4.3.p3.2.m2.2.2.2" xref="S3.SS4.3.p3.2.m2.2.2.2.cmml"><mrow id="S3.SS4.3.p3.2.m2.1.1.1.1" xref="S3.SS4.3.p3.2.m2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.2.m2.1.1.1.1.3" xref="S3.SS4.3.p3.2.m2.1.1.1.1.3.cmml">ℳ</mi><mo id="S3.SS4.3.p3.2.m2.1.1.1.1.2" xref="S3.SS4.3.p3.2.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.2" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.3" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.3.p3.2.m2.2.2.2.3" stretchy="false" xref="S3.SS4.3.p3.2.m2.2.2.2.3.cmml">→</mo><mrow id="S3.SS4.3.p3.2.m2.2.2.2.2" xref="S3.SS4.3.p3.2.m2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.2.m2.2.2.2.2.3" xref="S3.SS4.3.p3.2.m2.2.2.2.2.3.cmml">ℳ</mi><mo id="S3.SS4.3.p3.2.m2.2.2.2.2.2" xref="S3.SS4.3.p3.2.m2.2.2.2.2.2.cmml">⁢</mo><mrow id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.cmml"><mo id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.2" stretchy="false" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.cmml">(</mo><msup id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.2" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.2.cmml">ℬ</mi><mi id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.3" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.3" stretchy="false" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.2.m2.2b"><apply id="S3.SS4.3.p3.2.m2.2.2.cmml" xref="S3.SS4.3.p3.2.m2.2.2"><ci id="S3.SS4.3.p3.2.m2.2.2.3.cmml" xref="S3.SS4.3.p3.2.m2.2.2.3">:</ci><apply id="S3.SS4.3.p3.2.m2.2.2.4.cmml" xref="S3.SS4.3.p3.2.m2.2.2.4"><times id="S3.SS4.3.p3.2.m2.2.2.4.1.cmml" xref="S3.SS4.3.p3.2.m2.2.2.4.1"></times><ci id="S3.SS4.3.p3.2.m2.2.2.4.2.cmml" xref="S3.SS4.3.p3.2.m2.2.2.4.2">𝜎</ci><ci id="S3.SS4.3.p3.2.m2.2.2.4.3.cmml" xref="S3.SS4.3.p3.2.m2.2.2.4.3">𝑀</ci></apply><apply id="S3.SS4.3.p3.2.m2.2.2.2.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2"><ci id="S3.SS4.3.p3.2.m2.2.2.2.3.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2.3">→</ci><apply id="S3.SS4.3.p3.2.m2.1.1.1.1.cmml" xref="S3.SS4.3.p3.2.m2.1.1.1.1"><times id="S3.SS4.3.p3.2.m2.1.1.1.1.2.cmml" xref="S3.SS4.3.p3.2.m2.1.1.1.1.2"></times><ci id="S3.SS4.3.p3.2.m2.1.1.1.1.3.cmml" xref="S3.SS4.3.p3.2.m2.1.1.1.1.3">ℳ</ci><apply id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.2">𝒜</ci><ci id="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.3.cmml" xref="S3.SS4.3.p3.2.m2.1.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S3.SS4.3.p3.2.m2.2.2.2.2.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2.2"><times id="S3.SS4.3.p3.2.m2.2.2.2.2.2.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2.2.2"></times><ci id="S3.SS4.3.p3.2.m2.2.2.2.2.3.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2.2.3">ℳ</ci><apply id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.1.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1">superscript</csymbol><ci id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.2.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.2">ℬ</ci><ci id="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.3.cmml" xref="S3.SS4.3.p3.2.m2.2.2.2.2.1.1.1.3">ℤ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.2.m2.2c">\sigma M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal B^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.2.m2.2d">italic_σ italic_M : caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> is defined (see Definition-Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm6" title="Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.6</span></a> (2)). <span class="ltx_text ltx_inline-block" id="S3.SS4.3.p3.3.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S3.SS4.3.p3.3.1.m1.1"><semantics id="S3.SS4.3.p3.3.1.m1.1a"><mo id="S3.SS4.3.p3.3.1.m1.1.1" xref="S3.SS4.3.p3.3.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.3.1.m1.1b"><csymbol cd="latexml" id="S3.SS4.3.p3.3.1.m1.1.1.cmml" xref="S3.SS4.3.p3.3.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.3.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.3.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S3.SS4.3.p3.4.m3.1"><semantics id="S3.SS4.3.p3.4.m3.1a"><mo id="S3.SS4.3.p3.4.m3.1.1" xref="S3.SS4.3.p3.4.m3.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S3.SS4.3.p3.4.m3.1b"><csymbol cd="latexml" id="S3.SS4.3.p3.4.m3.1.1.cmml" xref="S3.SS4.3.p3.4.m3.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.3.p3.4.m3.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.3.p3.4.m3.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S3.Thmthm9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm9.1.1.1">Remark 3.9</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm9.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm9.p1"> <p class="ltx_p" id="S3.Thmthm9.p1.8">(1) Using the density of the set of weighted characteristic measures <math alttext="\lambda\mu_{w}" class="ltx_Math" display="inline" id="S3.Thmthm9.p1.1.m1.1"><semantics id="S3.Thmthm9.p1.1.m1.1a"><mrow id="S3.Thmthm9.p1.1.m1.1.1" xref="S3.Thmthm9.p1.1.m1.1.1.cmml"><mi id="S3.Thmthm9.p1.1.m1.1.1.2" xref="S3.Thmthm9.p1.1.m1.1.1.2.cmml">λ</mi><mo id="S3.Thmthm9.p1.1.m1.1.1.1" xref="S3.Thmthm9.p1.1.m1.1.1.1.cmml">⁢</mo><msub id="S3.Thmthm9.p1.1.m1.1.1.3" xref="S3.Thmthm9.p1.1.m1.1.1.3.cmml"><mi id="S3.Thmthm9.p1.1.m1.1.1.3.2" xref="S3.Thmthm9.p1.1.m1.1.1.3.2.cmml">μ</mi><mi id="S3.Thmthm9.p1.1.m1.1.1.3.3" xref="S3.Thmthm9.p1.1.m1.1.1.3.3.cmml">w</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p1.1.m1.1b"><apply id="S3.Thmthm9.p1.1.m1.1.1.cmml" xref="S3.Thmthm9.p1.1.m1.1.1"><times id="S3.Thmthm9.p1.1.m1.1.1.1.cmml" xref="S3.Thmthm9.p1.1.m1.1.1.1"></times><ci id="S3.Thmthm9.p1.1.m1.1.1.2.cmml" xref="S3.Thmthm9.p1.1.m1.1.1.2">𝜆</ci><apply id="S3.Thmthm9.p1.1.m1.1.1.3.cmml" xref="S3.Thmthm9.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm9.p1.1.m1.1.1.3.1.cmml" xref="S3.Thmthm9.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.Thmthm9.p1.1.m1.1.1.3.2.cmml" xref="S3.Thmthm9.p1.1.m1.1.1.3.2">𝜇</ci><ci id="S3.Thmthm9.p1.1.m1.1.1.3.3.cmml" xref="S3.Thmthm9.p1.1.m1.1.1.3.3">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p1.1.m1.1c">\lambda\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p1.1.m1.1d">italic_λ italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> (see Section <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS1" title="2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.1</span></a>) within <math alttext="\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.Thmthm9.p1.2.m2.1"><semantics id="S3.Thmthm9.p1.2.m2.1a"><mrow id="S3.Thmthm9.p1.2.m2.1.1" xref="S3.Thmthm9.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm9.p1.2.m2.1.1.3" xref="S3.Thmthm9.p1.2.m2.1.1.3.cmml">ℳ</mi><mo id="S3.Thmthm9.p1.2.m2.1.1.2" xref="S3.Thmthm9.p1.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.Thmthm9.p1.2.m2.1.1.1.1" xref="S3.Thmthm9.p1.2.m2.1.1.1.1.1.cmml"><mo id="S3.Thmthm9.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.Thmthm9.p1.2.m2.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmthm9.p1.2.m2.1.1.1.1.1" xref="S3.Thmthm9.p1.2.m2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm9.p1.2.m2.1.1.1.1.1.2" xref="S3.Thmthm9.p1.2.m2.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.Thmthm9.p1.2.m2.1.1.1.1.1.3" xref="S3.Thmthm9.p1.2.m2.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm9.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S3.Thmthm9.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p1.2.m2.1b"><apply id="S3.Thmthm9.p1.2.m2.1.1.cmml" xref="S3.Thmthm9.p1.2.m2.1.1"><times id="S3.Thmthm9.p1.2.m2.1.1.2.cmml" xref="S3.Thmthm9.p1.2.m2.1.1.2"></times><ci id="S3.Thmthm9.p1.2.m2.1.1.3.cmml" xref="S3.Thmthm9.p1.2.m2.1.1.3">ℳ</ci><apply id="S3.Thmthm9.p1.2.m2.1.1.1.1.1.cmml" xref="S3.Thmthm9.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm9.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.Thmthm9.p1.2.m2.1.1.1.1">superscript</csymbol><ci id="S3.Thmthm9.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.Thmthm9.p1.2.m2.1.1.1.1.1.2">𝒜</ci><ci id="S3.Thmthm9.p1.2.m2.1.1.1.1.1.3.cmml" xref="S3.Thmthm9.p1.2.m2.1.1.1.1.1.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p1.2.m2.1c">\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p1.2.m2.1d">caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> on one hand and the continuity of the map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S3.Thmthm9.p1.3.m3.1"><semantics id="S3.Thmthm9.p1.3.m3.1a"><mrow id="S3.Thmthm9.p1.3.m3.1.1" xref="S3.Thmthm9.p1.3.m3.1.1.cmml"><mi id="S3.Thmthm9.p1.3.m3.1.1.2" xref="S3.Thmthm9.p1.3.m3.1.1.2.cmml">σ</mi><mo id="S3.Thmthm9.p1.3.m3.1.1.1" xref="S3.Thmthm9.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm9.p1.3.m3.1.1.3" xref="S3.Thmthm9.p1.3.m3.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p1.3.m3.1b"><apply id="S3.Thmthm9.p1.3.m3.1.1.cmml" xref="S3.Thmthm9.p1.3.m3.1.1"><times id="S3.Thmthm9.p1.3.m3.1.1.1.cmml" xref="S3.Thmthm9.p1.3.m3.1.1.1"></times><ci id="S3.Thmthm9.p1.3.m3.1.1.2.cmml" xref="S3.Thmthm9.p1.3.m3.1.1.2">𝜎</ci><ci id="S3.Thmthm9.p1.3.m3.1.1.3.cmml" xref="S3.Thmthm9.p1.3.m3.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p1.3.m3.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p1.3.m3.1d">italic_σ italic_M</annotation></semantics></math> on the other, one can use statement (d) of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm7" title="Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.7</span></a> in order to alternatively determine the image <math alttext="\sigma M(\mu)" class="ltx_Math" display="inline" id="S3.Thmthm9.p1.4.m4.1"><semantics id="S3.Thmthm9.p1.4.m4.1a"><mrow id="S3.Thmthm9.p1.4.m4.1.2" xref="S3.Thmthm9.p1.4.m4.1.2.cmml"><mi id="S3.Thmthm9.p1.4.m4.1.2.2" xref="S3.Thmthm9.p1.4.m4.1.2.2.cmml">σ</mi><mo id="S3.Thmthm9.p1.4.m4.1.2.1" xref="S3.Thmthm9.p1.4.m4.1.2.1.cmml">⁢</mo><mi id="S3.Thmthm9.p1.4.m4.1.2.3" xref="S3.Thmthm9.p1.4.m4.1.2.3.cmml">M</mi><mo id="S3.Thmthm9.p1.4.m4.1.2.1a" xref="S3.Thmthm9.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm9.p1.4.m4.1.2.4.2" xref="S3.Thmthm9.p1.4.m4.1.2.cmml"><mo id="S3.Thmthm9.p1.4.m4.1.2.4.2.1" stretchy="false" xref="S3.Thmthm9.p1.4.m4.1.2.cmml">(</mo><mi id="S3.Thmthm9.p1.4.m4.1.1" xref="S3.Thmthm9.p1.4.m4.1.1.cmml">μ</mi><mo id="S3.Thmthm9.p1.4.m4.1.2.4.2.2" stretchy="false" xref="S3.Thmthm9.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p1.4.m4.1b"><apply id="S3.Thmthm9.p1.4.m4.1.2.cmml" xref="S3.Thmthm9.p1.4.m4.1.2"><times id="S3.Thmthm9.p1.4.m4.1.2.1.cmml" xref="S3.Thmthm9.p1.4.m4.1.2.1"></times><ci id="S3.Thmthm9.p1.4.m4.1.2.2.cmml" xref="S3.Thmthm9.p1.4.m4.1.2.2">𝜎</ci><ci id="S3.Thmthm9.p1.4.m4.1.2.3.cmml" xref="S3.Thmthm9.p1.4.m4.1.2.3">𝑀</ci><ci id="S3.Thmthm9.p1.4.m4.1.1.cmml" xref="S3.Thmthm9.p1.4.m4.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p1.4.m4.1c">\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p1.4.m4.1d">italic_σ italic_M ( italic_μ )</annotation></semantics></math> for any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm9.p1.5.m5.1"><semantics id="S3.Thmthm9.p1.5.m5.1a"><mi id="S3.Thmthm9.p1.5.m5.1.1" xref="S3.Thmthm9.p1.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p1.5.m5.1b"><ci id="S3.Thmthm9.p1.5.m5.1.1.cmml" xref="S3.Thmthm9.p1.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p1.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p1.5.m5.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm9.p1.6.m6.1"><semantics id="S3.Thmthm9.p1.6.m6.1a"><msup id="S3.Thmthm9.p1.6.m6.1.1" xref="S3.Thmthm9.p1.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm9.p1.6.m6.1.1.2" xref="S3.Thmthm9.p1.6.m6.1.1.2.cmml">𝒜</mi><mi id="S3.Thmthm9.p1.6.m6.1.1.3" xref="S3.Thmthm9.p1.6.m6.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p1.6.m6.1b"><apply id="S3.Thmthm9.p1.6.m6.1.1.cmml" xref="S3.Thmthm9.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.Thmthm9.p1.6.m6.1.1.1.cmml" xref="S3.Thmthm9.p1.6.m6.1.1">superscript</csymbol><ci id="S3.Thmthm9.p1.6.m6.1.1.2.cmml" xref="S3.Thmthm9.p1.6.m6.1.1.2">𝒜</ci><ci id="S3.Thmthm9.p1.6.m6.1.1.3.cmml" xref="S3.Thmthm9.p1.6.m6.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p1.6.m6.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p1.6.m6.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> as limit of weighted characteristic measures. This methods has been used for instance in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib11" title="">11</a>]</cite> to define the map induced by an automorphism of a free group <math alttext="F_{N}" class="ltx_Math" display="inline" id="S3.Thmthm9.p1.7.m7.1"><semantics id="S3.Thmthm9.p1.7.m7.1a"><msub id="S3.Thmthm9.p1.7.m7.1.1" xref="S3.Thmthm9.p1.7.m7.1.1.cmml"><mi id="S3.Thmthm9.p1.7.m7.1.1.2" xref="S3.Thmthm9.p1.7.m7.1.1.2.cmml">F</mi><mi id="S3.Thmthm9.p1.7.m7.1.1.3" xref="S3.Thmthm9.p1.7.m7.1.1.3.cmml">N</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p1.7.m7.1b"><apply id="S3.Thmthm9.p1.7.m7.1.1.cmml" xref="S3.Thmthm9.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.Thmthm9.p1.7.m7.1.1.1.cmml" xref="S3.Thmthm9.p1.7.m7.1.1">subscript</csymbol><ci id="S3.Thmthm9.p1.7.m7.1.1.2.cmml" xref="S3.Thmthm9.p1.7.m7.1.1.2">𝐹</ci><ci id="S3.Thmthm9.p1.7.m7.1.1.3.cmml" xref="S3.Thmthm9.p1.7.m7.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p1.7.m7.1c">F_{N}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p1.7.m7.1d">italic_F start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math> on the space of currents on <math alttext="F_{N}" class="ltx_Math" display="inline" id="S3.Thmthm9.p1.8.m8.1"><semantics id="S3.Thmthm9.p1.8.m8.1a"><msub id="S3.Thmthm9.p1.8.m8.1.1" xref="S3.Thmthm9.p1.8.m8.1.1.cmml"><mi id="S3.Thmthm9.p1.8.m8.1.1.2" xref="S3.Thmthm9.p1.8.m8.1.1.2.cmml">F</mi><mi id="S3.Thmthm9.p1.8.m8.1.1.3" xref="S3.Thmthm9.p1.8.m8.1.1.3.cmml">N</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p1.8.m8.1b"><apply id="S3.Thmthm9.p1.8.m8.1.1.cmml" xref="S3.Thmthm9.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S3.Thmthm9.p1.8.m8.1.1.1.cmml" xref="S3.Thmthm9.p1.8.m8.1.1">subscript</csymbol><ci id="S3.Thmthm9.p1.8.m8.1.1.2.cmml" xref="S3.Thmthm9.p1.8.m8.1.1.2">𝐹</ci><ci id="S3.Thmthm9.p1.8.m8.1.1.3.cmml" xref="S3.Thmthm9.p1.8.m8.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p1.8.m8.1c">F_{N}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p1.8.m8.1d">italic_F start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math>. The approach presented here, however, has many practical advantages; for instance it is more efficient for most computations.</p> </div> <div class="ltx_para ltx_noindent" id="S3.Thmthm9.p2"> <p class="ltx_p" id="S3.Thmthm9.p2.4">(2) A third alternative to determine the image <math alttext="\sigma M(\mu)" class="ltx_Math" display="inline" id="S3.Thmthm9.p2.1.m1.1"><semantics id="S3.Thmthm9.p2.1.m1.1a"><mrow id="S3.Thmthm9.p2.1.m1.1.2" xref="S3.Thmthm9.p2.1.m1.1.2.cmml"><mi id="S3.Thmthm9.p2.1.m1.1.2.2" xref="S3.Thmthm9.p2.1.m1.1.2.2.cmml">σ</mi><mo id="S3.Thmthm9.p2.1.m1.1.2.1" xref="S3.Thmthm9.p2.1.m1.1.2.1.cmml">⁢</mo><mi id="S3.Thmthm9.p2.1.m1.1.2.3" xref="S3.Thmthm9.p2.1.m1.1.2.3.cmml">M</mi><mo id="S3.Thmthm9.p2.1.m1.1.2.1a" xref="S3.Thmthm9.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm9.p2.1.m1.1.2.4.2" xref="S3.Thmthm9.p2.1.m1.1.2.cmml"><mo id="S3.Thmthm9.p2.1.m1.1.2.4.2.1" stretchy="false" xref="S3.Thmthm9.p2.1.m1.1.2.cmml">(</mo><mi id="S3.Thmthm9.p2.1.m1.1.1" xref="S3.Thmthm9.p2.1.m1.1.1.cmml">μ</mi><mo id="S3.Thmthm9.p2.1.m1.1.2.4.2.2" stretchy="false" xref="S3.Thmthm9.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p2.1.m1.1b"><apply id="S3.Thmthm9.p2.1.m1.1.2.cmml" xref="S3.Thmthm9.p2.1.m1.1.2"><times id="S3.Thmthm9.p2.1.m1.1.2.1.cmml" xref="S3.Thmthm9.p2.1.m1.1.2.1"></times><ci id="S3.Thmthm9.p2.1.m1.1.2.2.cmml" xref="S3.Thmthm9.p2.1.m1.1.2.2">𝜎</ci><ci id="S3.Thmthm9.p2.1.m1.1.2.3.cmml" xref="S3.Thmthm9.p2.1.m1.1.2.3">𝑀</ci><ci id="S3.Thmthm9.p2.1.m1.1.1.cmml" xref="S3.Thmthm9.p2.1.m1.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p2.1.m1.1c">\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p2.1.m1.1d">italic_σ italic_M ( italic_μ )</annotation></semantics></math> for any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm9.p2.2.m2.1"><semantics id="S3.Thmthm9.p2.2.m2.1a"><mi id="S3.Thmthm9.p2.2.m2.1.1" xref="S3.Thmthm9.p2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p2.2.m2.1b"><ci id="S3.Thmthm9.p2.2.m2.1.1.cmml" xref="S3.Thmthm9.p2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p2.2.m2.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm9.p2.3.m3.1"><semantics id="S3.Thmthm9.p2.3.m3.1a"><msup id="S3.Thmthm9.p2.3.m3.1.1" xref="S3.Thmthm9.p2.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm9.p2.3.m3.1.1.2" xref="S3.Thmthm9.p2.3.m3.1.1.2.cmml">𝒜</mi><mi id="S3.Thmthm9.p2.3.m3.1.1.3" xref="S3.Thmthm9.p2.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p2.3.m3.1b"><apply id="S3.Thmthm9.p2.3.m3.1.1.cmml" xref="S3.Thmthm9.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmthm9.p2.3.m3.1.1.1.cmml" xref="S3.Thmthm9.p2.3.m3.1.1">superscript</csymbol><ci id="S3.Thmthm9.p2.3.m3.1.1.2.cmml" xref="S3.Thmthm9.p2.3.m3.1.1.2">𝒜</ci><ci id="S3.Thmthm9.p2.3.m3.1.1.3.cmml" xref="S3.Thmthm9.p2.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p2.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p2.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is given in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite> by means of everywhere growing <math alttext="S" class="ltx_Math" display="inline" id="S3.Thmthm9.p2.4.m4.1"><semantics id="S3.Thmthm9.p2.4.m4.1a"><mi id="S3.Thmthm9.p2.4.m4.1.1" xref="S3.Thmthm9.p2.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm9.p2.4.m4.1b"><ci id="S3.Thmthm9.p2.4.m4.1.1.cmml" xref="S3.Thmthm9.p2.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm9.p2.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm9.p2.4.m4.1d">italic_S</annotation></semantics></math>-adic developments and vector towers over them.</p> </div> </div> <div class="ltx_para" id="S3.SS4.p4"> <p class="ltx_p" id="S3.SS4.p4.1">We also observe:</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="S3.Thmthm10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm10.1.1.1">Lemma 3.10</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm10.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm10.p1"> <p class="ltx_p" id="S3.Thmthm10.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmthm10.p1.6.6">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm10.p1.1.1.m1.1"><semantics id="S3.Thmthm10.p1.1.1.m1.1a"><mrow id="S3.Thmthm10.p1.1.1.m1.1.1" xref="S3.Thmthm10.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmthm10.p1.1.1.m1.1.1.2" xref="S3.Thmthm10.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S3.Thmthm10.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm10.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm10.p1.1.1.m1.1.1.3" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.cmml"><msup id="S3.Thmthm10.p1.1.1.m1.1.1.3.2" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm10.p1.1.1.m1.1.1.3.2.2" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm10.p1.1.1.m1.1.1.3.2.3" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm10.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.Thmthm10.p1.1.1.m1.1.1.3.3" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm10.p1.1.1.m1.1.1.3.3.2" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.Thmthm10.p1.1.1.m1.1.1.3.3.3" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm10.p1.1.1.m1.1b"><apply id="S3.Thmthm10.p1.1.1.m1.1.1.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1"><ci id="S3.Thmthm10.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.1">:</ci><ci id="S3.Thmthm10.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S3.Thmthm10.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3"><ci id="S3.Thmthm10.p1.1.1.m1.1.1.3.1.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm10.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm10.p1.1.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm10.p1.1.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm10.p1.1.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm10.p1.1.1.m1.1.1.3.3.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm10.p1.1.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.Thmthm10.p1.1.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.Thmthm10.p1.1.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm10.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm10.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm10.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-erasing monoid morphism, and let <math alttext="\mu\in\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S3.Thmthm10.p1.2.2.m2.1"><semantics id="S3.Thmthm10.p1.2.2.m2.1a"><mrow id="S3.Thmthm10.p1.2.2.m2.1.1" xref="S3.Thmthm10.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmthm10.p1.2.2.m2.1.1.3" xref="S3.Thmthm10.p1.2.2.m2.1.1.3.cmml">μ</mi><mo id="S3.Thmthm10.p1.2.2.m2.1.1.2" xref="S3.Thmthm10.p1.2.2.m2.1.1.2.cmml">∈</mo><mrow id="S3.Thmthm10.p1.2.2.m2.1.1.1" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm10.p1.2.2.m2.1.1.1.3" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.3.cmml">ℳ</mi><mo id="S3.Thmthm10.p1.2.2.m2.1.1.1.2" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.2" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.3" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm10.p1.2.2.m2.1b"><apply id="S3.Thmthm10.p1.2.2.m2.1.1.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1"><in id="S3.Thmthm10.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.2"></in><ci id="S3.Thmthm10.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.3">𝜇</ci><apply id="S3.Thmthm10.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.1"><times id="S3.Thmthm10.p1.2.2.m2.1.1.1.2.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.2"></times><ci id="S3.Thmthm10.p1.2.2.m2.1.1.1.3.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.3">ℳ</ci><apply id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.2">𝒜</ci><ci id="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm10.p1.2.2.m2.1.1.1.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm10.p1.2.2.m2.1c">\mu\in\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm10.p1.2.2.m2.1d">italic_μ ∈ caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> be an invariant non-zero measure with support contained in some subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm10.p1.3.3.m3.1"><semantics id="S3.Thmthm10.p1.3.3.m3.1a"><mrow id="S3.Thmthm10.p1.3.3.m3.1.1" xref="S3.Thmthm10.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmthm10.p1.3.3.m3.1.1.2" xref="S3.Thmthm10.p1.3.3.m3.1.1.2.cmml">X</mi><mo id="S3.Thmthm10.p1.3.3.m3.1.1.1" xref="S3.Thmthm10.p1.3.3.m3.1.1.1.cmml">⊆</mo><msup id="S3.Thmthm10.p1.3.3.m3.1.1.3" xref="S3.Thmthm10.p1.3.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm10.p1.3.3.m3.1.1.3.2" xref="S3.Thmthm10.p1.3.3.m3.1.1.3.2.cmml">𝒜</mi><mi id="S3.Thmthm10.p1.3.3.m3.1.1.3.3" xref="S3.Thmthm10.p1.3.3.m3.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm10.p1.3.3.m3.1b"><apply id="S3.Thmthm10.p1.3.3.m3.1.1.cmml" xref="S3.Thmthm10.p1.3.3.m3.1.1"><subset id="S3.Thmthm10.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmthm10.p1.3.3.m3.1.1.1"></subset><ci id="S3.Thmthm10.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmthm10.p1.3.3.m3.1.1.2">𝑋</ci><apply id="S3.Thmthm10.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmthm10.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm10.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmthm10.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmthm10.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmthm10.p1.3.3.m3.1.1.3.2">𝒜</ci><ci id="S3.Thmthm10.p1.3.3.m3.1.1.3.3.cmml" xref="S3.Thmthm10.p1.3.3.m3.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm10.p1.3.3.m3.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm10.p1.3.3.m3.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> that has image subshift <math alttext="Y:=\sigma(X)\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm10.p1.4.4.m4.1"><semantics id="S3.Thmthm10.p1.4.4.m4.1a"><mrow id="S3.Thmthm10.p1.4.4.m4.1.2" xref="S3.Thmthm10.p1.4.4.m4.1.2.cmml"><mi id="S3.Thmthm10.p1.4.4.m4.1.2.2" xref="S3.Thmthm10.p1.4.4.m4.1.2.2.cmml">Y</mi><mo id="S3.Thmthm10.p1.4.4.m4.1.2.3" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm10.p1.4.4.m4.1.2.3.cmml">:=</mo><mrow id="S3.Thmthm10.p1.4.4.m4.1.2.4" xref="S3.Thmthm10.p1.4.4.m4.1.2.4.cmml"><mi id="S3.Thmthm10.p1.4.4.m4.1.2.4.2" xref="S3.Thmthm10.p1.4.4.m4.1.2.4.2.cmml">σ</mi><mo id="S3.Thmthm10.p1.4.4.m4.1.2.4.1" xref="S3.Thmthm10.p1.4.4.m4.1.2.4.1.cmml">⁢</mo><mrow id="S3.Thmthm10.p1.4.4.m4.1.2.4.3.2" xref="S3.Thmthm10.p1.4.4.m4.1.2.4.cmml"><mo id="S3.Thmthm10.p1.4.4.m4.1.2.4.3.2.1" stretchy="false" xref="S3.Thmthm10.p1.4.4.m4.1.2.4.cmml">(</mo><mi id="S3.Thmthm10.p1.4.4.m4.1.1" xref="S3.Thmthm10.p1.4.4.m4.1.1.cmml">X</mi><mo id="S3.Thmthm10.p1.4.4.m4.1.2.4.3.2.2" stretchy="false" xref="S3.Thmthm10.p1.4.4.m4.1.2.4.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm10.p1.4.4.m4.1.2.5" xref="S3.Thmthm10.p1.4.4.m4.1.2.5.cmml">⊆</mo><msup id="S3.Thmthm10.p1.4.4.m4.1.2.6" xref="S3.Thmthm10.p1.4.4.m4.1.2.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm10.p1.4.4.m4.1.2.6.2" xref="S3.Thmthm10.p1.4.4.m4.1.2.6.2.cmml">ℬ</mi><mi id="S3.Thmthm10.p1.4.4.m4.1.2.6.3" xref="S3.Thmthm10.p1.4.4.m4.1.2.6.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm10.p1.4.4.m4.1b"><apply id="S3.Thmthm10.p1.4.4.m4.1.2.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2"><and id="S3.Thmthm10.p1.4.4.m4.1.2a.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2"></and><apply id="S3.Thmthm10.p1.4.4.m4.1.2b.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2"><csymbol cd="latexml" id="S3.Thmthm10.p1.4.4.m4.1.2.3.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.3">assign</csymbol><ci id="S3.Thmthm10.p1.4.4.m4.1.2.2.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.2">𝑌</ci><apply id="S3.Thmthm10.p1.4.4.m4.1.2.4.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.4"><times id="S3.Thmthm10.p1.4.4.m4.1.2.4.1.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.4.1"></times><ci id="S3.Thmthm10.p1.4.4.m4.1.2.4.2.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.4.2">𝜎</ci><ci id="S3.Thmthm10.p1.4.4.m4.1.1.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.1">𝑋</ci></apply></apply><apply id="S3.Thmthm10.p1.4.4.m4.1.2c.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2"><subset id="S3.Thmthm10.p1.4.4.m4.1.2.5.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.5"></subset><share href="https://arxiv.org/html/2211.11234v4#S3.Thmthm10.p1.4.4.m4.1.2.4.cmml" id="S3.Thmthm10.p1.4.4.m4.1.2d.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2"></share><apply id="S3.Thmthm10.p1.4.4.m4.1.2.6.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.6"><csymbol cd="ambiguous" id="S3.Thmthm10.p1.4.4.m4.1.2.6.1.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.6">superscript</csymbol><ci id="S3.Thmthm10.p1.4.4.m4.1.2.6.2.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.6.2">ℬ</ci><ci id="S3.Thmthm10.p1.4.4.m4.1.2.6.3.cmml" xref="S3.Thmthm10.p1.4.4.m4.1.2.6.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm10.p1.4.4.m4.1c">Y:=\sigma(X)\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm10.p1.4.4.m4.1d">italic_Y := italic_σ ( italic_X ) ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Then the transferred measure <math alttext="\sigma M(\mu)\,\,[=\mu^{\sigma}]" class="ltx_Math" display="inline" id="S3.Thmthm10.p1.5.5.m5.2"><semantics id="S3.Thmthm10.p1.5.5.m5.2a"><mrow id="S3.Thmthm10.p1.5.5.m5.2.2" xref="S3.Thmthm10.p1.5.5.m5.2.2.cmml"><mrow id="S3.Thmthm10.p1.5.5.m5.2.2.3" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.cmml"><mi id="S3.Thmthm10.p1.5.5.m5.2.2.3.2" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.2.cmml">σ</mi><mo id="S3.Thmthm10.p1.5.5.m5.2.2.3.1" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.1.cmml">⁢</mo><mi id="S3.Thmthm10.p1.5.5.m5.2.2.3.3" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.3.cmml">M</mi><mo id="S3.Thmthm10.p1.5.5.m5.2.2.3.1a" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm10.p1.5.5.m5.2.2.3.4.2" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.cmml"><mo id="S3.Thmthm10.p1.5.5.m5.2.2.3.4.2.1" stretchy="false" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.cmml">(</mo><mi id="S3.Thmthm10.p1.5.5.m5.1.1" xref="S3.Thmthm10.p1.5.5.m5.1.1.cmml">μ</mi><mo id="S3.Thmthm10.p1.5.5.m5.2.2.3.4.2.2" stretchy="false" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.cmml">)</mo></mrow></mrow><mspace id="S3.Thmthm10.p1.5.5.m5.2.2a" width="0.719em" xref="S3.Thmthm10.p1.5.5.m5.2.2.cmml"></mspace><mrow id="S3.Thmthm10.p1.5.5.m5.2.2.1.1" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.2.cmml"><mo id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.2" stretchy="false" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.2.1.cmml">[</mo><mrow id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.cmml"><mi id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.2" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.2.cmml"></mi><mo id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.1" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.1.cmml">=</mo><msup id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.cmml"><mi id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.2" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.2.cmml">μ</mi><mi id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.3" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.3.cmml">σ</mi></msup></mrow><mo id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.3" stretchy="false" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm10.p1.5.5.m5.2b"><apply id="S3.Thmthm10.p1.5.5.m5.2.2.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2"><csymbol cd="latexml" id="S3.Thmthm10.p1.5.5.m5.2.2.2.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2">annotated</csymbol><apply id="S3.Thmthm10.p1.5.5.m5.2.2.3.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.3"><times id="S3.Thmthm10.p1.5.5.m5.2.2.3.1.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.1"></times><ci id="S3.Thmthm10.p1.5.5.m5.2.2.3.2.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.2">𝜎</ci><ci id="S3.Thmthm10.p1.5.5.m5.2.2.3.3.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.3.3">𝑀</ci><ci id="S3.Thmthm10.p1.5.5.m5.1.1.cmml" xref="S3.Thmthm10.p1.5.5.m5.1.1">𝜇</ci></apply><apply id="S3.Thmthm10.p1.5.5.m5.2.2.1.2.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1"><csymbol cd="latexml" id="S3.Thmthm10.p1.5.5.m5.2.2.1.2.1.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1"><eq id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.1.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.1"></eq><csymbol cd="latexml" id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.2.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.2">absent</csymbol><apply id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.1.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3">superscript</csymbol><ci id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.2.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.2">𝜇</ci><ci id="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.3.cmml" xref="S3.Thmthm10.p1.5.5.m5.2.2.1.1.1.3.3">𝜎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm10.p1.5.5.m5.2c">\sigma M(\mu)\,\,[=\mu^{\sigma}]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm10.p1.5.5.m5.2d">italic_σ italic_M ( italic_μ ) [ = italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ]</annotation></semantics></math> has support in <math alttext="Y" class="ltx_Math" display="inline" id="S3.Thmthm10.p1.6.6.m6.1"><semantics id="S3.Thmthm10.p1.6.6.m6.1a"><mi id="S3.Thmthm10.p1.6.6.m6.1.1" xref="S3.Thmthm10.p1.6.6.m6.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm10.p1.6.6.m6.1b"><ci id="S3.Thmthm10.p1.6.6.m6.1.1.cmml" xref="S3.Thmthm10.p1.6.6.m6.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm10.p1.6.6.m6.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm10.p1.6.6.m6.1d">italic_Y</annotation></semantics></math>. More precisely, using the terminology from (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E7" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.7</span></a>), we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(3.7)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mbox{Supp}(\sigma M(\mu))=\sigma^{\Sigma}(\mbox{Supp}(\mu))\,." class="ltx_Math" display="block" id="S3.E7.m1.3"><semantics id="S3.E7.m1.3a"><mrow id="S3.E7.m1.3.3.1" xref="S3.E7.m1.3.3.1.1.cmml"><mrow id="S3.E7.m1.3.3.1.1" xref="S3.E7.m1.3.3.1.1.cmml"><mrow id="S3.E7.m1.3.3.1.1.1" xref="S3.E7.m1.3.3.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S3.E7.m1.3.3.1.1.1.3" xref="S3.E7.m1.3.3.1.1.1.3a.cmml">Supp</mtext><mo id="S3.E7.m1.3.3.1.1.1.2" xref="S3.E7.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.E7.m1.3.3.1.1.1.1.1" xref="S3.E7.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S3.E7.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.E7.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E7.m1.3.3.1.1.1.1.1.1" xref="S3.E7.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.E7.m1.3.3.1.1.1.1.1.1.2" xref="S3.E7.m1.3.3.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S3.E7.m1.3.3.1.1.1.1.1.1.1" xref="S3.E7.m1.3.3.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E7.m1.3.3.1.1.1.1.1.1.3" xref="S3.E7.m1.3.3.1.1.1.1.1.1.3.cmml">M</mi><mo id="S3.E7.m1.3.3.1.1.1.1.1.1.1a" xref="S3.E7.m1.3.3.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.E7.m1.3.3.1.1.1.1.1.1.4.2" xref="S3.E7.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S3.E7.m1.3.3.1.1.1.1.1.1.4.2.1" stretchy="false" xref="S3.E7.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mi id="S3.E7.m1.1.1" xref="S3.E7.m1.1.1.cmml">μ</mi><mo id="S3.E7.m1.3.3.1.1.1.1.1.1.4.2.2" stretchy="false" xref="S3.E7.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E7.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S3.E7.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E7.m1.3.3.1.1.3" xref="S3.E7.m1.3.3.1.1.3.cmml">=</mo><mrow id="S3.E7.m1.3.3.1.1.2" xref="S3.E7.m1.3.3.1.1.2.cmml"><msup id="S3.E7.m1.3.3.1.1.2.3" xref="S3.E7.m1.3.3.1.1.2.3.cmml"><mi id="S3.E7.m1.3.3.1.1.2.3.2" xref="S3.E7.m1.3.3.1.1.2.3.2.cmml">σ</mi><mi id="S3.E7.m1.3.3.1.1.2.3.3" mathvariant="normal" xref="S3.E7.m1.3.3.1.1.2.3.3.cmml">Σ</mi></msup><mo id="S3.E7.m1.3.3.1.1.2.2" xref="S3.E7.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="S3.E7.m1.3.3.1.1.2.1.1" xref="S3.E7.m1.3.3.1.1.2.1.1.1.cmml"><mo id="S3.E7.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="S3.E7.m1.3.3.1.1.2.1.1.1.cmml">(</mo><mrow id="S3.E7.m1.3.3.1.1.2.1.1.1" xref="S3.E7.m1.3.3.1.1.2.1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S3.E7.m1.3.3.1.1.2.1.1.1.2" xref="S3.E7.m1.3.3.1.1.2.1.1.1.2a.cmml">Supp</mtext><mo id="S3.E7.m1.3.3.1.1.2.1.1.1.1" xref="S3.E7.m1.3.3.1.1.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.E7.m1.3.3.1.1.2.1.1.1.3.2" xref="S3.E7.m1.3.3.1.1.2.1.1.1.cmml"><mo id="S3.E7.m1.3.3.1.1.2.1.1.1.3.2.1" stretchy="false" xref="S3.E7.m1.3.3.1.1.2.1.1.1.cmml">(</mo><mi id="S3.E7.m1.2.2" xref="S3.E7.m1.2.2.cmml">μ</mi><mo id="S3.E7.m1.3.3.1.1.2.1.1.1.3.2.2" stretchy="false" xref="S3.E7.m1.3.3.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E7.m1.3.3.1.1.2.1.1.3" stretchy="false" xref="S3.E7.m1.3.3.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.E7.m1.3.3.1.2" lspace="0.170em" xref="S3.E7.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E7.m1.3b"><apply id="S3.E7.m1.3.3.1.1.cmml" xref="S3.E7.m1.3.3.1"><eq id="S3.E7.m1.3.3.1.1.3.cmml" xref="S3.E7.m1.3.3.1.1.3"></eq><apply id="S3.E7.m1.3.3.1.1.1.cmml" xref="S3.E7.m1.3.3.1.1.1"><times id="S3.E7.m1.3.3.1.1.1.2.cmml" xref="S3.E7.m1.3.3.1.1.1.2"></times><ci id="S3.E7.m1.3.3.1.1.1.3a.cmml" xref="S3.E7.m1.3.3.1.1.1.3"><mtext class="ltx_mathvariant_italic" id="S3.E7.m1.3.3.1.1.1.3.cmml" xref="S3.E7.m1.3.3.1.1.1.3">Supp</mtext></ci><apply id="S3.E7.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.E7.m1.3.3.1.1.1.1.1"><times id="S3.E7.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S3.E7.m1.3.3.1.1.1.1.1.1.1"></times><ci id="S3.E7.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S3.E7.m1.3.3.1.1.1.1.1.1.2">𝜎</ci><ci id="S3.E7.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S3.E7.m1.3.3.1.1.1.1.1.1.3">𝑀</ci><ci id="S3.E7.m1.1.1.cmml" xref="S3.E7.m1.1.1">𝜇</ci></apply></apply><apply id="S3.E7.m1.3.3.1.1.2.cmml" xref="S3.E7.m1.3.3.1.1.2"><times id="S3.E7.m1.3.3.1.1.2.2.cmml" xref="S3.E7.m1.3.3.1.1.2.2"></times><apply id="S3.E7.m1.3.3.1.1.2.3.cmml" xref="S3.E7.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S3.E7.m1.3.3.1.1.2.3.1.cmml" xref="S3.E7.m1.3.3.1.1.2.3">superscript</csymbol><ci id="S3.E7.m1.3.3.1.1.2.3.2.cmml" xref="S3.E7.m1.3.3.1.1.2.3.2">𝜎</ci><ci id="S3.E7.m1.3.3.1.1.2.3.3.cmml" xref="S3.E7.m1.3.3.1.1.2.3.3">Σ</ci></apply><apply id="S3.E7.m1.3.3.1.1.2.1.1.1.cmml" xref="S3.E7.m1.3.3.1.1.2.1.1"><times id="S3.E7.m1.3.3.1.1.2.1.1.1.1.cmml" xref="S3.E7.m1.3.3.1.1.2.1.1.1.1"></times><ci id="S3.E7.m1.3.3.1.1.2.1.1.1.2a.cmml" xref="S3.E7.m1.3.3.1.1.2.1.1.1.2"><mtext class="ltx_mathvariant_italic" id="S3.E7.m1.3.3.1.1.2.1.1.1.2.cmml" xref="S3.E7.m1.3.3.1.1.2.1.1.1.2">Supp</mtext></ci><ci id="S3.E7.m1.2.2.cmml" xref="S3.E7.m1.2.2">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.3c">\mbox{Supp}(\sigma M(\mu))=\sigma^{\Sigma}(\mbox{Supp}(\mu))\,.</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.3d">Supp ( italic_σ italic_M ( italic_μ ) ) = italic_σ start_POSTSUPERSCRIPT roman_Σ end_POSTSUPERSCRIPT ( Supp ( italic_μ ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmthm10.p1.8"><span class="ltx_text ltx_font_italic" id="S3.Thmthm10.p1.8.2">In particular, the morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.Thmthm10.p1.7.1.m1.1"><semantics id="S3.Thmthm10.p1.7.1.m1.1a"><mi id="S3.Thmthm10.p1.7.1.m1.1.1" xref="S3.Thmthm10.p1.7.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm10.p1.7.1.m1.1b"><ci id="S3.Thmthm10.p1.7.1.m1.1.1.cmml" xref="S3.Thmthm10.p1.7.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm10.p1.7.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm10.p1.7.1.m1.1d">italic_σ</annotation></semantics></math> induces for any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm10.p1.8.2.m2.1"><semantics id="S3.Thmthm10.p1.8.2.m2.1a"><mrow id="S3.Thmthm10.p1.8.2.m2.1.1" xref="S3.Thmthm10.p1.8.2.m2.1.1.cmml"><mi id="S3.Thmthm10.p1.8.2.m2.1.1.2" xref="S3.Thmthm10.p1.8.2.m2.1.1.2.cmml">X</mi><mo id="S3.Thmthm10.p1.8.2.m2.1.1.1" xref="S3.Thmthm10.p1.8.2.m2.1.1.1.cmml">⊆</mo><msup id="S3.Thmthm10.p1.8.2.m2.1.1.3" xref="S3.Thmthm10.p1.8.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm10.p1.8.2.m2.1.1.3.2" xref="S3.Thmthm10.p1.8.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S3.Thmthm10.p1.8.2.m2.1.1.3.3" xref="S3.Thmthm10.p1.8.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm10.p1.8.2.m2.1b"><apply id="S3.Thmthm10.p1.8.2.m2.1.1.cmml" xref="S3.Thmthm10.p1.8.2.m2.1.1"><subset id="S3.Thmthm10.p1.8.2.m2.1.1.1.cmml" xref="S3.Thmthm10.p1.8.2.m2.1.1.1"></subset><ci id="S3.Thmthm10.p1.8.2.m2.1.1.2.cmml" xref="S3.Thmthm10.p1.8.2.m2.1.1.2">𝑋</ci><apply id="S3.Thmthm10.p1.8.2.m2.1.1.3.cmml" xref="S3.Thmthm10.p1.8.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm10.p1.8.2.m2.1.1.3.1.cmml" xref="S3.Thmthm10.p1.8.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmthm10.p1.8.2.m2.1.1.3.2.cmml" xref="S3.Thmthm10.p1.8.2.m2.1.1.3.2">𝒜</ci><ci id="S3.Thmthm10.p1.8.2.m2.1.1.3.3.cmml" xref="S3.Thmthm10.p1.8.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm10.p1.8.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm10.p1.8.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> a well defined map</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma M_{X}:\cal M(X)\to\cal M(\sigma(X))" class="ltx_Math" display="block" id="S3.Ex7.m1.3"><semantics id="S3.Ex7.m1.3a"><mrow id="S3.Ex7.m1.3.3" xref="S3.Ex7.m1.3.3.cmml"><mrow id="S3.Ex7.m1.3.3.3" xref="S3.Ex7.m1.3.3.3.cmml"><mi id="S3.Ex7.m1.3.3.3.2" xref="S3.Ex7.m1.3.3.3.2.cmml">σ</mi><mo id="S3.Ex7.m1.3.3.3.1" xref="S3.Ex7.m1.3.3.3.1.cmml">⁢</mo><msub id="S3.Ex7.m1.3.3.3.3" xref="S3.Ex7.m1.3.3.3.3.cmml"><mi id="S3.Ex7.m1.3.3.3.3.2" xref="S3.Ex7.m1.3.3.3.3.2.cmml">M</mi><mi id="S3.Ex7.m1.3.3.3.3.3" xref="S3.Ex7.m1.3.3.3.3.3.cmml">X</mi></msub></mrow><mo id="S3.Ex7.m1.3.3.2" lspace="0.278em" rspace="0.278em" xref="S3.Ex7.m1.3.3.2.cmml">:</mo><mrow id="S3.Ex7.m1.3.3.1" xref="S3.Ex7.m1.3.3.1.cmml"><mrow id="S3.Ex7.m1.3.3.1.3" xref="S3.Ex7.m1.3.3.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex7.m1.3.3.1.3.2" xref="S3.Ex7.m1.3.3.1.3.2.cmml">ℳ</mi><mo id="S3.Ex7.m1.3.3.1.3.1" xref="S3.Ex7.m1.3.3.1.3.1.cmml">⁢</mo><mrow id="S3.Ex7.m1.3.3.1.3.3.2" xref="S3.Ex7.m1.3.3.1.3.cmml"><mo id="S3.Ex7.m1.3.3.1.3.3.2.1" stretchy="false" xref="S3.Ex7.m1.3.3.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex7.m1.1.1" xref="S3.Ex7.m1.1.1.cmml">𝒳</mi><mo id="S3.Ex7.m1.3.3.1.3.3.2.2" stretchy="false" xref="S3.Ex7.m1.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex7.m1.3.3.1.2" stretchy="false" xref="S3.Ex7.m1.3.3.1.2.cmml">→</mo><mrow id="S3.Ex7.m1.3.3.1.1" xref="S3.Ex7.m1.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex7.m1.3.3.1.1.3" xref="S3.Ex7.m1.3.3.1.1.3.cmml">ℳ</mi><mo id="S3.Ex7.m1.3.3.1.1.2" xref="S3.Ex7.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S3.Ex7.m1.3.3.1.1.1.1" xref="S3.Ex7.m1.3.3.1.1.1.1.1.cmml"><mo id="S3.Ex7.m1.3.3.1.1.1.1.2" stretchy="false" xref="S3.Ex7.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex7.m1.3.3.1.1.1.1.1" xref="S3.Ex7.m1.3.3.1.1.1.1.1.cmml"><mi id="S3.Ex7.m1.3.3.1.1.1.1.1.2" xref="S3.Ex7.m1.3.3.1.1.1.1.1.2.cmml">σ</mi><mo id="S3.Ex7.m1.3.3.1.1.1.1.1.1" xref="S3.Ex7.m1.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Ex7.m1.3.3.1.1.1.1.1.3.2" xref="S3.Ex7.m1.3.3.1.1.1.1.1.cmml"><mo id="S3.Ex7.m1.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.Ex7.m1.3.3.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex7.m1.2.2" xref="S3.Ex7.m1.2.2.cmml">𝒳</mi><mo id="S3.Ex7.m1.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.Ex7.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex7.m1.3.3.1.1.1.1.3" stretchy="false" xref="S3.Ex7.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex7.m1.3b"><apply id="S3.Ex7.m1.3.3.cmml" xref="S3.Ex7.m1.3.3"><ci id="S3.Ex7.m1.3.3.2.cmml" xref="S3.Ex7.m1.3.3.2">:</ci><apply id="S3.Ex7.m1.3.3.3.cmml" xref="S3.Ex7.m1.3.3.3"><times id="S3.Ex7.m1.3.3.3.1.cmml" xref="S3.Ex7.m1.3.3.3.1"></times><ci id="S3.Ex7.m1.3.3.3.2.cmml" xref="S3.Ex7.m1.3.3.3.2">𝜎</ci><apply id="S3.Ex7.m1.3.3.3.3.cmml" xref="S3.Ex7.m1.3.3.3.3"><csymbol cd="ambiguous" id="S3.Ex7.m1.3.3.3.3.1.cmml" xref="S3.Ex7.m1.3.3.3.3">subscript</csymbol><ci id="S3.Ex7.m1.3.3.3.3.2.cmml" xref="S3.Ex7.m1.3.3.3.3.2">𝑀</ci><ci id="S3.Ex7.m1.3.3.3.3.3.cmml" xref="S3.Ex7.m1.3.3.3.3.3">𝑋</ci></apply></apply><apply id="S3.Ex7.m1.3.3.1.cmml" xref="S3.Ex7.m1.3.3.1"><ci id="S3.Ex7.m1.3.3.1.2.cmml" xref="S3.Ex7.m1.3.3.1.2">→</ci><apply id="S3.Ex7.m1.3.3.1.3.cmml" xref="S3.Ex7.m1.3.3.1.3"><times id="S3.Ex7.m1.3.3.1.3.1.cmml" xref="S3.Ex7.m1.3.3.1.3.1"></times><ci id="S3.Ex7.m1.3.3.1.3.2.cmml" xref="S3.Ex7.m1.3.3.1.3.2">ℳ</ci><ci id="S3.Ex7.m1.1.1.cmml" xref="S3.Ex7.m1.1.1">𝒳</ci></apply><apply id="S3.Ex7.m1.3.3.1.1.cmml" xref="S3.Ex7.m1.3.3.1.1"><times id="S3.Ex7.m1.3.3.1.1.2.cmml" xref="S3.Ex7.m1.3.3.1.1.2"></times><ci id="S3.Ex7.m1.3.3.1.1.3.cmml" xref="S3.Ex7.m1.3.3.1.1.3">ℳ</ci><apply id="S3.Ex7.m1.3.3.1.1.1.1.1.cmml" xref="S3.Ex7.m1.3.3.1.1.1.1"><times id="S3.Ex7.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.3.3.1.1.1.1.1.1"></times><ci id="S3.Ex7.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.Ex7.m1.3.3.1.1.1.1.1.2">𝜎</ci><ci id="S3.Ex7.m1.2.2.cmml" xref="S3.Ex7.m1.2.2">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m1.3c">\sigma M_{X}:\cal M(X)\to\cal M(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m1.3d">italic_σ italic_M start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmthm10.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmthm10.p1.9.1">which satisfies all the properties analogous to statements (a) - (e) of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm7" title="Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.7</span></a>, as well as to Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm8" title="Lemma 3.8. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.8</span></a>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS4.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS4.4.p1"> <p class="ltx_p" id="S3.SS4.4.p1.1">Recall first from equivalence (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E6" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.6</span></a>) that the language of the support of any shift-invariant measure is given by all words with positive measure of the associated cylinder.</p> </div> <div class="ltx_para" id="S3.SS4.5.p2"> <p class="ltx_p" id="S3.SS4.5.p2.13">In particular, any <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S3.SS4.5.p2.1.m1.1"><semantics id="S3.SS4.5.p2.1.m1.1a"><mrow id="S3.SS4.5.p2.1.m1.1.1" xref="S3.SS4.5.p2.1.m1.1.1.cmml"><mi id="S3.SS4.5.p2.1.m1.1.1.2" xref="S3.SS4.5.p2.1.m1.1.1.2.cmml">w</mi><mo id="S3.SS4.5.p2.1.m1.1.1.1" xref="S3.SS4.5.p2.1.m1.1.1.1.cmml">∈</mo><msup id="S3.SS4.5.p2.1.m1.1.1.3" xref="S3.SS4.5.p2.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p2.1.m1.1.1.3.2" xref="S3.SS4.5.p2.1.m1.1.1.3.2.cmml">𝒜</mi><mo id="S3.SS4.5.p2.1.m1.1.1.3.3" xref="S3.SS4.5.p2.1.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.1.m1.1b"><apply id="S3.SS4.5.p2.1.m1.1.1.cmml" xref="S3.SS4.5.p2.1.m1.1.1"><in id="S3.SS4.5.p2.1.m1.1.1.1.cmml" xref="S3.SS4.5.p2.1.m1.1.1.1"></in><ci id="S3.SS4.5.p2.1.m1.1.1.2.cmml" xref="S3.SS4.5.p2.1.m1.1.1.2">𝑤</ci><apply id="S3.SS4.5.p2.1.m1.1.1.3.cmml" xref="S3.SS4.5.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.5.p2.1.m1.1.1.3.1.cmml" xref="S3.SS4.5.p2.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS4.5.p2.1.m1.1.1.3.2.cmml" xref="S3.SS4.5.p2.1.m1.1.1.3.2">𝒜</ci><times id="S3.SS4.5.p2.1.m1.1.1.3.3.cmml" xref="S3.SS4.5.p2.1.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.1.m1.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.1.m1.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> belongs to <math alttext="\cal L(\mbox{Supp}(\mu))" class="ltx_Math" display="inline" id="S3.SS4.5.p2.2.m2.2"><semantics id="S3.SS4.5.p2.2.m2.2a"><mrow id="S3.SS4.5.p2.2.m2.2.2" xref="S3.SS4.5.p2.2.m2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p2.2.m2.2.2.3" xref="S3.SS4.5.p2.2.m2.2.2.3.cmml">ℒ</mi><mo id="S3.SS4.5.p2.2.m2.2.2.2" xref="S3.SS4.5.p2.2.m2.2.2.2.cmml">⁢</mo><mrow id="S3.SS4.5.p2.2.m2.2.2.1.1" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.cmml"><mo id="S3.SS4.5.p2.2.m2.2.2.1.1.2" stretchy="false" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS4.5.p2.2.m2.2.2.1.1.1" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.cmml"><mtext id="S3.SS4.5.p2.2.m2.2.2.1.1.1.2" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.2a.cmml">Supp</mtext><mo id="S3.SS4.5.p2.2.m2.2.2.1.1.1.1" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.2.m2.2.2.1.1.1.3.2" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.cmml"><mo id="S3.SS4.5.p2.2.m2.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.cmml">(</mo><mi id="S3.SS4.5.p2.2.m2.1.1" xref="S3.SS4.5.p2.2.m2.1.1.cmml">μ</mi><mo id="S3.SS4.5.p2.2.m2.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.2.m2.2.2.1.1.3" stretchy="false" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.2.m2.2b"><apply id="S3.SS4.5.p2.2.m2.2.2.cmml" xref="S3.SS4.5.p2.2.m2.2.2"><times id="S3.SS4.5.p2.2.m2.2.2.2.cmml" xref="S3.SS4.5.p2.2.m2.2.2.2"></times><ci id="S3.SS4.5.p2.2.m2.2.2.3.cmml" xref="S3.SS4.5.p2.2.m2.2.2.3">ℒ</ci><apply id="S3.SS4.5.p2.2.m2.2.2.1.1.1.cmml" xref="S3.SS4.5.p2.2.m2.2.2.1.1"><times id="S3.SS4.5.p2.2.m2.2.2.1.1.1.1.cmml" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.1"></times><ci id="S3.SS4.5.p2.2.m2.2.2.1.1.1.2a.cmml" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.2"><mtext id="S3.SS4.5.p2.2.m2.2.2.1.1.1.2.cmml" xref="S3.SS4.5.p2.2.m2.2.2.1.1.1.2">Supp</mtext></ci><ci id="S3.SS4.5.p2.2.m2.1.1.cmml" xref="S3.SS4.5.p2.2.m2.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.2.m2.2c">\cal L(\mbox{Supp}(\mu))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.2.m2.2d">caligraphic_L ( Supp ( italic_μ ) )</annotation></semantics></math> if and only if <math alttext="\mu(w)>0" class="ltx_Math" display="inline" id="S3.SS4.5.p2.3.m3.1"><semantics id="S3.SS4.5.p2.3.m3.1a"><mrow id="S3.SS4.5.p2.3.m3.1.2" xref="S3.SS4.5.p2.3.m3.1.2.cmml"><mrow id="S3.SS4.5.p2.3.m3.1.2.2" xref="S3.SS4.5.p2.3.m3.1.2.2.cmml"><mi id="S3.SS4.5.p2.3.m3.1.2.2.2" xref="S3.SS4.5.p2.3.m3.1.2.2.2.cmml">μ</mi><mo id="S3.SS4.5.p2.3.m3.1.2.2.1" xref="S3.SS4.5.p2.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.3.m3.1.2.2.3.2" xref="S3.SS4.5.p2.3.m3.1.2.2.cmml"><mo id="S3.SS4.5.p2.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.SS4.5.p2.3.m3.1.2.2.cmml">(</mo><mi id="S3.SS4.5.p2.3.m3.1.1" xref="S3.SS4.5.p2.3.m3.1.1.cmml">w</mi><mo id="S3.SS4.5.p2.3.m3.1.2.2.3.2.2" stretchy="false" xref="S3.SS4.5.p2.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.3.m3.1.2.1" xref="S3.SS4.5.p2.3.m3.1.2.1.cmml">></mo><mn id="S3.SS4.5.p2.3.m3.1.2.3" xref="S3.SS4.5.p2.3.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.3.m3.1b"><apply id="S3.SS4.5.p2.3.m3.1.2.cmml" xref="S3.SS4.5.p2.3.m3.1.2"><gt id="S3.SS4.5.p2.3.m3.1.2.1.cmml" xref="S3.SS4.5.p2.3.m3.1.2.1"></gt><apply id="S3.SS4.5.p2.3.m3.1.2.2.cmml" xref="S3.SS4.5.p2.3.m3.1.2.2"><times id="S3.SS4.5.p2.3.m3.1.2.2.1.cmml" xref="S3.SS4.5.p2.3.m3.1.2.2.1"></times><ci id="S3.SS4.5.p2.3.m3.1.2.2.2.cmml" xref="S3.SS4.5.p2.3.m3.1.2.2.2">𝜇</ci><ci id="S3.SS4.5.p2.3.m3.1.1.cmml" xref="S3.SS4.5.p2.3.m3.1.1">𝑤</ci></apply><cn id="S3.SS4.5.p2.3.m3.1.2.3.cmml" type="integer" xref="S3.SS4.5.p2.3.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.3.m3.1c">\mu(w)>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.3.m3.1d">italic_μ ( italic_w ) > 0</annotation></semantics></math>. In this case <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S3.SS4.5.p2.4.m4.1"><semantics id="S3.SS4.5.p2.4.m4.1a"><mrow id="S3.SS4.5.p2.4.m4.1.2" xref="S3.SS4.5.p2.4.m4.1.2.cmml"><mi id="S3.SS4.5.p2.4.m4.1.2.2" xref="S3.SS4.5.p2.4.m4.1.2.2.cmml">σ</mi><mo id="S3.SS4.5.p2.4.m4.1.2.1" xref="S3.SS4.5.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.4.m4.1.2.3.2" xref="S3.SS4.5.p2.4.m4.1.2.cmml"><mo id="S3.SS4.5.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S3.SS4.5.p2.4.m4.1.2.cmml">(</mo><mi id="S3.SS4.5.p2.4.m4.1.1" xref="S3.SS4.5.p2.4.m4.1.1.cmml">w</mi><mo id="S3.SS4.5.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S3.SS4.5.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.4.m4.1b"><apply id="S3.SS4.5.p2.4.m4.1.2.cmml" xref="S3.SS4.5.p2.4.m4.1.2"><times id="S3.SS4.5.p2.4.m4.1.2.1.cmml" xref="S3.SS4.5.p2.4.m4.1.2.1"></times><ci id="S3.SS4.5.p2.4.m4.1.2.2.cmml" xref="S3.SS4.5.p2.4.m4.1.2.2">𝜎</ci><ci id="S3.SS4.5.p2.4.m4.1.1.cmml" xref="S3.SS4.5.p2.4.m4.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.4.m4.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.4.m4.1d">italic_σ ( italic_w )</annotation></semantics></math> belongs to <math alttext="\cal L(\sigma^{\Sigma}(\mbox{Supp}(\mu)))" class="ltx_Math" display="inline" id="S3.SS4.5.p2.5.m5.2"><semantics id="S3.SS4.5.p2.5.m5.2a"><mrow id="S3.SS4.5.p2.5.m5.2.2" xref="S3.SS4.5.p2.5.m5.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p2.5.m5.2.2.3" xref="S3.SS4.5.p2.5.m5.2.2.3.cmml">ℒ</mi><mo id="S3.SS4.5.p2.5.m5.2.2.2" xref="S3.SS4.5.p2.5.m5.2.2.2.cmml">⁢</mo><mrow id="S3.SS4.5.p2.5.m5.2.2.1.1" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.cmml"><mo id="S3.SS4.5.p2.5.m5.2.2.1.1.2" stretchy="false" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.cmml">(</mo><mrow id="S3.SS4.5.p2.5.m5.2.2.1.1.1" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.cmml"><msup id="S3.SS4.5.p2.5.m5.2.2.1.1.1.3" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.cmml"><mi id="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.2" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.2.cmml">σ</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.3" mathvariant="script" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.3.cmml">Σ</mi></msup><mo id="S3.SS4.5.p2.5.m5.2.2.1.1.1.2" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.cmml"><mo id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.cmml"><mtext id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.2" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.2a.cmml">Supp</mtext><mo id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.1" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.3.2" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.cmml"><mo id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.cmml">(</mo><mi id="S3.SS4.5.p2.5.m5.1.1" xref="S3.SS4.5.p2.5.m5.1.1.cmml">μ</mi><mo id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.5.m5.2.2.1.1.3" stretchy="false" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.5.m5.2b"><apply id="S3.SS4.5.p2.5.m5.2.2.cmml" xref="S3.SS4.5.p2.5.m5.2.2"><times id="S3.SS4.5.p2.5.m5.2.2.2.cmml" xref="S3.SS4.5.p2.5.m5.2.2.2"></times><ci id="S3.SS4.5.p2.5.m5.2.2.3.cmml" xref="S3.SS4.5.p2.5.m5.2.2.3">ℒ</ci><apply id="S3.SS4.5.p2.5.m5.2.2.1.1.1.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1"><times id="S3.SS4.5.p2.5.m5.2.2.1.1.1.2.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.2"></times><apply id="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.1.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.3">superscript</csymbol><ci id="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.2.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.2">𝜎</ci><ci id="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.3.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.3.3">script-Σ</ci></apply><apply id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1"><times id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.1.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.1"></times><ci id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.2a.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.2"><mtext id="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.2.cmml" xref="S3.SS4.5.p2.5.m5.2.2.1.1.1.1.1.1.2">Supp</mtext></ci><ci id="S3.SS4.5.p2.5.m5.1.1.cmml" xref="S3.SS4.5.p2.5.m5.1.1">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.5.m5.2c">\cal L(\sigma^{\Sigma}(\mbox{Supp}(\mu)))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.5.m5.2d">caligraphic_L ( italic_σ start_POSTSUPERSCRIPT caligraphic_Σ end_POSTSUPERSCRIPT ( Supp ( italic_μ ) ) )</annotation></semantics></math>, and any <math alttext="w^{\prime}\in\cal L(\sigma^{\Sigma}(\mbox{Supp}(\mu)))" class="ltx_Math" display="inline" id="S3.SS4.5.p2.6.m6.2"><semantics id="S3.SS4.5.p2.6.m6.2a"><mrow id="S3.SS4.5.p2.6.m6.2.2" xref="S3.SS4.5.p2.6.m6.2.2.cmml"><msup id="S3.SS4.5.p2.6.m6.2.2.3" xref="S3.SS4.5.p2.6.m6.2.2.3.cmml"><mi id="S3.SS4.5.p2.6.m6.2.2.3.2" xref="S3.SS4.5.p2.6.m6.2.2.3.2.cmml">w</mi><mo id="S3.SS4.5.p2.6.m6.2.2.3.3" xref="S3.SS4.5.p2.6.m6.2.2.3.3.cmml">′</mo></msup><mo id="S3.SS4.5.p2.6.m6.2.2.2" xref="S3.SS4.5.p2.6.m6.2.2.2.cmml">∈</mo><mrow id="S3.SS4.5.p2.6.m6.2.2.1" xref="S3.SS4.5.p2.6.m6.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p2.6.m6.2.2.1.3" xref="S3.SS4.5.p2.6.m6.2.2.1.3.cmml">ℒ</mi><mo id="S3.SS4.5.p2.6.m6.2.2.1.2" xref="S3.SS4.5.p2.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS4.5.p2.6.m6.2.2.1.1.1" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.cmml"><mo id="S3.SS4.5.p2.6.m6.2.2.1.1.1.2" stretchy="false" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.cmml"><msup id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.cmml"><mi id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.2" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.2.cmml">σ</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.3" mathvariant="script" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.3.cmml">Σ</mi></msup><mo id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.2" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.cmml"><mtext id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.2" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.2a.cmml">Supp</mtext><mo id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.1" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.3.2" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.cmml">(</mo><mi id="S3.SS4.5.p2.6.m6.1.1" xref="S3.SS4.5.p2.6.m6.1.1.cmml">μ</mi><mo id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.6.m6.2.2.1.1.1.3" stretchy="false" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.6.m6.2b"><apply id="S3.SS4.5.p2.6.m6.2.2.cmml" xref="S3.SS4.5.p2.6.m6.2.2"><in id="S3.SS4.5.p2.6.m6.2.2.2.cmml" xref="S3.SS4.5.p2.6.m6.2.2.2"></in><apply id="S3.SS4.5.p2.6.m6.2.2.3.cmml" xref="S3.SS4.5.p2.6.m6.2.2.3"><csymbol cd="ambiguous" id="S3.SS4.5.p2.6.m6.2.2.3.1.cmml" xref="S3.SS4.5.p2.6.m6.2.2.3">superscript</csymbol><ci id="S3.SS4.5.p2.6.m6.2.2.3.2.cmml" xref="S3.SS4.5.p2.6.m6.2.2.3.2">𝑤</ci><ci id="S3.SS4.5.p2.6.m6.2.2.3.3.cmml" xref="S3.SS4.5.p2.6.m6.2.2.3.3">′</ci></apply><apply id="S3.SS4.5.p2.6.m6.2.2.1.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1"><times id="S3.SS4.5.p2.6.m6.2.2.1.2.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.2"></times><ci id="S3.SS4.5.p2.6.m6.2.2.1.3.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.3">ℒ</ci><apply id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1"><times id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.2.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.2"></times><apply id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.1.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3">superscript</csymbol><ci id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.2.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.2">𝜎</ci><ci id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.3.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.3.3">script-Σ</ci></apply><apply id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1"><times id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.1"></times><ci id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.2a.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.2"><mtext id="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.5.p2.6.m6.2.2.1.1.1.1.1.1.1.2">Supp</mtext></ci><ci id="S3.SS4.5.p2.6.m6.1.1.cmml" xref="S3.SS4.5.p2.6.m6.1.1">𝜇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.6.m6.2c">w^{\prime}\in\cal L(\sigma^{\Sigma}(\mbox{Supp}(\mu)))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.6.m6.2d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_L ( italic_σ start_POSTSUPERSCRIPT caligraphic_Σ end_POSTSUPERSCRIPT ( Supp ( italic_μ ) ) )</annotation></semantics></math> is a factor of some such <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S3.SS4.5.p2.7.m7.1"><semantics id="S3.SS4.5.p2.7.m7.1a"><mrow id="S3.SS4.5.p2.7.m7.1.2" xref="S3.SS4.5.p2.7.m7.1.2.cmml"><mi id="S3.SS4.5.p2.7.m7.1.2.2" xref="S3.SS4.5.p2.7.m7.1.2.2.cmml">σ</mi><mo id="S3.SS4.5.p2.7.m7.1.2.1" xref="S3.SS4.5.p2.7.m7.1.2.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.7.m7.1.2.3.2" xref="S3.SS4.5.p2.7.m7.1.2.cmml"><mo id="S3.SS4.5.p2.7.m7.1.2.3.2.1" stretchy="false" xref="S3.SS4.5.p2.7.m7.1.2.cmml">(</mo><mi id="S3.SS4.5.p2.7.m7.1.1" xref="S3.SS4.5.p2.7.m7.1.1.cmml">w</mi><mo id="S3.SS4.5.p2.7.m7.1.2.3.2.2" stretchy="false" xref="S3.SS4.5.p2.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.7.m7.1b"><apply id="S3.SS4.5.p2.7.m7.1.2.cmml" xref="S3.SS4.5.p2.7.m7.1.2"><times id="S3.SS4.5.p2.7.m7.1.2.1.cmml" xref="S3.SS4.5.p2.7.m7.1.2.1"></times><ci id="S3.SS4.5.p2.7.m7.1.2.2.cmml" xref="S3.SS4.5.p2.7.m7.1.2.2">𝜎</ci><ci id="S3.SS4.5.p2.7.m7.1.1.cmml" xref="S3.SS4.5.p2.7.m7.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.7.m7.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.7.m7.1d">italic_σ ( italic_w )</annotation></semantics></math>. From (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E6" title="In item (e) ‣ Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.6</span></a>) we know <math alttext="\mu^{\sigma}(\sigma(w))\geq\mu(w)" class="ltx_Math" display="inline" id="S3.SS4.5.p2.8.m8.3"><semantics id="S3.SS4.5.p2.8.m8.3a"><mrow id="S3.SS4.5.p2.8.m8.3.3" xref="S3.SS4.5.p2.8.m8.3.3.cmml"><mrow id="S3.SS4.5.p2.8.m8.3.3.1" xref="S3.SS4.5.p2.8.m8.3.3.1.cmml"><msup id="S3.SS4.5.p2.8.m8.3.3.1.3" xref="S3.SS4.5.p2.8.m8.3.3.1.3.cmml"><mi id="S3.SS4.5.p2.8.m8.3.3.1.3.2" xref="S3.SS4.5.p2.8.m8.3.3.1.3.2.cmml">μ</mi><mi id="S3.SS4.5.p2.8.m8.3.3.1.3.3" xref="S3.SS4.5.p2.8.m8.3.3.1.3.3.cmml">σ</mi></msup><mo id="S3.SS4.5.p2.8.m8.3.3.1.2" xref="S3.SS4.5.p2.8.m8.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS4.5.p2.8.m8.3.3.1.1.1" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.cmml"><mo id="S3.SS4.5.p2.8.m8.3.3.1.1.1.2" stretchy="false" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.cmml"><mi id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.2" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.2.cmml">σ</mi><mo id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.1" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.3.2" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.cmml"><mo id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.cmml">(</mo><mi id="S3.SS4.5.p2.8.m8.1.1" xref="S3.SS4.5.p2.8.m8.1.1.cmml">w</mi><mo id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.8.m8.3.3.1.1.1.3" stretchy="false" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.8.m8.3.3.2" xref="S3.SS4.5.p2.8.m8.3.3.2.cmml">≥</mo><mrow id="S3.SS4.5.p2.8.m8.3.3.3" xref="S3.SS4.5.p2.8.m8.3.3.3.cmml"><mi id="S3.SS4.5.p2.8.m8.3.3.3.2" xref="S3.SS4.5.p2.8.m8.3.3.3.2.cmml">μ</mi><mo id="S3.SS4.5.p2.8.m8.3.3.3.1" xref="S3.SS4.5.p2.8.m8.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.8.m8.3.3.3.3.2" xref="S3.SS4.5.p2.8.m8.3.3.3.cmml"><mo id="S3.SS4.5.p2.8.m8.3.3.3.3.2.1" stretchy="false" xref="S3.SS4.5.p2.8.m8.3.3.3.cmml">(</mo><mi id="S3.SS4.5.p2.8.m8.2.2" xref="S3.SS4.5.p2.8.m8.2.2.cmml">w</mi><mo id="S3.SS4.5.p2.8.m8.3.3.3.3.2.2" stretchy="false" xref="S3.SS4.5.p2.8.m8.3.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.8.m8.3b"><apply id="S3.SS4.5.p2.8.m8.3.3.cmml" xref="S3.SS4.5.p2.8.m8.3.3"><geq id="S3.SS4.5.p2.8.m8.3.3.2.cmml" xref="S3.SS4.5.p2.8.m8.3.3.2"></geq><apply id="S3.SS4.5.p2.8.m8.3.3.1.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1"><times id="S3.SS4.5.p2.8.m8.3.3.1.2.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1.2"></times><apply id="S3.SS4.5.p2.8.m8.3.3.1.3.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1.3"><csymbol cd="ambiguous" id="S3.SS4.5.p2.8.m8.3.3.1.3.1.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1.3">superscript</csymbol><ci id="S3.SS4.5.p2.8.m8.3.3.1.3.2.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1.3.2">𝜇</ci><ci id="S3.SS4.5.p2.8.m8.3.3.1.3.3.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1.3.3">𝜎</ci></apply><apply id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1"><times id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.1.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.1"></times><ci id="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.2.cmml" xref="S3.SS4.5.p2.8.m8.3.3.1.1.1.1.2">𝜎</ci><ci id="S3.SS4.5.p2.8.m8.1.1.cmml" xref="S3.SS4.5.p2.8.m8.1.1">𝑤</ci></apply></apply><apply id="S3.SS4.5.p2.8.m8.3.3.3.cmml" xref="S3.SS4.5.p2.8.m8.3.3.3"><times id="S3.SS4.5.p2.8.m8.3.3.3.1.cmml" xref="S3.SS4.5.p2.8.m8.3.3.3.1"></times><ci id="S3.SS4.5.p2.8.m8.3.3.3.2.cmml" xref="S3.SS4.5.p2.8.m8.3.3.3.2">𝜇</ci><ci id="S3.SS4.5.p2.8.m8.2.2.cmml" xref="S3.SS4.5.p2.8.m8.2.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.8.m8.3c">\mu^{\sigma}(\sigma(w))\geq\mu(w)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.8.m8.3d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_σ ( italic_w ) ) ≥ italic_μ ( italic_w )</annotation></semantics></math>, which implies that <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S3.SS4.5.p2.9.m9.1"><semantics id="S3.SS4.5.p2.9.m9.1a"><mrow id="S3.SS4.5.p2.9.m9.1.2" xref="S3.SS4.5.p2.9.m9.1.2.cmml"><mi id="S3.SS4.5.p2.9.m9.1.2.2" xref="S3.SS4.5.p2.9.m9.1.2.2.cmml">σ</mi><mo id="S3.SS4.5.p2.9.m9.1.2.1" xref="S3.SS4.5.p2.9.m9.1.2.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.9.m9.1.2.3.2" xref="S3.SS4.5.p2.9.m9.1.2.cmml"><mo id="S3.SS4.5.p2.9.m9.1.2.3.2.1" stretchy="false" xref="S3.SS4.5.p2.9.m9.1.2.cmml">(</mo><mi id="S3.SS4.5.p2.9.m9.1.1" xref="S3.SS4.5.p2.9.m9.1.1.cmml">w</mi><mo id="S3.SS4.5.p2.9.m9.1.2.3.2.2" stretchy="false" xref="S3.SS4.5.p2.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.9.m9.1b"><apply id="S3.SS4.5.p2.9.m9.1.2.cmml" xref="S3.SS4.5.p2.9.m9.1.2"><times id="S3.SS4.5.p2.9.m9.1.2.1.cmml" xref="S3.SS4.5.p2.9.m9.1.2.1"></times><ci id="S3.SS4.5.p2.9.m9.1.2.2.cmml" xref="S3.SS4.5.p2.9.m9.1.2.2">𝜎</ci><ci id="S3.SS4.5.p2.9.m9.1.1.cmml" xref="S3.SS4.5.p2.9.m9.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.9.m9.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.9.m9.1d">italic_σ ( italic_w )</annotation></semantics></math> belongs to <math alttext="\cal L(\mbox{Supp}(\mu^{\sigma})))" class="ltx_math_unparsed" display="inline" id="S3.SS4.5.p2.10.m10.1"><semantics id="S3.SS4.5.p2.10.m10.1a"><mrow id="S3.SS4.5.p2.10.m10.1b"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.5.p2.10.m10.1.1">ℒ</mi><mrow id="S3.SS4.5.p2.10.m10.1.2"><mo id="S3.SS4.5.p2.10.m10.1.2.1" stretchy="false">(</mo><mtext id="S3.SS4.5.p2.10.m10.1.2.2">Supp</mtext><mrow id="S3.SS4.5.p2.10.m10.1.2.3"><mo id="S3.SS4.5.p2.10.m10.1.2.3.1" stretchy="false">(</mo><msup id="S3.SS4.5.p2.10.m10.1.2.3.2"><mi id="S3.SS4.5.p2.10.m10.1.2.3.2.2">μ</mi><mi id="S3.SS4.5.p2.10.m10.1.2.3.2.3">σ</mi></msup><mo id="S3.SS4.5.p2.10.m10.1.2.3.3" stretchy="false">)</mo></mrow><mo id="S3.SS4.5.p2.10.m10.1.2.4" stretchy="false">)</mo></mrow><mo id="S3.SS4.5.p2.10.m10.1.3" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S3.SS4.5.p2.10.m10.1c">\cal L(\mbox{Supp}(\mu^{\sigma})))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.10.m10.1d">caligraphic_L ( Supp ( italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ) ) )</annotation></semantics></math>. Since <math alttext="\mu^{\sigma}(w^{\prime})\geq\mu^{\sigma}(\sigma(w))" class="ltx_Math" display="inline" id="S3.SS4.5.p2.11.m11.3"><semantics id="S3.SS4.5.p2.11.m11.3a"><mrow id="S3.SS4.5.p2.11.m11.3.3" xref="S3.SS4.5.p2.11.m11.3.3.cmml"><mrow id="S3.SS4.5.p2.11.m11.2.2.1" xref="S3.SS4.5.p2.11.m11.2.2.1.cmml"><msup id="S3.SS4.5.p2.11.m11.2.2.1.3" xref="S3.SS4.5.p2.11.m11.2.2.1.3.cmml"><mi id="S3.SS4.5.p2.11.m11.2.2.1.3.2" xref="S3.SS4.5.p2.11.m11.2.2.1.3.2.cmml">μ</mi><mi id="S3.SS4.5.p2.11.m11.2.2.1.3.3" xref="S3.SS4.5.p2.11.m11.2.2.1.3.3.cmml">σ</mi></msup><mo id="S3.SS4.5.p2.11.m11.2.2.1.2" xref="S3.SS4.5.p2.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS4.5.p2.11.m11.2.2.1.1.1" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.cmml"><mo id="S3.SS4.5.p2.11.m11.2.2.1.1.1.2" stretchy="false" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.cmml">(</mo><msup id="S3.SS4.5.p2.11.m11.2.2.1.1.1.1" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.cmml"><mi id="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.2" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.2.cmml">w</mi><mo id="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.3" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS4.5.p2.11.m11.2.2.1.1.1.3" stretchy="false" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.11.m11.3.3.3" xref="S3.SS4.5.p2.11.m11.3.3.3.cmml">≥</mo><mrow id="S3.SS4.5.p2.11.m11.3.3.2" xref="S3.SS4.5.p2.11.m11.3.3.2.cmml"><msup id="S3.SS4.5.p2.11.m11.3.3.2.3" xref="S3.SS4.5.p2.11.m11.3.3.2.3.cmml"><mi id="S3.SS4.5.p2.11.m11.3.3.2.3.2" xref="S3.SS4.5.p2.11.m11.3.3.2.3.2.cmml">μ</mi><mi id="S3.SS4.5.p2.11.m11.3.3.2.3.3" xref="S3.SS4.5.p2.11.m11.3.3.2.3.3.cmml">σ</mi></msup><mo id="S3.SS4.5.p2.11.m11.3.3.2.2" xref="S3.SS4.5.p2.11.m11.3.3.2.2.cmml">⁢</mo><mrow id="S3.SS4.5.p2.11.m11.3.3.2.1.1" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.cmml"><mo id="S3.SS4.5.p2.11.m11.3.3.2.1.1.2" stretchy="false" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.cmml">(</mo><mrow id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.cmml"><mi id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.2" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.2.cmml">σ</mi><mo id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.1" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.3.2" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.cmml"><mo id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.cmml">(</mo><mi id="S3.SS4.5.p2.11.m11.1.1" xref="S3.SS4.5.p2.11.m11.1.1.cmml">w</mi><mo id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.5.p2.11.m11.3.3.2.1.1.3" stretchy="false" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.11.m11.3b"><apply id="S3.SS4.5.p2.11.m11.3.3.cmml" xref="S3.SS4.5.p2.11.m11.3.3"><geq id="S3.SS4.5.p2.11.m11.3.3.3.cmml" xref="S3.SS4.5.p2.11.m11.3.3.3"></geq><apply id="S3.SS4.5.p2.11.m11.2.2.1.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1"><times id="S3.SS4.5.p2.11.m11.2.2.1.2.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.2"></times><apply id="S3.SS4.5.p2.11.m11.2.2.1.3.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.3"><csymbol cd="ambiguous" id="S3.SS4.5.p2.11.m11.2.2.1.3.1.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.3">superscript</csymbol><ci id="S3.SS4.5.p2.11.m11.2.2.1.3.2.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.3.2">𝜇</ci><ci id="S3.SS4.5.p2.11.m11.2.2.1.3.3.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.3.3">𝜎</ci></apply><apply id="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.1.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1">superscript</csymbol><ci id="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.2.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.2">𝑤</ci><ci id="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.3.cmml" xref="S3.SS4.5.p2.11.m11.2.2.1.1.1.1.3">′</ci></apply></apply><apply id="S3.SS4.5.p2.11.m11.3.3.2.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2"><times id="S3.SS4.5.p2.11.m11.3.3.2.2.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2.2"></times><apply id="S3.SS4.5.p2.11.m11.3.3.2.3.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2.3"><csymbol cd="ambiguous" id="S3.SS4.5.p2.11.m11.3.3.2.3.1.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2.3">superscript</csymbol><ci id="S3.SS4.5.p2.11.m11.3.3.2.3.2.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2.3.2">𝜇</ci><ci id="S3.SS4.5.p2.11.m11.3.3.2.3.3.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2.3.3">𝜎</ci></apply><apply id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1"><times id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.1.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.1"></times><ci id="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.2.cmml" xref="S3.SS4.5.p2.11.m11.3.3.2.1.1.1.2">𝜎</ci><ci id="S3.SS4.5.p2.11.m11.1.1.cmml" xref="S3.SS4.5.p2.11.m11.1.1">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.11.m11.3c">\mu^{\sigma}(w^{\prime})\geq\mu^{\sigma}(\sigma(w))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.11.m11.3d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≥ italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_σ ( italic_w ) )</annotation></semantics></math>, the same applies to any factor <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S3.SS4.5.p2.12.m12.1"><semantics id="S3.SS4.5.p2.12.m12.1a"><msup id="S3.SS4.5.p2.12.m12.1.1" xref="S3.SS4.5.p2.12.m12.1.1.cmml"><mi id="S3.SS4.5.p2.12.m12.1.1.2" xref="S3.SS4.5.p2.12.m12.1.1.2.cmml">w</mi><mo id="S3.SS4.5.p2.12.m12.1.1.3" xref="S3.SS4.5.p2.12.m12.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.12.m12.1b"><apply id="S3.SS4.5.p2.12.m12.1.1.cmml" xref="S3.SS4.5.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S3.SS4.5.p2.12.m12.1.1.1.cmml" xref="S3.SS4.5.p2.12.m12.1.1">superscript</csymbol><ci id="S3.SS4.5.p2.12.m12.1.1.2.cmml" xref="S3.SS4.5.p2.12.m12.1.1.2">𝑤</ci><ci id="S3.SS4.5.p2.12.m12.1.1.3.cmml" xref="S3.SS4.5.p2.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.12.m12.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.12.m12.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S3.SS4.5.p2.13.m13.1"><semantics id="S3.SS4.5.p2.13.m13.1a"><mrow id="S3.SS4.5.p2.13.m13.1.2" xref="S3.SS4.5.p2.13.m13.1.2.cmml"><mi id="S3.SS4.5.p2.13.m13.1.2.2" xref="S3.SS4.5.p2.13.m13.1.2.2.cmml">σ</mi><mo id="S3.SS4.5.p2.13.m13.1.2.1" xref="S3.SS4.5.p2.13.m13.1.2.1.cmml">⁢</mo><mrow id="S3.SS4.5.p2.13.m13.1.2.3.2" xref="S3.SS4.5.p2.13.m13.1.2.cmml"><mo id="S3.SS4.5.p2.13.m13.1.2.3.2.1" stretchy="false" xref="S3.SS4.5.p2.13.m13.1.2.cmml">(</mo><mi id="S3.SS4.5.p2.13.m13.1.1" xref="S3.SS4.5.p2.13.m13.1.1.cmml">w</mi><mo id="S3.SS4.5.p2.13.m13.1.2.3.2.2" stretchy="false" xref="S3.SS4.5.p2.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.5.p2.13.m13.1b"><apply id="S3.SS4.5.p2.13.m13.1.2.cmml" xref="S3.SS4.5.p2.13.m13.1.2"><times id="S3.SS4.5.p2.13.m13.1.2.1.cmml" xref="S3.SS4.5.p2.13.m13.1.2.1"></times><ci id="S3.SS4.5.p2.13.m13.1.2.2.cmml" xref="S3.SS4.5.p2.13.m13.1.2.2">𝜎</ci><ci id="S3.SS4.5.p2.13.m13.1.1.cmml" xref="S3.SS4.5.p2.13.m13.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.5.p2.13.m13.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.5.p2.13.m13.1d">italic_σ ( italic_w )</annotation></semantics></math>. We thus obtain</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\cal L(\sigma^{\Sigma}(\mbox{Supp}(\mu)))\subseteq\cal L(\mbox{Supp}(\mu^{% \sigma}))\,." class="ltx_Math" display="block" id="S3.Ex8.m1.2"><semantics id="S3.Ex8.m1.2a"><mrow id="S3.Ex8.m1.2.2.1" xref="S3.Ex8.m1.2.2.1.1.cmml"><mrow id="S3.Ex8.m1.2.2.1.1" xref="S3.Ex8.m1.2.2.1.1.cmml"><mrow id="S3.Ex8.m1.2.2.1.1.1" xref="S3.Ex8.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex8.m1.2.2.1.1.1.3" xref="S3.Ex8.m1.2.2.1.1.1.3.cmml">ℒ</mi><mo id="S3.Ex8.m1.2.2.1.1.1.2" xref="S3.Ex8.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex8.m1.2.2.1.1.1.1.1" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S3.Ex8.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex8.m1.2.2.1.1.1.1.1.1" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.cmml"><msup id="S3.Ex8.m1.2.2.1.1.1.1.1.1.3" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.2" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.2.cmml">σ</mi><mi class="ltx_font_mathcaligraphic" id="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.3" mathvariant="script" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.3.cmml">Σ</mi></msup><mo id="S3.Ex8.m1.2.2.1.1.1.1.1.1.2" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mtext id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.2a.cmml">Supp</mtext><mo id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S3.Ex8.m1.1.1" xref="S3.Ex8.m1.1.1.cmml">μ</mi><mo id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex8.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex8.m1.2.2.1.1.3" xref="S3.Ex8.m1.2.2.1.1.3.cmml">⊆</mo><mrow id="S3.Ex8.m1.2.2.1.1.2" xref="S3.Ex8.m1.2.2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex8.m1.2.2.1.1.2.3" xref="S3.Ex8.m1.2.2.1.1.2.3.cmml">ℒ</mi><mo id="S3.Ex8.m1.2.2.1.1.2.2" xref="S3.Ex8.m1.2.2.1.1.2.2.cmml">⁢</mo><mrow id="S3.Ex8.m1.2.2.1.1.2.1.1" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.cmml"><mo id="S3.Ex8.m1.2.2.1.1.2.1.1.2" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.cmml">(</mo><mrow id="S3.Ex8.m1.2.2.1.1.2.1.1.1" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.cmml"><mtext id="S3.Ex8.m1.2.2.1.1.2.1.1.1.3" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.3a.cmml">Supp</mtext><mo id="S3.Ex8.m1.2.2.1.1.2.1.1.1.2" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.cmml"><mo id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.2" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.cmml"><mi id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.2" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.3" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.3.cmml">σ</mi></msup><mo id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.3" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex8.m1.2.2.1.1.2.1.1.3" stretchy="false" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex8.m1.2.2.1.2" lspace="0.170em" xref="S3.Ex8.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex8.m1.2b"><apply id="S3.Ex8.m1.2.2.1.1.cmml" xref="S3.Ex8.m1.2.2.1"><subset id="S3.Ex8.m1.2.2.1.1.3.cmml" xref="S3.Ex8.m1.2.2.1.1.3"></subset><apply id="S3.Ex8.m1.2.2.1.1.1.cmml" xref="S3.Ex8.m1.2.2.1.1.1"><times id="S3.Ex8.m1.2.2.1.1.1.2.cmml" xref="S3.Ex8.m1.2.2.1.1.1.2"></times><ci id="S3.Ex8.m1.2.2.1.1.1.3.cmml" xref="S3.Ex8.m1.2.2.1.1.1.3">ℒ</ci><apply id="S3.Ex8.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1"><times id="S3.Ex8.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.2"></times><apply id="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.2">𝜎</ci><ci id="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.3.3">script-Σ</ci></apply><apply id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1"><times id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.1"></times><ci id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.2a.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.2"><mtext id="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex8.m1.2.2.1.1.1.1.1.1.1.1.1.2">Supp</mtext></ci><ci id="S3.Ex8.m1.1.1.cmml" xref="S3.Ex8.m1.1.1">𝜇</ci></apply></apply></apply><apply id="S3.Ex8.m1.2.2.1.1.2.cmml" xref="S3.Ex8.m1.2.2.1.1.2"><times id="S3.Ex8.m1.2.2.1.1.2.2.cmml" xref="S3.Ex8.m1.2.2.1.1.2.2"></times><ci id="S3.Ex8.m1.2.2.1.1.2.3.cmml" xref="S3.Ex8.m1.2.2.1.1.2.3">ℒ</ci><apply id="S3.Ex8.m1.2.2.1.1.2.1.1.1.cmml" xref="S3.Ex8.m1.2.2.1.1.2.1.1"><times id="S3.Ex8.m1.2.2.1.1.2.1.1.1.2.cmml" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.2"></times><ci id="S3.Ex8.m1.2.2.1.1.2.1.1.1.3a.cmml" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.3"><mtext id="S3.Ex8.m1.2.2.1.1.2.1.1.1.3.cmml" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.3">Supp</mtext></ci><apply id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.cmml" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.1.cmml" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1">superscript</csymbol><ci id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.2.cmml" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.2">𝜇</ci><ci id="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.3.cmml" xref="S3.Ex8.m1.2.2.1.1.2.1.1.1.1.1.1.3">𝜎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex8.m1.2c">\cal L(\sigma^{\Sigma}(\mbox{Supp}(\mu)))\subseteq\cal L(\mbox{Supp}(\mu^{% \sigma}))\,.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m1.2d">caligraphic_L ( italic_σ start_POSTSUPERSCRIPT caligraphic_Σ end_POSTSUPERSCRIPT ( Supp ( italic_μ ) ) ) ⊆ caligraphic_L ( Supp ( italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS4.6.p3"> <p class="ltx_p" id="S3.SS4.6.p3.14">Conversely (again using (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E6" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.6</span></a>)), any <math alttext="w^{\prime}\in\cal B^{*}" class="ltx_Math" display="inline" id="S3.SS4.6.p3.1.m1.1"><semantics id="S3.SS4.6.p3.1.m1.1a"><mrow id="S3.SS4.6.p3.1.m1.1.1" xref="S3.SS4.6.p3.1.m1.1.1.cmml"><msup id="S3.SS4.6.p3.1.m1.1.1.2" xref="S3.SS4.6.p3.1.m1.1.1.2.cmml"><mi id="S3.SS4.6.p3.1.m1.1.1.2.2" xref="S3.SS4.6.p3.1.m1.1.1.2.2.cmml">w</mi><mo id="S3.SS4.6.p3.1.m1.1.1.2.3" xref="S3.SS4.6.p3.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS4.6.p3.1.m1.1.1.1" xref="S3.SS4.6.p3.1.m1.1.1.1.cmml">∈</mo><msup id="S3.SS4.6.p3.1.m1.1.1.3" xref="S3.SS4.6.p3.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p3.1.m1.1.1.3.2" xref="S3.SS4.6.p3.1.m1.1.1.3.2.cmml">ℬ</mi><mo id="S3.SS4.6.p3.1.m1.1.1.3.3" xref="S3.SS4.6.p3.1.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.1.m1.1b"><apply id="S3.SS4.6.p3.1.m1.1.1.cmml" xref="S3.SS4.6.p3.1.m1.1.1"><in id="S3.SS4.6.p3.1.m1.1.1.1.cmml" xref="S3.SS4.6.p3.1.m1.1.1.1"></in><apply id="S3.SS4.6.p3.1.m1.1.1.2.cmml" xref="S3.SS4.6.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS4.6.p3.1.m1.1.1.2.1.cmml" xref="S3.SS4.6.p3.1.m1.1.1.2">superscript</csymbol><ci id="S3.SS4.6.p3.1.m1.1.1.2.2.cmml" xref="S3.SS4.6.p3.1.m1.1.1.2.2">𝑤</ci><ci id="S3.SS4.6.p3.1.m1.1.1.2.3.cmml" xref="S3.SS4.6.p3.1.m1.1.1.2.3">′</ci></apply><apply id="S3.SS4.6.p3.1.m1.1.1.3.cmml" xref="S3.SS4.6.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.6.p3.1.m1.1.1.3.1.cmml" xref="S3.SS4.6.p3.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS4.6.p3.1.m1.1.1.3.2.cmml" xref="S3.SS4.6.p3.1.m1.1.1.3.2">ℬ</ci><times id="S3.SS4.6.p3.1.m1.1.1.3.3.cmml" xref="S3.SS4.6.p3.1.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.1.m1.1c">w^{\prime}\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> belongs to <math alttext="\cal L(\mbox{Supp}(\mu^{\sigma}))" class="ltx_Math" display="inline" id="S3.SS4.6.p3.2.m2.1"><semantics id="S3.SS4.6.p3.2.m2.1a"><mrow id="S3.SS4.6.p3.2.m2.1.1" xref="S3.SS4.6.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p3.2.m2.1.1.3" xref="S3.SS4.6.p3.2.m2.1.1.3.cmml">ℒ</mi><mo id="S3.SS4.6.p3.2.m2.1.1.2" xref="S3.SS4.6.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.6.p3.2.m2.1.1.1.1" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.6.p3.2.m2.1.1.1.1.1" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.cmml"><mtext id="S3.SS4.6.p3.2.m2.1.1.1.1.1.3" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.3a.cmml">Supp</mtext><mo id="S3.SS4.6.p3.2.m2.1.1.1.1.1.2" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.2" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.3" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.3.cmml">σ</mi></msup><mo id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.2.m2.1b"><apply id="S3.SS4.6.p3.2.m2.1.1.cmml" xref="S3.SS4.6.p3.2.m2.1.1"><times id="S3.SS4.6.p3.2.m2.1.1.2.cmml" xref="S3.SS4.6.p3.2.m2.1.1.2"></times><ci id="S3.SS4.6.p3.2.m2.1.1.3.cmml" xref="S3.SS4.6.p3.2.m2.1.1.3">ℒ</ci><apply id="S3.SS4.6.p3.2.m2.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.2.m2.1.1.1.1"><times id="S3.SS4.6.p3.2.m2.1.1.1.1.1.2.cmml" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.2"></times><ci id="S3.SS4.6.p3.2.m2.1.1.1.1.1.3a.cmml" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.3"><mtext id="S3.SS4.6.p3.2.m2.1.1.1.1.1.3.cmml" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.3">Supp</mtext></ci><apply id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.2">𝜇</ci><ci id="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.3.cmml" xref="S3.SS4.6.p3.2.m2.1.1.1.1.1.1.1.1.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.2.m2.1c">\cal L(\mbox{Supp}(\mu^{\sigma}))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.2.m2.1d">caligraphic_L ( Supp ( italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ) )</annotation></semantics></math> if and only if <math alttext="\mu^{\sigma}(w^{\prime})>0" class="ltx_Math" display="inline" id="S3.SS4.6.p3.3.m3.1"><semantics id="S3.SS4.6.p3.3.m3.1a"><mrow id="S3.SS4.6.p3.3.m3.1.1" xref="S3.SS4.6.p3.3.m3.1.1.cmml"><mrow id="S3.SS4.6.p3.3.m3.1.1.1" xref="S3.SS4.6.p3.3.m3.1.1.1.cmml"><msup id="S3.SS4.6.p3.3.m3.1.1.1.3" xref="S3.SS4.6.p3.3.m3.1.1.1.3.cmml"><mi id="S3.SS4.6.p3.3.m3.1.1.1.3.2" xref="S3.SS4.6.p3.3.m3.1.1.1.3.2.cmml">μ</mi><mi id="S3.SS4.6.p3.3.m3.1.1.1.3.3" xref="S3.SS4.6.p3.3.m3.1.1.1.3.3.cmml">σ</mi></msup><mo id="S3.SS4.6.p3.3.m3.1.1.1.2" xref="S3.SS4.6.p3.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.6.p3.3.m3.1.1.1.1.1" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.6.p3.3.m3.1.1.1.1.1.1" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.cmml"><mi id="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.2" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.2.cmml">w</mi><mo id="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.3" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.SS4.6.p3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.3.m3.1.1.2" xref="S3.SS4.6.p3.3.m3.1.1.2.cmml">></mo><mn id="S3.SS4.6.p3.3.m3.1.1.3" xref="S3.SS4.6.p3.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.3.m3.1b"><apply id="S3.SS4.6.p3.3.m3.1.1.cmml" xref="S3.SS4.6.p3.3.m3.1.1"><gt id="S3.SS4.6.p3.3.m3.1.1.2.cmml" xref="S3.SS4.6.p3.3.m3.1.1.2"></gt><apply id="S3.SS4.6.p3.3.m3.1.1.1.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1"><times id="S3.SS4.6.p3.3.m3.1.1.1.2.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.2"></times><apply id="S3.SS4.6.p3.3.m3.1.1.1.3.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.6.p3.3.m3.1.1.1.3.1.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.3">superscript</csymbol><ci id="S3.SS4.6.p3.3.m3.1.1.1.3.2.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.3.2">𝜇</ci><ci id="S3.SS4.6.p3.3.m3.1.1.1.3.3.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.3.3">𝜎</ci></apply><apply id="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.2.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.2">𝑤</ci><ci id="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.3.cmml" xref="S3.SS4.6.p3.3.m3.1.1.1.1.1.1.3">′</ci></apply></apply><cn id="S3.SS4.6.p3.3.m3.1.1.3.cmml" type="integer" xref="S3.SS4.6.p3.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.3.m3.1c">\mu^{\sigma}(w^{\prime})>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.3.m3.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) > 0</annotation></semantics></math>. In this case there exists a word <math alttext="w\in\cal A_{\sigma}^{*}" class="ltx_Math" display="inline" id="S3.SS4.6.p3.4.m4.1"><semantics id="S3.SS4.6.p3.4.m4.1a"><mrow id="S3.SS4.6.p3.4.m4.1.1" xref="S3.SS4.6.p3.4.m4.1.1.cmml"><mi id="S3.SS4.6.p3.4.m4.1.1.2" xref="S3.SS4.6.p3.4.m4.1.1.2.cmml">w</mi><mo id="S3.SS4.6.p3.4.m4.1.1.1" xref="S3.SS4.6.p3.4.m4.1.1.1.cmml">∈</mo><msubsup id="S3.SS4.6.p3.4.m4.1.1.3" xref="S3.SS4.6.p3.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p3.4.m4.1.1.3.2.2" xref="S3.SS4.6.p3.4.m4.1.1.3.2.2.cmml">𝒜</mi><mi id="S3.SS4.6.p3.4.m4.1.1.3.2.3" xref="S3.SS4.6.p3.4.m4.1.1.3.2.3.cmml">σ</mi><mo id="S3.SS4.6.p3.4.m4.1.1.3.3" xref="S3.SS4.6.p3.4.m4.1.1.3.3.cmml">∗</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.4.m4.1b"><apply id="S3.SS4.6.p3.4.m4.1.1.cmml" xref="S3.SS4.6.p3.4.m4.1.1"><in id="S3.SS4.6.p3.4.m4.1.1.1.cmml" xref="S3.SS4.6.p3.4.m4.1.1.1"></in><ci id="S3.SS4.6.p3.4.m4.1.1.2.cmml" xref="S3.SS4.6.p3.4.m4.1.1.2">𝑤</ci><apply id="S3.SS4.6.p3.4.m4.1.1.3.cmml" xref="S3.SS4.6.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.6.p3.4.m4.1.1.3.1.cmml" xref="S3.SS4.6.p3.4.m4.1.1.3">superscript</csymbol><apply id="S3.SS4.6.p3.4.m4.1.1.3.2.cmml" xref="S3.SS4.6.p3.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.6.p3.4.m4.1.1.3.2.1.cmml" xref="S3.SS4.6.p3.4.m4.1.1.3">subscript</csymbol><ci id="S3.SS4.6.p3.4.m4.1.1.3.2.2.cmml" xref="S3.SS4.6.p3.4.m4.1.1.3.2.2">𝒜</ci><ci id="S3.SS4.6.p3.4.m4.1.1.3.2.3.cmml" xref="S3.SS4.6.p3.4.m4.1.1.3.2.3">𝜎</ci></apply><times id="S3.SS4.6.p3.4.m4.1.1.3.3.cmml" xref="S3.SS4.6.p3.4.m4.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.4.m4.1c">w\in\cal A_{\sigma}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.4.m4.1d">italic_w ∈ caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\alpha_{\sigma}(w)=w^{\prime}" class="ltx_Math" display="inline" id="S3.SS4.6.p3.5.m5.1"><semantics id="S3.SS4.6.p3.5.m5.1a"><mrow id="S3.SS4.6.p3.5.m5.1.2" xref="S3.SS4.6.p3.5.m5.1.2.cmml"><mrow id="S3.SS4.6.p3.5.m5.1.2.2" xref="S3.SS4.6.p3.5.m5.1.2.2.cmml"><msub id="S3.SS4.6.p3.5.m5.1.2.2.2" xref="S3.SS4.6.p3.5.m5.1.2.2.2.cmml"><mi id="S3.SS4.6.p3.5.m5.1.2.2.2.2" xref="S3.SS4.6.p3.5.m5.1.2.2.2.2.cmml">α</mi><mi id="S3.SS4.6.p3.5.m5.1.2.2.2.3" xref="S3.SS4.6.p3.5.m5.1.2.2.2.3.cmml">σ</mi></msub><mo id="S3.SS4.6.p3.5.m5.1.2.2.1" xref="S3.SS4.6.p3.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS4.6.p3.5.m5.1.2.2.3.2" xref="S3.SS4.6.p3.5.m5.1.2.2.cmml"><mo id="S3.SS4.6.p3.5.m5.1.2.2.3.2.1" stretchy="false" xref="S3.SS4.6.p3.5.m5.1.2.2.cmml">(</mo><mi id="S3.SS4.6.p3.5.m5.1.1" xref="S3.SS4.6.p3.5.m5.1.1.cmml">w</mi><mo id="S3.SS4.6.p3.5.m5.1.2.2.3.2.2" stretchy="false" xref="S3.SS4.6.p3.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.5.m5.1.2.1" xref="S3.SS4.6.p3.5.m5.1.2.1.cmml">=</mo><msup id="S3.SS4.6.p3.5.m5.1.2.3" xref="S3.SS4.6.p3.5.m5.1.2.3.cmml"><mi id="S3.SS4.6.p3.5.m5.1.2.3.2" xref="S3.SS4.6.p3.5.m5.1.2.3.2.cmml">w</mi><mo id="S3.SS4.6.p3.5.m5.1.2.3.3" xref="S3.SS4.6.p3.5.m5.1.2.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.5.m5.1b"><apply id="S3.SS4.6.p3.5.m5.1.2.cmml" xref="S3.SS4.6.p3.5.m5.1.2"><eq id="S3.SS4.6.p3.5.m5.1.2.1.cmml" xref="S3.SS4.6.p3.5.m5.1.2.1"></eq><apply id="S3.SS4.6.p3.5.m5.1.2.2.cmml" xref="S3.SS4.6.p3.5.m5.1.2.2"><times id="S3.SS4.6.p3.5.m5.1.2.2.1.cmml" xref="S3.SS4.6.p3.5.m5.1.2.2.1"></times><apply id="S3.SS4.6.p3.5.m5.1.2.2.2.cmml" xref="S3.SS4.6.p3.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.6.p3.5.m5.1.2.2.2.1.cmml" xref="S3.SS4.6.p3.5.m5.1.2.2.2">subscript</csymbol><ci id="S3.SS4.6.p3.5.m5.1.2.2.2.2.cmml" xref="S3.SS4.6.p3.5.m5.1.2.2.2.2">𝛼</ci><ci id="S3.SS4.6.p3.5.m5.1.2.2.2.3.cmml" xref="S3.SS4.6.p3.5.m5.1.2.2.2.3">𝜎</ci></apply><ci id="S3.SS4.6.p3.5.m5.1.1.cmml" xref="S3.SS4.6.p3.5.m5.1.1">𝑤</ci></apply><apply id="S3.SS4.6.p3.5.m5.1.2.3.cmml" xref="S3.SS4.6.p3.5.m5.1.2.3"><csymbol cd="ambiguous" id="S3.SS4.6.p3.5.m5.1.2.3.1.cmml" xref="S3.SS4.6.p3.5.m5.1.2.3">superscript</csymbol><ci id="S3.SS4.6.p3.5.m5.1.2.3.2.cmml" xref="S3.SS4.6.p3.5.m5.1.2.3.2">𝑤</ci><ci id="S3.SS4.6.p3.5.m5.1.2.3.3.cmml" xref="S3.SS4.6.p3.5.m5.1.2.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.5.m5.1c">\alpha_{\sigma}(w)=w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.5.m5.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_w ) = italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and with <math alttext="\mu_{\ell_{\sigma}}(w)>0" class="ltx_Math" display="inline" id="S3.SS4.6.p3.6.m6.1"><semantics id="S3.SS4.6.p3.6.m6.1a"><mrow id="S3.SS4.6.p3.6.m6.1.2" xref="S3.SS4.6.p3.6.m6.1.2.cmml"><mrow id="S3.SS4.6.p3.6.m6.1.2.2" xref="S3.SS4.6.p3.6.m6.1.2.2.cmml"><msub id="S3.SS4.6.p3.6.m6.1.2.2.2" xref="S3.SS4.6.p3.6.m6.1.2.2.2.cmml"><mi id="S3.SS4.6.p3.6.m6.1.2.2.2.2" xref="S3.SS4.6.p3.6.m6.1.2.2.2.2.cmml">μ</mi><msub id="S3.SS4.6.p3.6.m6.1.2.2.2.3" xref="S3.SS4.6.p3.6.m6.1.2.2.2.3.cmml"><mi id="S3.SS4.6.p3.6.m6.1.2.2.2.3.2" mathvariant="normal" xref="S3.SS4.6.p3.6.m6.1.2.2.2.3.2.cmml">ℓ</mi><mi id="S3.SS4.6.p3.6.m6.1.2.2.2.3.3" xref="S3.SS4.6.p3.6.m6.1.2.2.2.3.3.cmml">σ</mi></msub></msub><mo id="S3.SS4.6.p3.6.m6.1.2.2.1" xref="S3.SS4.6.p3.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS4.6.p3.6.m6.1.2.2.3.2" xref="S3.SS4.6.p3.6.m6.1.2.2.cmml"><mo id="S3.SS4.6.p3.6.m6.1.2.2.3.2.1" stretchy="false" xref="S3.SS4.6.p3.6.m6.1.2.2.cmml">(</mo><mi id="S3.SS4.6.p3.6.m6.1.1" xref="S3.SS4.6.p3.6.m6.1.1.cmml">w</mi><mo id="S3.SS4.6.p3.6.m6.1.2.2.3.2.2" stretchy="false" xref="S3.SS4.6.p3.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.6.m6.1.2.1" xref="S3.SS4.6.p3.6.m6.1.2.1.cmml">></mo><mn id="S3.SS4.6.p3.6.m6.1.2.3" xref="S3.SS4.6.p3.6.m6.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.6.m6.1b"><apply id="S3.SS4.6.p3.6.m6.1.2.cmml" xref="S3.SS4.6.p3.6.m6.1.2"><gt id="S3.SS4.6.p3.6.m6.1.2.1.cmml" xref="S3.SS4.6.p3.6.m6.1.2.1"></gt><apply id="S3.SS4.6.p3.6.m6.1.2.2.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2"><times id="S3.SS4.6.p3.6.m6.1.2.2.1.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2.1"></times><apply id="S3.SS4.6.p3.6.m6.1.2.2.2.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.6.p3.6.m6.1.2.2.2.1.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2.2">subscript</csymbol><ci id="S3.SS4.6.p3.6.m6.1.2.2.2.2.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2.2.2">𝜇</ci><apply id="S3.SS4.6.p3.6.m6.1.2.2.2.3.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS4.6.p3.6.m6.1.2.2.2.3.1.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2.2.3">subscript</csymbol><ci id="S3.SS4.6.p3.6.m6.1.2.2.2.3.2.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2.2.3.2">ℓ</ci><ci id="S3.SS4.6.p3.6.m6.1.2.2.2.3.3.cmml" xref="S3.SS4.6.p3.6.m6.1.2.2.2.3.3">𝜎</ci></apply></apply><ci id="S3.SS4.6.p3.6.m6.1.1.cmml" xref="S3.SS4.6.p3.6.m6.1.1">𝑤</ci></apply><cn id="S3.SS4.6.p3.6.m6.1.2.3.cmml" type="integer" xref="S3.SS4.6.p3.6.m6.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.6.m6.1c">\mu_{\ell_{\sigma}}(w)>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.6.m6.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_w ) > 0</annotation></semantics></math>. Hence (see formula (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E5" title="In Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.5</span></a>)) there is a word <math alttext="\widehat{w}\in\cal A^{*}" class="ltx_Math" display="inline" id="S3.SS4.6.p3.7.m7.1"><semantics id="S3.SS4.6.p3.7.m7.1a"><mrow id="S3.SS4.6.p3.7.m7.1.1" xref="S3.SS4.6.p3.7.m7.1.1.cmml"><mover accent="true" id="S3.SS4.6.p3.7.m7.1.1.2" xref="S3.SS4.6.p3.7.m7.1.1.2.cmml"><mi id="S3.SS4.6.p3.7.m7.1.1.2.2" xref="S3.SS4.6.p3.7.m7.1.1.2.2.cmml">w</mi><mo id="S3.SS4.6.p3.7.m7.1.1.2.1" xref="S3.SS4.6.p3.7.m7.1.1.2.1.cmml">^</mo></mover><mo id="S3.SS4.6.p3.7.m7.1.1.1" xref="S3.SS4.6.p3.7.m7.1.1.1.cmml">∈</mo><msup id="S3.SS4.6.p3.7.m7.1.1.3" xref="S3.SS4.6.p3.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p3.7.m7.1.1.3.2" xref="S3.SS4.6.p3.7.m7.1.1.3.2.cmml">𝒜</mi><mo id="S3.SS4.6.p3.7.m7.1.1.3.3" xref="S3.SS4.6.p3.7.m7.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.7.m7.1b"><apply id="S3.SS4.6.p3.7.m7.1.1.cmml" xref="S3.SS4.6.p3.7.m7.1.1"><in id="S3.SS4.6.p3.7.m7.1.1.1.cmml" xref="S3.SS4.6.p3.7.m7.1.1.1"></in><apply id="S3.SS4.6.p3.7.m7.1.1.2.cmml" xref="S3.SS4.6.p3.7.m7.1.1.2"><ci id="S3.SS4.6.p3.7.m7.1.1.2.1.cmml" xref="S3.SS4.6.p3.7.m7.1.1.2.1">^</ci><ci id="S3.SS4.6.p3.7.m7.1.1.2.2.cmml" xref="S3.SS4.6.p3.7.m7.1.1.2.2">𝑤</ci></apply><apply id="S3.SS4.6.p3.7.m7.1.1.3.cmml" xref="S3.SS4.6.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.6.p3.7.m7.1.1.3.1.cmml" xref="S3.SS4.6.p3.7.m7.1.1.3">superscript</csymbol><ci id="S3.SS4.6.p3.7.m7.1.1.3.2.cmml" xref="S3.SS4.6.p3.7.m7.1.1.3.2">𝒜</ci><times id="S3.SS4.6.p3.7.m7.1.1.3.3.cmml" xref="S3.SS4.6.p3.7.m7.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.7.m7.1c">\widehat{w}\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.7.m7.1d">over^ start_ARG italic_w end_ARG ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\mu(\widehat{w})>0" class="ltx_Math" display="inline" id="S3.SS4.6.p3.8.m8.1"><semantics id="S3.SS4.6.p3.8.m8.1a"><mrow id="S3.SS4.6.p3.8.m8.1.2" xref="S3.SS4.6.p3.8.m8.1.2.cmml"><mrow id="S3.SS4.6.p3.8.m8.1.2.2" xref="S3.SS4.6.p3.8.m8.1.2.2.cmml"><mi id="S3.SS4.6.p3.8.m8.1.2.2.2" xref="S3.SS4.6.p3.8.m8.1.2.2.2.cmml">μ</mi><mo id="S3.SS4.6.p3.8.m8.1.2.2.1" xref="S3.SS4.6.p3.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS4.6.p3.8.m8.1.2.2.3.2" xref="S3.SS4.6.p3.8.m8.1.1.cmml"><mo id="S3.SS4.6.p3.8.m8.1.2.2.3.2.1" stretchy="false" xref="S3.SS4.6.p3.8.m8.1.1.cmml">(</mo><mover accent="true" id="S3.SS4.6.p3.8.m8.1.1" xref="S3.SS4.6.p3.8.m8.1.1.cmml"><mi id="S3.SS4.6.p3.8.m8.1.1.2" xref="S3.SS4.6.p3.8.m8.1.1.2.cmml">w</mi><mo id="S3.SS4.6.p3.8.m8.1.1.1" xref="S3.SS4.6.p3.8.m8.1.1.1.cmml">^</mo></mover><mo id="S3.SS4.6.p3.8.m8.1.2.2.3.2.2" stretchy="false" xref="S3.SS4.6.p3.8.m8.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.8.m8.1.2.1" xref="S3.SS4.6.p3.8.m8.1.2.1.cmml">></mo><mn id="S3.SS4.6.p3.8.m8.1.2.3" xref="S3.SS4.6.p3.8.m8.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.8.m8.1b"><apply id="S3.SS4.6.p3.8.m8.1.2.cmml" xref="S3.SS4.6.p3.8.m8.1.2"><gt id="S3.SS4.6.p3.8.m8.1.2.1.cmml" xref="S3.SS4.6.p3.8.m8.1.2.1"></gt><apply id="S3.SS4.6.p3.8.m8.1.2.2.cmml" xref="S3.SS4.6.p3.8.m8.1.2.2"><times id="S3.SS4.6.p3.8.m8.1.2.2.1.cmml" xref="S3.SS4.6.p3.8.m8.1.2.2.1"></times><ci id="S3.SS4.6.p3.8.m8.1.2.2.2.cmml" xref="S3.SS4.6.p3.8.m8.1.2.2.2">𝜇</ci><apply id="S3.SS4.6.p3.8.m8.1.1.cmml" xref="S3.SS4.6.p3.8.m8.1.2.2.3.2"><ci id="S3.SS4.6.p3.8.m8.1.1.1.cmml" xref="S3.SS4.6.p3.8.m8.1.1.1">^</ci><ci id="S3.SS4.6.p3.8.m8.1.1.2.cmml" xref="S3.SS4.6.p3.8.m8.1.1.2">𝑤</ci></apply></apply><cn id="S3.SS4.6.p3.8.m8.1.2.3.cmml" type="integer" xref="S3.SS4.6.p3.8.m8.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.8.m8.1c">\mu(\widehat{w})>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.8.m8.1d">italic_μ ( over^ start_ARG italic_w end_ARG ) > 0</annotation></semantics></math> such that <math alttext="\pi_{\sigma}(\widehat{w})" class="ltx_Math" display="inline" id="S3.SS4.6.p3.9.m9.1"><semantics id="S3.SS4.6.p3.9.m9.1a"><mrow id="S3.SS4.6.p3.9.m9.1.2" xref="S3.SS4.6.p3.9.m9.1.2.cmml"><msub id="S3.SS4.6.p3.9.m9.1.2.2" xref="S3.SS4.6.p3.9.m9.1.2.2.cmml"><mi id="S3.SS4.6.p3.9.m9.1.2.2.2" xref="S3.SS4.6.p3.9.m9.1.2.2.2.cmml">π</mi><mi id="S3.SS4.6.p3.9.m9.1.2.2.3" xref="S3.SS4.6.p3.9.m9.1.2.2.3.cmml">σ</mi></msub><mo id="S3.SS4.6.p3.9.m9.1.2.1" xref="S3.SS4.6.p3.9.m9.1.2.1.cmml">⁢</mo><mrow id="S3.SS4.6.p3.9.m9.1.2.3.2" xref="S3.SS4.6.p3.9.m9.1.1.cmml"><mo id="S3.SS4.6.p3.9.m9.1.2.3.2.1" stretchy="false" xref="S3.SS4.6.p3.9.m9.1.1.cmml">(</mo><mover accent="true" id="S3.SS4.6.p3.9.m9.1.1" xref="S3.SS4.6.p3.9.m9.1.1.cmml"><mi id="S3.SS4.6.p3.9.m9.1.1.2" xref="S3.SS4.6.p3.9.m9.1.1.2.cmml">w</mi><mo id="S3.SS4.6.p3.9.m9.1.1.1" xref="S3.SS4.6.p3.9.m9.1.1.1.cmml">^</mo></mover><mo id="S3.SS4.6.p3.9.m9.1.2.3.2.2" stretchy="false" xref="S3.SS4.6.p3.9.m9.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.9.m9.1b"><apply id="S3.SS4.6.p3.9.m9.1.2.cmml" xref="S3.SS4.6.p3.9.m9.1.2"><times id="S3.SS4.6.p3.9.m9.1.2.1.cmml" xref="S3.SS4.6.p3.9.m9.1.2.1"></times><apply id="S3.SS4.6.p3.9.m9.1.2.2.cmml" xref="S3.SS4.6.p3.9.m9.1.2.2"><csymbol cd="ambiguous" id="S3.SS4.6.p3.9.m9.1.2.2.1.cmml" xref="S3.SS4.6.p3.9.m9.1.2.2">subscript</csymbol><ci id="S3.SS4.6.p3.9.m9.1.2.2.2.cmml" xref="S3.SS4.6.p3.9.m9.1.2.2.2">𝜋</ci><ci id="S3.SS4.6.p3.9.m9.1.2.2.3.cmml" xref="S3.SS4.6.p3.9.m9.1.2.2.3">𝜎</ci></apply><apply id="S3.SS4.6.p3.9.m9.1.1.cmml" xref="S3.SS4.6.p3.9.m9.1.2.3.2"><ci id="S3.SS4.6.p3.9.m9.1.1.1.cmml" xref="S3.SS4.6.p3.9.m9.1.1.1">^</ci><ci id="S3.SS4.6.p3.9.m9.1.1.2.cmml" xref="S3.SS4.6.p3.9.m9.1.1.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.9.m9.1c">\pi_{\sigma}(\widehat{w})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.9.m9.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over^ start_ARG italic_w end_ARG )</annotation></semantics></math> contains <math alttext="w" class="ltx_Math" display="inline" id="S3.SS4.6.p3.10.m10.1"><semantics id="S3.SS4.6.p3.10.m10.1a"><mi id="S3.SS4.6.p3.10.m10.1.1" xref="S3.SS4.6.p3.10.m10.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.10.m10.1b"><ci id="S3.SS4.6.p3.10.m10.1.1.cmml" xref="S3.SS4.6.p3.10.m10.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.10.m10.1c">w</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.10.m10.1d">italic_w</annotation></semantics></math> as factor. It follows that <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S3.SS4.6.p3.11.m11.1"><semantics id="S3.SS4.6.p3.11.m11.1a"><msup id="S3.SS4.6.p3.11.m11.1.1" xref="S3.SS4.6.p3.11.m11.1.1.cmml"><mi id="S3.SS4.6.p3.11.m11.1.1.2" xref="S3.SS4.6.p3.11.m11.1.1.2.cmml">w</mi><mo id="S3.SS4.6.p3.11.m11.1.1.3" xref="S3.SS4.6.p3.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.11.m11.1b"><apply id="S3.SS4.6.p3.11.m11.1.1.cmml" xref="S3.SS4.6.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S3.SS4.6.p3.11.m11.1.1.1.cmml" xref="S3.SS4.6.p3.11.m11.1.1">superscript</csymbol><ci id="S3.SS4.6.p3.11.m11.1.1.2.cmml" xref="S3.SS4.6.p3.11.m11.1.1.2">𝑤</ci><ci id="S3.SS4.6.p3.11.m11.1.1.3.cmml" xref="S3.SS4.6.p3.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.11.m11.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.11.m11.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is a factor of <math alttext="\sigma(\widehat{w})" class="ltx_Math" display="inline" id="S3.SS4.6.p3.12.m12.1"><semantics id="S3.SS4.6.p3.12.m12.1a"><mrow id="S3.SS4.6.p3.12.m12.1.2" xref="S3.SS4.6.p3.12.m12.1.2.cmml"><mi id="S3.SS4.6.p3.12.m12.1.2.2" xref="S3.SS4.6.p3.12.m12.1.2.2.cmml">σ</mi><mo id="S3.SS4.6.p3.12.m12.1.2.1" xref="S3.SS4.6.p3.12.m12.1.2.1.cmml">⁢</mo><mrow id="S3.SS4.6.p3.12.m12.1.2.3.2" xref="S3.SS4.6.p3.12.m12.1.1.cmml"><mo id="S3.SS4.6.p3.12.m12.1.2.3.2.1" stretchy="false" xref="S3.SS4.6.p3.12.m12.1.1.cmml">(</mo><mover accent="true" id="S3.SS4.6.p3.12.m12.1.1" xref="S3.SS4.6.p3.12.m12.1.1.cmml"><mi id="S3.SS4.6.p3.12.m12.1.1.2" xref="S3.SS4.6.p3.12.m12.1.1.2.cmml">w</mi><mo id="S3.SS4.6.p3.12.m12.1.1.1" xref="S3.SS4.6.p3.12.m12.1.1.1.cmml">^</mo></mover><mo id="S3.SS4.6.p3.12.m12.1.2.3.2.2" stretchy="false" xref="S3.SS4.6.p3.12.m12.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.12.m12.1b"><apply id="S3.SS4.6.p3.12.m12.1.2.cmml" xref="S3.SS4.6.p3.12.m12.1.2"><times id="S3.SS4.6.p3.12.m12.1.2.1.cmml" xref="S3.SS4.6.p3.12.m12.1.2.1"></times><ci id="S3.SS4.6.p3.12.m12.1.2.2.cmml" xref="S3.SS4.6.p3.12.m12.1.2.2">𝜎</ci><apply id="S3.SS4.6.p3.12.m12.1.1.cmml" xref="S3.SS4.6.p3.12.m12.1.2.3.2"><ci id="S3.SS4.6.p3.12.m12.1.1.1.cmml" xref="S3.SS4.6.p3.12.m12.1.1.1">^</ci><ci id="S3.SS4.6.p3.12.m12.1.1.2.cmml" xref="S3.SS4.6.p3.12.m12.1.1.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.12.m12.1c">\sigma(\widehat{w})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.12.m12.1d">italic_σ ( over^ start_ARG italic_w end_ARG )</annotation></semantics></math> and that <math alttext="\widehat{w}\in\cal L(\mbox{Supp}(\mu))" class="ltx_Math" display="inline" id="S3.SS4.6.p3.13.m13.2"><semantics id="S3.SS4.6.p3.13.m13.2a"><mrow id="S3.SS4.6.p3.13.m13.2.2" xref="S3.SS4.6.p3.13.m13.2.2.cmml"><mover accent="true" id="S3.SS4.6.p3.13.m13.2.2.3" xref="S3.SS4.6.p3.13.m13.2.2.3.cmml"><mi id="S3.SS4.6.p3.13.m13.2.2.3.2" xref="S3.SS4.6.p3.13.m13.2.2.3.2.cmml">w</mi><mo id="S3.SS4.6.p3.13.m13.2.2.3.1" xref="S3.SS4.6.p3.13.m13.2.2.3.1.cmml">^</mo></mover><mo id="S3.SS4.6.p3.13.m13.2.2.2" xref="S3.SS4.6.p3.13.m13.2.2.2.cmml">∈</mo><mrow id="S3.SS4.6.p3.13.m13.2.2.1" xref="S3.SS4.6.p3.13.m13.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p3.13.m13.2.2.1.3" xref="S3.SS4.6.p3.13.m13.2.2.1.3.cmml">ℒ</mi><mo id="S3.SS4.6.p3.13.m13.2.2.1.2" xref="S3.SS4.6.p3.13.m13.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS4.6.p3.13.m13.2.2.1.1.1" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.13.m13.2.2.1.1.1.2" stretchy="false" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.cmml"><mtext id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.2" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.2a.cmml">Supp</mtext><mo id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.1" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.3.2" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.cmml">(</mo><mi id="S3.SS4.6.p3.13.m13.1.1" xref="S3.SS4.6.p3.13.m13.1.1.cmml">μ</mi><mo id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.13.m13.2.2.1.1.1.3" stretchy="false" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.13.m13.2b"><apply id="S3.SS4.6.p3.13.m13.2.2.cmml" xref="S3.SS4.6.p3.13.m13.2.2"><in id="S3.SS4.6.p3.13.m13.2.2.2.cmml" xref="S3.SS4.6.p3.13.m13.2.2.2"></in><apply id="S3.SS4.6.p3.13.m13.2.2.3.cmml" xref="S3.SS4.6.p3.13.m13.2.2.3"><ci id="S3.SS4.6.p3.13.m13.2.2.3.1.cmml" xref="S3.SS4.6.p3.13.m13.2.2.3.1">^</ci><ci id="S3.SS4.6.p3.13.m13.2.2.3.2.cmml" xref="S3.SS4.6.p3.13.m13.2.2.3.2">𝑤</ci></apply><apply id="S3.SS4.6.p3.13.m13.2.2.1.cmml" xref="S3.SS4.6.p3.13.m13.2.2.1"><times id="S3.SS4.6.p3.13.m13.2.2.1.2.cmml" xref="S3.SS4.6.p3.13.m13.2.2.1.2"></times><ci id="S3.SS4.6.p3.13.m13.2.2.1.3.cmml" xref="S3.SS4.6.p3.13.m13.2.2.1.3">ℒ</ci><apply id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.cmml" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1"><times id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.1"></times><ci id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.2a.cmml" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.2"><mtext id="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.2.cmml" xref="S3.SS4.6.p3.13.m13.2.2.1.1.1.1.2">Supp</mtext></ci><ci id="S3.SS4.6.p3.13.m13.1.1.cmml" xref="S3.SS4.6.p3.13.m13.1.1">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.13.m13.2c">\widehat{w}\in\cal L(\mbox{Supp}(\mu))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.13.m13.2d">over^ start_ARG italic_w end_ARG ∈ caligraphic_L ( Supp ( italic_μ ) )</annotation></semantics></math>. Since <math alttext="\sigma(\cal L(\mbox{Supp}(\mu)))\subseteq\cal L(\sigma^{\Sigma}(\mbox{Supp}(% \mu)))" class="ltx_Math" display="inline" id="S3.SS4.6.p3.14.m14.4"><semantics id="S3.SS4.6.p3.14.m14.4a"><mrow id="S3.SS4.6.p3.14.m14.4.4" xref="S3.SS4.6.p3.14.m14.4.4.cmml"><mrow id="S3.SS4.6.p3.14.m14.3.3.1" xref="S3.SS4.6.p3.14.m14.3.3.1.cmml"><mi id="S3.SS4.6.p3.14.m14.3.3.1.3" xref="S3.SS4.6.p3.14.m14.3.3.1.3.cmml">σ</mi><mo id="S3.SS4.6.p3.14.m14.3.3.1.2" xref="S3.SS4.6.p3.14.m14.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS4.6.p3.14.m14.3.3.1.1.1" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.14.m14.3.3.1.1.1.2" stretchy="false" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.3" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.3.cmml">ℒ</mi><mo id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.2" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.cmml"><mtext id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.2" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.2a.cmml">Supp</mtext><mo id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.1" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.3.2" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.cmml">(</mo><mi id="S3.SS4.6.p3.14.m14.1.1" xref="S3.SS4.6.p3.14.m14.1.1.cmml">μ</mi><mo id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.14.m14.3.3.1.1.1.3" stretchy="false" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.14.m14.4.4.3" xref="S3.SS4.6.p3.14.m14.4.4.3.cmml">⊆</mo><mrow id="S3.SS4.6.p3.14.m14.4.4.2" xref="S3.SS4.6.p3.14.m14.4.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p3.14.m14.4.4.2.3" xref="S3.SS4.6.p3.14.m14.4.4.2.3.cmml">ℒ</mi><mo id="S3.SS4.6.p3.14.m14.4.4.2.2" xref="S3.SS4.6.p3.14.m14.4.4.2.2.cmml">⁢</mo><mrow id="S3.SS4.6.p3.14.m14.4.4.2.1.1" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.cmml"><mo id="S3.SS4.6.p3.14.m14.4.4.2.1.1.2" stretchy="false" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.cmml">(</mo><mrow id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.cmml"><msup id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.cmml"><mi id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.2" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.2.cmml">σ</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.3" mathvariant="script" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.3.cmml">Σ</mi></msup><mo id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.2" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.cmml"><mtext id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.2" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.2a.cmml">Supp</mtext><mo id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.1" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.3.2" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.cmml"><mo id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.cmml">(</mo><mi id="S3.SS4.6.p3.14.m14.2.2" xref="S3.SS4.6.p3.14.m14.2.2.cmml">μ</mi><mo id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.6.p3.14.m14.4.4.2.1.1.3" stretchy="false" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.6.p3.14.m14.4b"><apply id="S3.SS4.6.p3.14.m14.4.4.cmml" xref="S3.SS4.6.p3.14.m14.4.4"><subset id="S3.SS4.6.p3.14.m14.4.4.3.cmml" xref="S3.SS4.6.p3.14.m14.4.4.3"></subset><apply id="S3.SS4.6.p3.14.m14.3.3.1.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1"><times id="S3.SS4.6.p3.14.m14.3.3.1.2.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.2"></times><ci id="S3.SS4.6.p3.14.m14.3.3.1.3.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.3">𝜎</ci><apply id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1"><times id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.2.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.2"></times><ci id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.3.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.3">ℒ</ci><apply id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1"><times id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.1"></times><ci id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.2a.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.2"><mtext id="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.6.p3.14.m14.3.3.1.1.1.1.1.1.1.2">Supp</mtext></ci><ci id="S3.SS4.6.p3.14.m14.1.1.cmml" xref="S3.SS4.6.p3.14.m14.1.1">𝜇</ci></apply></apply></apply><apply id="S3.SS4.6.p3.14.m14.4.4.2.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2"><times id="S3.SS4.6.p3.14.m14.4.4.2.2.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.2"></times><ci id="S3.SS4.6.p3.14.m14.4.4.2.3.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.3">ℒ</ci><apply id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1"><times id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.2.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.2"></times><apply id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.1.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3">superscript</csymbol><ci id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.2.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.2">𝜎</ci><ci id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.3.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.3.3">script-Σ</ci></apply><apply id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1"><times id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.1.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.1"></times><ci id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.2a.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.2"><mtext id="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.2.cmml" xref="S3.SS4.6.p3.14.m14.4.4.2.1.1.1.1.1.1.2">Supp</mtext></ci><ci id="S3.SS4.6.p3.14.m14.2.2.cmml" xref="S3.SS4.6.p3.14.m14.2.2">𝜇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.6.p3.14.m14.4c">\sigma(\cal L(\mbox{Supp}(\mu)))\subseteq\cal L(\sigma^{\Sigma}(\mbox{Supp}(% \mu)))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.6.p3.14.m14.4d">italic_σ ( caligraphic_L ( Supp ( italic_μ ) ) ) ⊆ caligraphic_L ( italic_σ start_POSTSUPERSCRIPT caligraphic_Σ end_POSTSUPERSCRIPT ( Supp ( italic_μ ) ) )</annotation></semantics></math>, this implies</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\cal L(\mbox{Supp}(\mu^{\sigma}))\subseteq\cal L(\sigma^{\Sigma}(\mbox{Supp}(% \mu)))\,." class="ltx_Math" display="block" id="S3.Ex9.m1.2"><semantics id="S3.Ex9.m1.2a"><mrow id="S3.Ex9.m1.2.2.1" xref="S3.Ex9.m1.2.2.1.1.cmml"><mrow id="S3.Ex9.m1.2.2.1.1" xref="S3.Ex9.m1.2.2.1.1.cmml"><mrow id="S3.Ex9.m1.2.2.1.1.1" xref="S3.Ex9.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex9.m1.2.2.1.1.1.3" xref="S3.Ex9.m1.2.2.1.1.1.3.cmml">ℒ</mi><mo id="S3.Ex9.m1.2.2.1.1.1.2" xref="S3.Ex9.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex9.m1.2.2.1.1.1.1.1" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S3.Ex9.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex9.m1.2.2.1.1.1.1.1.1" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.cmml"><mtext id="S3.Ex9.m1.2.2.1.1.1.1.1.1.3" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.3a.cmml">Supp</mtext><mo id="S3.Ex9.m1.2.2.1.1.1.1.1.1.2" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.3.cmml">σ</mi></msup><mo id="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.2.2.1.1.1.1.1.3" stretchy="false" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.2.2.1.1.3" xref="S3.Ex9.m1.2.2.1.1.3.cmml">⊆</mo><mrow id="S3.Ex9.m1.2.2.1.1.2" xref="S3.Ex9.m1.2.2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex9.m1.2.2.1.1.2.3" xref="S3.Ex9.m1.2.2.1.1.2.3.cmml">ℒ</mi><mo id="S3.Ex9.m1.2.2.1.1.2.2" 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id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.2" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.2a.cmml">Supp</mtext><mo id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.1" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.3.2" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.cmml"><mo id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.cmml">(</mo><mi id="S3.Ex9.m1.1.1" xref="S3.Ex9.m1.1.1.cmml">μ</mi><mo id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.3" stretchy="false" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.2.2.1.1.2.1.1.3" stretchy="false" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex9.m1.2.2.1.2" lspace="0.170em" xref="S3.Ex9.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex9.m1.2b"><apply 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id="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex9.m1.2.2.1.1.1.1.1.1.1.1.1.3">𝜎</ci></apply></apply></apply><apply id="S3.Ex9.m1.2.2.1.1.2.cmml" xref="S3.Ex9.m1.2.2.1.1.2"><times id="S3.Ex9.m1.2.2.1.1.2.2.cmml" xref="S3.Ex9.m1.2.2.1.1.2.2"></times><ci id="S3.Ex9.m1.2.2.1.1.2.3.cmml" xref="S3.Ex9.m1.2.2.1.1.2.3">ℒ</ci><apply id="S3.Ex9.m1.2.2.1.1.2.1.1.1.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1"><times id="S3.Ex9.m1.2.2.1.1.2.1.1.1.2.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.2"></times><apply id="S3.Ex9.m1.2.2.1.1.2.1.1.1.3.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex9.m1.2.2.1.1.2.1.1.1.3.1.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.3">superscript</csymbol><ci id="S3.Ex9.m1.2.2.1.1.2.1.1.1.3.2.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.3.2">𝜎</ci><ci id="S3.Ex9.m1.2.2.1.1.2.1.1.1.3.3.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.3.3">script-Σ</ci></apply><apply id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1"><times id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.1.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.1"></times><ci id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.2a.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.2"><mtext id="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.2.cmml" xref="S3.Ex9.m1.2.2.1.1.2.1.1.1.1.1.1.2">Supp</mtext></ci><ci id="S3.Ex9.m1.1.1.cmml" xref="S3.Ex9.m1.1.1">𝜇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex9.m1.2c">\cal L(\mbox{Supp}(\mu^{\sigma}))\subseteq\cal L(\sigma^{\Sigma}(\mbox{Supp}(% \mu)))\,.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex9.m1.2d">caligraphic_L ( Supp ( italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ) ) ⊆ caligraphic_L ( italic_σ start_POSTSUPERSCRIPT caligraphic_Σ end_POSTSUPERSCRIPT ( Supp ( italic_μ ) ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS4.7.p4"> <p class="ltx_p" id="S3.SS4.7.p4.3">Hence we have <math alttext="\cal L(\mbox{Supp}(\mu^{\sigma}))=\cal L(\sigma^{\Sigma}(\mbox{Supp}(\mu)))" class="ltx_Math" display="inline" id="S3.SS4.7.p4.1.m1.3"><semantics id="S3.SS4.7.p4.1.m1.3a"><mrow id="S3.SS4.7.p4.1.m1.3.3" xref="S3.SS4.7.p4.1.m1.3.3.cmml"><mrow id="S3.SS4.7.p4.1.m1.2.2.1" xref="S3.SS4.7.p4.1.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.7.p4.1.m1.2.2.1.3" xref="S3.SS4.7.p4.1.m1.2.2.1.3.cmml">ℒ</mi><mo id="S3.SS4.7.p4.1.m1.2.2.1.2" xref="S3.SS4.7.p4.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS4.7.p4.1.m1.2.2.1.1.1" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.cmml"><mo id="S3.SS4.7.p4.1.m1.2.2.1.1.1.2" stretchy="false" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.cmml"><mtext id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.3" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.3a.cmml">Supp</mtext><mo id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.2" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.2" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.3" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.3.cmml">σ</mi></msup><mo id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.7.p4.1.m1.2.2.1.1.1.3" stretchy="false" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.7.p4.1.m1.3.3.3" xref="S3.SS4.7.p4.1.m1.3.3.3.cmml">=</mo><mrow id="S3.SS4.7.p4.1.m1.3.3.2" xref="S3.SS4.7.p4.1.m1.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.7.p4.1.m1.3.3.2.3" xref="S3.SS4.7.p4.1.m1.3.3.2.3.cmml">ℒ</mi><mo id="S3.SS4.7.p4.1.m1.3.3.2.2" xref="S3.SS4.7.p4.1.m1.3.3.2.2.cmml">⁢</mo><mrow id="S3.SS4.7.p4.1.m1.3.3.2.1.1" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.cmml"><mo id="S3.SS4.7.p4.1.m1.3.3.2.1.1.2" stretchy="false" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.cmml">(</mo><mrow id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.cmml"><msup id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.cmml"><mi id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.2" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.2.cmml">σ</mi><mi class="ltx_font_mathcaligraphic" id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.3" mathvariant="script" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.3.cmml">Σ</mi></msup><mo id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.2" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.cmml"><mo id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.cmml"><mtext id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.2" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.2a.cmml">Supp</mtext><mo id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.1" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.3.2" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.cmml"><mo id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.cmml">(</mo><mi id="S3.SS4.7.p4.1.m1.1.1" xref="S3.SS4.7.p4.1.m1.1.1.cmml">μ</mi><mo id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.7.p4.1.m1.3.3.2.1.1.3" stretchy="false" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p4.1.m1.3b"><apply id="S3.SS4.7.p4.1.m1.3.3.cmml" xref="S3.SS4.7.p4.1.m1.3.3"><eq id="S3.SS4.7.p4.1.m1.3.3.3.cmml" xref="S3.SS4.7.p4.1.m1.3.3.3"></eq><apply id="S3.SS4.7.p4.1.m1.2.2.1.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1"><times id="S3.SS4.7.p4.1.m1.2.2.1.2.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.2"></times><ci id="S3.SS4.7.p4.1.m1.2.2.1.3.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.3">ℒ</ci><apply id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1"><times id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.2.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.2"></times><ci id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.3a.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.3"><mtext id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.3.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.3">Supp</mtext></ci><apply id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1">superscript</csymbol><ci id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.2">𝜇</ci><ci id="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S3.SS4.7.p4.1.m1.2.2.1.1.1.1.1.1.1.3">𝜎</ci></apply></apply></apply><apply id="S3.SS4.7.p4.1.m1.3.3.2.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2"><times id="S3.SS4.7.p4.1.m1.3.3.2.2.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.2"></times><ci id="S3.SS4.7.p4.1.m1.3.3.2.3.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.3">ℒ</ci><apply id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1"><times id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.2.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.2"></times><apply id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.1.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3">superscript</csymbol><ci id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.2.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.2">𝜎</ci><ci id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.3.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.3.3">script-Σ</ci></apply><apply id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1"><times id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.1.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.1"></times><ci id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.2a.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.2"><mtext id="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.2.cmml" xref="S3.SS4.7.p4.1.m1.3.3.2.1.1.1.1.1.1.2">Supp</mtext></ci><ci id="S3.SS4.7.p4.1.m1.1.1.cmml" xref="S3.SS4.7.p4.1.m1.1.1">𝜇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p4.1.m1.3c">\cal L(\mbox{Supp}(\mu^{\sigma}))=\cal L(\sigma^{\Sigma}(\mbox{Supp}(\mu)))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p4.1.m1.3d">caligraphic_L ( Supp ( italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ) ) = caligraphic_L ( italic_σ start_POSTSUPERSCRIPT caligraphic_Σ end_POSTSUPERSCRIPT ( Supp ( italic_μ ) ) )</annotation></semantics></math>, which implies the equality (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E7" title="In Lemma 3.10. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.7</span></a>). <span class="ltx_text ltx_inline-block" id="S3.SS4.7.p4.2.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S3.SS4.7.p4.2.1.m1.1"><semantics id="S3.SS4.7.p4.2.1.m1.1a"><mo id="S3.SS4.7.p4.2.1.m1.1.1" xref="S3.SS4.7.p4.2.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p4.2.1.m1.1b"><csymbol cd="latexml" id="S3.SS4.7.p4.2.1.m1.1.1.cmml" xref="S3.SS4.7.p4.2.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p4.2.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p4.2.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S3.SS4.7.p4.3.m2.1"><semantics id="S3.SS4.7.p4.3.m2.1a"><mo id="S3.SS4.7.p4.3.m2.1.1" xref="S3.SS4.7.p4.3.m2.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S3.SS4.7.p4.3.m2.1b"><csymbol cd="latexml" id="S3.SS4.7.p4.3.m2.1.1.cmml" xref="S3.SS4.7.p4.3.m2.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.7.p4.3.m2.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.7.p4.3.m2.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S3.SS4.p5"> <p class="ltx_p" id="S3.SS4.p5.1">The following discussion, suggested to us by a comment of a referee, concerns a potentially alternative approach to the measure transfer map. We use here the terminology from §3 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib7" title="">7</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_rem" id="S3.Thmthm11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmthm11.1.1.1">Remark 3.11</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmthm11.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmthm11.p1"> <p class="ltx_p" id="S3.Thmthm11.p1.8">Given a subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.1.m1.1"><semantics id="S3.Thmthm11.p1.1.m1.1a"><mrow id="S3.Thmthm11.p1.1.m1.1.1" xref="S3.Thmthm11.p1.1.m1.1.1.cmml"><mi id="S3.Thmthm11.p1.1.m1.1.1.2" xref="S3.Thmthm11.p1.1.m1.1.1.2.cmml">X</mi><mo id="S3.Thmthm11.p1.1.m1.1.1.1" xref="S3.Thmthm11.p1.1.m1.1.1.1.cmml">⊆</mo><msup id="S3.Thmthm11.p1.1.m1.1.1.3" xref="S3.Thmthm11.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p1.1.m1.1.1.3.2" xref="S3.Thmthm11.p1.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S3.Thmthm11.p1.1.m1.1.1.3.3" xref="S3.Thmthm11.p1.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.1.m1.1b"><apply id="S3.Thmthm11.p1.1.m1.1.1.cmml" xref="S3.Thmthm11.p1.1.m1.1.1"><subset id="S3.Thmthm11.p1.1.m1.1.1.1.cmml" xref="S3.Thmthm11.p1.1.m1.1.1.1"></subset><ci id="S3.Thmthm11.p1.1.m1.1.1.2.cmml" xref="S3.Thmthm11.p1.1.m1.1.1.2">𝑋</ci><apply id="S3.Thmthm11.p1.1.m1.1.1.3.cmml" xref="S3.Thmthm11.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p1.1.m1.1.1.3.1.cmml" xref="S3.Thmthm11.p1.1.m1.1.1.3">superscript</csymbol><ci id="S3.Thmthm11.p1.1.m1.1.1.3.2.cmml" xref="S3.Thmthm11.p1.1.m1.1.1.3.2">𝒜</ci><ci id="S3.Thmthm11.p1.1.m1.1.1.3.3.cmml" xref="S3.Thmthm11.p1.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.1.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.1.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, one may consider the group <math alttext="H(X,\mathbb{Z})=C(X,\mathbb{Z})/\partial_{T}C(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.2.m2.6"><semantics id="S3.Thmthm11.p1.2.m2.6a"><mrow id="S3.Thmthm11.p1.2.m2.6.7" xref="S3.Thmthm11.p1.2.m2.6.7.cmml"><mrow id="S3.Thmthm11.p1.2.m2.6.7.2" xref="S3.Thmthm11.p1.2.m2.6.7.2.cmml"><mi id="S3.Thmthm11.p1.2.m2.6.7.2.2" xref="S3.Thmthm11.p1.2.m2.6.7.2.2.cmml">H</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.2.1" xref="S3.Thmthm11.p1.2.m2.6.7.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p1.2.m2.6.7.2.3.2" xref="S3.Thmthm11.p1.2.m2.6.7.2.3.1.cmml"><mo id="S3.Thmthm11.p1.2.m2.6.7.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p1.2.m2.6.7.2.3.1.cmml">(</mo><mi id="S3.Thmthm11.p1.2.m2.1.1" xref="S3.Thmthm11.p1.2.m2.1.1.cmml">X</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.2.3.2.2" xref="S3.Thmthm11.p1.2.m2.6.7.2.3.1.cmml">,</mo><mi id="S3.Thmthm11.p1.2.m2.2.2" xref="S3.Thmthm11.p1.2.m2.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.2.3.2.3" stretchy="false" xref="S3.Thmthm11.p1.2.m2.6.7.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p1.2.m2.6.7.1" xref="S3.Thmthm11.p1.2.m2.6.7.1.cmml">=</mo><mrow id="S3.Thmthm11.p1.2.m2.6.7.3" xref="S3.Thmthm11.p1.2.m2.6.7.3.cmml"><mrow id="S3.Thmthm11.p1.2.m2.6.7.3.2" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.cmml"><mi id="S3.Thmthm11.p1.2.m2.6.7.3.2.2" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.2.cmml">C</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.3.2.1" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p1.2.m2.6.7.3.2.3.2" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.3.1.cmml"><mo id="S3.Thmthm11.p1.2.m2.6.7.3.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.3.1.cmml">(</mo><mi id="S3.Thmthm11.p1.2.m2.3.3" xref="S3.Thmthm11.p1.2.m2.3.3.cmml">X</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.3.2.3.2.2" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.3.1.cmml">,</mo><mi id="S3.Thmthm11.p1.2.m2.4.4" xref="S3.Thmthm11.p1.2.m2.4.4.cmml">ℤ</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.3.2.3.2.3" stretchy="false" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p1.2.m2.6.7.3.1" xref="S3.Thmthm11.p1.2.m2.6.7.3.1.cmml">/</mo><mrow id="S3.Thmthm11.p1.2.m2.6.7.3.3" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.cmml"><msub id="S3.Thmthm11.p1.2.m2.6.7.3.3.1" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.1.cmml"><mo id="S3.Thmthm11.p1.2.m2.6.7.3.3.1.2" lspace="0em" rspace="0em" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.1.2.cmml">∂</mo><mi id="S3.Thmthm11.p1.2.m2.6.7.3.3.1.3" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.1.3.cmml">T</mi></msub><mrow id="S3.Thmthm11.p1.2.m2.6.7.3.3.2" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.cmml"><mi id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.2" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.2.cmml">C</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.1" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.2" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.1.cmml"><mo id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.1.cmml">(</mo><mi id="S3.Thmthm11.p1.2.m2.5.5" xref="S3.Thmthm11.p1.2.m2.5.5.cmml">X</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.2.2" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.1.cmml">,</mo><mi id="S3.Thmthm11.p1.2.m2.6.6" xref="S3.Thmthm11.p1.2.m2.6.6.cmml">ℤ</mi><mo id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.2.3" stretchy="false" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.2.m2.6b"><apply id="S3.Thmthm11.p1.2.m2.6.7.cmml" xref="S3.Thmthm11.p1.2.m2.6.7"><eq id="S3.Thmthm11.p1.2.m2.6.7.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.1"></eq><apply id="S3.Thmthm11.p1.2.m2.6.7.2.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.2"><times id="S3.Thmthm11.p1.2.m2.6.7.2.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.2.1"></times><ci id="S3.Thmthm11.p1.2.m2.6.7.2.2.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.2.2">𝐻</ci><interval closure="open" id="S3.Thmthm11.p1.2.m2.6.7.2.3.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.2.3.2"><ci id="S3.Thmthm11.p1.2.m2.1.1.cmml" xref="S3.Thmthm11.p1.2.m2.1.1">𝑋</ci><ci id="S3.Thmthm11.p1.2.m2.2.2.cmml" xref="S3.Thmthm11.p1.2.m2.2.2">ℤ</ci></interval></apply><apply id="S3.Thmthm11.p1.2.m2.6.7.3.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3"><divide id="S3.Thmthm11.p1.2.m2.6.7.3.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.1"></divide><apply id="S3.Thmthm11.p1.2.m2.6.7.3.2.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.2"><times id="S3.Thmthm11.p1.2.m2.6.7.3.2.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.1"></times><ci id="S3.Thmthm11.p1.2.m2.6.7.3.2.2.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p1.2.m2.6.7.3.2.3.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.2.3.2"><ci id="S3.Thmthm11.p1.2.m2.3.3.cmml" xref="S3.Thmthm11.p1.2.m2.3.3">𝑋</ci><ci id="S3.Thmthm11.p1.2.m2.4.4.cmml" xref="S3.Thmthm11.p1.2.m2.4.4">ℤ</ci></interval></apply><apply id="S3.Thmthm11.p1.2.m2.6.7.3.3.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3"><apply id="S3.Thmthm11.p1.2.m2.6.7.3.3.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p1.2.m2.6.7.3.3.1.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.1">subscript</csymbol><partialdiff id="S3.Thmthm11.p1.2.m2.6.7.3.3.1.2.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.1.2"></partialdiff><ci id="S3.Thmthm11.p1.2.m2.6.7.3.3.1.3.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.1.3">𝑇</ci></apply><apply id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2"><times id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.1"></times><ci id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.2.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.1.cmml" xref="S3.Thmthm11.p1.2.m2.6.7.3.3.2.3.2"><ci id="S3.Thmthm11.p1.2.m2.5.5.cmml" xref="S3.Thmthm11.p1.2.m2.5.5">𝑋</ci><ci id="S3.Thmthm11.p1.2.m2.6.6.cmml" xref="S3.Thmthm11.p1.2.m2.6.6">ℤ</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.2.m2.6c">H(X,\mathbb{Z})=C(X,\mathbb{Z})/\partial_{T}C(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.2.m2.6d">italic_H ( italic_X , blackboard_Z ) = italic_C ( italic_X , blackboard_Z ) / ∂ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_C ( italic_X , blackboard_Z )</annotation></semantics></math>, where <math alttext="C(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.3.m3.2"><semantics id="S3.Thmthm11.p1.3.m3.2a"><mrow id="S3.Thmthm11.p1.3.m3.2.3" xref="S3.Thmthm11.p1.3.m3.2.3.cmml"><mi id="S3.Thmthm11.p1.3.m3.2.3.2" xref="S3.Thmthm11.p1.3.m3.2.3.2.cmml">C</mi><mo id="S3.Thmthm11.p1.3.m3.2.3.1" xref="S3.Thmthm11.p1.3.m3.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p1.3.m3.2.3.3.2" xref="S3.Thmthm11.p1.3.m3.2.3.3.1.cmml"><mo id="S3.Thmthm11.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p1.3.m3.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p1.3.m3.1.1" xref="S3.Thmthm11.p1.3.m3.1.1.cmml">X</mi><mo id="S3.Thmthm11.p1.3.m3.2.3.3.2.2" xref="S3.Thmthm11.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p1.3.m3.2.2" xref="S3.Thmthm11.p1.3.m3.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.3.m3.2b"><apply id="S3.Thmthm11.p1.3.m3.2.3.cmml" xref="S3.Thmthm11.p1.3.m3.2.3"><times id="S3.Thmthm11.p1.3.m3.2.3.1.cmml" xref="S3.Thmthm11.p1.3.m3.2.3.1"></times><ci id="S3.Thmthm11.p1.3.m3.2.3.2.cmml" xref="S3.Thmthm11.p1.3.m3.2.3.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p1.3.m3.2.3.3.1.cmml" xref="S3.Thmthm11.p1.3.m3.2.3.3.2"><ci id="S3.Thmthm11.p1.3.m3.1.1.cmml" xref="S3.Thmthm11.p1.3.m3.1.1">𝑋</ci><ci id="S3.Thmthm11.p1.3.m3.2.2.cmml" xref="S3.Thmthm11.p1.3.m3.2.2">ℤ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.3.m3.2c">C(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.3.m3.2d">italic_C ( italic_X , blackboard_Z )</annotation></semantics></math> denotes the abelian group of continuous integer valued functions on <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.4.m4.1"><semantics id="S3.Thmthm11.p1.4.m4.1a"><mi id="S3.Thmthm11.p1.4.m4.1.1" xref="S3.Thmthm11.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.4.m4.1b"><ci id="S3.Thmthm11.p1.4.m4.1.1.cmml" xref="S3.Thmthm11.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.4.m4.1d">italic_X</annotation></semantics></math>, and <math alttext="\partial_{T}(X)" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.5.m5.1"><semantics id="S3.Thmthm11.p1.5.m5.1a"><mrow id="S3.Thmthm11.p1.5.m5.1.2" xref="S3.Thmthm11.p1.5.m5.1.2.cmml"><msub id="S3.Thmthm11.p1.5.m5.1.2.1" xref="S3.Thmthm11.p1.5.m5.1.2.1.cmml"><mo id="S3.Thmthm11.p1.5.m5.1.2.1.2" xref="S3.Thmthm11.p1.5.m5.1.2.1.2.cmml">∂</mo><mi id="S3.Thmthm11.p1.5.m5.1.2.1.3" xref="S3.Thmthm11.p1.5.m5.1.2.1.3.cmml">T</mi></msub><mrow id="S3.Thmthm11.p1.5.m5.1.2.2.2" xref="S3.Thmthm11.p1.5.m5.1.2.cmml"><mo id="S3.Thmthm11.p1.5.m5.1.2.2.2.1" lspace="0em" stretchy="false" xref="S3.Thmthm11.p1.5.m5.1.2.cmml">(</mo><mi id="S3.Thmthm11.p1.5.m5.1.1" xref="S3.Thmthm11.p1.5.m5.1.1.cmml">X</mi><mo id="S3.Thmthm11.p1.5.m5.1.2.2.2.2" stretchy="false" xref="S3.Thmthm11.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.5.m5.1b"><apply id="S3.Thmthm11.p1.5.m5.1.2.cmml" xref="S3.Thmthm11.p1.5.m5.1.2"><apply id="S3.Thmthm11.p1.5.m5.1.2.1.cmml" xref="S3.Thmthm11.p1.5.m5.1.2.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p1.5.m5.1.2.1.1.cmml" xref="S3.Thmthm11.p1.5.m5.1.2.1">subscript</csymbol><partialdiff id="S3.Thmthm11.p1.5.m5.1.2.1.2.cmml" xref="S3.Thmthm11.p1.5.m5.1.2.1.2"></partialdiff><ci id="S3.Thmthm11.p1.5.m5.1.2.1.3.cmml" xref="S3.Thmthm11.p1.5.m5.1.2.1.3">𝑇</ci></apply><ci id="S3.Thmthm11.p1.5.m5.1.1.cmml" xref="S3.Thmthm11.p1.5.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.5.m5.1c">\partial_{T}(X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.5.m5.1d">∂ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math> is the subgroup generated by functions of the form <math alttext="f\circ T-f" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.6.m6.1"><semantics id="S3.Thmthm11.p1.6.m6.1a"><mrow id="S3.Thmthm11.p1.6.m6.1.1" xref="S3.Thmthm11.p1.6.m6.1.1.cmml"><mrow id="S3.Thmthm11.p1.6.m6.1.1.2" xref="S3.Thmthm11.p1.6.m6.1.1.2.cmml"><mi id="S3.Thmthm11.p1.6.m6.1.1.2.2" xref="S3.Thmthm11.p1.6.m6.1.1.2.2.cmml">f</mi><mo id="S3.Thmthm11.p1.6.m6.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p1.6.m6.1.1.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p1.6.m6.1.1.2.3" xref="S3.Thmthm11.p1.6.m6.1.1.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p1.6.m6.1.1.1" xref="S3.Thmthm11.p1.6.m6.1.1.1.cmml">−</mo><mi id="S3.Thmthm11.p1.6.m6.1.1.3" xref="S3.Thmthm11.p1.6.m6.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.6.m6.1b"><apply id="S3.Thmthm11.p1.6.m6.1.1.cmml" xref="S3.Thmthm11.p1.6.m6.1.1"><minus id="S3.Thmthm11.p1.6.m6.1.1.1.cmml" xref="S3.Thmthm11.p1.6.m6.1.1.1"></minus><apply id="S3.Thmthm11.p1.6.m6.1.1.2.cmml" xref="S3.Thmthm11.p1.6.m6.1.1.2"><compose id="S3.Thmthm11.p1.6.m6.1.1.2.1.cmml" xref="S3.Thmthm11.p1.6.m6.1.1.2.1"></compose><ci id="S3.Thmthm11.p1.6.m6.1.1.2.2.cmml" xref="S3.Thmthm11.p1.6.m6.1.1.2.2">𝑓</ci><ci id="S3.Thmthm11.p1.6.m6.1.1.2.3.cmml" xref="S3.Thmthm11.p1.6.m6.1.1.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p1.6.m6.1.1.3.cmml" xref="S3.Thmthm11.p1.6.m6.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.6.m6.1c">f\circ T-f</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.6.m6.1d">italic_f ∘ italic_T - italic_f</annotation></semantics></math>. To every invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.7.m7.1"><semantics id="S3.Thmthm11.p1.7.m7.1a"><mi id="S3.Thmthm11.p1.7.m7.1.1" xref="S3.Thmthm11.p1.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.7.m7.1b"><ci id="S3.Thmthm11.p1.7.m7.1.1.cmml" xref="S3.Thmthm11.p1.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.7.m7.1d">italic_μ</annotation></semantics></math> on <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.8.m8.1"><semantics id="S3.Thmthm11.p1.8.m8.1a"><mi id="S3.Thmthm11.p1.8.m8.1.1" xref="S3.Thmthm11.p1.8.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.8.m8.1b"><ci id="S3.Thmthm11.p1.8.m8.1.1.cmml" xref="S3.Thmthm11.p1.8.m8.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.8.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.8.m8.1d">italic_X</annotation></semantics></math> one associates the linear form</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\alpha_{\mu}:H(X,\mathbb{Z})\to\mathbb{R}\,,\,\,f\mapsto\int fd\mu\,," class="ltx_Math" display="block" id="S3.Ex10.m1.3"><semantics id="S3.Ex10.m1.3a"><mrow id="S3.Ex10.m1.3.3.1" xref="S3.Ex10.m1.3.3.1.1.cmml"><mrow id="S3.Ex10.m1.3.3.1.1" xref="S3.Ex10.m1.3.3.1.1.cmml"><msub id="S3.Ex10.m1.3.3.1.1.4" xref="S3.Ex10.m1.3.3.1.1.4.cmml"><mi id="S3.Ex10.m1.3.3.1.1.4.2" xref="S3.Ex10.m1.3.3.1.1.4.2.cmml">α</mi><mi id="S3.Ex10.m1.3.3.1.1.4.3" xref="S3.Ex10.m1.3.3.1.1.4.3.cmml">μ</mi></msub><mo id="S3.Ex10.m1.3.3.1.1.3" lspace="0.278em" rspace="0.278em" xref="S3.Ex10.m1.3.3.1.1.3.cmml">:</mo><mrow id="S3.Ex10.m1.3.3.1.1.2.2" xref="S3.Ex10.m1.3.3.1.1.2.3.cmml"><mrow id="S3.Ex10.m1.3.3.1.1.1.1.1" xref="S3.Ex10.m1.3.3.1.1.1.1.1.cmml"><mrow id="S3.Ex10.m1.3.3.1.1.1.1.1.2" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.cmml"><mi id="S3.Ex10.m1.3.3.1.1.1.1.1.2.2" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.2.cmml">H</mi><mo id="S3.Ex10.m1.3.3.1.1.1.1.1.2.1" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.2" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.1.cmml"><mo id="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.1.cmml">(</mo><mi id="S3.Ex10.m1.1.1" xref="S3.Ex10.m1.1.1.cmml">X</mi><mo id="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.2.2" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.1.cmml">,</mo><mi id="S3.Ex10.m1.2.2" xref="S3.Ex10.m1.2.2.cmml">ℤ</mi><mo id="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.2.3" stretchy="false" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex10.m1.3.3.1.1.1.1.1.1" stretchy="false" xref="S3.Ex10.m1.3.3.1.1.1.1.1.1.cmml">→</mo><mi id="S3.Ex10.m1.3.3.1.1.1.1.1.3" xref="S3.Ex10.m1.3.3.1.1.1.1.1.3.cmml">ℝ</mi></mrow><mo id="S3.Ex10.m1.3.3.1.1.2.2.3" lspace="0.170em" rspace="0.497em" xref="S3.Ex10.m1.3.3.1.1.2.3a.cmml">,</mo><mrow id="S3.Ex10.m1.3.3.1.1.2.2.2" xref="S3.Ex10.m1.3.3.1.1.2.2.2.cmml"><mi id="S3.Ex10.m1.3.3.1.1.2.2.2.2" xref="S3.Ex10.m1.3.3.1.1.2.2.2.2.cmml">f</mi><mo id="S3.Ex10.m1.3.3.1.1.2.2.2.1" rspace="0.111em" stretchy="false" xref="S3.Ex10.m1.3.3.1.1.2.2.2.1.cmml">↦</mo><mrow id="S3.Ex10.m1.3.3.1.1.2.2.2.3" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.cmml"><mo id="S3.Ex10.m1.3.3.1.1.2.2.2.3.1" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.1.cmml">∫</mo><mrow id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.cmml"><mi id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.2" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.2.cmml">f</mi><mo id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.1" lspace="0em" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.cmml"><mo id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.1" rspace="0em" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.1.cmml">𝑑</mo><mi id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.2" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.2.cmml">μ</mi></mrow></mrow></mrow></mrow></mrow></mrow><mo id="S3.Ex10.m1.3.3.1.2" lspace="0.170em" xref="S3.Ex10.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex10.m1.3b"><apply id="S3.Ex10.m1.3.3.1.1.cmml" xref="S3.Ex10.m1.3.3.1"><ci id="S3.Ex10.m1.3.3.1.1.3.cmml" xref="S3.Ex10.m1.3.3.1.1.3">:</ci><apply id="S3.Ex10.m1.3.3.1.1.4.cmml" xref="S3.Ex10.m1.3.3.1.1.4"><csymbol cd="ambiguous" id="S3.Ex10.m1.3.3.1.1.4.1.cmml" xref="S3.Ex10.m1.3.3.1.1.4">subscript</csymbol><ci id="S3.Ex10.m1.3.3.1.1.4.2.cmml" xref="S3.Ex10.m1.3.3.1.1.4.2">𝛼</ci><ci id="S3.Ex10.m1.3.3.1.1.4.3.cmml" xref="S3.Ex10.m1.3.3.1.1.4.3">𝜇</ci></apply><apply id="S3.Ex10.m1.3.3.1.1.2.3.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex10.m1.3.3.1.1.2.3a.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.3">formulae-sequence</csymbol><apply id="S3.Ex10.m1.3.3.1.1.1.1.1.cmml" xref="S3.Ex10.m1.3.3.1.1.1.1.1"><ci id="S3.Ex10.m1.3.3.1.1.1.1.1.1.cmml" xref="S3.Ex10.m1.3.3.1.1.1.1.1.1">→</ci><apply id="S3.Ex10.m1.3.3.1.1.1.1.1.2.cmml" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2"><times id="S3.Ex10.m1.3.3.1.1.1.1.1.2.1.cmml" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.1"></times><ci id="S3.Ex10.m1.3.3.1.1.1.1.1.2.2.cmml" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.2">𝐻</ci><interval closure="open" id="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.1.cmml" xref="S3.Ex10.m1.3.3.1.1.1.1.1.2.3.2"><ci id="S3.Ex10.m1.1.1.cmml" xref="S3.Ex10.m1.1.1">𝑋</ci><ci id="S3.Ex10.m1.2.2.cmml" xref="S3.Ex10.m1.2.2">ℤ</ci></interval></apply><ci id="S3.Ex10.m1.3.3.1.1.1.1.1.3.cmml" xref="S3.Ex10.m1.3.3.1.1.1.1.1.3">ℝ</ci></apply><apply id="S3.Ex10.m1.3.3.1.1.2.2.2.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2"><csymbol cd="latexml" id="S3.Ex10.m1.3.3.1.1.2.2.2.1.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.1">maps-to</csymbol><ci id="S3.Ex10.m1.3.3.1.1.2.2.2.2.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.2">𝑓</ci><apply id="S3.Ex10.m1.3.3.1.1.2.2.2.3.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3"><int id="S3.Ex10.m1.3.3.1.1.2.2.2.3.1.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.1"></int><apply id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2"><times id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.1.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.1"></times><ci id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.2.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.2">𝑓</ci><apply id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3"><csymbol cd="latexml" id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.1.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.1">differential-d</csymbol><ci id="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.2.cmml" xref="S3.Ex10.m1.3.3.1.1.2.2.2.3.2.3.2">𝜇</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.3c">\alpha_{\mu}:H(X,\mathbb{Z})\to\mathbb{R}\,,\,\,f\mapsto\int fd\mu\,,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.3d">italic_α start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT : italic_H ( italic_X , blackboard_Z ) → blackboard_R , italic_f ↦ ∫ italic_f italic_d italic_μ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmthm11.p1.9">i.e. an element in the dual group <math alttext="H(X,\mathbb{R})^{*}:={\rm Hom}(H(X,\mathbb{Z}),\mathbb{R})" class="ltx_Math" display="inline" id="S3.Thmthm11.p1.9.m1.6"><semantics id="S3.Thmthm11.p1.9.m1.6a"><mrow id="S3.Thmthm11.p1.9.m1.6.6" xref="S3.Thmthm11.p1.9.m1.6.6.cmml"><mrow id="S3.Thmthm11.p1.9.m1.6.6.3" xref="S3.Thmthm11.p1.9.m1.6.6.3.cmml"><mi id="S3.Thmthm11.p1.9.m1.6.6.3.2" xref="S3.Thmthm11.p1.9.m1.6.6.3.2.cmml">H</mi><mo id="S3.Thmthm11.p1.9.m1.6.6.3.1" xref="S3.Thmthm11.p1.9.m1.6.6.3.1.cmml">⁢</mo><msup id="S3.Thmthm11.p1.9.m1.6.6.3.3" xref="S3.Thmthm11.p1.9.m1.6.6.3.3.cmml"><mrow id="S3.Thmthm11.p1.9.m1.6.6.3.3.2.2" xref="S3.Thmthm11.p1.9.m1.6.6.3.3.2.1.cmml"><mo id="S3.Thmthm11.p1.9.m1.6.6.3.3.2.2.1" stretchy="false" xref="S3.Thmthm11.p1.9.m1.6.6.3.3.2.1.cmml">(</mo><mi id="S3.Thmthm11.p1.9.m1.1.1" xref="S3.Thmthm11.p1.9.m1.1.1.cmml">X</mi><mo id="S3.Thmthm11.p1.9.m1.6.6.3.3.2.2.2" xref="S3.Thmthm11.p1.9.m1.6.6.3.3.2.1.cmml">,</mo><mi id="S3.Thmthm11.p1.9.m1.2.2" xref="S3.Thmthm11.p1.9.m1.2.2.cmml">ℝ</mi><mo id="S3.Thmthm11.p1.9.m1.6.6.3.3.2.2.3" rspace="0.278em" stretchy="false" xref="S3.Thmthm11.p1.9.m1.6.6.3.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p1.9.m1.6.6.3.3.3" xref="S3.Thmthm11.p1.9.m1.6.6.3.3.3.cmml">∗</mo></msup></mrow><mo id="S3.Thmthm11.p1.9.m1.6.6.2" rspace="0.278em" xref="S3.Thmthm11.p1.9.m1.6.6.2.cmml">:=</mo><mrow id="S3.Thmthm11.p1.9.m1.6.6.1" xref="S3.Thmthm11.p1.9.m1.6.6.1.cmml"><mi id="S3.Thmthm11.p1.9.m1.6.6.1.3" xref="S3.Thmthm11.p1.9.m1.6.6.1.3.cmml">Hom</mi><mo id="S3.Thmthm11.p1.9.m1.6.6.1.2" xref="S3.Thmthm11.p1.9.m1.6.6.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p1.9.m1.6.6.1.1.1" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.2.cmml"><mo id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.2.cmml">(</mo><mrow id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.cmml"><mi id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.2" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.2.cmml">H</mi><mo id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.1" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.2" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.1.cmml"><mo id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.2.1" stretchy="false" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.1.cmml">(</mo><mi id="S3.Thmthm11.p1.9.m1.3.3" xref="S3.Thmthm11.p1.9.m1.3.3.cmml">X</mi><mo id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.2.2" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.1.cmml">,</mo><mi id="S3.Thmthm11.p1.9.m1.4.4" xref="S3.Thmthm11.p1.9.m1.4.4.cmml">ℤ</mi><mo id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.2.3" stretchy="false" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.3" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.2.cmml">,</mo><mi id="S3.Thmthm11.p1.9.m1.5.5" xref="S3.Thmthm11.p1.9.m1.5.5.cmml">ℝ</mi><mo id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.4" stretchy="false" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p1.9.m1.6b"><apply id="S3.Thmthm11.p1.9.m1.6.6.cmml" xref="S3.Thmthm11.p1.9.m1.6.6"><csymbol cd="latexml" id="S3.Thmthm11.p1.9.m1.6.6.2.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.2">assign</csymbol><apply id="S3.Thmthm11.p1.9.m1.6.6.3.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.3"><times id="S3.Thmthm11.p1.9.m1.6.6.3.1.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.3.1"></times><ci id="S3.Thmthm11.p1.9.m1.6.6.3.2.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.3.2">𝐻</ci><apply id="S3.Thmthm11.p1.9.m1.6.6.3.3.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.3.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p1.9.m1.6.6.3.3.1.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.3.3">superscript</csymbol><interval closure="open" id="S3.Thmthm11.p1.9.m1.6.6.3.3.2.1.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.3.3.2.2"><ci id="S3.Thmthm11.p1.9.m1.1.1.cmml" xref="S3.Thmthm11.p1.9.m1.1.1">𝑋</ci><ci id="S3.Thmthm11.p1.9.m1.2.2.cmml" xref="S3.Thmthm11.p1.9.m1.2.2">ℝ</ci></interval><times id="S3.Thmthm11.p1.9.m1.6.6.3.3.3.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.3.3.3"></times></apply></apply><apply id="S3.Thmthm11.p1.9.m1.6.6.1.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.1"><times id="S3.Thmthm11.p1.9.m1.6.6.1.2.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.1.2"></times><ci id="S3.Thmthm11.p1.9.m1.6.6.1.3.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.1.3">Hom</ci><interval closure="open" id="S3.Thmthm11.p1.9.m1.6.6.1.1.2.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1"><apply id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1"><times id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.1.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.1"></times><ci id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.2.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.2">𝐻</ci><interval closure="open" id="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.1.cmml" xref="S3.Thmthm11.p1.9.m1.6.6.1.1.1.1.3.2"><ci id="S3.Thmthm11.p1.9.m1.3.3.cmml" xref="S3.Thmthm11.p1.9.m1.3.3">𝑋</ci><ci id="S3.Thmthm11.p1.9.m1.4.4.cmml" xref="S3.Thmthm11.p1.9.m1.4.4">ℤ</ci></interval></apply><ci id="S3.Thmthm11.p1.9.m1.5.5.cmml" xref="S3.Thmthm11.p1.9.m1.5.5">ℝ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p1.9.m1.6c">H(X,\mathbb{R})^{*}:={\rm Hom}(H(X,\mathbb{Z}),\mathbb{R})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p1.9.m1.6d">italic_H ( italic_X , blackboard_R ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT := roman_Hom ( italic_H ( italic_X , blackboard_Z ) , blackboard_R )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.Thmthm11.p2"> <p class="ltx_p" id="S3.Thmthm11.p2.19">The group <math alttext="H(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.1.m1.2"><semantics id="S3.Thmthm11.p2.1.m1.2a"><mrow id="S3.Thmthm11.p2.1.m1.2.3" xref="S3.Thmthm11.p2.1.m1.2.3.cmml"><mi id="S3.Thmthm11.p2.1.m1.2.3.2" xref="S3.Thmthm11.p2.1.m1.2.3.2.cmml">H</mi><mo id="S3.Thmthm11.p2.1.m1.2.3.1" xref="S3.Thmthm11.p2.1.m1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.1.m1.2.3.3.2" xref="S3.Thmthm11.p2.1.m1.2.3.3.1.cmml"><mo id="S3.Thmthm11.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p2.1.m1.1.1" xref="S3.Thmthm11.p2.1.m1.1.1.cmml">X</mi><mo id="S3.Thmthm11.p2.1.m1.2.3.3.2.2" xref="S3.Thmthm11.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p2.1.m1.2.2" xref="S3.Thmthm11.p2.1.m1.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.1.m1.2b"><apply id="S3.Thmthm11.p2.1.m1.2.3.cmml" xref="S3.Thmthm11.p2.1.m1.2.3"><times id="S3.Thmthm11.p2.1.m1.2.3.1.cmml" xref="S3.Thmthm11.p2.1.m1.2.3.1"></times><ci id="S3.Thmthm11.p2.1.m1.2.3.2.cmml" xref="S3.Thmthm11.p2.1.m1.2.3.2">𝐻</ci><interval closure="open" id="S3.Thmthm11.p2.1.m1.2.3.3.1.cmml" xref="S3.Thmthm11.p2.1.m1.2.3.3.2"><ci id="S3.Thmthm11.p2.1.m1.1.1.cmml" xref="S3.Thmthm11.p2.1.m1.1.1">𝑋</ci><ci id="S3.Thmthm11.p2.1.m1.2.2.cmml" xref="S3.Thmthm11.p2.1.m1.2.2">ℤ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.1.m1.2c">H(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.1.m1.2d">italic_H ( italic_X , blackboard_Z )</annotation></semantics></math> inherits from <math alttext="C(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.2.m2.2"><semantics id="S3.Thmthm11.p2.2.m2.2a"><mrow id="S3.Thmthm11.p2.2.m2.2.3" xref="S3.Thmthm11.p2.2.m2.2.3.cmml"><mi id="S3.Thmthm11.p2.2.m2.2.3.2" xref="S3.Thmthm11.p2.2.m2.2.3.2.cmml">C</mi><mo id="S3.Thmthm11.p2.2.m2.2.3.1" xref="S3.Thmthm11.p2.2.m2.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.2.m2.2.3.3.2" xref="S3.Thmthm11.p2.2.m2.2.3.3.1.cmml"><mo id="S3.Thmthm11.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.2.m2.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p2.2.m2.1.1" xref="S3.Thmthm11.p2.2.m2.1.1.cmml">X</mi><mo id="S3.Thmthm11.p2.2.m2.2.3.3.2.2" xref="S3.Thmthm11.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p2.2.m2.2.2" xref="S3.Thmthm11.p2.2.m2.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.2.m2.2b"><apply id="S3.Thmthm11.p2.2.m2.2.3.cmml" xref="S3.Thmthm11.p2.2.m2.2.3"><times id="S3.Thmthm11.p2.2.m2.2.3.1.cmml" xref="S3.Thmthm11.p2.2.m2.2.3.1"></times><ci id="S3.Thmthm11.p2.2.m2.2.3.2.cmml" xref="S3.Thmthm11.p2.2.m2.2.3.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p2.2.m2.2.3.3.1.cmml" xref="S3.Thmthm11.p2.2.m2.2.3.3.2"><ci id="S3.Thmthm11.p2.2.m2.1.1.cmml" xref="S3.Thmthm11.p2.2.m2.1.1">𝑋</ci><ci id="S3.Thmthm11.p2.2.m2.2.2.cmml" xref="S3.Thmthm11.p2.2.m2.2.2">ℤ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.2.m2.2c">C(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.2.m2.2d">italic_C ( italic_X , blackboard_Z )</annotation></semantics></math> the canonical order as well as the “order unit” given by the characteristic function <math alttext="{\bf 1}_{X}" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.3.m3.1"><semantics id="S3.Thmthm11.p2.3.m3.1a"><msub id="S3.Thmthm11.p2.3.m3.1.1" xref="S3.Thmthm11.p2.3.m3.1.1.cmml"><mn id="S3.Thmthm11.p2.3.m3.1.1.2" xref="S3.Thmthm11.p2.3.m3.1.1.2.cmml">𝟏</mn><mi id="S3.Thmthm11.p2.3.m3.1.1.3" xref="S3.Thmthm11.p2.3.m3.1.1.3.cmml">X</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.3.m3.1b"><apply id="S3.Thmthm11.p2.3.m3.1.1.cmml" xref="S3.Thmthm11.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.3.m3.1.1.1.cmml" xref="S3.Thmthm11.p2.3.m3.1.1">subscript</csymbol><cn id="S3.Thmthm11.p2.3.m3.1.1.2.cmml" type="integer" xref="S3.Thmthm11.p2.3.m3.1.1.2">1</cn><ci id="S3.Thmthm11.p2.3.m3.1.1.3.cmml" xref="S3.Thmthm11.p2.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.3.m3.1c">{\bf 1}_{X}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.3.m3.1d">bold_1 start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> (see §2.1.1 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib7" title="">7</a>]</cite>). Under the mild additional assumption that <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.4.m4.1"><semantics id="S3.Thmthm11.p2.4.m4.1a"><mi id="S3.Thmthm11.p2.4.m4.1.1" xref="S3.Thmthm11.p2.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.4.m4.1b"><ci id="S3.Thmthm11.p2.4.m4.1.1.cmml" xref="S3.Thmthm11.p2.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.4.m4.1d">italic_X</annotation></semantics></math> is recurrent (i.e. there exists some <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.5.m5.1"><semantics id="S3.Thmthm11.p2.5.m5.1a"><mrow id="S3.Thmthm11.p2.5.m5.1.1" xref="S3.Thmthm11.p2.5.m5.1.1.cmml"><mi id="S3.Thmthm11.p2.5.m5.1.1.2" xref="S3.Thmthm11.p2.5.m5.1.1.2.cmml">𝐱</mi><mo id="S3.Thmthm11.p2.5.m5.1.1.1" xref="S3.Thmthm11.p2.5.m5.1.1.1.cmml">∈</mo><mi id="S3.Thmthm11.p2.5.m5.1.1.3" xref="S3.Thmthm11.p2.5.m5.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.5.m5.1b"><apply id="S3.Thmthm11.p2.5.m5.1.1.cmml" xref="S3.Thmthm11.p2.5.m5.1.1"><in id="S3.Thmthm11.p2.5.m5.1.1.1.cmml" xref="S3.Thmthm11.p2.5.m5.1.1.1"></in><ci id="S3.Thmthm11.p2.5.m5.1.1.2.cmml" xref="S3.Thmthm11.p2.5.m5.1.1.2">𝐱</ci><ci id="S3.Thmthm11.p2.5.m5.1.1.3.cmml" xref="S3.Thmthm11.p2.5.m5.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.5.m5.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.5.m5.1d">bold_x ∈ italic_X</annotation></semantics></math> with positive half-word <math alttext="{\bf x}_{[1,+\infty)}" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.6.m6.2"><semantics id="S3.Thmthm11.p2.6.m6.2a"><msub id="S3.Thmthm11.p2.6.m6.2.3" xref="S3.Thmthm11.p2.6.m6.2.3.cmml"><mi id="S3.Thmthm11.p2.6.m6.2.3.2" xref="S3.Thmthm11.p2.6.m6.2.3.2.cmml">𝐱</mi><mrow id="S3.Thmthm11.p2.6.m6.2.2.2.2" xref="S3.Thmthm11.p2.6.m6.2.2.2.3.cmml"><mo id="S3.Thmthm11.p2.6.m6.2.2.2.2.2" stretchy="false" xref="S3.Thmthm11.p2.6.m6.2.2.2.3.cmml">[</mo><mn id="S3.Thmthm11.p2.6.m6.1.1.1.1" xref="S3.Thmthm11.p2.6.m6.1.1.1.1.cmml">1</mn><mo id="S3.Thmthm11.p2.6.m6.2.2.2.2.3" xref="S3.Thmthm11.p2.6.m6.2.2.2.3.cmml">,</mo><mrow id="S3.Thmthm11.p2.6.m6.2.2.2.2.1" xref="S3.Thmthm11.p2.6.m6.2.2.2.2.1.cmml"><mo id="S3.Thmthm11.p2.6.m6.2.2.2.2.1a" xref="S3.Thmthm11.p2.6.m6.2.2.2.2.1.cmml">+</mo><mi id="S3.Thmthm11.p2.6.m6.2.2.2.2.1.2" mathvariant="normal" xref="S3.Thmthm11.p2.6.m6.2.2.2.2.1.2.cmml">∞</mi></mrow><mo id="S3.Thmthm11.p2.6.m6.2.2.2.2.4" stretchy="false" xref="S3.Thmthm11.p2.6.m6.2.2.2.3.cmml">)</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.6.m6.2b"><apply id="S3.Thmthm11.p2.6.m6.2.3.cmml" xref="S3.Thmthm11.p2.6.m6.2.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.6.m6.2.3.1.cmml" xref="S3.Thmthm11.p2.6.m6.2.3">subscript</csymbol><ci id="S3.Thmthm11.p2.6.m6.2.3.2.cmml" xref="S3.Thmthm11.p2.6.m6.2.3.2">𝐱</ci><interval closure="closed-open" id="S3.Thmthm11.p2.6.m6.2.2.2.3.cmml" xref="S3.Thmthm11.p2.6.m6.2.2.2.2"><cn id="S3.Thmthm11.p2.6.m6.1.1.1.1.cmml" type="integer" xref="S3.Thmthm11.p2.6.m6.1.1.1.1">1</cn><apply id="S3.Thmthm11.p2.6.m6.2.2.2.2.1.cmml" xref="S3.Thmthm11.p2.6.m6.2.2.2.2.1"><plus id="S3.Thmthm11.p2.6.m6.2.2.2.2.1.1.cmml" xref="S3.Thmthm11.p2.6.m6.2.2.2.2.1"></plus><infinity id="S3.Thmthm11.p2.6.m6.2.2.2.2.1.2.cmml" xref="S3.Thmthm11.p2.6.m6.2.2.2.2.1.2"></infinity></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.6.m6.2c">{\bf x}_{[1,+\infty)}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.6.m6.2d">bold_x start_POSTSUBSCRIPT [ 1 , + ∞ ) end_POSTSUBSCRIPT</annotation></semantics></math> that is dense in <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.7.m7.1"><semantics id="S3.Thmthm11.p2.7.m7.1a"><mi id="S3.Thmthm11.p2.7.m7.1.1" xref="S3.Thmthm11.p2.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.7.m7.1b"><ci id="S3.Thmthm11.p2.7.m7.1.1.cmml" xref="S3.Thmthm11.p2.7.m7.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.7.m7.1d">italic_X</annotation></semantics></math>), the group <math alttext="H(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.8.m8.2"><semantics id="S3.Thmthm11.p2.8.m8.2a"><mrow id="S3.Thmthm11.p2.8.m8.2.3" xref="S3.Thmthm11.p2.8.m8.2.3.cmml"><mi id="S3.Thmthm11.p2.8.m8.2.3.2" xref="S3.Thmthm11.p2.8.m8.2.3.2.cmml">H</mi><mo id="S3.Thmthm11.p2.8.m8.2.3.1" xref="S3.Thmthm11.p2.8.m8.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.8.m8.2.3.3.2" xref="S3.Thmthm11.p2.8.m8.2.3.3.1.cmml"><mo id="S3.Thmthm11.p2.8.m8.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.8.m8.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p2.8.m8.1.1" xref="S3.Thmthm11.p2.8.m8.1.1.cmml">X</mi><mo id="S3.Thmthm11.p2.8.m8.2.3.3.2.2" xref="S3.Thmthm11.p2.8.m8.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p2.8.m8.2.2" xref="S3.Thmthm11.p2.8.m8.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p2.8.m8.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p2.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.8.m8.2b"><apply id="S3.Thmthm11.p2.8.m8.2.3.cmml" xref="S3.Thmthm11.p2.8.m8.2.3"><times id="S3.Thmthm11.p2.8.m8.2.3.1.cmml" xref="S3.Thmthm11.p2.8.m8.2.3.1"></times><ci id="S3.Thmthm11.p2.8.m8.2.3.2.cmml" xref="S3.Thmthm11.p2.8.m8.2.3.2">𝐻</ci><interval closure="open" id="S3.Thmthm11.p2.8.m8.2.3.3.1.cmml" xref="S3.Thmthm11.p2.8.m8.2.3.3.2"><ci id="S3.Thmthm11.p2.8.m8.1.1.cmml" xref="S3.Thmthm11.p2.8.m8.1.1">𝑋</ci><ci id="S3.Thmthm11.p2.8.m8.2.2.cmml" xref="S3.Thmthm11.p2.8.m8.2.2">ℤ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.8.m8.2c">H(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.8.m8.2d">italic_H ( italic_X , blackboard_Z )</annotation></semantics></math> is a “unital ordered group”, sometimes denoted by <math alttext="K^{0}(X,T)" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.9.m9.2"><semantics id="S3.Thmthm11.p2.9.m9.2a"><mrow id="S3.Thmthm11.p2.9.m9.2.3" xref="S3.Thmthm11.p2.9.m9.2.3.cmml"><msup id="S3.Thmthm11.p2.9.m9.2.3.2" xref="S3.Thmthm11.p2.9.m9.2.3.2.cmml"><mi id="S3.Thmthm11.p2.9.m9.2.3.2.2" xref="S3.Thmthm11.p2.9.m9.2.3.2.2.cmml">K</mi><mn id="S3.Thmthm11.p2.9.m9.2.3.2.3" xref="S3.Thmthm11.p2.9.m9.2.3.2.3.cmml">0</mn></msup><mo id="S3.Thmthm11.p2.9.m9.2.3.1" xref="S3.Thmthm11.p2.9.m9.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.9.m9.2.3.3.2" xref="S3.Thmthm11.p2.9.m9.2.3.3.1.cmml"><mo id="S3.Thmthm11.p2.9.m9.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.9.m9.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p2.9.m9.1.1" xref="S3.Thmthm11.p2.9.m9.1.1.cmml">X</mi><mo id="S3.Thmthm11.p2.9.m9.2.3.3.2.2" xref="S3.Thmthm11.p2.9.m9.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p2.9.m9.2.2" xref="S3.Thmthm11.p2.9.m9.2.2.cmml">T</mi><mo id="S3.Thmthm11.p2.9.m9.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p2.9.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.9.m9.2b"><apply id="S3.Thmthm11.p2.9.m9.2.3.cmml" xref="S3.Thmthm11.p2.9.m9.2.3"><times id="S3.Thmthm11.p2.9.m9.2.3.1.cmml" xref="S3.Thmthm11.p2.9.m9.2.3.1"></times><apply id="S3.Thmthm11.p2.9.m9.2.3.2.cmml" xref="S3.Thmthm11.p2.9.m9.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.9.m9.2.3.2.1.cmml" xref="S3.Thmthm11.p2.9.m9.2.3.2">superscript</csymbol><ci id="S3.Thmthm11.p2.9.m9.2.3.2.2.cmml" xref="S3.Thmthm11.p2.9.m9.2.3.2.2">𝐾</ci><cn id="S3.Thmthm11.p2.9.m9.2.3.2.3.cmml" type="integer" xref="S3.Thmthm11.p2.9.m9.2.3.2.3">0</cn></apply><interval closure="open" id="S3.Thmthm11.p2.9.m9.2.3.3.1.cmml" xref="S3.Thmthm11.p2.9.m9.2.3.3.2"><ci id="S3.Thmthm11.p2.9.m9.1.1.cmml" xref="S3.Thmthm11.p2.9.m9.1.1">𝑋</ci><ci id="S3.Thmthm11.p2.9.m9.2.2.cmml" xref="S3.Thmthm11.p2.9.m9.2.2">𝑇</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.9.m9.2c">K^{0}(X,T)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.9.m9.2d">italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_X , italic_T )</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib7" title="">7</a>]</cite>, §3.3). An order- and unit-preserving morphism from <math alttext="K^{0}(X,T)" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.10.m10.2"><semantics id="S3.Thmthm11.p2.10.m10.2a"><mrow id="S3.Thmthm11.p2.10.m10.2.3" xref="S3.Thmthm11.p2.10.m10.2.3.cmml"><msup id="S3.Thmthm11.p2.10.m10.2.3.2" xref="S3.Thmthm11.p2.10.m10.2.3.2.cmml"><mi id="S3.Thmthm11.p2.10.m10.2.3.2.2" xref="S3.Thmthm11.p2.10.m10.2.3.2.2.cmml">K</mi><mn id="S3.Thmthm11.p2.10.m10.2.3.2.3" xref="S3.Thmthm11.p2.10.m10.2.3.2.3.cmml">0</mn></msup><mo id="S3.Thmthm11.p2.10.m10.2.3.1" xref="S3.Thmthm11.p2.10.m10.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.10.m10.2.3.3.2" xref="S3.Thmthm11.p2.10.m10.2.3.3.1.cmml"><mo id="S3.Thmthm11.p2.10.m10.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.10.m10.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p2.10.m10.1.1" xref="S3.Thmthm11.p2.10.m10.1.1.cmml">X</mi><mo id="S3.Thmthm11.p2.10.m10.2.3.3.2.2" xref="S3.Thmthm11.p2.10.m10.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p2.10.m10.2.2" xref="S3.Thmthm11.p2.10.m10.2.2.cmml">T</mi><mo id="S3.Thmthm11.p2.10.m10.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p2.10.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.10.m10.2b"><apply id="S3.Thmthm11.p2.10.m10.2.3.cmml" xref="S3.Thmthm11.p2.10.m10.2.3"><times id="S3.Thmthm11.p2.10.m10.2.3.1.cmml" xref="S3.Thmthm11.p2.10.m10.2.3.1"></times><apply id="S3.Thmthm11.p2.10.m10.2.3.2.cmml" xref="S3.Thmthm11.p2.10.m10.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.10.m10.2.3.2.1.cmml" xref="S3.Thmthm11.p2.10.m10.2.3.2">superscript</csymbol><ci id="S3.Thmthm11.p2.10.m10.2.3.2.2.cmml" xref="S3.Thmthm11.p2.10.m10.2.3.2.2">𝐾</ci><cn id="S3.Thmthm11.p2.10.m10.2.3.2.3.cmml" type="integer" xref="S3.Thmthm11.p2.10.m10.2.3.2.3">0</cn></apply><interval closure="open" id="S3.Thmthm11.p2.10.m10.2.3.3.1.cmml" xref="S3.Thmthm11.p2.10.m10.2.3.3.2"><ci id="S3.Thmthm11.p2.10.m10.1.1.cmml" xref="S3.Thmthm11.p2.10.m10.1.1">𝑋</ci><ci id="S3.Thmthm11.p2.10.m10.2.2.cmml" xref="S3.Thmthm11.p2.10.m10.2.2">𝑇</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.10.m10.2c">K^{0}(X,T)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.10.m10.2d">italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_X , italic_T )</annotation></semantics></math> to <math alttext="\mathbb{R}" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.11.m11.1"><semantics id="S3.Thmthm11.p2.11.m11.1a"><mi id="S3.Thmthm11.p2.11.m11.1.1" xref="S3.Thmthm11.p2.11.m11.1.1.cmml">ℝ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.11.m11.1b"><ci id="S3.Thmthm11.p2.11.m11.1.1.cmml" xref="S3.Thmthm11.p2.11.m11.1.1">ℝ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.11.m11.1c">\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.11.m11.1d">blackboard_R</annotation></semantics></math> is called a <span class="ltx_text ltx_font_italic" id="S3.Thmthm11.p2.19.1">state</span>. The above map <math alttext="\mu\mapsto\alpha_{\mu}" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.12.m12.1"><semantics id="S3.Thmthm11.p2.12.m12.1a"><mrow id="S3.Thmthm11.p2.12.m12.1.1" xref="S3.Thmthm11.p2.12.m12.1.1.cmml"><mi id="S3.Thmthm11.p2.12.m12.1.1.2" xref="S3.Thmthm11.p2.12.m12.1.1.2.cmml">μ</mi><mo id="S3.Thmthm11.p2.12.m12.1.1.1" stretchy="false" xref="S3.Thmthm11.p2.12.m12.1.1.1.cmml">↦</mo><msub id="S3.Thmthm11.p2.12.m12.1.1.3" xref="S3.Thmthm11.p2.12.m12.1.1.3.cmml"><mi id="S3.Thmthm11.p2.12.m12.1.1.3.2" xref="S3.Thmthm11.p2.12.m12.1.1.3.2.cmml">α</mi><mi id="S3.Thmthm11.p2.12.m12.1.1.3.3" xref="S3.Thmthm11.p2.12.m12.1.1.3.3.cmml">μ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.12.m12.1b"><apply id="S3.Thmthm11.p2.12.m12.1.1.cmml" xref="S3.Thmthm11.p2.12.m12.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p2.12.m12.1.1.1.cmml" xref="S3.Thmthm11.p2.12.m12.1.1.1">maps-to</csymbol><ci id="S3.Thmthm11.p2.12.m12.1.1.2.cmml" xref="S3.Thmthm11.p2.12.m12.1.1.2">𝜇</ci><apply id="S3.Thmthm11.p2.12.m12.1.1.3.cmml" xref="S3.Thmthm11.p2.12.m12.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.12.m12.1.1.3.1.cmml" xref="S3.Thmthm11.p2.12.m12.1.1.3">subscript</csymbol><ci id="S3.Thmthm11.p2.12.m12.1.1.3.2.cmml" xref="S3.Thmthm11.p2.12.m12.1.1.3.2">𝛼</ci><ci id="S3.Thmthm11.p2.12.m12.1.1.3.3.cmml" xref="S3.Thmthm11.p2.12.m12.1.1.3.3">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.12.m12.1c">\mu\mapsto\alpha_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.12.m12.1d">italic_μ ↦ italic_α start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> gives a well known bijection (see Theorem 3.9.3 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib7" title="">7</a>]</cite>) from the set of invariant probability measures on <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.13.m13.1"><semantics id="S3.Thmthm11.p2.13.m13.1a"><mi id="S3.Thmthm11.p2.13.m13.1.1" xref="S3.Thmthm11.p2.13.m13.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.13.m13.1b"><ci id="S3.Thmthm11.p2.13.m13.1.1.cmml" xref="S3.Thmthm11.p2.13.m13.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.13.m13.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.13.m13.1d">italic_X</annotation></semantics></math> to the set of states on <math alttext="K^{0}(X,T)" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.14.m14.2"><semantics id="S3.Thmthm11.p2.14.m14.2a"><mrow id="S3.Thmthm11.p2.14.m14.2.3" xref="S3.Thmthm11.p2.14.m14.2.3.cmml"><msup id="S3.Thmthm11.p2.14.m14.2.3.2" xref="S3.Thmthm11.p2.14.m14.2.3.2.cmml"><mi id="S3.Thmthm11.p2.14.m14.2.3.2.2" xref="S3.Thmthm11.p2.14.m14.2.3.2.2.cmml">K</mi><mn id="S3.Thmthm11.p2.14.m14.2.3.2.3" xref="S3.Thmthm11.p2.14.m14.2.3.2.3.cmml">0</mn></msup><mo id="S3.Thmthm11.p2.14.m14.2.3.1" xref="S3.Thmthm11.p2.14.m14.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.14.m14.2.3.3.2" xref="S3.Thmthm11.p2.14.m14.2.3.3.1.cmml"><mo id="S3.Thmthm11.p2.14.m14.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.14.m14.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p2.14.m14.1.1" xref="S3.Thmthm11.p2.14.m14.1.1.cmml">X</mi><mo id="S3.Thmthm11.p2.14.m14.2.3.3.2.2" xref="S3.Thmthm11.p2.14.m14.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p2.14.m14.2.2" xref="S3.Thmthm11.p2.14.m14.2.2.cmml">T</mi><mo id="S3.Thmthm11.p2.14.m14.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p2.14.m14.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.14.m14.2b"><apply id="S3.Thmthm11.p2.14.m14.2.3.cmml" xref="S3.Thmthm11.p2.14.m14.2.3"><times id="S3.Thmthm11.p2.14.m14.2.3.1.cmml" xref="S3.Thmthm11.p2.14.m14.2.3.1"></times><apply id="S3.Thmthm11.p2.14.m14.2.3.2.cmml" xref="S3.Thmthm11.p2.14.m14.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.14.m14.2.3.2.1.cmml" xref="S3.Thmthm11.p2.14.m14.2.3.2">superscript</csymbol><ci id="S3.Thmthm11.p2.14.m14.2.3.2.2.cmml" xref="S3.Thmthm11.p2.14.m14.2.3.2.2">𝐾</ci><cn id="S3.Thmthm11.p2.14.m14.2.3.2.3.cmml" type="integer" xref="S3.Thmthm11.p2.14.m14.2.3.2.3">0</cn></apply><interval closure="open" id="S3.Thmthm11.p2.14.m14.2.3.3.1.cmml" xref="S3.Thmthm11.p2.14.m14.2.3.3.2"><ci id="S3.Thmthm11.p2.14.m14.1.1.cmml" xref="S3.Thmthm11.p2.14.m14.1.1">𝑋</ci><ci id="S3.Thmthm11.p2.14.m14.2.2.cmml" xref="S3.Thmthm11.p2.14.m14.2.2">𝑇</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.14.m14.2c">K^{0}(X,T)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.14.m14.2d">italic_K start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_X , italic_T )</annotation></semantics></math>. This bijection extends to an <math alttext="\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.15.m15.1"><semantics id="S3.Thmthm11.p2.15.m15.1a"><msub id="S3.Thmthm11.p2.15.m15.1.1" xref="S3.Thmthm11.p2.15.m15.1.1.cmml"><mi id="S3.Thmthm11.p2.15.m15.1.1.2" xref="S3.Thmthm11.p2.15.m15.1.1.2.cmml">ℝ</mi><mrow id="S3.Thmthm11.p2.15.m15.1.1.3" xref="S3.Thmthm11.p2.15.m15.1.1.3.cmml"><mi id="S3.Thmthm11.p2.15.m15.1.1.3.2" xref="S3.Thmthm11.p2.15.m15.1.1.3.2.cmml"></mi><mo id="S3.Thmthm11.p2.15.m15.1.1.3.1" xref="S3.Thmthm11.p2.15.m15.1.1.3.1.cmml">≥</mo><mn id="S3.Thmthm11.p2.15.m15.1.1.3.3" xref="S3.Thmthm11.p2.15.m15.1.1.3.3.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.15.m15.1b"><apply id="S3.Thmthm11.p2.15.m15.1.1.cmml" xref="S3.Thmthm11.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.15.m15.1.1.1.cmml" xref="S3.Thmthm11.p2.15.m15.1.1">subscript</csymbol><ci id="S3.Thmthm11.p2.15.m15.1.1.2.cmml" xref="S3.Thmthm11.p2.15.m15.1.1.2">ℝ</ci><apply id="S3.Thmthm11.p2.15.m15.1.1.3.cmml" xref="S3.Thmthm11.p2.15.m15.1.1.3"><geq id="S3.Thmthm11.p2.15.m15.1.1.3.1.cmml" xref="S3.Thmthm11.p2.15.m15.1.1.3.1"></geq><csymbol cd="latexml" id="S3.Thmthm11.p2.15.m15.1.1.3.2.cmml" xref="S3.Thmthm11.p2.15.m15.1.1.3.2">absent</csymbol><cn id="S3.Thmthm11.p2.15.m15.1.1.3.3.cmml" type="integer" xref="S3.Thmthm11.p2.15.m15.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.15.m15.1c">\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.15.m15.1d">blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math>-linear isomorphism <math alttext="i_{X}M" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.16.m16.1"><semantics id="S3.Thmthm11.p2.16.m16.1a"><mrow id="S3.Thmthm11.p2.16.m16.1.1" xref="S3.Thmthm11.p2.16.m16.1.1.cmml"><msub id="S3.Thmthm11.p2.16.m16.1.1.2" xref="S3.Thmthm11.p2.16.m16.1.1.2.cmml"><mi id="S3.Thmthm11.p2.16.m16.1.1.2.2" xref="S3.Thmthm11.p2.16.m16.1.1.2.2.cmml">i</mi><mi id="S3.Thmthm11.p2.16.m16.1.1.2.3" xref="S3.Thmthm11.p2.16.m16.1.1.2.3.cmml">X</mi></msub><mo id="S3.Thmthm11.p2.16.m16.1.1.1" xref="S3.Thmthm11.p2.16.m16.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p2.16.m16.1.1.3" xref="S3.Thmthm11.p2.16.m16.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.16.m16.1b"><apply id="S3.Thmthm11.p2.16.m16.1.1.cmml" xref="S3.Thmthm11.p2.16.m16.1.1"><times id="S3.Thmthm11.p2.16.m16.1.1.1.cmml" xref="S3.Thmthm11.p2.16.m16.1.1.1"></times><apply id="S3.Thmthm11.p2.16.m16.1.1.2.cmml" xref="S3.Thmthm11.p2.16.m16.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.16.m16.1.1.2.1.cmml" xref="S3.Thmthm11.p2.16.m16.1.1.2">subscript</csymbol><ci id="S3.Thmthm11.p2.16.m16.1.1.2.2.cmml" xref="S3.Thmthm11.p2.16.m16.1.1.2.2">𝑖</ci><ci id="S3.Thmthm11.p2.16.m16.1.1.2.3.cmml" xref="S3.Thmthm11.p2.16.m16.1.1.2.3">𝑋</ci></apply><ci id="S3.Thmthm11.p2.16.m16.1.1.3.cmml" xref="S3.Thmthm11.p2.16.m16.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.16.m16.1c">i_{X}M</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.16.m16.1d">italic_i start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> from <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.17.m17.1"><semantics id="S3.Thmthm11.p2.17.m17.1a"><mrow id="S3.Thmthm11.p2.17.m17.1.2" xref="S3.Thmthm11.p2.17.m17.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p2.17.m17.1.2.2" xref="S3.Thmthm11.p2.17.m17.1.2.2.cmml">ℳ</mi><mo id="S3.Thmthm11.p2.17.m17.1.2.1" xref="S3.Thmthm11.p2.17.m17.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.17.m17.1.2.3.2" xref="S3.Thmthm11.p2.17.m17.1.2.cmml"><mo id="S3.Thmthm11.p2.17.m17.1.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.17.m17.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p2.17.m17.1.1" xref="S3.Thmthm11.p2.17.m17.1.1.cmml">𝒳</mi><mo id="S3.Thmthm11.p2.17.m17.1.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p2.17.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.17.m17.1b"><apply id="S3.Thmthm11.p2.17.m17.1.2.cmml" xref="S3.Thmthm11.p2.17.m17.1.2"><times id="S3.Thmthm11.p2.17.m17.1.2.1.cmml" xref="S3.Thmthm11.p2.17.m17.1.2.1"></times><ci id="S3.Thmthm11.p2.17.m17.1.2.2.cmml" xref="S3.Thmthm11.p2.17.m17.1.2.2">ℳ</ci><ci id="S3.Thmthm11.p2.17.m17.1.1.cmml" xref="S3.Thmthm11.p2.17.m17.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.17.m17.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.17.m17.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> to the non-negative cone <math alttext="H_{+}^{*}(X,\mathbb{R})" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.18.m18.2"><semantics id="S3.Thmthm11.p2.18.m18.2a"><mrow id="S3.Thmthm11.p2.18.m18.2.3" xref="S3.Thmthm11.p2.18.m18.2.3.cmml"><msubsup id="S3.Thmthm11.p2.18.m18.2.3.2" xref="S3.Thmthm11.p2.18.m18.2.3.2.cmml"><mi id="S3.Thmthm11.p2.18.m18.2.3.2.2.2" xref="S3.Thmthm11.p2.18.m18.2.3.2.2.2.cmml">H</mi><mo id="S3.Thmthm11.p2.18.m18.2.3.2.2.3" xref="S3.Thmthm11.p2.18.m18.2.3.2.2.3.cmml">+</mo><mo id="S3.Thmthm11.p2.18.m18.2.3.2.3" xref="S3.Thmthm11.p2.18.m18.2.3.2.3.cmml">∗</mo></msubsup><mo id="S3.Thmthm11.p2.18.m18.2.3.1" xref="S3.Thmthm11.p2.18.m18.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.18.m18.2.3.3.2" xref="S3.Thmthm11.p2.18.m18.2.3.3.1.cmml"><mo id="S3.Thmthm11.p2.18.m18.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.18.m18.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p2.18.m18.1.1" xref="S3.Thmthm11.p2.18.m18.1.1.cmml">X</mi><mo id="S3.Thmthm11.p2.18.m18.2.3.3.2.2" xref="S3.Thmthm11.p2.18.m18.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p2.18.m18.2.2" xref="S3.Thmthm11.p2.18.m18.2.2.cmml">ℝ</mi><mo id="S3.Thmthm11.p2.18.m18.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p2.18.m18.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.18.m18.2b"><apply id="S3.Thmthm11.p2.18.m18.2.3.cmml" xref="S3.Thmthm11.p2.18.m18.2.3"><times id="S3.Thmthm11.p2.18.m18.2.3.1.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.1"></times><apply id="S3.Thmthm11.p2.18.m18.2.3.2.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.18.m18.2.3.2.1.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.2">superscript</csymbol><apply id="S3.Thmthm11.p2.18.m18.2.3.2.2.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.18.m18.2.3.2.2.1.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.2">subscript</csymbol><ci id="S3.Thmthm11.p2.18.m18.2.3.2.2.2.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.2.2.2">𝐻</ci><plus id="S3.Thmthm11.p2.18.m18.2.3.2.2.3.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.2.2.3"></plus></apply><times id="S3.Thmthm11.p2.18.m18.2.3.2.3.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.2.3"></times></apply><interval closure="open" id="S3.Thmthm11.p2.18.m18.2.3.3.1.cmml" xref="S3.Thmthm11.p2.18.m18.2.3.3.2"><ci id="S3.Thmthm11.p2.18.m18.1.1.cmml" xref="S3.Thmthm11.p2.18.m18.1.1">𝑋</ci><ci id="S3.Thmthm11.p2.18.m18.2.2.cmml" xref="S3.Thmthm11.p2.18.m18.2.2">ℝ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.18.m18.2c">H_{+}^{*}(X,\mathbb{R})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.18.m18.2d">italic_H start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X , blackboard_R )</annotation></semantics></math> in the dual group <math alttext="H^{*}(X,\mathbb{R})" class="ltx_Math" display="inline" id="S3.Thmthm11.p2.19.m19.2"><semantics id="S3.Thmthm11.p2.19.m19.2a"><mrow id="S3.Thmthm11.p2.19.m19.2.3" xref="S3.Thmthm11.p2.19.m19.2.3.cmml"><msup id="S3.Thmthm11.p2.19.m19.2.3.2" xref="S3.Thmthm11.p2.19.m19.2.3.2.cmml"><mi id="S3.Thmthm11.p2.19.m19.2.3.2.2" xref="S3.Thmthm11.p2.19.m19.2.3.2.2.cmml">H</mi><mo id="S3.Thmthm11.p2.19.m19.2.3.2.3" xref="S3.Thmthm11.p2.19.m19.2.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm11.p2.19.m19.2.3.1" xref="S3.Thmthm11.p2.19.m19.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p2.19.m19.2.3.3.2" xref="S3.Thmthm11.p2.19.m19.2.3.3.1.cmml"><mo id="S3.Thmthm11.p2.19.m19.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p2.19.m19.2.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p2.19.m19.1.1" xref="S3.Thmthm11.p2.19.m19.1.1.cmml">X</mi><mo id="S3.Thmthm11.p2.19.m19.2.3.3.2.2" xref="S3.Thmthm11.p2.19.m19.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p2.19.m19.2.2" xref="S3.Thmthm11.p2.19.m19.2.2.cmml">ℝ</mi><mo id="S3.Thmthm11.p2.19.m19.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p2.19.m19.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p2.19.m19.2b"><apply id="S3.Thmthm11.p2.19.m19.2.3.cmml" xref="S3.Thmthm11.p2.19.m19.2.3"><times id="S3.Thmthm11.p2.19.m19.2.3.1.cmml" xref="S3.Thmthm11.p2.19.m19.2.3.1"></times><apply id="S3.Thmthm11.p2.19.m19.2.3.2.cmml" xref="S3.Thmthm11.p2.19.m19.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p2.19.m19.2.3.2.1.cmml" xref="S3.Thmthm11.p2.19.m19.2.3.2">superscript</csymbol><ci id="S3.Thmthm11.p2.19.m19.2.3.2.2.cmml" xref="S3.Thmthm11.p2.19.m19.2.3.2.2">𝐻</ci><times id="S3.Thmthm11.p2.19.m19.2.3.2.3.cmml" xref="S3.Thmthm11.p2.19.m19.2.3.2.3"></times></apply><interval closure="open" id="S3.Thmthm11.p2.19.m19.2.3.3.1.cmml" xref="S3.Thmthm11.p2.19.m19.2.3.3.2"><ci id="S3.Thmthm11.p2.19.m19.1.1.cmml" xref="S3.Thmthm11.p2.19.m19.1.1">𝑋</ci><ci id="S3.Thmthm11.p2.19.m19.2.2.cmml" xref="S3.Thmthm11.p2.19.m19.2.2">ℝ</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p2.19.m19.2c">H^{*}(X,\mathbb{R})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p2.19.m19.2d">italic_H start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X , blackboard_R )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.Thmthm11.p3"> <p class="ltx_p" id="S3.Thmthm11.p3.5">Given now a morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm11.p3.1.m1.1"><semantics id="S3.Thmthm11.p3.1.m1.1a"><mrow id="S3.Thmthm11.p3.1.m1.1.1" xref="S3.Thmthm11.p3.1.m1.1.1.cmml"><mi id="S3.Thmthm11.p3.1.m1.1.1.2" xref="S3.Thmthm11.p3.1.m1.1.1.2.cmml">σ</mi><mo id="S3.Thmthm11.p3.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm11.p3.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm11.p3.1.m1.1.1.3" xref="S3.Thmthm11.p3.1.m1.1.1.3.cmml"><msup id="S3.Thmthm11.p3.1.m1.1.1.3.2" xref="S3.Thmthm11.p3.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p3.1.m1.1.1.3.2.2" xref="S3.Thmthm11.p3.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm11.p3.1.m1.1.1.3.2.3" xref="S3.Thmthm11.p3.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm11.p3.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm11.p3.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.Thmthm11.p3.1.m1.1.1.3.3" xref="S3.Thmthm11.p3.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p3.1.m1.1.1.3.3.2" xref="S3.Thmthm11.p3.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.Thmthm11.p3.1.m1.1.1.3.3.3" xref="S3.Thmthm11.p3.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p3.1.m1.1b"><apply id="S3.Thmthm11.p3.1.m1.1.1.cmml" xref="S3.Thmthm11.p3.1.m1.1.1"><ci id="S3.Thmthm11.p3.1.m1.1.1.1.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.1">:</ci><ci id="S3.Thmthm11.p3.1.m1.1.1.2.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.2">𝜎</ci><apply id="S3.Thmthm11.p3.1.m1.1.1.3.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3"><ci id="S3.Thmthm11.p3.1.m1.1.1.3.1.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm11.p3.1.m1.1.1.3.2.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm11.p3.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm11.p3.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm11.p3.1.m1.1.1.3.3.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.Thmthm11.p3.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.Thmthm11.p3.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm11.p3.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p3.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p3.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, then for the image subshift <math alttext="Y=\sigma(X)" class="ltx_Math" display="inline" id="S3.Thmthm11.p3.2.m2.1"><semantics id="S3.Thmthm11.p3.2.m2.1a"><mrow id="S3.Thmthm11.p3.2.m2.1.2" xref="S3.Thmthm11.p3.2.m2.1.2.cmml"><mi id="S3.Thmthm11.p3.2.m2.1.2.2" xref="S3.Thmthm11.p3.2.m2.1.2.2.cmml">Y</mi><mo id="S3.Thmthm11.p3.2.m2.1.2.1" xref="S3.Thmthm11.p3.2.m2.1.2.1.cmml">=</mo><mrow id="S3.Thmthm11.p3.2.m2.1.2.3" xref="S3.Thmthm11.p3.2.m2.1.2.3.cmml"><mi id="S3.Thmthm11.p3.2.m2.1.2.3.2" xref="S3.Thmthm11.p3.2.m2.1.2.3.2.cmml">σ</mi><mo id="S3.Thmthm11.p3.2.m2.1.2.3.1" xref="S3.Thmthm11.p3.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p3.2.m2.1.2.3.3.2" xref="S3.Thmthm11.p3.2.m2.1.2.3.cmml"><mo id="S3.Thmthm11.p3.2.m2.1.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p3.2.m2.1.2.3.cmml">(</mo><mi id="S3.Thmthm11.p3.2.m2.1.1" xref="S3.Thmthm11.p3.2.m2.1.1.cmml">X</mi><mo id="S3.Thmthm11.p3.2.m2.1.2.3.3.2.2" stretchy="false" xref="S3.Thmthm11.p3.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p3.2.m2.1b"><apply id="S3.Thmthm11.p3.2.m2.1.2.cmml" xref="S3.Thmthm11.p3.2.m2.1.2"><eq id="S3.Thmthm11.p3.2.m2.1.2.1.cmml" xref="S3.Thmthm11.p3.2.m2.1.2.1"></eq><ci id="S3.Thmthm11.p3.2.m2.1.2.2.cmml" xref="S3.Thmthm11.p3.2.m2.1.2.2">𝑌</ci><apply id="S3.Thmthm11.p3.2.m2.1.2.3.cmml" xref="S3.Thmthm11.p3.2.m2.1.2.3"><times id="S3.Thmthm11.p3.2.m2.1.2.3.1.cmml" xref="S3.Thmthm11.p3.2.m2.1.2.3.1"></times><ci id="S3.Thmthm11.p3.2.m2.1.2.3.2.cmml" xref="S3.Thmthm11.p3.2.m2.1.2.3.2">𝜎</ci><ci id="S3.Thmthm11.p3.2.m2.1.1.cmml" xref="S3.Thmthm11.p3.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p3.2.m2.1c">Y=\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p3.2.m2.1d">italic_Y = italic_σ ( italic_X )</annotation></semantics></math> the measure transfer map defines canonically a map <math alttext="i_{Y}M\circ\sigma_{X}M\circ(i_{X}M)^{-1}:H_{+}^{*}(X,\mathbb{R})\to H_{+}^{*}(% Y,\mathbb{R})" class="ltx_Math" display="inline" id="S3.Thmthm11.p3.3.m3.5"><semantics id="S3.Thmthm11.p3.3.m3.5a"><mrow id="S3.Thmthm11.p3.3.m3.5.5" xref="S3.Thmthm11.p3.3.m3.5.5.cmml"><mrow id="S3.Thmthm11.p3.3.m3.5.5.1" xref="S3.Thmthm11.p3.3.m3.5.5.1.cmml"><mrow id="S3.Thmthm11.p3.3.m3.5.5.1.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.cmml"><mrow id="S3.Thmthm11.p3.3.m3.5.5.1.3.2" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.cmml"><mrow id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.cmml"><msub id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.cmml"><mi id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.2.cmml">i</mi><mi id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.3.cmml">Y</mi></msub><mo id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.1" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.1.cmml">⁢</mo><mi id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.3.cmml">M</mi></mrow><mo id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.1.cmml">∘</mo><msub id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.cmml"><mi id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.2" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.2.cmml">σ</mi><mi id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.3.cmml">X</mi></msub></mrow><mo id="S3.Thmthm11.p3.3.m3.5.5.1.3.1" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p3.3.m3.5.5.1.3.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.3.cmml">M</mi></mrow><mo id="S3.Thmthm11.p3.3.m3.5.5.1.2" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p3.3.m3.5.5.1.2.cmml">∘</mo><msup id="S3.Thmthm11.p3.3.m3.5.5.1.1" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.cmml"><mrow id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.cmml"><mo id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.cmml"><msub id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.2.cmml">i</mi><mi id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.3.cmml">X</mi></msub><mo id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.1" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.3.cmml">M</mi></mrow><mo id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.Thmthm11.p3.3.m3.5.5.1.1.3" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.3.cmml"><mo id="S3.Thmthm11.p3.3.m3.5.5.1.1.3a" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.3.cmml">−</mo><mn id="S3.Thmthm11.p3.3.m3.5.5.1.1.3.2" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S3.Thmthm11.p3.3.m3.5.5.2" rspace="0.278em" xref="S3.Thmthm11.p3.3.m3.5.5.2.cmml">:</mo><mrow id="S3.Thmthm11.p3.3.m3.5.5.3" xref="S3.Thmthm11.p3.3.m3.5.5.3.cmml"><mrow id="S3.Thmthm11.p3.3.m3.5.5.3.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.cmml"><msubsup id="S3.Thmthm11.p3.3.m3.5.5.3.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2.cmml"><mi id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.2.cmml">H</mi><mo id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.3" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.3.cmml">+</mo><mo id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.3" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2.3.cmml">∗</mo></msubsup><mo id="S3.Thmthm11.p3.3.m3.5.5.3.2.1" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p3.3.m3.5.5.3.2.3.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.3.1.cmml"><mo id="S3.Thmthm11.p3.3.m3.5.5.3.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.3.1.cmml">(</mo><mi id="S3.Thmthm11.p3.3.m3.1.1" xref="S3.Thmthm11.p3.3.m3.1.1.cmml">X</mi><mo id="S3.Thmthm11.p3.3.m3.5.5.3.2.3.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.3.1.cmml">,</mo><mi id="S3.Thmthm11.p3.3.m3.2.2" xref="S3.Thmthm11.p3.3.m3.2.2.cmml">ℝ</mi><mo id="S3.Thmthm11.p3.3.m3.5.5.3.2.3.2.3" stretchy="false" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p3.3.m3.5.5.3.1" stretchy="false" xref="S3.Thmthm11.p3.3.m3.5.5.3.1.cmml">→</mo><mrow id="S3.Thmthm11.p3.3.m3.5.5.3.3" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.cmml"><msubsup id="S3.Thmthm11.p3.3.m3.5.5.3.3.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2.cmml"><mi id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.2.cmml">H</mi><mo id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.3" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.3.cmml">+</mo><mo id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.3" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2.3.cmml">∗</mo></msubsup><mo id="S3.Thmthm11.p3.3.m3.5.5.3.3.1" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p3.3.m3.5.5.3.3.3.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.3.1.cmml"><mo id="S3.Thmthm11.p3.3.m3.5.5.3.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p3.3.m3.3.3" xref="S3.Thmthm11.p3.3.m3.3.3.cmml">Y</mi><mo id="S3.Thmthm11.p3.3.m3.5.5.3.3.3.2.2" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p3.3.m3.4.4" xref="S3.Thmthm11.p3.3.m3.4.4.cmml">ℝ</mi><mo id="S3.Thmthm11.p3.3.m3.5.5.3.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p3.3.m3.5b"><apply id="S3.Thmthm11.p3.3.m3.5.5.cmml" xref="S3.Thmthm11.p3.3.m3.5.5"><ci id="S3.Thmthm11.p3.3.m3.5.5.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.2">:</ci><apply id="S3.Thmthm11.p3.3.m3.5.5.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1"><compose id="S3.Thmthm11.p3.3.m3.5.5.1.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.2"></compose><apply id="S3.Thmthm11.p3.3.m3.5.5.1.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3"><times id="S3.Thmthm11.p3.3.m3.5.5.1.3.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.1"></times><apply id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2"><compose id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.1"></compose><apply id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2"><times id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.1"></times><apply id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2">subscript</csymbol><ci id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.2">𝑖</ci><ci id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.2.3">𝑌</ci></apply><ci id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.2.3">𝑀</ci></apply><apply id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3">subscript</csymbol><ci id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.2">𝜎</ci><ci id="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.2.3.3">𝑋</ci></apply></apply><ci id="S3.Thmthm11.p3.3.m3.5.5.1.3.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.3.3">𝑀</ci></apply><apply id="S3.Thmthm11.p3.3.m3.5.5.1.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.3.m3.5.5.1.1.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1">superscript</csymbol><apply id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1"><times id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.1"></times><apply id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2">subscript</csymbol><ci id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.2">𝑖</ci><ci id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.2.3">𝑋</ci></apply><ci id="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.1.1.1.3">𝑀</ci></apply><apply id="S3.Thmthm11.p3.3.m3.5.5.1.1.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.3"><minus id="S3.Thmthm11.p3.3.m3.5.5.1.1.3.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.3"></minus><cn id="S3.Thmthm11.p3.3.m3.5.5.1.1.3.2.cmml" type="integer" xref="S3.Thmthm11.p3.3.m3.5.5.1.1.3.2">1</cn></apply></apply></apply><apply id="S3.Thmthm11.p3.3.m3.5.5.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3"><ci id="S3.Thmthm11.p3.3.m3.5.5.3.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.1">→</ci><apply id="S3.Thmthm11.p3.3.m3.5.5.3.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2"><times id="S3.Thmthm11.p3.3.m3.5.5.3.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.1"></times><apply id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2">superscript</csymbol><apply id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2">subscript</csymbol><ci id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.2">𝐻</ci><plus id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2.2.3"></plus></apply><times id="S3.Thmthm11.p3.3.m3.5.5.3.2.2.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.2.3"></times></apply><interval closure="open" id="S3.Thmthm11.p3.3.m3.5.5.3.2.3.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.2.3.2"><ci id="S3.Thmthm11.p3.3.m3.1.1.cmml" xref="S3.Thmthm11.p3.3.m3.1.1">𝑋</ci><ci id="S3.Thmthm11.p3.3.m3.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.2.2">ℝ</ci></interval></apply><apply id="S3.Thmthm11.p3.3.m3.5.5.3.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3"><times id="S3.Thmthm11.p3.3.m3.5.5.3.3.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.1"></times><apply id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2">superscript</csymbol><apply id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2">subscript</csymbol><ci id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.2.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.2">𝐻</ci><plus id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2.2.3"></plus></apply><times id="S3.Thmthm11.p3.3.m3.5.5.3.3.2.3.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.2.3"></times></apply><interval closure="open" id="S3.Thmthm11.p3.3.m3.5.5.3.3.3.1.cmml" xref="S3.Thmthm11.p3.3.m3.5.5.3.3.3.2"><ci id="S3.Thmthm11.p3.3.m3.3.3.cmml" xref="S3.Thmthm11.p3.3.m3.3.3">𝑌</ci><ci id="S3.Thmthm11.p3.3.m3.4.4.cmml" xref="S3.Thmthm11.p3.3.m3.4.4">ℝ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p3.3.m3.5c">i_{Y}M\circ\sigma_{X}M\circ(i_{X}M)^{-1}:H_{+}^{*}(X,\mathbb{R})\to H_{+}^{*}(% Y,\mathbb{R})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p3.3.m3.5d">italic_i start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT italic_M ∘ italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M ∘ ( italic_i start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT : italic_H start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_X , blackboard_R ) → italic_H start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_Y , blackboard_R )</annotation></semantics></math>, and hence, after normalization, a map from the set of states for <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmthm11.p3.4.m4.1"><semantics id="S3.Thmthm11.p3.4.m4.1a"><mi id="S3.Thmthm11.p3.4.m4.1.1" xref="S3.Thmthm11.p3.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p3.4.m4.1b"><ci id="S3.Thmthm11.p3.4.m4.1.1.cmml" xref="S3.Thmthm11.p3.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p3.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p3.4.m4.1d">italic_X</annotation></semantics></math> to the set of states for <math alttext="Y" class="ltx_Math" display="inline" id="S3.Thmthm11.p3.5.m5.1"><semantics id="S3.Thmthm11.p3.5.m5.1a"><mi id="S3.Thmthm11.p3.5.m5.1.1" xref="S3.Thmthm11.p3.5.m5.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p3.5.m5.1b"><ci id="S3.Thmthm11.p3.5.m5.1.1.cmml" xref="S3.Thmthm11.p3.5.m5.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p3.5.m5.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p3.5.m5.1d">italic_Y</annotation></semantics></math>. The natural question arises, whether this map can be defined alternatively by a more direct approach.</p> </div> <div class="ltx_para" id="S3.Thmthm11.p4"> <p class="ltx_p" id="S3.Thmthm11.p4.5">Indeed, if <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.1.m1.1"><semantics id="S3.Thmthm11.p4.1.m1.1a"><mrow id="S3.Thmthm11.p4.1.m1.1.1" xref="S3.Thmthm11.p4.1.m1.1.1.cmml"><mi id="S3.Thmthm11.p4.1.m1.1.1.2" xref="S3.Thmthm11.p4.1.m1.1.1.2.cmml">σ</mi><mo id="S3.Thmthm11.p4.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm11.p4.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm11.p4.1.m1.1.1.3" xref="S3.Thmthm11.p4.1.m1.1.1.3.cmml"><msup id="S3.Thmthm11.p4.1.m1.1.1.3.2" xref="S3.Thmthm11.p4.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p4.1.m1.1.1.3.2.2" xref="S3.Thmthm11.p4.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm11.p4.1.m1.1.1.3.2.3" xref="S3.Thmthm11.p4.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm11.p4.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm11.p4.1.m1.1.1.3.1.cmml">→</mo><msup id="S3.Thmthm11.p4.1.m1.1.1.3.3" xref="S3.Thmthm11.p4.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p4.1.m1.1.1.3.3.2" xref="S3.Thmthm11.p4.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.Thmthm11.p4.1.m1.1.1.3.3.3" xref="S3.Thmthm11.p4.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.1.m1.1b"><apply id="S3.Thmthm11.p4.1.m1.1.1.cmml" xref="S3.Thmthm11.p4.1.m1.1.1"><ci id="S3.Thmthm11.p4.1.m1.1.1.1.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.1">:</ci><ci id="S3.Thmthm11.p4.1.m1.1.1.2.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.2">𝜎</ci><apply id="S3.Thmthm11.p4.1.m1.1.1.3.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3"><ci id="S3.Thmthm11.p4.1.m1.1.1.3.1.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.1">→</ci><apply id="S3.Thmthm11.p4.1.m1.1.1.3.2.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.1.m1.1.1.3.2.1.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm11.p4.1.m1.1.1.3.2.2.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm11.p4.1.m1.1.1.3.2.3.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.2.3"></times></apply><apply id="S3.Thmthm11.p4.1.m1.1.1.3.3.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.1.m1.1.1.3.3.1.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.Thmthm11.p4.1.m1.1.1.3.3.2.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.3.2">ℬ</ci><times id="S3.Thmthm11.p4.1.m1.1.1.3.3.3.cmml" xref="S3.Thmthm11.p4.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is letter-to-letter, then for two subshifts <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.2.m2.1"><semantics id="S3.Thmthm11.p4.2.m2.1a"><mrow id="S3.Thmthm11.p4.2.m2.1.1" xref="S3.Thmthm11.p4.2.m2.1.1.cmml"><mi id="S3.Thmthm11.p4.2.m2.1.1.2" xref="S3.Thmthm11.p4.2.m2.1.1.2.cmml">X</mi><mo id="S3.Thmthm11.p4.2.m2.1.1.1" xref="S3.Thmthm11.p4.2.m2.1.1.1.cmml">⊆</mo><msup id="S3.Thmthm11.p4.2.m2.1.1.3" xref="S3.Thmthm11.p4.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p4.2.m2.1.1.3.2" xref="S3.Thmthm11.p4.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S3.Thmthm11.p4.2.m2.1.1.3.3" xref="S3.Thmthm11.p4.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.2.m2.1b"><apply id="S3.Thmthm11.p4.2.m2.1.1.cmml" xref="S3.Thmthm11.p4.2.m2.1.1"><subset id="S3.Thmthm11.p4.2.m2.1.1.1.cmml" xref="S3.Thmthm11.p4.2.m2.1.1.1"></subset><ci id="S3.Thmthm11.p4.2.m2.1.1.2.cmml" xref="S3.Thmthm11.p4.2.m2.1.1.2">𝑋</ci><apply id="S3.Thmthm11.p4.2.m2.1.1.3.cmml" xref="S3.Thmthm11.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.2.m2.1.1.3.1.cmml" xref="S3.Thmthm11.p4.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmthm11.p4.2.m2.1.1.3.2.cmml" xref="S3.Thmthm11.p4.2.m2.1.1.3.2">𝒜</ci><ci id="S3.Thmthm11.p4.2.m2.1.1.3.3.cmml" xref="S3.Thmthm11.p4.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="Y=\sigma(X)=\sigma^{\mathbb{Z}}(X)" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.3.m3.2"><semantics id="S3.Thmthm11.p4.3.m3.2a"><mrow id="S3.Thmthm11.p4.3.m3.2.3" xref="S3.Thmthm11.p4.3.m3.2.3.cmml"><mi id="S3.Thmthm11.p4.3.m3.2.3.2" xref="S3.Thmthm11.p4.3.m3.2.3.2.cmml">Y</mi><mo id="S3.Thmthm11.p4.3.m3.2.3.3" xref="S3.Thmthm11.p4.3.m3.2.3.3.cmml">=</mo><mrow id="S3.Thmthm11.p4.3.m3.2.3.4" xref="S3.Thmthm11.p4.3.m3.2.3.4.cmml"><mi id="S3.Thmthm11.p4.3.m3.2.3.4.2" xref="S3.Thmthm11.p4.3.m3.2.3.4.2.cmml">σ</mi><mo id="S3.Thmthm11.p4.3.m3.2.3.4.1" xref="S3.Thmthm11.p4.3.m3.2.3.4.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.3.m3.2.3.4.3.2" xref="S3.Thmthm11.p4.3.m3.2.3.4.cmml"><mo id="S3.Thmthm11.p4.3.m3.2.3.4.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.3.m3.2.3.4.cmml">(</mo><mi id="S3.Thmthm11.p4.3.m3.1.1" xref="S3.Thmthm11.p4.3.m3.1.1.cmml">X</mi><mo id="S3.Thmthm11.p4.3.m3.2.3.4.3.2.2" stretchy="false" xref="S3.Thmthm11.p4.3.m3.2.3.4.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p4.3.m3.2.3.5" xref="S3.Thmthm11.p4.3.m3.2.3.5.cmml">=</mo><mrow id="S3.Thmthm11.p4.3.m3.2.3.6" xref="S3.Thmthm11.p4.3.m3.2.3.6.cmml"><msup id="S3.Thmthm11.p4.3.m3.2.3.6.2" xref="S3.Thmthm11.p4.3.m3.2.3.6.2.cmml"><mi id="S3.Thmthm11.p4.3.m3.2.3.6.2.2" xref="S3.Thmthm11.p4.3.m3.2.3.6.2.2.cmml">σ</mi><mi id="S3.Thmthm11.p4.3.m3.2.3.6.2.3" xref="S3.Thmthm11.p4.3.m3.2.3.6.2.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm11.p4.3.m3.2.3.6.1" xref="S3.Thmthm11.p4.3.m3.2.3.6.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.3.m3.2.3.6.3.2" xref="S3.Thmthm11.p4.3.m3.2.3.6.cmml"><mo id="S3.Thmthm11.p4.3.m3.2.3.6.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.3.m3.2.3.6.cmml">(</mo><mi id="S3.Thmthm11.p4.3.m3.2.2" xref="S3.Thmthm11.p4.3.m3.2.2.cmml">X</mi><mo id="S3.Thmthm11.p4.3.m3.2.3.6.3.2.2" stretchy="false" xref="S3.Thmthm11.p4.3.m3.2.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.3.m3.2b"><apply id="S3.Thmthm11.p4.3.m3.2.3.cmml" xref="S3.Thmthm11.p4.3.m3.2.3"><and id="S3.Thmthm11.p4.3.m3.2.3a.cmml" xref="S3.Thmthm11.p4.3.m3.2.3"></and><apply id="S3.Thmthm11.p4.3.m3.2.3b.cmml" xref="S3.Thmthm11.p4.3.m3.2.3"><eq id="S3.Thmthm11.p4.3.m3.2.3.3.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.3"></eq><ci id="S3.Thmthm11.p4.3.m3.2.3.2.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.2">𝑌</ci><apply id="S3.Thmthm11.p4.3.m3.2.3.4.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.4"><times id="S3.Thmthm11.p4.3.m3.2.3.4.1.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.4.1"></times><ci id="S3.Thmthm11.p4.3.m3.2.3.4.2.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.4.2">𝜎</ci><ci id="S3.Thmthm11.p4.3.m3.1.1.cmml" xref="S3.Thmthm11.p4.3.m3.1.1">𝑋</ci></apply></apply><apply id="S3.Thmthm11.p4.3.m3.2.3c.cmml" xref="S3.Thmthm11.p4.3.m3.2.3"><eq id="S3.Thmthm11.p4.3.m3.2.3.5.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S3.Thmthm11.p4.3.m3.2.3.4.cmml" id="S3.Thmthm11.p4.3.m3.2.3d.cmml" xref="S3.Thmthm11.p4.3.m3.2.3"></share><apply id="S3.Thmthm11.p4.3.m3.2.3.6.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.6"><times id="S3.Thmthm11.p4.3.m3.2.3.6.1.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.6.1"></times><apply id="S3.Thmthm11.p4.3.m3.2.3.6.2.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.6.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.3.m3.2.3.6.2.1.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.6.2">superscript</csymbol><ci id="S3.Thmthm11.p4.3.m3.2.3.6.2.2.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.6.2.2">𝜎</ci><ci id="S3.Thmthm11.p4.3.m3.2.3.6.2.3.cmml" xref="S3.Thmthm11.p4.3.m3.2.3.6.2.3">ℤ</ci></apply><ci id="S3.Thmthm11.p4.3.m3.2.2.cmml" xref="S3.Thmthm11.p4.3.m3.2.2">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.3.m3.2c">Y=\sigma(X)=\sigma^{\mathbb{Z}}(X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.3.m3.2d">italic_Y = italic_σ ( italic_X ) = italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( italic_X )</annotation></semantics></math> any function <math alttext="g\in\partial_{T}C(Y,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.4.m4.2"><semantics id="S3.Thmthm11.p4.4.m4.2a"><mrow id="S3.Thmthm11.p4.4.m4.2.3" xref="S3.Thmthm11.p4.4.m4.2.3.cmml"><mi id="S3.Thmthm11.p4.4.m4.2.3.2" xref="S3.Thmthm11.p4.4.m4.2.3.2.cmml">g</mi><mo id="S3.Thmthm11.p4.4.m4.2.3.1" rspace="0.1389em" xref="S3.Thmthm11.p4.4.m4.2.3.1.cmml">∈</mo><mrow id="S3.Thmthm11.p4.4.m4.2.3.3" xref="S3.Thmthm11.p4.4.m4.2.3.3.cmml"><msub id="S3.Thmthm11.p4.4.m4.2.3.3.1" xref="S3.Thmthm11.p4.4.m4.2.3.3.1.cmml"><mo id="S3.Thmthm11.p4.4.m4.2.3.3.1.2" lspace="0.1389em" rspace="0em" xref="S3.Thmthm11.p4.4.m4.2.3.3.1.2.cmml">∂</mo><mi id="S3.Thmthm11.p4.4.m4.2.3.3.1.3" xref="S3.Thmthm11.p4.4.m4.2.3.3.1.3.cmml">T</mi></msub><mrow id="S3.Thmthm11.p4.4.m4.2.3.3.2" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.cmml"><mi id="S3.Thmthm11.p4.4.m4.2.3.3.2.2" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.2.cmml">C</mi><mo id="S3.Thmthm11.p4.4.m4.2.3.3.2.1" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.4.m4.2.3.3.2.3.2" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.3.1.cmml"><mo id="S3.Thmthm11.p4.4.m4.2.3.3.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.3.1.cmml">(</mo><mi id="S3.Thmthm11.p4.4.m4.1.1" xref="S3.Thmthm11.p4.4.m4.1.1.cmml">Y</mi><mo id="S3.Thmthm11.p4.4.m4.2.3.3.2.3.2.2" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.3.1.cmml">,</mo><mi id="S3.Thmthm11.p4.4.m4.2.2" xref="S3.Thmthm11.p4.4.m4.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p4.4.m4.2.3.3.2.3.2.3" stretchy="false" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.4.m4.2b"><apply id="S3.Thmthm11.p4.4.m4.2.3.cmml" xref="S3.Thmthm11.p4.4.m4.2.3"><in id="S3.Thmthm11.p4.4.m4.2.3.1.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.1"></in><ci id="S3.Thmthm11.p4.4.m4.2.3.2.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.2">𝑔</ci><apply id="S3.Thmthm11.p4.4.m4.2.3.3.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3"><apply id="S3.Thmthm11.p4.4.m4.2.3.3.1.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.4.m4.2.3.3.1.1.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3.1">subscript</csymbol><partialdiff id="S3.Thmthm11.p4.4.m4.2.3.3.1.2.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3.1.2"></partialdiff><ci id="S3.Thmthm11.p4.4.m4.2.3.3.1.3.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3.1.3">𝑇</ci></apply><apply id="S3.Thmthm11.p4.4.m4.2.3.3.2.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3.2"><times id="S3.Thmthm11.p4.4.m4.2.3.3.2.1.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.1"></times><ci id="S3.Thmthm11.p4.4.m4.2.3.3.2.2.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p4.4.m4.2.3.3.2.3.1.cmml" xref="S3.Thmthm11.p4.4.m4.2.3.3.2.3.2"><ci id="S3.Thmthm11.p4.4.m4.1.1.cmml" xref="S3.Thmthm11.p4.4.m4.1.1">𝑌</ci><ci id="S3.Thmthm11.p4.4.m4.2.2.cmml" xref="S3.Thmthm11.p4.4.m4.2.2">ℤ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.4.m4.2c">g\in\partial_{T}C(Y,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.4.m4.2d">italic_g ∈ ∂ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_C ( italic_Y , blackboard_Z )</annotation></semantics></math> defines a function <math alttext="g\circ\sigma^{\mathbb{Z}}\in C(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.5.m5.2"><semantics id="S3.Thmthm11.p4.5.m5.2a"><mrow id="S3.Thmthm11.p4.5.m5.2.3" xref="S3.Thmthm11.p4.5.m5.2.3.cmml"><mrow id="S3.Thmthm11.p4.5.m5.2.3.2" xref="S3.Thmthm11.p4.5.m5.2.3.2.cmml"><mi id="S3.Thmthm11.p4.5.m5.2.3.2.2" xref="S3.Thmthm11.p4.5.m5.2.3.2.2.cmml">g</mi><mo id="S3.Thmthm11.p4.5.m5.2.3.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p4.5.m5.2.3.2.1.cmml">∘</mo><msup id="S3.Thmthm11.p4.5.m5.2.3.2.3" xref="S3.Thmthm11.p4.5.m5.2.3.2.3.cmml"><mi id="S3.Thmthm11.p4.5.m5.2.3.2.3.2" xref="S3.Thmthm11.p4.5.m5.2.3.2.3.2.cmml">σ</mi><mi id="S3.Thmthm11.p4.5.m5.2.3.2.3.3" xref="S3.Thmthm11.p4.5.m5.2.3.2.3.3.cmml">ℤ</mi></msup></mrow><mo id="S3.Thmthm11.p4.5.m5.2.3.1" xref="S3.Thmthm11.p4.5.m5.2.3.1.cmml">∈</mo><mrow id="S3.Thmthm11.p4.5.m5.2.3.3" xref="S3.Thmthm11.p4.5.m5.2.3.3.cmml"><mi id="S3.Thmthm11.p4.5.m5.2.3.3.2" xref="S3.Thmthm11.p4.5.m5.2.3.3.2.cmml">C</mi><mo id="S3.Thmthm11.p4.5.m5.2.3.3.1" xref="S3.Thmthm11.p4.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.5.m5.2.3.3.3.2" xref="S3.Thmthm11.p4.5.m5.2.3.3.3.1.cmml"><mo id="S3.Thmthm11.p4.5.m5.2.3.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.5.m5.2.3.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p4.5.m5.1.1" xref="S3.Thmthm11.p4.5.m5.1.1.cmml">X</mi><mo id="S3.Thmthm11.p4.5.m5.2.3.3.3.2.2" xref="S3.Thmthm11.p4.5.m5.2.3.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p4.5.m5.2.2" xref="S3.Thmthm11.p4.5.m5.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p4.5.m5.2.3.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p4.5.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.5.m5.2b"><apply id="S3.Thmthm11.p4.5.m5.2.3.cmml" xref="S3.Thmthm11.p4.5.m5.2.3"><in id="S3.Thmthm11.p4.5.m5.2.3.1.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.1"></in><apply id="S3.Thmthm11.p4.5.m5.2.3.2.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.2"><compose id="S3.Thmthm11.p4.5.m5.2.3.2.1.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.2.1"></compose><ci id="S3.Thmthm11.p4.5.m5.2.3.2.2.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.2.2">𝑔</ci><apply id="S3.Thmthm11.p4.5.m5.2.3.2.3.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.2.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.5.m5.2.3.2.3.1.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.2.3">superscript</csymbol><ci id="S3.Thmthm11.p4.5.m5.2.3.2.3.2.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.2.3.2">𝜎</ci><ci id="S3.Thmthm11.p4.5.m5.2.3.2.3.3.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.2.3.3">ℤ</ci></apply></apply><apply id="S3.Thmthm11.p4.5.m5.2.3.3.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.3"><times id="S3.Thmthm11.p4.5.m5.2.3.3.1.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.3.1"></times><ci id="S3.Thmthm11.p4.5.m5.2.3.3.2.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.3.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p4.5.m5.2.3.3.3.1.cmml" xref="S3.Thmthm11.p4.5.m5.2.3.3.3.2"><ci id="S3.Thmthm11.p4.5.m5.1.1.cmml" xref="S3.Thmthm11.p4.5.m5.1.1">𝑋</ci><ci id="S3.Thmthm11.p4.5.m5.2.2.cmml" xref="S3.Thmthm11.p4.5.m5.2.2">ℤ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.5.m5.2c">g\circ\sigma^{\mathbb{Z}}\in C(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.5.m5.2d">italic_g ∘ italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ∈ italic_C ( italic_X , blackboard_Z )</annotation></semantics></math> which also satisfies</p> <table class="ltx_equation ltx_eqn_table" id="S3.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(3.8)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g\circ\sigma^{\mathbb{Z}}\in\partial_{T}C(X,\mathbb{Z})\,." class="ltx_Math" display="block" id="S3.E8.m1.3"><semantics id="S3.E8.m1.3a"><mrow id="S3.E8.m1.3.3.1" xref="S3.E8.m1.3.3.1.1.cmml"><mrow id="S3.E8.m1.3.3.1.1" xref="S3.E8.m1.3.3.1.1.cmml"><mrow id="S3.E8.m1.3.3.1.1.2" xref="S3.E8.m1.3.3.1.1.2.cmml"><mi id="S3.E8.m1.3.3.1.1.2.2" xref="S3.E8.m1.3.3.1.1.2.2.cmml">g</mi><mo id="S3.E8.m1.3.3.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.E8.m1.3.3.1.1.2.1.cmml">∘</mo><msup id="S3.E8.m1.3.3.1.1.2.3" xref="S3.E8.m1.3.3.1.1.2.3.cmml"><mi id="S3.E8.m1.3.3.1.1.2.3.2" xref="S3.E8.m1.3.3.1.1.2.3.2.cmml">σ</mi><mi id="S3.E8.m1.3.3.1.1.2.3.3" xref="S3.E8.m1.3.3.1.1.2.3.3.cmml">ℤ</mi></msup></mrow><mo id="S3.E8.m1.3.3.1.1.1" rspace="0.1389em" xref="S3.E8.m1.3.3.1.1.1.cmml">∈</mo><mrow id="S3.E8.m1.3.3.1.1.3" xref="S3.E8.m1.3.3.1.1.3.cmml"><msub id="S3.E8.m1.3.3.1.1.3.1" xref="S3.E8.m1.3.3.1.1.3.1.cmml"><mo id="S3.E8.m1.3.3.1.1.3.1.2" lspace="0.1389em" rspace="0em" xref="S3.E8.m1.3.3.1.1.3.1.2.cmml">∂</mo><mi id="S3.E8.m1.3.3.1.1.3.1.3" xref="S3.E8.m1.3.3.1.1.3.1.3.cmml">T</mi></msub><mrow id="S3.E8.m1.3.3.1.1.3.2" xref="S3.E8.m1.3.3.1.1.3.2.cmml"><mi id="S3.E8.m1.3.3.1.1.3.2.2" xref="S3.E8.m1.3.3.1.1.3.2.2.cmml">C</mi><mo id="S3.E8.m1.3.3.1.1.3.2.1" xref="S3.E8.m1.3.3.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.E8.m1.3.3.1.1.3.2.3.2" xref="S3.E8.m1.3.3.1.1.3.2.3.1.cmml"><mo id="S3.E8.m1.3.3.1.1.3.2.3.2.1" stretchy="false" xref="S3.E8.m1.3.3.1.1.3.2.3.1.cmml">(</mo><mi id="S3.E8.m1.1.1" xref="S3.E8.m1.1.1.cmml">X</mi><mo id="S3.E8.m1.3.3.1.1.3.2.3.2.2" xref="S3.E8.m1.3.3.1.1.3.2.3.1.cmml">,</mo><mi id="S3.E8.m1.2.2" xref="S3.E8.m1.2.2.cmml">ℤ</mi><mo id="S3.E8.m1.3.3.1.1.3.2.3.2.3" stretchy="false" xref="S3.E8.m1.3.3.1.1.3.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.E8.m1.3.3.1.2" lspace="0.170em" xref="S3.E8.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E8.m1.3b"><apply id="S3.E8.m1.3.3.1.1.cmml" xref="S3.E8.m1.3.3.1"><in id="S3.E8.m1.3.3.1.1.1.cmml" xref="S3.E8.m1.3.3.1.1.1"></in><apply id="S3.E8.m1.3.3.1.1.2.cmml" xref="S3.E8.m1.3.3.1.1.2"><compose id="S3.E8.m1.3.3.1.1.2.1.cmml" xref="S3.E8.m1.3.3.1.1.2.1"></compose><ci id="S3.E8.m1.3.3.1.1.2.2.cmml" xref="S3.E8.m1.3.3.1.1.2.2">𝑔</ci><apply id="S3.E8.m1.3.3.1.1.2.3.cmml" xref="S3.E8.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S3.E8.m1.3.3.1.1.2.3.1.cmml" xref="S3.E8.m1.3.3.1.1.2.3">superscript</csymbol><ci id="S3.E8.m1.3.3.1.1.2.3.2.cmml" xref="S3.E8.m1.3.3.1.1.2.3.2">𝜎</ci><ci id="S3.E8.m1.3.3.1.1.2.3.3.cmml" xref="S3.E8.m1.3.3.1.1.2.3.3">ℤ</ci></apply></apply><apply id="S3.E8.m1.3.3.1.1.3.cmml" xref="S3.E8.m1.3.3.1.1.3"><apply id="S3.E8.m1.3.3.1.1.3.1.cmml" xref="S3.E8.m1.3.3.1.1.3.1"><csymbol cd="ambiguous" id="S3.E8.m1.3.3.1.1.3.1.1.cmml" xref="S3.E8.m1.3.3.1.1.3.1">subscript</csymbol><partialdiff id="S3.E8.m1.3.3.1.1.3.1.2.cmml" xref="S3.E8.m1.3.3.1.1.3.1.2"></partialdiff><ci id="S3.E8.m1.3.3.1.1.3.1.3.cmml" xref="S3.E8.m1.3.3.1.1.3.1.3">𝑇</ci></apply><apply id="S3.E8.m1.3.3.1.1.3.2.cmml" xref="S3.E8.m1.3.3.1.1.3.2"><times id="S3.E8.m1.3.3.1.1.3.2.1.cmml" xref="S3.E8.m1.3.3.1.1.3.2.1"></times><ci id="S3.E8.m1.3.3.1.1.3.2.2.cmml" xref="S3.E8.m1.3.3.1.1.3.2.2">𝐶</ci><interval closure="open" id="S3.E8.m1.3.3.1.1.3.2.3.1.cmml" xref="S3.E8.m1.3.3.1.1.3.2.3.2"><ci id="S3.E8.m1.1.1.cmml" xref="S3.E8.m1.1.1">𝑋</ci><ci id="S3.E8.m1.2.2.cmml" xref="S3.E8.m1.2.2">ℤ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m1.3c">g\circ\sigma^{\mathbb{Z}}\in\partial_{T}C(X,\mathbb{Z})\,.</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.3d">italic_g ∘ italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ∈ ∂ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_C ( italic_X , blackboard_Z ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmthm11.p4.14">Thus <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.6.m1.1"><semantics id="S3.Thmthm11.p4.6.m1.1a"><mi id="S3.Thmthm11.p4.6.m1.1.1" xref="S3.Thmthm11.p4.6.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.6.m1.1b"><ci id="S3.Thmthm11.p4.6.m1.1.1.cmml" xref="S3.Thmthm11.p4.6.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.6.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.6.m1.1d">italic_σ</annotation></semantics></math> defines a contravariant morphism <math alttext="H(\sigma):H(Y,\mathbb{Z})\to H(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.7.m2.5"><semantics id="S3.Thmthm11.p4.7.m2.5a"><mrow id="S3.Thmthm11.p4.7.m2.5.6" xref="S3.Thmthm11.p4.7.m2.5.6.cmml"><mrow id="S3.Thmthm11.p4.7.m2.5.6.2" xref="S3.Thmthm11.p4.7.m2.5.6.2.cmml"><mi id="S3.Thmthm11.p4.7.m2.5.6.2.2" xref="S3.Thmthm11.p4.7.m2.5.6.2.2.cmml">H</mi><mo id="S3.Thmthm11.p4.7.m2.5.6.2.1" xref="S3.Thmthm11.p4.7.m2.5.6.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.7.m2.5.6.2.3.2" xref="S3.Thmthm11.p4.7.m2.5.6.2.cmml"><mo id="S3.Thmthm11.p4.7.m2.5.6.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.7.m2.5.6.2.cmml">(</mo><mi id="S3.Thmthm11.p4.7.m2.1.1" xref="S3.Thmthm11.p4.7.m2.1.1.cmml">σ</mi><mo id="S3.Thmthm11.p4.7.m2.5.6.2.3.2.2" rspace="0.278em" stretchy="false" xref="S3.Thmthm11.p4.7.m2.5.6.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p4.7.m2.5.6.1" rspace="0.278em" xref="S3.Thmthm11.p4.7.m2.5.6.1.cmml">:</mo><mrow id="S3.Thmthm11.p4.7.m2.5.6.3" xref="S3.Thmthm11.p4.7.m2.5.6.3.cmml"><mrow id="S3.Thmthm11.p4.7.m2.5.6.3.2" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.cmml"><mi id="S3.Thmthm11.p4.7.m2.5.6.3.2.2" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.2.cmml">H</mi><mo id="S3.Thmthm11.p4.7.m2.5.6.3.2.1" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.7.m2.5.6.3.2.3.2" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.3.1.cmml"><mo id="S3.Thmthm11.p4.7.m2.5.6.3.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.3.1.cmml">(</mo><mi id="S3.Thmthm11.p4.7.m2.2.2" xref="S3.Thmthm11.p4.7.m2.2.2.cmml">Y</mi><mo id="S3.Thmthm11.p4.7.m2.5.6.3.2.3.2.2" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.3.1.cmml">,</mo><mi id="S3.Thmthm11.p4.7.m2.3.3" xref="S3.Thmthm11.p4.7.m2.3.3.cmml">ℤ</mi><mo id="S3.Thmthm11.p4.7.m2.5.6.3.2.3.2.3" stretchy="false" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p4.7.m2.5.6.3.1" stretchy="false" xref="S3.Thmthm11.p4.7.m2.5.6.3.1.cmml">→</mo><mrow id="S3.Thmthm11.p4.7.m2.5.6.3.3" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.cmml"><mi id="S3.Thmthm11.p4.7.m2.5.6.3.3.2" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.2.cmml">H</mi><mo id="S3.Thmthm11.p4.7.m2.5.6.3.3.1" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.7.m2.5.6.3.3.3.2" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.3.1.cmml"><mo id="S3.Thmthm11.p4.7.m2.5.6.3.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p4.7.m2.4.4" xref="S3.Thmthm11.p4.7.m2.4.4.cmml">X</mi><mo id="S3.Thmthm11.p4.7.m2.5.6.3.3.3.2.2" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p4.7.m2.5.5" xref="S3.Thmthm11.p4.7.m2.5.5.cmml">ℤ</mi><mo id="S3.Thmthm11.p4.7.m2.5.6.3.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.7.m2.5b"><apply id="S3.Thmthm11.p4.7.m2.5.6.cmml" xref="S3.Thmthm11.p4.7.m2.5.6"><ci id="S3.Thmthm11.p4.7.m2.5.6.1.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.1">:</ci><apply id="S3.Thmthm11.p4.7.m2.5.6.2.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.2"><times id="S3.Thmthm11.p4.7.m2.5.6.2.1.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.2.1"></times><ci id="S3.Thmthm11.p4.7.m2.5.6.2.2.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.2.2">𝐻</ci><ci id="S3.Thmthm11.p4.7.m2.1.1.cmml" xref="S3.Thmthm11.p4.7.m2.1.1">𝜎</ci></apply><apply id="S3.Thmthm11.p4.7.m2.5.6.3.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3"><ci id="S3.Thmthm11.p4.7.m2.5.6.3.1.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.1">→</ci><apply id="S3.Thmthm11.p4.7.m2.5.6.3.2.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.2"><times id="S3.Thmthm11.p4.7.m2.5.6.3.2.1.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.1"></times><ci id="S3.Thmthm11.p4.7.m2.5.6.3.2.2.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.2">𝐻</ci><interval closure="open" id="S3.Thmthm11.p4.7.m2.5.6.3.2.3.1.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.2.3.2"><ci id="S3.Thmthm11.p4.7.m2.2.2.cmml" xref="S3.Thmthm11.p4.7.m2.2.2">𝑌</ci><ci id="S3.Thmthm11.p4.7.m2.3.3.cmml" xref="S3.Thmthm11.p4.7.m2.3.3">ℤ</ci></interval></apply><apply id="S3.Thmthm11.p4.7.m2.5.6.3.3.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.3"><times id="S3.Thmthm11.p4.7.m2.5.6.3.3.1.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.1"></times><ci id="S3.Thmthm11.p4.7.m2.5.6.3.3.2.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.2">𝐻</ci><interval closure="open" id="S3.Thmthm11.p4.7.m2.5.6.3.3.3.1.cmml" xref="S3.Thmthm11.p4.7.m2.5.6.3.3.3.2"><ci id="S3.Thmthm11.p4.7.m2.4.4.cmml" xref="S3.Thmthm11.p4.7.m2.4.4">𝑋</ci><ci id="S3.Thmthm11.p4.7.m2.5.5.cmml" xref="S3.Thmthm11.p4.7.m2.5.5">ℤ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.7.m2.5c">H(\sigma):H(Y,\mathbb{Z})\to H(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.7.m2.5d">italic_H ( italic_σ ) : italic_H ( italic_Y , blackboard_Z ) → italic_H ( italic_X , blackboard_Z )</annotation></semantics></math>, and hence an induced dual map <math alttext="H(\sigma)^{*}:H(X,\mathbb{R})^{*}\to H(Y,\mathbb{R})^{*}" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.8.m3.5"><semantics id="S3.Thmthm11.p4.8.m3.5a"><mrow id="S3.Thmthm11.p4.8.m3.5.6" xref="S3.Thmthm11.p4.8.m3.5.6.cmml"><mrow id="S3.Thmthm11.p4.8.m3.5.6.2" xref="S3.Thmthm11.p4.8.m3.5.6.2.cmml"><mi id="S3.Thmthm11.p4.8.m3.5.6.2.2" xref="S3.Thmthm11.p4.8.m3.5.6.2.2.cmml">H</mi><mo id="S3.Thmthm11.p4.8.m3.5.6.2.1" xref="S3.Thmthm11.p4.8.m3.5.6.2.1.cmml">⁢</mo><msup id="S3.Thmthm11.p4.8.m3.5.6.2.3" xref="S3.Thmthm11.p4.8.m3.5.6.2.3.cmml"><mrow id="S3.Thmthm11.p4.8.m3.5.6.2.3.2.2" xref="S3.Thmthm11.p4.8.m3.5.6.2.3.cmml"><mo id="S3.Thmthm11.p4.8.m3.5.6.2.3.2.2.1" stretchy="false" xref="S3.Thmthm11.p4.8.m3.5.6.2.3.cmml">(</mo><mi id="S3.Thmthm11.p4.8.m3.1.1" xref="S3.Thmthm11.p4.8.m3.1.1.cmml">σ</mi><mo id="S3.Thmthm11.p4.8.m3.5.6.2.3.2.2.2" rspace="0.278em" stretchy="false" xref="S3.Thmthm11.p4.8.m3.5.6.2.3.cmml">)</mo></mrow><mo id="S3.Thmthm11.p4.8.m3.5.6.2.3.3" xref="S3.Thmthm11.p4.8.m3.5.6.2.3.3.cmml">∗</mo></msup></mrow><mo id="S3.Thmthm11.p4.8.m3.5.6.1" rspace="0.278em" xref="S3.Thmthm11.p4.8.m3.5.6.1.cmml">:</mo><mrow id="S3.Thmthm11.p4.8.m3.5.6.3" xref="S3.Thmthm11.p4.8.m3.5.6.3.cmml"><mrow id="S3.Thmthm11.p4.8.m3.5.6.3.2" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.cmml"><mi id="S3.Thmthm11.p4.8.m3.5.6.3.2.2" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.2.cmml">H</mi><mo id="S3.Thmthm11.p4.8.m3.5.6.3.2.1" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.1.cmml">⁢</mo><msup id="S3.Thmthm11.p4.8.m3.5.6.3.2.3" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3.cmml"><mrow id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.2" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.1.cmml"><mo id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.2.1" stretchy="false" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.1.cmml">(</mo><mi id="S3.Thmthm11.p4.8.m3.2.2" xref="S3.Thmthm11.p4.8.m3.2.2.cmml">X</mi><mo id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.2.2" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.1.cmml">,</mo><mi id="S3.Thmthm11.p4.8.m3.3.3" xref="S3.Thmthm11.p4.8.m3.3.3.cmml">ℝ</mi><mo id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.2.3" stretchy="false" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.3" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3.3.cmml">∗</mo></msup></mrow><mo id="S3.Thmthm11.p4.8.m3.5.6.3.1" stretchy="false" xref="S3.Thmthm11.p4.8.m3.5.6.3.1.cmml">→</mo><mrow id="S3.Thmthm11.p4.8.m3.5.6.3.3" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.cmml"><mi id="S3.Thmthm11.p4.8.m3.5.6.3.3.2" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.2.cmml">H</mi><mo id="S3.Thmthm11.p4.8.m3.5.6.3.3.1" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.1.cmml">⁢</mo><msup id="S3.Thmthm11.p4.8.m3.5.6.3.3.3" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3.cmml"><mrow id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.2" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.1.cmml"><mo id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.2.1" stretchy="false" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.1.cmml">(</mo><mi id="S3.Thmthm11.p4.8.m3.4.4" xref="S3.Thmthm11.p4.8.m3.4.4.cmml">Y</mi><mo id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.2.2" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.1.cmml">,</mo><mi id="S3.Thmthm11.p4.8.m3.5.5" xref="S3.Thmthm11.p4.8.m3.5.5.cmml">ℝ</mi><mo id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.2.3" stretchy="false" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.3" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3.3.cmml">∗</mo></msup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.8.m3.5b"><apply id="S3.Thmthm11.p4.8.m3.5.6.cmml" xref="S3.Thmthm11.p4.8.m3.5.6"><ci id="S3.Thmthm11.p4.8.m3.5.6.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.1">:</ci><apply id="S3.Thmthm11.p4.8.m3.5.6.2.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.2"><times id="S3.Thmthm11.p4.8.m3.5.6.2.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.2.1"></times><ci id="S3.Thmthm11.p4.8.m3.5.6.2.2.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.2.2">𝐻</ci><apply id="S3.Thmthm11.p4.8.m3.5.6.2.3.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.2.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.8.m3.5.6.2.3.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.2.3">superscript</csymbol><ci id="S3.Thmthm11.p4.8.m3.1.1.cmml" xref="S3.Thmthm11.p4.8.m3.1.1">𝜎</ci><times id="S3.Thmthm11.p4.8.m3.5.6.2.3.3.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.2.3.3"></times></apply></apply><apply id="S3.Thmthm11.p4.8.m3.5.6.3.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3"><ci id="S3.Thmthm11.p4.8.m3.5.6.3.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.1">→</ci><apply id="S3.Thmthm11.p4.8.m3.5.6.3.2.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.2"><times id="S3.Thmthm11.p4.8.m3.5.6.3.2.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.1"></times><ci id="S3.Thmthm11.p4.8.m3.5.6.3.2.2.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.2">𝐻</ci><apply id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3">superscript</csymbol><interval closure="open" id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3.2.2"><ci id="S3.Thmthm11.p4.8.m3.2.2.cmml" xref="S3.Thmthm11.p4.8.m3.2.2">𝑋</ci><ci id="S3.Thmthm11.p4.8.m3.3.3.cmml" xref="S3.Thmthm11.p4.8.m3.3.3">ℝ</ci></interval><times id="S3.Thmthm11.p4.8.m3.5.6.3.2.3.3.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.2.3.3"></times></apply></apply><apply id="S3.Thmthm11.p4.8.m3.5.6.3.3.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.3"><times id="S3.Thmthm11.p4.8.m3.5.6.3.3.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.1"></times><ci id="S3.Thmthm11.p4.8.m3.5.6.3.3.2.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.2">𝐻</ci><apply id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3">superscript</csymbol><interval closure="open" id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.1.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3.2.2"><ci id="S3.Thmthm11.p4.8.m3.4.4.cmml" xref="S3.Thmthm11.p4.8.m3.4.4">𝑌</ci><ci id="S3.Thmthm11.p4.8.m3.5.5.cmml" xref="S3.Thmthm11.p4.8.m3.5.5">ℝ</ci></interval><times id="S3.Thmthm11.p4.8.m3.5.6.3.3.3.3.cmml" xref="S3.Thmthm11.p4.8.m3.5.6.3.3.3.3"></times></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.8.m3.5c">H(\sigma)^{*}:H(X,\mathbb{R})^{*}\to H(Y,\mathbb{R})^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.8.m3.5d">italic_H ( italic_σ ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT : italic_H ( italic_X , blackboard_R ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → italic_H ( italic_Y , blackboard_R ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. Since <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.9.m4.1"><semantics id="S3.Thmthm11.p4.9.m4.1a"><mi id="S3.Thmthm11.p4.9.m4.1.1" xref="S3.Thmthm11.p4.9.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.9.m4.1b"><ci id="S3.Thmthm11.p4.9.m4.1.1.cmml" xref="S3.Thmthm11.p4.9.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.9.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.9.m4.1d">italic_X</annotation></semantics></math> is assumed to be recurrent, we obtain via the above isomorphisms <math alttext="i_{X}M" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.10.m5.1"><semantics id="S3.Thmthm11.p4.10.m5.1a"><mrow id="S3.Thmthm11.p4.10.m5.1.1" xref="S3.Thmthm11.p4.10.m5.1.1.cmml"><msub id="S3.Thmthm11.p4.10.m5.1.1.2" xref="S3.Thmthm11.p4.10.m5.1.1.2.cmml"><mi id="S3.Thmthm11.p4.10.m5.1.1.2.2" xref="S3.Thmthm11.p4.10.m5.1.1.2.2.cmml">i</mi><mi id="S3.Thmthm11.p4.10.m5.1.1.2.3" xref="S3.Thmthm11.p4.10.m5.1.1.2.3.cmml">X</mi></msub><mo id="S3.Thmthm11.p4.10.m5.1.1.1" xref="S3.Thmthm11.p4.10.m5.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p4.10.m5.1.1.3" xref="S3.Thmthm11.p4.10.m5.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.10.m5.1b"><apply id="S3.Thmthm11.p4.10.m5.1.1.cmml" xref="S3.Thmthm11.p4.10.m5.1.1"><times id="S3.Thmthm11.p4.10.m5.1.1.1.cmml" xref="S3.Thmthm11.p4.10.m5.1.1.1"></times><apply id="S3.Thmthm11.p4.10.m5.1.1.2.cmml" xref="S3.Thmthm11.p4.10.m5.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.10.m5.1.1.2.1.cmml" xref="S3.Thmthm11.p4.10.m5.1.1.2">subscript</csymbol><ci id="S3.Thmthm11.p4.10.m5.1.1.2.2.cmml" xref="S3.Thmthm11.p4.10.m5.1.1.2.2">𝑖</ci><ci id="S3.Thmthm11.p4.10.m5.1.1.2.3.cmml" xref="S3.Thmthm11.p4.10.m5.1.1.2.3">𝑋</ci></apply><ci id="S3.Thmthm11.p4.10.m5.1.1.3.cmml" xref="S3.Thmthm11.p4.10.m5.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.10.m5.1c">i_{X}M</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.10.m5.1d">italic_i start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> and <math alttext="i_{Y}M" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.11.m6.1"><semantics id="S3.Thmthm11.p4.11.m6.1a"><mrow id="S3.Thmthm11.p4.11.m6.1.1" xref="S3.Thmthm11.p4.11.m6.1.1.cmml"><msub id="S3.Thmthm11.p4.11.m6.1.1.2" xref="S3.Thmthm11.p4.11.m6.1.1.2.cmml"><mi id="S3.Thmthm11.p4.11.m6.1.1.2.2" xref="S3.Thmthm11.p4.11.m6.1.1.2.2.cmml">i</mi><mi id="S3.Thmthm11.p4.11.m6.1.1.2.3" xref="S3.Thmthm11.p4.11.m6.1.1.2.3.cmml">Y</mi></msub><mo id="S3.Thmthm11.p4.11.m6.1.1.1" xref="S3.Thmthm11.p4.11.m6.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p4.11.m6.1.1.3" xref="S3.Thmthm11.p4.11.m6.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.11.m6.1b"><apply id="S3.Thmthm11.p4.11.m6.1.1.cmml" xref="S3.Thmthm11.p4.11.m6.1.1"><times id="S3.Thmthm11.p4.11.m6.1.1.1.cmml" xref="S3.Thmthm11.p4.11.m6.1.1.1"></times><apply id="S3.Thmthm11.p4.11.m6.1.1.2.cmml" xref="S3.Thmthm11.p4.11.m6.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.11.m6.1.1.2.1.cmml" xref="S3.Thmthm11.p4.11.m6.1.1.2">subscript</csymbol><ci id="S3.Thmthm11.p4.11.m6.1.1.2.2.cmml" xref="S3.Thmthm11.p4.11.m6.1.1.2.2">𝑖</ci><ci id="S3.Thmthm11.p4.11.m6.1.1.2.3.cmml" xref="S3.Thmthm11.p4.11.m6.1.1.2.3">𝑌</ci></apply><ci id="S3.Thmthm11.p4.11.m6.1.1.3.cmml" xref="S3.Thmthm11.p4.11.m6.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.11.m6.1c">i_{Y}M</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.11.m6.1d">italic_i start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT italic_M</annotation></semantics></math> a map <math alttext="(i_{Y}M)^{-1}\circ H(\sigma)^{*}\circ i_{X}M:\cal M(X)\to\cal M(Y)" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.12.m7.4"><semantics id="S3.Thmthm11.p4.12.m7.4a"><mrow id="S3.Thmthm11.p4.12.m7.4.4" xref="S3.Thmthm11.p4.12.m7.4.4.cmml"><mrow id="S3.Thmthm11.p4.12.m7.4.4.1" xref="S3.Thmthm11.p4.12.m7.4.4.1.cmml"><mrow id="S3.Thmthm11.p4.12.m7.4.4.1.1" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.cmml"><mrow id="S3.Thmthm11.p4.12.m7.4.4.1.1.1" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.cmml"><mrow id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.cmml"><msup id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.cmml"><msub id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.2" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.2.cmml">i</mi><mi id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.3.cmml">Y</mi></msub><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.1" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.3.cmml">M</mi></mrow><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3.cmml"><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3a" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3.cmml">−</mo><mn id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3.2" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.2.cmml">∘</mo><mi id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.3.cmml">H</mi></mrow><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.2" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.2.cmml">⁢</mo><msup id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.cmml"><mrow id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.2.2" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.cmml"><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.2.2.1" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.cmml">(</mo><mi id="S3.Thmthm11.p4.12.m7.1.1" xref="S3.Thmthm11.p4.12.m7.1.1.cmml">σ</mi><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.2.2.2" rspace="0.055em" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.cmml">)</mo></mrow><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S3.Thmthm11.p4.12.m7.4.4.1.1.2" rspace="0.222em" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.2.cmml">∘</mo><msub id="S3.Thmthm11.p4.12.m7.4.4.1.1.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.3.cmml"><mi id="S3.Thmthm11.p4.12.m7.4.4.1.1.3.2" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.3.2.cmml">i</mi><mi id="S3.Thmthm11.p4.12.m7.4.4.1.1.3.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.3.3.cmml">X</mi></msub></mrow><mo id="S3.Thmthm11.p4.12.m7.4.4.1.2" xref="S3.Thmthm11.p4.12.m7.4.4.1.2.cmml">⁢</mo><mi id="S3.Thmthm11.p4.12.m7.4.4.1.3" xref="S3.Thmthm11.p4.12.m7.4.4.1.3.cmml">M</mi></mrow><mo id="S3.Thmthm11.p4.12.m7.4.4.2" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm11.p4.12.m7.4.4.2.cmml">:</mo><mrow id="S3.Thmthm11.p4.12.m7.4.4.3" xref="S3.Thmthm11.p4.12.m7.4.4.3.cmml"><mrow id="S3.Thmthm11.p4.12.m7.4.4.3.2" xref="S3.Thmthm11.p4.12.m7.4.4.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p4.12.m7.4.4.3.2.2" xref="S3.Thmthm11.p4.12.m7.4.4.3.2.2.cmml">ℳ</mi><mo id="S3.Thmthm11.p4.12.m7.4.4.3.2.1" xref="S3.Thmthm11.p4.12.m7.4.4.3.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.12.m7.4.4.3.2.3.2" xref="S3.Thmthm11.p4.12.m7.4.4.3.2.cmml"><mo id="S3.Thmthm11.p4.12.m7.4.4.3.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p4.12.m7.2.2" xref="S3.Thmthm11.p4.12.m7.2.2.cmml">𝒳</mi><mo id="S3.Thmthm11.p4.12.m7.4.4.3.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p4.12.m7.4.4.3.1" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.3.1.cmml">→</mo><mrow id="S3.Thmthm11.p4.12.m7.4.4.3.3" xref="S3.Thmthm11.p4.12.m7.4.4.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p4.12.m7.4.4.3.3.2" xref="S3.Thmthm11.p4.12.m7.4.4.3.3.2.cmml">ℳ</mi><mo id="S3.Thmthm11.p4.12.m7.4.4.3.3.1" xref="S3.Thmthm11.p4.12.m7.4.4.3.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p4.12.m7.4.4.3.3.3.2" xref="S3.Thmthm11.p4.12.m7.4.4.3.3.cmml"><mo id="S3.Thmthm11.p4.12.m7.4.4.3.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p4.12.m7.3.3" xref="S3.Thmthm11.p4.12.m7.3.3.cmml">𝒴</mi><mo id="S3.Thmthm11.p4.12.m7.4.4.3.3.3.2.2" stretchy="false" xref="S3.Thmthm11.p4.12.m7.4.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.12.m7.4b"><apply id="S3.Thmthm11.p4.12.m7.4.4.cmml" xref="S3.Thmthm11.p4.12.m7.4.4"><ci id="S3.Thmthm11.p4.12.m7.4.4.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.2">:</ci><apply id="S3.Thmthm11.p4.12.m7.4.4.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1"><times id="S3.Thmthm11.p4.12.m7.4.4.1.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.2"></times><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1"><compose id="S3.Thmthm11.p4.12.m7.4.4.1.1.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.2"></compose><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1"><times id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.2"></times><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1"><compose id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.2"></compose><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1">superscript</csymbol><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1"><times id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.1"></times><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.2">𝑖</ci><ci id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.2.3">𝑌</ci></apply><ci id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.1.1.1.3">𝑀</ci></apply><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3"><minus id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3"></minus><cn id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.1.3.2">1</cn></apply></apply><ci id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.1.3">𝐻</ci></apply><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3">superscript</csymbol><ci id="S3.Thmthm11.p4.12.m7.1.1.cmml" xref="S3.Thmthm11.p4.12.m7.1.1">𝜎</ci><times id="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.1.3.3"></times></apply></apply><apply id="S3.Thmthm11.p4.12.m7.4.4.1.1.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.12.m7.4.4.1.1.3.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.3">subscript</csymbol><ci id="S3.Thmthm11.p4.12.m7.4.4.1.1.3.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.3.2">𝑖</ci><ci id="S3.Thmthm11.p4.12.m7.4.4.1.1.3.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.1.3.3">𝑋</ci></apply></apply><ci id="S3.Thmthm11.p4.12.m7.4.4.1.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.1.3">𝑀</ci></apply><apply id="S3.Thmthm11.p4.12.m7.4.4.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.3"><ci id="S3.Thmthm11.p4.12.m7.4.4.3.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.3.1">→</ci><apply id="S3.Thmthm11.p4.12.m7.4.4.3.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.3.2"><times id="S3.Thmthm11.p4.12.m7.4.4.3.2.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.3.2.1"></times><ci id="S3.Thmthm11.p4.12.m7.4.4.3.2.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.3.2.2">ℳ</ci><ci id="S3.Thmthm11.p4.12.m7.2.2.cmml" xref="S3.Thmthm11.p4.12.m7.2.2">𝒳</ci></apply><apply id="S3.Thmthm11.p4.12.m7.4.4.3.3.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.3.3"><times id="S3.Thmthm11.p4.12.m7.4.4.3.3.1.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.3.3.1"></times><ci id="S3.Thmthm11.p4.12.m7.4.4.3.3.2.cmml" xref="S3.Thmthm11.p4.12.m7.4.4.3.3.2">ℳ</ci><ci id="S3.Thmthm11.p4.12.m7.3.3.cmml" xref="S3.Thmthm11.p4.12.m7.3.3">𝒴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.12.m7.4c">(i_{Y}M)^{-1}\circ H(\sigma)^{*}\circ i_{X}M:\cal M(X)\to\cal M(Y)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.12.m7.4d">( italic_i start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT italic_M ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∘ italic_H ( italic_σ ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∘ italic_i start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( caligraphic_Y )</annotation></semantics></math>, which agrees with the map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S3.Thmthm11.p4.13.m8.1"><semantics id="S3.Thmthm11.p4.13.m8.1a"><mrow id="S3.Thmthm11.p4.13.m8.1.1" xref="S3.Thmthm11.p4.13.m8.1.1.cmml"><mi id="S3.Thmthm11.p4.13.m8.1.1.2" xref="S3.Thmthm11.p4.13.m8.1.1.2.cmml">σ</mi><mo id="S3.Thmthm11.p4.13.m8.1.1.1" xref="S3.Thmthm11.p4.13.m8.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p4.13.m8.1.1.3" xref="S3.Thmthm11.p4.13.m8.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.13.m8.1b"><apply id="S3.Thmthm11.p4.13.m8.1.1.cmml" xref="S3.Thmthm11.p4.13.m8.1.1"><times id="S3.Thmthm11.p4.13.m8.1.1.1.cmml" xref="S3.Thmthm11.p4.13.m8.1.1.1"></times><ci id="S3.Thmthm11.p4.13.m8.1.1.2.cmml" xref="S3.Thmthm11.p4.13.m8.1.1.2">𝜎</ci><ci id="S3.Thmthm11.p4.13.m8.1.1.3.cmml" xref="S3.Thmthm11.p4.13.m8.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.13.m8.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.13.m8.1d">italic_σ italic_M</annotation></semantics></math>, since in this letter-to-letter case both coincide with the push-forward map <math alttext="\sigma^{\mathbb{Z}}_{*}\," class="ltx_Math" display="inline" id="S3.Thmthm11.p4.14.m9.1"><semantics id="S3.Thmthm11.p4.14.m9.1a"><msubsup id="S3.Thmthm11.p4.14.m9.1.1" xref="S3.Thmthm11.p4.14.m9.1.1.cmml"><mi id="S3.Thmthm11.p4.14.m9.1.1.2.2" xref="S3.Thmthm11.p4.14.m9.1.1.2.2.cmml">σ</mi><mo id="S3.Thmthm11.p4.14.m9.1.1.3" xref="S3.Thmthm11.p4.14.m9.1.1.3.cmml">∗</mo><mi id="S3.Thmthm11.p4.14.m9.1.1.2.3" xref="S3.Thmthm11.p4.14.m9.1.1.2.3.cmml">ℤ</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p4.14.m9.1b"><apply id="S3.Thmthm11.p4.14.m9.1.1.cmml" xref="S3.Thmthm11.p4.14.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.14.m9.1.1.1.cmml" xref="S3.Thmthm11.p4.14.m9.1.1">subscript</csymbol><apply id="S3.Thmthm11.p4.14.m9.1.1.2.cmml" xref="S3.Thmthm11.p4.14.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p4.14.m9.1.1.2.1.cmml" xref="S3.Thmthm11.p4.14.m9.1.1">superscript</csymbol><ci id="S3.Thmthm11.p4.14.m9.1.1.2.2.cmml" xref="S3.Thmthm11.p4.14.m9.1.1.2.2">𝜎</ci><ci id="S3.Thmthm11.p4.14.m9.1.1.2.3.cmml" xref="S3.Thmthm11.p4.14.m9.1.1.2.3">ℤ</ci></apply><times id="S3.Thmthm11.p4.14.m9.1.1.3.cmml" xref="S3.Thmthm11.p4.14.m9.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p4.14.m9.1c">\sigma^{\mathbb{Z}}_{*}\,</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p4.14.m9.1d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.Thmthm11.p5"> <p class="ltx_p" id="S3.Thmthm11.p5.1">However, for any attempt to push the above approach further to get an alternative “coinvariant” description of the measure transfer map, there is a problem at the very base, in that the property (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E8" title="In Remark 3.11. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.8</span></a>) fails to hold in most cases where <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.Thmthm11.p5.1.m1.1"><semantics id="S3.Thmthm11.p5.1.m1.1a"><mi id="S3.Thmthm11.p5.1.m1.1.1" xref="S3.Thmthm11.p5.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p5.1.m1.1b"><ci id="S3.Thmthm11.p5.1.m1.1.1.cmml" xref="S3.Thmthm11.p5.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p5.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p5.1.m1.1d">italic_σ</annotation></semantics></math> is not letter-to-letter. We describe below a simple example which highlights the basic problem:</p> </div> <div class="ltx_para" id="S3.Thmthm11.p6"> <p class="ltx_p" id="S3.Thmthm11.p6.8">Set <math alttext="\cal A=\{a,b,c\}" class="ltx_Math" display="inline" id="S3.Thmthm11.p6.1.m1.3"><semantics id="S3.Thmthm11.p6.1.m1.3a"><mrow id="S3.Thmthm11.p6.1.m1.3.4" xref="S3.Thmthm11.p6.1.m1.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p6.1.m1.3.4.2" xref="S3.Thmthm11.p6.1.m1.3.4.2.cmml">𝒜</mi><mo id="S3.Thmthm11.p6.1.m1.3.4.1" xref="S3.Thmthm11.p6.1.m1.3.4.1.cmml">=</mo><mrow id="S3.Thmthm11.p6.1.m1.3.4.3.2" xref="S3.Thmthm11.p6.1.m1.3.4.3.1.cmml"><mo id="S3.Thmthm11.p6.1.m1.3.4.3.2.1" stretchy="false" xref="S3.Thmthm11.p6.1.m1.3.4.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p6.1.m1.1.1" xref="S3.Thmthm11.p6.1.m1.1.1.cmml">𝒶</mi><mo id="S3.Thmthm11.p6.1.m1.3.4.3.2.2" xref="S3.Thmthm11.p6.1.m1.3.4.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p6.1.m1.2.2" xref="S3.Thmthm11.p6.1.m1.2.2.cmml">𝒷</mi><mo id="S3.Thmthm11.p6.1.m1.3.4.3.2.3" xref="S3.Thmthm11.p6.1.m1.3.4.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p6.1.m1.3.3" xref="S3.Thmthm11.p6.1.m1.3.3.cmml">𝒸</mi><mo id="S3.Thmthm11.p6.1.m1.3.4.3.2.4" stretchy="false" xref="S3.Thmthm11.p6.1.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p6.1.m1.3b"><apply id="S3.Thmthm11.p6.1.m1.3.4.cmml" xref="S3.Thmthm11.p6.1.m1.3.4"><eq id="S3.Thmthm11.p6.1.m1.3.4.1.cmml" xref="S3.Thmthm11.p6.1.m1.3.4.1"></eq><ci id="S3.Thmthm11.p6.1.m1.3.4.2.cmml" xref="S3.Thmthm11.p6.1.m1.3.4.2">𝒜</ci><set id="S3.Thmthm11.p6.1.m1.3.4.3.1.cmml" xref="S3.Thmthm11.p6.1.m1.3.4.3.2"><ci id="S3.Thmthm11.p6.1.m1.1.1.cmml" xref="S3.Thmthm11.p6.1.m1.1.1">𝒶</ci><ci id="S3.Thmthm11.p6.1.m1.2.2.cmml" xref="S3.Thmthm11.p6.1.m1.2.2">𝒷</ci><ci id="S3.Thmthm11.p6.1.m1.3.3.cmml" xref="S3.Thmthm11.p6.1.m1.3.3">𝒸</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p6.1.m1.3c">\cal A=\{a,b,c\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p6.1.m1.3d">caligraphic_A = { caligraphic_a , caligraphic_b , caligraphic_c }</annotation></semantics></math> and <math alttext="B=\{d,e\}" class="ltx_Math" display="inline" id="S3.Thmthm11.p6.2.m2.2"><semantics id="S3.Thmthm11.p6.2.m2.2a"><mrow id="S3.Thmthm11.p6.2.m2.2.3" xref="S3.Thmthm11.p6.2.m2.2.3.cmml"><mi id="S3.Thmthm11.p6.2.m2.2.3.2" xref="S3.Thmthm11.p6.2.m2.2.3.2.cmml">B</mi><mo id="S3.Thmthm11.p6.2.m2.2.3.1" xref="S3.Thmthm11.p6.2.m2.2.3.1.cmml">=</mo><mrow id="S3.Thmthm11.p6.2.m2.2.3.3.2" xref="S3.Thmthm11.p6.2.m2.2.3.3.1.cmml"><mo id="S3.Thmthm11.p6.2.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p6.2.m2.2.3.3.1.cmml">{</mo><mi id="S3.Thmthm11.p6.2.m2.1.1" xref="S3.Thmthm11.p6.2.m2.1.1.cmml">d</mi><mo id="S3.Thmthm11.p6.2.m2.2.3.3.2.2" xref="S3.Thmthm11.p6.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p6.2.m2.2.2" xref="S3.Thmthm11.p6.2.m2.2.2.cmml">e</mi><mo id="S3.Thmthm11.p6.2.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p6.2.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p6.2.m2.2b"><apply id="S3.Thmthm11.p6.2.m2.2.3.cmml" xref="S3.Thmthm11.p6.2.m2.2.3"><eq id="S3.Thmthm11.p6.2.m2.2.3.1.cmml" xref="S3.Thmthm11.p6.2.m2.2.3.1"></eq><ci id="S3.Thmthm11.p6.2.m2.2.3.2.cmml" xref="S3.Thmthm11.p6.2.m2.2.3.2">𝐵</ci><set id="S3.Thmthm11.p6.2.m2.2.3.3.1.cmml" xref="S3.Thmthm11.p6.2.m2.2.3.3.2"><ci id="S3.Thmthm11.p6.2.m2.1.1.cmml" xref="S3.Thmthm11.p6.2.m2.1.1">𝑑</ci><ci id="S3.Thmthm11.p6.2.m2.2.2.cmml" xref="S3.Thmthm11.p6.2.m2.2.2">𝑒</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p6.2.m2.2c">B=\{d,e\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p6.2.m2.2d">italic_B = { italic_d , italic_e }</annotation></semantics></math>, and define <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S3.Thmthm11.p6.3.m3.1"><semantics id="S3.Thmthm11.p6.3.m3.1a"><mrow id="S3.Thmthm11.p6.3.m3.1.1" xref="S3.Thmthm11.p6.3.m3.1.1.cmml"><mi id="S3.Thmthm11.p6.3.m3.1.1.2" xref="S3.Thmthm11.p6.3.m3.1.1.2.cmml">σ</mi><mo id="S3.Thmthm11.p6.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm11.p6.3.m3.1.1.1.cmml">:</mo><mrow id="S3.Thmthm11.p6.3.m3.1.1.3" xref="S3.Thmthm11.p6.3.m3.1.1.3.cmml"><msup id="S3.Thmthm11.p6.3.m3.1.1.3.2" xref="S3.Thmthm11.p6.3.m3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p6.3.m3.1.1.3.2.2" xref="S3.Thmthm11.p6.3.m3.1.1.3.2.2.cmml">𝒜</mi><mo id="S3.Thmthm11.p6.3.m3.1.1.3.2.3" xref="S3.Thmthm11.p6.3.m3.1.1.3.2.3.cmml">∗</mo></msup><mo id="S3.Thmthm11.p6.3.m3.1.1.3.1" stretchy="false" xref="S3.Thmthm11.p6.3.m3.1.1.3.1.cmml">→</mo><msup id="S3.Thmthm11.p6.3.m3.1.1.3.3" xref="S3.Thmthm11.p6.3.m3.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p6.3.m3.1.1.3.3.2" xref="S3.Thmthm11.p6.3.m3.1.1.3.3.2.cmml">ℬ</mi><mo id="S3.Thmthm11.p6.3.m3.1.1.3.3.3" xref="S3.Thmthm11.p6.3.m3.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p6.3.m3.1b"><apply id="S3.Thmthm11.p6.3.m3.1.1.cmml" xref="S3.Thmthm11.p6.3.m3.1.1"><ci id="S3.Thmthm11.p6.3.m3.1.1.1.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.1">:</ci><ci id="S3.Thmthm11.p6.3.m3.1.1.2.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.2">𝜎</ci><apply id="S3.Thmthm11.p6.3.m3.1.1.3.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3"><ci id="S3.Thmthm11.p6.3.m3.1.1.3.1.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.1">→</ci><apply id="S3.Thmthm11.p6.3.m3.1.1.3.2.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p6.3.m3.1.1.3.2.1.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.2">superscript</csymbol><ci id="S3.Thmthm11.p6.3.m3.1.1.3.2.2.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.2.2">𝒜</ci><times id="S3.Thmthm11.p6.3.m3.1.1.3.2.3.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.2.3"></times></apply><apply id="S3.Thmthm11.p6.3.m3.1.1.3.3.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p6.3.m3.1.1.3.3.1.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.3">superscript</csymbol><ci id="S3.Thmthm11.p6.3.m3.1.1.3.3.2.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.3.2">ℬ</ci><times id="S3.Thmthm11.p6.3.m3.1.1.3.3.3.cmml" xref="S3.Thmthm11.p6.3.m3.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p6.3.m3.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p6.3.m3.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> by setting <math alttext="\sigma(a)=d\,,\,\,\sigma(b)=e" class="ltx_Math" display="inline" id="S3.Thmthm11.p6.4.m4.4"><semantics id="S3.Thmthm11.p6.4.m4.4a"><mrow id="S3.Thmthm11.p6.4.m4.4.4.2" xref="S3.Thmthm11.p6.4.m4.4.4.3.cmml"><mrow id="S3.Thmthm11.p6.4.m4.3.3.1.1" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.cmml"><mrow id="S3.Thmthm11.p6.4.m4.3.3.1.1.2" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2.cmml"><mi id="S3.Thmthm11.p6.4.m4.3.3.1.1.2.2" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2.2.cmml">σ</mi><mo id="S3.Thmthm11.p6.4.m4.3.3.1.1.2.1" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p6.4.m4.3.3.1.1.2.3.2" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2.cmml"><mo id="S3.Thmthm11.p6.4.m4.3.3.1.1.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2.cmml">(</mo><mi id="S3.Thmthm11.p6.4.m4.1.1" xref="S3.Thmthm11.p6.4.m4.1.1.cmml">a</mi><mo id="S3.Thmthm11.p6.4.m4.3.3.1.1.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p6.4.m4.3.3.1.1.1" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.1.cmml">=</mo><mi id="S3.Thmthm11.p6.4.m4.3.3.1.1.3" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p6.4.m4.4.4.2.3" lspace="0.170em" rspace="0.497em" xref="S3.Thmthm11.p6.4.m4.4.4.3a.cmml">,</mo><mrow id="S3.Thmthm11.p6.4.m4.4.4.2.2" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.cmml"><mrow id="S3.Thmthm11.p6.4.m4.4.4.2.2.2" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2.cmml"><mi id="S3.Thmthm11.p6.4.m4.4.4.2.2.2.2" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2.2.cmml">σ</mi><mo id="S3.Thmthm11.p6.4.m4.4.4.2.2.2.1" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p6.4.m4.4.4.2.2.2.3.2" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2.cmml"><mo id="S3.Thmthm11.p6.4.m4.4.4.2.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2.cmml">(</mo><mi id="S3.Thmthm11.p6.4.m4.2.2" xref="S3.Thmthm11.p6.4.m4.2.2.cmml">b</mi><mo id="S3.Thmthm11.p6.4.m4.4.4.2.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p6.4.m4.4.4.2.2.1" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.1.cmml">=</mo><mi id="S3.Thmthm11.p6.4.m4.4.4.2.2.3" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.3.cmml">e</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p6.4.m4.4b"><apply id="S3.Thmthm11.p6.4.m4.4.4.3.cmml" xref="S3.Thmthm11.p6.4.m4.4.4.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p6.4.m4.4.4.3a.cmml" xref="S3.Thmthm11.p6.4.m4.4.4.2.3">formulae-sequence</csymbol><apply id="S3.Thmthm11.p6.4.m4.3.3.1.1.cmml" xref="S3.Thmthm11.p6.4.m4.3.3.1.1"><eq id="S3.Thmthm11.p6.4.m4.3.3.1.1.1.cmml" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.1"></eq><apply id="S3.Thmthm11.p6.4.m4.3.3.1.1.2.cmml" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2"><times id="S3.Thmthm11.p6.4.m4.3.3.1.1.2.1.cmml" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2.1"></times><ci id="S3.Thmthm11.p6.4.m4.3.3.1.1.2.2.cmml" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.2.2">𝜎</ci><ci id="S3.Thmthm11.p6.4.m4.1.1.cmml" xref="S3.Thmthm11.p6.4.m4.1.1">𝑎</ci></apply><ci id="S3.Thmthm11.p6.4.m4.3.3.1.1.3.cmml" xref="S3.Thmthm11.p6.4.m4.3.3.1.1.3">𝑑</ci></apply><apply id="S3.Thmthm11.p6.4.m4.4.4.2.2.cmml" xref="S3.Thmthm11.p6.4.m4.4.4.2.2"><eq id="S3.Thmthm11.p6.4.m4.4.4.2.2.1.cmml" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.1"></eq><apply id="S3.Thmthm11.p6.4.m4.4.4.2.2.2.cmml" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2"><times id="S3.Thmthm11.p6.4.m4.4.4.2.2.2.1.cmml" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2.1"></times><ci id="S3.Thmthm11.p6.4.m4.4.4.2.2.2.2.cmml" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.2.2">𝜎</ci><ci id="S3.Thmthm11.p6.4.m4.2.2.cmml" xref="S3.Thmthm11.p6.4.m4.2.2">𝑏</ci></apply><ci id="S3.Thmthm11.p6.4.m4.4.4.2.2.3.cmml" xref="S3.Thmthm11.p6.4.m4.4.4.2.2.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p6.4.m4.4c">\sigma(a)=d\,,\,\,\sigma(b)=e</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p6.4.m4.4d">italic_σ ( italic_a ) = italic_d , italic_σ ( italic_b ) = italic_e</annotation></semantics></math> and <math alttext="\sigma(c)=de" class="ltx_Math" display="inline" id="S3.Thmthm11.p6.5.m5.1"><semantics id="S3.Thmthm11.p6.5.m5.1a"><mrow id="S3.Thmthm11.p6.5.m5.1.2" xref="S3.Thmthm11.p6.5.m5.1.2.cmml"><mrow id="S3.Thmthm11.p6.5.m5.1.2.2" xref="S3.Thmthm11.p6.5.m5.1.2.2.cmml"><mi id="S3.Thmthm11.p6.5.m5.1.2.2.2" xref="S3.Thmthm11.p6.5.m5.1.2.2.2.cmml">σ</mi><mo id="S3.Thmthm11.p6.5.m5.1.2.2.1" xref="S3.Thmthm11.p6.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p6.5.m5.1.2.2.3.2" xref="S3.Thmthm11.p6.5.m5.1.2.2.cmml"><mo id="S3.Thmthm11.p6.5.m5.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p6.5.m5.1.2.2.cmml">(</mo><mi id="S3.Thmthm11.p6.5.m5.1.1" xref="S3.Thmthm11.p6.5.m5.1.1.cmml">c</mi><mo id="S3.Thmthm11.p6.5.m5.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p6.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p6.5.m5.1.2.1" xref="S3.Thmthm11.p6.5.m5.1.2.1.cmml">=</mo><mrow id="S3.Thmthm11.p6.5.m5.1.2.3" xref="S3.Thmthm11.p6.5.m5.1.2.3.cmml"><mi id="S3.Thmthm11.p6.5.m5.1.2.3.2" xref="S3.Thmthm11.p6.5.m5.1.2.3.2.cmml">d</mi><mo id="S3.Thmthm11.p6.5.m5.1.2.3.1" xref="S3.Thmthm11.p6.5.m5.1.2.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p6.5.m5.1.2.3.3" xref="S3.Thmthm11.p6.5.m5.1.2.3.3.cmml">e</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p6.5.m5.1b"><apply id="S3.Thmthm11.p6.5.m5.1.2.cmml" xref="S3.Thmthm11.p6.5.m5.1.2"><eq id="S3.Thmthm11.p6.5.m5.1.2.1.cmml" xref="S3.Thmthm11.p6.5.m5.1.2.1"></eq><apply id="S3.Thmthm11.p6.5.m5.1.2.2.cmml" xref="S3.Thmthm11.p6.5.m5.1.2.2"><times id="S3.Thmthm11.p6.5.m5.1.2.2.1.cmml" xref="S3.Thmthm11.p6.5.m5.1.2.2.1"></times><ci id="S3.Thmthm11.p6.5.m5.1.2.2.2.cmml" xref="S3.Thmthm11.p6.5.m5.1.2.2.2">𝜎</ci><ci id="S3.Thmthm11.p6.5.m5.1.1.cmml" xref="S3.Thmthm11.p6.5.m5.1.1">𝑐</ci></apply><apply id="S3.Thmthm11.p6.5.m5.1.2.3.cmml" xref="S3.Thmthm11.p6.5.m5.1.2.3"><times id="S3.Thmthm11.p6.5.m5.1.2.3.1.cmml" xref="S3.Thmthm11.p6.5.m5.1.2.3.1"></times><ci id="S3.Thmthm11.p6.5.m5.1.2.3.2.cmml" xref="S3.Thmthm11.p6.5.m5.1.2.3.2">𝑑</ci><ci id="S3.Thmthm11.p6.5.m5.1.2.3.3.cmml" xref="S3.Thmthm11.p6.5.m5.1.2.3.3">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p6.5.m5.1c">\sigma(c)=de</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p6.5.m5.1d">italic_σ ( italic_c ) = italic_d italic_e</annotation></semantics></math>. Set <math alttext="X=\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm11.p6.6.m6.1"><semantics id="S3.Thmthm11.p6.6.m6.1a"><mrow id="S3.Thmthm11.p6.6.m6.1.1" xref="S3.Thmthm11.p6.6.m6.1.1.cmml"><mi id="S3.Thmthm11.p6.6.m6.1.1.2" xref="S3.Thmthm11.p6.6.m6.1.1.2.cmml">X</mi><mo id="S3.Thmthm11.p6.6.m6.1.1.1" xref="S3.Thmthm11.p6.6.m6.1.1.1.cmml">=</mo><msup id="S3.Thmthm11.p6.6.m6.1.1.3" xref="S3.Thmthm11.p6.6.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p6.6.m6.1.1.3.2" xref="S3.Thmthm11.p6.6.m6.1.1.3.2.cmml">𝒜</mi><mi id="S3.Thmthm11.p6.6.m6.1.1.3.3" xref="S3.Thmthm11.p6.6.m6.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p6.6.m6.1b"><apply id="S3.Thmthm11.p6.6.m6.1.1.cmml" xref="S3.Thmthm11.p6.6.m6.1.1"><eq id="S3.Thmthm11.p6.6.m6.1.1.1.cmml" xref="S3.Thmthm11.p6.6.m6.1.1.1"></eq><ci id="S3.Thmthm11.p6.6.m6.1.1.2.cmml" xref="S3.Thmthm11.p6.6.m6.1.1.2">𝑋</ci><apply id="S3.Thmthm11.p6.6.m6.1.1.3.cmml" xref="S3.Thmthm11.p6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p6.6.m6.1.1.3.1.cmml" xref="S3.Thmthm11.p6.6.m6.1.1.3">superscript</csymbol><ci id="S3.Thmthm11.p6.6.m6.1.1.3.2.cmml" xref="S3.Thmthm11.p6.6.m6.1.1.3.2">𝒜</ci><ci id="S3.Thmthm11.p6.6.m6.1.1.3.3.cmml" xref="S3.Thmthm11.p6.6.m6.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p6.6.m6.1c">X=\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p6.6.m6.1d">italic_X = caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="Y=\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S3.Thmthm11.p6.7.m7.1"><semantics id="S3.Thmthm11.p6.7.m7.1a"><mrow id="S3.Thmthm11.p6.7.m7.1.1" xref="S3.Thmthm11.p6.7.m7.1.1.cmml"><mi id="S3.Thmthm11.p6.7.m7.1.1.2" xref="S3.Thmthm11.p6.7.m7.1.1.2.cmml">Y</mi><mo id="S3.Thmthm11.p6.7.m7.1.1.1" xref="S3.Thmthm11.p6.7.m7.1.1.1.cmml">=</mo><msup id="S3.Thmthm11.p6.7.m7.1.1.3" xref="S3.Thmthm11.p6.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmthm11.p6.7.m7.1.1.3.2" xref="S3.Thmthm11.p6.7.m7.1.1.3.2.cmml">ℬ</mi><mi id="S3.Thmthm11.p6.7.m7.1.1.3.3" xref="S3.Thmthm11.p6.7.m7.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p6.7.m7.1b"><apply id="S3.Thmthm11.p6.7.m7.1.1.cmml" xref="S3.Thmthm11.p6.7.m7.1.1"><eq id="S3.Thmthm11.p6.7.m7.1.1.1.cmml" xref="S3.Thmthm11.p6.7.m7.1.1.1"></eq><ci id="S3.Thmthm11.p6.7.m7.1.1.2.cmml" xref="S3.Thmthm11.p6.7.m7.1.1.2">𝑌</ci><apply id="S3.Thmthm11.p6.7.m7.1.1.3.cmml" xref="S3.Thmthm11.p6.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p6.7.m7.1.1.3.1.cmml" xref="S3.Thmthm11.p6.7.m7.1.1.3">superscript</csymbol><ci id="S3.Thmthm11.p6.7.m7.1.1.3.2.cmml" xref="S3.Thmthm11.p6.7.m7.1.1.3.2">ℬ</ci><ci id="S3.Thmthm11.p6.7.m7.1.1.3.3.cmml" xref="S3.Thmthm11.p6.7.m7.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p6.7.m7.1c">Y=\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p6.7.m7.1d">italic_Y = caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and notice <math alttext="Y=\sigma(X)" class="ltx_Math" display="inline" id="S3.Thmthm11.p6.8.m8.1"><semantics id="S3.Thmthm11.p6.8.m8.1a"><mrow id="S3.Thmthm11.p6.8.m8.1.2" xref="S3.Thmthm11.p6.8.m8.1.2.cmml"><mi id="S3.Thmthm11.p6.8.m8.1.2.2" xref="S3.Thmthm11.p6.8.m8.1.2.2.cmml">Y</mi><mo id="S3.Thmthm11.p6.8.m8.1.2.1" xref="S3.Thmthm11.p6.8.m8.1.2.1.cmml">=</mo><mrow id="S3.Thmthm11.p6.8.m8.1.2.3" xref="S3.Thmthm11.p6.8.m8.1.2.3.cmml"><mi id="S3.Thmthm11.p6.8.m8.1.2.3.2" xref="S3.Thmthm11.p6.8.m8.1.2.3.2.cmml">σ</mi><mo id="S3.Thmthm11.p6.8.m8.1.2.3.1" xref="S3.Thmthm11.p6.8.m8.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p6.8.m8.1.2.3.3.2" xref="S3.Thmthm11.p6.8.m8.1.2.3.cmml"><mo id="S3.Thmthm11.p6.8.m8.1.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p6.8.m8.1.2.3.cmml">(</mo><mi id="S3.Thmthm11.p6.8.m8.1.1" xref="S3.Thmthm11.p6.8.m8.1.1.cmml">X</mi><mo id="S3.Thmthm11.p6.8.m8.1.2.3.3.2.2" stretchy="false" xref="S3.Thmthm11.p6.8.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p6.8.m8.1b"><apply id="S3.Thmthm11.p6.8.m8.1.2.cmml" xref="S3.Thmthm11.p6.8.m8.1.2"><eq id="S3.Thmthm11.p6.8.m8.1.2.1.cmml" xref="S3.Thmthm11.p6.8.m8.1.2.1"></eq><ci id="S3.Thmthm11.p6.8.m8.1.2.2.cmml" xref="S3.Thmthm11.p6.8.m8.1.2.2">𝑌</ci><apply id="S3.Thmthm11.p6.8.m8.1.2.3.cmml" xref="S3.Thmthm11.p6.8.m8.1.2.3"><times id="S3.Thmthm11.p6.8.m8.1.2.3.1.cmml" xref="S3.Thmthm11.p6.8.m8.1.2.3.1"></times><ci id="S3.Thmthm11.p6.8.m8.1.2.3.2.cmml" xref="S3.Thmthm11.p6.8.m8.1.2.3.2">𝜎</ci><ci id="S3.Thmthm11.p6.8.m8.1.1.cmml" xref="S3.Thmthm11.p6.8.m8.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p6.8.m8.1c">Y=\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p6.8.m8.1d">italic_Y = italic_σ ( italic_X )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.Thmthm11.p7"> <p class="ltx_p" id="S3.Thmthm11.p7.22">Consider now the function <math alttext="g:Y\to\mathbb{Z}" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.1.m1.1"><semantics id="S3.Thmthm11.p7.1.m1.1a"><mrow id="S3.Thmthm11.p7.1.m1.1.1" xref="S3.Thmthm11.p7.1.m1.1.1.cmml"><mi id="S3.Thmthm11.p7.1.m1.1.1.2" xref="S3.Thmthm11.p7.1.m1.1.1.2.cmml">g</mi><mo id="S3.Thmthm11.p7.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmthm11.p7.1.m1.1.1.1.cmml">:</mo><mrow id="S3.Thmthm11.p7.1.m1.1.1.3" xref="S3.Thmthm11.p7.1.m1.1.1.3.cmml"><mi id="S3.Thmthm11.p7.1.m1.1.1.3.2" xref="S3.Thmthm11.p7.1.m1.1.1.3.2.cmml">Y</mi><mo id="S3.Thmthm11.p7.1.m1.1.1.3.1" stretchy="false" xref="S3.Thmthm11.p7.1.m1.1.1.3.1.cmml">→</mo><mi id="S3.Thmthm11.p7.1.m1.1.1.3.3" xref="S3.Thmthm11.p7.1.m1.1.1.3.3.cmml">ℤ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.1.m1.1b"><apply id="S3.Thmthm11.p7.1.m1.1.1.cmml" xref="S3.Thmthm11.p7.1.m1.1.1"><ci id="S3.Thmthm11.p7.1.m1.1.1.1.cmml" xref="S3.Thmthm11.p7.1.m1.1.1.1">:</ci><ci id="S3.Thmthm11.p7.1.m1.1.1.2.cmml" xref="S3.Thmthm11.p7.1.m1.1.1.2">𝑔</ci><apply id="S3.Thmthm11.p7.1.m1.1.1.3.cmml" xref="S3.Thmthm11.p7.1.m1.1.1.3"><ci id="S3.Thmthm11.p7.1.m1.1.1.3.1.cmml" xref="S3.Thmthm11.p7.1.m1.1.1.3.1">→</ci><ci id="S3.Thmthm11.p7.1.m1.1.1.3.2.cmml" xref="S3.Thmthm11.p7.1.m1.1.1.3.2">𝑌</ci><ci id="S3.Thmthm11.p7.1.m1.1.1.3.3.cmml" xref="S3.Thmthm11.p7.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.1.m1.1c">g:Y\to\mathbb{Z}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.1.m1.1d">italic_g : italic_Y → blackboard_Z</annotation></semantics></math> defined by <math alttext="g({\bf y})=0" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.2.m2.1"><semantics id="S3.Thmthm11.p7.2.m2.1a"><mrow id="S3.Thmthm11.p7.2.m2.1.2" xref="S3.Thmthm11.p7.2.m2.1.2.cmml"><mrow id="S3.Thmthm11.p7.2.m2.1.2.2" xref="S3.Thmthm11.p7.2.m2.1.2.2.cmml"><mi id="S3.Thmthm11.p7.2.m2.1.2.2.2" xref="S3.Thmthm11.p7.2.m2.1.2.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.2.m2.1.2.2.1" xref="S3.Thmthm11.p7.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.2.m2.1.2.2.3.2" xref="S3.Thmthm11.p7.2.m2.1.2.2.cmml"><mo id="S3.Thmthm11.p7.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.2.m2.1.2.2.cmml">(</mo><mi id="S3.Thmthm11.p7.2.m2.1.1" xref="S3.Thmthm11.p7.2.m2.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.2.m2.1.2.1" xref="S3.Thmthm11.p7.2.m2.1.2.1.cmml">=</mo><mn id="S3.Thmthm11.p7.2.m2.1.2.3" xref="S3.Thmthm11.p7.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.2.m2.1b"><apply id="S3.Thmthm11.p7.2.m2.1.2.cmml" xref="S3.Thmthm11.p7.2.m2.1.2"><eq id="S3.Thmthm11.p7.2.m2.1.2.1.cmml" xref="S3.Thmthm11.p7.2.m2.1.2.1"></eq><apply id="S3.Thmthm11.p7.2.m2.1.2.2.cmml" xref="S3.Thmthm11.p7.2.m2.1.2.2"><times id="S3.Thmthm11.p7.2.m2.1.2.2.1.cmml" xref="S3.Thmthm11.p7.2.m2.1.2.2.1"></times><ci id="S3.Thmthm11.p7.2.m2.1.2.2.2.cmml" xref="S3.Thmthm11.p7.2.m2.1.2.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.2.m2.1.1.cmml" xref="S3.Thmthm11.p7.2.m2.1.1">𝐲</ci></apply><cn id="S3.Thmthm11.p7.2.m2.1.2.3.cmml" type="integer" xref="S3.Thmthm11.p7.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.2.m2.1c">g({\bf y})=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.2.m2.1d">italic_g ( bold_y ) = 0</annotation></semantics></math> if <math alttext="{\bf y}\in[d]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.3.m3.1"><semantics id="S3.Thmthm11.p7.3.m3.1a"><mrow id="S3.Thmthm11.p7.3.m3.1.2" xref="S3.Thmthm11.p7.3.m3.1.2.cmml"><mi id="S3.Thmthm11.p7.3.m3.1.2.2" xref="S3.Thmthm11.p7.3.m3.1.2.2.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.3.m3.1.2.1" xref="S3.Thmthm11.p7.3.m3.1.2.1.cmml">∈</mo><mrow id="S3.Thmthm11.p7.3.m3.1.2.3.2" xref="S3.Thmthm11.p7.3.m3.1.2.3.1.cmml"><mo id="S3.Thmthm11.p7.3.m3.1.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.3.m3.1.2.3.1.1.cmml">[</mo><mi id="S3.Thmthm11.p7.3.m3.1.1" xref="S3.Thmthm11.p7.3.m3.1.1.cmml">d</mi><mo id="S3.Thmthm11.p7.3.m3.1.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.3.m3.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.3.m3.1b"><apply id="S3.Thmthm11.p7.3.m3.1.2.cmml" xref="S3.Thmthm11.p7.3.m3.1.2"><in id="S3.Thmthm11.p7.3.m3.1.2.1.cmml" xref="S3.Thmthm11.p7.3.m3.1.2.1"></in><ci id="S3.Thmthm11.p7.3.m3.1.2.2.cmml" xref="S3.Thmthm11.p7.3.m3.1.2.2">𝐲</ci><apply id="S3.Thmthm11.p7.3.m3.1.2.3.1.cmml" xref="S3.Thmthm11.p7.3.m3.1.2.3.2"><csymbol cd="latexml" id="S3.Thmthm11.p7.3.m3.1.2.3.1.1.cmml" xref="S3.Thmthm11.p7.3.m3.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.Thmthm11.p7.3.m3.1.1.cmml" xref="S3.Thmthm11.p7.3.m3.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.3.m3.1c">{\bf y}\in[d]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.3.m3.1d">bold_y ∈ [ italic_d ]</annotation></semantics></math> and <math alttext="g({\bf y})=1" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.4.m4.1"><semantics id="S3.Thmthm11.p7.4.m4.1a"><mrow id="S3.Thmthm11.p7.4.m4.1.2" xref="S3.Thmthm11.p7.4.m4.1.2.cmml"><mrow id="S3.Thmthm11.p7.4.m4.1.2.2" xref="S3.Thmthm11.p7.4.m4.1.2.2.cmml"><mi id="S3.Thmthm11.p7.4.m4.1.2.2.2" xref="S3.Thmthm11.p7.4.m4.1.2.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.4.m4.1.2.2.1" xref="S3.Thmthm11.p7.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.4.m4.1.2.2.3.2" xref="S3.Thmthm11.p7.4.m4.1.2.2.cmml"><mo id="S3.Thmthm11.p7.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.4.m4.1.2.2.cmml">(</mo><mi id="S3.Thmthm11.p7.4.m4.1.1" xref="S3.Thmthm11.p7.4.m4.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.4.m4.1.2.1" xref="S3.Thmthm11.p7.4.m4.1.2.1.cmml">=</mo><mn id="S3.Thmthm11.p7.4.m4.1.2.3" xref="S3.Thmthm11.p7.4.m4.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.4.m4.1b"><apply id="S3.Thmthm11.p7.4.m4.1.2.cmml" xref="S3.Thmthm11.p7.4.m4.1.2"><eq id="S3.Thmthm11.p7.4.m4.1.2.1.cmml" xref="S3.Thmthm11.p7.4.m4.1.2.1"></eq><apply id="S3.Thmthm11.p7.4.m4.1.2.2.cmml" xref="S3.Thmthm11.p7.4.m4.1.2.2"><times id="S3.Thmthm11.p7.4.m4.1.2.2.1.cmml" xref="S3.Thmthm11.p7.4.m4.1.2.2.1"></times><ci id="S3.Thmthm11.p7.4.m4.1.2.2.2.cmml" xref="S3.Thmthm11.p7.4.m4.1.2.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.4.m4.1.1.cmml" xref="S3.Thmthm11.p7.4.m4.1.1">𝐲</ci></apply><cn id="S3.Thmthm11.p7.4.m4.1.2.3.cmml" type="integer" xref="S3.Thmthm11.p7.4.m4.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.4.m4.1c">g({\bf y})=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.4.m4.1d">italic_g ( bold_y ) = 1</annotation></semantics></math> if <math alttext="{\bf y}\in[e]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.5.m5.1"><semantics id="S3.Thmthm11.p7.5.m5.1a"><mrow id="S3.Thmthm11.p7.5.m5.1.2" xref="S3.Thmthm11.p7.5.m5.1.2.cmml"><mi id="S3.Thmthm11.p7.5.m5.1.2.2" xref="S3.Thmthm11.p7.5.m5.1.2.2.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.5.m5.1.2.1" xref="S3.Thmthm11.p7.5.m5.1.2.1.cmml">∈</mo><mrow id="S3.Thmthm11.p7.5.m5.1.2.3.2" xref="S3.Thmthm11.p7.5.m5.1.2.3.1.cmml"><mo id="S3.Thmthm11.p7.5.m5.1.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.5.m5.1.2.3.1.1.cmml">[</mo><mi id="S3.Thmthm11.p7.5.m5.1.1" xref="S3.Thmthm11.p7.5.m5.1.1.cmml">e</mi><mo id="S3.Thmthm11.p7.5.m5.1.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.5.m5.1.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.5.m5.1b"><apply id="S3.Thmthm11.p7.5.m5.1.2.cmml" xref="S3.Thmthm11.p7.5.m5.1.2"><in id="S3.Thmthm11.p7.5.m5.1.2.1.cmml" xref="S3.Thmthm11.p7.5.m5.1.2.1"></in><ci id="S3.Thmthm11.p7.5.m5.1.2.2.cmml" xref="S3.Thmthm11.p7.5.m5.1.2.2">𝐲</ci><apply id="S3.Thmthm11.p7.5.m5.1.2.3.1.cmml" xref="S3.Thmthm11.p7.5.m5.1.2.3.2"><csymbol cd="latexml" id="S3.Thmthm11.p7.5.m5.1.2.3.1.1.cmml" xref="S3.Thmthm11.p7.5.m5.1.2.3.2.1">delimited-[]</csymbol><ci id="S3.Thmthm11.p7.5.m5.1.1.cmml" xref="S3.Thmthm11.p7.5.m5.1.1">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.5.m5.1c">{\bf y}\in[e]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.5.m5.1d">bold_y ∈ [ italic_e ]</annotation></semantics></math>, which gives <math alttext="g({\bf y})=0" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.6.m6.1"><semantics id="S3.Thmthm11.p7.6.m6.1a"><mrow id="S3.Thmthm11.p7.6.m6.1.2" xref="S3.Thmthm11.p7.6.m6.1.2.cmml"><mrow id="S3.Thmthm11.p7.6.m6.1.2.2" xref="S3.Thmthm11.p7.6.m6.1.2.2.cmml"><mi id="S3.Thmthm11.p7.6.m6.1.2.2.2" xref="S3.Thmthm11.p7.6.m6.1.2.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.6.m6.1.2.2.1" xref="S3.Thmthm11.p7.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.6.m6.1.2.2.3.2" xref="S3.Thmthm11.p7.6.m6.1.2.2.cmml"><mo id="S3.Thmthm11.p7.6.m6.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.6.m6.1.2.2.cmml">(</mo><mi id="S3.Thmthm11.p7.6.m6.1.1" xref="S3.Thmthm11.p7.6.m6.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.6.m6.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.6.m6.1.2.1" xref="S3.Thmthm11.p7.6.m6.1.2.1.cmml">=</mo><mn id="S3.Thmthm11.p7.6.m6.1.2.3" xref="S3.Thmthm11.p7.6.m6.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.6.m6.1b"><apply id="S3.Thmthm11.p7.6.m6.1.2.cmml" xref="S3.Thmthm11.p7.6.m6.1.2"><eq id="S3.Thmthm11.p7.6.m6.1.2.1.cmml" xref="S3.Thmthm11.p7.6.m6.1.2.1"></eq><apply id="S3.Thmthm11.p7.6.m6.1.2.2.cmml" xref="S3.Thmthm11.p7.6.m6.1.2.2"><times id="S3.Thmthm11.p7.6.m6.1.2.2.1.cmml" xref="S3.Thmthm11.p7.6.m6.1.2.2.1"></times><ci id="S3.Thmthm11.p7.6.m6.1.2.2.2.cmml" xref="S3.Thmthm11.p7.6.m6.1.2.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.6.m6.1.1.cmml" xref="S3.Thmthm11.p7.6.m6.1.1">𝐲</ci></apply><cn id="S3.Thmthm11.p7.6.m6.1.2.3.cmml" type="integer" xref="S3.Thmthm11.p7.6.m6.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.6.m6.1c">g({\bf y})=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.6.m6.1d">italic_g ( bold_y ) = 0</annotation></semantics></math> if <math alttext="{\bf y}\in[dd]\cup[de]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.7.m7.2"><semantics id="S3.Thmthm11.p7.7.m7.2a"><mrow id="S3.Thmthm11.p7.7.m7.2.2" xref="S3.Thmthm11.p7.7.m7.2.2.cmml"><mi id="S3.Thmthm11.p7.7.m7.2.2.4" xref="S3.Thmthm11.p7.7.m7.2.2.4.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.7.m7.2.2.3" xref="S3.Thmthm11.p7.7.m7.2.2.3.cmml">∈</mo><mrow id="S3.Thmthm11.p7.7.m7.2.2.2" xref="S3.Thmthm11.p7.7.m7.2.2.2.cmml"><mrow id="S3.Thmthm11.p7.7.m7.1.1.1.1.1" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.2" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.3" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.Thmthm11.p7.7.m7.2.2.2.3" xref="S3.Thmthm11.p7.7.m7.2.2.2.3.cmml">∪</mo><mrow id="S3.Thmthm11.p7.7.m7.2.2.2.2.1" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.2.cmml"><mo id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.2" stretchy="false" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.cmml"><mi id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.2" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.1" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.3" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.3" stretchy="false" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.7.m7.2b"><apply id="S3.Thmthm11.p7.7.m7.2.2.cmml" xref="S3.Thmthm11.p7.7.m7.2.2"><in id="S3.Thmthm11.p7.7.m7.2.2.3.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.3"></in><ci id="S3.Thmthm11.p7.7.m7.2.2.4.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.4">𝐲</ci><apply id="S3.Thmthm11.p7.7.m7.2.2.2.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.2"><union id="S3.Thmthm11.p7.7.m7.2.2.2.3.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.2.3"></union><apply id="S3.Thmthm11.p7.7.m7.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.7.m7.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1"><times id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.7.m7.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.Thmthm11.p7.7.m7.2.2.2.2.2.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.7.m7.2.2.2.2.2.1.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1"><times id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.1.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.1"></times><ci id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.2.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.3.cmml" xref="S3.Thmthm11.p7.7.m7.2.2.2.2.1.1.3">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.7.m7.2c">{\bf y}\in[dd]\cup[de]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.7.m7.2d">bold_y ∈ [ italic_d italic_d ] ∪ [ italic_d italic_e ]</annotation></semantics></math> as well as <math alttext="g({\bf y})=1" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.8.m8.1"><semantics id="S3.Thmthm11.p7.8.m8.1a"><mrow id="S3.Thmthm11.p7.8.m8.1.2" xref="S3.Thmthm11.p7.8.m8.1.2.cmml"><mrow id="S3.Thmthm11.p7.8.m8.1.2.2" xref="S3.Thmthm11.p7.8.m8.1.2.2.cmml"><mi id="S3.Thmthm11.p7.8.m8.1.2.2.2" xref="S3.Thmthm11.p7.8.m8.1.2.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.8.m8.1.2.2.1" xref="S3.Thmthm11.p7.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.8.m8.1.2.2.3.2" xref="S3.Thmthm11.p7.8.m8.1.2.2.cmml"><mo id="S3.Thmthm11.p7.8.m8.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.8.m8.1.2.2.cmml">(</mo><mi id="S3.Thmthm11.p7.8.m8.1.1" xref="S3.Thmthm11.p7.8.m8.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.8.m8.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.8.m8.1.2.1" xref="S3.Thmthm11.p7.8.m8.1.2.1.cmml">=</mo><mn id="S3.Thmthm11.p7.8.m8.1.2.3" xref="S3.Thmthm11.p7.8.m8.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.8.m8.1b"><apply id="S3.Thmthm11.p7.8.m8.1.2.cmml" xref="S3.Thmthm11.p7.8.m8.1.2"><eq id="S3.Thmthm11.p7.8.m8.1.2.1.cmml" xref="S3.Thmthm11.p7.8.m8.1.2.1"></eq><apply id="S3.Thmthm11.p7.8.m8.1.2.2.cmml" xref="S3.Thmthm11.p7.8.m8.1.2.2"><times id="S3.Thmthm11.p7.8.m8.1.2.2.1.cmml" xref="S3.Thmthm11.p7.8.m8.1.2.2.1"></times><ci id="S3.Thmthm11.p7.8.m8.1.2.2.2.cmml" xref="S3.Thmthm11.p7.8.m8.1.2.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.8.m8.1.1.cmml" xref="S3.Thmthm11.p7.8.m8.1.1">𝐲</ci></apply><cn id="S3.Thmthm11.p7.8.m8.1.2.3.cmml" type="integer" xref="S3.Thmthm11.p7.8.m8.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.8.m8.1c">g({\bf y})=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.8.m8.1d">italic_g ( bold_y ) = 1</annotation></semantics></math> if <math alttext="{\bf y}\in[ed]\cup[ee]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.9.m9.2"><semantics id="S3.Thmthm11.p7.9.m9.2a"><mrow id="S3.Thmthm11.p7.9.m9.2.2" xref="S3.Thmthm11.p7.9.m9.2.2.cmml"><mi id="S3.Thmthm11.p7.9.m9.2.2.4" xref="S3.Thmthm11.p7.9.m9.2.2.4.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.9.m9.2.2.3" xref="S3.Thmthm11.p7.9.m9.2.2.3.cmml">∈</mo><mrow id="S3.Thmthm11.p7.9.m9.2.2.2" xref="S3.Thmthm11.p7.9.m9.2.2.2.cmml"><mrow id="S3.Thmthm11.p7.9.m9.1.1.1.1.1" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.2" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.2.cmml">e</mi><mo id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.3" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.Thmthm11.p7.9.m9.2.2.2.3" xref="S3.Thmthm11.p7.9.m9.2.2.2.3.cmml">∪</mo><mrow id="S3.Thmthm11.p7.9.m9.2.2.2.2.1" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.2.cmml"><mo id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.2" stretchy="false" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.cmml"><mi id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.2" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.2.cmml">e</mi><mo id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.1" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.3" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.3" stretchy="false" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.9.m9.2b"><apply id="S3.Thmthm11.p7.9.m9.2.2.cmml" xref="S3.Thmthm11.p7.9.m9.2.2"><in id="S3.Thmthm11.p7.9.m9.2.2.3.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.3"></in><ci id="S3.Thmthm11.p7.9.m9.2.2.4.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.4">𝐲</ci><apply id="S3.Thmthm11.p7.9.m9.2.2.2.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.2"><union id="S3.Thmthm11.p7.9.m9.2.2.2.3.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.2.3"></union><apply id="S3.Thmthm11.p7.9.m9.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.9.m9.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1"><times id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.2">𝑒</ci><ci id="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.9.m9.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.Thmthm11.p7.9.m9.2.2.2.2.2.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.9.m9.2.2.2.2.2.1.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1"><times id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.1.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.1"></times><ci id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.2.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.2">𝑒</ci><ci id="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.3.cmml" xref="S3.Thmthm11.p7.9.m9.2.2.2.2.1.1.3">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.9.m9.2c">{\bf y}\in[ed]\cup[ee]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.9.m9.2d">bold_y ∈ [ italic_e italic_d ] ∪ [ italic_e italic_e ]</annotation></semantics></math>. Since <math alttext="T([dd]\cup[ed])=[d]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.10.m10.2"><semantics id="S3.Thmthm11.p7.10.m10.2a"><mrow id="S3.Thmthm11.p7.10.m10.2.2" xref="S3.Thmthm11.p7.10.m10.2.2.cmml"><mrow id="S3.Thmthm11.p7.10.m10.2.2.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.cmml"><mi id="S3.Thmthm11.p7.10.m10.2.2.1.3" xref="S3.Thmthm11.p7.10.m10.2.2.1.3.cmml">T</mi><mo id="S3.Thmthm11.p7.10.m10.2.2.1.2" xref="S3.Thmthm11.p7.10.m10.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.10.m10.2.2.1.1.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.cmml"><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.2" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.3" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.3" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.3.cmml">∪</mo><mrow id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.2.cmml"><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.2" stretchy="false" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.cmml"><mi id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.2" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.2.cmml">e</mi><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.1" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.3" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.3" stretchy="false" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.2.1.cmml">]</mo></mrow></mrow><mo id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.10.m10.2.2.2" xref="S3.Thmthm11.p7.10.m10.2.2.2.cmml">=</mo><mrow id="S3.Thmthm11.p7.10.m10.2.2.3.2" xref="S3.Thmthm11.p7.10.m10.2.2.3.1.cmml"><mo id="S3.Thmthm11.p7.10.m10.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.10.m10.2.2.3.1.1.cmml">[</mo><mi id="S3.Thmthm11.p7.10.m10.1.1" xref="S3.Thmthm11.p7.10.m10.1.1.cmml">d</mi><mo id="S3.Thmthm11.p7.10.m10.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.10.m10.2.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.10.m10.2b"><apply id="S3.Thmthm11.p7.10.m10.2.2.cmml" xref="S3.Thmthm11.p7.10.m10.2.2"><eq id="S3.Thmthm11.p7.10.m10.2.2.2.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.2"></eq><apply id="S3.Thmthm11.p7.10.m10.2.2.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1"><times id="S3.Thmthm11.p7.10.m10.2.2.1.2.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.2"></times><ci id="S3.Thmthm11.p7.10.m10.2.2.1.3.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.3">𝑇</ci><apply id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1"><union id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.3"></union><apply id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1"><times id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.2.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1"><times id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.1"></times><ci id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.2.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.2">𝑒</ci><ci id="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.3.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.1.1.1.1.2.1.1.3">𝑑</ci></apply></apply></apply></apply><apply id="S3.Thmthm11.p7.10.m10.2.2.3.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.3.2"><csymbol cd="latexml" id="S3.Thmthm11.p7.10.m10.2.2.3.1.1.cmml" xref="S3.Thmthm11.p7.10.m10.2.2.3.2.1">delimited-[]</csymbol><ci id="S3.Thmthm11.p7.10.m10.1.1.cmml" xref="S3.Thmthm11.p7.10.m10.1.1">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.10.m10.2c">T([dd]\cup[ed])=[d]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.10.m10.2d">italic_T ( [ italic_d italic_d ] ∪ [ italic_e italic_d ] ) = [ italic_d ]</annotation></semantics></math> and <math alttext="T([de]\cup[ee])=[e]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.11.m11.2"><semantics id="S3.Thmthm11.p7.11.m11.2a"><mrow id="S3.Thmthm11.p7.11.m11.2.2" xref="S3.Thmthm11.p7.11.m11.2.2.cmml"><mrow id="S3.Thmthm11.p7.11.m11.2.2.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.cmml"><mi id="S3.Thmthm11.p7.11.m11.2.2.1.3" xref="S3.Thmthm11.p7.11.m11.2.2.1.3.cmml">T</mi><mo id="S3.Thmthm11.p7.11.m11.2.2.1.2" xref="S3.Thmthm11.p7.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.11.m11.2.2.1.1.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.cmml"><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.2" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.3" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.3" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.3.cmml">∪</mo><mrow id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.2.cmml"><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.2" stretchy="false" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.cmml"><mi id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.2" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.2.cmml">e</mi><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.1" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.3" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.3" stretchy="false" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.2.1.cmml">]</mo></mrow></mrow><mo id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.11.m11.2.2.2" xref="S3.Thmthm11.p7.11.m11.2.2.2.cmml">=</mo><mrow id="S3.Thmthm11.p7.11.m11.2.2.3.2" xref="S3.Thmthm11.p7.11.m11.2.2.3.1.cmml"><mo id="S3.Thmthm11.p7.11.m11.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.11.m11.2.2.3.1.1.cmml">[</mo><mi id="S3.Thmthm11.p7.11.m11.1.1" xref="S3.Thmthm11.p7.11.m11.1.1.cmml">e</mi><mo id="S3.Thmthm11.p7.11.m11.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.11.m11.2.2.3.1.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.11.m11.2b"><apply id="S3.Thmthm11.p7.11.m11.2.2.cmml" xref="S3.Thmthm11.p7.11.m11.2.2"><eq id="S3.Thmthm11.p7.11.m11.2.2.2.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.2"></eq><apply id="S3.Thmthm11.p7.11.m11.2.2.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1"><times id="S3.Thmthm11.p7.11.m11.2.2.1.2.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.2"></times><ci id="S3.Thmthm11.p7.11.m11.2.2.1.3.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.3">𝑇</ci><apply id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1"><union id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.3"></union><apply id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1"><times id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.1.1.1.3">𝑒</ci></apply></apply><apply id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.2.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1"><times id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.1"></times><ci id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.2.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.2">𝑒</ci><ci id="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.3.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.1.1.1.1.2.1.1.3">𝑒</ci></apply></apply></apply></apply><apply id="S3.Thmthm11.p7.11.m11.2.2.3.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.3.2"><csymbol cd="latexml" id="S3.Thmthm11.p7.11.m11.2.2.3.1.1.cmml" xref="S3.Thmthm11.p7.11.m11.2.2.3.2.1">delimited-[]</csymbol><ci id="S3.Thmthm11.p7.11.m11.1.1.cmml" xref="S3.Thmthm11.p7.11.m11.1.1">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.11.m11.2c">T([de]\cup[ee])=[e]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.11.m11.2d">italic_T ( [ italic_d italic_e ] ∪ [ italic_e italic_e ] ) = [ italic_e ]</annotation></semantics></math>, we obtain <math alttext="g\circ T({\bf y})=0" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.12.m12.1"><semantics id="S3.Thmthm11.p7.12.m12.1a"><mrow id="S3.Thmthm11.p7.12.m12.1.2" xref="S3.Thmthm11.p7.12.m12.1.2.cmml"><mrow id="S3.Thmthm11.p7.12.m12.1.2.2" xref="S3.Thmthm11.p7.12.m12.1.2.2.cmml"><mrow id="S3.Thmthm11.p7.12.m12.1.2.2.2" xref="S3.Thmthm11.p7.12.m12.1.2.2.2.cmml"><mi id="S3.Thmthm11.p7.12.m12.1.2.2.2.2" xref="S3.Thmthm11.p7.12.m12.1.2.2.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.12.m12.1.2.2.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p7.12.m12.1.2.2.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p7.12.m12.1.2.2.2.3" xref="S3.Thmthm11.p7.12.m12.1.2.2.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p7.12.m12.1.2.2.1" xref="S3.Thmthm11.p7.12.m12.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.12.m12.1.2.2.3.2" xref="S3.Thmthm11.p7.12.m12.1.2.2.cmml"><mo id="S3.Thmthm11.p7.12.m12.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.12.m12.1.2.2.cmml">(</mo><mi id="S3.Thmthm11.p7.12.m12.1.1" xref="S3.Thmthm11.p7.12.m12.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.12.m12.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.12.m12.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.12.m12.1.2.1" xref="S3.Thmthm11.p7.12.m12.1.2.1.cmml">=</mo><mn id="S3.Thmthm11.p7.12.m12.1.2.3" xref="S3.Thmthm11.p7.12.m12.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.12.m12.1b"><apply id="S3.Thmthm11.p7.12.m12.1.2.cmml" xref="S3.Thmthm11.p7.12.m12.1.2"><eq id="S3.Thmthm11.p7.12.m12.1.2.1.cmml" xref="S3.Thmthm11.p7.12.m12.1.2.1"></eq><apply id="S3.Thmthm11.p7.12.m12.1.2.2.cmml" xref="S3.Thmthm11.p7.12.m12.1.2.2"><times id="S3.Thmthm11.p7.12.m12.1.2.2.1.cmml" xref="S3.Thmthm11.p7.12.m12.1.2.2.1"></times><apply id="S3.Thmthm11.p7.12.m12.1.2.2.2.cmml" xref="S3.Thmthm11.p7.12.m12.1.2.2.2"><compose id="S3.Thmthm11.p7.12.m12.1.2.2.2.1.cmml" xref="S3.Thmthm11.p7.12.m12.1.2.2.2.1"></compose><ci id="S3.Thmthm11.p7.12.m12.1.2.2.2.2.cmml" xref="S3.Thmthm11.p7.12.m12.1.2.2.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.12.m12.1.2.2.2.3.cmml" xref="S3.Thmthm11.p7.12.m12.1.2.2.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p7.12.m12.1.1.cmml" xref="S3.Thmthm11.p7.12.m12.1.1">𝐲</ci></apply><cn id="S3.Thmthm11.p7.12.m12.1.2.3.cmml" type="integer" xref="S3.Thmthm11.p7.12.m12.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.12.m12.1c">g\circ T({\bf y})=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.12.m12.1d">italic_g ∘ italic_T ( bold_y ) = 0</annotation></semantics></math> if <math alttext="{\bf y}\in[dd]\cup[ed]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.13.m13.2"><semantics id="S3.Thmthm11.p7.13.m13.2a"><mrow id="S3.Thmthm11.p7.13.m13.2.2" xref="S3.Thmthm11.p7.13.m13.2.2.cmml"><mi id="S3.Thmthm11.p7.13.m13.2.2.4" xref="S3.Thmthm11.p7.13.m13.2.2.4.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.13.m13.2.2.3" xref="S3.Thmthm11.p7.13.m13.2.2.3.cmml">∈</mo><mrow id="S3.Thmthm11.p7.13.m13.2.2.2" xref="S3.Thmthm11.p7.13.m13.2.2.2.cmml"><mrow id="S3.Thmthm11.p7.13.m13.1.1.1.1.1" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.2" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.3" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.Thmthm11.p7.13.m13.2.2.2.3" xref="S3.Thmthm11.p7.13.m13.2.2.2.3.cmml">∪</mo><mrow id="S3.Thmthm11.p7.13.m13.2.2.2.2.1" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.2.cmml"><mo id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.2" stretchy="false" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.cmml"><mi id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.2" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.2.cmml">e</mi><mo id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.1" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.3" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.3" stretchy="false" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.13.m13.2b"><apply id="S3.Thmthm11.p7.13.m13.2.2.cmml" xref="S3.Thmthm11.p7.13.m13.2.2"><in id="S3.Thmthm11.p7.13.m13.2.2.3.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.3"></in><ci id="S3.Thmthm11.p7.13.m13.2.2.4.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.4">𝐲</ci><apply id="S3.Thmthm11.p7.13.m13.2.2.2.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.2"><union id="S3.Thmthm11.p7.13.m13.2.2.2.3.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.2.3"></union><apply id="S3.Thmthm11.p7.13.m13.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.13.m13.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1"><times id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.13.m13.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.Thmthm11.p7.13.m13.2.2.2.2.2.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.13.m13.2.2.2.2.2.1.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1"><times id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.1.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.1"></times><ci id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.2.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.2">𝑒</ci><ci id="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.3.cmml" xref="S3.Thmthm11.p7.13.m13.2.2.2.2.1.1.3">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.13.m13.2c">{\bf y}\in[dd]\cup[ed]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.13.m13.2d">bold_y ∈ [ italic_d italic_d ] ∪ [ italic_e italic_d ]</annotation></semantics></math> and <math alttext="g\circ T({\bf y})=1" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.14.m14.1"><semantics id="S3.Thmthm11.p7.14.m14.1a"><mrow id="S3.Thmthm11.p7.14.m14.1.2" xref="S3.Thmthm11.p7.14.m14.1.2.cmml"><mrow id="S3.Thmthm11.p7.14.m14.1.2.2" xref="S3.Thmthm11.p7.14.m14.1.2.2.cmml"><mrow id="S3.Thmthm11.p7.14.m14.1.2.2.2" xref="S3.Thmthm11.p7.14.m14.1.2.2.2.cmml"><mi id="S3.Thmthm11.p7.14.m14.1.2.2.2.2" xref="S3.Thmthm11.p7.14.m14.1.2.2.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.14.m14.1.2.2.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p7.14.m14.1.2.2.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p7.14.m14.1.2.2.2.3" xref="S3.Thmthm11.p7.14.m14.1.2.2.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p7.14.m14.1.2.2.1" xref="S3.Thmthm11.p7.14.m14.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.14.m14.1.2.2.3.2" xref="S3.Thmthm11.p7.14.m14.1.2.2.cmml"><mo id="S3.Thmthm11.p7.14.m14.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.14.m14.1.2.2.cmml">(</mo><mi id="S3.Thmthm11.p7.14.m14.1.1" xref="S3.Thmthm11.p7.14.m14.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.14.m14.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.14.m14.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.14.m14.1.2.1" xref="S3.Thmthm11.p7.14.m14.1.2.1.cmml">=</mo><mn id="S3.Thmthm11.p7.14.m14.1.2.3" xref="S3.Thmthm11.p7.14.m14.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.14.m14.1b"><apply id="S3.Thmthm11.p7.14.m14.1.2.cmml" xref="S3.Thmthm11.p7.14.m14.1.2"><eq id="S3.Thmthm11.p7.14.m14.1.2.1.cmml" xref="S3.Thmthm11.p7.14.m14.1.2.1"></eq><apply id="S3.Thmthm11.p7.14.m14.1.2.2.cmml" xref="S3.Thmthm11.p7.14.m14.1.2.2"><times id="S3.Thmthm11.p7.14.m14.1.2.2.1.cmml" xref="S3.Thmthm11.p7.14.m14.1.2.2.1"></times><apply id="S3.Thmthm11.p7.14.m14.1.2.2.2.cmml" xref="S3.Thmthm11.p7.14.m14.1.2.2.2"><compose id="S3.Thmthm11.p7.14.m14.1.2.2.2.1.cmml" xref="S3.Thmthm11.p7.14.m14.1.2.2.2.1"></compose><ci id="S3.Thmthm11.p7.14.m14.1.2.2.2.2.cmml" xref="S3.Thmthm11.p7.14.m14.1.2.2.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.14.m14.1.2.2.2.3.cmml" xref="S3.Thmthm11.p7.14.m14.1.2.2.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p7.14.m14.1.1.cmml" xref="S3.Thmthm11.p7.14.m14.1.1">𝐲</ci></apply><cn id="S3.Thmthm11.p7.14.m14.1.2.3.cmml" type="integer" xref="S3.Thmthm11.p7.14.m14.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.14.m14.1c">g\circ T({\bf y})=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.14.m14.1d">italic_g ∘ italic_T ( bold_y ) = 1</annotation></semantics></math> if <math alttext="{\bf y}\in[de]\cup[ee]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.15.m15.2"><semantics id="S3.Thmthm11.p7.15.m15.2a"><mrow id="S3.Thmthm11.p7.15.m15.2.2" xref="S3.Thmthm11.p7.15.m15.2.2.cmml"><mi id="S3.Thmthm11.p7.15.m15.2.2.4" xref="S3.Thmthm11.p7.15.m15.2.2.4.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.15.m15.2.2.3" xref="S3.Thmthm11.p7.15.m15.2.2.3.cmml">∈</mo><mrow id="S3.Thmthm11.p7.15.m15.2.2.2" xref="S3.Thmthm11.p7.15.m15.2.2.2.cmml"><mrow id="S3.Thmthm11.p7.15.m15.1.1.1.1.1" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.2" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.3" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.Thmthm11.p7.15.m15.2.2.2.3" xref="S3.Thmthm11.p7.15.m15.2.2.2.3.cmml">∪</mo><mrow id="S3.Thmthm11.p7.15.m15.2.2.2.2.1" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.2.cmml"><mo id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.2" stretchy="false" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.cmml"><mi id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.2" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.2.cmml">e</mi><mo id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.1" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.3" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.3" stretchy="false" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.15.m15.2b"><apply id="S3.Thmthm11.p7.15.m15.2.2.cmml" xref="S3.Thmthm11.p7.15.m15.2.2"><in id="S3.Thmthm11.p7.15.m15.2.2.3.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.3"></in><ci id="S3.Thmthm11.p7.15.m15.2.2.4.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.4">𝐲</ci><apply id="S3.Thmthm11.p7.15.m15.2.2.2.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.2"><union id="S3.Thmthm11.p7.15.m15.2.2.2.3.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.2.3"></union><apply id="S3.Thmthm11.p7.15.m15.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.15.m15.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1"><times id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.15.m15.1.1.1.1.1.1.3">𝑒</ci></apply></apply><apply id="S3.Thmthm11.p7.15.m15.2.2.2.2.2.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.15.m15.2.2.2.2.2.1.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1"><times id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.1.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.1"></times><ci id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.2.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.2">𝑒</ci><ci id="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.3.cmml" xref="S3.Thmthm11.p7.15.m15.2.2.2.2.1.1.3">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.15.m15.2c">{\bf y}\in[de]\cup[ee]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.15.m15.2d">bold_y ∈ [ italic_d italic_e ] ∪ [ italic_e italic_e ]</annotation></semantics></math>. Hence the function <math alttext="g\circ T-g\in\partial_{T}C(Y,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.16.m16.2"><semantics id="S3.Thmthm11.p7.16.m16.2a"><mrow id="S3.Thmthm11.p7.16.m16.2.3" xref="S3.Thmthm11.p7.16.m16.2.3.cmml"><mrow id="S3.Thmthm11.p7.16.m16.2.3.2" xref="S3.Thmthm11.p7.16.m16.2.3.2.cmml"><mrow id="S3.Thmthm11.p7.16.m16.2.3.2.2" xref="S3.Thmthm11.p7.16.m16.2.3.2.2.cmml"><mi id="S3.Thmthm11.p7.16.m16.2.3.2.2.2" xref="S3.Thmthm11.p7.16.m16.2.3.2.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.16.m16.2.3.2.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p7.16.m16.2.3.2.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p7.16.m16.2.3.2.2.3" xref="S3.Thmthm11.p7.16.m16.2.3.2.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p7.16.m16.2.3.2.1" xref="S3.Thmthm11.p7.16.m16.2.3.2.1.cmml">−</mo><mi id="S3.Thmthm11.p7.16.m16.2.3.2.3" xref="S3.Thmthm11.p7.16.m16.2.3.2.3.cmml">g</mi></mrow><mo id="S3.Thmthm11.p7.16.m16.2.3.1" rspace="0.1389em" xref="S3.Thmthm11.p7.16.m16.2.3.1.cmml">∈</mo><mrow id="S3.Thmthm11.p7.16.m16.2.3.3" xref="S3.Thmthm11.p7.16.m16.2.3.3.cmml"><msub id="S3.Thmthm11.p7.16.m16.2.3.3.1" xref="S3.Thmthm11.p7.16.m16.2.3.3.1.cmml"><mo id="S3.Thmthm11.p7.16.m16.2.3.3.1.2" lspace="0.1389em" rspace="0em" xref="S3.Thmthm11.p7.16.m16.2.3.3.1.2.cmml">∂</mo><mi id="S3.Thmthm11.p7.16.m16.2.3.3.1.3" xref="S3.Thmthm11.p7.16.m16.2.3.3.1.3.cmml">T</mi></msub><mrow id="S3.Thmthm11.p7.16.m16.2.3.3.2" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.cmml"><mi id="S3.Thmthm11.p7.16.m16.2.3.3.2.2" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.2.cmml">C</mi><mo id="S3.Thmthm11.p7.16.m16.2.3.3.2.1" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.16.m16.2.3.3.2.3.2" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.3.1.cmml"><mo id="S3.Thmthm11.p7.16.m16.2.3.3.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.3.1.cmml">(</mo><mi id="S3.Thmthm11.p7.16.m16.1.1" xref="S3.Thmthm11.p7.16.m16.1.1.cmml">Y</mi><mo id="S3.Thmthm11.p7.16.m16.2.3.3.2.3.2.2" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.3.1.cmml">,</mo><mi id="S3.Thmthm11.p7.16.m16.2.2" xref="S3.Thmthm11.p7.16.m16.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p7.16.m16.2.3.3.2.3.2.3" stretchy="false" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.16.m16.2b"><apply id="S3.Thmthm11.p7.16.m16.2.3.cmml" xref="S3.Thmthm11.p7.16.m16.2.3"><in id="S3.Thmthm11.p7.16.m16.2.3.1.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.1"></in><apply id="S3.Thmthm11.p7.16.m16.2.3.2.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.2"><minus id="S3.Thmthm11.p7.16.m16.2.3.2.1.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.2.1"></minus><apply id="S3.Thmthm11.p7.16.m16.2.3.2.2.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.2.2"><compose id="S3.Thmthm11.p7.16.m16.2.3.2.2.1.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.2.2.1"></compose><ci id="S3.Thmthm11.p7.16.m16.2.3.2.2.2.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.2.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.16.m16.2.3.2.2.3.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.2.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p7.16.m16.2.3.2.3.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.2.3">𝑔</ci></apply><apply id="S3.Thmthm11.p7.16.m16.2.3.3.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3"><apply id="S3.Thmthm11.p7.16.m16.2.3.3.1.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p7.16.m16.2.3.3.1.1.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3.1">subscript</csymbol><partialdiff id="S3.Thmthm11.p7.16.m16.2.3.3.1.2.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3.1.2"></partialdiff><ci id="S3.Thmthm11.p7.16.m16.2.3.3.1.3.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3.1.3">𝑇</ci></apply><apply id="S3.Thmthm11.p7.16.m16.2.3.3.2.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3.2"><times id="S3.Thmthm11.p7.16.m16.2.3.3.2.1.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.1"></times><ci id="S3.Thmthm11.p7.16.m16.2.3.3.2.2.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p7.16.m16.2.3.3.2.3.1.cmml" xref="S3.Thmthm11.p7.16.m16.2.3.3.2.3.2"><ci id="S3.Thmthm11.p7.16.m16.1.1.cmml" xref="S3.Thmthm11.p7.16.m16.1.1">𝑌</ci><ci id="S3.Thmthm11.p7.16.m16.2.2.cmml" xref="S3.Thmthm11.p7.16.m16.2.2">ℤ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.16.m16.2c">g\circ T-g\in\partial_{T}C(Y,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.16.m16.2d">italic_g ∘ italic_T - italic_g ∈ ∂ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_C ( italic_Y , blackboard_Z )</annotation></semantics></math> satisfies <math alttext="(g\circ T-g)({\bf y})=0" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.17.m17.2"><semantics id="S3.Thmthm11.p7.17.m17.2a"><mrow id="S3.Thmthm11.p7.17.m17.2.2" xref="S3.Thmthm11.p7.17.m17.2.2.cmml"><mrow id="S3.Thmthm11.p7.17.m17.2.2.1" xref="S3.Thmthm11.p7.17.m17.2.2.1.cmml"><mrow id="S3.Thmthm11.p7.17.m17.2.2.1.1.1" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.cmml"><mo id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.2" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.3" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.1" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.1.cmml">−</mo><mi id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.3" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p7.17.m17.2.2.1.2" xref="S3.Thmthm11.p7.17.m17.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.17.m17.2.2.1.3.2" xref="S3.Thmthm11.p7.17.m17.2.2.1.cmml"><mo id="S3.Thmthm11.p7.17.m17.2.2.1.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.17.m17.2.2.1.cmml">(</mo><mi id="S3.Thmthm11.p7.17.m17.1.1" xref="S3.Thmthm11.p7.17.m17.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.17.m17.2.2.1.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.17.m17.2.2.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.17.m17.2.2.2" xref="S3.Thmthm11.p7.17.m17.2.2.2.cmml">=</mo><mn id="S3.Thmthm11.p7.17.m17.2.2.3" xref="S3.Thmthm11.p7.17.m17.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.17.m17.2b"><apply id="S3.Thmthm11.p7.17.m17.2.2.cmml" xref="S3.Thmthm11.p7.17.m17.2.2"><eq id="S3.Thmthm11.p7.17.m17.2.2.2.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.2"></eq><apply id="S3.Thmthm11.p7.17.m17.2.2.1.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1"><times id="S3.Thmthm11.p7.17.m17.2.2.1.2.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1.2"></times><apply id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1"><minus id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.1"></minus><apply id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2"><compose id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.1"></compose><ci id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.17.m17.2.2.1.1.1.1.3">𝑔</ci></apply><ci id="S3.Thmthm11.p7.17.m17.1.1.cmml" xref="S3.Thmthm11.p7.17.m17.1.1">𝐲</ci></apply><cn id="S3.Thmthm11.p7.17.m17.2.2.3.cmml" type="integer" xref="S3.Thmthm11.p7.17.m17.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.17.m17.2c">(g\circ T-g)({\bf y})=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.17.m17.2d">( italic_g ∘ italic_T - italic_g ) ( bold_y ) = 0</annotation></semantics></math> if <math alttext="{\bf y}\in[dd]\cup[ee]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.18.m18.2"><semantics id="S3.Thmthm11.p7.18.m18.2a"><mrow id="S3.Thmthm11.p7.18.m18.2.2" xref="S3.Thmthm11.p7.18.m18.2.2.cmml"><mi id="S3.Thmthm11.p7.18.m18.2.2.4" xref="S3.Thmthm11.p7.18.m18.2.2.4.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.18.m18.2.2.3" xref="S3.Thmthm11.p7.18.m18.2.2.3.cmml">∈</mo><mrow id="S3.Thmthm11.p7.18.m18.2.2.2" xref="S3.Thmthm11.p7.18.m18.2.2.2.cmml"><mrow id="S3.Thmthm11.p7.18.m18.1.1.1.1.1" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.2" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.1" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.3" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S3.Thmthm11.p7.18.m18.2.2.2.3" xref="S3.Thmthm11.p7.18.m18.2.2.2.3.cmml">∪</mo><mrow id="S3.Thmthm11.p7.18.m18.2.2.2.2.1" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.2.cmml"><mo id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.2" stretchy="false" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.cmml"><mi id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.2" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.2.cmml">e</mi><mo id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.1" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.3" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.3" stretchy="false" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.2.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.18.m18.2b"><apply id="S3.Thmthm11.p7.18.m18.2.2.cmml" xref="S3.Thmthm11.p7.18.m18.2.2"><in id="S3.Thmthm11.p7.18.m18.2.2.3.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.3"></in><ci id="S3.Thmthm11.p7.18.m18.2.2.4.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.4">𝐲</ci><apply id="S3.Thmthm11.p7.18.m18.2.2.2.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.2"><union id="S3.Thmthm11.p7.18.m18.2.2.2.3.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.2.3"></union><apply id="S3.Thmthm11.p7.18.m18.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.18.m18.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1"><times id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.18.m18.1.1.1.1.1.1.3">𝑑</ci></apply></apply><apply id="S3.Thmthm11.p7.18.m18.2.2.2.2.2.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.18.m18.2.2.2.2.2.1.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1"><times id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.1.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.1"></times><ci id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.2.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.2">𝑒</ci><ci id="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.3.cmml" xref="S3.Thmthm11.p7.18.m18.2.2.2.2.1.1.3">𝑒</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.18.m18.2c">{\bf y}\in[dd]\cup[ee]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.18.m18.2d">bold_y ∈ [ italic_d italic_d ] ∪ [ italic_e italic_e ]</annotation></semantics></math>, <math alttext="(g\circ T-g)({\bf y})=1" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.19.m19.2"><semantics id="S3.Thmthm11.p7.19.m19.2a"><mrow id="S3.Thmthm11.p7.19.m19.2.2" xref="S3.Thmthm11.p7.19.m19.2.2.cmml"><mrow id="S3.Thmthm11.p7.19.m19.2.2.1" xref="S3.Thmthm11.p7.19.m19.2.2.1.cmml"><mrow id="S3.Thmthm11.p7.19.m19.2.2.1.1.1" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.cmml"><mo id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.2" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.3" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.1" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.1.cmml">−</mo><mi id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.3" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p7.19.m19.2.2.1.2" xref="S3.Thmthm11.p7.19.m19.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.19.m19.2.2.1.3.2" xref="S3.Thmthm11.p7.19.m19.2.2.1.cmml"><mo id="S3.Thmthm11.p7.19.m19.2.2.1.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.19.m19.2.2.1.cmml">(</mo><mi id="S3.Thmthm11.p7.19.m19.1.1" xref="S3.Thmthm11.p7.19.m19.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.19.m19.2.2.1.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.19.m19.2.2.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.19.m19.2.2.2" xref="S3.Thmthm11.p7.19.m19.2.2.2.cmml">=</mo><mn id="S3.Thmthm11.p7.19.m19.2.2.3" xref="S3.Thmthm11.p7.19.m19.2.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.19.m19.2b"><apply id="S3.Thmthm11.p7.19.m19.2.2.cmml" xref="S3.Thmthm11.p7.19.m19.2.2"><eq id="S3.Thmthm11.p7.19.m19.2.2.2.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.2"></eq><apply id="S3.Thmthm11.p7.19.m19.2.2.1.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1"><times id="S3.Thmthm11.p7.19.m19.2.2.1.2.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1.2"></times><apply id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1"><minus id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.1"></minus><apply id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2"><compose id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.1"></compose><ci id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.19.m19.2.2.1.1.1.1.3">𝑔</ci></apply><ci id="S3.Thmthm11.p7.19.m19.1.1.cmml" xref="S3.Thmthm11.p7.19.m19.1.1">𝐲</ci></apply><cn id="S3.Thmthm11.p7.19.m19.2.2.3.cmml" type="integer" xref="S3.Thmthm11.p7.19.m19.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.19.m19.2c">(g\circ T-g)({\bf y})=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.19.m19.2d">( italic_g ∘ italic_T - italic_g ) ( bold_y ) = 1</annotation></semantics></math> if <math alttext="{\bf y}\in[de]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.20.m20.1"><semantics id="S3.Thmthm11.p7.20.m20.1a"><mrow id="S3.Thmthm11.p7.20.m20.1.1" xref="S3.Thmthm11.p7.20.m20.1.1.cmml"><mi id="S3.Thmthm11.p7.20.m20.1.1.3" xref="S3.Thmthm11.p7.20.m20.1.1.3.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.20.m20.1.1.2" xref="S3.Thmthm11.p7.20.m20.1.1.2.cmml">∈</mo><mrow id="S3.Thmthm11.p7.20.m20.1.1.1.1" xref="S3.Thmthm11.p7.20.m20.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.20.m20.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.20.m20.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.20.m20.1.1.1.1.1" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.20.m20.1.1.1.1.1.2" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p7.20.m20.1.1.1.1.1.1" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.20.m20.1.1.1.1.1.3" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p7.20.m20.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.20.m20.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.20.m20.1b"><apply id="S3.Thmthm11.p7.20.m20.1.1.cmml" xref="S3.Thmthm11.p7.20.m20.1.1"><in id="S3.Thmthm11.p7.20.m20.1.1.2.cmml" xref="S3.Thmthm11.p7.20.m20.1.1.2"></in><ci id="S3.Thmthm11.p7.20.m20.1.1.3.cmml" xref="S3.Thmthm11.p7.20.m20.1.1.3">𝐲</ci><apply id="S3.Thmthm11.p7.20.m20.1.1.1.2.cmml" xref="S3.Thmthm11.p7.20.m20.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.20.m20.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.20.m20.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.1"><times id="S3.Thmthm11.p7.20.m20.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.20.m20.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p7.20.m20.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.20.m20.1.1.1.1.1.3">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.20.m20.1c">{\bf y}\in[de]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.20.m20.1d">bold_y ∈ [ italic_d italic_e ]</annotation></semantics></math> and <math alttext="(g\circ T-g)({\bf y})=-1" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.21.m21.2"><semantics id="S3.Thmthm11.p7.21.m21.2a"><mrow id="S3.Thmthm11.p7.21.m21.2.2" xref="S3.Thmthm11.p7.21.m21.2.2.cmml"><mrow id="S3.Thmthm11.p7.21.m21.2.2.1" xref="S3.Thmthm11.p7.21.m21.2.2.1.cmml"><mrow id="S3.Thmthm11.p7.21.m21.2.2.1.1.1" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.cmml"><mo id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.2" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.2.cmml">g</mi><mo id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.3" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.1" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.1.cmml">−</mo><mi id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.3" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p7.21.m21.2.2.1.2" xref="S3.Thmthm11.p7.21.m21.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p7.21.m21.2.2.1.3.2" xref="S3.Thmthm11.p7.21.m21.2.2.1.cmml"><mo id="S3.Thmthm11.p7.21.m21.2.2.1.3.2.1" stretchy="false" xref="S3.Thmthm11.p7.21.m21.2.2.1.cmml">(</mo><mi id="S3.Thmthm11.p7.21.m21.1.1" xref="S3.Thmthm11.p7.21.m21.1.1.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.21.m21.2.2.1.3.2.2" stretchy="false" xref="S3.Thmthm11.p7.21.m21.2.2.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p7.21.m21.2.2.2" xref="S3.Thmthm11.p7.21.m21.2.2.2.cmml">=</mo><mrow id="S3.Thmthm11.p7.21.m21.2.2.3" xref="S3.Thmthm11.p7.21.m21.2.2.3.cmml"><mo id="S3.Thmthm11.p7.21.m21.2.2.3a" xref="S3.Thmthm11.p7.21.m21.2.2.3.cmml">−</mo><mn id="S3.Thmthm11.p7.21.m21.2.2.3.2" xref="S3.Thmthm11.p7.21.m21.2.2.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.21.m21.2b"><apply id="S3.Thmthm11.p7.21.m21.2.2.cmml" xref="S3.Thmthm11.p7.21.m21.2.2"><eq id="S3.Thmthm11.p7.21.m21.2.2.2.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.2"></eq><apply id="S3.Thmthm11.p7.21.m21.2.2.1.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1"><times id="S3.Thmthm11.p7.21.m21.2.2.1.2.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1.2"></times><apply id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1"><minus id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.1"></minus><apply id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2"><compose id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.1"></compose><ci id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.2">𝑔</ci><ci id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.1.1.1.1.3">𝑔</ci></apply><ci id="S3.Thmthm11.p7.21.m21.1.1.cmml" xref="S3.Thmthm11.p7.21.m21.1.1">𝐲</ci></apply><apply id="S3.Thmthm11.p7.21.m21.2.2.3.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.3"><minus id="S3.Thmthm11.p7.21.m21.2.2.3.1.cmml" xref="S3.Thmthm11.p7.21.m21.2.2.3"></minus><cn id="S3.Thmthm11.p7.21.m21.2.2.3.2.cmml" type="integer" xref="S3.Thmthm11.p7.21.m21.2.2.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.21.m21.2c">(g\circ T-g)({\bf y})=-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.21.m21.2d">( italic_g ∘ italic_T - italic_g ) ( bold_y ) = - 1</annotation></semantics></math> if <math alttext="{\bf y}\in[ed]" class="ltx_Math" display="inline" id="S3.Thmthm11.p7.22.m22.1"><semantics id="S3.Thmthm11.p7.22.m22.1a"><mrow id="S3.Thmthm11.p7.22.m22.1.1" xref="S3.Thmthm11.p7.22.m22.1.1.cmml"><mi id="S3.Thmthm11.p7.22.m22.1.1.3" xref="S3.Thmthm11.p7.22.m22.1.1.3.cmml">𝐲</mi><mo id="S3.Thmthm11.p7.22.m22.1.1.2" xref="S3.Thmthm11.p7.22.m22.1.1.2.cmml">∈</mo><mrow id="S3.Thmthm11.p7.22.m22.1.1.1.1" xref="S3.Thmthm11.p7.22.m22.1.1.1.2.cmml"><mo id="S3.Thmthm11.p7.22.m22.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p7.22.m22.1.1.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p7.22.m22.1.1.1.1.1" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.1.cmml"><mi id="S3.Thmthm11.p7.22.m22.1.1.1.1.1.2" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.1.2.cmml">e</mi><mo id="S3.Thmthm11.p7.22.m22.1.1.1.1.1.1" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p7.22.m22.1.1.1.1.1.3" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S3.Thmthm11.p7.22.m22.1.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p7.22.m22.1.1.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p7.22.m22.1b"><apply id="S3.Thmthm11.p7.22.m22.1.1.cmml" xref="S3.Thmthm11.p7.22.m22.1.1"><in id="S3.Thmthm11.p7.22.m22.1.1.2.cmml" xref="S3.Thmthm11.p7.22.m22.1.1.2"></in><ci id="S3.Thmthm11.p7.22.m22.1.1.3.cmml" xref="S3.Thmthm11.p7.22.m22.1.1.3">𝐲</ci><apply id="S3.Thmthm11.p7.22.m22.1.1.1.2.cmml" xref="S3.Thmthm11.p7.22.m22.1.1.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p7.22.m22.1.1.1.2.1.cmml" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p7.22.m22.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.1"><times id="S3.Thmthm11.p7.22.m22.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.1.1"></times><ci id="S3.Thmthm11.p7.22.m22.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.1.2">𝑒</ci><ci id="S3.Thmthm11.p7.22.m22.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p7.22.m22.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p7.22.m22.1c">{\bf y}\in[ed]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p7.22.m22.1d">bold_y ∈ [ italic_e italic_d ]</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.Thmthm11.p8"> <p class="ltx_p" id="S3.Thmthm11.p8.13">Let us now consider the function <math alttext="(g\circ T-g)\circ\sigma^{\mathbb{Z}}\in C(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.1.m1.3"><semantics id="S3.Thmthm11.p8.1.m1.3a"><mrow id="S3.Thmthm11.p8.1.m1.3.3" xref="S3.Thmthm11.p8.1.m1.3.3.cmml"><mrow id="S3.Thmthm11.p8.1.m1.3.3.1" xref="S3.Thmthm11.p8.1.m1.3.3.1.cmml"><mrow id="S3.Thmthm11.p8.1.m1.3.3.1.1.1" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.cmml"><mo id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.2" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.2.cmml">g</mi><mo id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.3" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.1" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.1.cmml">−</mo><mi id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.3" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p8.1.m1.3.3.1.2" rspace="0.222em" xref="S3.Thmthm11.p8.1.m1.3.3.1.2.cmml">∘</mo><msup id="S3.Thmthm11.p8.1.m1.3.3.1.3" xref="S3.Thmthm11.p8.1.m1.3.3.1.3.cmml"><mi id="S3.Thmthm11.p8.1.m1.3.3.1.3.2" xref="S3.Thmthm11.p8.1.m1.3.3.1.3.2.cmml">σ</mi><mi id="S3.Thmthm11.p8.1.m1.3.3.1.3.3" xref="S3.Thmthm11.p8.1.m1.3.3.1.3.3.cmml">ℤ</mi></msup></mrow><mo id="S3.Thmthm11.p8.1.m1.3.3.2" xref="S3.Thmthm11.p8.1.m1.3.3.2.cmml">∈</mo><mrow id="S3.Thmthm11.p8.1.m1.3.3.3" xref="S3.Thmthm11.p8.1.m1.3.3.3.cmml"><mi id="S3.Thmthm11.p8.1.m1.3.3.3.2" xref="S3.Thmthm11.p8.1.m1.3.3.3.2.cmml">C</mi><mo id="S3.Thmthm11.p8.1.m1.3.3.3.1" xref="S3.Thmthm11.p8.1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p8.1.m1.3.3.3.3.2" xref="S3.Thmthm11.p8.1.m1.3.3.3.3.1.cmml"><mo id="S3.Thmthm11.p8.1.m1.3.3.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p8.1.m1.3.3.3.3.1.cmml">(</mo><mi id="S3.Thmthm11.p8.1.m1.1.1" xref="S3.Thmthm11.p8.1.m1.1.1.cmml">X</mi><mo id="S3.Thmthm11.p8.1.m1.3.3.3.3.2.2" xref="S3.Thmthm11.p8.1.m1.3.3.3.3.1.cmml">,</mo><mi id="S3.Thmthm11.p8.1.m1.2.2" xref="S3.Thmthm11.p8.1.m1.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p8.1.m1.3.3.3.3.2.3" stretchy="false" xref="S3.Thmthm11.p8.1.m1.3.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.1.m1.3b"><apply id="S3.Thmthm11.p8.1.m1.3.3.cmml" xref="S3.Thmthm11.p8.1.m1.3.3"><in id="S3.Thmthm11.p8.1.m1.3.3.2.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.2"></in><apply id="S3.Thmthm11.p8.1.m1.3.3.1.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1"><compose id="S3.Thmthm11.p8.1.m1.3.3.1.2.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.2"></compose><apply id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1"><minus id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.1.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.1"></minus><apply id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2"><compose id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.1"></compose><ci id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.2">𝑔</ci><ci id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.3.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.1.1.1.3">𝑔</ci></apply><apply id="S3.Thmthm11.p8.1.m1.3.3.1.3.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.1.m1.3.3.1.3.1.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.3">superscript</csymbol><ci id="S3.Thmthm11.p8.1.m1.3.3.1.3.2.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.3.2">𝜎</ci><ci id="S3.Thmthm11.p8.1.m1.3.3.1.3.3.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.1.3.3">ℤ</ci></apply></apply><apply id="S3.Thmthm11.p8.1.m1.3.3.3.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.3"><times id="S3.Thmthm11.p8.1.m1.3.3.3.1.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.3.1"></times><ci id="S3.Thmthm11.p8.1.m1.3.3.3.2.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.3.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p8.1.m1.3.3.3.3.1.cmml" xref="S3.Thmthm11.p8.1.m1.3.3.3.3.2"><ci id="S3.Thmthm11.p8.1.m1.1.1.cmml" xref="S3.Thmthm11.p8.1.m1.1.1">𝑋</ci><ci id="S3.Thmthm11.p8.1.m1.2.2.cmml" xref="S3.Thmthm11.p8.1.m1.2.2">ℤ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.1.m1.3c">(g\circ T-g)\circ\sigma^{\mathbb{Z}}\in C(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.1.m1.3d">( italic_g ∘ italic_T - italic_g ) ∘ italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ∈ italic_C ( italic_X , blackboard_Z )</annotation></semantics></math>, and its value on the periodic word <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.2.m2.1"><semantics id="S3.Thmthm11.p8.2.m2.1a"><mrow id="S3.Thmthm11.p8.2.m2.1.1" xref="S3.Thmthm11.p8.2.m2.1.1.cmml"><mi id="S3.Thmthm11.p8.2.m2.1.1.2" xref="S3.Thmthm11.p8.2.m2.1.1.2.cmml">𝐱</mi><mo id="S3.Thmthm11.p8.2.m2.1.1.1" xref="S3.Thmthm11.p8.2.m2.1.1.1.cmml">∈</mo><mi id="S3.Thmthm11.p8.2.m2.1.1.3" xref="S3.Thmthm11.p8.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.2.m2.1b"><apply id="S3.Thmthm11.p8.2.m2.1.1.cmml" xref="S3.Thmthm11.p8.2.m2.1.1"><in id="S3.Thmthm11.p8.2.m2.1.1.1.cmml" xref="S3.Thmthm11.p8.2.m2.1.1.1"></in><ci id="S3.Thmthm11.p8.2.m2.1.1.2.cmml" xref="S3.Thmthm11.p8.2.m2.1.1.2">𝐱</ci><ci id="S3.Thmthm11.p8.2.m2.1.1.3.cmml" xref="S3.Thmthm11.p8.2.m2.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.2.m2.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.2.m2.1d">bold_x ∈ italic_X</annotation></semantics></math> given by <math alttext="{\bf x}=\ldots ccc\ldots" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.3.m3.1"><semantics id="S3.Thmthm11.p8.3.m3.1a"><mrow id="S3.Thmthm11.p8.3.m3.1.1" xref="S3.Thmthm11.p8.3.m3.1.1.cmml"><mi id="S3.Thmthm11.p8.3.m3.1.1.2" xref="S3.Thmthm11.p8.3.m3.1.1.2.cmml">𝐱</mi><mo id="S3.Thmthm11.p8.3.m3.1.1.1" xref="S3.Thmthm11.p8.3.m3.1.1.1.cmml">=</mo><mrow id="S3.Thmthm11.p8.3.m3.1.1.3" xref="S3.Thmthm11.p8.3.m3.1.1.3.cmml"><mi id="S3.Thmthm11.p8.3.m3.1.1.3.2" mathvariant="normal" xref="S3.Thmthm11.p8.3.m3.1.1.3.2.cmml">…</mi><mo id="S3.Thmthm11.p8.3.m3.1.1.3.1" xref="S3.Thmthm11.p8.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.3.m3.1.1.3.3" xref="S3.Thmthm11.p8.3.m3.1.1.3.3.cmml">c</mi><mo id="S3.Thmthm11.p8.3.m3.1.1.3.1a" xref="S3.Thmthm11.p8.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.3.m3.1.1.3.4" xref="S3.Thmthm11.p8.3.m3.1.1.3.4.cmml">c</mi><mo id="S3.Thmthm11.p8.3.m3.1.1.3.1b" xref="S3.Thmthm11.p8.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.3.m3.1.1.3.5" xref="S3.Thmthm11.p8.3.m3.1.1.3.5.cmml">c</mi><mo id="S3.Thmthm11.p8.3.m3.1.1.3.1c" xref="S3.Thmthm11.p8.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.3.m3.1.1.3.6" mathvariant="normal" xref="S3.Thmthm11.p8.3.m3.1.1.3.6.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.3.m3.1b"><apply id="S3.Thmthm11.p8.3.m3.1.1.cmml" xref="S3.Thmthm11.p8.3.m3.1.1"><eq id="S3.Thmthm11.p8.3.m3.1.1.1.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.1"></eq><ci id="S3.Thmthm11.p8.3.m3.1.1.2.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.2">𝐱</ci><apply id="S3.Thmthm11.p8.3.m3.1.1.3.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.3"><times id="S3.Thmthm11.p8.3.m3.1.1.3.1.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.3.1"></times><ci id="S3.Thmthm11.p8.3.m3.1.1.3.2.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.3.2">…</ci><ci id="S3.Thmthm11.p8.3.m3.1.1.3.3.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.3.3">𝑐</ci><ci id="S3.Thmthm11.p8.3.m3.1.1.3.4.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.3.4">𝑐</ci><ci id="S3.Thmthm11.p8.3.m3.1.1.3.5.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.3.5">𝑐</ci><ci id="S3.Thmthm11.p8.3.m3.1.1.3.6.cmml" xref="S3.Thmthm11.p8.3.m3.1.1.3.6">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.3.m3.1c">{\bf x}=\ldots ccc\ldots</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.3.m3.1d">bold_x = … italic_c italic_c italic_c …</annotation></semantics></math>. We have <math alttext="{\bf x}_{[1,+\infty)}=ccc\ldots" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.4.m4.2"><semantics id="S3.Thmthm11.p8.4.m4.2a"><mrow id="S3.Thmthm11.p8.4.m4.2.3" xref="S3.Thmthm11.p8.4.m4.2.3.cmml"><msub id="S3.Thmthm11.p8.4.m4.2.3.2" xref="S3.Thmthm11.p8.4.m4.2.3.2.cmml"><mi id="S3.Thmthm11.p8.4.m4.2.3.2.2" xref="S3.Thmthm11.p8.4.m4.2.3.2.2.cmml">𝐱</mi><mrow id="S3.Thmthm11.p8.4.m4.2.2.2.2" xref="S3.Thmthm11.p8.4.m4.2.2.2.3.cmml"><mo id="S3.Thmthm11.p8.4.m4.2.2.2.2.2" stretchy="false" xref="S3.Thmthm11.p8.4.m4.2.2.2.3.cmml">[</mo><mn id="S3.Thmthm11.p8.4.m4.1.1.1.1" xref="S3.Thmthm11.p8.4.m4.1.1.1.1.cmml">1</mn><mo id="S3.Thmthm11.p8.4.m4.2.2.2.2.3" xref="S3.Thmthm11.p8.4.m4.2.2.2.3.cmml">,</mo><mrow id="S3.Thmthm11.p8.4.m4.2.2.2.2.1" xref="S3.Thmthm11.p8.4.m4.2.2.2.2.1.cmml"><mo id="S3.Thmthm11.p8.4.m4.2.2.2.2.1a" xref="S3.Thmthm11.p8.4.m4.2.2.2.2.1.cmml">+</mo><mi id="S3.Thmthm11.p8.4.m4.2.2.2.2.1.2" mathvariant="normal" xref="S3.Thmthm11.p8.4.m4.2.2.2.2.1.2.cmml">∞</mi></mrow><mo id="S3.Thmthm11.p8.4.m4.2.2.2.2.4" stretchy="false" xref="S3.Thmthm11.p8.4.m4.2.2.2.3.cmml">)</mo></mrow></msub><mo id="S3.Thmthm11.p8.4.m4.2.3.1" xref="S3.Thmthm11.p8.4.m4.2.3.1.cmml">=</mo><mrow id="S3.Thmthm11.p8.4.m4.2.3.3" xref="S3.Thmthm11.p8.4.m4.2.3.3.cmml"><mi id="S3.Thmthm11.p8.4.m4.2.3.3.2" xref="S3.Thmthm11.p8.4.m4.2.3.3.2.cmml">c</mi><mo id="S3.Thmthm11.p8.4.m4.2.3.3.1" xref="S3.Thmthm11.p8.4.m4.2.3.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.4.m4.2.3.3.3" xref="S3.Thmthm11.p8.4.m4.2.3.3.3.cmml">c</mi><mo id="S3.Thmthm11.p8.4.m4.2.3.3.1a" xref="S3.Thmthm11.p8.4.m4.2.3.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.4.m4.2.3.3.4" xref="S3.Thmthm11.p8.4.m4.2.3.3.4.cmml">c</mi><mo id="S3.Thmthm11.p8.4.m4.2.3.3.1b" xref="S3.Thmthm11.p8.4.m4.2.3.3.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.4.m4.2.3.3.5" mathvariant="normal" xref="S3.Thmthm11.p8.4.m4.2.3.3.5.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.4.m4.2b"><apply id="S3.Thmthm11.p8.4.m4.2.3.cmml" xref="S3.Thmthm11.p8.4.m4.2.3"><eq id="S3.Thmthm11.p8.4.m4.2.3.1.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.1"></eq><apply id="S3.Thmthm11.p8.4.m4.2.3.2.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.4.m4.2.3.2.1.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.2">subscript</csymbol><ci id="S3.Thmthm11.p8.4.m4.2.3.2.2.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.2.2">𝐱</ci><interval closure="closed-open" id="S3.Thmthm11.p8.4.m4.2.2.2.3.cmml" xref="S3.Thmthm11.p8.4.m4.2.2.2.2"><cn id="S3.Thmthm11.p8.4.m4.1.1.1.1.cmml" type="integer" xref="S3.Thmthm11.p8.4.m4.1.1.1.1">1</cn><apply id="S3.Thmthm11.p8.4.m4.2.2.2.2.1.cmml" xref="S3.Thmthm11.p8.4.m4.2.2.2.2.1"><plus id="S3.Thmthm11.p8.4.m4.2.2.2.2.1.1.cmml" xref="S3.Thmthm11.p8.4.m4.2.2.2.2.1"></plus><infinity id="S3.Thmthm11.p8.4.m4.2.2.2.2.1.2.cmml" xref="S3.Thmthm11.p8.4.m4.2.2.2.2.1.2"></infinity></apply></interval></apply><apply id="S3.Thmthm11.p8.4.m4.2.3.3.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.3"><times id="S3.Thmthm11.p8.4.m4.2.3.3.1.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.3.1"></times><ci id="S3.Thmthm11.p8.4.m4.2.3.3.2.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.3.2">𝑐</ci><ci id="S3.Thmthm11.p8.4.m4.2.3.3.3.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.3.3">𝑐</ci><ci id="S3.Thmthm11.p8.4.m4.2.3.3.4.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.3.4">𝑐</ci><ci id="S3.Thmthm11.p8.4.m4.2.3.3.5.cmml" xref="S3.Thmthm11.p8.4.m4.2.3.3.5">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.4.m4.2c">{\bf x}_{[1,+\infty)}=ccc\ldots</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.4.m4.2d">bold_x start_POSTSUBSCRIPT [ 1 , + ∞ ) end_POSTSUBSCRIPT = italic_c italic_c italic_c …</annotation></semantics></math> and thus <math alttext="\sigma^{\mathbb{Z}}({\bf x})_{[1,+\infty)}=\sigma({\bf x}_{[1,+\infty)})=dedede\ldots" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.5.m5.6"><semantics id="S3.Thmthm11.p8.5.m5.6a"><mrow id="S3.Thmthm11.p8.5.m5.6.6" xref="S3.Thmthm11.p8.5.m5.6.6.cmml"><mrow id="S3.Thmthm11.p8.5.m5.6.6.3" xref="S3.Thmthm11.p8.5.m5.6.6.3.cmml"><msup id="S3.Thmthm11.p8.5.m5.6.6.3.2" xref="S3.Thmthm11.p8.5.m5.6.6.3.2.cmml"><mi id="S3.Thmthm11.p8.5.m5.6.6.3.2.2" xref="S3.Thmthm11.p8.5.m5.6.6.3.2.2.cmml">σ</mi><mi id="S3.Thmthm11.p8.5.m5.6.6.3.2.3" xref="S3.Thmthm11.p8.5.m5.6.6.3.2.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm11.p8.5.m5.6.6.3.1" xref="S3.Thmthm11.p8.5.m5.6.6.3.1.cmml">⁢</mo><msub id="S3.Thmthm11.p8.5.m5.6.6.3.3" xref="S3.Thmthm11.p8.5.m5.6.6.3.3.cmml"><mrow id="S3.Thmthm11.p8.5.m5.6.6.3.3.2.2" xref="S3.Thmthm11.p8.5.m5.6.6.3.3.cmml"><mo id="S3.Thmthm11.p8.5.m5.6.6.3.3.2.2.1" stretchy="false" xref="S3.Thmthm11.p8.5.m5.6.6.3.3.cmml">(</mo><mi id="S3.Thmthm11.p8.5.m5.5.5" xref="S3.Thmthm11.p8.5.m5.5.5.cmml">𝐱</mi><mo id="S3.Thmthm11.p8.5.m5.6.6.3.3.2.2.2" stretchy="false" xref="S3.Thmthm11.p8.5.m5.6.6.3.3.cmml">)</mo></mrow><mrow id="S3.Thmthm11.p8.5.m5.2.2.2.2" xref="S3.Thmthm11.p8.5.m5.2.2.2.3.cmml"><mo id="S3.Thmthm11.p8.5.m5.2.2.2.2.2" stretchy="false" xref="S3.Thmthm11.p8.5.m5.2.2.2.3.cmml">[</mo><mn id="S3.Thmthm11.p8.5.m5.1.1.1.1" xref="S3.Thmthm11.p8.5.m5.1.1.1.1.cmml">1</mn><mo id="S3.Thmthm11.p8.5.m5.2.2.2.2.3" xref="S3.Thmthm11.p8.5.m5.2.2.2.3.cmml">,</mo><mrow id="S3.Thmthm11.p8.5.m5.2.2.2.2.1" xref="S3.Thmthm11.p8.5.m5.2.2.2.2.1.cmml"><mo id="S3.Thmthm11.p8.5.m5.2.2.2.2.1a" xref="S3.Thmthm11.p8.5.m5.2.2.2.2.1.cmml">+</mo><mi id="S3.Thmthm11.p8.5.m5.2.2.2.2.1.2" mathvariant="normal" xref="S3.Thmthm11.p8.5.m5.2.2.2.2.1.2.cmml">∞</mi></mrow><mo id="S3.Thmthm11.p8.5.m5.2.2.2.2.4" stretchy="false" xref="S3.Thmthm11.p8.5.m5.2.2.2.3.cmml">)</mo></mrow></msub></mrow><mo id="S3.Thmthm11.p8.5.m5.6.6.4" xref="S3.Thmthm11.p8.5.m5.6.6.4.cmml">=</mo><mrow id="S3.Thmthm11.p8.5.m5.6.6.1" xref="S3.Thmthm11.p8.5.m5.6.6.1.cmml"><mi id="S3.Thmthm11.p8.5.m5.6.6.1.3" xref="S3.Thmthm11.p8.5.m5.6.6.1.3.cmml">σ</mi><mo id="S3.Thmthm11.p8.5.m5.6.6.1.2" xref="S3.Thmthm11.p8.5.m5.6.6.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p8.5.m5.6.6.1.1.1" xref="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.cmml"><mo id="S3.Thmthm11.p8.5.m5.6.6.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.cmml">(</mo><msub id="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1" xref="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.cmml"><mi id="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.2" xref="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.2.cmml">𝐱</mi><mrow id="S3.Thmthm11.p8.5.m5.4.4.2.2" xref="S3.Thmthm11.p8.5.m5.4.4.2.3.cmml"><mo id="S3.Thmthm11.p8.5.m5.4.4.2.2.2" stretchy="false" xref="S3.Thmthm11.p8.5.m5.4.4.2.3.cmml">[</mo><mn id="S3.Thmthm11.p8.5.m5.3.3.1.1" xref="S3.Thmthm11.p8.5.m5.3.3.1.1.cmml">1</mn><mo id="S3.Thmthm11.p8.5.m5.4.4.2.2.3" xref="S3.Thmthm11.p8.5.m5.4.4.2.3.cmml">,</mo><mrow id="S3.Thmthm11.p8.5.m5.4.4.2.2.1" xref="S3.Thmthm11.p8.5.m5.4.4.2.2.1.cmml"><mo id="S3.Thmthm11.p8.5.m5.4.4.2.2.1a" xref="S3.Thmthm11.p8.5.m5.4.4.2.2.1.cmml">+</mo><mi id="S3.Thmthm11.p8.5.m5.4.4.2.2.1.2" mathvariant="normal" xref="S3.Thmthm11.p8.5.m5.4.4.2.2.1.2.cmml">∞</mi></mrow><mo id="S3.Thmthm11.p8.5.m5.4.4.2.2.4" stretchy="false" xref="S3.Thmthm11.p8.5.m5.4.4.2.3.cmml">)</mo></mrow></msub><mo id="S3.Thmthm11.p8.5.m5.6.6.1.1.1.3" stretchy="false" xref="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p8.5.m5.6.6.5" xref="S3.Thmthm11.p8.5.m5.6.6.5.cmml">=</mo><mrow id="S3.Thmthm11.p8.5.m5.6.6.6" xref="S3.Thmthm11.p8.5.m5.6.6.6.cmml"><mi id="S3.Thmthm11.p8.5.m5.6.6.6.2" xref="S3.Thmthm11.p8.5.m5.6.6.6.2.cmml">d</mi><mo id="S3.Thmthm11.p8.5.m5.6.6.6.1" xref="S3.Thmthm11.p8.5.m5.6.6.6.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.5.m5.6.6.6.3" xref="S3.Thmthm11.p8.5.m5.6.6.6.3.cmml">e</mi><mo id="S3.Thmthm11.p8.5.m5.6.6.6.1a" xref="S3.Thmthm11.p8.5.m5.6.6.6.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.5.m5.6.6.6.4" xref="S3.Thmthm11.p8.5.m5.6.6.6.4.cmml">d</mi><mo id="S3.Thmthm11.p8.5.m5.6.6.6.1b" xref="S3.Thmthm11.p8.5.m5.6.6.6.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.5.m5.6.6.6.5" xref="S3.Thmthm11.p8.5.m5.6.6.6.5.cmml">e</mi><mo id="S3.Thmthm11.p8.5.m5.6.6.6.1c" xref="S3.Thmthm11.p8.5.m5.6.6.6.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.5.m5.6.6.6.6" xref="S3.Thmthm11.p8.5.m5.6.6.6.6.cmml">d</mi><mo id="S3.Thmthm11.p8.5.m5.6.6.6.1d" xref="S3.Thmthm11.p8.5.m5.6.6.6.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.5.m5.6.6.6.7" xref="S3.Thmthm11.p8.5.m5.6.6.6.7.cmml">e</mi><mo id="S3.Thmthm11.p8.5.m5.6.6.6.1e" xref="S3.Thmthm11.p8.5.m5.6.6.6.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.5.m5.6.6.6.8" mathvariant="normal" xref="S3.Thmthm11.p8.5.m5.6.6.6.8.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.5.m5.6b"><apply id="S3.Thmthm11.p8.5.m5.6.6.cmml" xref="S3.Thmthm11.p8.5.m5.6.6"><and id="S3.Thmthm11.p8.5.m5.6.6a.cmml" xref="S3.Thmthm11.p8.5.m5.6.6"></and><apply id="S3.Thmthm11.p8.5.m5.6.6b.cmml" xref="S3.Thmthm11.p8.5.m5.6.6"><eq id="S3.Thmthm11.p8.5.m5.6.6.4.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.4"></eq><apply id="S3.Thmthm11.p8.5.m5.6.6.3.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.3"><times id="S3.Thmthm11.p8.5.m5.6.6.3.1.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.3.1"></times><apply id="S3.Thmthm11.p8.5.m5.6.6.3.2.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.5.m5.6.6.3.2.1.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.3.2">superscript</csymbol><ci id="S3.Thmthm11.p8.5.m5.6.6.3.2.2.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.3.2.2">𝜎</ci><ci id="S3.Thmthm11.p8.5.m5.6.6.3.2.3.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.3.2.3">ℤ</ci></apply><apply id="S3.Thmthm11.p8.5.m5.6.6.3.3.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.3.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.5.m5.6.6.3.3.1.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.3.3">subscript</csymbol><ci id="S3.Thmthm11.p8.5.m5.5.5.cmml" xref="S3.Thmthm11.p8.5.m5.5.5">𝐱</ci><interval closure="closed-open" id="S3.Thmthm11.p8.5.m5.2.2.2.3.cmml" xref="S3.Thmthm11.p8.5.m5.2.2.2.2"><cn id="S3.Thmthm11.p8.5.m5.1.1.1.1.cmml" type="integer" xref="S3.Thmthm11.p8.5.m5.1.1.1.1">1</cn><apply id="S3.Thmthm11.p8.5.m5.2.2.2.2.1.cmml" xref="S3.Thmthm11.p8.5.m5.2.2.2.2.1"><plus id="S3.Thmthm11.p8.5.m5.2.2.2.2.1.1.cmml" xref="S3.Thmthm11.p8.5.m5.2.2.2.2.1"></plus><infinity id="S3.Thmthm11.p8.5.m5.2.2.2.2.1.2.cmml" xref="S3.Thmthm11.p8.5.m5.2.2.2.2.1.2"></infinity></apply></interval></apply></apply><apply id="S3.Thmthm11.p8.5.m5.6.6.1.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.1"><times id="S3.Thmthm11.p8.5.m5.6.6.1.2.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.1.2"></times><ci id="S3.Thmthm11.p8.5.m5.6.6.1.3.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.1.3">𝜎</ci><apply id="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.1.1.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.1.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.1.1.1">subscript</csymbol><ci id="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.2.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.1.1.1.1.2">𝐱</ci><interval closure="closed-open" id="S3.Thmthm11.p8.5.m5.4.4.2.3.cmml" xref="S3.Thmthm11.p8.5.m5.4.4.2.2"><cn id="S3.Thmthm11.p8.5.m5.3.3.1.1.cmml" type="integer" xref="S3.Thmthm11.p8.5.m5.3.3.1.1">1</cn><apply id="S3.Thmthm11.p8.5.m5.4.4.2.2.1.cmml" xref="S3.Thmthm11.p8.5.m5.4.4.2.2.1"><plus id="S3.Thmthm11.p8.5.m5.4.4.2.2.1.1.cmml" xref="S3.Thmthm11.p8.5.m5.4.4.2.2.1"></plus><infinity id="S3.Thmthm11.p8.5.m5.4.4.2.2.1.2.cmml" xref="S3.Thmthm11.p8.5.m5.4.4.2.2.1.2"></infinity></apply></interval></apply></apply></apply><apply id="S3.Thmthm11.p8.5.m5.6.6c.cmml" xref="S3.Thmthm11.p8.5.m5.6.6"><eq id="S3.Thmthm11.p8.5.m5.6.6.5.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S3.Thmthm11.p8.5.m5.6.6.1.cmml" id="S3.Thmthm11.p8.5.m5.6.6d.cmml" xref="S3.Thmthm11.p8.5.m5.6.6"></share><apply id="S3.Thmthm11.p8.5.m5.6.6.6.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6"><times id="S3.Thmthm11.p8.5.m5.6.6.6.1.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6.1"></times><ci id="S3.Thmthm11.p8.5.m5.6.6.6.2.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6.2">𝑑</ci><ci id="S3.Thmthm11.p8.5.m5.6.6.6.3.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6.3">𝑒</ci><ci id="S3.Thmthm11.p8.5.m5.6.6.6.4.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6.4">𝑑</ci><ci id="S3.Thmthm11.p8.5.m5.6.6.6.5.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6.5">𝑒</ci><ci id="S3.Thmthm11.p8.5.m5.6.6.6.6.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6.6">𝑑</ci><ci id="S3.Thmthm11.p8.5.m5.6.6.6.7.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6.7">𝑒</ci><ci id="S3.Thmthm11.p8.5.m5.6.6.6.8.cmml" xref="S3.Thmthm11.p8.5.m5.6.6.6.8">…</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.5.m5.6c">\sigma^{\mathbb{Z}}({\bf x})_{[1,+\infty)}=\sigma({\bf x}_{[1,+\infty)})=dedede\ldots</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.5.m5.6d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x ) start_POSTSUBSCRIPT [ 1 , + ∞ ) end_POSTSUBSCRIPT = italic_σ ( bold_x start_POSTSUBSCRIPT [ 1 , + ∞ ) end_POSTSUBSCRIPT ) = italic_d italic_e italic_d italic_e italic_d italic_e …</annotation></semantics></math>, so that <math alttext="\sigma^{\mathbb{Z}}({\bf x})\in[de]" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.6.m6.2"><semantics id="S3.Thmthm11.p8.6.m6.2a"><mrow id="S3.Thmthm11.p8.6.m6.2.2" xref="S3.Thmthm11.p8.6.m6.2.2.cmml"><mrow id="S3.Thmthm11.p8.6.m6.2.2.3" xref="S3.Thmthm11.p8.6.m6.2.2.3.cmml"><msup id="S3.Thmthm11.p8.6.m6.2.2.3.2" xref="S3.Thmthm11.p8.6.m6.2.2.3.2.cmml"><mi id="S3.Thmthm11.p8.6.m6.2.2.3.2.2" xref="S3.Thmthm11.p8.6.m6.2.2.3.2.2.cmml">σ</mi><mi id="S3.Thmthm11.p8.6.m6.2.2.3.2.3" xref="S3.Thmthm11.p8.6.m6.2.2.3.2.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm11.p8.6.m6.2.2.3.1" xref="S3.Thmthm11.p8.6.m6.2.2.3.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p8.6.m6.2.2.3.3.2" xref="S3.Thmthm11.p8.6.m6.2.2.3.cmml"><mo id="S3.Thmthm11.p8.6.m6.2.2.3.3.2.1" stretchy="false" xref="S3.Thmthm11.p8.6.m6.2.2.3.cmml">(</mo><mi id="S3.Thmthm11.p8.6.m6.1.1" xref="S3.Thmthm11.p8.6.m6.1.1.cmml">𝐱</mi><mo id="S3.Thmthm11.p8.6.m6.2.2.3.3.2.2" stretchy="false" xref="S3.Thmthm11.p8.6.m6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p8.6.m6.2.2.2" xref="S3.Thmthm11.p8.6.m6.2.2.2.cmml">∈</mo><mrow id="S3.Thmthm11.p8.6.m6.2.2.1.1" xref="S3.Thmthm11.p8.6.m6.2.2.1.2.cmml"><mo id="S3.Thmthm11.p8.6.m6.2.2.1.1.2" stretchy="false" xref="S3.Thmthm11.p8.6.m6.2.2.1.2.1.cmml">[</mo><mrow id="S3.Thmthm11.p8.6.m6.2.2.1.1.1" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.1.cmml"><mi id="S3.Thmthm11.p8.6.m6.2.2.1.1.1.2" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.1.2.cmml">d</mi><mo id="S3.Thmthm11.p8.6.m6.2.2.1.1.1.1" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.1.1.cmml">⁢</mo><mi id="S3.Thmthm11.p8.6.m6.2.2.1.1.1.3" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.1.3.cmml">e</mi></mrow><mo id="S3.Thmthm11.p8.6.m6.2.2.1.1.3" stretchy="false" xref="S3.Thmthm11.p8.6.m6.2.2.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.6.m6.2b"><apply id="S3.Thmthm11.p8.6.m6.2.2.cmml" xref="S3.Thmthm11.p8.6.m6.2.2"><in id="S3.Thmthm11.p8.6.m6.2.2.2.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.2"></in><apply id="S3.Thmthm11.p8.6.m6.2.2.3.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.3"><times id="S3.Thmthm11.p8.6.m6.2.2.3.1.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.3.1"></times><apply id="S3.Thmthm11.p8.6.m6.2.2.3.2.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.3.2"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.6.m6.2.2.3.2.1.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.3.2">superscript</csymbol><ci id="S3.Thmthm11.p8.6.m6.2.2.3.2.2.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.3.2.2">𝜎</ci><ci id="S3.Thmthm11.p8.6.m6.2.2.3.2.3.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.3.2.3">ℤ</ci></apply><ci id="S3.Thmthm11.p8.6.m6.1.1.cmml" xref="S3.Thmthm11.p8.6.m6.1.1">𝐱</ci></apply><apply id="S3.Thmthm11.p8.6.m6.2.2.1.2.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.1.1"><csymbol cd="latexml" id="S3.Thmthm11.p8.6.m6.2.2.1.2.1.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.2">delimited-[]</csymbol><apply id="S3.Thmthm11.p8.6.m6.2.2.1.1.1.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.1"><times id="S3.Thmthm11.p8.6.m6.2.2.1.1.1.1.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.1.1"></times><ci id="S3.Thmthm11.p8.6.m6.2.2.1.1.1.2.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.1.2">𝑑</ci><ci id="S3.Thmthm11.p8.6.m6.2.2.1.1.1.3.cmml" xref="S3.Thmthm11.p8.6.m6.2.2.1.1.1.3">𝑒</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.6.m6.2c">\sigma^{\mathbb{Z}}({\bf x})\in[de]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.6.m6.2d">italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x ) ∈ [ italic_d italic_e ]</annotation></semantics></math>. As a consequence we obtain <math alttext="(g\circ T-g)\circ\sigma^{\mathbb{Z}}({\bf x})=1" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.7.m7.2"><semantics id="S3.Thmthm11.p8.7.m7.2a"><mrow id="S3.Thmthm11.p8.7.m7.2.2" xref="S3.Thmthm11.p8.7.m7.2.2.cmml"><mrow id="S3.Thmthm11.p8.7.m7.2.2.1" xref="S3.Thmthm11.p8.7.m7.2.2.1.cmml"><mrow id="S3.Thmthm11.p8.7.m7.2.2.1.1" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.cmml"><mrow id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.cmml"><mo id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.2" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.2.cmml">g</mi><mo id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.3" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.1" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.1.cmml">−</mo><mi id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.3" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p8.7.m7.2.2.1.1.2" rspace="0.222em" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.2.cmml">∘</mo><msup id="S3.Thmthm11.p8.7.m7.2.2.1.1.3" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.3.cmml"><mi id="S3.Thmthm11.p8.7.m7.2.2.1.1.3.2" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.3.2.cmml">σ</mi><mi id="S3.Thmthm11.p8.7.m7.2.2.1.1.3.3" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.3.3.cmml">ℤ</mi></msup></mrow><mo id="S3.Thmthm11.p8.7.m7.2.2.1.2" xref="S3.Thmthm11.p8.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p8.7.m7.2.2.1.3.2" xref="S3.Thmthm11.p8.7.m7.2.2.1.cmml"><mo id="S3.Thmthm11.p8.7.m7.2.2.1.3.2.1" stretchy="false" xref="S3.Thmthm11.p8.7.m7.2.2.1.cmml">(</mo><mi id="S3.Thmthm11.p8.7.m7.1.1" xref="S3.Thmthm11.p8.7.m7.1.1.cmml">𝐱</mi><mo id="S3.Thmthm11.p8.7.m7.2.2.1.3.2.2" stretchy="false" xref="S3.Thmthm11.p8.7.m7.2.2.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p8.7.m7.2.2.2" xref="S3.Thmthm11.p8.7.m7.2.2.2.cmml">=</mo><mn id="S3.Thmthm11.p8.7.m7.2.2.3" xref="S3.Thmthm11.p8.7.m7.2.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.7.m7.2b"><apply id="S3.Thmthm11.p8.7.m7.2.2.cmml" xref="S3.Thmthm11.p8.7.m7.2.2"><eq id="S3.Thmthm11.p8.7.m7.2.2.2.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.2"></eq><apply id="S3.Thmthm11.p8.7.m7.2.2.1.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1"><times id="S3.Thmthm11.p8.7.m7.2.2.1.2.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.2"></times><apply id="S3.Thmthm11.p8.7.m7.2.2.1.1.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1"><compose id="S3.Thmthm11.p8.7.m7.2.2.1.1.2.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.2"></compose><apply id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1"><minus id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.1"></minus><apply id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2"><compose id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.1"></compose><ci id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.2">𝑔</ci><ci id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.1.1.1.3">𝑔</ci></apply><apply id="S3.Thmthm11.p8.7.m7.2.2.1.1.3.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.7.m7.2.2.1.1.3.1.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.3">superscript</csymbol><ci id="S3.Thmthm11.p8.7.m7.2.2.1.1.3.2.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.3.2">𝜎</ci><ci id="S3.Thmthm11.p8.7.m7.2.2.1.1.3.3.cmml" xref="S3.Thmthm11.p8.7.m7.2.2.1.1.3.3">ℤ</ci></apply></apply><ci id="S3.Thmthm11.p8.7.m7.1.1.cmml" xref="S3.Thmthm11.p8.7.m7.1.1">𝐱</ci></apply><cn id="S3.Thmthm11.p8.7.m7.2.2.3.cmml" type="integer" xref="S3.Thmthm11.p8.7.m7.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.7.m7.2c">(g\circ T-g)\circ\sigma^{\mathbb{Z}}({\bf x})=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.7.m7.2d">( italic_g ∘ italic_T - italic_g ) ∘ italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x ) = 1</annotation></semantics></math>, and since <math alttext="T({\bf x})={\bf x}" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.8.m8.1"><semantics id="S3.Thmthm11.p8.8.m8.1a"><mrow id="S3.Thmthm11.p8.8.m8.1.2" xref="S3.Thmthm11.p8.8.m8.1.2.cmml"><mrow id="S3.Thmthm11.p8.8.m8.1.2.2" xref="S3.Thmthm11.p8.8.m8.1.2.2.cmml"><mi id="S3.Thmthm11.p8.8.m8.1.2.2.2" xref="S3.Thmthm11.p8.8.m8.1.2.2.2.cmml">T</mi><mo id="S3.Thmthm11.p8.8.m8.1.2.2.1" xref="S3.Thmthm11.p8.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p8.8.m8.1.2.2.3.2" xref="S3.Thmthm11.p8.8.m8.1.2.2.cmml"><mo id="S3.Thmthm11.p8.8.m8.1.2.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p8.8.m8.1.2.2.cmml">(</mo><mi id="S3.Thmthm11.p8.8.m8.1.1" xref="S3.Thmthm11.p8.8.m8.1.1.cmml">𝐱</mi><mo id="S3.Thmthm11.p8.8.m8.1.2.2.3.2.2" stretchy="false" xref="S3.Thmthm11.p8.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p8.8.m8.1.2.1" xref="S3.Thmthm11.p8.8.m8.1.2.1.cmml">=</mo><mi id="S3.Thmthm11.p8.8.m8.1.2.3" xref="S3.Thmthm11.p8.8.m8.1.2.3.cmml">𝐱</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.8.m8.1b"><apply id="S3.Thmthm11.p8.8.m8.1.2.cmml" xref="S3.Thmthm11.p8.8.m8.1.2"><eq id="S3.Thmthm11.p8.8.m8.1.2.1.cmml" xref="S3.Thmthm11.p8.8.m8.1.2.1"></eq><apply id="S3.Thmthm11.p8.8.m8.1.2.2.cmml" xref="S3.Thmthm11.p8.8.m8.1.2.2"><times id="S3.Thmthm11.p8.8.m8.1.2.2.1.cmml" xref="S3.Thmthm11.p8.8.m8.1.2.2.1"></times><ci id="S3.Thmthm11.p8.8.m8.1.2.2.2.cmml" xref="S3.Thmthm11.p8.8.m8.1.2.2.2">𝑇</ci><ci id="S3.Thmthm11.p8.8.m8.1.1.cmml" xref="S3.Thmthm11.p8.8.m8.1.1">𝐱</ci></apply><ci id="S3.Thmthm11.p8.8.m8.1.2.3.cmml" xref="S3.Thmthm11.p8.8.m8.1.2.3">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.8.m8.1c">T({\bf x})={\bf x}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.8.m8.1d">italic_T ( bold_x ) = bold_x</annotation></semantics></math>, also <math alttext="(g\circ T-g)\circ\sigma^{\mathbb{Z}}\circ T^{n}({\bf x})=1" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.9.m9.2"><semantics id="S3.Thmthm11.p8.9.m9.2a"><mrow id="S3.Thmthm11.p8.9.m9.2.2" xref="S3.Thmthm11.p8.9.m9.2.2.cmml"><mrow id="S3.Thmthm11.p8.9.m9.2.2.1" xref="S3.Thmthm11.p8.9.m9.2.2.1.cmml"><mrow id="S3.Thmthm11.p8.9.m9.2.2.1.1" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.cmml"><mrow id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.cmml"><mo id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.2" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.2.cmml">g</mi><mo id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.3" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.1" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.1.cmml">−</mo><mi id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.3" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p8.9.m9.2.2.1.1.2" rspace="0.222em" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.2.cmml">∘</mo><msup id="S3.Thmthm11.p8.9.m9.2.2.1.1.3" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.3.cmml"><mi id="S3.Thmthm11.p8.9.m9.2.2.1.1.3.2" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.3.2.cmml">σ</mi><mi id="S3.Thmthm11.p8.9.m9.2.2.1.1.3.3" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.3.3.cmml">ℤ</mi></msup><mo id="S3.Thmthm11.p8.9.m9.2.2.1.1.2a" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.2.cmml">∘</mo><msup id="S3.Thmthm11.p8.9.m9.2.2.1.1.4" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.4.cmml"><mi id="S3.Thmthm11.p8.9.m9.2.2.1.1.4.2" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.4.2.cmml">T</mi><mi id="S3.Thmthm11.p8.9.m9.2.2.1.1.4.3" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.4.3.cmml">n</mi></msup></mrow><mo id="S3.Thmthm11.p8.9.m9.2.2.1.2" xref="S3.Thmthm11.p8.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmthm11.p8.9.m9.2.2.1.3.2" xref="S3.Thmthm11.p8.9.m9.2.2.1.cmml"><mo id="S3.Thmthm11.p8.9.m9.2.2.1.3.2.1" stretchy="false" xref="S3.Thmthm11.p8.9.m9.2.2.1.cmml">(</mo><mi id="S3.Thmthm11.p8.9.m9.1.1" xref="S3.Thmthm11.p8.9.m9.1.1.cmml">𝐱</mi><mo id="S3.Thmthm11.p8.9.m9.2.2.1.3.2.2" stretchy="false" xref="S3.Thmthm11.p8.9.m9.2.2.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmthm11.p8.9.m9.2.2.2" xref="S3.Thmthm11.p8.9.m9.2.2.2.cmml">=</mo><mn id="S3.Thmthm11.p8.9.m9.2.2.3" xref="S3.Thmthm11.p8.9.m9.2.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.9.m9.2b"><apply id="S3.Thmthm11.p8.9.m9.2.2.cmml" xref="S3.Thmthm11.p8.9.m9.2.2"><eq id="S3.Thmthm11.p8.9.m9.2.2.2.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.2"></eq><apply id="S3.Thmthm11.p8.9.m9.2.2.1.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1"><times id="S3.Thmthm11.p8.9.m9.2.2.1.2.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.2"></times><apply id="S3.Thmthm11.p8.9.m9.2.2.1.1.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1"><compose id="S3.Thmthm11.p8.9.m9.2.2.1.1.2.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.2"></compose><apply id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1"><minus id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.1.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.1"></minus><apply id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2"><compose id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.1"></compose><ci id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.2">𝑔</ci><ci id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.3.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.1.1.1.3">𝑔</ci></apply><apply id="S3.Thmthm11.p8.9.m9.2.2.1.1.3.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.9.m9.2.2.1.1.3.1.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.3">superscript</csymbol><ci id="S3.Thmthm11.p8.9.m9.2.2.1.1.3.2.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.3.2">𝜎</ci><ci id="S3.Thmthm11.p8.9.m9.2.2.1.1.3.3.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.3.3">ℤ</ci></apply><apply id="S3.Thmthm11.p8.9.m9.2.2.1.1.4.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.4"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.9.m9.2.2.1.1.4.1.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.4">superscript</csymbol><ci id="S3.Thmthm11.p8.9.m9.2.2.1.1.4.2.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.4.2">𝑇</ci><ci id="S3.Thmthm11.p8.9.m9.2.2.1.1.4.3.cmml" xref="S3.Thmthm11.p8.9.m9.2.2.1.1.4.3">𝑛</ci></apply></apply><ci id="S3.Thmthm11.p8.9.m9.1.1.cmml" xref="S3.Thmthm11.p8.9.m9.1.1">𝐱</ci></apply><cn id="S3.Thmthm11.p8.9.m9.2.2.3.cmml" type="integer" xref="S3.Thmthm11.p8.9.m9.2.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.9.m9.2c">(g\circ T-g)\circ\sigma^{\mathbb{Z}}\circ T^{n}({\bf x})=1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.9.m9.2d">( italic_g ∘ italic_T - italic_g ) ∘ italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ∘ italic_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( bold_x ) = 1</annotation></semantics></math> for any integer <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.10.m10.1"><semantics id="S3.Thmthm11.p8.10.m10.1a"><mrow id="S3.Thmthm11.p8.10.m10.1.1" xref="S3.Thmthm11.p8.10.m10.1.1.cmml"><mi id="S3.Thmthm11.p8.10.m10.1.1.2" xref="S3.Thmthm11.p8.10.m10.1.1.2.cmml">n</mi><mo id="S3.Thmthm11.p8.10.m10.1.1.1" xref="S3.Thmthm11.p8.10.m10.1.1.1.cmml">≥</mo><mn id="S3.Thmthm11.p8.10.m10.1.1.3" xref="S3.Thmthm11.p8.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.10.m10.1b"><apply id="S3.Thmthm11.p8.10.m10.1.1.cmml" xref="S3.Thmthm11.p8.10.m10.1.1"><geq id="S3.Thmthm11.p8.10.m10.1.1.1.cmml" xref="S3.Thmthm11.p8.10.m10.1.1.1"></geq><ci id="S3.Thmthm11.p8.10.m10.1.1.2.cmml" xref="S3.Thmthm11.p8.10.m10.1.1.2">𝑛</ci><cn id="S3.Thmthm11.p8.10.m10.1.1.3.cmml" type="integer" xref="S3.Thmthm11.p8.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.10.m10.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.10.m10.1d">italic_n ≥ 0</annotation></semantics></math>. However, from Proposition 3.2.1 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib7" title="">7</a>]</cite> we know that for any function <math alttext="f\in\partial_{T}(C,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.11.m11.2"><semantics id="S3.Thmthm11.p8.11.m11.2a"><mrow id="S3.Thmthm11.p8.11.m11.2.3" xref="S3.Thmthm11.p8.11.m11.2.3.cmml"><mi id="S3.Thmthm11.p8.11.m11.2.3.2" xref="S3.Thmthm11.p8.11.m11.2.3.2.cmml">f</mi><mo id="S3.Thmthm11.p8.11.m11.2.3.1" rspace="0.1389em" xref="S3.Thmthm11.p8.11.m11.2.3.1.cmml">∈</mo><mrow id="S3.Thmthm11.p8.11.m11.2.3.3" xref="S3.Thmthm11.p8.11.m11.2.3.3.cmml"><msub id="S3.Thmthm11.p8.11.m11.2.3.3.1" xref="S3.Thmthm11.p8.11.m11.2.3.3.1.cmml"><mo id="S3.Thmthm11.p8.11.m11.2.3.3.1.2" lspace="0.1389em" rspace="0em" xref="S3.Thmthm11.p8.11.m11.2.3.3.1.2.cmml">∂</mo><mi id="S3.Thmthm11.p8.11.m11.2.3.3.1.3" xref="S3.Thmthm11.p8.11.m11.2.3.3.1.3.cmml">T</mi></msub><mrow id="S3.Thmthm11.p8.11.m11.2.3.3.2.2" xref="S3.Thmthm11.p8.11.m11.2.3.3.2.1.cmml"><mo id="S3.Thmthm11.p8.11.m11.2.3.3.2.2.1" stretchy="false" xref="S3.Thmthm11.p8.11.m11.2.3.3.2.1.cmml">(</mo><mi id="S3.Thmthm11.p8.11.m11.1.1" xref="S3.Thmthm11.p8.11.m11.1.1.cmml">C</mi><mo id="S3.Thmthm11.p8.11.m11.2.3.3.2.2.2" xref="S3.Thmthm11.p8.11.m11.2.3.3.2.1.cmml">,</mo><mi id="S3.Thmthm11.p8.11.m11.2.2" xref="S3.Thmthm11.p8.11.m11.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p8.11.m11.2.3.3.2.2.3" stretchy="false" xref="S3.Thmthm11.p8.11.m11.2.3.3.2.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.11.m11.2b"><apply id="S3.Thmthm11.p8.11.m11.2.3.cmml" xref="S3.Thmthm11.p8.11.m11.2.3"><in id="S3.Thmthm11.p8.11.m11.2.3.1.cmml" xref="S3.Thmthm11.p8.11.m11.2.3.1"></in><ci id="S3.Thmthm11.p8.11.m11.2.3.2.cmml" xref="S3.Thmthm11.p8.11.m11.2.3.2">𝑓</ci><apply id="S3.Thmthm11.p8.11.m11.2.3.3.cmml" xref="S3.Thmthm11.p8.11.m11.2.3.3"><apply id="S3.Thmthm11.p8.11.m11.2.3.3.1.cmml" xref="S3.Thmthm11.p8.11.m11.2.3.3.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.11.m11.2.3.3.1.1.cmml" xref="S3.Thmthm11.p8.11.m11.2.3.3.1">subscript</csymbol><partialdiff id="S3.Thmthm11.p8.11.m11.2.3.3.1.2.cmml" xref="S3.Thmthm11.p8.11.m11.2.3.3.1.2"></partialdiff><ci id="S3.Thmthm11.p8.11.m11.2.3.3.1.3.cmml" xref="S3.Thmthm11.p8.11.m11.2.3.3.1.3">𝑇</ci></apply><interval closure="open" id="S3.Thmthm11.p8.11.m11.2.3.3.2.1.cmml" xref="S3.Thmthm11.p8.11.m11.2.3.3.2.2"><ci id="S3.Thmthm11.p8.11.m11.1.1.cmml" xref="S3.Thmthm11.p8.11.m11.1.1">𝐶</ci><ci id="S3.Thmthm11.p8.11.m11.2.2.cmml" xref="S3.Thmthm11.p8.11.m11.2.2">ℤ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.11.m11.2c">f\in\partial_{T}(C,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.11.m11.2d">italic_f ∈ ∂ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT ( italic_C , blackboard_Z )</annotation></semantics></math> the sequence of function <math alttext="f+f\circ T+\ldots+f\circ T^{n}" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.12.m12.1"><semantics id="S3.Thmthm11.p8.12.m12.1a"><mrow id="S3.Thmthm11.p8.12.m12.1.1" xref="S3.Thmthm11.p8.12.m12.1.1.cmml"><mi id="S3.Thmthm11.p8.12.m12.1.1.2" xref="S3.Thmthm11.p8.12.m12.1.1.2.cmml">f</mi><mo id="S3.Thmthm11.p8.12.m12.1.1.1" xref="S3.Thmthm11.p8.12.m12.1.1.1.cmml">+</mo><mrow id="S3.Thmthm11.p8.12.m12.1.1.3" xref="S3.Thmthm11.p8.12.m12.1.1.3.cmml"><mi id="S3.Thmthm11.p8.12.m12.1.1.3.2" xref="S3.Thmthm11.p8.12.m12.1.1.3.2.cmml">f</mi><mo id="S3.Thmthm11.p8.12.m12.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p8.12.m12.1.1.3.1.cmml">∘</mo><mi id="S3.Thmthm11.p8.12.m12.1.1.3.3" xref="S3.Thmthm11.p8.12.m12.1.1.3.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p8.12.m12.1.1.1a" xref="S3.Thmthm11.p8.12.m12.1.1.1.cmml">+</mo><mi id="S3.Thmthm11.p8.12.m12.1.1.4" mathvariant="normal" xref="S3.Thmthm11.p8.12.m12.1.1.4.cmml">…</mi><mo id="S3.Thmthm11.p8.12.m12.1.1.1b" xref="S3.Thmthm11.p8.12.m12.1.1.1.cmml">+</mo><mrow id="S3.Thmthm11.p8.12.m12.1.1.5" xref="S3.Thmthm11.p8.12.m12.1.1.5.cmml"><mi id="S3.Thmthm11.p8.12.m12.1.1.5.2" xref="S3.Thmthm11.p8.12.m12.1.1.5.2.cmml">f</mi><mo id="S3.Thmthm11.p8.12.m12.1.1.5.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p8.12.m12.1.1.5.1.cmml">∘</mo><msup id="S3.Thmthm11.p8.12.m12.1.1.5.3" xref="S3.Thmthm11.p8.12.m12.1.1.5.3.cmml"><mi id="S3.Thmthm11.p8.12.m12.1.1.5.3.2" xref="S3.Thmthm11.p8.12.m12.1.1.5.3.2.cmml">T</mi><mi id="S3.Thmthm11.p8.12.m12.1.1.5.3.3" xref="S3.Thmthm11.p8.12.m12.1.1.5.3.3.cmml">n</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.12.m12.1b"><apply id="S3.Thmthm11.p8.12.m12.1.1.cmml" xref="S3.Thmthm11.p8.12.m12.1.1"><plus id="S3.Thmthm11.p8.12.m12.1.1.1.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.1"></plus><ci id="S3.Thmthm11.p8.12.m12.1.1.2.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.2">𝑓</ci><apply id="S3.Thmthm11.p8.12.m12.1.1.3.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.3"><compose id="S3.Thmthm11.p8.12.m12.1.1.3.1.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.3.1"></compose><ci id="S3.Thmthm11.p8.12.m12.1.1.3.2.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.3.2">𝑓</ci><ci id="S3.Thmthm11.p8.12.m12.1.1.3.3.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.3.3">𝑇</ci></apply><ci id="S3.Thmthm11.p8.12.m12.1.1.4.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.4">…</ci><apply id="S3.Thmthm11.p8.12.m12.1.1.5.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.5"><compose id="S3.Thmthm11.p8.12.m12.1.1.5.1.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.5.1"></compose><ci id="S3.Thmthm11.p8.12.m12.1.1.5.2.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.5.2">𝑓</ci><apply id="S3.Thmthm11.p8.12.m12.1.1.5.3.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.5.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.12.m12.1.1.5.3.1.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.5.3">superscript</csymbol><ci id="S3.Thmthm11.p8.12.m12.1.1.5.3.2.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.5.3.2">𝑇</ci><ci id="S3.Thmthm11.p8.12.m12.1.1.5.3.3.cmml" xref="S3.Thmthm11.p8.12.m12.1.1.5.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.12.m12.1c">f+f\circ T+\ldots+f\circ T^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.12.m12.1d">italic_f + italic_f ∘ italic_T + … + italic_f ∘ italic_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is bounded uniformly. It follows that <math alttext="(g\circ T-g)\circ\sigma^{\mathbb{Z}}\notin\partial_{T}C(X,\mathbb{Z})" class="ltx_Math" display="inline" id="S3.Thmthm11.p8.13.m13.3"><semantics id="S3.Thmthm11.p8.13.m13.3a"><mrow id="S3.Thmthm11.p8.13.m13.3.3" xref="S3.Thmthm11.p8.13.m13.3.3.cmml"><mrow id="S3.Thmthm11.p8.13.m13.3.3.1" xref="S3.Thmthm11.p8.13.m13.3.3.1.cmml"><mrow id="S3.Thmthm11.p8.13.m13.3.3.1.1.1" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.cmml"><mo id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.2" stretchy="false" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.cmml"><mrow id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.cmml"><mi id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.2" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.2.cmml">g</mi><mo id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.1.cmml">∘</mo><mi id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.3" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.3.cmml">T</mi></mrow><mo id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.1" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.1.cmml">−</mo><mi id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.3" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.3" rspace="0.055em" stretchy="false" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmthm11.p8.13.m13.3.3.1.2" rspace="0.222em" xref="S3.Thmthm11.p8.13.m13.3.3.1.2.cmml">∘</mo><msup id="S3.Thmthm11.p8.13.m13.3.3.1.3" xref="S3.Thmthm11.p8.13.m13.3.3.1.3.cmml"><mi id="S3.Thmthm11.p8.13.m13.3.3.1.3.2" xref="S3.Thmthm11.p8.13.m13.3.3.1.3.2.cmml">σ</mi><mi id="S3.Thmthm11.p8.13.m13.3.3.1.3.3" xref="S3.Thmthm11.p8.13.m13.3.3.1.3.3.cmml">ℤ</mi></msup></mrow><mo id="S3.Thmthm11.p8.13.m13.3.3.2" rspace="0.1389em" xref="S3.Thmthm11.p8.13.m13.3.3.2.cmml">∉</mo><mrow id="S3.Thmthm11.p8.13.m13.3.3.3" xref="S3.Thmthm11.p8.13.m13.3.3.3.cmml"><msub id="S3.Thmthm11.p8.13.m13.3.3.3.1" xref="S3.Thmthm11.p8.13.m13.3.3.3.1.cmml"><mo id="S3.Thmthm11.p8.13.m13.3.3.3.1.2" lspace="0.1389em" rspace="0em" xref="S3.Thmthm11.p8.13.m13.3.3.3.1.2.cmml">∂</mo><mi id="S3.Thmthm11.p8.13.m13.3.3.3.1.3" xref="S3.Thmthm11.p8.13.m13.3.3.3.1.3.cmml">T</mi></msub><mrow id="S3.Thmthm11.p8.13.m13.3.3.3.2" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.cmml"><mi id="S3.Thmthm11.p8.13.m13.3.3.3.2.2" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.2.cmml">C</mi><mo id="S3.Thmthm11.p8.13.m13.3.3.3.2.1" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.1.cmml">⁢</mo><mrow id="S3.Thmthm11.p8.13.m13.3.3.3.2.3.2" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.3.1.cmml"><mo id="S3.Thmthm11.p8.13.m13.3.3.3.2.3.2.1" stretchy="false" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.3.1.cmml">(</mo><mi id="S3.Thmthm11.p8.13.m13.1.1" xref="S3.Thmthm11.p8.13.m13.1.1.cmml">X</mi><mo id="S3.Thmthm11.p8.13.m13.3.3.3.2.3.2.2" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.3.1.cmml">,</mo><mi id="S3.Thmthm11.p8.13.m13.2.2" xref="S3.Thmthm11.p8.13.m13.2.2.cmml">ℤ</mi><mo id="S3.Thmthm11.p8.13.m13.3.3.3.2.3.2.3" stretchy="false" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmthm11.p8.13.m13.3b"><apply id="S3.Thmthm11.p8.13.m13.3.3.cmml" xref="S3.Thmthm11.p8.13.m13.3.3"><notin id="S3.Thmthm11.p8.13.m13.3.3.2.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.2"></notin><apply id="S3.Thmthm11.p8.13.m13.3.3.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1"><compose id="S3.Thmthm11.p8.13.m13.3.3.1.2.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.2"></compose><apply id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1"><minus id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.1"></minus><apply id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2"><compose id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.1"></compose><ci id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.2.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.2">𝑔</ci><ci id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.3.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.2.3">𝑇</ci></apply><ci id="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.3.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.1.1.1.3">𝑔</ci></apply><apply id="S3.Thmthm11.p8.13.m13.3.3.1.3.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.3"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.13.m13.3.3.1.3.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.3">superscript</csymbol><ci id="S3.Thmthm11.p8.13.m13.3.3.1.3.2.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.3.2">𝜎</ci><ci id="S3.Thmthm11.p8.13.m13.3.3.1.3.3.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.1.3.3">ℤ</ci></apply></apply><apply id="S3.Thmthm11.p8.13.m13.3.3.3.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3"><apply id="S3.Thmthm11.p8.13.m13.3.3.3.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3.1"><csymbol cd="ambiguous" id="S3.Thmthm11.p8.13.m13.3.3.3.1.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3.1">subscript</csymbol><partialdiff id="S3.Thmthm11.p8.13.m13.3.3.3.1.2.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3.1.2"></partialdiff><ci id="S3.Thmthm11.p8.13.m13.3.3.3.1.3.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3.1.3">𝑇</ci></apply><apply id="S3.Thmthm11.p8.13.m13.3.3.3.2.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3.2"><times id="S3.Thmthm11.p8.13.m13.3.3.3.2.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.1"></times><ci id="S3.Thmthm11.p8.13.m13.3.3.3.2.2.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.2">𝐶</ci><interval closure="open" id="S3.Thmthm11.p8.13.m13.3.3.3.2.3.1.cmml" xref="S3.Thmthm11.p8.13.m13.3.3.3.2.3.2"><ci id="S3.Thmthm11.p8.13.m13.1.1.cmml" xref="S3.Thmthm11.p8.13.m13.1.1">𝑋</ci><ci id="S3.Thmthm11.p8.13.m13.2.2.cmml" xref="S3.Thmthm11.p8.13.m13.2.2">ℤ</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmthm11.p8.13.m13.3c">(g\circ T-g)\circ\sigma^{\mathbb{Z}}\notin\partial_{T}C(X,\mathbb{Z})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmthm11.p8.13.m13.3d">( italic_g ∘ italic_T - italic_g ) ∘ italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ∉ ∂ start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT italic_C ( italic_X , blackboard_Z )</annotation></semantics></math>, so that property (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E8" title="In Remark 3.11. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.8</span></a>) fails.</p> </div> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4. </span>Evaluation of the transferred measure <math alttext="\sigma M(\mu)" class="ltx_Math" display="inline" id="S4.1.m1.1"><semantics id="S4.1.m1.1b"><mrow id="S4.1.m1.1.2" xref="S4.1.m1.1.2.cmml"><mi id="S4.1.m1.1.2.2" xref="S4.1.m1.1.2.2.cmml">σ</mi><mo id="S4.1.m1.1.2.1" xref="S4.1.m1.1.2.1.cmml">⁢</mo><mi id="S4.1.m1.1.2.3" xref="S4.1.m1.1.2.3.cmml">M</mi><mo id="S4.1.m1.1.2.1b" xref="S4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.1.m1.1.2.4.2" xref="S4.1.m1.1.2.cmml"><mo id="S4.1.m1.1.2.4.2.1" stretchy="false" xref="S4.1.m1.1.2.cmml">(</mo><mi id="S4.1.m1.1.1" xref="S4.1.m1.1.1.cmml">μ</mi><mo id="S4.1.m1.1.2.4.2.2" stretchy="false" xref="S4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.1.m1.1c"><apply id="S4.1.m1.1.2.cmml" xref="S4.1.m1.1.2"><times id="S4.1.m1.1.2.1.cmml" xref="S4.1.m1.1.2.1"></times><ci id="S4.1.m1.1.2.2.cmml" xref="S4.1.m1.1.2.2">𝜎</ci><ci id="S4.1.m1.1.2.3.cmml" xref="S4.1.m1.1.2.3">𝑀</ci><ci id="S4.1.m1.1.1.cmml" xref="S4.1.m1.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.1.m1.1d">\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S4.1.m1.1e">italic_σ italic_M ( italic_μ )</annotation></semantics></math> </h2> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1. </span>A first example for the measure transfer</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1"></p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.7">We will illustrate in this subsection the induced measure transfer map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mrow id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">σ</mi><mo id="S4.SS1.p2.1.m1.1.1.1" xref="S4.SS1.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p2.1.m1.1.1.3" xref="S4.SS1.p2.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1"><times id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1.1"></times><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝜎</ci><ci id="S4.SS1.p2.1.m1.1.1.3.cmml" xref="S4.SS1.p2.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_σ italic_M</annotation></semantics></math> from Definition-Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm6" title="Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.6</span></a> for an explicitely given morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mrow id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml"><mi id="S4.SS1.p2.2.m2.1.1.2" xref="S4.SS1.p2.2.m2.1.1.2.cmml">σ</mi><mo id="S4.SS1.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S4.SS1.p2.2.m2.1.1.3" xref="S4.SS1.p2.2.m2.1.1.3.cmml"><msup id="S4.SS1.p2.2.m2.1.1.3.2" xref="S4.SS1.p2.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.2.m2.1.1.3.2.2" xref="S4.SS1.p2.2.m2.1.1.3.2.2.cmml">𝒜</mi><mo id="S4.SS1.p2.2.m2.1.1.3.2.3" xref="S4.SS1.p2.2.m2.1.1.3.2.3.cmml">∗</mo></msup><mo id="S4.SS1.p2.2.m2.1.1.3.1" stretchy="false" xref="S4.SS1.p2.2.m2.1.1.3.1.cmml">→</mo><msup id="S4.SS1.p2.2.m2.1.1.3.3" xref="S4.SS1.p2.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.2.m2.1.1.3.3.2" xref="S4.SS1.p2.2.m2.1.1.3.3.2.cmml">ℬ</mi><mo id="S4.SS1.p2.2.m2.1.1.3.3.3" xref="S4.SS1.p2.2.m2.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><apply id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1"><ci id="S4.SS1.p2.2.m2.1.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1.1">:</ci><ci id="S4.SS1.p2.2.m2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.1.1.2">𝜎</ci><apply id="S4.SS1.p2.2.m2.1.1.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3"><ci id="S4.SS1.p2.2.m2.1.1.3.1.cmml" xref="S4.SS1.p2.2.m2.1.1.3.1">→</ci><apply id="S4.SS1.p2.2.m2.1.1.3.2.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.3.2.1.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2">superscript</csymbol><ci id="S4.SS1.p2.2.m2.1.1.3.2.2.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2.2">𝒜</ci><times id="S4.SS1.p2.2.m2.1.1.3.2.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3.2.3"></times></apply><apply id="S4.SS1.p2.2.m2.1.1.3.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.1.1.3.3.1.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3">superscript</csymbol><ci id="S4.SS1.p2.2.m2.1.1.3.3.2.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3.2">ℬ</ci><times id="S4.SS1.p2.2.m2.1.1.3.3.3.cmml" xref="S4.SS1.p2.2.m2.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. For this example we will carry through in all detail, for any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mi id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><msup id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.4.m4.1.1.2" xref="S4.SS1.p2.4.m4.1.1.2.cmml">𝒜</mi><mi id="S4.SS1.p2.4.m4.1.1.3" xref="S4.SS1.p2.4.m4.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><apply id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1">superscript</csymbol><ci id="S4.SS1.p2.4.m4.1.1.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2">𝒜</ci><ci id="S4.SS1.p2.4.m4.1.1.3.cmml" xref="S4.SS1.p2.4.m4.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, the computation of the values of <math alttext="\sigma M(w)" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mrow id="S4.SS1.p2.5.m5.1.2" xref="S4.SS1.p2.5.m5.1.2.cmml"><mi id="S4.SS1.p2.5.m5.1.2.2" xref="S4.SS1.p2.5.m5.1.2.2.cmml">σ</mi><mo id="S4.SS1.p2.5.m5.1.2.1" xref="S4.SS1.p2.5.m5.1.2.1.cmml">⁢</mo><mi id="S4.SS1.p2.5.m5.1.2.3" xref="S4.SS1.p2.5.m5.1.2.3.cmml">M</mi><mo id="S4.SS1.p2.5.m5.1.2.1a" xref="S4.SS1.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.5.m5.1.2.4.2" xref="S4.SS1.p2.5.m5.1.2.cmml"><mo id="S4.SS1.p2.5.m5.1.2.4.2.1" stretchy="false" xref="S4.SS1.p2.5.m5.1.2.cmml">(</mo><mi id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">w</mi><mo id="S4.SS1.p2.5.m5.1.2.4.2.2" stretchy="false" xref="S4.SS1.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><apply id="S4.SS1.p2.5.m5.1.2.cmml" xref="S4.SS1.p2.5.m5.1.2"><times id="S4.SS1.p2.5.m5.1.2.1.cmml" xref="S4.SS1.p2.5.m5.1.2.1"></times><ci id="S4.SS1.p2.5.m5.1.2.2.cmml" xref="S4.SS1.p2.5.m5.1.2.2">𝜎</ci><ci id="S4.SS1.p2.5.m5.1.2.3.cmml" xref="S4.SS1.p2.5.m5.1.2.3">𝑀</ci><ci id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">\sigma M(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_σ italic_M ( italic_w )</annotation></semantics></math> for any word <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.1"><semantics id="S4.SS1.p2.6.m6.1a"><mrow id="S4.SS1.p2.6.m6.1.1" xref="S4.SS1.p2.6.m6.1.1.cmml"><mi id="S4.SS1.p2.6.m6.1.1.2" xref="S4.SS1.p2.6.m6.1.1.2.cmml">w</mi><mo id="S4.SS1.p2.6.m6.1.1.1" xref="S4.SS1.p2.6.m6.1.1.1.cmml">∈</mo><msup id="S4.SS1.p2.6.m6.1.1.3" xref="S4.SS1.p2.6.m6.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p2.6.m6.1.1.3.2" xref="S4.SS1.p2.6.m6.1.1.3.2.cmml">𝒜</mi><mo id="S4.SS1.p2.6.m6.1.1.3.3" xref="S4.SS1.p2.6.m6.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.1b"><apply id="S4.SS1.p2.6.m6.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1"><in id="S4.SS1.p2.6.m6.1.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1.1"></in><ci id="S4.SS1.p2.6.m6.1.1.2.cmml" xref="S4.SS1.p2.6.m6.1.1.2">𝑤</ci><apply id="S4.SS1.p2.6.m6.1.1.3.cmml" xref="S4.SS1.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p2.6.m6.1.1.3.1.cmml" xref="S4.SS1.p2.6.m6.1.1.3">superscript</csymbol><ci id="S4.SS1.p2.6.m6.1.1.3.2.cmml" xref="S4.SS1.p2.6.m6.1.1.3.2">𝒜</ci><times id="S4.SS1.p2.6.m6.1.1.3.3.cmml" xref="S4.SS1.p2.6.m6.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of length <math alttext="|w|\leq 2" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.1"><semantics id="S4.SS1.p2.7.m7.1a"><mrow id="S4.SS1.p2.7.m7.1.2" xref="S4.SS1.p2.7.m7.1.2.cmml"><mrow id="S4.SS1.p2.7.m7.1.2.2.2" xref="S4.SS1.p2.7.m7.1.2.2.1.cmml"><mo id="S4.SS1.p2.7.m7.1.2.2.2.1" stretchy="false" xref="S4.SS1.p2.7.m7.1.2.2.1.1.cmml">|</mo><mi id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml">w</mi><mo id="S4.SS1.p2.7.m7.1.2.2.2.2" stretchy="false" xref="S4.SS1.p2.7.m7.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS1.p2.7.m7.1.2.1" xref="S4.SS1.p2.7.m7.1.2.1.cmml">≤</mo><mn id="S4.SS1.p2.7.m7.1.2.3" xref="S4.SS1.p2.7.m7.1.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.1b"><apply id="S4.SS1.p2.7.m7.1.2.cmml" xref="S4.SS1.p2.7.m7.1.2"><leq id="S4.SS1.p2.7.m7.1.2.1.cmml" xref="S4.SS1.p2.7.m7.1.2.1"></leq><apply id="S4.SS1.p2.7.m7.1.2.2.1.cmml" xref="S4.SS1.p2.7.m7.1.2.2.2"><abs id="S4.SS1.p2.7.m7.1.2.2.1.1.cmml" xref="S4.SS1.p2.7.m7.1.2.2.2.1"></abs><ci id="S4.SS1.p2.7.m7.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1">𝑤</ci></apply><cn id="S4.SS1.p2.7.m7.1.2.3.cmml" type="integer" xref="S4.SS1.p2.7.m7.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.1c">|w|\leq 2</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.1d">| italic_w | ≤ 2</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_convention" id="S4.Thmthm1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmthm1.1.1.1">Convention 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmthm1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmthm1.p1"> <p class="ltx_p" id="S4.Thmthm1.p1.7">We simplify here (as well as in some other concrete computations below) the notation used before by writing <math alttext="a,b,c,\ldots" class="ltx_Math" display="inline" id="S4.Thmthm1.p1.1.m1.4"><semantics id="S4.Thmthm1.p1.1.m1.4a"><mrow id="S4.Thmthm1.p1.1.m1.4.5.2" xref="S4.Thmthm1.p1.1.m1.4.5.1.cmml"><mi id="S4.Thmthm1.p1.1.m1.1.1" xref="S4.Thmthm1.p1.1.m1.1.1.cmml">a</mi><mo id="S4.Thmthm1.p1.1.m1.4.5.2.1" xref="S4.Thmthm1.p1.1.m1.4.5.1.cmml">,</mo><mi id="S4.Thmthm1.p1.1.m1.2.2" xref="S4.Thmthm1.p1.1.m1.2.2.cmml">b</mi><mo id="S4.Thmthm1.p1.1.m1.4.5.2.2" xref="S4.Thmthm1.p1.1.m1.4.5.1.cmml">,</mo><mi id="S4.Thmthm1.p1.1.m1.3.3" xref="S4.Thmthm1.p1.1.m1.3.3.cmml">c</mi><mo id="S4.Thmthm1.p1.1.m1.4.5.2.3" xref="S4.Thmthm1.p1.1.m1.4.5.1.cmml">,</mo><mi id="S4.Thmthm1.p1.1.m1.4.4" mathvariant="normal" xref="S4.Thmthm1.p1.1.m1.4.4.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm1.p1.1.m1.4b"><list id="S4.Thmthm1.p1.1.m1.4.5.1.cmml" xref="S4.Thmthm1.p1.1.m1.4.5.2"><ci id="S4.Thmthm1.p1.1.m1.1.1.cmml" xref="S4.Thmthm1.p1.1.m1.1.1">𝑎</ci><ci id="S4.Thmthm1.p1.1.m1.2.2.cmml" xref="S4.Thmthm1.p1.1.m1.2.2">𝑏</ci><ci id="S4.Thmthm1.p1.1.m1.3.3.cmml" xref="S4.Thmthm1.p1.1.m1.3.3">𝑐</ci><ci id="S4.Thmthm1.p1.1.m1.4.4.cmml" xref="S4.Thmthm1.p1.1.m1.4.4">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm1.p1.1.m1.4c">a,b,c,\ldots</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm1.p1.1.m1.4d">italic_a , italic_b , italic_c , …</annotation></semantics></math> instead of <math alttext="a_{1},a_{2},a_{3},\ldots" class="ltx_Math" display="inline" id="S4.Thmthm1.p1.2.m2.4"><semantics id="S4.Thmthm1.p1.2.m2.4a"><mrow id="S4.Thmthm1.p1.2.m2.4.4.3" xref="S4.Thmthm1.p1.2.m2.4.4.4.cmml"><msub id="S4.Thmthm1.p1.2.m2.2.2.1.1" xref="S4.Thmthm1.p1.2.m2.2.2.1.1.cmml"><mi id="S4.Thmthm1.p1.2.m2.2.2.1.1.2" xref="S4.Thmthm1.p1.2.m2.2.2.1.1.2.cmml">a</mi><mn id="S4.Thmthm1.p1.2.m2.2.2.1.1.3" xref="S4.Thmthm1.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S4.Thmthm1.p1.2.m2.4.4.3.4" xref="S4.Thmthm1.p1.2.m2.4.4.4.cmml">,</mo><msub id="S4.Thmthm1.p1.2.m2.3.3.2.2" xref="S4.Thmthm1.p1.2.m2.3.3.2.2.cmml"><mi id="S4.Thmthm1.p1.2.m2.3.3.2.2.2" xref="S4.Thmthm1.p1.2.m2.3.3.2.2.2.cmml">a</mi><mn id="S4.Thmthm1.p1.2.m2.3.3.2.2.3" xref="S4.Thmthm1.p1.2.m2.3.3.2.2.3.cmml">2</mn></msub><mo id="S4.Thmthm1.p1.2.m2.4.4.3.5" xref="S4.Thmthm1.p1.2.m2.4.4.4.cmml">,</mo><msub id="S4.Thmthm1.p1.2.m2.4.4.3.3" xref="S4.Thmthm1.p1.2.m2.4.4.3.3.cmml"><mi id="S4.Thmthm1.p1.2.m2.4.4.3.3.2" xref="S4.Thmthm1.p1.2.m2.4.4.3.3.2.cmml">a</mi><mn id="S4.Thmthm1.p1.2.m2.4.4.3.3.3" xref="S4.Thmthm1.p1.2.m2.4.4.3.3.3.cmml">3</mn></msub><mo id="S4.Thmthm1.p1.2.m2.4.4.3.6" xref="S4.Thmthm1.p1.2.m2.4.4.4.cmml">,</mo><mi id="S4.Thmthm1.p1.2.m2.1.1" mathvariant="normal" xref="S4.Thmthm1.p1.2.m2.1.1.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm1.p1.2.m2.4b"><list id="S4.Thmthm1.p1.2.m2.4.4.4.cmml" xref="S4.Thmthm1.p1.2.m2.4.4.3"><apply id="S4.Thmthm1.p1.2.m2.2.2.1.1.cmml" xref="S4.Thmthm1.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S4.Thmthm1.p1.2.m2.2.2.1.1.1.cmml" xref="S4.Thmthm1.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S4.Thmthm1.p1.2.m2.2.2.1.1.2.cmml" xref="S4.Thmthm1.p1.2.m2.2.2.1.1.2">𝑎</ci><cn id="S4.Thmthm1.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S4.Thmthm1.p1.2.m2.2.2.1.1.3">1</cn></apply><apply id="S4.Thmthm1.p1.2.m2.3.3.2.2.cmml" xref="S4.Thmthm1.p1.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S4.Thmthm1.p1.2.m2.3.3.2.2.1.cmml" xref="S4.Thmthm1.p1.2.m2.3.3.2.2">subscript</csymbol><ci id="S4.Thmthm1.p1.2.m2.3.3.2.2.2.cmml" xref="S4.Thmthm1.p1.2.m2.3.3.2.2.2">𝑎</ci><cn id="S4.Thmthm1.p1.2.m2.3.3.2.2.3.cmml" type="integer" xref="S4.Thmthm1.p1.2.m2.3.3.2.2.3">2</cn></apply><apply id="S4.Thmthm1.p1.2.m2.4.4.3.3.cmml" xref="S4.Thmthm1.p1.2.m2.4.4.3.3"><csymbol cd="ambiguous" id="S4.Thmthm1.p1.2.m2.4.4.3.3.1.cmml" xref="S4.Thmthm1.p1.2.m2.4.4.3.3">subscript</csymbol><ci id="S4.Thmthm1.p1.2.m2.4.4.3.3.2.cmml" xref="S4.Thmthm1.p1.2.m2.4.4.3.3.2">𝑎</ci><cn id="S4.Thmthm1.p1.2.m2.4.4.3.3.3.cmml" type="integer" xref="S4.Thmthm1.p1.2.m2.4.4.3.3.3">3</cn></apply><ci id="S4.Thmthm1.p1.2.m2.1.1.cmml" xref="S4.Thmthm1.p1.2.m2.1.1">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm1.p1.2.m2.4c">a_{1},a_{2},a_{3},\ldots</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm1.p1.2.m2.4d">italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , …</annotation></semantics></math> for the elements of the given alphabet <math alttext="\cal A" class="ltx_Math" display="inline" id="S4.Thmthm1.p1.3.m3.1"><semantics id="S4.Thmthm1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm1.p1.3.m3.1.1" xref="S4.Thmthm1.p1.3.m3.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S4.Thmthm1.p1.3.m3.1b"><ci id="S4.Thmthm1.p1.3.m3.1.1.cmml" xref="S4.Thmthm1.p1.3.m3.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm1.p1.3.m3.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm1.p1.3.m3.1d">caligraphic_A</annotation></semantics></math>, and we write <math alttext="a_{k}" class="ltx_Math" display="inline" id="S4.Thmthm1.p1.4.m4.1"><semantics id="S4.Thmthm1.p1.4.m4.1a"><msub id="S4.Thmthm1.p1.4.m4.1.1" xref="S4.Thmthm1.p1.4.m4.1.1.cmml"><mi id="S4.Thmthm1.p1.4.m4.1.1.2" xref="S4.Thmthm1.p1.4.m4.1.1.2.cmml">a</mi><mi id="S4.Thmthm1.p1.4.m4.1.1.3" xref="S4.Thmthm1.p1.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmthm1.p1.4.m4.1b"><apply id="S4.Thmthm1.p1.4.m4.1.1.cmml" xref="S4.Thmthm1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.Thmthm1.p1.4.m4.1.1.1.cmml" xref="S4.Thmthm1.p1.4.m4.1.1">subscript</csymbol><ci id="S4.Thmthm1.p1.4.m4.1.1.2.cmml" xref="S4.Thmthm1.p1.4.m4.1.1.2">𝑎</ci><ci id="S4.Thmthm1.p1.4.m4.1.1.3.cmml" xref="S4.Thmthm1.p1.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm1.p1.4.m4.1c">a_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm1.p1.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="b_{k}" class="ltx_Math" display="inline" id="S4.Thmthm1.p1.5.m5.1"><semantics id="S4.Thmthm1.p1.5.m5.1a"><msub id="S4.Thmthm1.p1.5.m5.1.1" xref="S4.Thmthm1.p1.5.m5.1.1.cmml"><mi id="S4.Thmthm1.p1.5.m5.1.1.2" xref="S4.Thmthm1.p1.5.m5.1.1.2.cmml">b</mi><mi id="S4.Thmthm1.p1.5.m5.1.1.3" xref="S4.Thmthm1.p1.5.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmthm1.p1.5.m5.1b"><apply id="S4.Thmthm1.p1.5.m5.1.1.cmml" xref="S4.Thmthm1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.Thmthm1.p1.5.m5.1.1.1.cmml" xref="S4.Thmthm1.p1.5.m5.1.1">subscript</csymbol><ci id="S4.Thmthm1.p1.5.m5.1.1.2.cmml" xref="S4.Thmthm1.p1.5.m5.1.1.2">𝑏</ci><ci id="S4.Thmthm1.p1.5.m5.1.1.3.cmml" xref="S4.Thmthm1.p1.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm1.p1.5.m5.1c">b_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm1.p1.5.m5.1d">italic_b start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> instead of <math alttext="a(k)" class="ltx_Math" display="inline" id="S4.Thmthm1.p1.6.m6.1"><semantics id="S4.Thmthm1.p1.6.m6.1a"><mrow id="S4.Thmthm1.p1.6.m6.1.2" xref="S4.Thmthm1.p1.6.m6.1.2.cmml"><mi id="S4.Thmthm1.p1.6.m6.1.2.2" xref="S4.Thmthm1.p1.6.m6.1.2.2.cmml">a</mi><mo id="S4.Thmthm1.p1.6.m6.1.2.1" xref="S4.Thmthm1.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S4.Thmthm1.p1.6.m6.1.2.3.2" xref="S4.Thmthm1.p1.6.m6.1.2.cmml"><mo id="S4.Thmthm1.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S4.Thmthm1.p1.6.m6.1.2.cmml">(</mo><mi id="S4.Thmthm1.p1.6.m6.1.1" xref="S4.Thmthm1.p1.6.m6.1.1.cmml">k</mi><mo id="S4.Thmthm1.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S4.Thmthm1.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm1.p1.6.m6.1b"><apply id="S4.Thmthm1.p1.6.m6.1.2.cmml" xref="S4.Thmthm1.p1.6.m6.1.2"><times id="S4.Thmthm1.p1.6.m6.1.2.1.cmml" xref="S4.Thmthm1.p1.6.m6.1.2.1"></times><ci id="S4.Thmthm1.p1.6.m6.1.2.2.cmml" xref="S4.Thmthm1.p1.6.m6.1.2.2">𝑎</ci><ci id="S4.Thmthm1.p1.6.m6.1.1.cmml" xref="S4.Thmthm1.p1.6.m6.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm1.p1.6.m6.1c">a(k)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm1.p1.6.m6.1d">italic_a ( italic_k )</annotation></semantics></math> or <math alttext="b(k)" class="ltx_Math" display="inline" id="S4.Thmthm1.p1.7.m7.1"><semantics id="S4.Thmthm1.p1.7.m7.1a"><mrow id="S4.Thmthm1.p1.7.m7.1.2" xref="S4.Thmthm1.p1.7.m7.1.2.cmml"><mi id="S4.Thmthm1.p1.7.m7.1.2.2" xref="S4.Thmthm1.p1.7.m7.1.2.2.cmml">b</mi><mo id="S4.Thmthm1.p1.7.m7.1.2.1" xref="S4.Thmthm1.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S4.Thmthm1.p1.7.m7.1.2.3.2" xref="S4.Thmthm1.p1.7.m7.1.2.cmml"><mo id="S4.Thmthm1.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S4.Thmthm1.p1.7.m7.1.2.cmml">(</mo><mi id="S4.Thmthm1.p1.7.m7.1.1" xref="S4.Thmthm1.p1.7.m7.1.1.cmml">k</mi><mo id="S4.Thmthm1.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S4.Thmthm1.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm1.p1.7.m7.1b"><apply id="S4.Thmthm1.p1.7.m7.1.2.cmml" xref="S4.Thmthm1.p1.7.m7.1.2"><times id="S4.Thmthm1.p1.7.m7.1.2.1.cmml" xref="S4.Thmthm1.p1.7.m7.1.2.1"></times><ci id="S4.Thmthm1.p1.7.m7.1.2.2.cmml" xref="S4.Thmthm1.p1.7.m7.1.2.2">𝑏</ci><ci id="S4.Thmthm1.p1.7.m7.1.1.cmml" xref="S4.Thmthm1.p1.7.m7.1.1">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm1.p1.7.m7.1c">b(k)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm1.p1.7.m7.1d">italic_b ( italic_k )</annotation></semantics></math> for the letters of a corresponding subdivision alphabet.</p> </div> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.2">For the alphabets <math alttext="\cal A=\{a,b\}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.2"><semantics id="S4.SS1.p3.1.m1.2a"><mrow id="S4.SS1.p3.1.m1.2.3" xref="S4.SS1.p3.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.1.m1.2.3.2" xref="S4.SS1.p3.1.m1.2.3.2.cmml">𝒜</mi><mo id="S4.SS1.p3.1.m1.2.3.1" xref="S4.SS1.p3.1.m1.2.3.1.cmml">=</mo><mrow id="S4.SS1.p3.1.m1.2.3.3.2" xref="S4.SS1.p3.1.m1.2.3.3.1.cmml"><mo id="S4.SS1.p3.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS1.p3.1.m1.2.3.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">𝒶</mi><mo id="S4.SS1.p3.1.m1.2.3.3.2.2" xref="S4.SS1.p3.1.m1.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.1.m1.2.2" xref="S4.SS1.p3.1.m1.2.2.cmml">𝒷</mi><mo id="S4.SS1.p3.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS1.p3.1.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.2b"><apply id="S4.SS1.p3.1.m1.2.3.cmml" xref="S4.SS1.p3.1.m1.2.3"><eq id="S4.SS1.p3.1.m1.2.3.1.cmml" xref="S4.SS1.p3.1.m1.2.3.1"></eq><ci id="S4.SS1.p3.1.m1.2.3.2.cmml" xref="S4.SS1.p3.1.m1.2.3.2">𝒜</ci><set id="S4.SS1.p3.1.m1.2.3.3.1.cmml" xref="S4.SS1.p3.1.m1.2.3.3.2"><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">𝒶</ci><ci id="S4.SS1.p3.1.m1.2.2.cmml" xref="S4.SS1.p3.1.m1.2.2">𝒷</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.2c">\cal A=\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.2d">caligraphic_A = { caligraphic_a , caligraphic_b }</annotation></semantics></math> and <math alttext="\cal B=\{c,d\}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.2"><semantics id="S4.SS1.p3.2.m2.2a"><mrow id="S4.SS1.p3.2.m2.2.3" xref="S4.SS1.p3.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.2.m2.2.3.2" xref="S4.SS1.p3.2.m2.2.3.2.cmml">ℬ</mi><mo id="S4.SS1.p3.2.m2.2.3.1" xref="S4.SS1.p3.2.m2.2.3.1.cmml">=</mo><mrow id="S4.SS1.p3.2.m2.2.3.3.2" xref="S4.SS1.p3.2.m2.2.3.3.1.cmml"><mo id="S4.SS1.p3.2.m2.2.3.3.2.1" stretchy="false" xref="S4.SS1.p3.2.m2.2.3.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml">𝒸</mi><mo id="S4.SS1.p3.2.m2.2.3.3.2.2" xref="S4.SS1.p3.2.m2.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.2.m2.2.2" xref="S4.SS1.p3.2.m2.2.2.cmml">𝒹</mi><mo id="S4.SS1.p3.2.m2.2.3.3.2.3" stretchy="false" xref="S4.SS1.p3.2.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.2b"><apply id="S4.SS1.p3.2.m2.2.3.cmml" xref="S4.SS1.p3.2.m2.2.3"><eq id="S4.SS1.p3.2.m2.2.3.1.cmml" xref="S4.SS1.p3.2.m2.2.3.1"></eq><ci id="S4.SS1.p3.2.m2.2.3.2.cmml" xref="S4.SS1.p3.2.m2.2.3.2">ℬ</ci><set id="S4.SS1.p3.2.m2.2.3.3.1.cmml" xref="S4.SS1.p3.2.m2.2.3.3.2"><ci id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">𝒸</ci><ci id="S4.SS1.p3.2.m2.2.2.cmml" xref="S4.SS1.p3.2.m2.2.2">𝒹</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.2c">\cal B=\{c,d\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.2d">caligraphic_B = { caligraphic_c , caligraphic_d }</annotation></semantics></math> consider the morphism given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma:\cal A^{*}\to\cal B^{*}\,,\,\,\,a\mapsto cdc,\,b\mapsto dcc\,." class="ltx_Math" display="block" id="S4.Ex1.m1.1"><semantics id="S4.Ex1.m1.1a"><mrow id="S4.Ex1.m1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.cmml"><mrow id="S4.Ex1.m1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.cmml"><mi id="S4.Ex1.m1.1.1.1.1.4" xref="S4.Ex1.m1.1.1.1.1.4.cmml">σ</mi><mo id="S4.Ex1.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S4.Ex1.m1.1.1.1.1.3.cmml">:</mo><mrow id="S4.Ex1.m1.1.1.1.1.2.2" xref="S4.Ex1.m1.1.1.1.1.2.3.cmml"><mrow id="S4.Ex1.m1.1.1.1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.1.cmml"><msup id="S4.Ex1.m1.1.1.1.1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.cmml">𝒜</mi><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.2.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.3.cmml">∗</mo></msup><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.1" stretchy="false" xref="S4.Ex1.m1.1.1.1.1.1.1.1.1.cmml">→</mo><msup id="S4.Ex1.m1.1.1.1.1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex1.m1.1.1.1.1.1.1.1.3.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.2.cmml">ℬ</mi><mo id="S4.Ex1.m1.1.1.1.1.1.1.1.3.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.3.cmml">∗</mo></msup></mrow><mo id="S4.Ex1.m1.1.1.1.1.2.2.3" rspace="0.667em" xref="S4.Ex1.m1.1.1.1.1.2.3a.cmml">,</mo><mrow id="S4.Ex1.m1.1.1.1.1.2.2.2.2" xref="S4.Ex1.m1.1.1.1.1.2.2.2.3.cmml"><mrow id="S4.Ex1.m1.1.1.1.1.2.2.2.1.1" xref="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.2" xref="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.2.cmml">𝒶</mi><mo 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id="S4.Ex1.m1.1b"><apply id="S4.Ex1.m1.1.1.1.1.cmml" xref="S4.Ex1.m1.1.1.1"><ci id="S4.Ex1.m1.1.1.1.1.3.cmml" xref="S4.Ex1.m1.1.1.1.1.3">:</ci><ci id="S4.Ex1.m1.1.1.1.1.4.cmml" xref="S4.Ex1.m1.1.1.1.1.4">𝜎</ci><apply id="S4.Ex1.m1.1.1.1.1.2.3.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.Ex1.m1.1.1.1.1.2.3a.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.3">formulae-sequence</csymbol><apply id="S4.Ex1.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex1.m1.1.1.1.1.1.1.1"><ci id="S4.Ex1.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex1.m1.1.1.1.1.1.1.1.1">→</ci><apply id="S4.Ex1.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex1.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2">superscript</csymbol><ci id="S4.Ex1.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.2">𝒜</ci><times id="S4.Ex1.m1.1.1.1.1.1.1.1.2.3.cmml" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.3"></times></apply><apply id="S4.Ex1.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3"><csymbol 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id="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.3.2.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.3.2">𝒸</ci><ci id="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.3.3.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.3.3">𝒹</ci><ci id="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.3.4.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.1.1.3.4">𝒸</ci></apply></apply><apply id="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.2.2"><csymbol cd="latexml" id="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.1.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.1">maps-to</csymbol><ci id="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.2.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.2">𝒷</ci><apply id="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3"><times id="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.1.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.1"></times><ci id="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.2.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.2">𝒹</ci><ci id="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.3.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.3">𝒸</ci><ci id="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.4.cmml" xref="S4.Ex1.m1.1.1.1.1.2.2.2.2.2.3.4">𝒸</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}\,,\,\,\,a\mapsto cdc,\,b\mapsto dcc\,.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , caligraphic_a ↦ caligraphic_c caligraphic_d caligraphic_c , caligraphic_b ↦ caligraphic_d caligraphic_c caligraphic_c .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.5">We derive <math alttext="\ell_{\sigma}(a)=\ell_{\sigma}(b)=3" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m1.2"><semantics id="S4.SS1.p3.3.m1.2a"><mrow id="S4.SS1.p3.3.m1.2.3" xref="S4.SS1.p3.3.m1.2.3.cmml"><mrow id="S4.SS1.p3.3.m1.2.3.2" xref="S4.SS1.p3.3.m1.2.3.2.cmml"><msub id="S4.SS1.p3.3.m1.2.3.2.2" xref="S4.SS1.p3.3.m1.2.3.2.2.cmml"><mi id="S4.SS1.p3.3.m1.2.3.2.2.2" mathvariant="normal" xref="S4.SS1.p3.3.m1.2.3.2.2.2.cmml">ℓ</mi><mi id="S4.SS1.p3.3.m1.2.3.2.2.3" xref="S4.SS1.p3.3.m1.2.3.2.2.3.cmml">σ</mi></msub><mo id="S4.SS1.p3.3.m1.2.3.2.1" xref="S4.SS1.p3.3.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS1.p3.3.m1.2.3.2.3.2" xref="S4.SS1.p3.3.m1.2.3.2.cmml"><mo id="S4.SS1.p3.3.m1.2.3.2.3.2.1" stretchy="false" xref="S4.SS1.p3.3.m1.2.3.2.cmml">(</mo><mi id="S4.SS1.p3.3.m1.1.1" xref="S4.SS1.p3.3.m1.1.1.cmml">a</mi><mo id="S4.SS1.p3.3.m1.2.3.2.3.2.2" stretchy="false" xref="S4.SS1.p3.3.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.3.m1.2.3.3" xref="S4.SS1.p3.3.m1.2.3.3.cmml">=</mo><mrow id="S4.SS1.p3.3.m1.2.3.4" xref="S4.SS1.p3.3.m1.2.3.4.cmml"><msub id="S4.SS1.p3.3.m1.2.3.4.2" xref="S4.SS1.p3.3.m1.2.3.4.2.cmml"><mi id="S4.SS1.p3.3.m1.2.3.4.2.2" mathvariant="normal" xref="S4.SS1.p3.3.m1.2.3.4.2.2.cmml">ℓ</mi><mi id="S4.SS1.p3.3.m1.2.3.4.2.3" xref="S4.SS1.p3.3.m1.2.3.4.2.3.cmml">σ</mi></msub><mo id="S4.SS1.p3.3.m1.2.3.4.1" xref="S4.SS1.p3.3.m1.2.3.4.1.cmml">⁢</mo><mrow id="S4.SS1.p3.3.m1.2.3.4.3.2" xref="S4.SS1.p3.3.m1.2.3.4.cmml"><mo id="S4.SS1.p3.3.m1.2.3.4.3.2.1" stretchy="false" xref="S4.SS1.p3.3.m1.2.3.4.cmml">(</mo><mi id="S4.SS1.p3.3.m1.2.2" xref="S4.SS1.p3.3.m1.2.2.cmml">b</mi><mo id="S4.SS1.p3.3.m1.2.3.4.3.2.2" stretchy="false" xref="S4.SS1.p3.3.m1.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p3.3.m1.2.3.5" xref="S4.SS1.p3.3.m1.2.3.5.cmml">=</mo><mn id="S4.SS1.p3.3.m1.2.3.6" xref="S4.SS1.p3.3.m1.2.3.6.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m1.2b"><apply id="S4.SS1.p3.3.m1.2.3.cmml" xref="S4.SS1.p3.3.m1.2.3"><and id="S4.SS1.p3.3.m1.2.3a.cmml" xref="S4.SS1.p3.3.m1.2.3"></and><apply id="S4.SS1.p3.3.m1.2.3b.cmml" xref="S4.SS1.p3.3.m1.2.3"><eq id="S4.SS1.p3.3.m1.2.3.3.cmml" xref="S4.SS1.p3.3.m1.2.3.3"></eq><apply id="S4.SS1.p3.3.m1.2.3.2.cmml" xref="S4.SS1.p3.3.m1.2.3.2"><times id="S4.SS1.p3.3.m1.2.3.2.1.cmml" xref="S4.SS1.p3.3.m1.2.3.2.1"></times><apply id="S4.SS1.p3.3.m1.2.3.2.2.cmml" xref="S4.SS1.p3.3.m1.2.3.2.2"><csymbol cd="ambiguous" id="S4.SS1.p3.3.m1.2.3.2.2.1.cmml" xref="S4.SS1.p3.3.m1.2.3.2.2">subscript</csymbol><ci id="S4.SS1.p3.3.m1.2.3.2.2.2.cmml" xref="S4.SS1.p3.3.m1.2.3.2.2.2">ℓ</ci><ci id="S4.SS1.p3.3.m1.2.3.2.2.3.cmml" xref="S4.SS1.p3.3.m1.2.3.2.2.3">𝜎</ci></apply><ci id="S4.SS1.p3.3.m1.1.1.cmml" xref="S4.SS1.p3.3.m1.1.1">𝑎</ci></apply><apply id="S4.SS1.p3.3.m1.2.3.4.cmml" xref="S4.SS1.p3.3.m1.2.3.4"><times id="S4.SS1.p3.3.m1.2.3.4.1.cmml" xref="S4.SS1.p3.3.m1.2.3.4.1"></times><apply id="S4.SS1.p3.3.m1.2.3.4.2.cmml" xref="S4.SS1.p3.3.m1.2.3.4.2"><csymbol cd="ambiguous" id="S4.SS1.p3.3.m1.2.3.4.2.1.cmml" xref="S4.SS1.p3.3.m1.2.3.4.2">subscript</csymbol><ci id="S4.SS1.p3.3.m1.2.3.4.2.2.cmml" xref="S4.SS1.p3.3.m1.2.3.4.2.2">ℓ</ci><ci id="S4.SS1.p3.3.m1.2.3.4.2.3.cmml" xref="S4.SS1.p3.3.m1.2.3.4.2.3">𝜎</ci></apply><ci id="S4.SS1.p3.3.m1.2.2.cmml" xref="S4.SS1.p3.3.m1.2.2">𝑏</ci></apply></apply><apply id="S4.SS1.p3.3.m1.2.3c.cmml" xref="S4.SS1.p3.3.m1.2.3"><eq id="S4.SS1.p3.3.m1.2.3.5.cmml" xref="S4.SS1.p3.3.m1.2.3.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S4.SS1.p3.3.m1.2.3.4.cmml" id="S4.SS1.p3.3.m1.2.3d.cmml" xref="S4.SS1.p3.3.m1.2.3"></share><cn id="S4.SS1.p3.3.m1.2.3.6.cmml" type="integer" xref="S4.SS1.p3.3.m1.2.3.6">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m1.2c">\ell_{\sigma}(a)=\ell_{\sigma}(b)=3</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m1.2d">roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a ) = roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b ) = 3</annotation></semantics></math> and <math alttext="\cal A_{\sigma}=\{a_{1},a_{2},a_{3},b_{1},b_{2},b_{3}\}" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m2.6"><semantics id="S4.SS1.p3.4.m2.6a"><mrow id="S4.SS1.p3.4.m2.6.6" xref="S4.SS1.p3.4.m2.6.6.cmml"><msub id="S4.SS1.p3.4.m2.6.6.8" xref="S4.SS1.p3.4.m2.6.6.8.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.6.6.8.2" xref="S4.SS1.p3.4.m2.6.6.8.2.cmml">𝒜</mi><mi id="S4.SS1.p3.4.m2.6.6.8.3" xref="S4.SS1.p3.4.m2.6.6.8.3.cmml">σ</mi></msub><mo id="S4.SS1.p3.4.m2.6.6.7" xref="S4.SS1.p3.4.m2.6.6.7.cmml">=</mo><mrow id="S4.SS1.p3.4.m2.6.6.6.6" xref="S4.SS1.p3.4.m2.6.6.6.7.cmml"><mo id="S4.SS1.p3.4.m2.6.6.6.6.7" stretchy="false" xref="S4.SS1.p3.4.m2.6.6.6.7.cmml">{</mo><msub id="S4.SS1.p3.4.m2.1.1.1.1.1" xref="S4.SS1.p3.4.m2.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.1.1.1.1.1.2" xref="S4.SS1.p3.4.m2.1.1.1.1.1.2.cmml">𝒶</mi><mn class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.1.1.1.1.1.3" mathvariant="script" xref="S4.SS1.p3.4.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p3.4.m2.6.6.6.6.8" xref="S4.SS1.p3.4.m2.6.6.6.7.cmml">,</mo><msub id="S4.SS1.p3.4.m2.2.2.2.2.2" xref="S4.SS1.p3.4.m2.2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.2.2.2.2.2.2" xref="S4.SS1.p3.4.m2.2.2.2.2.2.2.cmml">𝒶</mi><mn class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.2.2.2.2.2.3" mathvariant="script" xref="S4.SS1.p3.4.m2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.SS1.p3.4.m2.6.6.6.6.9" xref="S4.SS1.p3.4.m2.6.6.6.7.cmml">,</mo><msub id="S4.SS1.p3.4.m2.3.3.3.3.3" xref="S4.SS1.p3.4.m2.3.3.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.3.3.3.3.3.2" xref="S4.SS1.p3.4.m2.3.3.3.3.3.2.cmml">𝒶</mi><mn class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.3.3.3.3.3.3" mathvariant="script" xref="S4.SS1.p3.4.m2.3.3.3.3.3.3.cmml">3</mn></msub><mo id="S4.SS1.p3.4.m2.6.6.6.6.10" xref="S4.SS1.p3.4.m2.6.6.6.7.cmml">,</mo><msub id="S4.SS1.p3.4.m2.4.4.4.4.4" xref="S4.SS1.p3.4.m2.4.4.4.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.4.4.4.4.4.2" xref="S4.SS1.p3.4.m2.4.4.4.4.4.2.cmml">𝒷</mi><mn class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.4.4.4.4.4.3" mathvariant="script" xref="S4.SS1.p3.4.m2.4.4.4.4.4.3.cmml">1</mn></msub><mo id="S4.SS1.p3.4.m2.6.6.6.6.11" xref="S4.SS1.p3.4.m2.6.6.6.7.cmml">,</mo><msub id="S4.SS1.p3.4.m2.5.5.5.5.5" xref="S4.SS1.p3.4.m2.5.5.5.5.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.5.5.5.5.5.2" xref="S4.SS1.p3.4.m2.5.5.5.5.5.2.cmml">𝒷</mi><mn class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.5.5.5.5.5.3" mathvariant="script" xref="S4.SS1.p3.4.m2.5.5.5.5.5.3.cmml">2</mn></msub><mo id="S4.SS1.p3.4.m2.6.6.6.6.12" xref="S4.SS1.p3.4.m2.6.6.6.7.cmml">,</mo><msub id="S4.SS1.p3.4.m2.6.6.6.6.6" xref="S4.SS1.p3.4.m2.6.6.6.6.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.6.6.6.6.6.2" xref="S4.SS1.p3.4.m2.6.6.6.6.6.2.cmml">𝒷</mi><mn class="ltx_font_mathcaligraphic" id="S4.SS1.p3.4.m2.6.6.6.6.6.3" mathvariant="script" xref="S4.SS1.p3.4.m2.6.6.6.6.6.3.cmml">3</mn></msub><mo id="S4.SS1.p3.4.m2.6.6.6.6.13" stretchy="false" xref="S4.SS1.p3.4.m2.6.6.6.7.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m2.6b"><apply id="S4.SS1.p3.4.m2.6.6.cmml" xref="S4.SS1.p3.4.m2.6.6"><eq id="S4.SS1.p3.4.m2.6.6.7.cmml" xref="S4.SS1.p3.4.m2.6.6.7"></eq><apply id="S4.SS1.p3.4.m2.6.6.8.cmml" xref="S4.SS1.p3.4.m2.6.6.8"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m2.6.6.8.1.cmml" xref="S4.SS1.p3.4.m2.6.6.8">subscript</csymbol><ci id="S4.SS1.p3.4.m2.6.6.8.2.cmml" xref="S4.SS1.p3.4.m2.6.6.8.2">𝒜</ci><ci id="S4.SS1.p3.4.m2.6.6.8.3.cmml" xref="S4.SS1.p3.4.m2.6.6.8.3">𝜎</ci></apply><set id="S4.SS1.p3.4.m2.6.6.6.7.cmml" xref="S4.SS1.p3.4.m2.6.6.6.6"><apply id="S4.SS1.p3.4.m2.1.1.1.1.1.cmml" xref="S4.SS1.p3.4.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m2.1.1.1.1.1.1.cmml" xref="S4.SS1.p3.4.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p3.4.m2.1.1.1.1.1.2.cmml" xref="S4.SS1.p3.4.m2.1.1.1.1.1.2">𝒶</ci><cn id="S4.SS1.p3.4.m2.1.1.1.1.1.3.cmml" type="integer" 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id="S4.SS1.p3.4.m2.4.4.4.4.4.2.cmml" xref="S4.SS1.p3.4.m2.4.4.4.4.4.2">𝒷</ci><cn id="S4.SS1.p3.4.m2.4.4.4.4.4.3.cmml" type="integer" xref="S4.SS1.p3.4.m2.4.4.4.4.4.3">1</cn></apply><apply id="S4.SS1.p3.4.m2.5.5.5.5.5.cmml" xref="S4.SS1.p3.4.m2.5.5.5.5.5"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m2.5.5.5.5.5.1.cmml" xref="S4.SS1.p3.4.m2.5.5.5.5.5">subscript</csymbol><ci id="S4.SS1.p3.4.m2.5.5.5.5.5.2.cmml" xref="S4.SS1.p3.4.m2.5.5.5.5.5.2">𝒷</ci><cn id="S4.SS1.p3.4.m2.5.5.5.5.5.3.cmml" type="integer" xref="S4.SS1.p3.4.m2.5.5.5.5.5.3">2</cn></apply><apply id="S4.SS1.p3.4.m2.6.6.6.6.6.cmml" xref="S4.SS1.p3.4.m2.6.6.6.6.6"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m2.6.6.6.6.6.1.cmml" xref="S4.SS1.p3.4.m2.6.6.6.6.6">subscript</csymbol><ci id="S4.SS1.p3.4.m2.6.6.6.6.6.2.cmml" xref="S4.SS1.p3.4.m2.6.6.6.6.6.2">𝒷</ci><cn id="S4.SS1.p3.4.m2.6.6.6.6.6.3.cmml" type="integer" xref="S4.SS1.p3.4.m2.6.6.6.6.6.3">3</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m2.6c">\cal A_{\sigma}=\{a_{1},a_{2},a_{3},b_{1},b_{2},b_{3}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m2.6d">caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = { caligraphic_a start_POSTSUBSCRIPT caligraphic_1 end_POSTSUBSCRIPT , caligraphic_a start_POSTSUBSCRIPT caligraphic_2 end_POSTSUBSCRIPT , caligraphic_a start_POSTSUBSCRIPT caligraphic_3 end_POSTSUBSCRIPT , caligraphic_b start_POSTSUBSCRIPT caligraphic_1 end_POSTSUBSCRIPT , caligraphic_b start_POSTSUBSCRIPT caligraphic_2 end_POSTSUBSCRIPT , caligraphic_b start_POSTSUBSCRIPT caligraphic_3 end_POSTSUBSCRIPT }</annotation></semantics></math> as well as the corresponding subdivision morphisms <math alttext="\pi_{\sigma}:\cal A^{*}\to\cal A_{\sigma}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m3.1"><semantics id="S4.SS1.p3.5.m3.1a"><mrow id="S4.SS1.p3.5.m3.1.1" xref="S4.SS1.p3.5.m3.1.1.cmml"><msub id="S4.SS1.p3.5.m3.1.1.2" xref="S4.SS1.p3.5.m3.1.1.2.cmml"><mi id="S4.SS1.p3.5.m3.1.1.2.2" xref="S4.SS1.p3.5.m3.1.1.2.2.cmml">π</mi><mi id="S4.SS1.p3.5.m3.1.1.2.3" xref="S4.SS1.p3.5.m3.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS1.p3.5.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p3.5.m3.1.1.1.cmml">:</mo><mrow id="S4.SS1.p3.5.m3.1.1.3" xref="S4.SS1.p3.5.m3.1.1.3.cmml"><msup id="S4.SS1.p3.5.m3.1.1.3.2" xref="S4.SS1.p3.5.m3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.5.m3.1.1.3.2.2" xref="S4.SS1.p3.5.m3.1.1.3.2.2.cmml">𝒜</mi><mo id="S4.SS1.p3.5.m3.1.1.3.2.3" xref="S4.SS1.p3.5.m3.1.1.3.2.3.cmml">∗</mo></msup><mo id="S4.SS1.p3.5.m3.1.1.3.1" stretchy="false" xref="S4.SS1.p3.5.m3.1.1.3.1.cmml">→</mo><msubsup id="S4.SS1.p3.5.m3.1.1.3.3" xref="S4.SS1.p3.5.m3.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.5.m3.1.1.3.3.2.2" xref="S4.SS1.p3.5.m3.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S4.SS1.p3.5.m3.1.1.3.3.2.3" xref="S4.SS1.p3.5.m3.1.1.3.3.2.3.cmml">σ</mi><mo id="S4.SS1.p3.5.m3.1.1.3.3.3" xref="S4.SS1.p3.5.m3.1.1.3.3.3.cmml">∗</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m3.1b"><apply id="S4.SS1.p3.5.m3.1.1.cmml" xref="S4.SS1.p3.5.m3.1.1"><ci id="S4.SS1.p3.5.m3.1.1.1.cmml" xref="S4.SS1.p3.5.m3.1.1.1">:</ci><apply id="S4.SS1.p3.5.m3.1.1.2.cmml" xref="S4.SS1.p3.5.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m3.1.1.2.1.cmml" xref="S4.SS1.p3.5.m3.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.5.m3.1.1.2.2.cmml" xref="S4.SS1.p3.5.m3.1.1.2.2">𝜋</ci><ci id="S4.SS1.p3.5.m3.1.1.2.3.cmml" xref="S4.SS1.p3.5.m3.1.1.2.3">𝜎</ci></apply><apply id="S4.SS1.p3.5.m3.1.1.3.cmml" xref="S4.SS1.p3.5.m3.1.1.3"><ci id="S4.SS1.p3.5.m3.1.1.3.1.cmml" xref="S4.SS1.p3.5.m3.1.1.3.1">→</ci><apply id="S4.SS1.p3.5.m3.1.1.3.2.cmml" xref="S4.SS1.p3.5.m3.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m3.1.1.3.2.1.cmml" xref="S4.SS1.p3.5.m3.1.1.3.2">superscript</csymbol><ci id="S4.SS1.p3.5.m3.1.1.3.2.2.cmml" xref="S4.SS1.p3.5.m3.1.1.3.2.2">𝒜</ci><times id="S4.SS1.p3.5.m3.1.1.3.2.3.cmml" xref="S4.SS1.p3.5.m3.1.1.3.2.3"></times></apply><apply id="S4.SS1.p3.5.m3.1.1.3.3.cmml" xref="S4.SS1.p3.5.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m3.1.1.3.3.1.cmml" xref="S4.SS1.p3.5.m3.1.1.3.3">superscript</csymbol><apply id="S4.SS1.p3.5.m3.1.1.3.3.2.cmml" xref="S4.SS1.p3.5.m3.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m3.1.1.3.3.2.1.cmml" xref="S4.SS1.p3.5.m3.1.1.3.3">subscript</csymbol><ci id="S4.SS1.p3.5.m3.1.1.3.3.2.2.cmml" xref="S4.SS1.p3.5.m3.1.1.3.3.2.2">𝒜</ci><ci id="S4.SS1.p3.5.m3.1.1.3.3.2.3.cmml" xref="S4.SS1.p3.5.m3.1.1.3.3.2.3">𝜎</ci></apply><times id="S4.SS1.p3.5.m3.1.1.3.3.3.cmml" xref="S4.SS1.p3.5.m3.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m3.1c">\pi_{\sigma}:\cal A^{*}\to\cal A_{\sigma}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m3.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\pi_{\sigma}(a)=a_{1}a_{2}a_{3},\,\pi_{\sigma}(b)=b_{1}b_{2}b_{3}\,." class="ltx_Math" display="block" id="S4.Ex2.m1.3"><semantics id="S4.Ex2.m1.3a"><mrow id="S4.Ex2.m1.3.3.1"><mrow id="S4.Ex2.m1.3.3.1.1.2" xref="S4.Ex2.m1.3.3.1.1.3.cmml"><mrow id="S4.Ex2.m1.3.3.1.1.1.1" xref="S4.Ex2.m1.3.3.1.1.1.1.cmml"><mrow id="S4.Ex2.m1.3.3.1.1.1.1.2" xref="S4.Ex2.m1.3.3.1.1.1.1.2.cmml"><msub id="S4.Ex2.m1.3.3.1.1.1.1.2.2" xref="S4.Ex2.m1.3.3.1.1.1.1.2.2.cmml"><mi id="S4.Ex2.m1.3.3.1.1.1.1.2.2.2" xref="S4.Ex2.m1.3.3.1.1.1.1.2.2.2.cmml">π</mi><mi id="S4.Ex2.m1.3.3.1.1.1.1.2.2.3" xref="S4.Ex2.m1.3.3.1.1.1.1.2.2.3.cmml">σ</mi></msub><mo id="S4.Ex2.m1.3.3.1.1.1.1.2.1" xref="S4.Ex2.m1.3.3.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex2.m1.3.3.1.1.1.1.2.3.2" xref="S4.Ex2.m1.3.3.1.1.1.1.2.cmml"><mo id="S4.Ex2.m1.3.3.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.Ex2.m1.3.3.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex2.m1.1.1" xref="S4.Ex2.m1.1.1.cmml">a</mi><mo id="S4.Ex2.m1.3.3.1.1.1.1.2.3.2.2" stretchy="false" xref="S4.Ex2.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex2.m1.3.3.1.1.1.1.1" xref="S4.Ex2.m1.3.3.1.1.1.1.1.cmml">=</mo><mrow id="S4.Ex2.m1.3.3.1.1.1.1.3" xref="S4.Ex2.m1.3.3.1.1.1.1.3.cmml"><msub id="S4.Ex2.m1.3.3.1.1.1.1.3.2" xref="S4.Ex2.m1.3.3.1.1.1.1.3.2.cmml"><mi id="S4.Ex2.m1.3.3.1.1.1.1.3.2.2" xref="S4.Ex2.m1.3.3.1.1.1.1.3.2.2.cmml">a</mi><mn id="S4.Ex2.m1.3.3.1.1.1.1.3.2.3" xref="S4.Ex2.m1.3.3.1.1.1.1.3.2.3.cmml">1</mn></msub><mo id="S4.Ex2.m1.3.3.1.1.1.1.3.1" xref="S4.Ex2.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><msub id="S4.Ex2.m1.3.3.1.1.1.1.3.3" 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xref="S4.Ex2.m1.3.3.1.1.2.2.2.2.3.cmml">σ</mi></msub><mo id="S4.Ex2.m1.3.3.1.1.2.2.2.1" xref="S4.Ex2.m1.3.3.1.1.2.2.2.1.cmml">⁢</mo><mrow id="S4.Ex2.m1.3.3.1.1.2.2.2.3.2" xref="S4.Ex2.m1.3.3.1.1.2.2.2.cmml"><mo id="S4.Ex2.m1.3.3.1.1.2.2.2.3.2.1" stretchy="false" xref="S4.Ex2.m1.3.3.1.1.2.2.2.cmml">(</mo><mi id="S4.Ex2.m1.2.2" xref="S4.Ex2.m1.2.2.cmml">b</mi><mo id="S4.Ex2.m1.3.3.1.1.2.2.2.3.2.2" stretchy="false" xref="S4.Ex2.m1.3.3.1.1.2.2.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex2.m1.3.3.1.1.2.2.1" xref="S4.Ex2.m1.3.3.1.1.2.2.1.cmml">=</mo><mrow id="S4.Ex2.m1.3.3.1.1.2.2.3" xref="S4.Ex2.m1.3.3.1.1.2.2.3.cmml"><msub id="S4.Ex2.m1.3.3.1.1.2.2.3.2" xref="S4.Ex2.m1.3.3.1.1.2.2.3.2.cmml"><mi id="S4.Ex2.m1.3.3.1.1.2.2.3.2.2" xref="S4.Ex2.m1.3.3.1.1.2.2.3.2.2.cmml">b</mi><mn id="S4.Ex2.m1.3.3.1.1.2.2.3.2.3" xref="S4.Ex2.m1.3.3.1.1.2.2.3.2.3.cmml">1</mn></msub><mo id="S4.Ex2.m1.3.3.1.1.2.2.3.1" xref="S4.Ex2.m1.3.3.1.1.2.2.3.1.cmml">⁢</mo><msub id="S4.Ex2.m1.3.3.1.1.2.2.3.3" 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xref="S4.Ex2.m1.3.3.1.1.2.2.3.2">subscript</csymbol><ci id="S4.Ex2.m1.3.3.1.1.2.2.3.2.2.cmml" xref="S4.Ex2.m1.3.3.1.1.2.2.3.2.2">𝑏</ci><cn id="S4.Ex2.m1.3.3.1.1.2.2.3.2.3.cmml" type="integer" xref="S4.Ex2.m1.3.3.1.1.2.2.3.2.3">1</cn></apply><apply id="S4.Ex2.m1.3.3.1.1.2.2.3.3.cmml" xref="S4.Ex2.m1.3.3.1.1.2.2.3.3"><csymbol cd="ambiguous" id="S4.Ex2.m1.3.3.1.1.2.2.3.3.1.cmml" xref="S4.Ex2.m1.3.3.1.1.2.2.3.3">subscript</csymbol><ci id="S4.Ex2.m1.3.3.1.1.2.2.3.3.2.cmml" xref="S4.Ex2.m1.3.3.1.1.2.2.3.3.2">𝑏</ci><cn id="S4.Ex2.m1.3.3.1.1.2.2.3.3.3.cmml" type="integer" xref="S4.Ex2.m1.3.3.1.1.2.2.3.3.3">2</cn></apply><apply id="S4.Ex2.m1.3.3.1.1.2.2.3.4.cmml" xref="S4.Ex2.m1.3.3.1.1.2.2.3.4"><csymbol cd="ambiguous" id="S4.Ex2.m1.3.3.1.1.2.2.3.4.1.cmml" xref="S4.Ex2.m1.3.3.1.1.2.2.3.4">subscript</csymbol><ci id="S4.Ex2.m1.3.3.1.1.2.2.3.4.2.cmml" xref="S4.Ex2.m1.3.3.1.1.2.2.3.4.2">𝑏</ci><cn id="S4.Ex2.m1.3.3.1.1.2.2.3.4.3.cmml" type="integer" xref="S4.Ex2.m1.3.3.1.1.2.2.3.4.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m1.3c">\pi_{\sigma}(a)=a_{1}a_{2}a_{3},\,\pi_{\sigma}(b)=b_{1}b_{2}b_{3}\,.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex2.m1.3d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a ) = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b ) = italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.6">Similarly, the corresponding letter-to-letter morphism <math alttext="\alpha_{\sigma}:\cal A_{\sigma}^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m1.1"><semantics id="S4.SS1.p3.6.m1.1a"><mrow id="S4.SS1.p3.6.m1.1.1" xref="S4.SS1.p3.6.m1.1.1.cmml"><msub id="S4.SS1.p3.6.m1.1.1.2" xref="S4.SS1.p3.6.m1.1.1.2.cmml"><mi id="S4.SS1.p3.6.m1.1.1.2.2" xref="S4.SS1.p3.6.m1.1.1.2.2.cmml">α</mi><mi id="S4.SS1.p3.6.m1.1.1.2.3" xref="S4.SS1.p3.6.m1.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS1.p3.6.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p3.6.m1.1.1.1.cmml">:</mo><mrow id="S4.SS1.p3.6.m1.1.1.3" xref="S4.SS1.p3.6.m1.1.1.3.cmml"><msubsup id="S4.SS1.p3.6.m1.1.1.3.2" xref="S4.SS1.p3.6.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.6.m1.1.1.3.2.2.2" xref="S4.SS1.p3.6.m1.1.1.3.2.2.2.cmml">𝒜</mi><mi id="S4.SS1.p3.6.m1.1.1.3.2.2.3" xref="S4.SS1.p3.6.m1.1.1.3.2.2.3.cmml">σ</mi><mo id="S4.SS1.p3.6.m1.1.1.3.2.3" xref="S4.SS1.p3.6.m1.1.1.3.2.3.cmml">∗</mo></msubsup><mo id="S4.SS1.p3.6.m1.1.1.3.1" stretchy="false" xref="S4.SS1.p3.6.m1.1.1.3.1.cmml">→</mo><msup id="S4.SS1.p3.6.m1.1.1.3.3" xref="S4.SS1.p3.6.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p3.6.m1.1.1.3.3.2" xref="S4.SS1.p3.6.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S4.SS1.p3.6.m1.1.1.3.3.3" xref="S4.SS1.p3.6.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m1.1b"><apply id="S4.SS1.p3.6.m1.1.1.cmml" xref="S4.SS1.p3.6.m1.1.1"><ci id="S4.SS1.p3.6.m1.1.1.1.cmml" xref="S4.SS1.p3.6.m1.1.1.1">:</ci><apply id="S4.SS1.p3.6.m1.1.1.2.cmml" xref="S4.SS1.p3.6.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m1.1.1.2.1.cmml" xref="S4.SS1.p3.6.m1.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.6.m1.1.1.2.2.cmml" xref="S4.SS1.p3.6.m1.1.1.2.2">𝛼</ci><ci id="S4.SS1.p3.6.m1.1.1.2.3.cmml" xref="S4.SS1.p3.6.m1.1.1.2.3">𝜎</ci></apply><apply id="S4.SS1.p3.6.m1.1.1.3.cmml" xref="S4.SS1.p3.6.m1.1.1.3"><ci id="S4.SS1.p3.6.m1.1.1.3.1.cmml" xref="S4.SS1.p3.6.m1.1.1.3.1">→</ci><apply id="S4.SS1.p3.6.m1.1.1.3.2.cmml" xref="S4.SS1.p3.6.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m1.1.1.3.2.1.cmml" xref="S4.SS1.p3.6.m1.1.1.3.2">superscript</csymbol><apply id="S4.SS1.p3.6.m1.1.1.3.2.2.cmml" xref="S4.SS1.p3.6.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m1.1.1.3.2.2.1.cmml" xref="S4.SS1.p3.6.m1.1.1.3.2">subscript</csymbol><ci id="S4.SS1.p3.6.m1.1.1.3.2.2.2.cmml" xref="S4.SS1.p3.6.m1.1.1.3.2.2.2">𝒜</ci><ci id="S4.SS1.p3.6.m1.1.1.3.2.2.3.cmml" xref="S4.SS1.p3.6.m1.1.1.3.2.2.3">𝜎</ci></apply><times id="S4.SS1.p3.6.m1.1.1.3.2.3.cmml" xref="S4.SS1.p3.6.m1.1.1.3.2.3"></times></apply><apply id="S4.SS1.p3.6.m1.1.1.3.3.cmml" xref="S4.SS1.p3.6.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.6.m1.1.1.3.3.1.cmml" xref="S4.SS1.p3.6.m1.1.1.3.3">superscript</csymbol><ci id="S4.SS1.p3.6.m1.1.1.3.3.2.cmml" xref="S4.SS1.p3.6.m1.1.1.3.3.2">ℬ</ci><times id="S4.SS1.p3.6.m1.1.1.3.3.3.cmml" xref="S4.SS1.p3.6.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m1.1c">\alpha_{\sigma}:\cal A_{\sigma}^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m1.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\alpha_{\sigma}(a_{1})=\alpha_{\sigma}(a_{3})=\alpha_{\sigma}(b_{2})=\alpha_{% \sigma}(b_{3})=c,\,\,\alpha_{\sigma}(a_{2})=\alpha_{\sigma}(b_{1})=d\,." class="ltx_Math" display="block" id="S4.Ex3.m1.1"><semantics id="S4.Ex3.m1.1a"><mrow id="S4.Ex3.m1.1.1.1"><mrow id="S4.Ex3.m1.1.1.1.1.2" 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stretchy="false" xref="S4.Ex3.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex3.m1.1.1.1.1.1.1.6" xref="S4.Ex3.m1.1.1.1.1.1.1.6.cmml">=</mo><mrow id="S4.Ex3.m1.1.1.1.1.1.1.2" xref="S4.Ex3.m1.1.1.1.1.1.1.2.cmml"><msub id="S4.Ex3.m1.1.1.1.1.1.1.2.3" xref="S4.Ex3.m1.1.1.1.1.1.1.2.3.cmml"><mi id="S4.Ex3.m1.1.1.1.1.1.1.2.3.2" xref="S4.Ex3.m1.1.1.1.1.1.1.2.3.2.cmml">α</mi><mi id="S4.Ex3.m1.1.1.1.1.1.1.2.3.3" xref="S4.Ex3.m1.1.1.1.1.1.1.2.3.3.cmml">σ</mi></msub><mo id="S4.Ex3.m1.1.1.1.1.1.1.2.2" xref="S4.Ex3.m1.1.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex3.m1.1.1.1.1.1.1.2.1.1" xref="S4.Ex3.m1.1.1.1.1.1.1.2.1.1.1.cmml"><mo id="S4.Ex3.m1.1.1.1.1.1.1.2.1.1.2" stretchy="false" xref="S4.Ex3.m1.1.1.1.1.1.1.2.1.1.1.cmml">(</mo><msub id="S4.Ex3.m1.1.1.1.1.1.1.2.1.1.1" xref="S4.Ex3.m1.1.1.1.1.1.1.2.1.1.1.cmml"><mi id="S4.Ex3.m1.1.1.1.1.1.1.2.1.1.1.2" xref="S4.Ex3.m1.1.1.1.1.1.1.2.1.1.1.2.cmml">a</mi><mn id="S4.Ex3.m1.1.1.1.1.1.1.2.1.1.1.3" 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id="S4.Ex3.m1.1c">\alpha_{\sigma}(a_{1})=\alpha_{\sigma}(a_{3})=\alpha_{\sigma}(b_{2})=\alpha_{% \sigma}(b_{3})=c,\,\,\alpha_{\sigma}(a_{2})=\alpha_{\sigma}(b_{1})=d\,.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex3.m1.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_c , italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_d .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.7">We obtain (for <math alttext="\mu_{\sigma}:=\mu_{\ell_{\sigma}}" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m1.1"><semantics id="S4.SS1.p3.7.m1.1a"><mrow id="S4.SS1.p3.7.m1.1.1" xref="S4.SS1.p3.7.m1.1.1.cmml"><msub id="S4.SS1.p3.7.m1.1.1.2" xref="S4.SS1.p3.7.m1.1.1.2.cmml"><mi id="S4.SS1.p3.7.m1.1.1.2.2" xref="S4.SS1.p3.7.m1.1.1.2.2.cmml">μ</mi><mi id="S4.SS1.p3.7.m1.1.1.2.3" xref="S4.SS1.p3.7.m1.1.1.2.3.cmml">σ</mi></msub><mo id="S4.SS1.p3.7.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p3.7.m1.1.1.1.cmml">:=</mo><msub id="S4.SS1.p3.7.m1.1.1.3" xref="S4.SS1.p3.7.m1.1.1.3.cmml"><mi id="S4.SS1.p3.7.m1.1.1.3.2" xref="S4.SS1.p3.7.m1.1.1.3.2.cmml">μ</mi><msub id="S4.SS1.p3.7.m1.1.1.3.3" xref="S4.SS1.p3.7.m1.1.1.3.3.cmml"><mi id="S4.SS1.p3.7.m1.1.1.3.3.2" mathvariant="normal" xref="S4.SS1.p3.7.m1.1.1.3.3.2.cmml">ℓ</mi><mi id="S4.SS1.p3.7.m1.1.1.3.3.3" xref="S4.SS1.p3.7.m1.1.1.3.3.3.cmml">σ</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m1.1b"><apply id="S4.SS1.p3.7.m1.1.1.cmml" xref="S4.SS1.p3.7.m1.1.1"><csymbol cd="latexml" id="S4.SS1.p3.7.m1.1.1.1.cmml" xref="S4.SS1.p3.7.m1.1.1.1">assign</csymbol><apply id="S4.SS1.p3.7.m1.1.1.2.cmml" xref="S4.SS1.p3.7.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m1.1.1.2.1.cmml" xref="S4.SS1.p3.7.m1.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.7.m1.1.1.2.2.cmml" xref="S4.SS1.p3.7.m1.1.1.2.2">𝜇</ci><ci id="S4.SS1.p3.7.m1.1.1.2.3.cmml" xref="S4.SS1.p3.7.m1.1.1.2.3">𝜎</ci></apply><apply id="S4.SS1.p3.7.m1.1.1.3.cmml" xref="S4.SS1.p3.7.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m1.1.1.3.1.cmml" xref="S4.SS1.p3.7.m1.1.1.3">subscript</csymbol><ci id="S4.SS1.p3.7.m1.1.1.3.2.cmml" xref="S4.SS1.p3.7.m1.1.1.3.2">𝜇</ci><apply id="S4.SS1.p3.7.m1.1.1.3.3.cmml" xref="S4.SS1.p3.7.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m1.1.1.3.3.1.cmml" xref="S4.SS1.p3.7.m1.1.1.3.3">subscript</csymbol><ci id="S4.SS1.p3.7.m1.1.1.3.3.2.cmml" xref="S4.SS1.p3.7.m1.1.1.3.3.2">ℓ</ci><ci id="S4.SS1.p3.7.m1.1.1.3.3.3.cmml" xref="S4.SS1.p3.7.m1.1.1.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m1.1c">\mu_{\sigma}:=\mu_{\ell_{\sigma}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m1.1d">italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT := italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>)</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu_{\sigma}(a_{1})=\mu_{\sigma}(a_{2})=\mu_{\sigma}(a_{3})=\mu(a),\,\,\mu_{% \sigma}(b_{1})=\mu_{\sigma}(b_{2})=\mu_{\sigma}(b_{3})=\mu(b)\,," 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id="S4.Ex4.m1.3.3.1.1.1.1.1.1.1.1.3" xref="S4.Ex4.m1.3.3.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Ex4.m1.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex4.m1.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex4.m1.3.3.1.1.1.1.5" xref="S4.Ex4.m1.3.3.1.1.1.1.5.cmml">=</mo><mrow id="S4.Ex4.m1.3.3.1.1.1.1.2" xref="S4.Ex4.m1.3.3.1.1.1.1.2.cmml"><msub id="S4.Ex4.m1.3.3.1.1.1.1.2.3" xref="S4.Ex4.m1.3.3.1.1.1.1.2.3.cmml"><mi id="S4.Ex4.m1.3.3.1.1.1.1.2.3.2" xref="S4.Ex4.m1.3.3.1.1.1.1.2.3.2.cmml">μ</mi><mi id="S4.Ex4.m1.3.3.1.1.1.1.2.3.3" xref="S4.Ex4.m1.3.3.1.1.1.1.2.3.3.cmml">σ</mi></msub><mo id="S4.Ex4.m1.3.3.1.1.1.1.2.2" xref="S4.Ex4.m1.3.3.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex4.m1.3.3.1.1.1.1.2.1.1" xref="S4.Ex4.m1.3.3.1.1.1.1.2.1.1.1.cmml"><mo id="S4.Ex4.m1.3.3.1.1.1.1.2.1.1.2" stretchy="false" xref="S4.Ex4.m1.3.3.1.1.1.1.2.1.1.1.cmml">(</mo><msub id="S4.Ex4.m1.3.3.1.1.1.1.2.1.1.1" xref="S4.Ex4.m1.3.3.1.1.1.1.2.1.1.1.cmml"><mi id="S4.Ex4.m1.3.3.1.1.1.1.2.1.1.1.2" 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end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_μ ( italic_a ) , italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_μ ( italic_b ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.8">and thus (following (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E3" title="In 3.2. Letter-to-letter morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.3</span></a>))</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{array}[]{ccccc}(\alpha_{\sigma})_{*}(\mu_{\sigma})(c)&=&\mu_{\sigma}(a_% {1})+\mu_{\sigma}(a_{3})+\mu_{\sigma}(b_{2})+\mu_{\sigma}(b_{3})&=&2\mu(a)+2% \mu(b)\,,\\ (\alpha_{\sigma})_{*}(\mu_{\sigma})(d)&=&\mu_{\sigma}(a_{2})+\mu_{\sigma}(b_{1% })&=&\mu(a)+\mu(b)\,.\end{array}" class="ltx_Math" display="block" id="S4.Ex5.m1.18"><semantics id="S4.Ex5.m1.18a"><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex5.m1.18.18" rowspacing="0pt" xref="S4.Ex5.m1.18.18.cmml"><mtr id="S4.Ex5.m1.18.18a" xref="S4.Ex5.m1.18.18.cmml"><mtd id="S4.Ex5.m1.18.18b" xref="S4.Ex5.m1.18.18.cmml"><mrow 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xref="S4.Ex5.m1.7.7.7.7.4.cmml"><mrow id="S4.Ex5.m1.4.4.4.4.1.1" xref="S4.Ex5.m1.4.4.4.4.1.1.cmml"><msub id="S4.Ex5.m1.4.4.4.4.1.1.3" xref="S4.Ex5.m1.4.4.4.4.1.1.3.cmml"><mi id="S4.Ex5.m1.4.4.4.4.1.1.3.2" xref="S4.Ex5.m1.4.4.4.4.1.1.3.2.cmml">μ</mi><mi id="S4.Ex5.m1.4.4.4.4.1.1.3.3" xref="S4.Ex5.m1.4.4.4.4.1.1.3.3.cmml">σ</mi></msub><mo id="S4.Ex5.m1.4.4.4.4.1.1.2" xref="S4.Ex5.m1.4.4.4.4.1.1.2.cmml">⁢</mo><mrow id="S4.Ex5.m1.4.4.4.4.1.1.1.1" xref="S4.Ex5.m1.4.4.4.4.1.1.1.1.1.cmml"><mo id="S4.Ex5.m1.4.4.4.4.1.1.1.1.2" stretchy="false" xref="S4.Ex5.m1.4.4.4.4.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex5.m1.4.4.4.4.1.1.1.1.1" xref="S4.Ex5.m1.4.4.4.4.1.1.1.1.1.cmml"><mi id="S4.Ex5.m1.4.4.4.4.1.1.1.1.1.2" xref="S4.Ex5.m1.4.4.4.4.1.1.1.1.1.2.cmml">a</mi><mn id="S4.Ex5.m1.4.4.4.4.1.1.1.1.1.3" xref="S4.Ex5.m1.4.4.4.4.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Ex5.m1.4.4.4.4.1.1.1.1.3" stretchy="false" xref="S4.Ex5.m1.4.4.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex5.m1.7.7.7.7.4.5" xref="S4.Ex5.m1.7.7.7.7.4.5.cmml">+</mo><mrow id="S4.Ex5.m1.5.5.5.5.2.2" xref="S4.Ex5.m1.5.5.5.5.2.2.cmml"><msub id="S4.Ex5.m1.5.5.5.5.2.2.3" xref="S4.Ex5.m1.5.5.5.5.2.2.3.cmml"><mi id="S4.Ex5.m1.5.5.5.5.2.2.3.2" xref="S4.Ex5.m1.5.5.5.5.2.2.3.2.cmml">μ</mi><mi id="S4.Ex5.m1.5.5.5.5.2.2.3.3" xref="S4.Ex5.m1.5.5.5.5.2.2.3.3.cmml">σ</mi></msub><mo id="S4.Ex5.m1.5.5.5.5.2.2.2" xref="S4.Ex5.m1.5.5.5.5.2.2.2.cmml">⁢</mo><mrow id="S4.Ex5.m1.5.5.5.5.2.2.1.1" xref="S4.Ex5.m1.5.5.5.5.2.2.1.1.1.cmml"><mo id="S4.Ex5.m1.5.5.5.5.2.2.1.1.2" stretchy="false" xref="S4.Ex5.m1.5.5.5.5.2.2.1.1.1.cmml">(</mo><msub id="S4.Ex5.m1.5.5.5.5.2.2.1.1.1" xref="S4.Ex5.m1.5.5.5.5.2.2.1.1.1.cmml"><mi id="S4.Ex5.m1.5.5.5.5.2.2.1.1.1.2" xref="S4.Ex5.m1.5.5.5.5.2.2.1.1.1.2.cmml">a</mi><mn id="S4.Ex5.m1.5.5.5.5.2.2.1.1.1.3" xref="S4.Ex5.m1.5.5.5.5.2.2.1.1.1.3.cmml">3</mn></msub><mo id="S4.Ex5.m1.5.5.5.5.2.2.1.1.3" stretchy="false" xref="S4.Ex5.m1.5.5.5.5.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex5.m1.7.7.7.7.4.5a" xref="S4.Ex5.m1.7.7.7.7.4.5.cmml">+</mo><mrow id="S4.Ex5.m1.6.6.6.6.3.3" xref="S4.Ex5.m1.6.6.6.6.3.3.cmml"><msub id="S4.Ex5.m1.6.6.6.6.3.3.3" xref="S4.Ex5.m1.6.6.6.6.3.3.3.cmml"><mi id="S4.Ex5.m1.6.6.6.6.3.3.3.2" xref="S4.Ex5.m1.6.6.6.6.3.3.3.2.cmml">μ</mi><mi id="S4.Ex5.m1.6.6.6.6.3.3.3.3" xref="S4.Ex5.m1.6.6.6.6.3.3.3.3.cmml">σ</mi></msub><mo id="S4.Ex5.m1.6.6.6.6.3.3.2" xref="S4.Ex5.m1.6.6.6.6.3.3.2.cmml">⁢</mo><mrow id="S4.Ex5.m1.6.6.6.6.3.3.1.1" xref="S4.Ex5.m1.6.6.6.6.3.3.1.1.1.cmml"><mo id="S4.Ex5.m1.6.6.6.6.3.3.1.1.2" stretchy="false" xref="S4.Ex5.m1.6.6.6.6.3.3.1.1.1.cmml">(</mo><msub id="S4.Ex5.m1.6.6.6.6.3.3.1.1.1" xref="S4.Ex5.m1.6.6.6.6.3.3.1.1.1.cmml"><mi id="S4.Ex5.m1.6.6.6.6.3.3.1.1.1.2" xref="S4.Ex5.m1.6.6.6.6.3.3.1.1.1.2.cmml">b</mi><mn id="S4.Ex5.m1.6.6.6.6.3.3.1.1.1.3" xref="S4.Ex5.m1.6.6.6.6.3.3.1.1.1.3.cmml">2</mn></msub><mo id="S4.Ex5.m1.6.6.6.6.3.3.1.1.3" stretchy="false" xref="S4.Ex5.m1.6.6.6.6.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex5.m1.7.7.7.7.4.5b" xref="S4.Ex5.m1.7.7.7.7.4.5.cmml">+</mo><mrow id="S4.Ex5.m1.7.7.7.7.4.4" xref="S4.Ex5.m1.7.7.7.7.4.4.cmml"><msub id="S4.Ex5.m1.7.7.7.7.4.4.3" xref="S4.Ex5.m1.7.7.7.7.4.4.3.cmml"><mi id="S4.Ex5.m1.7.7.7.7.4.4.3.2" xref="S4.Ex5.m1.7.7.7.7.4.4.3.2.cmml">μ</mi><mi id="S4.Ex5.m1.7.7.7.7.4.4.3.3" xref="S4.Ex5.m1.7.7.7.7.4.4.3.3.cmml">σ</mi></msub><mo id="S4.Ex5.m1.7.7.7.7.4.4.2" xref="S4.Ex5.m1.7.7.7.7.4.4.2.cmml">⁢</mo><mrow id="S4.Ex5.m1.7.7.7.7.4.4.1.1" xref="S4.Ex5.m1.7.7.7.7.4.4.1.1.1.cmml"><mo id="S4.Ex5.m1.7.7.7.7.4.4.1.1.2" stretchy="false" xref="S4.Ex5.m1.7.7.7.7.4.4.1.1.1.cmml">(</mo><msub id="S4.Ex5.m1.7.7.7.7.4.4.1.1.1" xref="S4.Ex5.m1.7.7.7.7.4.4.1.1.1.cmml"><mi id="S4.Ex5.m1.7.7.7.7.4.4.1.1.1.2" xref="S4.Ex5.m1.7.7.7.7.4.4.1.1.1.2.cmml">b</mi><mn id="S4.Ex5.m1.7.7.7.7.4.4.1.1.1.3" xref="S4.Ex5.m1.7.7.7.7.4.4.1.1.1.3.cmml">3</mn></msub><mo id="S4.Ex5.m1.7.7.7.7.4.4.1.1.3" stretchy="false" xref="S4.Ex5.m1.7.7.7.7.4.4.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd><mtd id="S4.Ex5.m1.18.18e" xref="S4.Ex5.m1.18.18.cmml"><mo id="S4.Ex5.m1.10.10.10.12.1" xref="S4.Ex5.m1.10.10.10.12.1.cmml">=</mo></mtd><mtd id="S4.Ex5.m1.18.18f" xref="S4.Ex5.m1.18.18.cmml"><mrow id="S4.Ex5.m1.10.10.10.10.3.3" xref="S4.Ex5.m1.10.10.10.10.3.3.1.cmml"><mrow id="S4.Ex5.m1.10.10.10.10.3.3.1" xref="S4.Ex5.m1.10.10.10.10.3.3.1.cmml"><mrow id="S4.Ex5.m1.10.10.10.10.3.3.1.2" xref="S4.Ex5.m1.10.10.10.10.3.3.1.2.cmml"><mn id="S4.Ex5.m1.10.10.10.10.3.3.1.2.2" xref="S4.Ex5.m1.10.10.10.10.3.3.1.2.2.cmml">2</mn><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.2.1" xref="S4.Ex5.m1.10.10.10.10.3.3.1.2.1.cmml">⁢</mo><mi id="S4.Ex5.m1.10.10.10.10.3.3.1.2.3" xref="S4.Ex5.m1.10.10.10.10.3.3.1.2.3.cmml">μ</mi><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.2.1a" xref="S4.Ex5.m1.10.10.10.10.3.3.1.2.1.cmml">⁢</mo><mrow id="S4.Ex5.m1.10.10.10.10.3.3.1.2.4.2" xref="S4.Ex5.m1.10.10.10.10.3.3.1.2.cmml"><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.2.4.2.1" stretchy="false" xref="S4.Ex5.m1.10.10.10.10.3.3.1.2.cmml">(</mo><mi id="S4.Ex5.m1.8.8.8.8.1.1" xref="S4.Ex5.m1.8.8.8.8.1.1.cmml">a</mi><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.2.4.2.2" stretchy="false" xref="S4.Ex5.m1.10.10.10.10.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.1" xref="S4.Ex5.m1.10.10.10.10.3.3.1.1.cmml">+</mo><mrow id="S4.Ex5.m1.10.10.10.10.3.3.1.3" xref="S4.Ex5.m1.10.10.10.10.3.3.1.3.cmml"><mn id="S4.Ex5.m1.10.10.10.10.3.3.1.3.2" xref="S4.Ex5.m1.10.10.10.10.3.3.1.3.2.cmml">2</mn><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.3.1" xref="S4.Ex5.m1.10.10.10.10.3.3.1.3.1.cmml">⁢</mo><mi id="S4.Ex5.m1.10.10.10.10.3.3.1.3.3" xref="S4.Ex5.m1.10.10.10.10.3.3.1.3.3.cmml">μ</mi><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.3.1a" xref="S4.Ex5.m1.10.10.10.10.3.3.1.3.1.cmml">⁢</mo><mrow id="S4.Ex5.m1.10.10.10.10.3.3.1.3.4.2" xref="S4.Ex5.m1.10.10.10.10.3.3.1.3.cmml"><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.3.4.2.1" stretchy="false" xref="S4.Ex5.m1.10.10.10.10.3.3.1.3.cmml">(</mo><mi id="S4.Ex5.m1.9.9.9.9.2.2" xref="S4.Ex5.m1.9.9.9.9.2.2.cmml">b</mi><mo id="S4.Ex5.m1.10.10.10.10.3.3.1.3.4.2.2" rspace="0.170em" stretchy="false" xref="S4.Ex5.m1.10.10.10.10.3.3.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex5.m1.10.10.10.10.3.3.2" xref="S4.Ex5.m1.10.10.10.10.3.3.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S4.Ex5.m1.18.18g" xref="S4.Ex5.m1.18.18.cmml"><mtd id="S4.Ex5.m1.18.18h" xref="S4.Ex5.m1.18.18.cmml"><mrow id="S4.Ex5.m1.13.13.13.3.3" xref="S4.Ex5.m1.13.13.13.3.3.cmml"><msub id="S4.Ex5.m1.12.12.12.2.2.2" xref="S4.Ex5.m1.12.12.12.2.2.2.cmml"><mrow id="S4.Ex5.m1.12.12.12.2.2.2.1.1" xref="S4.Ex5.m1.12.12.12.2.2.2.1.1.1.cmml"><mo id="S4.Ex5.m1.12.12.12.2.2.2.1.1.2" stretchy="false" xref="S4.Ex5.m1.12.12.12.2.2.2.1.1.1.cmml">(</mo><msub id="S4.Ex5.m1.12.12.12.2.2.2.1.1.1" xref="S4.Ex5.m1.12.12.12.2.2.2.1.1.1.cmml"><mi id="S4.Ex5.m1.12.12.12.2.2.2.1.1.1.2" xref="S4.Ex5.m1.12.12.12.2.2.2.1.1.1.2.cmml">α</mi><mi id="S4.Ex5.m1.12.12.12.2.2.2.1.1.1.3" xref="S4.Ex5.m1.12.12.12.2.2.2.1.1.1.3.cmml">σ</mi></msub><mo id="S4.Ex5.m1.12.12.12.2.2.2.1.1.3" stretchy="false" xref="S4.Ex5.m1.12.12.12.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex5.m1.12.12.12.2.2.2.3" xref="S4.Ex5.m1.12.12.12.2.2.2.3.cmml">∗</mo></msub><mo id="S4.Ex5.m1.13.13.13.3.3.4" xref="S4.Ex5.m1.13.13.13.3.3.4.cmml">⁢</mo><mrow id="S4.Ex5.m1.13.13.13.3.3.3.1" xref="S4.Ex5.m1.13.13.13.3.3.3.1.1.cmml"><mo id="S4.Ex5.m1.13.13.13.3.3.3.1.2" stretchy="false" xref="S4.Ex5.m1.13.13.13.3.3.3.1.1.cmml">(</mo><msub id="S4.Ex5.m1.13.13.13.3.3.3.1.1" xref="S4.Ex5.m1.13.13.13.3.3.3.1.1.cmml"><mi id="S4.Ex5.m1.13.13.13.3.3.3.1.1.2" xref="S4.Ex5.m1.13.13.13.3.3.3.1.1.2.cmml">μ</mi><mi id="S4.Ex5.m1.13.13.13.3.3.3.1.1.3" xref="S4.Ex5.m1.13.13.13.3.3.3.1.1.3.cmml">σ</mi></msub><mo id="S4.Ex5.m1.13.13.13.3.3.3.1.3" stretchy="false" xref="S4.Ex5.m1.13.13.13.3.3.3.1.1.cmml">)</mo></mrow><mo id="S4.Ex5.m1.13.13.13.3.3.4a" xref="S4.Ex5.m1.13.13.13.3.3.4.cmml">⁢</mo><mrow id="S4.Ex5.m1.13.13.13.3.3.5.2" xref="S4.Ex5.m1.13.13.13.3.3.cmml"><mo id="S4.Ex5.m1.13.13.13.3.3.5.2.1" stretchy="false" xref="S4.Ex5.m1.13.13.13.3.3.cmml">(</mo><mi id="S4.Ex5.m1.11.11.11.1.1.1" xref="S4.Ex5.m1.11.11.11.1.1.1.cmml">d</mi><mo id="S4.Ex5.m1.13.13.13.3.3.5.2.2" stretchy="false" xref="S4.Ex5.m1.13.13.13.3.3.cmml">)</mo></mrow></mrow></mtd><mtd id="S4.Ex5.m1.18.18i" xref="S4.Ex5.m1.18.18.cmml"><mo id="S4.Ex5.m1.18.18.18.9.1" xref="S4.Ex5.m1.18.18.18.9.1.cmml">=</mo></mtd><mtd id="S4.Ex5.m1.18.18j" xref="S4.Ex5.m1.18.18.cmml"><mrow id="S4.Ex5.m1.15.15.15.5.2" xref="S4.Ex5.m1.15.15.15.5.2.cmml"><mrow id="S4.Ex5.m1.14.14.14.4.1.1" xref="S4.Ex5.m1.14.14.14.4.1.1.cmml"><msub id="S4.Ex5.m1.14.14.14.4.1.1.3" xref="S4.Ex5.m1.14.14.14.4.1.1.3.cmml"><mi id="S4.Ex5.m1.14.14.14.4.1.1.3.2" xref="S4.Ex5.m1.14.14.14.4.1.1.3.2.cmml">μ</mi><mi id="S4.Ex5.m1.14.14.14.4.1.1.3.3" xref="S4.Ex5.m1.14.14.14.4.1.1.3.3.cmml">σ</mi></msub><mo id="S4.Ex5.m1.14.14.14.4.1.1.2" xref="S4.Ex5.m1.14.14.14.4.1.1.2.cmml">⁢</mo><mrow id="S4.Ex5.m1.14.14.14.4.1.1.1.1" 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start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) ( italic_c ) end_CELL start_CELL = end_CELL start_CELL italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_CELL start_CELL = end_CELL start_CELL 2 italic_μ ( italic_a ) + 2 italic_μ ( italic_b ) , end_CELL end_ROW start_ROW start_CELL ( italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) ( italic_d ) end_CELL start_CELL = end_CELL start_CELL italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL = end_CELL start_CELL italic_μ ( italic_a ) + italic_μ ( italic_b ) . end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.4">Similarly, one computes</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{array}[]{c}\mu_{\sigma}(a_{1}a_{2})=\mu_{\sigma}(a_{2}a_{3})=\mu(a),\,% \,\mu_{\sigma}(b_{1}b_{2})=\mu_{\sigma}(b_{2}b_{3})=\mu(b)\,,\\ 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2 end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_μ ( italic_a ) , italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = italic_μ ( italic_b ) , end_CELL end_ROW start_ROW start_CELL italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_μ ( italic_a italic_a ) , italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_μ ( italic_a italic_b ) , italic_μ 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xref="S4.SS1.p4.2.m2.1.1.2.cmml">w</mi><mo id="S4.SS1.p4.2.m2.1.1.1" xref="S4.SS1.p4.2.m2.1.1.1.cmml">∈</mo><msup id="S4.SS1.p4.2.m2.1.1.3" xref="S4.SS1.p4.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p4.2.m2.1.1.3.2" xref="S4.SS1.p4.2.m2.1.1.3.2.cmml">𝒜</mi><mo id="S4.SS1.p4.2.m2.1.1.3.3" xref="S4.SS1.p4.2.m2.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><apply id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1"><in id="S4.SS1.p4.2.m2.1.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1.1"></in><ci id="S4.SS1.p4.2.m2.1.1.2.cmml" xref="S4.SS1.p4.2.m2.1.1.2">𝑤</ci><apply id="S4.SS1.p4.2.m2.1.1.3.cmml" xref="S4.SS1.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p4.2.m2.1.1.3.1.cmml" xref="S4.SS1.p4.2.m2.1.1.3">superscript</csymbol><ci id="S4.SS1.p4.2.m2.1.1.3.2.cmml" xref="S4.SS1.p4.2.m2.1.1.3.2">𝒜</ci><times id="S4.SS1.p4.2.m2.1.1.3.3.cmml" xref="S4.SS1.p4.2.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of length <math alttext="|w|=2" class="ltx_Math" display="inline" id="S4.SS1.p4.3.m3.1"><semantics id="S4.SS1.p4.3.m3.1a"><mrow id="S4.SS1.p4.3.m3.1.2" xref="S4.SS1.p4.3.m3.1.2.cmml"><mrow id="S4.SS1.p4.3.m3.1.2.2.2" xref="S4.SS1.p4.3.m3.1.2.2.1.cmml"><mo id="S4.SS1.p4.3.m3.1.2.2.2.1" stretchy="false" xref="S4.SS1.p4.3.m3.1.2.2.1.1.cmml">|</mo><mi id="S4.SS1.p4.3.m3.1.1" xref="S4.SS1.p4.3.m3.1.1.cmml">w</mi><mo id="S4.SS1.p4.3.m3.1.2.2.2.2" stretchy="false" xref="S4.SS1.p4.3.m3.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS1.p4.3.m3.1.2.1" xref="S4.SS1.p4.3.m3.1.2.1.cmml">=</mo><mn id="S4.SS1.p4.3.m3.1.2.3" xref="S4.SS1.p4.3.m3.1.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.3.m3.1b"><apply id="S4.SS1.p4.3.m3.1.2.cmml" xref="S4.SS1.p4.3.m3.1.2"><eq id="S4.SS1.p4.3.m3.1.2.1.cmml" xref="S4.SS1.p4.3.m3.1.2.1"></eq><apply id="S4.SS1.p4.3.m3.1.2.2.1.cmml" xref="S4.SS1.p4.3.m3.1.2.2.2"><abs id="S4.SS1.p4.3.m3.1.2.2.1.1.cmml" xref="S4.SS1.p4.3.m3.1.2.2.2.1"></abs><ci id="S4.SS1.p4.3.m3.1.1.cmml" xref="S4.SS1.p4.3.m3.1.1">𝑤</ci></apply><cn id="S4.SS1.p4.3.m3.1.2.3.cmml" type="integer" xref="S4.SS1.p4.3.m3.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.3.m3.1c">|w|=2</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.3.m3.1d">| italic_w | = 2</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p5"> <p class="ltx_p" id="S4.SS1.p5.1">Correspondingly, one obtains</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{array}[]{clc}(\alpha_{\sigma})_{*}(\mu_{\sigma})(cc)=&[\mu_{\sigma}(a_{% 1}a_{1})+\mu_{\sigma}(a_{1}a_{3})+\mu_{\sigma}(a_{1}b_{2})+\mu_{\sigma}(a_{1}b% _{3})]\\ &+[\mu_{\sigma}(a_{3}a_{1})+\mu_{\sigma}(a_{3}a_{3})+\mu_{\sigma}(a_{3}b_{2})+% \mu_{\sigma}(a_{3}b_{3})]\\ &+[\mu_{\sigma}(b_{2}a_{1})+\mu_{\sigma}(b_{2}a_{3})+\mu_{\sigma}(b_{2}b_{2})+% \mu_{\sigma}(b_{2}b_{3})]\\ &+[\mu_{\sigma}(b_{3}a_{1})+\mu_{\sigma}(b_{3}a_{3})+\mu_{\sigma}(b_{3}b_{2})+% \mu_{\sigma}(b_{3}b_{3})]\\ \qquad\qquad\qquad\,\,\,=&\mu(aa)+\mu(b)+\mu(ba)\,,\end{array}" class="ltx_Math" display="block" id="S4.Ex7.m1.9"><semantics id="S4.Ex7.m1.9a"><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex7.m1.9.9" rowspacing="0pt" xref="S4.Ex7.m1.9.9.cmml"><mtr id="S4.Ex7.m1.9.9a" xref="S4.Ex7.m1.9.9.cmml"><mtd id="S4.Ex7.m1.9.9b" xref="S4.Ex7.m1.9.9.cmml"><mrow id="S4.Ex7.m1.3.3.3.3.3" xref="S4.Ex7.m1.3.3.3.3.3.cmml"><mrow id="S4.Ex7.m1.3.3.3.3.3.3" xref="S4.Ex7.m1.3.3.3.3.3.3.cmml"><msub id="S4.Ex7.m1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex7.m1.1.1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">α</mi><mi id="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex7.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Ex7.m1.1.1.1.1.1.1.1.3" xref="S4.Ex7.m1.1.1.1.1.1.1.1.3.cmml">∗</mo></msub><mo id="S4.Ex7.m1.3.3.3.3.3.3.4" xref="S4.Ex7.m1.3.3.3.3.3.3.4.cmml">⁢</mo><mrow id="S4.Ex7.m1.2.2.2.2.2.2.2.1" xref="S4.Ex7.m1.2.2.2.2.2.2.2.1.1.cmml"><mo id="S4.Ex7.m1.2.2.2.2.2.2.2.1.2" stretchy="false" xref="S4.Ex7.m1.2.2.2.2.2.2.2.1.1.cmml">(</mo><msub id="S4.Ex7.m1.2.2.2.2.2.2.2.1.1" xref="S4.Ex7.m1.2.2.2.2.2.2.2.1.1.cmml"><mi id="S4.Ex7.m1.2.2.2.2.2.2.2.1.1.2" xref="S4.Ex7.m1.2.2.2.2.2.2.2.1.1.2.cmml">μ</mi><mi id="S4.Ex7.m1.2.2.2.2.2.2.2.1.1.3" xref="S4.Ex7.m1.2.2.2.2.2.2.2.1.1.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.2.2.2.2.2.2.2.1.3" stretchy="false" xref="S4.Ex7.m1.2.2.2.2.2.2.2.1.1.cmml">)</mo></mrow><mo id="S4.Ex7.m1.3.3.3.3.3.3.4a" xref="S4.Ex7.m1.3.3.3.3.3.3.4.cmml">⁢</mo><mrow id="S4.Ex7.m1.3.3.3.3.3.3.3.1" xref="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.cmml"><mo id="S4.Ex7.m1.3.3.3.3.3.3.3.1.2" stretchy="false" xref="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.cmml">(</mo><mrow id="S4.Ex7.m1.3.3.3.3.3.3.3.1.1" xref="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.cmml"><mi id="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.2" xref="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.2.cmml">c</mi><mo id="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.1" xref="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.1.cmml">⁢</mo><mi id="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.3" xref="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.3.cmml">c</mi></mrow><mo id="S4.Ex7.m1.3.3.3.3.3.3.3.1.3" stretchy="false" xref="S4.Ex7.m1.3.3.3.3.3.3.3.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.3.3.3.3.3.4" xref="S4.Ex7.m1.3.3.3.3.3.4.cmml">=</mo><mi id="S4.Ex7.m1.3.3.3.3.3.5" xref="S4.Ex7.m1.3.3.3.3.3.5.cmml"></mi></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex7.m1.9.9c" xref="S4.Ex7.m1.9.9.cmml"><mrow id="S4.Ex7.m1.4.4.4.4.1.1" xref="S4.Ex7.m1.4.4.4.4.1.2.cmml"><mo id="S4.Ex7.m1.4.4.4.4.1.1.2" stretchy="false" xref="S4.Ex7.m1.4.4.4.4.1.2.1.cmml">[</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.cmml"><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.cmml"><msub id="S4.Ex7.m1.4.4.4.4.1.1.1.1.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.3.cmml"><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.1.3.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.3.2.cmml">μ</mi><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.1.3.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.1.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.cmml"><msub id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.2.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.2.2.cmml">a</mi><mn id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.2.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.3.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.3.2.cmml">a</mi><mn id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.3.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex7.m1.4.4.4.4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.5" xref="S4.Ex7.m1.4.4.4.4.1.1.1.5.cmml">+</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.cmml"><msub id="S4.Ex7.m1.4.4.4.4.1.1.1.2.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.3.cmml"><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.2.3.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.3.2.cmml">μ</mi><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.2.3.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.3.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.2.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.cmml"><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.2" stretchy="false" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.cmml">(</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.cmml"><msub id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.2.cmml"><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.2.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.2.2.cmml">a</mi><mn id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.2.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.1.cmml">⁢</mo><msub id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.3.cmml"><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.3.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.3.2.cmml">a</mi><mn id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.3.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.3.3.cmml">3</mn></msub></mrow><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.3" stretchy="false" xref="S4.Ex7.m1.4.4.4.4.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.5a" xref="S4.Ex7.m1.4.4.4.4.1.1.1.5.cmml">+</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.3.cmml"><msub id="S4.Ex7.m1.4.4.4.4.1.1.1.3.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.3.3.cmml"><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.3.3.2" 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id="S4.Ex7.m1.4.4.4.4.1.1.1.3.1.1.1.3.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.3.1.1.1.3.2.cmml">b</mi><mn id="S4.Ex7.m1.4.4.4.4.1.1.1.3.1.1.1.3.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.3.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.3.1.1.3" stretchy="false" xref="S4.Ex7.m1.4.4.4.4.1.1.1.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.5b" xref="S4.Ex7.m1.4.4.4.4.1.1.1.5.cmml">+</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.4" xref="S4.Ex7.m1.4.4.4.4.1.1.1.4.cmml"><msub id="S4.Ex7.m1.4.4.4.4.1.1.1.4.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.4.3.cmml"><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.4.3.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.4.3.2.cmml">μ</mi><mi id="S4.Ex7.m1.4.4.4.4.1.1.1.4.3.3" xref="S4.Ex7.m1.4.4.4.4.1.1.1.4.3.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.4.2" xref="S4.Ex7.m1.4.4.4.4.1.1.1.4.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.4.4.4.4.1.1.1.4.1.1" xref="S4.Ex7.m1.4.4.4.4.1.1.1.4.1.1.1.cmml"><mo id="S4.Ex7.m1.4.4.4.4.1.1.1.4.1.1.2" stretchy="false" 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stretchy="false" xref="S4.Ex7.m1.4.4.4.4.1.2.1.cmml">]</mo></mrow></mtd><mtd id="S4.Ex7.m1.9.9d" xref="S4.Ex7.m1.9.9.cmml"></mtd></mtr><mtr id="S4.Ex7.m1.9.9e" xref="S4.Ex7.m1.9.9.cmml"><mtd id="S4.Ex7.m1.9.9f" xref="S4.Ex7.m1.9.9.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex7.m1.9.9g" xref="S4.Ex7.m1.9.9.cmml"><mrow id="S4.Ex7.m1.5.5.5.1.1" xref="S4.Ex7.m1.5.5.5.1.1.cmml"><mo id="S4.Ex7.m1.5.5.5.1.1a" xref="S4.Ex7.m1.5.5.5.1.1.cmml">+</mo><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.2.cmml"><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.2" stretchy="false" xref="S4.Ex7.m1.5.5.5.1.1.1.2.1.cmml">[</mo><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.cmml"><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.cmml"><msub id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.3.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.3.2.cmml">μ</mi><mi id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.3.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.cmml"><msub id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.2.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.2.2.cmml">a</mi><mn id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.2.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.2.3.cmml">3</mn></msub><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.2.cmml">a</mi><mn id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.5" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.5.cmml">+</mo><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1.1.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.cmml"><msub id="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.3.cmml"><mi id="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.3.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.3.2.cmml">μ</mi><mi id="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.3.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.3.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.1.1.1.cmml"><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.2.1.1.2" 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id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.3.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.3.2.cmml">μ</mi><mi id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.3.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.3.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.cmml"><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.2" stretchy="false" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.cmml">(</mo><mrow id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.cmml"><msub id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.2.cmml"><mi id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.2.2" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.2.2.cmml">a</mi><mn id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.2.3" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.2.3.cmml">3</mn></msub><mo id="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.1" xref="S4.Ex7.m1.5.5.5.1.1.1.1.1.4.1.1.1.1.cmml">⁢</mo><msub 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xref="S4.Ex7.m1.6.6.6.1.1.1.2.1.cmml">]</mo></mrow></mrow></mtd><mtd id="S4.Ex7.m1.9.9l" xref="S4.Ex7.m1.9.9.cmml"></mtd></mtr><mtr id="S4.Ex7.m1.9.9m" xref="S4.Ex7.m1.9.9.cmml"><mtd id="S4.Ex7.m1.9.9n" xref="S4.Ex7.m1.9.9.cmml"></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex7.m1.9.9o" xref="S4.Ex7.m1.9.9.cmml"><mrow id="S4.Ex7.m1.7.7.7.1.1" xref="S4.Ex7.m1.7.7.7.1.1.cmml"><mo id="S4.Ex7.m1.7.7.7.1.1a" xref="S4.Ex7.m1.7.7.7.1.1.cmml">+</mo><mrow id="S4.Ex7.m1.7.7.7.1.1.1.1" xref="S4.Ex7.m1.7.7.7.1.1.1.2.cmml"><mo id="S4.Ex7.m1.7.7.7.1.1.1.1.2" stretchy="false" xref="S4.Ex7.m1.7.7.7.1.1.1.2.1.cmml">[</mo><mrow id="S4.Ex7.m1.7.7.7.1.1.1.1.1" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.cmml"><mrow id="S4.Ex7.m1.7.7.7.1.1.1.1.1.1" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.1.cmml"><msub id="S4.Ex7.m1.7.7.7.1.1.1.1.1.1.3" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex7.m1.7.7.7.1.1.1.1.1.1.3.2" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.1.3.2.cmml">μ</mi><mi id="S4.Ex7.m1.7.7.7.1.1.1.1.1.1.3.3" 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id="S4.Ex7.m1.7.7.7.1.1.1.1.1.5a" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.5.cmml">+</mo><mrow id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.cmml"><msub id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.3" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.3.cmml"><mi id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.3.2" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.3.2.cmml">μ</mi><mi id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.3.3" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.3.3.cmml">σ</mi></msub><mo id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.2" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.2.cmml">⁢</mo><mrow id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1.1.cmml"><mo id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1.2" stretchy="false" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1.1.cmml">(</mo><mrow id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1.1" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1.1.cmml"><msub id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1.1.2" xref="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1.1.2.cmml"><mi id="S4.Ex7.m1.7.7.7.1.1.1.1.1.3.1.1.1.2.2" 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italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) ( italic_c italic_c ) = end_CELL start_CELL [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL italic_μ ( italic_a italic_a ) + italic_μ ( italic_b ) + italic_μ ( italic_b italic_a ) , end_CELL start_CELL end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p5.2">further</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{array}[]{c}(\alpha_{\sigma})_{*}(\mu_{\sigma})(cd)=[\mu_{\sigma}(a_{1}a% _{2})+\mu_{\sigma}(a_{1}b_{1})]+[\mu_{\sigma}(a_{3}a_{2})+\mu_{\sigma}(a_{3}b_% {1})]\\ +[\mu_{\sigma}(b_{2}a_{2})+\mu_{\sigma}(b_{2}b_{1})]+[\mu_{\sigma}(b_{3}a_{2})% +\mu_{\sigma}(b_{3}b_{1})]\\ =\mu(a)+\mu(ab)+\mu(bb)\,,\end{array}" class="ltx_Math" display="block" id="S4.Ex8.m1.9"><semantics id="S4.Ex8.m1.9a"><mtable displaystyle="true" id="S4.Ex8.m1.9.9" rowspacing="0pt" 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id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.1.1.1.1.3" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.1.1.1.1.3.2" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.1.1.1.1.3.2.cmml">a</mi><mn id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.1.1.1.1.3.3" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.3" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.3.cmml">+</mo><mrow id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2.cmml"><msub id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2.3" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2.3.cmml"><mi id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2.3.2" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2.3.2.cmml">μ</mi><mi id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2.3.3" xref="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2.3.3.cmml">σ</mi></msub><mo id="S4.Ex8.m1.4.4.4.4.4.4.1.1.1.2.2" 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id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.3.2" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.3.2.cmml">μ</mi><mi id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.3.3" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.2" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.2.2" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.2.2.cmml">b</mi><mn id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.2.3" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.1.1.1.1.1" 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id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.2" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.cmml"><mo id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.2" stretchy="false" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.cmml"><msub id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.2" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.2.cmml"><mi id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.2.2" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.2.2.cmml">b</mi><mn id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.2.3" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.2.3.cmml">2</mn></msub><mo id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.1" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.1.cmml">⁢</mo><msub id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.3" xref="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.3.cmml"><mi id="S4.Ex8.m1.6.6.6.1.1.1.1.1.1.2.1.1.1.3.2" 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id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.3.3" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.3.3.cmml">σ</mi></msub><mo id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.2" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.cmml"><mo id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.cmml"><msub id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.2" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.2.2" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.2.2.cmml">b</mi><mn id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.2.3" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.2.3.cmml">3</mn></msub><mo id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.1" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.3" xref="S4.Ex8.m1.7.7.7.2.2.2.1.1.1.1.1.1.3.cmml"><mi 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xref="S4.Ex8.m1.9.9.9.2.2.2.1.2.2.1.1.1.2">𝑏</ci><ci id="S4.Ex8.m1.9.9.9.2.2.2.1.2.2.1.1.1.3.cmml" xref="S4.Ex8.m1.9.9.9.2.2.2.1.2.2.1.1.1.3">𝑏</ci></apply></apply></apply></apply></matrixrow></matrix></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m1.9c">\begin{array}[]{c}(\alpha_{\sigma})_{*}(\mu_{\sigma})(cd)=[\mu_{\sigma}(a_{1}a% _{2})+\mu_{\sigma}(a_{1}b_{1})]+[\mu_{\sigma}(a_{3}a_{2})+\mu_{\sigma}(a_{3}b_% {1})]\\ +[\mu_{\sigma}(b_{2}a_{2})+\mu_{\sigma}(b_{2}b_{1})]+[\mu_{\sigma}(b_{3}a_{2})% +\mu_{\sigma}(b_{3}b_{1})]\\ =\mu(a)+\mu(ab)+\mu(bb)\,,\end{array}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.9d">start_ARRAY start_ROW start_CELL ( italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) ( italic_c italic_d ) = [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] + [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] end_CELL end_ROW start_ROW start_CELL + [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] + [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] end_CELL end_ROW start_ROW start_CELL = italic_μ ( italic_a ) + italic_μ ( italic_a italic_b ) + italic_μ ( italic_b italic_b ) , end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S4.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{array}[]{c}(\alpha_{\sigma})_{*}(\mu_{\sigma})(dc)=[\mu_{\sigma}(a_{2}a% _{1})+\mu_{\sigma}(a_{2}a_{3})+\mu_{\sigma}(a_{2}b_{2})+\mu_{\sigma}(a_{2}b_{3% })]\\ +[\mu_{\sigma}(b_{1}a_{1})+\mu_{\sigma}(b_{1}a_{3})+\mu_{\sigma}(b_{1}b_{2})+% \mu_{\sigma}(b_{1}b_{3})]\\ =\mu(a)+\mu(b)\,,\end{array}" class="ltx_Math" display="block" id="S4.Ex9.m1.8"><semantics id="S4.Ex9.m1.8a"><mtable displaystyle="true" id="S4.Ex9.m1.8.8" rowspacing="0pt" xref="S4.Ex9.m1.8.8.cmml"><mtr id="S4.Ex9.m1.8.8a" xref="S4.Ex9.m1.8.8.cmml"><mtd id="S4.Ex9.m1.8.8b" xref="S4.Ex9.m1.8.8.cmml"><mrow id="S4.Ex9.m1.4.4.4.4.4" xref="S4.Ex9.m1.4.4.4.4.4.cmml"><mrow id="S4.Ex9.m1.3.3.3.3.3.3" xref="S4.Ex9.m1.3.3.3.3.3.3.cmml"><msub id="S4.Ex9.m1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.1.cmml"><mrow id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">α</mi><mi id="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S4.Ex9.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">σ</mi></msub><mo 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start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] end_CELL end_ROW start_ROW start_CELL + [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ] end_CELL end_ROW start_ROW start_CELL = italic_μ ( italic_a ) + italic_μ ( italic_b ) , end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p5.3">and finally</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{array}[]{c}(\alpha_{\sigma})_{*}(\mu_{\sigma})(dd)=[\mu_{\sigma}(a_{2}a% _{2})+\mu_{\sigma}(a_{2}b_{1})]+[\mu_{\sigma}(b_{1}a_{2})+\mu_{\sigma}(b_{1}b_% {1})]=0\,.\end{array}" class="ltx_Math" display="block" id="S4.Ex10.m1.1"><semantics id="S4.Ex10.m1.1a"><mtable displaystyle="true" id="S4.Ex10.m1.1.1" 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id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.2.cmml"><mi id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.2.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.2.2.cmml">a</mi><mn id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.2.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.1" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.3.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.3.2.cmml">a</mi><mn id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.3.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.3.cmml">+</mo><mrow id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.cmml"><msub id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.3.cmml"><mi id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.3.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.3.2.cmml">μ</mi><mi id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.3.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.3.3.cmml">σ</mi></msub><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.2.cmml">⁢</mo><mrow id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.cmml"><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.2" stretchy="false" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.cmml">(</mo><mrow id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.cmml"><msub id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.2.cmml"><mi id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.2.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.2.2.cmml">a</mi><mn id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.2.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.2.3.cmml">2</mn></msub><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.1" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.1.cmml">⁢</mo><msub id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.3.cmml"><mi id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.3.2" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.3.2.cmml">b</mi><mn id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.3.3" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.3" stretchy="false" xref="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex10.m1.1.1.1.1.1.1.1.4.1.1.3" stretchy="false" 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italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] + [ italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ] = 0 . end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS1.p6"> <p class="ltx_p" id="S4.SS1.p6.1">Since by definition we have <math alttext="\sigma M(\mu)=(\alpha_{\sigma})_{*}(\mu_{\sigma})" class="ltx_Math" display="inline" id="S4.SS1.p6.1.m1.3"><semantics id="S4.SS1.p6.1.m1.3a"><mrow id="S4.SS1.p6.1.m1.3.3" xref="S4.SS1.p6.1.m1.3.3.cmml"><mrow id="S4.SS1.p6.1.m1.3.3.4" xref="S4.SS1.p6.1.m1.3.3.4.cmml"><mi id="S4.SS1.p6.1.m1.3.3.4.2" xref="S4.SS1.p6.1.m1.3.3.4.2.cmml">σ</mi><mo 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xref="S4.SS1.p6.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p6.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S4.SS1.p6.1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p6.1.m1.2.2.1.1.1.1.1.2.cmml" xref="S4.SS1.p6.1.m1.2.2.1.1.1.1.1.2">𝛼</ci><ci id="S4.SS1.p6.1.m1.2.2.1.1.1.1.1.3.cmml" xref="S4.SS1.p6.1.m1.2.2.1.1.1.1.1.3">𝜎</ci></apply><times id="S4.SS1.p6.1.m1.2.2.1.1.3.cmml" xref="S4.SS1.p6.1.m1.2.2.1.1.3"></times></apply><apply id="S4.SS1.p6.1.m1.3.3.2.2.1.1.cmml" xref="S4.SS1.p6.1.m1.3.3.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.p6.1.m1.3.3.2.2.1.1.1.cmml" xref="S4.SS1.p6.1.m1.3.3.2.2.1">subscript</csymbol><ci id="S4.SS1.p6.1.m1.3.3.2.2.1.1.2.cmml" xref="S4.SS1.p6.1.m1.3.3.2.2.1.1.2">𝜇</ci><ci id="S4.SS1.p6.1.m1.3.3.2.2.1.1.3.cmml" xref="S4.SS1.p6.1.m1.3.3.2.2.1.1.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p6.1.m1.3c">\sigma M(\mu)=(\alpha_{\sigma})_{*}(\mu_{\sigma})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p6.1.m1.3d">italic_σ italic_M ( italic_μ ) = ( italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT )</annotation></semantics></math>, we have computed</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\begin{array}[]{cclc}\sigma M(\mu)(c)&=&2(\mu(a)+\mu(b))\\ \sigma M(\mu)(d)&=&\mu(a)+\mu(b)\\ \sigma M(\mu)(cc)&=&\mu(b)+\mu(aa)+\mu(ba)\\ \sigma M(\mu)(cd)&=&\mu(a)+\mu(ab)+\mu(bb)\\ \sigma M(\mu)(dc)&=&\mu(a)+\mu(b)\\ \sigma M(\mu)(dd)&=&0\,.\end{array}" class="ltx_Math" display="block" id="S4.Ex11.m1.26"><semantics id="S4.Ex11.m1.26a"><mtable columnspacing="5pt" displaystyle="true" id="S4.Ex11.m1.26.26" rowspacing="0pt" xref="S4.Ex11.m1.26.26.cmml"><mtr id="S4.Ex11.m1.26.26a" 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xref="S4.Ex11.m1.2.2.2.2.2.cmml">(</mo><mi id="S4.Ex11.m1.2.2.2.2.2.2" xref="S4.Ex11.m1.2.2.2.2.2.2.cmml">c</mi><mo id="S4.Ex11.m1.2.2.2.2.2.7.2.2" stretchy="false" xref="S4.Ex11.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></mtd><mtd id="S4.Ex11.m1.26.26c" xref="S4.Ex11.m1.26.26.cmml"><mo id="S4.Ex11.m1.5.5.5.6.1" xref="S4.Ex11.m1.5.5.5.6.1.cmml">=</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex11.m1.26.26d" xref="S4.Ex11.m1.26.26.cmml"><mrow id="S4.Ex11.m1.5.5.5.5.3" xref="S4.Ex11.m1.5.5.5.5.3.cmml"><mn id="S4.Ex11.m1.5.5.5.5.3.5" xref="S4.Ex11.m1.5.5.5.5.3.5.cmml">2</mn><mo id="S4.Ex11.m1.5.5.5.5.3.4" xref="S4.Ex11.m1.5.5.5.5.3.4.cmml">⁢</mo><mrow id="S4.Ex11.m1.5.5.5.5.3.3.1" xref="S4.Ex11.m1.5.5.5.5.3.3.1.1.cmml"><mo id="S4.Ex11.m1.5.5.5.5.3.3.1.2" stretchy="false" xref="S4.Ex11.m1.5.5.5.5.3.3.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.5.5.5.5.3.3.1.1" xref="S4.Ex11.m1.5.5.5.5.3.3.1.1.cmml"><mrow id="S4.Ex11.m1.5.5.5.5.3.3.1.1.2" xref="S4.Ex11.m1.5.5.5.5.3.3.1.1.2.cmml"><mi 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xref="S4.Ex11.m1.9.9.9.4.2.cmml"><mrow id="S4.Ex11.m1.9.9.9.4.2.4" xref="S4.Ex11.m1.9.9.9.4.2.4.cmml"><mi id="S4.Ex11.m1.9.9.9.4.2.4.2" xref="S4.Ex11.m1.9.9.9.4.2.4.2.cmml">μ</mi><mo id="S4.Ex11.m1.9.9.9.4.2.4.1" xref="S4.Ex11.m1.9.9.9.4.2.4.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.9.9.9.4.2.4.3.2" xref="S4.Ex11.m1.9.9.9.4.2.4.cmml"><mo id="S4.Ex11.m1.9.9.9.4.2.4.3.2.1" stretchy="false" xref="S4.Ex11.m1.9.9.9.4.2.4.cmml">(</mo><mi id="S4.Ex11.m1.8.8.8.3.1.1" xref="S4.Ex11.m1.8.8.8.3.1.1.cmml">a</mi><mo id="S4.Ex11.m1.9.9.9.4.2.4.3.2.2" stretchy="false" xref="S4.Ex11.m1.9.9.9.4.2.4.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.9.9.9.4.2.3" xref="S4.Ex11.m1.9.9.9.4.2.3.cmml">+</mo><mrow id="S4.Ex11.m1.9.9.9.4.2.5" xref="S4.Ex11.m1.9.9.9.4.2.5.cmml"><mi id="S4.Ex11.m1.9.9.9.4.2.5.2" xref="S4.Ex11.m1.9.9.9.4.2.5.2.cmml">μ</mi><mo id="S4.Ex11.m1.9.9.9.4.2.5.1" xref="S4.Ex11.m1.9.9.9.4.2.5.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.9.9.9.4.2.5.3.2" xref="S4.Ex11.m1.9.9.9.4.2.5.cmml"><mo id="S4.Ex11.m1.9.9.9.4.2.5.3.2.1" stretchy="false" xref="S4.Ex11.m1.9.9.9.4.2.5.cmml">(</mo><mi id="S4.Ex11.m1.9.9.9.4.2.2" xref="S4.Ex11.m1.9.9.9.4.2.2.cmml">b</mi><mo id="S4.Ex11.m1.9.9.9.4.2.5.3.2.2" stretchy="false" xref="S4.Ex11.m1.9.9.9.4.2.5.cmml">)</mo></mrow></mrow></mrow></mtd><mtd id="S4.Ex11.m1.26.26j" xref="S4.Ex11.m1.26.26.cmml"></mtd></mtr><mtr id="S4.Ex11.m1.26.26k" xref="S4.Ex11.m1.26.26.cmml"><mtd id="S4.Ex11.m1.26.26l" xref="S4.Ex11.m1.26.26.cmml"><mrow id="S4.Ex11.m1.11.11.11.2.2" xref="S4.Ex11.m1.11.11.11.2.2.cmml"><mi id="S4.Ex11.m1.11.11.11.2.2.4" xref="S4.Ex11.m1.11.11.11.2.2.4.cmml">σ</mi><mo id="S4.Ex11.m1.11.11.11.2.2.3" xref="S4.Ex11.m1.11.11.11.2.2.3.cmml">⁢</mo><mi id="S4.Ex11.m1.11.11.11.2.2.5" xref="S4.Ex11.m1.11.11.11.2.2.5.cmml">M</mi><mo id="S4.Ex11.m1.11.11.11.2.2.3a" xref="S4.Ex11.m1.11.11.11.2.2.3.cmml">⁢</mo><mrow id="S4.Ex11.m1.11.11.11.2.2.6.2" xref="S4.Ex11.m1.11.11.11.2.2.cmml"><mo id="S4.Ex11.m1.11.11.11.2.2.6.2.1" stretchy="false" xref="S4.Ex11.m1.11.11.11.2.2.cmml">(</mo><mi id="S4.Ex11.m1.10.10.10.1.1.1" xref="S4.Ex11.m1.10.10.10.1.1.1.cmml">μ</mi><mo id="S4.Ex11.m1.11.11.11.2.2.6.2.2" stretchy="false" xref="S4.Ex11.m1.11.11.11.2.2.cmml">)</mo></mrow><mo id="S4.Ex11.m1.11.11.11.2.2.3b" xref="S4.Ex11.m1.11.11.11.2.2.3.cmml">⁢</mo><mrow id="S4.Ex11.m1.11.11.11.2.2.2.1" xref="S4.Ex11.m1.11.11.11.2.2.2.1.1.cmml"><mo id="S4.Ex11.m1.11.11.11.2.2.2.1.2" stretchy="false" xref="S4.Ex11.m1.11.11.11.2.2.2.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.11.11.11.2.2.2.1.1" xref="S4.Ex11.m1.11.11.11.2.2.2.1.1.cmml"><mi id="S4.Ex11.m1.11.11.11.2.2.2.1.1.2" xref="S4.Ex11.m1.11.11.11.2.2.2.1.1.2.cmml">c</mi><mo id="S4.Ex11.m1.11.11.11.2.2.2.1.1.1" xref="S4.Ex11.m1.11.11.11.2.2.2.1.1.1.cmml">⁢</mo><mi id="S4.Ex11.m1.11.11.11.2.2.2.1.1.3" xref="S4.Ex11.m1.11.11.11.2.2.2.1.1.3.cmml">c</mi></mrow><mo id="S4.Ex11.m1.11.11.11.2.2.2.1.3" stretchy="false" xref="S4.Ex11.m1.11.11.11.2.2.2.1.1.cmml">)</mo></mrow></mrow></mtd><mtd id="S4.Ex11.m1.26.26m" xref="S4.Ex11.m1.26.26.cmml"><mo id="S4.Ex11.m1.14.14.14.6.1" xref="S4.Ex11.m1.14.14.14.6.1.cmml">=</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex11.m1.26.26n" xref="S4.Ex11.m1.26.26.cmml"><mrow id="S4.Ex11.m1.14.14.14.5.3" xref="S4.Ex11.m1.14.14.14.5.3.cmml"><mrow id="S4.Ex11.m1.14.14.14.5.3.5" xref="S4.Ex11.m1.14.14.14.5.3.5.cmml"><mi id="S4.Ex11.m1.14.14.14.5.3.5.2" xref="S4.Ex11.m1.14.14.14.5.3.5.2.cmml">μ</mi><mo id="S4.Ex11.m1.14.14.14.5.3.5.1" xref="S4.Ex11.m1.14.14.14.5.3.5.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.14.14.14.5.3.5.3.2" xref="S4.Ex11.m1.14.14.14.5.3.5.cmml"><mo id="S4.Ex11.m1.14.14.14.5.3.5.3.2.1" stretchy="false" xref="S4.Ex11.m1.14.14.14.5.3.5.cmml">(</mo><mi id="S4.Ex11.m1.12.12.12.3.1.1" xref="S4.Ex11.m1.12.12.12.3.1.1.cmml">b</mi><mo id="S4.Ex11.m1.14.14.14.5.3.5.3.2.2" stretchy="false" xref="S4.Ex11.m1.14.14.14.5.3.5.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.14.14.14.5.3.4" xref="S4.Ex11.m1.14.14.14.5.3.4.cmml">+</mo><mrow id="S4.Ex11.m1.13.13.13.4.2.2" xref="S4.Ex11.m1.13.13.13.4.2.2.cmml"><mi id="S4.Ex11.m1.13.13.13.4.2.2.3" xref="S4.Ex11.m1.13.13.13.4.2.2.3.cmml">μ</mi><mo id="S4.Ex11.m1.13.13.13.4.2.2.2" xref="S4.Ex11.m1.13.13.13.4.2.2.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.13.13.13.4.2.2.1.1" xref="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.cmml"><mo id="S4.Ex11.m1.13.13.13.4.2.2.1.1.2" stretchy="false" xref="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.13.13.13.4.2.2.1.1.1" xref="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.cmml"><mi id="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.2" xref="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.2.cmml">a</mi><mo id="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.1" xref="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.3" xref="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.3.cmml">a</mi></mrow><mo id="S4.Ex11.m1.13.13.13.4.2.2.1.1.3" stretchy="false" xref="S4.Ex11.m1.13.13.13.4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.14.14.14.5.3.4a" xref="S4.Ex11.m1.14.14.14.5.3.4.cmml">+</mo><mrow id="S4.Ex11.m1.14.14.14.5.3.3" xref="S4.Ex11.m1.14.14.14.5.3.3.cmml"><mi id="S4.Ex11.m1.14.14.14.5.3.3.3" xref="S4.Ex11.m1.14.14.14.5.3.3.3.cmml">μ</mi><mo id="S4.Ex11.m1.14.14.14.5.3.3.2" xref="S4.Ex11.m1.14.14.14.5.3.3.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.14.14.14.5.3.3.1.1" xref="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.cmml"><mo id="S4.Ex11.m1.14.14.14.5.3.3.1.1.2" stretchy="false" xref="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.14.14.14.5.3.3.1.1.1" xref="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.cmml"><mi id="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.2" xref="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.2.cmml">b</mi><mo id="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.1" xref="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.3" xref="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.3.cmml">a</mi></mrow><mo id="S4.Ex11.m1.14.14.14.5.3.3.1.1.3" stretchy="false" xref="S4.Ex11.m1.14.14.14.5.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd><mtd id="S4.Ex11.m1.26.26o" xref="S4.Ex11.m1.26.26.cmml"></mtd></mtr><mtr id="S4.Ex11.m1.26.26p" xref="S4.Ex11.m1.26.26.cmml"><mtd id="S4.Ex11.m1.26.26q" xref="S4.Ex11.m1.26.26.cmml"><mrow id="S4.Ex11.m1.16.16.16.2.2" xref="S4.Ex11.m1.16.16.16.2.2.cmml"><mi id="S4.Ex11.m1.16.16.16.2.2.4" xref="S4.Ex11.m1.16.16.16.2.2.4.cmml">σ</mi><mo id="S4.Ex11.m1.16.16.16.2.2.3" xref="S4.Ex11.m1.16.16.16.2.2.3.cmml">⁢</mo><mi id="S4.Ex11.m1.16.16.16.2.2.5" xref="S4.Ex11.m1.16.16.16.2.2.5.cmml">M</mi><mo id="S4.Ex11.m1.16.16.16.2.2.3a" xref="S4.Ex11.m1.16.16.16.2.2.3.cmml">⁢</mo><mrow id="S4.Ex11.m1.16.16.16.2.2.6.2" xref="S4.Ex11.m1.16.16.16.2.2.cmml"><mo id="S4.Ex11.m1.16.16.16.2.2.6.2.1" stretchy="false" xref="S4.Ex11.m1.16.16.16.2.2.cmml">(</mo><mi id="S4.Ex11.m1.15.15.15.1.1.1" xref="S4.Ex11.m1.15.15.15.1.1.1.cmml">μ</mi><mo id="S4.Ex11.m1.16.16.16.2.2.6.2.2" stretchy="false" xref="S4.Ex11.m1.16.16.16.2.2.cmml">)</mo></mrow><mo id="S4.Ex11.m1.16.16.16.2.2.3b" xref="S4.Ex11.m1.16.16.16.2.2.3.cmml">⁢</mo><mrow id="S4.Ex11.m1.16.16.16.2.2.2.1" xref="S4.Ex11.m1.16.16.16.2.2.2.1.1.cmml"><mo id="S4.Ex11.m1.16.16.16.2.2.2.1.2" stretchy="false" xref="S4.Ex11.m1.16.16.16.2.2.2.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.16.16.16.2.2.2.1.1" xref="S4.Ex11.m1.16.16.16.2.2.2.1.1.cmml"><mi id="S4.Ex11.m1.16.16.16.2.2.2.1.1.2" xref="S4.Ex11.m1.16.16.16.2.2.2.1.1.2.cmml">c</mi><mo id="S4.Ex11.m1.16.16.16.2.2.2.1.1.1" xref="S4.Ex11.m1.16.16.16.2.2.2.1.1.1.cmml">⁢</mo><mi id="S4.Ex11.m1.16.16.16.2.2.2.1.1.3" xref="S4.Ex11.m1.16.16.16.2.2.2.1.1.3.cmml">d</mi></mrow><mo id="S4.Ex11.m1.16.16.16.2.2.2.1.3" stretchy="false" xref="S4.Ex11.m1.16.16.16.2.2.2.1.1.cmml">)</mo></mrow></mrow></mtd><mtd id="S4.Ex11.m1.26.26r" xref="S4.Ex11.m1.26.26.cmml"><mo id="S4.Ex11.m1.19.19.19.6.1" xref="S4.Ex11.m1.19.19.19.6.1.cmml">=</mo></mtd><mtd class="ltx_align_left" columnalign="left" id="S4.Ex11.m1.26.26s" xref="S4.Ex11.m1.26.26.cmml"><mrow id="S4.Ex11.m1.19.19.19.5.3" xref="S4.Ex11.m1.19.19.19.5.3.cmml"><mrow id="S4.Ex11.m1.19.19.19.5.3.5" xref="S4.Ex11.m1.19.19.19.5.3.5.cmml"><mi id="S4.Ex11.m1.19.19.19.5.3.5.2" xref="S4.Ex11.m1.19.19.19.5.3.5.2.cmml">μ</mi><mo id="S4.Ex11.m1.19.19.19.5.3.5.1" xref="S4.Ex11.m1.19.19.19.5.3.5.1.cmml">⁢</mo><mrow id="S4.Ex11.m1.19.19.19.5.3.5.3.2" xref="S4.Ex11.m1.19.19.19.5.3.5.cmml"><mo id="S4.Ex11.m1.19.19.19.5.3.5.3.2.1" stretchy="false" xref="S4.Ex11.m1.19.19.19.5.3.5.cmml">(</mo><mi id="S4.Ex11.m1.17.17.17.3.1.1" xref="S4.Ex11.m1.17.17.17.3.1.1.cmml">a</mi><mo id="S4.Ex11.m1.19.19.19.5.3.5.3.2.2" stretchy="false" xref="S4.Ex11.m1.19.19.19.5.3.5.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.19.19.19.5.3.4" xref="S4.Ex11.m1.19.19.19.5.3.4.cmml">+</mo><mrow id="S4.Ex11.m1.18.18.18.4.2.2" xref="S4.Ex11.m1.18.18.18.4.2.2.cmml"><mi id="S4.Ex11.m1.18.18.18.4.2.2.3" xref="S4.Ex11.m1.18.18.18.4.2.2.3.cmml">μ</mi><mo id="S4.Ex11.m1.18.18.18.4.2.2.2" xref="S4.Ex11.m1.18.18.18.4.2.2.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.18.18.18.4.2.2.1.1" xref="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.cmml"><mo id="S4.Ex11.m1.18.18.18.4.2.2.1.1.2" stretchy="false" xref="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.18.18.18.4.2.2.1.1.1" xref="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.cmml"><mi id="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.2" xref="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.2.cmml">a</mi><mo id="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.1" xref="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.3" xref="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.3.cmml">b</mi></mrow><mo id="S4.Ex11.m1.18.18.18.4.2.2.1.1.3" stretchy="false" xref="S4.Ex11.m1.18.18.18.4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex11.m1.19.19.19.5.3.4a" xref="S4.Ex11.m1.19.19.19.5.3.4.cmml">+</mo><mrow id="S4.Ex11.m1.19.19.19.5.3.3" xref="S4.Ex11.m1.19.19.19.5.3.3.cmml"><mi id="S4.Ex11.m1.19.19.19.5.3.3.3" xref="S4.Ex11.m1.19.19.19.5.3.3.3.cmml">μ</mi><mo id="S4.Ex11.m1.19.19.19.5.3.3.2" xref="S4.Ex11.m1.19.19.19.5.3.3.2.cmml">⁢</mo><mrow id="S4.Ex11.m1.19.19.19.5.3.3.1.1" xref="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.cmml"><mo id="S4.Ex11.m1.19.19.19.5.3.3.1.1.2" stretchy="false" xref="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.cmml">(</mo><mrow id="S4.Ex11.m1.19.19.19.5.3.3.1.1.1" xref="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.cmml"><mi id="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.2" xref="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.2.cmml">b</mi><mo id="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.1" xref="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.3" xref="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.3.cmml">b</mi></mrow><mo id="S4.Ex11.m1.19.19.19.5.3.3.1.1.3" stretchy="false" xref="S4.Ex11.m1.19.19.19.5.3.3.1.1.1.cmml">)</mo></mrow></mrow></mrow></mtd><mtd id="S4.Ex11.m1.26.26t" xref="S4.Ex11.m1.26.26.cmml"></mtd></mtr><mtr id="S4.Ex11.m1.26.26u" xref="S4.Ex11.m1.26.26.cmml"><mtd id="S4.Ex11.m1.26.26v" xref="S4.Ex11.m1.26.26.cmml"><mrow id="S4.Ex11.m1.21.21.21.2.2" xref="S4.Ex11.m1.21.21.21.2.2.cmml"><mi id="S4.Ex11.m1.21.21.21.2.2.4" xref="S4.Ex11.m1.21.21.21.2.2.4.cmml">σ</mi><mo id="S4.Ex11.m1.21.21.21.2.2.3" xref="S4.Ex11.m1.21.21.21.2.2.3.cmml">⁢</mo><mi id="S4.Ex11.m1.21.21.21.2.2.5" xref="S4.Ex11.m1.21.21.21.2.2.5.cmml">M</mi><mo id="S4.Ex11.m1.21.21.21.2.2.3a" xref="S4.Ex11.m1.21.21.21.2.2.3.cmml">⁢</mo><mrow id="S4.Ex11.m1.21.21.21.2.2.6.2" xref="S4.Ex11.m1.21.21.21.2.2.cmml"><mo id="S4.Ex11.m1.21.21.21.2.2.6.2.1" stretchy="false" xref="S4.Ex11.m1.21.21.21.2.2.cmml">(</mo><mi id="S4.Ex11.m1.20.20.20.1.1.1" xref="S4.Ex11.m1.20.20.20.1.1.1.cmml">μ</mi><mo id="S4.Ex11.m1.21.21.21.2.2.6.2.2" stretchy="false" xref="S4.Ex11.m1.21.21.21.2.2.cmml">)</mo></mrow><mo id="S4.Ex11.m1.21.21.21.2.2.3b" xref="S4.Ex11.m1.21.21.21.2.2.3.cmml">⁢</mo><mrow id="S4.Ex11.m1.21.21.21.2.2.2.1" xref="S4.Ex11.m1.21.21.21.2.2.2.1.1.cmml"><mo id="S4.Ex11.m1.21.21.21.2.2.2.1.2" stretchy="false" 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end_CELL start_CELL italic_μ ( italic_a ) + italic_μ ( italic_b ) end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_σ italic_M ( italic_μ ) ( italic_c italic_c ) end_CELL start_CELL = end_CELL start_CELL italic_μ ( italic_b ) + italic_μ ( italic_a italic_a ) + italic_μ ( italic_b italic_a ) end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_σ italic_M ( italic_μ ) ( italic_c italic_d ) end_CELL start_CELL = end_CELL start_CELL italic_μ ( italic_a ) + italic_μ ( italic_a italic_b ) + italic_μ ( italic_b italic_b ) end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_σ italic_M ( italic_μ ) ( italic_d italic_c ) end_CELL start_CELL = end_CELL start_CELL italic_μ ( italic_a ) + italic_μ ( italic_b ) end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_σ italic_M ( italic_μ ) ( italic_d italic_d ) end_CELL start_CELL = end_CELL start_CELL 0 . end_CELL start_CELL end_CELL end_ROW end_ARRAY</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2. </span>An alternative evaluation method</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1"></p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.3">As illustrated by the example considered in the previous subsection, already for fairly simple morphisms <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mi id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><ci id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_σ</annotation></semantics></math> the preimage set <math alttext="\alpha_{\sigma}^{-1}(w)" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1.2" xref="S4.SS2.p2.2.m2.1.2.cmml"><msubsup id="S4.SS2.p2.2.m2.1.2.2" xref="S4.SS2.p2.2.m2.1.2.2.cmml"><mi id="S4.SS2.p2.2.m2.1.2.2.2.2" xref="S4.SS2.p2.2.m2.1.2.2.2.2.cmml">α</mi><mi id="S4.SS2.p2.2.m2.1.2.2.2.3" xref="S4.SS2.p2.2.m2.1.2.2.2.3.cmml">σ</mi><mrow id="S4.SS2.p2.2.m2.1.2.2.3" xref="S4.SS2.p2.2.m2.1.2.2.3.cmml"><mo id="S4.SS2.p2.2.m2.1.2.2.3a" xref="S4.SS2.p2.2.m2.1.2.2.3.cmml">−</mo><mn id="S4.SS2.p2.2.m2.1.2.2.3.2" xref="S4.SS2.p2.2.m2.1.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS2.p2.2.m2.1.2.1" xref="S4.SS2.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p2.2.m2.1.2.3.2" xref="S4.SS2.p2.2.m2.1.2.cmml"><mo id="S4.SS2.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p2.2.m2.1.2.cmml">(</mo><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">w</mi><mo id="S4.SS2.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.2.cmml" xref="S4.SS2.p2.2.m2.1.2"><times id="S4.SS2.p2.2.m2.1.2.1.cmml" xref="S4.SS2.p2.2.m2.1.2.1"></times><apply id="S4.SS2.p2.2.m2.1.2.2.cmml" xref="S4.SS2.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.2.2.1.cmml" xref="S4.SS2.p2.2.m2.1.2.2">superscript</csymbol><apply id="S4.SS2.p2.2.m2.1.2.2.2.cmml" xref="S4.SS2.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.2.2.2.1.cmml" xref="S4.SS2.p2.2.m2.1.2.2">subscript</csymbol><ci id="S4.SS2.p2.2.m2.1.2.2.2.2.cmml" xref="S4.SS2.p2.2.m2.1.2.2.2.2">𝛼</ci><ci id="S4.SS2.p2.2.m2.1.2.2.2.3.cmml" xref="S4.SS2.p2.2.m2.1.2.2.2.3">𝜎</ci></apply><apply id="S4.SS2.p2.2.m2.1.2.2.3.cmml" xref="S4.SS2.p2.2.m2.1.2.2.3"><minus id="S4.SS2.p2.2.m2.1.2.2.3.1.cmml" xref="S4.SS2.p2.2.m2.1.2.2.3"></minus><cn id="S4.SS2.p2.2.m2.1.2.2.3.2.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.2.2.3.2">1</cn></apply></apply><ci id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">\alpha_{\sigma}^{-1}(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w )</annotation></semantics></math> may become rather large, even for small <math alttext="|w|" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><mrow id="S4.SS2.p2.3.m3.1.2.2" xref="S4.SS2.p2.3.m3.1.2.1.cmml"><mo id="S4.SS2.p2.3.m3.1.2.2.1" stretchy="false" xref="S4.SS2.p2.3.m3.1.2.1.1.cmml">|</mo><mi id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml">w</mi><mo id="S4.SS2.p2.3.m3.1.2.2.2" stretchy="false" xref="S4.SS2.p2.3.m3.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><apply id="S4.SS2.p2.3.m3.1.2.1.cmml" xref="S4.SS2.p2.3.m3.1.2.2"><abs id="S4.SS2.p2.3.m3.1.2.1.1.cmml" xref="S4.SS2.p2.3.m3.1.2.2.1"></abs><ci id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">|w|</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">| italic_w |</annotation></semantics></math>. In this subsection we explain how a more efficient evaluation technique is obtained (compare formulas (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.E3" title="In Proposition 4.2. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.3</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E5" title="In Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.5</span></a>)) and we give an example of a typical computation.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.10">Given a morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mrow id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.m1.1.1.2" xref="S4.SS2.p3.1.m1.1.1.2.cmml">σ</mi><mo id="S4.SS2.p3.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS2.p3.1.m1.1.1.1.cmml">:</mo><mrow id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml"><msup id="S4.SS2.p3.1.m1.1.1.3.2" xref="S4.SS2.p3.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.1.m1.1.1.3.2.2" xref="S4.SS2.p3.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S4.SS2.p3.1.m1.1.1.3.2.3" xref="S4.SS2.p3.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo 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cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.6.1.cmml" xref="S4.SS2.p3.2.m2.1.1.6">superscript</csymbol><ci id="S4.SS2.p3.2.m2.1.1.6.2.cmml" xref="S4.SS2.p3.2.m2.1.1.6.2">𝒜</ci><times id="S4.SS2.p3.2.m2.1.1.6.3.cmml" xref="S4.SS2.p3.2.m2.1.1.6.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">w=x_{1}\ldots x_{n}\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">italic_w = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and its image <math alttext="\sigma(w)=y_{1}\ldots y_{m}\in\cal B^{*}" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m3.1"><semantics id="S4.SS2.p3.3.m3.1a"><mrow id="S4.SS2.p3.3.m3.1.2" xref="S4.SS2.p3.3.m3.1.2.cmml"><mrow id="S4.SS2.p3.3.m3.1.2.2" xref="S4.SS2.p3.3.m3.1.2.2.cmml"><mi id="S4.SS2.p3.3.m3.1.2.2.2" xref="S4.SS2.p3.3.m3.1.2.2.2.cmml">σ</mi><mo id="S4.SS2.p3.3.m3.1.2.2.1" xref="S4.SS2.p3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.p3.3.m3.1.2.2.3.2" xref="S4.SS2.p3.3.m3.1.2.2.cmml"><mo id="S4.SS2.p3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S4.SS2.p3.3.m3.1.2.2.cmml">(</mo><mi id="S4.SS2.p3.3.m3.1.1" xref="S4.SS2.p3.3.m3.1.1.cmml">w</mi><mo id="S4.SS2.p3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S4.SS2.p3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p3.3.m3.1.2.3" xref="S4.SS2.p3.3.m3.1.2.3.cmml">=</mo><mrow id="S4.SS2.p3.3.m3.1.2.4" xref="S4.SS2.p3.3.m3.1.2.4.cmml"><msub id="S4.SS2.p3.3.m3.1.2.4.2" xref="S4.SS2.p3.3.m3.1.2.4.2.cmml"><mi id="S4.SS2.p3.3.m3.1.2.4.2.2" xref="S4.SS2.p3.3.m3.1.2.4.2.2.cmml">y</mi><mn id="S4.SS2.p3.3.m3.1.2.4.2.3" xref="S4.SS2.p3.3.m3.1.2.4.2.3.cmml">1</mn></msub><mo id="S4.SS2.p3.3.m3.1.2.4.1" xref="S4.SS2.p3.3.m3.1.2.4.1.cmml">⁢</mo><mi id="S4.SS2.p3.3.m3.1.2.4.3" mathvariant="normal" xref="S4.SS2.p3.3.m3.1.2.4.3.cmml">…</mi><mo id="S4.SS2.p3.3.m3.1.2.4.1a" xref="S4.SS2.p3.3.m3.1.2.4.1.cmml">⁢</mo><msub id="S4.SS2.p3.3.m3.1.2.4.4" xref="S4.SS2.p3.3.m3.1.2.4.4.cmml"><mi id="S4.SS2.p3.3.m3.1.2.4.4.2" xref="S4.SS2.p3.3.m3.1.2.4.4.2.cmml">y</mi><mi id="S4.SS2.p3.3.m3.1.2.4.4.3" xref="S4.SS2.p3.3.m3.1.2.4.4.3.cmml">m</mi></msub></mrow><mo id="S4.SS2.p3.3.m3.1.2.5" xref="S4.SS2.p3.3.m3.1.2.5.cmml">∈</mo><msup id="S4.SS2.p3.3.m3.1.2.6" xref="S4.SS2.p3.3.m3.1.2.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.3.m3.1.2.6.2" xref="S4.SS2.p3.3.m3.1.2.6.2.cmml">ℬ</mi><mo id="S4.SS2.p3.3.m3.1.2.6.3" xref="S4.SS2.p3.3.m3.1.2.6.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m3.1b"><apply id="S4.SS2.p3.3.m3.1.2.cmml" xref="S4.SS2.p3.3.m3.1.2"><and id="S4.SS2.p3.3.m3.1.2a.cmml" xref="S4.SS2.p3.3.m3.1.2"></and><apply id="S4.SS2.p3.3.m3.1.2b.cmml" xref="S4.SS2.p3.3.m3.1.2"><eq id="S4.SS2.p3.3.m3.1.2.3.cmml" xref="S4.SS2.p3.3.m3.1.2.3"></eq><apply id="S4.SS2.p3.3.m3.1.2.2.cmml" xref="S4.SS2.p3.3.m3.1.2.2"><times id="S4.SS2.p3.3.m3.1.2.2.1.cmml" xref="S4.SS2.p3.3.m3.1.2.2.1"></times><ci id="S4.SS2.p3.3.m3.1.2.2.2.cmml" xref="S4.SS2.p3.3.m3.1.2.2.2">𝜎</ci><ci id="S4.SS2.p3.3.m3.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1">𝑤</ci></apply><apply id="S4.SS2.p3.3.m3.1.2.4.cmml" xref="S4.SS2.p3.3.m3.1.2.4"><times id="S4.SS2.p3.3.m3.1.2.4.1.cmml" xref="S4.SS2.p3.3.m3.1.2.4.1"></times><apply id="S4.SS2.p3.3.m3.1.2.4.2.cmml" xref="S4.SS2.p3.3.m3.1.2.4.2"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m3.1.2.4.2.1.cmml" xref="S4.SS2.p3.3.m3.1.2.4.2">subscript</csymbol><ci id="S4.SS2.p3.3.m3.1.2.4.2.2.cmml" xref="S4.SS2.p3.3.m3.1.2.4.2.2">𝑦</ci><cn id="S4.SS2.p3.3.m3.1.2.4.2.3.cmml" type="integer" xref="S4.SS2.p3.3.m3.1.2.4.2.3">1</cn></apply><ci id="S4.SS2.p3.3.m3.1.2.4.3.cmml" xref="S4.SS2.p3.3.m3.1.2.4.3">…</ci><apply id="S4.SS2.p3.3.m3.1.2.4.4.cmml" xref="S4.SS2.p3.3.m3.1.2.4.4"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m3.1.2.4.4.1.cmml" xref="S4.SS2.p3.3.m3.1.2.4.4">subscript</csymbol><ci id="S4.SS2.p3.3.m3.1.2.4.4.2.cmml" xref="S4.SS2.p3.3.m3.1.2.4.4.2">𝑦</ci><ci id="S4.SS2.p3.3.m3.1.2.4.4.3.cmml" xref="S4.SS2.p3.3.m3.1.2.4.4.3">𝑚</ci></apply></apply></apply><apply id="S4.SS2.p3.3.m3.1.2c.cmml" xref="S4.SS2.p3.3.m3.1.2"><in id="S4.SS2.p3.3.m3.1.2.5.cmml" xref="S4.SS2.p3.3.m3.1.2.5"></in><share href="https://arxiv.org/html/2211.11234v4#S4.SS2.p3.3.m3.1.2.4.cmml" id="S4.SS2.p3.3.m3.1.2d.cmml" xref="S4.SS2.p3.3.m3.1.2"></share><apply id="S4.SS2.p3.3.m3.1.2.6.cmml" xref="S4.SS2.p3.3.m3.1.2.6"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m3.1.2.6.1.cmml" xref="S4.SS2.p3.3.m3.1.2.6">superscript</csymbol><ci id="S4.SS2.p3.3.m3.1.2.6.2.cmml" xref="S4.SS2.p3.3.m3.1.2.6.2">ℬ</ci><times id="S4.SS2.p3.3.m3.1.2.6.3.cmml" xref="S4.SS2.p3.3.m3.1.2.6.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m3.1c">\sigma(w)=y_{1}\ldots y_{m}\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.m3.1d">italic_σ ( italic_w ) = italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, with letters <math alttext="x_{i}\in\cal A" class="ltx_Math" display="inline" id="S4.SS2.p3.4.m4.1"><semantics id="S4.SS2.p3.4.m4.1a"><mrow id="S4.SS2.p3.4.m4.1.1" xref="S4.SS2.p3.4.m4.1.1.cmml"><msub id="S4.SS2.p3.4.m4.1.1.2" xref="S4.SS2.p3.4.m4.1.1.2.cmml"><mi id="S4.SS2.p3.4.m4.1.1.2.2" xref="S4.SS2.p3.4.m4.1.1.2.2.cmml">x</mi><mi id="S4.SS2.p3.4.m4.1.1.2.3" xref="S4.SS2.p3.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS2.p3.4.m4.1.1.1" xref="S4.SS2.p3.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.4.m4.1.1.3" xref="S4.SS2.p3.4.m4.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.4.m4.1b"><apply id="S4.SS2.p3.4.m4.1.1.cmml" xref="S4.SS2.p3.4.m4.1.1"><in id="S4.SS2.p3.4.m4.1.1.1.cmml" xref="S4.SS2.p3.4.m4.1.1.1"></in><apply id="S4.SS2.p3.4.m4.1.1.2.cmml" xref="S4.SS2.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.4.m4.1.1.2.1.cmml" xref="S4.SS2.p3.4.m4.1.1.2">subscript</csymbol><ci id="S4.SS2.p3.4.m4.1.1.2.2.cmml" xref="S4.SS2.p3.4.m4.1.1.2.2">𝑥</ci><ci id="S4.SS2.p3.4.m4.1.1.2.3.cmml" xref="S4.SS2.p3.4.m4.1.1.2.3">𝑖</ci></apply><ci id="S4.SS2.p3.4.m4.1.1.3.cmml" xref="S4.SS2.p3.4.m4.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.4.m4.1c">x_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> and <math alttext="y_{j}\in\cal B" class="ltx_Math" display="inline" id="S4.SS2.p3.5.m5.1"><semantics id="S4.SS2.p3.5.m5.1a"><mrow id="S4.SS2.p3.5.m5.1.1" xref="S4.SS2.p3.5.m5.1.1.cmml"><msub id="S4.SS2.p3.5.m5.1.1.2" xref="S4.SS2.p3.5.m5.1.1.2.cmml"><mi id="S4.SS2.p3.5.m5.1.1.2.2" xref="S4.SS2.p3.5.m5.1.1.2.2.cmml">y</mi><mi id="S4.SS2.p3.5.m5.1.1.2.3" xref="S4.SS2.p3.5.m5.1.1.2.3.cmml">j</mi></msub><mo id="S4.SS2.p3.5.m5.1.1.1" xref="S4.SS2.p3.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.5.m5.1.1.3" xref="S4.SS2.p3.5.m5.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.5.m5.1b"><apply id="S4.SS2.p3.5.m5.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1"><in id="S4.SS2.p3.5.m5.1.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1.1"></in><apply id="S4.SS2.p3.5.m5.1.1.2.cmml" xref="S4.SS2.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.5.m5.1.1.2.1.cmml" xref="S4.SS2.p3.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS2.p3.5.m5.1.1.2.2.cmml" xref="S4.SS2.p3.5.m5.1.1.2.2">𝑦</ci><ci id="S4.SS2.p3.5.m5.1.1.2.3.cmml" xref="S4.SS2.p3.5.m5.1.1.2.3">𝑗</ci></apply><ci id="S4.SS2.p3.5.m5.1.1.3.cmml" xref="S4.SS2.p3.5.m5.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.5.m5.1c">y_{j}\in\cal B</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.5.m5.1d">italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ caligraphic_B</annotation></semantics></math>. An <span class="ltx_text ltx_font_italic" id="S4.SS2.p3.7.2">occurrence of <math alttext="w^{\prime}\in\cal B^{*}" class="ltx_Math" display="inline" id="S4.SS2.p3.6.1.m1.1"><semantics id="S4.SS2.p3.6.1.m1.1a"><mrow id="S4.SS2.p3.6.1.m1.1.1" xref="S4.SS2.p3.6.1.m1.1.1.cmml"><msup id="S4.SS2.p3.6.1.m1.1.1.2" xref="S4.SS2.p3.6.1.m1.1.1.2.cmml"><mi id="S4.SS2.p3.6.1.m1.1.1.2.2" xref="S4.SS2.p3.6.1.m1.1.1.2.2.cmml">w</mi><mo id="S4.SS2.p3.6.1.m1.1.1.2.3" xref="S4.SS2.p3.6.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS2.p3.6.1.m1.1.1.1" xref="S4.SS2.p3.6.1.m1.1.1.1.cmml">∈</mo><msup id="S4.SS2.p3.6.1.m1.1.1.3" xref="S4.SS2.p3.6.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.6.1.m1.1.1.3.2" xref="S4.SS2.p3.6.1.m1.1.1.3.2.cmml">ℬ</mi><mo id="S4.SS2.p3.6.1.m1.1.1.3.3" xref="S4.SS2.p3.6.1.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.6.1.m1.1b"><apply id="S4.SS2.p3.6.1.m1.1.1.cmml" xref="S4.SS2.p3.6.1.m1.1.1"><in id="S4.SS2.p3.6.1.m1.1.1.1.cmml" xref="S4.SS2.p3.6.1.m1.1.1.1"></in><apply id="S4.SS2.p3.6.1.m1.1.1.2.cmml" xref="S4.SS2.p3.6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.6.1.m1.1.1.2.1.cmml" xref="S4.SS2.p3.6.1.m1.1.1.2">superscript</csymbol><ci id="S4.SS2.p3.6.1.m1.1.1.2.2.cmml" xref="S4.SS2.p3.6.1.m1.1.1.2.2">𝑤</ci><ci id="S4.SS2.p3.6.1.m1.1.1.2.3.cmml" xref="S4.SS2.p3.6.1.m1.1.1.2.3">′</ci></apply><apply id="S4.SS2.p3.6.1.m1.1.1.3.cmml" xref="S4.SS2.p3.6.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.6.1.m1.1.1.3.1.cmml" xref="S4.SS2.p3.6.1.m1.1.1.3">superscript</csymbol><ci id="S4.SS2.p3.6.1.m1.1.1.3.2.cmml" xref="S4.SS2.p3.6.1.m1.1.1.3.2">ℬ</ci><times id="S4.SS2.p3.6.1.m1.1.1.3.3.cmml" xref="S4.SS2.p3.6.1.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.6.1.m1.1c">w^{\prime}\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.6.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S4.SS2.p3.7.2.m2.1"><semantics id="S4.SS2.p3.7.2.m2.1a"><mrow id="S4.SS2.p3.7.2.m2.1.2" xref="S4.SS2.p3.7.2.m2.1.2.cmml"><mi id="S4.SS2.p3.7.2.m2.1.2.2" xref="S4.SS2.p3.7.2.m2.1.2.2.cmml">σ</mi><mo id="S4.SS2.p3.7.2.m2.1.2.1" xref="S4.SS2.p3.7.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p3.7.2.m2.1.2.3.2" xref="S4.SS2.p3.7.2.m2.1.2.cmml"><mo id="S4.SS2.p3.7.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p3.7.2.m2.1.2.cmml">(</mo><mi id="S4.SS2.p3.7.2.m2.1.1" xref="S4.SS2.p3.7.2.m2.1.1.cmml">w</mi><mo id="S4.SS2.p3.7.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p3.7.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.7.2.m2.1b"><apply id="S4.SS2.p3.7.2.m2.1.2.cmml" xref="S4.SS2.p3.7.2.m2.1.2"><times id="S4.SS2.p3.7.2.m2.1.2.1.cmml" xref="S4.SS2.p3.7.2.m2.1.2.1"></times><ci id="S4.SS2.p3.7.2.m2.1.2.2.cmml" xref="S4.SS2.p3.7.2.m2.1.2.2">𝜎</ci><ci id="S4.SS2.p3.7.2.m2.1.1.cmml" xref="S4.SS2.p3.7.2.m2.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.7.2.m2.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.7.2.m2.1d">italic_σ ( italic_w )</annotation></semantics></math></span> is a factor <math alttext="y_{k^{\prime}}\ldots y_{\ell^{\prime}}" class="ltx_Math" display="inline" id="S4.SS2.p3.8.m6.1"><semantics id="S4.SS2.p3.8.m6.1a"><mrow id="S4.SS2.p3.8.m6.1.1" xref="S4.SS2.p3.8.m6.1.1.cmml"><msub id="S4.SS2.p3.8.m6.1.1.2" xref="S4.SS2.p3.8.m6.1.1.2.cmml"><mi id="S4.SS2.p3.8.m6.1.1.2.2" xref="S4.SS2.p3.8.m6.1.1.2.2.cmml">y</mi><msup id="S4.SS2.p3.8.m6.1.1.2.3" xref="S4.SS2.p3.8.m6.1.1.2.3.cmml"><mi id="S4.SS2.p3.8.m6.1.1.2.3.2" xref="S4.SS2.p3.8.m6.1.1.2.3.2.cmml">k</mi><mo id="S4.SS2.p3.8.m6.1.1.2.3.3" xref="S4.SS2.p3.8.m6.1.1.2.3.3.cmml">′</mo></msup></msub><mo id="S4.SS2.p3.8.m6.1.1.1" xref="S4.SS2.p3.8.m6.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p3.8.m6.1.1.3" mathvariant="normal" xref="S4.SS2.p3.8.m6.1.1.3.cmml">…</mi><mo id="S4.SS2.p3.8.m6.1.1.1a" xref="S4.SS2.p3.8.m6.1.1.1.cmml">⁢</mo><msub id="S4.SS2.p3.8.m6.1.1.4" xref="S4.SS2.p3.8.m6.1.1.4.cmml"><mi id="S4.SS2.p3.8.m6.1.1.4.2" xref="S4.SS2.p3.8.m6.1.1.4.2.cmml">y</mi><msup id="S4.SS2.p3.8.m6.1.1.4.3" xref="S4.SS2.p3.8.m6.1.1.4.3.cmml"><mi id="S4.SS2.p3.8.m6.1.1.4.3.2" mathvariant="normal" xref="S4.SS2.p3.8.m6.1.1.4.3.2.cmml">ℓ</mi><mo id="S4.SS2.p3.8.m6.1.1.4.3.3" xref="S4.SS2.p3.8.m6.1.1.4.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.8.m6.1b"><apply id="S4.SS2.p3.8.m6.1.1.cmml" xref="S4.SS2.p3.8.m6.1.1"><times id="S4.SS2.p3.8.m6.1.1.1.cmml" xref="S4.SS2.p3.8.m6.1.1.1"></times><apply id="S4.SS2.p3.8.m6.1.1.2.cmml" xref="S4.SS2.p3.8.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.8.m6.1.1.2.1.cmml" xref="S4.SS2.p3.8.m6.1.1.2">subscript</csymbol><ci id="S4.SS2.p3.8.m6.1.1.2.2.cmml" xref="S4.SS2.p3.8.m6.1.1.2.2">𝑦</ci><apply id="S4.SS2.p3.8.m6.1.1.2.3.cmml" xref="S4.SS2.p3.8.m6.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS2.p3.8.m6.1.1.2.3.1.cmml" xref="S4.SS2.p3.8.m6.1.1.2.3">superscript</csymbol><ci id="S4.SS2.p3.8.m6.1.1.2.3.2.cmml" xref="S4.SS2.p3.8.m6.1.1.2.3.2">𝑘</ci><ci id="S4.SS2.p3.8.m6.1.1.2.3.3.cmml" xref="S4.SS2.p3.8.m6.1.1.2.3.3">′</ci></apply></apply><ci id="S4.SS2.p3.8.m6.1.1.3.cmml" xref="S4.SS2.p3.8.m6.1.1.3">…</ci><apply id="S4.SS2.p3.8.m6.1.1.4.cmml" xref="S4.SS2.p3.8.m6.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p3.8.m6.1.1.4.1.cmml" xref="S4.SS2.p3.8.m6.1.1.4">subscript</csymbol><ci id="S4.SS2.p3.8.m6.1.1.4.2.cmml" xref="S4.SS2.p3.8.m6.1.1.4.2">𝑦</ci><apply id="S4.SS2.p3.8.m6.1.1.4.3.cmml" xref="S4.SS2.p3.8.m6.1.1.4.3"><csymbol cd="ambiguous" id="S4.SS2.p3.8.m6.1.1.4.3.1.cmml" xref="S4.SS2.p3.8.m6.1.1.4.3">superscript</csymbol><ci id="S4.SS2.p3.8.m6.1.1.4.3.2.cmml" xref="S4.SS2.p3.8.m6.1.1.4.3.2">ℓ</ci><ci id="S4.SS2.p3.8.m6.1.1.4.3.3.cmml" xref="S4.SS2.p3.8.m6.1.1.4.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.8.m6.1c">y_{k^{\prime}}\ldots y_{\ell^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.8.m6.1d">italic_y start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT … italic_y start_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="y_{1}\ldots y_{m}" class="ltx_Math" display="inline" id="S4.SS2.p3.9.m7.1"><semantics id="S4.SS2.p3.9.m7.1a"><mrow id="S4.SS2.p3.9.m7.1.1" xref="S4.SS2.p3.9.m7.1.1.cmml"><msub id="S4.SS2.p3.9.m7.1.1.2" xref="S4.SS2.p3.9.m7.1.1.2.cmml"><mi id="S4.SS2.p3.9.m7.1.1.2.2" xref="S4.SS2.p3.9.m7.1.1.2.2.cmml">y</mi><mn id="S4.SS2.p3.9.m7.1.1.2.3" xref="S4.SS2.p3.9.m7.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.p3.9.m7.1.1.1" xref="S4.SS2.p3.9.m7.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p3.9.m7.1.1.3" mathvariant="normal" xref="S4.SS2.p3.9.m7.1.1.3.cmml">…</mi><mo id="S4.SS2.p3.9.m7.1.1.1a" xref="S4.SS2.p3.9.m7.1.1.1.cmml">⁢</mo><msub id="S4.SS2.p3.9.m7.1.1.4" xref="S4.SS2.p3.9.m7.1.1.4.cmml"><mi id="S4.SS2.p3.9.m7.1.1.4.2" xref="S4.SS2.p3.9.m7.1.1.4.2.cmml">y</mi><mi id="S4.SS2.p3.9.m7.1.1.4.3" xref="S4.SS2.p3.9.m7.1.1.4.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.9.m7.1b"><apply id="S4.SS2.p3.9.m7.1.1.cmml" xref="S4.SS2.p3.9.m7.1.1"><times id="S4.SS2.p3.9.m7.1.1.1.cmml" xref="S4.SS2.p3.9.m7.1.1.1"></times><apply id="S4.SS2.p3.9.m7.1.1.2.cmml" xref="S4.SS2.p3.9.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.9.m7.1.1.2.1.cmml" xref="S4.SS2.p3.9.m7.1.1.2">subscript</csymbol><ci id="S4.SS2.p3.9.m7.1.1.2.2.cmml" xref="S4.SS2.p3.9.m7.1.1.2.2">𝑦</ci><cn id="S4.SS2.p3.9.m7.1.1.2.3.cmml" type="integer" xref="S4.SS2.p3.9.m7.1.1.2.3">1</cn></apply><ci id="S4.SS2.p3.9.m7.1.1.3.cmml" xref="S4.SS2.p3.9.m7.1.1.3">…</ci><apply id="S4.SS2.p3.9.m7.1.1.4.cmml" xref="S4.SS2.p3.9.m7.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p3.9.m7.1.1.4.1.cmml" xref="S4.SS2.p3.9.m7.1.1.4">subscript</csymbol><ci id="S4.SS2.p3.9.m7.1.1.4.2.cmml" xref="S4.SS2.p3.9.m7.1.1.4.2">𝑦</ci><ci id="S4.SS2.p3.9.m7.1.1.4.3.cmml" xref="S4.SS2.p3.9.m7.1.1.4.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.9.m7.1c">y_{1}\ldots y_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.9.m7.1d">italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_y start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> which satisfies <math alttext="w^{\prime}=y_{k^{\prime}}\ldots y_{\ell^{\prime}}\,." class="ltx_Math" display="inline" id="S4.SS2.p3.10.m8.1"><semantics id="S4.SS2.p3.10.m8.1a"><mrow id="S4.SS2.p3.10.m8.1.1.1" xref="S4.SS2.p3.10.m8.1.1.1.1.cmml"><mrow id="S4.SS2.p3.10.m8.1.1.1.1" xref="S4.SS2.p3.10.m8.1.1.1.1.cmml"><msup id="S4.SS2.p3.10.m8.1.1.1.1.2" xref="S4.SS2.p3.10.m8.1.1.1.1.2.cmml"><mi id="S4.SS2.p3.10.m8.1.1.1.1.2.2" xref="S4.SS2.p3.10.m8.1.1.1.1.2.2.cmml">w</mi><mo id="S4.SS2.p3.10.m8.1.1.1.1.2.3" xref="S4.SS2.p3.10.m8.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS2.p3.10.m8.1.1.1.1.1" xref="S4.SS2.p3.10.m8.1.1.1.1.1.cmml">=</mo><mrow id="S4.SS2.p3.10.m8.1.1.1.1.3" xref="S4.SS2.p3.10.m8.1.1.1.1.3.cmml"><msub id="S4.SS2.p3.10.m8.1.1.1.1.3.2" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.cmml"><mi id="S4.SS2.p3.10.m8.1.1.1.1.3.2.2" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.2.cmml">y</mi><msup id="S4.SS2.p3.10.m8.1.1.1.1.3.2.3" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.cmml"><mi id="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.2" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.2.cmml">k</mi><mo id="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.3" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.3.cmml">′</mo></msup></msub><mo id="S4.SS2.p3.10.m8.1.1.1.1.3.1" xref="S4.SS2.p3.10.m8.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.SS2.p3.10.m8.1.1.1.1.3.3" mathvariant="normal" xref="S4.SS2.p3.10.m8.1.1.1.1.3.3.cmml">…</mi><mo id="S4.SS2.p3.10.m8.1.1.1.1.3.1a" xref="S4.SS2.p3.10.m8.1.1.1.1.3.1.cmml">⁢</mo><msub id="S4.SS2.p3.10.m8.1.1.1.1.3.4" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.cmml"><mi id="S4.SS2.p3.10.m8.1.1.1.1.3.4.2" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.2.cmml">y</mi><msup id="S4.SS2.p3.10.m8.1.1.1.1.3.4.3" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.cmml"><mi id="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.2" mathvariant="normal" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.2.cmml">ℓ</mi><mo id="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.3" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.3.cmml">′</mo></msup></msub></mrow></mrow><mo id="S4.SS2.p3.10.m8.1.1.1.2" lspace="0em" xref="S4.SS2.p3.10.m8.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.10.m8.1b"><apply id="S4.SS2.p3.10.m8.1.1.1.1.cmml" xref="S4.SS2.p3.10.m8.1.1.1"><eq id="S4.SS2.p3.10.m8.1.1.1.1.1.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.1"></eq><apply id="S4.SS2.p3.10.m8.1.1.1.1.2.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p3.10.m8.1.1.1.1.2.1.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.2">superscript</csymbol><ci id="S4.SS2.p3.10.m8.1.1.1.1.2.2.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.2.2">𝑤</ci><ci id="S4.SS2.p3.10.m8.1.1.1.1.2.3.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.2.3">′</ci></apply><apply id="S4.SS2.p3.10.m8.1.1.1.1.3.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3"><times id="S4.SS2.p3.10.m8.1.1.1.1.3.1.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.1"></times><apply id="S4.SS2.p3.10.m8.1.1.1.1.3.2.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p3.10.m8.1.1.1.1.3.2.1.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2">subscript</csymbol><ci id="S4.SS2.p3.10.m8.1.1.1.1.3.2.2.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.2">𝑦</ci><apply id="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.1.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.3">superscript</csymbol><ci id="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.2.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.2">𝑘</ci><ci id="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.3.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.2.3.3">′</ci></apply></apply><ci id="S4.SS2.p3.10.m8.1.1.1.1.3.3.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.3">…</ci><apply id="S4.SS2.p3.10.m8.1.1.1.1.3.4.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4"><csymbol cd="ambiguous" id="S4.SS2.p3.10.m8.1.1.1.1.3.4.1.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4">subscript</csymbol><ci id="S4.SS2.p3.10.m8.1.1.1.1.3.4.2.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.2">𝑦</ci><apply id="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.3"><csymbol cd="ambiguous" id="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.1.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.3">superscript</csymbol><ci id="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.2.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.2">ℓ</ci><ci id="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.3.cmml" xref="S4.SS2.p3.10.m8.1.1.1.1.3.4.3.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.10.m8.1c">w^{\prime}=y_{k^{\prime}}\ldots y_{\ell^{\prime}}\,.</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.10.m8.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_y start_POSTSUBSCRIPT italic_k start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT … italic_y start_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT .</annotation></semantics></math></p> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.21">An occurrence of <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.1"><semantics id="S4.SS2.p4.1.m1.1a"><msup id="S4.SS2.p4.1.m1.1.1" xref="S4.SS2.p4.1.m1.1.1.cmml"><mi id="S4.SS2.p4.1.m1.1.1.2" xref="S4.SS2.p4.1.m1.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.1.m1.1.1.3" xref="S4.SS2.p4.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><apply id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.1.m1.1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.p4.1.m1.1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.1.2">𝑤</ci><ci id="S4.SS2.p4.1.m1.1.1.3.cmml" xref="S4.SS2.p4.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S4.SS2.p4.2.m2.1"><semantics id="S4.SS2.p4.2.m2.1a"><mrow id="S4.SS2.p4.2.m2.1.2" xref="S4.SS2.p4.2.m2.1.2.cmml"><mi id="S4.SS2.p4.2.m2.1.2.2" xref="S4.SS2.p4.2.m2.1.2.2.cmml">σ</mi><mo id="S4.SS2.p4.2.m2.1.2.1" xref="S4.SS2.p4.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.2.m2.1.2.3.2" xref="S4.SS2.p4.2.m2.1.2.cmml"><mo id="S4.SS2.p4.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.2.m2.1.2.cmml">(</mo><mi id="S4.SS2.p4.2.m2.1.1" xref="S4.SS2.p4.2.m2.1.1.cmml">w</mi><mo id="S4.SS2.p4.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.2.m2.1b"><apply id="S4.SS2.p4.2.m2.1.2.cmml" xref="S4.SS2.p4.2.m2.1.2"><times id="S4.SS2.p4.2.m2.1.2.1.cmml" xref="S4.SS2.p4.2.m2.1.2.1"></times><ci id="S4.SS2.p4.2.m2.1.2.2.cmml" xref="S4.SS2.p4.2.m2.1.2.2">𝜎</ci><ci id="S4.SS2.p4.2.m2.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.2.m2.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.2.m2.1d">italic_σ ( italic_w )</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S4.SS2.p4.21.1">essential</span> if its first letter occurs in <math alttext="\sigma(x_{1})" class="ltx_Math" display="inline" id="S4.SS2.p4.3.m3.1"><semantics id="S4.SS2.p4.3.m3.1a"><mrow id="S4.SS2.p4.3.m3.1.1" xref="S4.SS2.p4.3.m3.1.1.cmml"><mi id="S4.SS2.p4.3.m3.1.1.3" xref="S4.SS2.p4.3.m3.1.1.3.cmml">σ</mi><mo id="S4.SS2.p4.3.m3.1.1.2" xref="S4.SS2.p4.3.m3.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.3.m3.1.1.1.1" xref="S4.SS2.p4.3.m3.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.3.m3.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.3.m3.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.3.m3.1.1.1.1.1" xref="S4.SS2.p4.3.m3.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.3.m3.1.1.1.1.1.2" xref="S4.SS2.p4.3.m3.1.1.1.1.1.2.cmml">x</mi><mn id="S4.SS2.p4.3.m3.1.1.1.1.1.3" xref="S4.SS2.p4.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.p4.3.m3.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.3.m3.1b"><apply id="S4.SS2.p4.3.m3.1.1.cmml" xref="S4.SS2.p4.3.m3.1.1"><times id="S4.SS2.p4.3.m3.1.1.2.cmml" xref="S4.SS2.p4.3.m3.1.1.2"></times><ci id="S4.SS2.p4.3.m3.1.1.3.cmml" xref="S4.SS2.p4.3.m3.1.1.3">𝜎</ci><apply id="S4.SS2.p4.3.m3.1.1.1.1.1.cmml" xref="S4.SS2.p4.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.3.m3.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.3.m3.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.3.m3.1.1.1.1.1.2">𝑥</ci><cn id="S4.SS2.p4.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p4.3.m3.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.3.m3.1c">\sigma(x_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.3.m3.1d">italic_σ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> and its last letter in <math alttext="\sigma(x_{n})" class="ltx_Math" display="inline" id="S4.SS2.p4.4.m4.1"><semantics id="S4.SS2.p4.4.m4.1a"><mrow id="S4.SS2.p4.4.m4.1.1" xref="S4.SS2.p4.4.m4.1.1.cmml"><mi id="S4.SS2.p4.4.m4.1.1.3" xref="S4.SS2.p4.4.m4.1.1.3.cmml">σ</mi><mo id="S4.SS2.p4.4.m4.1.1.2" xref="S4.SS2.p4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.4.m4.1.1.1.1" xref="S4.SS2.p4.4.m4.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.4.m4.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.4.m4.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.4.m4.1.1.1.1.1" xref="S4.SS2.p4.4.m4.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.4.m4.1.1.1.1.1.2" xref="S4.SS2.p4.4.m4.1.1.1.1.1.2.cmml">x</mi><mi id="S4.SS2.p4.4.m4.1.1.1.1.1.3" xref="S4.SS2.p4.4.m4.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.SS2.p4.4.m4.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.4.m4.1b"><apply id="S4.SS2.p4.4.m4.1.1.cmml" xref="S4.SS2.p4.4.m4.1.1"><times id="S4.SS2.p4.4.m4.1.1.2.cmml" xref="S4.SS2.p4.4.m4.1.1.2"></times><ci id="S4.SS2.p4.4.m4.1.1.3.cmml" xref="S4.SS2.p4.4.m4.1.1.3">𝜎</ci><apply id="S4.SS2.p4.4.m4.1.1.1.1.1.cmml" xref="S4.SS2.p4.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.4.m4.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.4.m4.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.4.m4.1.1.1.1.1.2">𝑥</ci><ci id="S4.SS2.p4.4.m4.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.4.m4.1.1.1.1.1.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.4.m4.1c">\sigma(x_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.4.m4.1d">italic_σ ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>. In particular, for <math alttext="|w|=1" class="ltx_Math" display="inline" id="S4.SS2.p4.5.m5.1"><semantics id="S4.SS2.p4.5.m5.1a"><mrow id="S4.SS2.p4.5.m5.1.2" xref="S4.SS2.p4.5.m5.1.2.cmml"><mrow id="S4.SS2.p4.5.m5.1.2.2.2" xref="S4.SS2.p4.5.m5.1.2.2.1.cmml"><mo id="S4.SS2.p4.5.m5.1.2.2.2.1" stretchy="false" xref="S4.SS2.p4.5.m5.1.2.2.1.1.cmml">|</mo><mi id="S4.SS2.p4.5.m5.1.1" xref="S4.SS2.p4.5.m5.1.1.cmml">w</mi><mo id="S4.SS2.p4.5.m5.1.2.2.2.2" stretchy="false" xref="S4.SS2.p4.5.m5.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS2.p4.5.m5.1.2.1" xref="S4.SS2.p4.5.m5.1.2.1.cmml">=</mo><mn id="S4.SS2.p4.5.m5.1.2.3" xref="S4.SS2.p4.5.m5.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.5.m5.1b"><apply id="S4.SS2.p4.5.m5.1.2.cmml" xref="S4.SS2.p4.5.m5.1.2"><eq id="S4.SS2.p4.5.m5.1.2.1.cmml" xref="S4.SS2.p4.5.m5.1.2.1"></eq><apply id="S4.SS2.p4.5.m5.1.2.2.1.cmml" xref="S4.SS2.p4.5.m5.1.2.2.2"><abs id="S4.SS2.p4.5.m5.1.2.2.1.1.cmml" xref="S4.SS2.p4.5.m5.1.2.2.2.1"></abs><ci id="S4.SS2.p4.5.m5.1.1.cmml" xref="S4.SS2.p4.5.m5.1.1">𝑤</ci></apply><cn id="S4.SS2.p4.5.m5.1.2.3.cmml" type="integer" xref="S4.SS2.p4.5.m5.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.5.m5.1c">|w|=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.5.m5.1d">| italic_w | = 1</annotation></semantics></math> any occurrence of <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p4.6.m6.1"><semantics id="S4.SS2.p4.6.m6.1a"><msup id="S4.SS2.p4.6.m6.1.1" xref="S4.SS2.p4.6.m6.1.1.cmml"><mi id="S4.SS2.p4.6.m6.1.1.2" xref="S4.SS2.p4.6.m6.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.6.m6.1.1.3" xref="S4.SS2.p4.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.6.m6.1b"><apply id="S4.SS2.p4.6.m6.1.1.cmml" xref="S4.SS2.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.6.m6.1.1.1.cmml" xref="S4.SS2.p4.6.m6.1.1">superscript</csymbol><ci id="S4.SS2.p4.6.m6.1.1.2.cmml" xref="S4.SS2.p4.6.m6.1.1.2">𝑤</ci><ci id="S4.SS2.p4.6.m6.1.1.3.cmml" xref="S4.SS2.p4.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.6.m6.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.6.m6.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S4.SS2.p4.7.m7.1"><semantics id="S4.SS2.p4.7.m7.1a"><mrow id="S4.SS2.p4.7.m7.1.2" xref="S4.SS2.p4.7.m7.1.2.cmml"><mi id="S4.SS2.p4.7.m7.1.2.2" xref="S4.SS2.p4.7.m7.1.2.2.cmml">σ</mi><mo id="S4.SS2.p4.7.m7.1.2.1" xref="S4.SS2.p4.7.m7.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.7.m7.1.2.3.2" xref="S4.SS2.p4.7.m7.1.2.cmml"><mo id="S4.SS2.p4.7.m7.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.7.m7.1.2.cmml">(</mo><mi id="S4.SS2.p4.7.m7.1.1" xref="S4.SS2.p4.7.m7.1.1.cmml">w</mi><mo id="S4.SS2.p4.7.m7.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.7.m7.1b"><apply id="S4.SS2.p4.7.m7.1.2.cmml" xref="S4.SS2.p4.7.m7.1.2"><times id="S4.SS2.p4.7.m7.1.2.1.cmml" xref="S4.SS2.p4.7.m7.1.2.1"></times><ci id="S4.SS2.p4.7.m7.1.2.2.cmml" xref="S4.SS2.p4.7.m7.1.2.2">𝜎</ci><ci id="S4.SS2.p4.7.m7.1.1.cmml" xref="S4.SS2.p4.7.m7.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.7.m7.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.7.m7.1d">italic_σ ( italic_w )</annotation></semantics></math> is essential. If <math alttext="w" class="ltx_Math" display="inline" id="S4.SS2.p4.8.m8.1"><semantics id="S4.SS2.p4.8.m8.1a"><mi id="S4.SS2.p4.8.m8.1.1" xref="S4.SS2.p4.8.m8.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.8.m8.1b"><ci id="S4.SS2.p4.8.m8.1.1.cmml" xref="S4.SS2.p4.8.m8.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.8.m8.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.8.m8.1d">italic_w</annotation></semantics></math> has length <math alttext="|w|=2" class="ltx_Math" display="inline" id="S4.SS2.p4.9.m9.1"><semantics id="S4.SS2.p4.9.m9.1a"><mrow id="S4.SS2.p4.9.m9.1.2" xref="S4.SS2.p4.9.m9.1.2.cmml"><mrow id="S4.SS2.p4.9.m9.1.2.2.2" xref="S4.SS2.p4.9.m9.1.2.2.1.cmml"><mo id="S4.SS2.p4.9.m9.1.2.2.2.1" stretchy="false" xref="S4.SS2.p4.9.m9.1.2.2.1.1.cmml">|</mo><mi id="S4.SS2.p4.9.m9.1.1" xref="S4.SS2.p4.9.m9.1.1.cmml">w</mi><mo id="S4.SS2.p4.9.m9.1.2.2.2.2" stretchy="false" xref="S4.SS2.p4.9.m9.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS2.p4.9.m9.1.2.1" xref="S4.SS2.p4.9.m9.1.2.1.cmml">=</mo><mn id="S4.SS2.p4.9.m9.1.2.3" xref="S4.SS2.p4.9.m9.1.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.9.m9.1b"><apply id="S4.SS2.p4.9.m9.1.2.cmml" xref="S4.SS2.p4.9.m9.1.2"><eq id="S4.SS2.p4.9.m9.1.2.1.cmml" xref="S4.SS2.p4.9.m9.1.2.1"></eq><apply id="S4.SS2.p4.9.m9.1.2.2.1.cmml" xref="S4.SS2.p4.9.m9.1.2.2.2"><abs id="S4.SS2.p4.9.m9.1.2.2.1.1.cmml" xref="S4.SS2.p4.9.m9.1.2.2.2.1"></abs><ci id="S4.SS2.p4.9.m9.1.1.cmml" xref="S4.SS2.p4.9.m9.1.1">𝑤</ci></apply><cn id="S4.SS2.p4.9.m9.1.2.3.cmml" type="integer" xref="S4.SS2.p4.9.m9.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.9.m9.1c">|w|=2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.9.m9.1d">| italic_w | = 2</annotation></semantics></math>, then an occurrence of <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p4.10.m10.1"><semantics id="S4.SS2.p4.10.m10.1a"><msup id="S4.SS2.p4.10.m10.1.1" xref="S4.SS2.p4.10.m10.1.1.cmml"><mi id="S4.SS2.p4.10.m10.1.1.2" xref="S4.SS2.p4.10.m10.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.10.m10.1.1.3" xref="S4.SS2.p4.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.10.m10.1b"><apply id="S4.SS2.p4.10.m10.1.1.cmml" xref="S4.SS2.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.10.m10.1.1.1.cmml" xref="S4.SS2.p4.10.m10.1.1">superscript</csymbol><ci id="S4.SS2.p4.10.m10.1.1.2.cmml" xref="S4.SS2.p4.10.m10.1.1.2">𝑤</ci><ci id="S4.SS2.p4.10.m10.1.1.3.cmml" xref="S4.SS2.p4.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.10.m10.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.10.m10.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S4.SS2.p4.11.m11.1"><semantics id="S4.SS2.p4.11.m11.1a"><mrow id="S4.SS2.p4.11.m11.1.2" xref="S4.SS2.p4.11.m11.1.2.cmml"><mi id="S4.SS2.p4.11.m11.1.2.2" xref="S4.SS2.p4.11.m11.1.2.2.cmml">σ</mi><mo id="S4.SS2.p4.11.m11.1.2.1" xref="S4.SS2.p4.11.m11.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.11.m11.1.2.3.2" xref="S4.SS2.p4.11.m11.1.2.cmml"><mo id="S4.SS2.p4.11.m11.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.11.m11.1.2.cmml">(</mo><mi id="S4.SS2.p4.11.m11.1.1" xref="S4.SS2.p4.11.m11.1.1.cmml">w</mi><mo id="S4.SS2.p4.11.m11.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.11.m11.1b"><apply id="S4.SS2.p4.11.m11.1.2.cmml" xref="S4.SS2.p4.11.m11.1.2"><times id="S4.SS2.p4.11.m11.1.2.1.cmml" xref="S4.SS2.p4.11.m11.1.2.1"></times><ci id="S4.SS2.p4.11.m11.1.2.2.cmml" xref="S4.SS2.p4.11.m11.1.2.2">𝜎</ci><ci id="S4.SS2.p4.11.m11.1.1.cmml" xref="S4.SS2.p4.11.m11.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.11.m11.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.11.m11.1d">italic_σ ( italic_w )</annotation></semantics></math> is essential if the factor <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p4.12.m12.1"><semantics id="S4.SS2.p4.12.m12.1a"><msup id="S4.SS2.p4.12.m12.1.1" xref="S4.SS2.p4.12.m12.1.1.cmml"><mi id="S4.SS2.p4.12.m12.1.1.2" xref="S4.SS2.p4.12.m12.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.12.m12.1.1.3" xref="S4.SS2.p4.12.m12.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.12.m12.1b"><apply id="S4.SS2.p4.12.m12.1.1.cmml" xref="S4.SS2.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.12.m12.1.1.1.cmml" xref="S4.SS2.p4.12.m12.1.1">superscript</csymbol><ci id="S4.SS2.p4.12.m12.1.1.2.cmml" xref="S4.SS2.p4.12.m12.1.1.2">𝑤</ci><ci id="S4.SS2.p4.12.m12.1.1.3.cmml" xref="S4.SS2.p4.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.12.m12.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.12.m12.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> overlaps from <math alttext="\sigma(x_{1})" class="ltx_Math" display="inline" id="S4.SS2.p4.13.m13.1"><semantics id="S4.SS2.p4.13.m13.1a"><mrow id="S4.SS2.p4.13.m13.1.1" xref="S4.SS2.p4.13.m13.1.1.cmml"><mi id="S4.SS2.p4.13.m13.1.1.3" xref="S4.SS2.p4.13.m13.1.1.3.cmml">σ</mi><mo id="S4.SS2.p4.13.m13.1.1.2" xref="S4.SS2.p4.13.m13.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.13.m13.1.1.1.1" xref="S4.SS2.p4.13.m13.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.13.m13.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.13.m13.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.13.m13.1.1.1.1.1" xref="S4.SS2.p4.13.m13.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.13.m13.1.1.1.1.1.2" xref="S4.SS2.p4.13.m13.1.1.1.1.1.2.cmml">x</mi><mn id="S4.SS2.p4.13.m13.1.1.1.1.1.3" xref="S4.SS2.p4.13.m13.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.p4.13.m13.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.13.m13.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.13.m13.1b"><apply id="S4.SS2.p4.13.m13.1.1.cmml" xref="S4.SS2.p4.13.m13.1.1"><times id="S4.SS2.p4.13.m13.1.1.2.cmml" xref="S4.SS2.p4.13.m13.1.1.2"></times><ci id="S4.SS2.p4.13.m13.1.1.3.cmml" xref="S4.SS2.p4.13.m13.1.1.3">𝜎</ci><apply id="S4.SS2.p4.13.m13.1.1.1.1.1.cmml" xref="S4.SS2.p4.13.m13.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.13.m13.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.13.m13.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.13.m13.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.13.m13.1.1.1.1.1.2">𝑥</ci><cn id="S4.SS2.p4.13.m13.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p4.13.m13.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.13.m13.1c">\sigma(x_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.13.m13.1d">italic_σ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> into <math alttext="\sigma(x_{2})" class="ltx_Math" display="inline" id="S4.SS2.p4.14.m14.1"><semantics id="S4.SS2.p4.14.m14.1a"><mrow id="S4.SS2.p4.14.m14.1.1" xref="S4.SS2.p4.14.m14.1.1.cmml"><mi id="S4.SS2.p4.14.m14.1.1.3" xref="S4.SS2.p4.14.m14.1.1.3.cmml">σ</mi><mo id="S4.SS2.p4.14.m14.1.1.2" xref="S4.SS2.p4.14.m14.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.14.m14.1.1.1.1" xref="S4.SS2.p4.14.m14.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.14.m14.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.14.m14.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p4.14.m14.1.1.1.1.1" xref="S4.SS2.p4.14.m14.1.1.1.1.1.cmml"><mi id="S4.SS2.p4.14.m14.1.1.1.1.1.2" xref="S4.SS2.p4.14.m14.1.1.1.1.1.2.cmml">x</mi><mn id="S4.SS2.p4.14.m14.1.1.1.1.1.3" xref="S4.SS2.p4.14.m14.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S4.SS2.p4.14.m14.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.14.m14.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.14.m14.1b"><apply id="S4.SS2.p4.14.m14.1.1.cmml" xref="S4.SS2.p4.14.m14.1.1"><times id="S4.SS2.p4.14.m14.1.1.2.cmml" xref="S4.SS2.p4.14.m14.1.1.2"></times><ci id="S4.SS2.p4.14.m14.1.1.3.cmml" xref="S4.SS2.p4.14.m14.1.1.3">𝜎</ci><apply id="S4.SS2.p4.14.m14.1.1.1.1.1.cmml" xref="S4.SS2.p4.14.m14.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.14.m14.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.14.m14.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p4.14.m14.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.14.m14.1.1.1.1.1.2">𝑥</ci><cn id="S4.SS2.p4.14.m14.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p4.14.m14.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.14.m14.1c">\sigma(x_{2})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.14.m14.1d">italic_σ ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. For <math alttext="|w|\geq 3" class="ltx_Math" display="inline" id="S4.SS2.p4.15.m15.1"><semantics id="S4.SS2.p4.15.m15.1a"><mrow id="S4.SS2.p4.15.m15.1.2" xref="S4.SS2.p4.15.m15.1.2.cmml"><mrow id="S4.SS2.p4.15.m15.1.2.2.2" xref="S4.SS2.p4.15.m15.1.2.2.1.cmml"><mo id="S4.SS2.p4.15.m15.1.2.2.2.1" stretchy="false" xref="S4.SS2.p4.15.m15.1.2.2.1.1.cmml">|</mo><mi id="S4.SS2.p4.15.m15.1.1" xref="S4.SS2.p4.15.m15.1.1.cmml">w</mi><mo id="S4.SS2.p4.15.m15.1.2.2.2.2" stretchy="false" xref="S4.SS2.p4.15.m15.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS2.p4.15.m15.1.2.1" xref="S4.SS2.p4.15.m15.1.2.1.cmml">≥</mo><mn id="S4.SS2.p4.15.m15.1.2.3" xref="S4.SS2.p4.15.m15.1.2.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.15.m15.1b"><apply id="S4.SS2.p4.15.m15.1.2.cmml" xref="S4.SS2.p4.15.m15.1.2"><geq id="S4.SS2.p4.15.m15.1.2.1.cmml" xref="S4.SS2.p4.15.m15.1.2.1"></geq><apply id="S4.SS2.p4.15.m15.1.2.2.1.cmml" xref="S4.SS2.p4.15.m15.1.2.2.2"><abs id="S4.SS2.p4.15.m15.1.2.2.1.1.cmml" xref="S4.SS2.p4.15.m15.1.2.2.2.1"></abs><ci id="S4.SS2.p4.15.m15.1.1.cmml" xref="S4.SS2.p4.15.m15.1.1">𝑤</ci></apply><cn id="S4.SS2.p4.15.m15.1.2.3.cmml" type="integer" xref="S4.SS2.p4.15.m15.1.2.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.15.m15.1c">|w|\geq 3</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.15.m15.1d">| italic_w | ≥ 3</annotation></semantics></math> a factor <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p4.16.m16.1"><semantics id="S4.SS2.p4.16.m16.1a"><msup id="S4.SS2.p4.16.m16.1.1" xref="S4.SS2.p4.16.m16.1.1.cmml"><mi id="S4.SS2.p4.16.m16.1.1.2" xref="S4.SS2.p4.16.m16.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.16.m16.1.1.3" xref="S4.SS2.p4.16.m16.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.16.m16.1b"><apply id="S4.SS2.p4.16.m16.1.1.cmml" xref="S4.SS2.p4.16.m16.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.16.m16.1.1.1.cmml" xref="S4.SS2.p4.16.m16.1.1">superscript</csymbol><ci id="S4.SS2.p4.16.m16.1.1.2.cmml" xref="S4.SS2.p4.16.m16.1.1.2">𝑤</ci><ci id="S4.SS2.p4.16.m16.1.1.3.cmml" xref="S4.SS2.p4.16.m16.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.16.m16.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.16.m16.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> of <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S4.SS2.p4.17.m17.1"><semantics id="S4.SS2.p4.17.m17.1a"><mrow id="S4.SS2.p4.17.m17.1.2" xref="S4.SS2.p4.17.m17.1.2.cmml"><mi id="S4.SS2.p4.17.m17.1.2.2" xref="S4.SS2.p4.17.m17.1.2.2.cmml">σ</mi><mo id="S4.SS2.p4.17.m17.1.2.1" xref="S4.SS2.p4.17.m17.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.17.m17.1.2.3.2" xref="S4.SS2.p4.17.m17.1.2.cmml"><mo id="S4.SS2.p4.17.m17.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.17.m17.1.2.cmml">(</mo><mi id="S4.SS2.p4.17.m17.1.1" xref="S4.SS2.p4.17.m17.1.1.cmml">w</mi><mo id="S4.SS2.p4.17.m17.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.17.m17.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.17.m17.1b"><apply id="S4.SS2.p4.17.m17.1.2.cmml" xref="S4.SS2.p4.17.m17.1.2"><times id="S4.SS2.p4.17.m17.1.2.1.cmml" xref="S4.SS2.p4.17.m17.1.2.1"></times><ci id="S4.SS2.p4.17.m17.1.2.2.cmml" xref="S4.SS2.p4.17.m17.1.2.2">𝜎</ci><ci id="S4.SS2.p4.17.m17.1.1.cmml" xref="S4.SS2.p4.17.m17.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.17.m17.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.17.m17.1d">italic_σ ( italic_w )</annotation></semantics></math> is an essential occurrence if <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p4.18.m18.1"><semantics id="S4.SS2.p4.18.m18.1a"><msup id="S4.SS2.p4.18.m18.1.1" xref="S4.SS2.p4.18.m18.1.1.cmml"><mi id="S4.SS2.p4.18.m18.1.1.2" xref="S4.SS2.p4.18.m18.1.1.2.cmml">w</mi><mo id="S4.SS2.p4.18.m18.1.1.3" xref="S4.SS2.p4.18.m18.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.18.m18.1b"><apply id="S4.SS2.p4.18.m18.1.1.cmml" xref="S4.SS2.p4.18.m18.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.18.m18.1.1.1.cmml" xref="S4.SS2.p4.18.m18.1.1">superscript</csymbol><ci id="S4.SS2.p4.18.m18.1.1.2.cmml" xref="S4.SS2.p4.18.m18.1.1.2">𝑤</ci><ci id="S4.SS2.p4.18.m18.1.1.3.cmml" xref="S4.SS2.p4.18.m18.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.18.m18.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.18.m18.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> contains the image <math alttext="\sigma(x_{2}\ldots x_{n-1})" class="ltx_Math" display="inline" id="S4.SS2.p4.19.m19.1"><semantics id="S4.SS2.p4.19.m19.1a"><mrow id="S4.SS2.p4.19.m19.1.1" xref="S4.SS2.p4.19.m19.1.1.cmml"><mi id="S4.SS2.p4.19.m19.1.1.3" xref="S4.SS2.p4.19.m19.1.1.3.cmml">σ</mi><mo id="S4.SS2.p4.19.m19.1.1.2" xref="S4.SS2.p4.19.m19.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p4.19.m19.1.1.1.1" xref="S4.SS2.p4.19.m19.1.1.1.1.1.cmml"><mo id="S4.SS2.p4.19.m19.1.1.1.1.2" stretchy="false" xref="S4.SS2.p4.19.m19.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p4.19.m19.1.1.1.1.1" xref="S4.SS2.p4.19.m19.1.1.1.1.1.cmml"><msub id="S4.SS2.p4.19.m19.1.1.1.1.1.2" xref="S4.SS2.p4.19.m19.1.1.1.1.1.2.cmml"><mi id="S4.SS2.p4.19.m19.1.1.1.1.1.2.2" xref="S4.SS2.p4.19.m19.1.1.1.1.1.2.2.cmml">x</mi><mn id="S4.SS2.p4.19.m19.1.1.1.1.1.2.3" xref="S4.SS2.p4.19.m19.1.1.1.1.1.2.3.cmml">2</mn></msub><mo id="S4.SS2.p4.19.m19.1.1.1.1.1.1" xref="S4.SS2.p4.19.m19.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p4.19.m19.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS2.p4.19.m19.1.1.1.1.1.3.cmml">…</mi><mo id="S4.SS2.p4.19.m19.1.1.1.1.1.1a" xref="S4.SS2.p4.19.m19.1.1.1.1.1.1.cmml">⁢</mo><msub id="S4.SS2.p4.19.m19.1.1.1.1.1.4" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.cmml"><mi id="S4.SS2.p4.19.m19.1.1.1.1.1.4.2" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.2.cmml">x</mi><mrow id="S4.SS2.p4.19.m19.1.1.1.1.1.4.3" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.cmml"><mi id="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.2" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.2.cmml">n</mi><mo id="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.1" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.1.cmml">−</mo><mn id="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.3" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S4.SS2.p4.19.m19.1.1.1.1.3" stretchy="false" xref="S4.SS2.p4.19.m19.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.19.m19.1b"><apply id="S4.SS2.p4.19.m19.1.1.cmml" xref="S4.SS2.p4.19.m19.1.1"><times id="S4.SS2.p4.19.m19.1.1.2.cmml" xref="S4.SS2.p4.19.m19.1.1.2"></times><ci id="S4.SS2.p4.19.m19.1.1.3.cmml" xref="S4.SS2.p4.19.m19.1.1.3">𝜎</ci><apply id="S4.SS2.p4.19.m19.1.1.1.1.1.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1"><times id="S4.SS2.p4.19.m19.1.1.1.1.1.1.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.1"></times><apply id="S4.SS2.p4.19.m19.1.1.1.1.1.2.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p4.19.m19.1.1.1.1.1.2.1.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.2">subscript</csymbol><ci id="S4.SS2.p4.19.m19.1.1.1.1.1.2.2.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.2.2">𝑥</ci><cn id="S4.SS2.p4.19.m19.1.1.1.1.1.2.3.cmml" type="integer" xref="S4.SS2.p4.19.m19.1.1.1.1.1.2.3">2</cn></apply><ci id="S4.SS2.p4.19.m19.1.1.1.1.1.3.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.3">…</ci><apply id="S4.SS2.p4.19.m19.1.1.1.1.1.4.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p4.19.m19.1.1.1.1.1.4.1.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4">subscript</csymbol><ci id="S4.SS2.p4.19.m19.1.1.1.1.1.4.2.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.2">𝑥</ci><apply id="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.3"><minus id="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.1.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.1"></minus><ci id="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.2.cmml" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.2">𝑛</ci><cn id="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.3.cmml" type="integer" xref="S4.SS2.p4.19.m19.1.1.1.1.1.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.19.m19.1c">\sigma(x_{2}\ldots x_{n-1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.19.m19.1d">italic_σ ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_x start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT )</annotation></semantics></math> of the “maximal inner factor” <math alttext="x_{2}\ldots x_{n-1}" class="ltx_Math" display="inline" id="S4.SS2.p4.20.m20.1"><semantics id="S4.SS2.p4.20.m20.1a"><mrow id="S4.SS2.p4.20.m20.1.1" xref="S4.SS2.p4.20.m20.1.1.cmml"><msub id="S4.SS2.p4.20.m20.1.1.2" xref="S4.SS2.p4.20.m20.1.1.2.cmml"><mi id="S4.SS2.p4.20.m20.1.1.2.2" xref="S4.SS2.p4.20.m20.1.1.2.2.cmml">x</mi><mn id="S4.SS2.p4.20.m20.1.1.2.3" xref="S4.SS2.p4.20.m20.1.1.2.3.cmml">2</mn></msub><mo id="S4.SS2.p4.20.m20.1.1.1" xref="S4.SS2.p4.20.m20.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p4.20.m20.1.1.3" mathvariant="normal" xref="S4.SS2.p4.20.m20.1.1.3.cmml">…</mi><mo id="S4.SS2.p4.20.m20.1.1.1a" xref="S4.SS2.p4.20.m20.1.1.1.cmml">⁢</mo><msub id="S4.SS2.p4.20.m20.1.1.4" xref="S4.SS2.p4.20.m20.1.1.4.cmml"><mi id="S4.SS2.p4.20.m20.1.1.4.2" xref="S4.SS2.p4.20.m20.1.1.4.2.cmml">x</mi><mrow id="S4.SS2.p4.20.m20.1.1.4.3" xref="S4.SS2.p4.20.m20.1.1.4.3.cmml"><mi id="S4.SS2.p4.20.m20.1.1.4.3.2" xref="S4.SS2.p4.20.m20.1.1.4.3.2.cmml">n</mi><mo id="S4.SS2.p4.20.m20.1.1.4.3.1" xref="S4.SS2.p4.20.m20.1.1.4.3.1.cmml">−</mo><mn id="S4.SS2.p4.20.m20.1.1.4.3.3" xref="S4.SS2.p4.20.m20.1.1.4.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.20.m20.1b"><apply id="S4.SS2.p4.20.m20.1.1.cmml" xref="S4.SS2.p4.20.m20.1.1"><times id="S4.SS2.p4.20.m20.1.1.1.cmml" xref="S4.SS2.p4.20.m20.1.1.1"></times><apply id="S4.SS2.p4.20.m20.1.1.2.cmml" xref="S4.SS2.p4.20.m20.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p4.20.m20.1.1.2.1.cmml" xref="S4.SS2.p4.20.m20.1.1.2">subscript</csymbol><ci id="S4.SS2.p4.20.m20.1.1.2.2.cmml" xref="S4.SS2.p4.20.m20.1.1.2.2">𝑥</ci><cn id="S4.SS2.p4.20.m20.1.1.2.3.cmml" type="integer" xref="S4.SS2.p4.20.m20.1.1.2.3">2</cn></apply><ci id="S4.SS2.p4.20.m20.1.1.3.cmml" xref="S4.SS2.p4.20.m20.1.1.3">…</ci><apply id="S4.SS2.p4.20.m20.1.1.4.cmml" xref="S4.SS2.p4.20.m20.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p4.20.m20.1.1.4.1.cmml" xref="S4.SS2.p4.20.m20.1.1.4">subscript</csymbol><ci id="S4.SS2.p4.20.m20.1.1.4.2.cmml" xref="S4.SS2.p4.20.m20.1.1.4.2">𝑥</ci><apply id="S4.SS2.p4.20.m20.1.1.4.3.cmml" xref="S4.SS2.p4.20.m20.1.1.4.3"><minus id="S4.SS2.p4.20.m20.1.1.4.3.1.cmml" xref="S4.SS2.p4.20.m20.1.1.4.3.1"></minus><ci id="S4.SS2.p4.20.m20.1.1.4.3.2.cmml" xref="S4.SS2.p4.20.m20.1.1.4.3.2">𝑛</ci><cn id="S4.SS2.p4.20.m20.1.1.4.3.3.cmml" type="integer" xref="S4.SS2.p4.20.m20.1.1.4.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.20.m20.1c">x_{2}\ldots x_{n-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.20.m20.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_x start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="w" class="ltx_Math" display="inline" id="S4.SS2.p4.21.m21.1"><semantics id="S4.SS2.p4.21.m21.1a"><mi id="S4.SS2.p4.21.m21.1.1" xref="S4.SS2.p4.21.m21.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.21.m21.1b"><ci id="S4.SS2.p4.21.m21.1.1.cmml" xref="S4.SS2.p4.21.m21.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.21.m21.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.21.m21.1d">italic_w</annotation></semantics></math> as factor, but not as prefix or suffix.</p> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.10">The number of all essential occurrences of <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p5.1.m1.1"><semantics id="S4.SS2.p5.1.m1.1a"><msup id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml"><mi id="S4.SS2.p5.1.m1.1.1.2" xref="S4.SS2.p5.1.m1.1.1.2.cmml">w</mi><mo id="S4.SS2.p5.1.m1.1.1.3" xref="S4.SS2.p5.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.1.m1.1b"><apply id="S4.SS2.p5.1.m1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.1.m1.1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.p5.1.m1.1.1.2.cmml" xref="S4.SS2.p5.1.m1.1.1.2">𝑤</ci><ci id="S4.SS2.p5.1.m1.1.1.3.cmml" xref="S4.SS2.p5.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S4.SS2.p5.2.m2.1"><semantics id="S4.SS2.p5.2.m2.1a"><mrow id="S4.SS2.p5.2.m2.1.2" xref="S4.SS2.p5.2.m2.1.2.cmml"><mi id="S4.SS2.p5.2.m2.1.2.2" xref="S4.SS2.p5.2.m2.1.2.2.cmml">σ</mi><mo id="S4.SS2.p5.2.m2.1.2.1" xref="S4.SS2.p5.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p5.2.m2.1.2.3.2" xref="S4.SS2.p5.2.m2.1.2.cmml"><mo id="S4.SS2.p5.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.p5.2.m2.1.2.cmml">(</mo><mi id="S4.SS2.p5.2.m2.1.1" xref="S4.SS2.p5.2.m2.1.1.cmml">w</mi><mo id="S4.SS2.p5.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.p5.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.2.m2.1b"><apply id="S4.SS2.p5.2.m2.1.2.cmml" xref="S4.SS2.p5.2.m2.1.2"><times id="S4.SS2.p5.2.m2.1.2.1.cmml" xref="S4.SS2.p5.2.m2.1.2.1"></times><ci id="S4.SS2.p5.2.m2.1.2.2.cmml" xref="S4.SS2.p5.2.m2.1.2.2">𝜎</ci><ci id="S4.SS2.p5.2.m2.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.2.m2.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.2.m2.1d">italic_σ ( italic_w )</annotation></semantics></math> will be denoted by <math alttext="\lfloor\sigma(w)\rfloor_{w^{\prime}}" class="ltx_Math" display="inline" id="S4.SS2.p5.3.m3.2"><semantics id="S4.SS2.p5.3.m3.2a"><msub id="S4.SS2.p5.3.m3.2.2" xref="S4.SS2.p5.3.m3.2.2.cmml"><mrow id="S4.SS2.p5.3.m3.2.2.1.1" xref="S4.SS2.p5.3.m3.2.2.1.2.cmml"><mo id="S4.SS2.p5.3.m3.2.2.1.1.2" stretchy="false" xref="S4.SS2.p5.3.m3.2.2.1.2.1.cmml">⌊</mo><mrow id="S4.SS2.p5.3.m3.2.2.1.1.1" xref="S4.SS2.p5.3.m3.2.2.1.1.1.cmml"><mi id="S4.SS2.p5.3.m3.2.2.1.1.1.2" xref="S4.SS2.p5.3.m3.2.2.1.1.1.2.cmml">σ</mi><mo id="S4.SS2.p5.3.m3.2.2.1.1.1.1" xref="S4.SS2.p5.3.m3.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS2.p5.3.m3.2.2.1.1.1.3.2" xref="S4.SS2.p5.3.m3.2.2.1.1.1.cmml"><mo id="S4.SS2.p5.3.m3.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.SS2.p5.3.m3.2.2.1.1.1.cmml">(</mo><mi id="S4.SS2.p5.3.m3.1.1" xref="S4.SS2.p5.3.m3.1.1.cmml">w</mi><mo id="S4.SS2.p5.3.m3.2.2.1.1.1.3.2.2" stretchy="false" xref="S4.SS2.p5.3.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p5.3.m3.2.2.1.1.3" stretchy="false" xref="S4.SS2.p5.3.m3.2.2.1.2.1.cmml">⌋</mo></mrow><msup id="S4.SS2.p5.3.m3.2.2.3" xref="S4.SS2.p5.3.m3.2.2.3.cmml"><mi id="S4.SS2.p5.3.m3.2.2.3.2" xref="S4.SS2.p5.3.m3.2.2.3.2.cmml">w</mi><mo id="S4.SS2.p5.3.m3.2.2.3.3" xref="S4.SS2.p5.3.m3.2.2.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.3.m3.2b"><apply id="S4.SS2.p5.3.m3.2.2.cmml" xref="S4.SS2.p5.3.m3.2.2"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.2.2.2.cmml" xref="S4.SS2.p5.3.m3.2.2">subscript</csymbol><apply id="S4.SS2.p5.3.m3.2.2.1.2.cmml" xref="S4.SS2.p5.3.m3.2.2.1.1"><floor id="S4.SS2.p5.3.m3.2.2.1.2.1.cmml" xref="S4.SS2.p5.3.m3.2.2.1.1.2"></floor><apply id="S4.SS2.p5.3.m3.2.2.1.1.1.cmml" xref="S4.SS2.p5.3.m3.2.2.1.1.1"><times id="S4.SS2.p5.3.m3.2.2.1.1.1.1.cmml" xref="S4.SS2.p5.3.m3.2.2.1.1.1.1"></times><ci id="S4.SS2.p5.3.m3.2.2.1.1.1.2.cmml" xref="S4.SS2.p5.3.m3.2.2.1.1.1.2">𝜎</ci><ci id="S4.SS2.p5.3.m3.1.1.cmml" xref="S4.SS2.p5.3.m3.1.1">𝑤</ci></apply></apply><apply id="S4.SS2.p5.3.m3.2.2.3.cmml" xref="S4.SS2.p5.3.m3.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.2.2.3.1.cmml" xref="S4.SS2.p5.3.m3.2.2.3">superscript</csymbol><ci id="S4.SS2.p5.3.m3.2.2.3.2.cmml" xref="S4.SS2.p5.3.m3.2.2.3.2">𝑤</ci><ci id="S4.SS2.p5.3.m3.2.2.3.3.cmml" xref="S4.SS2.p5.3.m3.2.2.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.3.m3.2c">\lfloor\sigma(w)\rfloor_{w^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.3.m3.2d">⌊ italic_σ ( italic_w ) ⌋ start_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>. It follows directly from these definitions that for any <math alttext="w^{\prime}\in\cal B^{*}" class="ltx_Math" display="inline" id="S4.SS2.p5.4.m4.1"><semantics id="S4.SS2.p5.4.m4.1a"><mrow id="S4.SS2.p5.4.m4.1.1" xref="S4.SS2.p5.4.m4.1.1.cmml"><msup id="S4.SS2.p5.4.m4.1.1.2" xref="S4.SS2.p5.4.m4.1.1.2.cmml"><mi id="S4.SS2.p5.4.m4.1.1.2.2" xref="S4.SS2.p5.4.m4.1.1.2.2.cmml">w</mi><mo id="S4.SS2.p5.4.m4.1.1.2.3" xref="S4.SS2.p5.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS2.p5.4.m4.1.1.1" xref="S4.SS2.p5.4.m4.1.1.1.cmml">∈</mo><msup id="S4.SS2.p5.4.m4.1.1.3" xref="S4.SS2.p5.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p5.4.m4.1.1.3.2" xref="S4.SS2.p5.4.m4.1.1.3.2.cmml">ℬ</mi><mo id="S4.SS2.p5.4.m4.1.1.3.3" xref="S4.SS2.p5.4.m4.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.4.m4.1b"><apply id="S4.SS2.p5.4.m4.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1"><in id="S4.SS2.p5.4.m4.1.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1.1"></in><apply id="S4.SS2.p5.4.m4.1.1.2.cmml" xref="S4.SS2.p5.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m4.1.1.2.1.cmml" xref="S4.SS2.p5.4.m4.1.1.2">superscript</csymbol><ci id="S4.SS2.p5.4.m4.1.1.2.2.cmml" xref="S4.SS2.p5.4.m4.1.1.2.2">𝑤</ci><ci id="S4.SS2.p5.4.m4.1.1.2.3.cmml" xref="S4.SS2.p5.4.m4.1.1.2.3">′</ci></apply><apply id="S4.SS2.p5.4.m4.1.1.3.cmml" xref="S4.SS2.p5.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m4.1.1.3.1.cmml" xref="S4.SS2.p5.4.m4.1.1.3">superscript</csymbol><ci id="S4.SS2.p5.4.m4.1.1.3.2.cmml" xref="S4.SS2.p5.4.m4.1.1.3.2">ℬ</ci><times id="S4.SS2.p5.4.m4.1.1.3.3.cmml" xref="S4.SS2.p5.4.m4.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.4.m4.1c">w^{\prime}\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.4.m4.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the set of all words <math alttext="w" class="ltx_Math" display="inline" id="S4.SS2.p5.5.m5.1"><semantics id="S4.SS2.p5.5.m5.1a"><mi id="S4.SS2.p5.5.m5.1.1" xref="S4.SS2.p5.5.m5.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.5.m5.1b"><ci id="S4.SS2.p5.5.m5.1.1.cmml" xref="S4.SS2.p5.5.m5.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.5.m5.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.5.m5.1d">italic_w</annotation></semantics></math> in <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S4.SS2.p5.6.m6.1"><semantics id="S4.SS2.p5.6.m6.1a"><msup id="S4.SS2.p5.6.m6.1.1" xref="S4.SS2.p5.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p5.6.m6.1.1.2" xref="S4.SS2.p5.6.m6.1.1.2.cmml">𝒜</mi><mo id="S4.SS2.p5.6.m6.1.1.3" xref="S4.SS2.p5.6.m6.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.6.m6.1b"><apply id="S4.SS2.p5.6.m6.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.6.m6.1.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1">superscript</csymbol><ci id="S4.SS2.p5.6.m6.1.1.2.cmml" xref="S4.SS2.p5.6.m6.1.1.2">𝒜</ci><times id="S4.SS2.p5.6.m6.1.1.3.cmml" xref="S4.SS2.p5.6.m6.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.6.m6.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.6.m6.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, for which <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S4.SS2.p5.7.m7.1"><semantics id="S4.SS2.p5.7.m7.1a"><mrow id="S4.SS2.p5.7.m7.1.2" xref="S4.SS2.p5.7.m7.1.2.cmml"><mi id="S4.SS2.p5.7.m7.1.2.2" xref="S4.SS2.p5.7.m7.1.2.2.cmml">σ</mi><mo id="S4.SS2.p5.7.m7.1.2.1" xref="S4.SS2.p5.7.m7.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p5.7.m7.1.2.3.2" xref="S4.SS2.p5.7.m7.1.2.cmml"><mo id="S4.SS2.p5.7.m7.1.2.3.2.1" stretchy="false" xref="S4.SS2.p5.7.m7.1.2.cmml">(</mo><mi id="S4.SS2.p5.7.m7.1.1" xref="S4.SS2.p5.7.m7.1.1.cmml">w</mi><mo id="S4.SS2.p5.7.m7.1.2.3.2.2" stretchy="false" xref="S4.SS2.p5.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.7.m7.1b"><apply id="S4.SS2.p5.7.m7.1.2.cmml" xref="S4.SS2.p5.7.m7.1.2"><times id="S4.SS2.p5.7.m7.1.2.1.cmml" xref="S4.SS2.p5.7.m7.1.2.1"></times><ci id="S4.SS2.p5.7.m7.1.2.2.cmml" xref="S4.SS2.p5.7.m7.1.2.2">𝜎</ci><ci id="S4.SS2.p5.7.m7.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.7.m7.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.7.m7.1d">italic_σ ( italic_w )</annotation></semantics></math> contains at least one essential occurrence of <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.p5.8.m8.1"><semantics id="S4.SS2.p5.8.m8.1a"><msup id="S4.SS2.p5.8.m8.1.1" xref="S4.SS2.p5.8.m8.1.1.cmml"><mi id="S4.SS2.p5.8.m8.1.1.2" xref="S4.SS2.p5.8.m8.1.1.2.cmml">w</mi><mo id="S4.SS2.p5.8.m8.1.1.3" xref="S4.SS2.p5.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.8.m8.1b"><apply id="S4.SS2.p5.8.m8.1.1.cmml" xref="S4.SS2.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.8.m8.1.1.1.cmml" xref="S4.SS2.p5.8.m8.1.1">superscript</csymbol><ci id="S4.SS2.p5.8.m8.1.1.2.cmml" xref="S4.SS2.p5.8.m8.1.1.2">𝑤</ci><ci id="S4.SS2.p5.8.m8.1.1.3.cmml" xref="S4.SS2.p5.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.8.m8.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.8.m8.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, is finite. Indeed, for <math alttext="|w^{\prime}|\geq 2" class="ltx_Math" display="inline" id="S4.SS2.p5.9.m9.1"><semantics id="S4.SS2.p5.9.m9.1a"><mrow id="S4.SS2.p5.9.m9.1.1" xref="S4.SS2.p5.9.m9.1.1.cmml"><mrow id="S4.SS2.p5.9.m9.1.1.1.1" xref="S4.SS2.p5.9.m9.1.1.1.2.cmml"><mo id="S4.SS2.p5.9.m9.1.1.1.1.2" stretchy="false" xref="S4.SS2.p5.9.m9.1.1.1.2.1.cmml">|</mo><msup id="S4.SS2.p5.9.m9.1.1.1.1.1" xref="S4.SS2.p5.9.m9.1.1.1.1.1.cmml"><mi id="S4.SS2.p5.9.m9.1.1.1.1.1.2" xref="S4.SS2.p5.9.m9.1.1.1.1.1.2.cmml">w</mi><mo id="S4.SS2.p5.9.m9.1.1.1.1.1.3" xref="S4.SS2.p5.9.m9.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.p5.9.m9.1.1.1.1.3" stretchy="false" xref="S4.SS2.p5.9.m9.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.SS2.p5.9.m9.1.1.2" xref="S4.SS2.p5.9.m9.1.1.2.cmml">≥</mo><mn id="S4.SS2.p5.9.m9.1.1.3" xref="S4.SS2.p5.9.m9.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.9.m9.1b"><apply id="S4.SS2.p5.9.m9.1.1.cmml" xref="S4.SS2.p5.9.m9.1.1"><geq id="S4.SS2.p5.9.m9.1.1.2.cmml" xref="S4.SS2.p5.9.m9.1.1.2"></geq><apply id="S4.SS2.p5.9.m9.1.1.1.2.cmml" xref="S4.SS2.p5.9.m9.1.1.1.1"><abs id="S4.SS2.p5.9.m9.1.1.1.2.1.cmml" xref="S4.SS2.p5.9.m9.1.1.1.1.2"></abs><apply id="S4.SS2.p5.9.m9.1.1.1.1.1.cmml" xref="S4.SS2.p5.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.9.m9.1.1.1.1.1.1.cmml" xref="S4.SS2.p5.9.m9.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.p5.9.m9.1.1.1.1.1.2.cmml" xref="S4.SS2.p5.9.m9.1.1.1.1.1.2">𝑤</ci><ci id="S4.SS2.p5.9.m9.1.1.1.1.1.3.cmml" xref="S4.SS2.p5.9.m9.1.1.1.1.1.3">′</ci></apply></apply><cn id="S4.SS2.p5.9.m9.1.1.3.cmml" type="integer" xref="S4.SS2.p5.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.9.m9.1c">|w^{\prime}|\geq 2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.9.m9.1d">| italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≥ 2</annotation></semantics></math> one easily verifies that any such <math alttext="w" class="ltx_Math" display="inline" id="S4.SS2.p5.10.m10.1"><semantics id="S4.SS2.p5.10.m10.1a"><mi id="S4.SS2.p5.10.m10.1.1" xref="S4.SS2.p5.10.m10.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.10.m10.1b"><ci id="S4.SS2.p5.10.m10.1.1.cmml" xref="S4.SS2.p5.10.m10.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.10.m10.1c">w</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.10.m10.1d">italic_w</annotation></semantics></math> must satisfy</p> <table class="ltx_equation ltx_eqn_table" id="S4.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(4.1)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{|w^{\prime}|}{||\sigma||}\leq|w|\leq\frac{|w^{\prime}|-2}{\langle\sigma% \rangle}+2\,," class="ltx_Math" display="block" id="S4.E1.m1.6"><semantics id="S4.E1.m1.6a"><mrow id="S4.E1.m1.6.6.1" xref="S4.E1.m1.6.6.1.1.cmml"><mrow id="S4.E1.m1.6.6.1.1" xref="S4.E1.m1.6.6.1.1.cmml"><mfrac id="S4.E1.m1.2.2" xref="S4.E1.m1.2.2.cmml"><mrow id="S4.E1.m1.1.1.1.1" xref="S4.E1.m1.1.1.1.2.cmml"><mo id="S4.E1.m1.1.1.1.1.2" stretchy="false" xref="S4.E1.m1.1.1.1.2.1.cmml">|</mo><msup id="S4.E1.m1.1.1.1.1.1" xref="S4.E1.m1.1.1.1.1.1.cmml"><mi id="S4.E1.m1.1.1.1.1.1.2" xref="S4.E1.m1.1.1.1.1.1.2.cmml">w</mi><mo id="S4.E1.m1.1.1.1.1.1.3" xref="S4.E1.m1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.E1.m1.1.1.1.1.3" stretchy="false" xref="S4.E1.m1.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S4.E1.m1.2.2.2.3" xref="S4.E1.m1.2.2.2.2.cmml"><mo id="S4.E1.m1.2.2.2.3.1" stretchy="false" xref="S4.E1.m1.2.2.2.2.1.cmml">‖</mo><mi id="S4.E1.m1.2.2.2.1" xref="S4.E1.m1.2.2.2.1.cmml">σ</mi><mo id="S4.E1.m1.2.2.2.3.2" stretchy="false" xref="S4.E1.m1.2.2.2.2.1.cmml">‖</mo></mrow></mfrac><mo id="S4.E1.m1.6.6.1.1.2" xref="S4.E1.m1.6.6.1.1.2.cmml">≤</mo><mrow id="S4.E1.m1.6.6.1.1.3.2" xref="S4.E1.m1.6.6.1.1.3.1.cmml"><mo id="S4.E1.m1.6.6.1.1.3.2.1" stretchy="false" xref="S4.E1.m1.6.6.1.1.3.1.1.cmml">|</mo><mi id="S4.E1.m1.5.5" xref="S4.E1.m1.5.5.cmml">w</mi><mo id="S4.E1.m1.6.6.1.1.3.2.2" stretchy="false" xref="S4.E1.m1.6.6.1.1.3.1.1.cmml">|</mo></mrow><mo id="S4.E1.m1.6.6.1.1.4" xref="S4.E1.m1.6.6.1.1.4.cmml">≤</mo><mrow id="S4.E1.m1.6.6.1.1.5" xref="S4.E1.m1.6.6.1.1.5.cmml"><mfrac id="S4.E1.m1.4.4" xref="S4.E1.m1.4.4.cmml"><mrow id="S4.E1.m1.3.3.1" xref="S4.E1.m1.3.3.1.cmml"><mrow id="S4.E1.m1.3.3.1.1.1" xref="S4.E1.m1.3.3.1.1.2.cmml"><mo id="S4.E1.m1.3.3.1.1.1.2" stretchy="false" xref="S4.E1.m1.3.3.1.1.2.1.cmml">|</mo><msup id="S4.E1.m1.3.3.1.1.1.1" xref="S4.E1.m1.3.3.1.1.1.1.cmml"><mi id="S4.E1.m1.3.3.1.1.1.1.2" xref="S4.E1.m1.3.3.1.1.1.1.2.cmml">w</mi><mo id="S4.E1.m1.3.3.1.1.1.1.3" xref="S4.E1.m1.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.E1.m1.3.3.1.1.1.3" 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id="S4.E1.m1.3.3.1.1.1.1.2.cmml" xref="S4.E1.m1.3.3.1.1.1.1.2">𝑤</ci><ci id="S4.E1.m1.3.3.1.1.1.1.3.cmml" xref="S4.E1.m1.3.3.1.1.1.1.3">′</ci></apply></apply><cn id="S4.E1.m1.3.3.1.3.cmml" type="integer" xref="S4.E1.m1.3.3.1.3">2</cn></apply><apply id="S4.E1.m1.4.4.2.2.cmml" xref="S4.E1.m1.4.4.2.3"><csymbol cd="latexml" id="S4.E1.m1.4.4.2.2.1.cmml" xref="S4.E1.m1.4.4.2.3.1">delimited-⟨⟩</csymbol><ci id="S4.E1.m1.4.4.2.1.cmml" xref="S4.E1.m1.4.4.2.1">𝜎</ci></apply></apply><cn id="S4.E1.m1.6.6.1.1.5.2.cmml" type="integer" xref="S4.E1.m1.6.6.1.1.5.2">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E1.m1.6c">\frac{|w^{\prime}|}{||\sigma||}\leq|w|\leq\frac{|w^{\prime}|-2}{\langle\sigma% \rangle}+2\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.6d">divide start_ARG | italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | end_ARG start_ARG | | italic_σ | | end_ARG ≤ | italic_w | ≤ divide start_ARG | italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | - 2 end_ARG start_ARG ⟨ italic_σ ⟩ end_ARG + 2 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.p5.13">where <math alttext="||\sigma||" class="ltx_Math" display="inline" id="S4.SS2.p5.11.m1.1"><semantics id="S4.SS2.p5.11.m1.1a"><mrow id="S4.SS2.p5.11.m1.1.2.2" xref="S4.SS2.p5.11.m1.1.2.1.cmml"><mo id="S4.SS2.p5.11.m1.1.2.2.1" stretchy="false" xref="S4.SS2.p5.11.m1.1.2.1.1.cmml">‖</mo><mi id="S4.SS2.p5.11.m1.1.1" xref="S4.SS2.p5.11.m1.1.1.cmml">σ</mi><mo id="S4.SS2.p5.11.m1.1.2.2.2" stretchy="false" xref="S4.SS2.p5.11.m1.1.2.1.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.11.m1.1b"><apply id="S4.SS2.p5.11.m1.1.2.1.cmml" xref="S4.SS2.p5.11.m1.1.2.2"><csymbol cd="latexml" id="S4.SS2.p5.11.m1.1.2.1.1.cmml" xref="S4.SS2.p5.11.m1.1.2.2.1">norm</csymbol><ci id="S4.SS2.p5.11.m1.1.1.cmml" xref="S4.SS2.p5.11.m1.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.11.m1.1c">||\sigma||</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.11.m1.1d">| | italic_σ | |</annotation></semantics></math> and <math alttext="\langle\sigma\rangle" class="ltx_Math" display="inline" id="S4.SS2.p5.12.m2.1"><semantics id="S4.SS2.p5.12.m2.1a"><mrow id="S4.SS2.p5.12.m2.1.2.2" xref="S4.SS2.p5.12.m2.1.2.1.cmml"><mo id="S4.SS2.p5.12.m2.1.2.2.1" stretchy="false" xref="S4.SS2.p5.12.m2.1.2.1.1.cmml">⟨</mo><mi id="S4.SS2.p5.12.m2.1.1" xref="S4.SS2.p5.12.m2.1.1.cmml">σ</mi><mo id="S4.SS2.p5.12.m2.1.2.2.2" stretchy="false" xref="S4.SS2.p5.12.m2.1.2.1.1.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.12.m2.1b"><apply id="S4.SS2.p5.12.m2.1.2.1.cmml" xref="S4.SS2.p5.12.m2.1.2.2"><csymbol cd="latexml" id="S4.SS2.p5.12.m2.1.2.1.1.cmml" xref="S4.SS2.p5.12.m2.1.2.2.1">delimited-⟨⟩</csymbol><ci id="S4.SS2.p5.12.m2.1.1.cmml" xref="S4.SS2.p5.12.m2.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.12.m2.1c">\langle\sigma\rangle</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.12.m2.1d">⟨ italic_σ ⟩</annotation></semantics></math> denote the biggest and smallest length respectively of any of the letter images <math alttext="\sigma(a_{i})" class="ltx_Math" display="inline" id="S4.SS2.p5.13.m3.1"><semantics id="S4.SS2.p5.13.m3.1a"><mrow id="S4.SS2.p5.13.m3.1.1" xref="S4.SS2.p5.13.m3.1.1.cmml"><mi id="S4.SS2.p5.13.m3.1.1.3" xref="S4.SS2.p5.13.m3.1.1.3.cmml">σ</mi><mo id="S4.SS2.p5.13.m3.1.1.2" xref="S4.SS2.p5.13.m3.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.p5.13.m3.1.1.1.1" xref="S4.SS2.p5.13.m3.1.1.1.1.1.cmml"><mo id="S4.SS2.p5.13.m3.1.1.1.1.2" stretchy="false" xref="S4.SS2.p5.13.m3.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.p5.13.m3.1.1.1.1.1" xref="S4.SS2.p5.13.m3.1.1.1.1.1.cmml"><mi id="S4.SS2.p5.13.m3.1.1.1.1.1.2" xref="S4.SS2.p5.13.m3.1.1.1.1.1.2.cmml">a</mi><mi id="S4.SS2.p5.13.m3.1.1.1.1.1.3" xref="S4.SS2.p5.13.m3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.SS2.p5.13.m3.1.1.1.1.3" stretchy="false" xref="S4.SS2.p5.13.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.13.m3.1b"><apply id="S4.SS2.p5.13.m3.1.1.cmml" xref="S4.SS2.p5.13.m3.1.1"><times id="S4.SS2.p5.13.m3.1.1.2.cmml" xref="S4.SS2.p5.13.m3.1.1.2"></times><ci id="S4.SS2.p5.13.m3.1.1.3.cmml" xref="S4.SS2.p5.13.m3.1.1.3">𝜎</ci><apply id="S4.SS2.p5.13.m3.1.1.1.1.1.cmml" xref="S4.SS2.p5.13.m3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.13.m3.1.1.1.1.1.1.cmml" xref="S4.SS2.p5.13.m3.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p5.13.m3.1.1.1.1.1.2.cmml" xref="S4.SS2.p5.13.m3.1.1.1.1.1.2">𝑎</ci><ci id="S4.SS2.p5.13.m3.1.1.1.1.1.3.cmml" xref="S4.SS2.p5.13.m3.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.13.m3.1c">\sigma(a_{i})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.13.m3.1d">italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_prop" id="S4.Thmthm2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmthm2.1.1.1">Proposition 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmthm2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmthm2.p1"> <p class="ltx_p" id="S4.Thmthm2.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmthm2.p1.6.6">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S4.Thmthm2.p1.1.1.m1.1"><semantics id="S4.Thmthm2.p1.1.1.m1.1a"><mrow id="S4.Thmthm2.p1.1.1.m1.1.1" xref="S4.Thmthm2.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmthm2.p1.1.1.m1.1.1.2" xref="S4.Thmthm2.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S4.Thmthm2.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmthm2.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S4.Thmthm2.p1.1.1.m1.1.1.3" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.cmml"><msup id="S4.Thmthm2.p1.1.1.m1.1.1.3.2" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm2.p1.1.1.m1.1.1.3.2.2" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S4.Thmthm2.p1.1.1.m1.1.1.3.2.3" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S4.Thmthm2.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S4.Thmthm2.p1.1.1.m1.1.1.3.3" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm2.p1.1.1.m1.1.1.3.3.2" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S4.Thmthm2.p1.1.1.m1.1.1.3.3.3" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm2.p1.1.1.m1.1b"><apply id="S4.Thmthm2.p1.1.1.m1.1.1.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1"><ci id="S4.Thmthm2.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.1">:</ci><ci id="S4.Thmthm2.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S4.Thmthm2.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3"><ci id="S4.Thmthm2.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.1">→</ci><apply id="S4.Thmthm2.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmthm2.p1.1.1.m1.1.1.3.2.1.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S4.Thmthm2.p1.1.1.m1.1.1.3.2.2.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S4.Thmthm2.p1.1.1.m1.1.1.3.2.3.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S4.Thmthm2.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmthm2.p1.1.1.m1.1.1.3.3.1.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S4.Thmthm2.p1.1.1.m1.1.1.3.3.2.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S4.Thmthm2.p1.1.1.m1.1.1.3.3.3.cmml" xref="S4.Thmthm2.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm2.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm2.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be any non-erasing morphism of free monoids, and let <math alttext="\mu" class="ltx_Math" display="inline" id="S4.Thmthm2.p1.2.2.m2.1"><semantics id="S4.Thmthm2.p1.2.2.m2.1a"><mi id="S4.Thmthm2.p1.2.2.m2.1.1" xref="S4.Thmthm2.p1.2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmthm2.p1.2.2.m2.1b"><ci id="S4.Thmthm2.p1.2.2.m2.1.1.cmml" xref="S4.Thmthm2.p1.2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm2.p1.2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm2.p1.2.2.m2.1d">italic_μ</annotation></semantics></math> be any invariant measure on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S4.Thmthm2.p1.3.3.m3.1"><semantics id="S4.Thmthm2.p1.3.3.m3.1a"><msup id="S4.Thmthm2.p1.3.3.m3.1.1" xref="S4.Thmthm2.p1.3.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm2.p1.3.3.m3.1.1.2" xref="S4.Thmthm2.p1.3.3.m3.1.1.2.cmml">𝒜</mi><mi id="S4.Thmthm2.p1.3.3.m3.1.1.3" xref="S4.Thmthm2.p1.3.3.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S4.Thmthm2.p1.3.3.m3.1b"><apply id="S4.Thmthm2.p1.3.3.m3.1.1.cmml" xref="S4.Thmthm2.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmthm2.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmthm2.p1.3.3.m3.1.1">superscript</csymbol><ci id="S4.Thmthm2.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmthm2.p1.3.3.m3.1.1.2">𝒜</ci><ci id="S4.Thmthm2.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmthm2.p1.3.3.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm2.p1.3.3.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm2.p1.3.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Then for any <math alttext="w^{\prime}\in\cal B^{*}" class="ltx_Math" display="inline" id="S4.Thmthm2.p1.4.4.m4.1"><semantics id="S4.Thmthm2.p1.4.4.m4.1a"><mrow id="S4.Thmthm2.p1.4.4.m4.1.1" xref="S4.Thmthm2.p1.4.4.m4.1.1.cmml"><msup id="S4.Thmthm2.p1.4.4.m4.1.1.2" xref="S4.Thmthm2.p1.4.4.m4.1.1.2.cmml"><mi id="S4.Thmthm2.p1.4.4.m4.1.1.2.2" xref="S4.Thmthm2.p1.4.4.m4.1.1.2.2.cmml">w</mi><mo id="S4.Thmthm2.p1.4.4.m4.1.1.2.3" xref="S4.Thmthm2.p1.4.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S4.Thmthm2.p1.4.4.m4.1.1.1" xref="S4.Thmthm2.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="S4.Thmthm2.p1.4.4.m4.1.1.3" xref="S4.Thmthm2.p1.4.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm2.p1.4.4.m4.1.1.3.2" xref="S4.Thmthm2.p1.4.4.m4.1.1.3.2.cmml">ℬ</mi><mo id="S4.Thmthm2.p1.4.4.m4.1.1.3.3" xref="S4.Thmthm2.p1.4.4.m4.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm2.p1.4.4.m4.1b"><apply id="S4.Thmthm2.p1.4.4.m4.1.1.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1"><in id="S4.Thmthm2.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.1"></in><apply id="S4.Thmthm2.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.Thmthm2.p1.4.4.m4.1.1.2.1.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.2">superscript</csymbol><ci id="S4.Thmthm2.p1.4.4.m4.1.1.2.2.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.2.2">𝑤</ci><ci id="S4.Thmthm2.p1.4.4.m4.1.1.2.3.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.2.3">′</ci></apply><apply id="S4.Thmthm2.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.Thmthm2.p1.4.4.m4.1.1.3.1.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S4.Thmthm2.p1.4.4.m4.1.1.3.2.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.3.2">ℬ</ci><times id="S4.Thmthm2.p1.4.4.m4.1.1.3.3.cmml" xref="S4.Thmthm2.p1.4.4.m4.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm2.p1.4.4.m4.1c">w^{\prime}\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm2.p1.4.4.m4.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the transferred measure <math alttext="\mu^{\sigma}=\sigma M(\mu)" class="ltx_Math" display="inline" id="S4.Thmthm2.p1.5.5.m5.1"><semantics id="S4.Thmthm2.p1.5.5.m5.1a"><mrow id="S4.Thmthm2.p1.5.5.m5.1.2" xref="S4.Thmthm2.p1.5.5.m5.1.2.cmml"><msup id="S4.Thmthm2.p1.5.5.m5.1.2.2" xref="S4.Thmthm2.p1.5.5.m5.1.2.2.cmml"><mi id="S4.Thmthm2.p1.5.5.m5.1.2.2.2" xref="S4.Thmthm2.p1.5.5.m5.1.2.2.2.cmml">μ</mi><mi id="S4.Thmthm2.p1.5.5.m5.1.2.2.3" xref="S4.Thmthm2.p1.5.5.m5.1.2.2.3.cmml">σ</mi></msup><mo id="S4.Thmthm2.p1.5.5.m5.1.2.1" xref="S4.Thmthm2.p1.5.5.m5.1.2.1.cmml">=</mo><mrow id="S4.Thmthm2.p1.5.5.m5.1.2.3" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.cmml"><mi id="S4.Thmthm2.p1.5.5.m5.1.2.3.2" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.2.cmml">σ</mi><mo id="S4.Thmthm2.p1.5.5.m5.1.2.3.1" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.1.cmml">⁢</mo><mi id="S4.Thmthm2.p1.5.5.m5.1.2.3.3" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.3.cmml">M</mi><mo id="S4.Thmthm2.p1.5.5.m5.1.2.3.1a" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S4.Thmthm2.p1.5.5.m5.1.2.3.4.2" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.cmml"><mo id="S4.Thmthm2.p1.5.5.m5.1.2.3.4.2.1" stretchy="false" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.cmml">(</mo><mi id="S4.Thmthm2.p1.5.5.m5.1.1" xref="S4.Thmthm2.p1.5.5.m5.1.1.cmml">μ</mi><mo id="S4.Thmthm2.p1.5.5.m5.1.2.3.4.2.2" stretchy="false" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm2.p1.5.5.m5.1b"><apply id="S4.Thmthm2.p1.5.5.m5.1.2.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2"><eq id="S4.Thmthm2.p1.5.5.m5.1.2.1.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.1"></eq><apply id="S4.Thmthm2.p1.5.5.m5.1.2.2.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S4.Thmthm2.p1.5.5.m5.1.2.2.1.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.2">superscript</csymbol><ci id="S4.Thmthm2.p1.5.5.m5.1.2.2.2.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.2.2">𝜇</ci><ci id="S4.Thmthm2.p1.5.5.m5.1.2.2.3.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.2.3">𝜎</ci></apply><apply id="S4.Thmthm2.p1.5.5.m5.1.2.3.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.3"><times id="S4.Thmthm2.p1.5.5.m5.1.2.3.1.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.1"></times><ci id="S4.Thmthm2.p1.5.5.m5.1.2.3.2.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.2">𝜎</ci><ci id="S4.Thmthm2.p1.5.5.m5.1.2.3.3.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.2.3.3">𝑀</ci><ci id="S4.Thmthm2.p1.5.5.m5.1.1.cmml" xref="S4.Thmthm2.p1.5.5.m5.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm2.p1.5.5.m5.1c">\mu^{\sigma}=\sigma M(\mu)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm2.p1.5.5.m5.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = italic_σ italic_M ( italic_μ )</annotation></semantics></math>, evaluated on the cylinder <math alttext="[w^{\prime}]" class="ltx_Math" display="inline" id="S4.Thmthm2.p1.6.6.m6.1"><semantics id="S4.Thmthm2.p1.6.6.m6.1a"><mrow id="S4.Thmthm2.p1.6.6.m6.1.1.1" xref="S4.Thmthm2.p1.6.6.m6.1.1.2.cmml"><mo id="S4.Thmthm2.p1.6.6.m6.1.1.1.2" stretchy="false" xref="S4.Thmthm2.p1.6.6.m6.1.1.2.1.cmml">[</mo><msup id="S4.Thmthm2.p1.6.6.m6.1.1.1.1" xref="S4.Thmthm2.p1.6.6.m6.1.1.1.1.cmml"><mi id="S4.Thmthm2.p1.6.6.m6.1.1.1.1.2" xref="S4.Thmthm2.p1.6.6.m6.1.1.1.1.2.cmml">w</mi><mo id="S4.Thmthm2.p1.6.6.m6.1.1.1.1.3" xref="S4.Thmthm2.p1.6.6.m6.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Thmthm2.p1.6.6.m6.1.1.1.3" stretchy="false" xref="S4.Thmthm2.p1.6.6.m6.1.1.2.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm2.p1.6.6.m6.1b"><apply id="S4.Thmthm2.p1.6.6.m6.1.1.2.cmml" xref="S4.Thmthm2.p1.6.6.m6.1.1.1"><csymbol cd="latexml" id="S4.Thmthm2.p1.6.6.m6.1.1.2.1.cmml" xref="S4.Thmthm2.p1.6.6.m6.1.1.1.2">delimited-[]</csymbol><apply id="S4.Thmthm2.p1.6.6.m6.1.1.1.1.cmml" xref="S4.Thmthm2.p1.6.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmthm2.p1.6.6.m6.1.1.1.1.1.cmml" xref="S4.Thmthm2.p1.6.6.m6.1.1.1.1">superscript</csymbol><ci id="S4.Thmthm2.p1.6.6.m6.1.1.1.1.2.cmml" xref="S4.Thmthm2.p1.6.6.m6.1.1.1.1.2">𝑤</ci><ci id="S4.Thmthm2.p1.6.6.m6.1.1.1.1.3.cmml" xref="S4.Thmthm2.p1.6.6.m6.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm2.p1.6.6.m6.1c">[w^{\prime}]</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm2.p1.6.6.m6.1d">[ italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ]</annotation></semantics></math>, has the value</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(4.2)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu^{\sigma}(w^{\prime})=\sum_{u\,\in\,\cal A^{*}}{\lfloor\sigma(u)\rfloor}_{w% ^{\prime}}\cdot\mu(u)\,." class="ltx_Math" display="block" id="S4.E2.m1.3"><semantics id="S4.E2.m1.3a"><mrow id="S4.E2.m1.3.3.1" xref="S4.E2.m1.3.3.1.1.cmml"><mrow id="S4.E2.m1.3.3.1.1" xref="S4.E2.m1.3.3.1.1.cmml"><mrow id="S4.E2.m1.3.3.1.1.1" xref="S4.E2.m1.3.3.1.1.1.cmml"><msup id="S4.E2.m1.3.3.1.1.1.3" xref="S4.E2.m1.3.3.1.1.1.3.cmml"><mi id="S4.E2.m1.3.3.1.1.1.3.2" xref="S4.E2.m1.3.3.1.1.1.3.2.cmml">μ</mi><mi id="S4.E2.m1.3.3.1.1.1.3.3" xref="S4.E2.m1.3.3.1.1.1.3.3.cmml">σ</mi></msup><mo id="S4.E2.m1.3.3.1.1.1.2" xref="S4.E2.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.E2.m1.3.3.1.1.1.1.1" xref="S4.E2.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S4.E2.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.E2.m1.3.3.1.1.1.1.1.1.cmml">(</mo><msup id="S4.E2.m1.3.3.1.1.1.1.1.1" xref="S4.E2.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S4.E2.m1.3.3.1.1.1.1.1.1.2" xref="S4.E2.m1.3.3.1.1.1.1.1.1.2.cmml">w</mi><mo id="S4.E2.m1.3.3.1.1.1.1.1.1.3" xref="S4.E2.m1.3.3.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.E2.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.E2.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E2.m1.3.3.1.1.3" rspace="0.111em" xref="S4.E2.m1.3.3.1.1.3.cmml">=</mo><mrow id="S4.E2.m1.3.3.1.1.2" xref="S4.E2.m1.3.3.1.1.2.cmml"><munder id="S4.E2.m1.3.3.1.1.2.2" xref="S4.E2.m1.3.3.1.1.2.2.cmml"><mo id="S4.E2.m1.3.3.1.1.2.2.2" movablelimits="false" rspace="0em" xref="S4.E2.m1.3.3.1.1.2.2.2.cmml">∑</mo><mrow id="S4.E2.m1.3.3.1.1.2.2.3" xref="S4.E2.m1.3.3.1.1.2.2.3.cmml"><mi id="S4.E2.m1.3.3.1.1.2.2.3.2" xref="S4.E2.m1.3.3.1.1.2.2.3.2.cmml">u</mi><mo id="S4.E2.m1.3.3.1.1.2.2.3.1" lspace="0.448em" rspace="0.448em" xref="S4.E2.m1.3.3.1.1.2.2.3.1.cmml">∈</mo><msup id="S4.E2.m1.3.3.1.1.2.2.3.3" xref="S4.E2.m1.3.3.1.1.2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E2.m1.3.3.1.1.2.2.3.3.2" xref="S4.E2.m1.3.3.1.1.2.2.3.3.2.cmml">𝒜</mi><mo id="S4.E2.m1.3.3.1.1.2.2.3.3.3" xref="S4.E2.m1.3.3.1.1.2.2.3.3.3.cmml">∗</mo></msup></mrow></munder><mrow id="S4.E2.m1.3.3.1.1.2.1" xref="S4.E2.m1.3.3.1.1.2.1.cmml"><mrow id="S4.E2.m1.3.3.1.1.2.1.1" xref="S4.E2.m1.3.3.1.1.2.1.1.cmml"><msub id="S4.E2.m1.3.3.1.1.2.1.1.1" xref="S4.E2.m1.3.3.1.1.2.1.1.1.cmml"><mrow id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.2.cmml"><mo id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.2" stretchy="false" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.2.1.cmml">⌊</mo><mrow id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.cmml"><mi id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.2" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.1" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.3.2" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.cmml"><mo id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.cmml">(</mo><mi id="S4.E2.m1.1.1" xref="S4.E2.m1.1.1.cmml">u</mi><mo id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.3" rspace="0.055em" stretchy="false" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.2.1.cmml">⌋</mo></mrow><msup id="S4.E2.m1.3.3.1.1.2.1.1.1.3" xref="S4.E2.m1.3.3.1.1.2.1.1.1.3.cmml"><mi id="S4.E2.m1.3.3.1.1.2.1.1.1.3.2" xref="S4.E2.m1.3.3.1.1.2.1.1.1.3.2.cmml">w</mi><mo id="S4.E2.m1.3.3.1.1.2.1.1.1.3.3" xref="S4.E2.m1.3.3.1.1.2.1.1.1.3.3.cmml">′</mo></msup></msub><mo id="S4.E2.m1.3.3.1.1.2.1.1.2" rspace="0.222em" xref="S4.E2.m1.3.3.1.1.2.1.1.2.cmml">⋅</mo><mi id="S4.E2.m1.3.3.1.1.2.1.1.3" xref="S4.E2.m1.3.3.1.1.2.1.1.3.cmml">μ</mi></mrow><mo id="S4.E2.m1.3.3.1.1.2.1.2" xref="S4.E2.m1.3.3.1.1.2.1.2.cmml">⁢</mo><mrow id="S4.E2.m1.3.3.1.1.2.1.3.2" xref="S4.E2.m1.3.3.1.1.2.1.cmml"><mo id="S4.E2.m1.3.3.1.1.2.1.3.2.1" stretchy="false" xref="S4.E2.m1.3.3.1.1.2.1.cmml">(</mo><mi id="S4.E2.m1.2.2" xref="S4.E2.m1.2.2.cmml">u</mi><mo id="S4.E2.m1.3.3.1.1.2.1.3.2.2" stretchy="false" xref="S4.E2.m1.3.3.1.1.2.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E2.m1.3.3.1.2" lspace="0.170em" xref="S4.E2.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E2.m1.3b"><apply id="S4.E2.m1.3.3.1.1.cmml" xref="S4.E2.m1.3.3.1"><eq id="S4.E2.m1.3.3.1.1.3.cmml" xref="S4.E2.m1.3.3.1.1.3"></eq><apply id="S4.E2.m1.3.3.1.1.1.cmml" xref="S4.E2.m1.3.3.1.1.1"><times id="S4.E2.m1.3.3.1.1.1.2.cmml" xref="S4.E2.m1.3.3.1.1.1.2"></times><apply id="S4.E2.m1.3.3.1.1.1.3.cmml" xref="S4.E2.m1.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S4.E2.m1.3.3.1.1.1.3.1.cmml" xref="S4.E2.m1.3.3.1.1.1.3">superscript</csymbol><ci id="S4.E2.m1.3.3.1.1.1.3.2.cmml" xref="S4.E2.m1.3.3.1.1.1.3.2">𝜇</ci><ci id="S4.E2.m1.3.3.1.1.1.3.3.cmml" xref="S4.E2.m1.3.3.1.1.1.3.3">𝜎</ci></apply><apply id="S4.E2.m1.3.3.1.1.1.1.1.1.cmml" xref="S4.E2.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E2.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E2.m1.3.3.1.1.1.1.1">superscript</csymbol><ci id="S4.E2.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S4.E2.m1.3.3.1.1.1.1.1.1.2">𝑤</ci><ci id="S4.E2.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S4.E2.m1.3.3.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S4.E2.m1.3.3.1.1.2.cmml" xref="S4.E2.m1.3.3.1.1.2"><apply id="S4.E2.m1.3.3.1.1.2.2.cmml" xref="S4.E2.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S4.E2.m1.3.3.1.1.2.2.1.cmml" xref="S4.E2.m1.3.3.1.1.2.2">subscript</csymbol><sum id="S4.E2.m1.3.3.1.1.2.2.2.cmml" xref="S4.E2.m1.3.3.1.1.2.2.2"></sum><apply id="S4.E2.m1.3.3.1.1.2.2.3.cmml" xref="S4.E2.m1.3.3.1.1.2.2.3"><in id="S4.E2.m1.3.3.1.1.2.2.3.1.cmml" xref="S4.E2.m1.3.3.1.1.2.2.3.1"></in><ci id="S4.E2.m1.3.3.1.1.2.2.3.2.cmml" xref="S4.E2.m1.3.3.1.1.2.2.3.2">𝑢</ci><apply id="S4.E2.m1.3.3.1.1.2.2.3.3.cmml" xref="S4.E2.m1.3.3.1.1.2.2.3.3"><csymbol cd="ambiguous" id="S4.E2.m1.3.3.1.1.2.2.3.3.1.cmml" xref="S4.E2.m1.3.3.1.1.2.2.3.3">superscript</csymbol><ci id="S4.E2.m1.3.3.1.1.2.2.3.3.2.cmml" xref="S4.E2.m1.3.3.1.1.2.2.3.3.2">𝒜</ci><times id="S4.E2.m1.3.3.1.1.2.2.3.3.3.cmml" xref="S4.E2.m1.3.3.1.1.2.2.3.3.3"></times></apply></apply></apply><apply id="S4.E2.m1.3.3.1.1.2.1.cmml" xref="S4.E2.m1.3.3.1.1.2.1"><times id="S4.E2.m1.3.3.1.1.2.1.2.cmml" xref="S4.E2.m1.3.3.1.1.2.1.2"></times><apply id="S4.E2.m1.3.3.1.1.2.1.1.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1"><ci id="S4.E2.m1.3.3.1.1.2.1.1.2.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.2">⋅</ci><apply id="S4.E2.m1.3.3.1.1.2.1.1.1.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S4.E2.m1.3.3.1.1.2.1.1.1.2.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1">subscript</csymbol><apply id="S4.E2.m1.3.3.1.1.2.1.1.1.1.2.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1"><floor id="S4.E2.m1.3.3.1.1.2.1.1.1.1.2.1.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.2"></floor><apply id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1"><times id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.1.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.1"></times><ci id="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.2.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.1.1.1.2">𝜎</ci><ci id="S4.E2.m1.1.1.cmml" xref="S4.E2.m1.1.1">𝑢</ci></apply></apply><apply id="S4.E2.m1.3.3.1.1.2.1.1.1.3.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.E2.m1.3.3.1.1.2.1.1.1.3.1.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.3">superscript</csymbol><ci id="S4.E2.m1.3.3.1.1.2.1.1.1.3.2.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.3.2">𝑤</ci><ci id="S4.E2.m1.3.3.1.1.2.1.1.1.3.3.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.1.3.3">′</ci></apply></apply><ci id="S4.E2.m1.3.3.1.1.2.1.1.3.cmml" xref="S4.E2.m1.3.3.1.1.2.1.1.3">𝜇</ci></apply><ci id="S4.E2.m1.2.2.cmml" xref="S4.E2.m1.2.2">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E2.m1.3c">\mu^{\sigma}(w^{\prime})=\sum_{u\,\in\,\cal A^{*}}{\lfloor\sigma(u)\rfloor}_{w% ^{\prime}}\cdot\mu(u)\,.</annotation><annotation encoding="application/x-llamapun" id="S4.E2.m1.3d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_u ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⌊ italic_σ ( italic_u ) ⌋ start_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⋅ italic_μ ( italic_u ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmthm2.p1.7"><span class="ltx_text ltx_font_italic" id="S4.Thmthm2.p1.7.1">In particular, for <math alttext="|w^{\prime}|\geq 2" class="ltx_Math" display="inline" id="S4.Thmthm2.p1.7.1.m1.1"><semantics id="S4.Thmthm2.p1.7.1.m1.1a"><mrow id="S4.Thmthm2.p1.7.1.m1.1.1" xref="S4.Thmthm2.p1.7.1.m1.1.1.cmml"><mrow id="S4.Thmthm2.p1.7.1.m1.1.1.1.1" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.2.cmml"><mo id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.2" stretchy="false" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.2.1.cmml">|</mo><msup id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.cmml"><mi id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.2" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.2.cmml">w</mi><mo id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.3" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.3" stretchy="false" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Thmthm2.p1.7.1.m1.1.1.2" xref="S4.Thmthm2.p1.7.1.m1.1.1.2.cmml">≥</mo><mn id="S4.Thmthm2.p1.7.1.m1.1.1.3" xref="S4.Thmthm2.p1.7.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm2.p1.7.1.m1.1b"><apply id="S4.Thmthm2.p1.7.1.m1.1.1.cmml" xref="S4.Thmthm2.p1.7.1.m1.1.1"><geq id="S4.Thmthm2.p1.7.1.m1.1.1.2.cmml" xref="S4.Thmthm2.p1.7.1.m1.1.1.2"></geq><apply id="S4.Thmthm2.p1.7.1.m1.1.1.1.2.cmml" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1"><abs id="S4.Thmthm2.p1.7.1.m1.1.1.1.2.1.cmml" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1.2"></abs><apply id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.cmml" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.1.cmml" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.2.cmml" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.2">𝑤</ci><ci id="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.3.cmml" xref="S4.Thmthm2.p1.7.1.m1.1.1.1.1.1.3">′</ci></apply></apply><cn id="S4.Thmthm2.p1.7.1.m1.1.1.3.cmml" type="integer" xref="S4.Thmthm2.p1.7.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm2.p1.7.1.m1.1c">|w^{\prime}|\geq 2</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm2.p1.7.1.m1.1d">| italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | ≥ 2</annotation></semantics></math>, we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(4.3)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu^{\sigma}(w^{\prime})=\sum_{{\big{\{}}u\,\in\,\cal A^{*}\,{\big{|}}\,|u|% \leq\frac{|w^{\prime}|-2}{\langle\sigma\rangle}+2{\big{\}}}}{\lfloor\sigma(u)% \rfloor}_{w^{\prime}}\cdot\mu(u)\,." class="ltx_Math" display="block" id="S4.E3.m1.8"><semantics id="S4.E3.m1.8a"><mrow id="S4.E3.m1.8.8.1" xref="S4.E3.m1.8.8.1.1.cmml"><mrow id="S4.E3.m1.8.8.1.1" xref="S4.E3.m1.8.8.1.1.cmml"><mrow id="S4.E3.m1.8.8.1.1.1" xref="S4.E3.m1.8.8.1.1.1.cmml"><msup id="S4.E3.m1.8.8.1.1.1.3" xref="S4.E3.m1.8.8.1.1.1.3.cmml"><mi id="S4.E3.m1.8.8.1.1.1.3.2" xref="S4.E3.m1.8.8.1.1.1.3.2.cmml">μ</mi><mi id="S4.E3.m1.8.8.1.1.1.3.3" xref="S4.E3.m1.8.8.1.1.1.3.3.cmml">σ</mi></msup><mo id="S4.E3.m1.8.8.1.1.1.2" xref="S4.E3.m1.8.8.1.1.1.2.cmml">⁢</mo><mrow 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xref="S4.E3.m1.5.5.5.5.2.2.1.1.cmml">|</mo></mrow><mo id="S4.E3.m1.5.5.5.5.2.1" xref="S4.E3.m1.5.5.5.5.2.1.cmml">≤</mo><mrow id="S4.E3.m1.5.5.5.5.2.3" xref="S4.E3.m1.5.5.5.5.2.3.cmml"><mfrac id="S4.E3.m1.2.2.2.2" xref="S4.E3.m1.2.2.2.2.cmml"><mrow id="S4.E3.m1.1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.1.cmml"><mrow id="S4.E3.m1.1.1.1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.1.1.2.cmml"><mo id="S4.E3.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.E3.m1.1.1.1.1.1.1.2.1.cmml">|</mo><msup id="S4.E3.m1.1.1.1.1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E3.m1.1.1.1.1.1.1.1.1.2" xref="S4.E3.m1.1.1.1.1.1.1.1.1.2.cmml">𝓌</mi><mo id="S4.E3.m1.1.1.1.1.1.1.1.1.3" xref="S4.E3.m1.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.E3.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.E3.m1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.E3.m1.1.1.1.1.1.2" xref="S4.E3.m1.1.1.1.1.1.2.cmml">−</mo><mn class="ltx_font_mathcaligraphic" id="S4.E3.m1.1.1.1.1.1.3" mathvariant="script" xref="S4.E3.m1.1.1.1.1.1.3.cmml">2</mn></mrow><mrow id="S4.E3.m1.2.2.2.2.2.3" xref="S4.E3.m1.2.2.2.2.2.2.cmml"><mo id="S4.E3.m1.2.2.2.2.2.3.1" stretchy="false" xref="S4.E3.m1.2.2.2.2.2.2.1.cmml">⟨</mo><mi id="S4.E3.m1.2.2.2.2.2.1" xref="S4.E3.m1.2.2.2.2.2.1.cmml">σ</mi><mo id="S4.E3.m1.2.2.2.2.2.3.2" stretchy="false" xref="S4.E3.m1.2.2.2.2.2.2.1.cmml">⟩</mo></mrow></mfrac><mo id="S4.E3.m1.5.5.5.5.2.3.1" xref="S4.E3.m1.5.5.5.5.2.3.1.cmml">+</mo><mn class="ltx_font_mathcaligraphic" id="S4.E3.m1.5.5.5.5.2.3.2" mathvariant="script" xref="S4.E3.m1.5.5.5.5.2.3.2.cmml">2</mn></mrow></mrow><mo id="S4.E3.m1.5.5.5.5.5" maxsize="171%" minsize="171%" xref="S4.E3.m1.5.5.5.6.1.cmml">}</mo></mrow></munder><mrow id="S4.E3.m1.8.8.1.1.2.1" xref="S4.E3.m1.8.8.1.1.2.1.cmml"><mrow id="S4.E3.m1.8.8.1.1.2.1.1" xref="S4.E3.m1.8.8.1.1.2.1.1.cmml"><msub id="S4.E3.m1.8.8.1.1.2.1.1.1" xref="S4.E3.m1.8.8.1.1.2.1.1.1.cmml"><mrow id="S4.E3.m1.8.8.1.1.2.1.1.1.1.1" xref="S4.E3.m1.8.8.1.1.2.1.1.1.1.2.cmml"><mo 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xref="S4.E3.m1.8.8.1.1.2.1.1.1.3.3">′</ci></apply></apply><ci id="S4.E3.m1.8.8.1.1.2.1.1.3.cmml" xref="S4.E3.m1.8.8.1.1.2.1.1.3">𝜇</ci></apply><ci id="S4.E3.m1.7.7.cmml" xref="S4.E3.m1.7.7">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m1.8c">\mu^{\sigma}(w^{\prime})=\sum_{{\big{\{}}u\,\in\,\cal A^{*}\,{\big{|}}\,|u|% \leq\frac{|w^{\prime}|-2}{\langle\sigma\rangle}+2{\big{\}}}}{\lfloor\sigma(u)% \rfloor}_{w^{\prime}}\cdot\mu(u)\,.</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.8d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT { italic_u ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | | caligraphic_u | ≤ divide start_ARG | caligraphic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | - caligraphic_2 end_ARG start_ARG ⟨ italic_σ ⟩ end_ARG + caligraphic_2 } end_POSTSUBSCRIPT ⌊ italic_σ ( italic_u ) ⌋ start_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ⋅ italic_μ ( italic_u ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.1.p1"> <p class="ltx_p" id="S4.SS2.1.p1.11">Recall that in formula (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E5" title="In Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.5</span></a>), for any <math alttext="u\in\alpha_{\sigma}^{-1}(w^{\prime})" class="ltx_Math" display="inline" id="S4.SS2.1.p1.1.m1.1"><semantics id="S4.SS2.1.p1.1.m1.1a"><mrow id="S4.SS2.1.p1.1.m1.1.1" xref="S4.SS2.1.p1.1.m1.1.1.cmml"><mi id="S4.SS2.1.p1.1.m1.1.1.3" xref="S4.SS2.1.p1.1.m1.1.1.3.cmml">u</mi><mo id="S4.SS2.1.p1.1.m1.1.1.2" xref="S4.SS2.1.p1.1.m1.1.1.2.cmml">∈</mo><mrow id="S4.SS2.1.p1.1.m1.1.1.1" xref="S4.SS2.1.p1.1.m1.1.1.1.cmml"><msubsup id="S4.SS2.1.p1.1.m1.1.1.1.3" xref="S4.SS2.1.p1.1.m1.1.1.1.3.cmml"><mi id="S4.SS2.1.p1.1.m1.1.1.1.3.2.2" xref="S4.SS2.1.p1.1.m1.1.1.1.3.2.2.cmml">α</mi><mi id="S4.SS2.1.p1.1.m1.1.1.1.3.2.3" xref="S4.SS2.1.p1.1.m1.1.1.1.3.2.3.cmml">σ</mi><mrow id="S4.SS2.1.p1.1.m1.1.1.1.3.3" xref="S4.SS2.1.p1.1.m1.1.1.1.3.3.cmml"><mo id="S4.SS2.1.p1.1.m1.1.1.1.3.3a" xref="S4.SS2.1.p1.1.m1.1.1.1.3.3.cmml">−</mo><mn 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xref="S4.SS2.1.p1.1.m1.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.SS2.1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.1.p1.1.m1.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.1.p1.1.m1.1.1.1.1.1.1.2">𝑤</ci><ci id="S4.SS2.1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.1.p1.1.m1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.1.m1.1c">u\in\alpha_{\sigma}^{-1}(w^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.1.m1.1d">italic_u ∈ italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>, either <math alttext="\widehat{u}" class="ltx_Math" display="inline" id="S4.SS2.1.p1.2.m2.1"><semantics id="S4.SS2.1.p1.2.m2.1a"><mover accent="true" id="S4.SS2.1.p1.2.m2.1.1" xref="S4.SS2.1.p1.2.m2.1.1.cmml"><mi id="S4.SS2.1.p1.2.m2.1.1.2" xref="S4.SS2.1.p1.2.m2.1.1.2.cmml">u</mi><mo id="S4.SS2.1.p1.2.m2.1.1.1" xref="S4.SS2.1.p1.2.m2.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.2.m2.1b"><apply id="S4.SS2.1.p1.2.m2.1.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1"><ci id="S4.SS2.1.p1.2.m2.1.1.1.cmml" xref="S4.SS2.1.p1.2.m2.1.1.1">^</ci><ci id="S4.SS2.1.p1.2.m2.1.1.2.cmml" xref="S4.SS2.1.p1.2.m2.1.1.2">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.2.m2.1c">\widehat{u}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.2.m2.1d">over^ start_ARG italic_u end_ARG</annotation></semantics></math> is defined as shortest word in <math alttext="\cal A^{*}" class="ltx_Math" display="inline" id="S4.SS2.1.p1.3.m3.1"><semantics id="S4.SS2.1.p1.3.m3.1a"><msup id="S4.SS2.1.p1.3.m3.1.1" xref="S4.SS2.1.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.1.p1.3.m3.1.1.2" xref="S4.SS2.1.p1.3.m3.1.1.2.cmml">𝒜</mi><mo id="S4.SS2.1.p1.3.m3.1.1.3" xref="S4.SS2.1.p1.3.m3.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.3.m3.1b"><apply id="S4.SS2.1.p1.3.m3.1.1.cmml" xref="S4.SS2.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.1.p1.3.m3.1.1.1.cmml" xref="S4.SS2.1.p1.3.m3.1.1">superscript</csymbol><ci id="S4.SS2.1.p1.3.m3.1.1.2.cmml" xref="S4.SS2.1.p1.3.m3.1.1.2">𝒜</ci><times id="S4.SS2.1.p1.3.m3.1.1.3.cmml" xref="S4.SS2.1.p1.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.3.m3.1c">\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.3.m3.1d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with the property that <math alttext="\pi_{\sigma}(\widehat{u})" class="ltx_Math" display="inline" id="S4.SS2.1.p1.4.m4.1"><semantics id="S4.SS2.1.p1.4.m4.1a"><mrow id="S4.SS2.1.p1.4.m4.1.2" xref="S4.SS2.1.p1.4.m4.1.2.cmml"><msub id="S4.SS2.1.p1.4.m4.1.2.2" xref="S4.SS2.1.p1.4.m4.1.2.2.cmml"><mi id="S4.SS2.1.p1.4.m4.1.2.2.2" xref="S4.SS2.1.p1.4.m4.1.2.2.2.cmml">π</mi><mi id="S4.SS2.1.p1.4.m4.1.2.2.3" xref="S4.SS2.1.p1.4.m4.1.2.2.3.cmml">σ</mi></msub><mo id="S4.SS2.1.p1.4.m4.1.2.1" xref="S4.SS2.1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.4.m4.1.2.3.2" xref="S4.SS2.1.p1.4.m4.1.1.cmml"><mo id="S4.SS2.1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S4.SS2.1.p1.4.m4.1.1.cmml">(</mo><mover accent="true" id="S4.SS2.1.p1.4.m4.1.1" xref="S4.SS2.1.p1.4.m4.1.1.cmml"><mi id="S4.SS2.1.p1.4.m4.1.1.2" xref="S4.SS2.1.p1.4.m4.1.1.2.cmml">u</mi><mo id="S4.SS2.1.p1.4.m4.1.1.1" xref="S4.SS2.1.p1.4.m4.1.1.1.cmml">^</mo></mover><mo id="S4.SS2.1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S4.SS2.1.p1.4.m4.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.4.m4.1b"><apply id="S4.SS2.1.p1.4.m4.1.2.cmml" xref="S4.SS2.1.p1.4.m4.1.2"><times id="S4.SS2.1.p1.4.m4.1.2.1.cmml" xref="S4.SS2.1.p1.4.m4.1.2.1"></times><apply id="S4.SS2.1.p1.4.m4.1.2.2.cmml" xref="S4.SS2.1.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.4.m4.1.2.2.1.cmml" xref="S4.SS2.1.p1.4.m4.1.2.2">subscript</csymbol><ci id="S4.SS2.1.p1.4.m4.1.2.2.2.cmml" xref="S4.SS2.1.p1.4.m4.1.2.2.2">𝜋</ci><ci id="S4.SS2.1.p1.4.m4.1.2.2.3.cmml" xref="S4.SS2.1.p1.4.m4.1.2.2.3">𝜎</ci></apply><apply id="S4.SS2.1.p1.4.m4.1.1.cmml" xref="S4.SS2.1.p1.4.m4.1.2.3.2"><ci id="S4.SS2.1.p1.4.m4.1.1.1.cmml" xref="S4.SS2.1.p1.4.m4.1.1.1">^</ci><ci id="S4.SS2.1.p1.4.m4.1.1.2.cmml" xref="S4.SS2.1.p1.4.m4.1.1.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.4.m4.1c">\pi_{\sigma}(\widehat{u})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.4.m4.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over^ start_ARG italic_u end_ARG )</annotation></semantics></math> contains <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.1.p1.5.m5.1"><semantics id="S4.SS2.1.p1.5.m5.1a"><mi id="S4.SS2.1.p1.5.m5.1.1" xref="S4.SS2.1.p1.5.m5.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.5.m5.1b"><ci id="S4.SS2.1.p1.5.m5.1.1.cmml" xref="S4.SS2.1.p1.5.m5.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.5.m5.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.5.m5.1d">italic_u</annotation></semantics></math> as factor, or else (if such <math alttext="\widehat{u}" class="ltx_Math" display="inline" id="S4.SS2.1.p1.6.m6.1"><semantics id="S4.SS2.1.p1.6.m6.1a"><mover accent="true" id="S4.SS2.1.p1.6.m6.1.1" xref="S4.SS2.1.p1.6.m6.1.1.cmml"><mi id="S4.SS2.1.p1.6.m6.1.1.2" xref="S4.SS2.1.p1.6.m6.1.1.2.cmml">u</mi><mo id="S4.SS2.1.p1.6.m6.1.1.1" xref="S4.SS2.1.p1.6.m6.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.6.m6.1b"><apply id="S4.SS2.1.p1.6.m6.1.1.cmml" xref="S4.SS2.1.p1.6.m6.1.1"><ci id="S4.SS2.1.p1.6.m6.1.1.1.cmml" xref="S4.SS2.1.p1.6.m6.1.1.1">^</ci><ci id="S4.SS2.1.p1.6.m6.1.1.2.cmml" xref="S4.SS2.1.p1.6.m6.1.1.2">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.6.m6.1c">\widehat{u}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.6.m6.1d">over^ start_ARG italic_u end_ARG</annotation></semantics></math> doesn’t exist) we have formally set <math alttext="\mu(\widehat{u})=\mu_{\ell_{\sigma}}(u)=0" class="ltx_Math" display="inline" id="S4.SS2.1.p1.7.m7.2"><semantics id="S4.SS2.1.p1.7.m7.2a"><mrow id="S4.SS2.1.p1.7.m7.2.3" xref="S4.SS2.1.p1.7.m7.2.3.cmml"><mrow id="S4.SS2.1.p1.7.m7.2.3.2" xref="S4.SS2.1.p1.7.m7.2.3.2.cmml"><mi id="S4.SS2.1.p1.7.m7.2.3.2.2" xref="S4.SS2.1.p1.7.m7.2.3.2.2.cmml">μ</mi><mo id="S4.SS2.1.p1.7.m7.2.3.2.1" xref="S4.SS2.1.p1.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.7.m7.2.3.2.3.2" xref="S4.SS2.1.p1.7.m7.1.1.cmml"><mo id="S4.SS2.1.p1.7.m7.2.3.2.3.2.1" stretchy="false" xref="S4.SS2.1.p1.7.m7.1.1.cmml">(</mo><mover accent="true" id="S4.SS2.1.p1.7.m7.1.1" xref="S4.SS2.1.p1.7.m7.1.1.cmml"><mi id="S4.SS2.1.p1.7.m7.1.1.2" xref="S4.SS2.1.p1.7.m7.1.1.2.cmml">u</mi><mo id="S4.SS2.1.p1.7.m7.1.1.1" xref="S4.SS2.1.p1.7.m7.1.1.1.cmml">^</mo></mover><mo id="S4.SS2.1.p1.7.m7.2.3.2.3.2.2" stretchy="false" xref="S4.SS2.1.p1.7.m7.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.1.p1.7.m7.2.3.3" xref="S4.SS2.1.p1.7.m7.2.3.3.cmml">=</mo><mrow id="S4.SS2.1.p1.7.m7.2.3.4" xref="S4.SS2.1.p1.7.m7.2.3.4.cmml"><msub id="S4.SS2.1.p1.7.m7.2.3.4.2" xref="S4.SS2.1.p1.7.m7.2.3.4.2.cmml"><mi id="S4.SS2.1.p1.7.m7.2.3.4.2.2" xref="S4.SS2.1.p1.7.m7.2.3.4.2.2.cmml">μ</mi><msub id="S4.SS2.1.p1.7.m7.2.3.4.2.3" xref="S4.SS2.1.p1.7.m7.2.3.4.2.3.cmml"><mi id="S4.SS2.1.p1.7.m7.2.3.4.2.3.2" mathvariant="normal" xref="S4.SS2.1.p1.7.m7.2.3.4.2.3.2.cmml">ℓ</mi><mi id="S4.SS2.1.p1.7.m7.2.3.4.2.3.3" xref="S4.SS2.1.p1.7.m7.2.3.4.2.3.3.cmml">σ</mi></msub></msub><mo id="S4.SS2.1.p1.7.m7.2.3.4.1" xref="S4.SS2.1.p1.7.m7.2.3.4.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.7.m7.2.3.4.3.2" xref="S4.SS2.1.p1.7.m7.2.3.4.cmml"><mo id="S4.SS2.1.p1.7.m7.2.3.4.3.2.1" stretchy="false" xref="S4.SS2.1.p1.7.m7.2.3.4.cmml">(</mo><mi id="S4.SS2.1.p1.7.m7.2.2" xref="S4.SS2.1.p1.7.m7.2.2.cmml">u</mi><mo id="S4.SS2.1.p1.7.m7.2.3.4.3.2.2" stretchy="false" xref="S4.SS2.1.p1.7.m7.2.3.4.cmml">)</mo></mrow></mrow><mo id="S4.SS2.1.p1.7.m7.2.3.5" xref="S4.SS2.1.p1.7.m7.2.3.5.cmml">=</mo><mn id="S4.SS2.1.p1.7.m7.2.3.6" xref="S4.SS2.1.p1.7.m7.2.3.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.7.m7.2b"><apply id="S4.SS2.1.p1.7.m7.2.3.cmml" xref="S4.SS2.1.p1.7.m7.2.3"><and id="S4.SS2.1.p1.7.m7.2.3a.cmml" xref="S4.SS2.1.p1.7.m7.2.3"></and><apply id="S4.SS2.1.p1.7.m7.2.3b.cmml" xref="S4.SS2.1.p1.7.m7.2.3"><eq id="S4.SS2.1.p1.7.m7.2.3.3.cmml" xref="S4.SS2.1.p1.7.m7.2.3.3"></eq><apply id="S4.SS2.1.p1.7.m7.2.3.2.cmml" xref="S4.SS2.1.p1.7.m7.2.3.2"><times id="S4.SS2.1.p1.7.m7.2.3.2.1.cmml" xref="S4.SS2.1.p1.7.m7.2.3.2.1"></times><ci id="S4.SS2.1.p1.7.m7.2.3.2.2.cmml" xref="S4.SS2.1.p1.7.m7.2.3.2.2">𝜇</ci><apply id="S4.SS2.1.p1.7.m7.1.1.cmml" xref="S4.SS2.1.p1.7.m7.2.3.2.3.2"><ci id="S4.SS2.1.p1.7.m7.1.1.1.cmml" xref="S4.SS2.1.p1.7.m7.1.1.1">^</ci><ci id="S4.SS2.1.p1.7.m7.1.1.2.cmml" xref="S4.SS2.1.p1.7.m7.1.1.2">𝑢</ci></apply></apply><apply id="S4.SS2.1.p1.7.m7.2.3.4.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4"><times id="S4.SS2.1.p1.7.m7.2.3.4.1.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4.1"></times><apply id="S4.SS2.1.p1.7.m7.2.3.4.2.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.7.m7.2.3.4.2.1.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4.2">subscript</csymbol><ci id="S4.SS2.1.p1.7.m7.2.3.4.2.2.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4.2.2">𝜇</ci><apply id="S4.SS2.1.p1.7.m7.2.3.4.2.3.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4.2.3"><csymbol cd="ambiguous" id="S4.SS2.1.p1.7.m7.2.3.4.2.3.1.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4.2.3">subscript</csymbol><ci id="S4.SS2.1.p1.7.m7.2.3.4.2.3.2.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4.2.3.2">ℓ</ci><ci id="S4.SS2.1.p1.7.m7.2.3.4.2.3.3.cmml" xref="S4.SS2.1.p1.7.m7.2.3.4.2.3.3">𝜎</ci></apply></apply><ci id="S4.SS2.1.p1.7.m7.2.2.cmml" xref="S4.SS2.1.p1.7.m7.2.2">𝑢</ci></apply></apply><apply id="S4.SS2.1.p1.7.m7.2.3c.cmml" xref="S4.SS2.1.p1.7.m7.2.3"><eq id="S4.SS2.1.p1.7.m7.2.3.5.cmml" xref="S4.SS2.1.p1.7.m7.2.3.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S4.SS2.1.p1.7.m7.2.3.4.cmml" id="S4.SS2.1.p1.7.m7.2.3d.cmml" xref="S4.SS2.1.p1.7.m7.2.3"></share><cn id="S4.SS2.1.p1.7.m7.2.3.6.cmml" type="integer" xref="S4.SS2.1.p1.7.m7.2.3.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.7.m7.2c">\mu(\widehat{u})=\mu_{\ell_{\sigma}}(u)=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.7.m7.2d">italic_μ ( over^ start_ARG italic_u end_ARG ) = italic_μ start_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_u ) = 0</annotation></semantics></math>. In the first case the factor <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.1.p1.8.m8.1"><semantics id="S4.SS2.1.p1.8.m8.1a"><mi id="S4.SS2.1.p1.8.m8.1.1" xref="S4.SS2.1.p1.8.m8.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.8.m8.1b"><ci id="S4.SS2.1.p1.8.m8.1.1.cmml" xref="S4.SS2.1.p1.8.m8.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.8.m8.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.8.m8.1d">italic_u</annotation></semantics></math> of <math alttext="\pi_{\sigma}(\widehat{u})" class="ltx_Math" display="inline" id="S4.SS2.1.p1.9.m9.1"><semantics id="S4.SS2.1.p1.9.m9.1a"><mrow id="S4.SS2.1.p1.9.m9.1.2" xref="S4.SS2.1.p1.9.m9.1.2.cmml"><msub id="S4.SS2.1.p1.9.m9.1.2.2" xref="S4.SS2.1.p1.9.m9.1.2.2.cmml"><mi id="S4.SS2.1.p1.9.m9.1.2.2.2" xref="S4.SS2.1.p1.9.m9.1.2.2.2.cmml">π</mi><mi id="S4.SS2.1.p1.9.m9.1.2.2.3" xref="S4.SS2.1.p1.9.m9.1.2.2.3.cmml">σ</mi></msub><mo id="S4.SS2.1.p1.9.m9.1.2.1" xref="S4.SS2.1.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.9.m9.1.2.3.2" xref="S4.SS2.1.p1.9.m9.1.1.cmml"><mo id="S4.SS2.1.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S4.SS2.1.p1.9.m9.1.1.cmml">(</mo><mover accent="true" id="S4.SS2.1.p1.9.m9.1.1" xref="S4.SS2.1.p1.9.m9.1.1.cmml"><mi id="S4.SS2.1.p1.9.m9.1.1.2" xref="S4.SS2.1.p1.9.m9.1.1.2.cmml">u</mi><mo id="S4.SS2.1.p1.9.m9.1.1.1" xref="S4.SS2.1.p1.9.m9.1.1.1.cmml">^</mo></mover><mo id="S4.SS2.1.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S4.SS2.1.p1.9.m9.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.9.m9.1b"><apply id="S4.SS2.1.p1.9.m9.1.2.cmml" xref="S4.SS2.1.p1.9.m9.1.2"><times id="S4.SS2.1.p1.9.m9.1.2.1.cmml" xref="S4.SS2.1.p1.9.m9.1.2.1"></times><apply id="S4.SS2.1.p1.9.m9.1.2.2.cmml" xref="S4.SS2.1.p1.9.m9.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.9.m9.1.2.2.1.cmml" xref="S4.SS2.1.p1.9.m9.1.2.2">subscript</csymbol><ci id="S4.SS2.1.p1.9.m9.1.2.2.2.cmml" xref="S4.SS2.1.p1.9.m9.1.2.2.2">𝜋</ci><ci id="S4.SS2.1.p1.9.m9.1.2.2.3.cmml" xref="S4.SS2.1.p1.9.m9.1.2.2.3">𝜎</ci></apply><apply id="S4.SS2.1.p1.9.m9.1.1.cmml" xref="S4.SS2.1.p1.9.m9.1.2.3.2"><ci id="S4.SS2.1.p1.9.m9.1.1.1.cmml" xref="S4.SS2.1.p1.9.m9.1.1.1">^</ci><ci id="S4.SS2.1.p1.9.m9.1.1.2.cmml" xref="S4.SS2.1.p1.9.m9.1.1.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.9.m9.1c">\pi_{\sigma}(\widehat{u})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.9.m9.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over^ start_ARG italic_u end_ARG )</annotation></semantics></math> defines an essential occurrence of <math alttext="w^{\prime}=\alpha_{\sigma}(u)" class="ltx_Math" display="inline" id="S4.SS2.1.p1.10.m10.1"><semantics id="S4.SS2.1.p1.10.m10.1a"><mrow id="S4.SS2.1.p1.10.m10.1.2" xref="S4.SS2.1.p1.10.m10.1.2.cmml"><msup id="S4.SS2.1.p1.10.m10.1.2.2" xref="S4.SS2.1.p1.10.m10.1.2.2.cmml"><mi id="S4.SS2.1.p1.10.m10.1.2.2.2" xref="S4.SS2.1.p1.10.m10.1.2.2.2.cmml">w</mi><mo id="S4.SS2.1.p1.10.m10.1.2.2.3" xref="S4.SS2.1.p1.10.m10.1.2.2.3.cmml">′</mo></msup><mo id="S4.SS2.1.p1.10.m10.1.2.1" xref="S4.SS2.1.p1.10.m10.1.2.1.cmml">=</mo><mrow id="S4.SS2.1.p1.10.m10.1.2.3" xref="S4.SS2.1.p1.10.m10.1.2.3.cmml"><msub id="S4.SS2.1.p1.10.m10.1.2.3.2" xref="S4.SS2.1.p1.10.m10.1.2.3.2.cmml"><mi id="S4.SS2.1.p1.10.m10.1.2.3.2.2" xref="S4.SS2.1.p1.10.m10.1.2.3.2.2.cmml">α</mi><mi id="S4.SS2.1.p1.10.m10.1.2.3.2.3" xref="S4.SS2.1.p1.10.m10.1.2.3.2.3.cmml">σ</mi></msub><mo id="S4.SS2.1.p1.10.m10.1.2.3.1" xref="S4.SS2.1.p1.10.m10.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.10.m10.1.2.3.3.2" xref="S4.SS2.1.p1.10.m10.1.2.3.cmml"><mo id="S4.SS2.1.p1.10.m10.1.2.3.3.2.1" stretchy="false" xref="S4.SS2.1.p1.10.m10.1.2.3.cmml">(</mo><mi id="S4.SS2.1.p1.10.m10.1.1" xref="S4.SS2.1.p1.10.m10.1.1.cmml">u</mi><mo id="S4.SS2.1.p1.10.m10.1.2.3.3.2.2" stretchy="false" xref="S4.SS2.1.p1.10.m10.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.10.m10.1b"><apply id="S4.SS2.1.p1.10.m10.1.2.cmml" xref="S4.SS2.1.p1.10.m10.1.2"><eq id="S4.SS2.1.p1.10.m10.1.2.1.cmml" xref="S4.SS2.1.p1.10.m10.1.2.1"></eq><apply id="S4.SS2.1.p1.10.m10.1.2.2.cmml" xref="S4.SS2.1.p1.10.m10.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.10.m10.1.2.2.1.cmml" xref="S4.SS2.1.p1.10.m10.1.2.2">superscript</csymbol><ci id="S4.SS2.1.p1.10.m10.1.2.2.2.cmml" xref="S4.SS2.1.p1.10.m10.1.2.2.2">𝑤</ci><ci id="S4.SS2.1.p1.10.m10.1.2.2.3.cmml" xref="S4.SS2.1.p1.10.m10.1.2.2.3">′</ci></apply><apply id="S4.SS2.1.p1.10.m10.1.2.3.cmml" xref="S4.SS2.1.p1.10.m10.1.2.3"><times id="S4.SS2.1.p1.10.m10.1.2.3.1.cmml" xref="S4.SS2.1.p1.10.m10.1.2.3.1"></times><apply id="S4.SS2.1.p1.10.m10.1.2.3.2.cmml" xref="S4.SS2.1.p1.10.m10.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.1.p1.10.m10.1.2.3.2.1.cmml" xref="S4.SS2.1.p1.10.m10.1.2.3.2">subscript</csymbol><ci id="S4.SS2.1.p1.10.m10.1.2.3.2.2.cmml" xref="S4.SS2.1.p1.10.m10.1.2.3.2.2">𝛼</ci><ci id="S4.SS2.1.p1.10.m10.1.2.3.2.3.cmml" xref="S4.SS2.1.p1.10.m10.1.2.3.2.3">𝜎</ci></apply><ci id="S4.SS2.1.p1.10.m10.1.1.cmml" xref="S4.SS2.1.p1.10.m10.1.1">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.10.m10.1c">w^{\prime}=\alpha_{\sigma}(u)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.10.m10.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_u )</annotation></semantics></math> in <math alttext="\sigma(\widehat{u})" class="ltx_Math" display="inline" id="S4.SS2.1.p1.11.m11.1"><semantics id="S4.SS2.1.p1.11.m11.1a"><mrow id="S4.SS2.1.p1.11.m11.1.2" xref="S4.SS2.1.p1.11.m11.1.2.cmml"><mi id="S4.SS2.1.p1.11.m11.1.2.2" xref="S4.SS2.1.p1.11.m11.1.2.2.cmml">σ</mi><mo id="S4.SS2.1.p1.11.m11.1.2.1" xref="S4.SS2.1.p1.11.m11.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.1.p1.11.m11.1.2.3.2" xref="S4.SS2.1.p1.11.m11.1.1.cmml"><mo id="S4.SS2.1.p1.11.m11.1.2.3.2.1" stretchy="false" xref="S4.SS2.1.p1.11.m11.1.1.cmml">(</mo><mover accent="true" id="S4.SS2.1.p1.11.m11.1.1" xref="S4.SS2.1.p1.11.m11.1.1.cmml"><mi id="S4.SS2.1.p1.11.m11.1.1.2" xref="S4.SS2.1.p1.11.m11.1.1.2.cmml">u</mi><mo id="S4.SS2.1.p1.11.m11.1.1.1" xref="S4.SS2.1.p1.11.m11.1.1.1.cmml">^</mo></mover><mo id="S4.SS2.1.p1.11.m11.1.2.3.2.2" stretchy="false" xref="S4.SS2.1.p1.11.m11.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.1.p1.11.m11.1b"><apply id="S4.SS2.1.p1.11.m11.1.2.cmml" xref="S4.SS2.1.p1.11.m11.1.2"><times id="S4.SS2.1.p1.11.m11.1.2.1.cmml" xref="S4.SS2.1.p1.11.m11.1.2.1"></times><ci id="S4.SS2.1.p1.11.m11.1.2.2.cmml" xref="S4.SS2.1.p1.11.m11.1.2.2">𝜎</ci><apply id="S4.SS2.1.p1.11.m11.1.1.cmml" xref="S4.SS2.1.p1.11.m11.1.2.3.2"><ci id="S4.SS2.1.p1.11.m11.1.1.1.cmml" xref="S4.SS2.1.p1.11.m11.1.1.1">^</ci><ci id="S4.SS2.1.p1.11.m11.1.1.2.cmml" xref="S4.SS2.1.p1.11.m11.1.1.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.1.p1.11.m11.1c">\sigma(\widehat{u})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.1.p1.11.m11.1d">italic_σ ( over^ start_ARG italic_u end_ARG )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.2.p2"> <p class="ltx_p" id="S4.SS2.2.p2.11">Conversely, every essential occurrence of <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.1.m1.1"><semantics id="S4.SS2.2.p2.1.m1.1a"><msup id="S4.SS2.2.p2.1.m1.1.1" xref="S4.SS2.2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.2.p2.1.m1.1.1.2" xref="S4.SS2.2.p2.1.m1.1.1.2.cmml">w</mi><mo id="S4.SS2.2.p2.1.m1.1.1.3" xref="S4.SS2.2.p2.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.1.m1.1b"><apply id="S4.SS2.2.p2.1.m1.1.1.cmml" xref="S4.SS2.2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.2.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.2.p2.1.m1.1.1.2">𝑤</ci><ci id="S4.SS2.2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.2.p2.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.1.m1.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\sigma(v)" class="ltx_Math" display="inline" id="S4.SS2.2.p2.2.m2.1"><semantics id="S4.SS2.2.p2.2.m2.1a"><mrow id="S4.SS2.2.p2.2.m2.1.2" xref="S4.SS2.2.p2.2.m2.1.2.cmml"><mi id="S4.SS2.2.p2.2.m2.1.2.2" xref="S4.SS2.2.p2.2.m2.1.2.2.cmml">σ</mi><mo id="S4.SS2.2.p2.2.m2.1.2.1" xref="S4.SS2.2.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.2.p2.2.m2.1.2.3.2" xref="S4.SS2.2.p2.2.m2.1.2.cmml"><mo id="S4.SS2.2.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S4.SS2.2.p2.2.m2.1.2.cmml">(</mo><mi id="S4.SS2.2.p2.2.m2.1.1" xref="S4.SS2.2.p2.2.m2.1.1.cmml">v</mi><mo id="S4.SS2.2.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S4.SS2.2.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.2.m2.1b"><apply id="S4.SS2.2.p2.2.m2.1.2.cmml" xref="S4.SS2.2.p2.2.m2.1.2"><times id="S4.SS2.2.p2.2.m2.1.2.1.cmml" xref="S4.SS2.2.p2.2.m2.1.2.1"></times><ci id="S4.SS2.2.p2.2.m2.1.2.2.cmml" xref="S4.SS2.2.p2.2.m2.1.2.2">𝜎</ci><ci id="S4.SS2.2.p2.2.m2.1.1.cmml" xref="S4.SS2.2.p2.2.m2.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.2.m2.1c">\sigma(v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.2.m2.1d">italic_σ ( italic_v )</annotation></semantics></math>, for any <math alttext="v\in\cal A^{*}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.3.m3.1"><semantics id="S4.SS2.2.p2.3.m3.1a"><mrow id="S4.SS2.2.p2.3.m3.1.1" xref="S4.SS2.2.p2.3.m3.1.1.cmml"><mi id="S4.SS2.2.p2.3.m3.1.1.2" xref="S4.SS2.2.p2.3.m3.1.1.2.cmml">v</mi><mo id="S4.SS2.2.p2.3.m3.1.1.1" xref="S4.SS2.2.p2.3.m3.1.1.1.cmml">∈</mo><msup id="S4.SS2.2.p2.3.m3.1.1.3" xref="S4.SS2.2.p2.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.2.p2.3.m3.1.1.3.2" xref="S4.SS2.2.p2.3.m3.1.1.3.2.cmml">𝒜</mi><mo id="S4.SS2.2.p2.3.m3.1.1.3.3" xref="S4.SS2.2.p2.3.m3.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.3.m3.1b"><apply id="S4.SS2.2.p2.3.m3.1.1.cmml" xref="S4.SS2.2.p2.3.m3.1.1"><in id="S4.SS2.2.p2.3.m3.1.1.1.cmml" xref="S4.SS2.2.p2.3.m3.1.1.1"></in><ci id="S4.SS2.2.p2.3.m3.1.1.2.cmml" xref="S4.SS2.2.p2.3.m3.1.1.2">𝑣</ci><apply id="S4.SS2.2.p2.3.m3.1.1.3.cmml" xref="S4.SS2.2.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.2.p2.3.m3.1.1.3.1.cmml" xref="S4.SS2.2.p2.3.m3.1.1.3">superscript</csymbol><ci id="S4.SS2.2.p2.3.m3.1.1.3.2.cmml" xref="S4.SS2.2.p2.3.m3.1.1.3.2">𝒜</ci><times id="S4.SS2.2.p2.3.m3.1.1.3.3.cmml" xref="S4.SS2.2.p2.3.m3.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.3.m3.1c">v\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.3.m3.1d">italic_v ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, defines a factor <math alttext="u" class="ltx_Math" display="inline" id="S4.SS2.2.p2.4.m4.1"><semantics id="S4.SS2.2.p2.4.m4.1a"><mi id="S4.SS2.2.p2.4.m4.1.1" xref="S4.SS2.2.p2.4.m4.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.4.m4.1b"><ci id="S4.SS2.2.p2.4.m4.1.1.cmml" xref="S4.SS2.2.p2.4.m4.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.4.m4.1c">u</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.4.m4.1d">italic_u</annotation></semantics></math> of <math alttext="\pi_{\sigma}(v)" class="ltx_Math" display="inline" id="S4.SS2.2.p2.5.m5.1"><semantics id="S4.SS2.2.p2.5.m5.1a"><mrow id="S4.SS2.2.p2.5.m5.1.2" xref="S4.SS2.2.p2.5.m5.1.2.cmml"><msub id="S4.SS2.2.p2.5.m5.1.2.2" xref="S4.SS2.2.p2.5.m5.1.2.2.cmml"><mi id="S4.SS2.2.p2.5.m5.1.2.2.2" xref="S4.SS2.2.p2.5.m5.1.2.2.2.cmml">π</mi><mi id="S4.SS2.2.p2.5.m5.1.2.2.3" xref="S4.SS2.2.p2.5.m5.1.2.2.3.cmml">σ</mi></msub><mo id="S4.SS2.2.p2.5.m5.1.2.1" xref="S4.SS2.2.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.2.p2.5.m5.1.2.3.2" xref="S4.SS2.2.p2.5.m5.1.2.cmml"><mo id="S4.SS2.2.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S4.SS2.2.p2.5.m5.1.2.cmml">(</mo><mi id="S4.SS2.2.p2.5.m5.1.1" xref="S4.SS2.2.p2.5.m5.1.1.cmml">v</mi><mo id="S4.SS2.2.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S4.SS2.2.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.5.m5.1b"><apply id="S4.SS2.2.p2.5.m5.1.2.cmml" xref="S4.SS2.2.p2.5.m5.1.2"><times id="S4.SS2.2.p2.5.m5.1.2.1.cmml" xref="S4.SS2.2.p2.5.m5.1.2.1"></times><apply id="S4.SS2.2.p2.5.m5.1.2.2.cmml" xref="S4.SS2.2.p2.5.m5.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.2.p2.5.m5.1.2.2.1.cmml" xref="S4.SS2.2.p2.5.m5.1.2.2">subscript</csymbol><ci id="S4.SS2.2.p2.5.m5.1.2.2.2.cmml" xref="S4.SS2.2.p2.5.m5.1.2.2.2">𝜋</ci><ci id="S4.SS2.2.p2.5.m5.1.2.2.3.cmml" xref="S4.SS2.2.p2.5.m5.1.2.2.3">𝜎</ci></apply><ci id="S4.SS2.2.p2.5.m5.1.1.cmml" xref="S4.SS2.2.p2.5.m5.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.5.m5.1c">\pi_{\sigma}(v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.5.m5.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_v )</annotation></semantics></math> with <math alttext="u\in\alpha_{\sigma}^{-1}(w^{\prime})" class="ltx_Math" display="inline" id="S4.SS2.2.p2.6.m6.1"><semantics id="S4.SS2.2.p2.6.m6.1a"><mrow id="S4.SS2.2.p2.6.m6.1.1" xref="S4.SS2.2.p2.6.m6.1.1.cmml"><mi id="S4.SS2.2.p2.6.m6.1.1.3" xref="S4.SS2.2.p2.6.m6.1.1.3.cmml">u</mi><mo id="S4.SS2.2.p2.6.m6.1.1.2" xref="S4.SS2.2.p2.6.m6.1.1.2.cmml">∈</mo><mrow id="S4.SS2.2.p2.6.m6.1.1.1" xref="S4.SS2.2.p2.6.m6.1.1.1.cmml"><msubsup id="S4.SS2.2.p2.6.m6.1.1.1.3" xref="S4.SS2.2.p2.6.m6.1.1.1.3.cmml"><mi id="S4.SS2.2.p2.6.m6.1.1.1.3.2.2" xref="S4.SS2.2.p2.6.m6.1.1.1.3.2.2.cmml">α</mi><mi id="S4.SS2.2.p2.6.m6.1.1.1.3.2.3" xref="S4.SS2.2.p2.6.m6.1.1.1.3.2.3.cmml">σ</mi><mrow id="S4.SS2.2.p2.6.m6.1.1.1.3.3" xref="S4.SS2.2.p2.6.m6.1.1.1.3.3.cmml"><mo id="S4.SS2.2.p2.6.m6.1.1.1.3.3a" xref="S4.SS2.2.p2.6.m6.1.1.1.3.3.cmml">−</mo><mn id="S4.SS2.2.p2.6.m6.1.1.1.3.3.2" xref="S4.SS2.2.p2.6.m6.1.1.1.3.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS2.2.p2.6.m6.1.1.1.2" xref="S4.SS2.2.p2.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.2.p2.6.m6.1.1.1.1.1" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.cmml"><mo id="S4.SS2.2.p2.6.m6.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.2.p2.6.m6.1.1.1.1.1.1" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.cmml"><mi id="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.2" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.2.cmml">w</mi><mo id="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.3" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S4.SS2.2.p2.6.m6.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.6.m6.1b"><apply id="S4.SS2.2.p2.6.m6.1.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1"><in id="S4.SS2.2.p2.6.m6.1.1.2.cmml" xref="S4.SS2.2.p2.6.m6.1.1.2"></in><ci id="S4.SS2.2.p2.6.m6.1.1.3.cmml" xref="S4.SS2.2.p2.6.m6.1.1.3">𝑢</ci><apply id="S4.SS2.2.p2.6.m6.1.1.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1"><times id="S4.SS2.2.p2.6.m6.1.1.1.2.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.2"></times><apply id="S4.SS2.2.p2.6.m6.1.1.1.3.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.2.p2.6.m6.1.1.1.3.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.3">superscript</csymbol><apply id="S4.SS2.2.p2.6.m6.1.1.1.3.2.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.2.p2.6.m6.1.1.1.3.2.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.3">subscript</csymbol><ci id="S4.SS2.2.p2.6.m6.1.1.1.3.2.2.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.3.2.2">𝛼</ci><ci id="S4.SS2.2.p2.6.m6.1.1.1.3.2.3.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.3.2.3">𝜎</ci></apply><apply id="S4.SS2.2.p2.6.m6.1.1.1.3.3.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.3.3"><minus id="S4.SS2.2.p2.6.m6.1.1.1.3.3.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.3.3"></minus><cn id="S4.SS2.2.p2.6.m6.1.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.2.p2.6.m6.1.1.1.3.3.2">1</cn></apply></apply><apply id="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.1.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.2.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.2">𝑤</ci><ci id="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.3.cmml" xref="S4.SS2.2.p2.6.m6.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.6.m6.1c">u\in\alpha_{\sigma}^{-1}(w^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.6.m6.1d">italic_u ∈ italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> for which we have <math alttext="\widehat{u}=v\," class="ltx_Math" display="inline" id="S4.SS2.2.p2.7.m7.1"><semantics id="S4.SS2.2.p2.7.m7.1a"><mrow id="S4.SS2.2.p2.7.m7.1.1" xref="S4.SS2.2.p2.7.m7.1.1.cmml"><mover accent="true" id="S4.SS2.2.p2.7.m7.1.1.2" xref="S4.SS2.2.p2.7.m7.1.1.2.cmml"><mi id="S4.SS2.2.p2.7.m7.1.1.2.2" xref="S4.SS2.2.p2.7.m7.1.1.2.2.cmml">u</mi><mo id="S4.SS2.2.p2.7.m7.1.1.2.1" xref="S4.SS2.2.p2.7.m7.1.1.2.1.cmml">^</mo></mover><mo id="S4.SS2.2.p2.7.m7.1.1.1" xref="S4.SS2.2.p2.7.m7.1.1.1.cmml">=</mo><mi id="S4.SS2.2.p2.7.m7.1.1.3" xref="S4.SS2.2.p2.7.m7.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.7.m7.1b"><apply id="S4.SS2.2.p2.7.m7.1.1.cmml" xref="S4.SS2.2.p2.7.m7.1.1"><eq id="S4.SS2.2.p2.7.m7.1.1.1.cmml" xref="S4.SS2.2.p2.7.m7.1.1.1"></eq><apply id="S4.SS2.2.p2.7.m7.1.1.2.cmml" xref="S4.SS2.2.p2.7.m7.1.1.2"><ci id="S4.SS2.2.p2.7.m7.1.1.2.1.cmml" xref="S4.SS2.2.p2.7.m7.1.1.2.1">^</ci><ci id="S4.SS2.2.p2.7.m7.1.1.2.2.cmml" xref="S4.SS2.2.p2.7.m7.1.1.2.2">𝑢</ci></apply><ci id="S4.SS2.2.p2.7.m7.1.1.3.cmml" xref="S4.SS2.2.p2.7.m7.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.7.m7.1c">\widehat{u}=v\,</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.7.m7.1d">over^ start_ARG italic_u end_ARG = italic_v</annotation></semantics></math>. Indeed, the word <math alttext="\widehat{u}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.8.m8.1"><semantics id="S4.SS2.2.p2.8.m8.1a"><mover accent="true" id="S4.SS2.2.p2.8.m8.1.1" xref="S4.SS2.2.p2.8.m8.1.1.cmml"><mi id="S4.SS2.2.p2.8.m8.1.1.2" xref="S4.SS2.2.p2.8.m8.1.1.2.cmml">u</mi><mo id="S4.SS2.2.p2.8.m8.1.1.1" xref="S4.SS2.2.p2.8.m8.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.8.m8.1b"><apply id="S4.SS2.2.p2.8.m8.1.1.cmml" xref="S4.SS2.2.p2.8.m8.1.1"><ci id="S4.SS2.2.p2.8.m8.1.1.1.cmml" xref="S4.SS2.2.p2.8.m8.1.1.1">^</ci><ci id="S4.SS2.2.p2.8.m8.1.1.2.cmml" xref="S4.SS2.2.p2.8.m8.1.1.2">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.8.m8.1c">\widehat{u}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.8.m8.1d">over^ start_ARG italic_u end_ARG</annotation></semantics></math> can not be a proper factor of <math alttext="v" class="ltx_Math" display="inline" id="S4.SS2.2.p2.9.m9.1"><semantics id="S4.SS2.2.p2.9.m9.1a"><mi id="S4.SS2.2.p2.9.m9.1.1" xref="S4.SS2.2.p2.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.9.m9.1b"><ci id="S4.SS2.2.p2.9.m9.1.1.cmml" xref="S4.SS2.2.p2.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.9.m9.1d">italic_v</annotation></semantics></math>, or else the given occurrence of <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.SS2.2.p2.10.m10.1"><semantics id="S4.SS2.2.p2.10.m10.1a"><msup id="S4.SS2.2.p2.10.m10.1.1" xref="S4.SS2.2.p2.10.m10.1.1.cmml"><mi id="S4.SS2.2.p2.10.m10.1.1.2" xref="S4.SS2.2.p2.10.m10.1.1.2.cmml">w</mi><mo id="S4.SS2.2.p2.10.m10.1.1.3" xref="S4.SS2.2.p2.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.10.m10.1b"><apply id="S4.SS2.2.p2.10.m10.1.1.cmml" xref="S4.SS2.2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S4.SS2.2.p2.10.m10.1.1.1.cmml" xref="S4.SS2.2.p2.10.m10.1.1">superscript</csymbol><ci id="S4.SS2.2.p2.10.m10.1.1.2.cmml" xref="S4.SS2.2.p2.10.m10.1.1.2">𝑤</ci><ci id="S4.SS2.2.p2.10.m10.1.1.3.cmml" xref="S4.SS2.2.p2.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.10.m10.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.10.m10.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> as factor of <math alttext="\sigma(v)" class="ltx_Math" display="inline" id="S4.SS2.2.p2.11.m11.1"><semantics id="S4.SS2.2.p2.11.m11.1a"><mrow id="S4.SS2.2.p2.11.m11.1.2" xref="S4.SS2.2.p2.11.m11.1.2.cmml"><mi id="S4.SS2.2.p2.11.m11.1.2.2" xref="S4.SS2.2.p2.11.m11.1.2.2.cmml">σ</mi><mo id="S4.SS2.2.p2.11.m11.1.2.1" xref="S4.SS2.2.p2.11.m11.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.2.p2.11.m11.1.2.3.2" xref="S4.SS2.2.p2.11.m11.1.2.cmml"><mo id="S4.SS2.2.p2.11.m11.1.2.3.2.1" stretchy="false" xref="S4.SS2.2.p2.11.m11.1.2.cmml">(</mo><mi id="S4.SS2.2.p2.11.m11.1.1" xref="S4.SS2.2.p2.11.m11.1.1.cmml">v</mi><mo id="S4.SS2.2.p2.11.m11.1.2.3.2.2" stretchy="false" xref="S4.SS2.2.p2.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.2.p2.11.m11.1b"><apply id="S4.SS2.2.p2.11.m11.1.2.cmml" xref="S4.SS2.2.p2.11.m11.1.2"><times id="S4.SS2.2.p2.11.m11.1.2.1.cmml" xref="S4.SS2.2.p2.11.m11.1.2.1"></times><ci id="S4.SS2.2.p2.11.m11.1.2.2.cmml" xref="S4.SS2.2.p2.11.m11.1.2.2">𝜎</ci><ci id="S4.SS2.2.p2.11.m11.1.1.cmml" xref="S4.SS2.2.p2.11.m11.1.1">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.2.p2.11.m11.1c">\sigma(v)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.2.p2.11.m11.1d">italic_σ ( italic_v )</annotation></semantics></math> would not have been essential.</p> </div> <div class="ltx_para" id="S4.SS2.3.p3"> <p class="ltx_p" id="S4.SS2.3.p3.2">It follows that the sums in the formulas (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E5" title="In Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.5</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.E3" title="In Proposition 4.2. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.3</span></a>) differ only in their organization of the indexing, so that the results of the right hand sides of (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E5" title="In Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.5</span></a>) and of (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.E3" title="In Proposition 4.2. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.3</span></a>) must be equal. <span class="ltx_text ltx_inline-block" id="S4.SS2.3.p3.1.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S4.SS2.3.p3.1.1.m1.1"><semantics id="S4.SS2.3.p3.1.1.m1.1a"><mo id="S4.SS2.3.p3.1.1.m1.1.1" xref="S4.SS2.3.p3.1.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.1.1.m1.1b"><csymbol cd="latexml" id="S4.SS2.3.p3.1.1.m1.1.1.cmml" xref="S4.SS2.3.p3.1.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.1.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.1.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S4.SS2.3.p3.2.m1.1"><semantics id="S4.SS2.3.p3.2.m1.1a"><mo id="S4.SS2.3.p3.2.m1.1.1" xref="S4.SS2.3.p3.2.m1.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.3.p3.2.m1.1b"><csymbol cd="latexml" id="S4.SS2.3.p3.2.m1.1.1.cmml" xref="S4.SS2.3.p3.2.m1.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.3.p3.2.m1.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.3.p3.2.m1.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S4.SS2.p6"> <p class="ltx_p" id="S4.SS2.p6.1">We will illustrate now that the new formula (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.E3" title="In Proposition 4.2. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.3</span></a>) is a lot more convenient in practice (in particular in view of the fact that in the example below the set <math alttext="\alpha_{\sigma}^{-1}(w)" class="ltx_Math" display="inline" id="S4.SS2.p6.1.m1.1"><semantics id="S4.SS2.p6.1.m1.1a"><mrow id="S4.SS2.p6.1.m1.1.2" xref="S4.SS2.p6.1.m1.1.2.cmml"><msubsup id="S4.SS2.p6.1.m1.1.2.2" xref="S4.SS2.p6.1.m1.1.2.2.cmml"><mi id="S4.SS2.p6.1.m1.1.2.2.2.2" xref="S4.SS2.p6.1.m1.1.2.2.2.2.cmml">α</mi><mi id="S4.SS2.p6.1.m1.1.2.2.2.3" xref="S4.SS2.p6.1.m1.1.2.2.2.3.cmml">σ</mi><mrow id="S4.SS2.p6.1.m1.1.2.2.3" xref="S4.SS2.p6.1.m1.1.2.2.3.cmml"><mo id="S4.SS2.p6.1.m1.1.2.2.3a" xref="S4.SS2.p6.1.m1.1.2.2.3.cmml">−</mo><mn id="S4.SS2.p6.1.m1.1.2.2.3.2" xref="S4.SS2.p6.1.m1.1.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S4.SS2.p6.1.m1.1.2.1" xref="S4.SS2.p6.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p6.1.m1.1.2.3.2" xref="S4.SS2.p6.1.m1.1.2.cmml"><mo id="S4.SS2.p6.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.p6.1.m1.1.2.cmml">(</mo><mi id="S4.SS2.p6.1.m1.1.1" xref="S4.SS2.p6.1.m1.1.1.cmml">w</mi><mo id="S4.SS2.p6.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.p6.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p6.1.m1.1b"><apply id="S4.SS2.p6.1.m1.1.2.cmml" xref="S4.SS2.p6.1.m1.1.2"><times id="S4.SS2.p6.1.m1.1.2.1.cmml" xref="S4.SS2.p6.1.m1.1.2.1"></times><apply id="S4.SS2.p6.1.m1.1.2.2.cmml" xref="S4.SS2.p6.1.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p6.1.m1.1.2.2.1.cmml" xref="S4.SS2.p6.1.m1.1.2.2">superscript</csymbol><apply id="S4.SS2.p6.1.m1.1.2.2.2.cmml" xref="S4.SS2.p6.1.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p6.1.m1.1.2.2.2.1.cmml" xref="S4.SS2.p6.1.m1.1.2.2">subscript</csymbol><ci id="S4.SS2.p6.1.m1.1.2.2.2.2.cmml" xref="S4.SS2.p6.1.m1.1.2.2.2.2">𝛼</ci><ci id="S4.SS2.p6.1.m1.1.2.2.2.3.cmml" xref="S4.SS2.p6.1.m1.1.2.2.2.3">𝜎</ci></apply><apply id="S4.SS2.p6.1.m1.1.2.2.3.cmml" xref="S4.SS2.p6.1.m1.1.2.2.3"><minus id="S4.SS2.p6.1.m1.1.2.2.3.1.cmml" xref="S4.SS2.p6.1.m1.1.2.2.3"></minus><cn id="S4.SS2.p6.1.m1.1.2.2.3.2.cmml" type="integer" xref="S4.SS2.p6.1.m1.1.2.2.3.2">1</cn></apply></apply><ci id="S4.SS2.p6.1.m1.1.1.cmml" xref="S4.SS2.p6.1.m1.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p6.1.m1.1c">\alpha_{\sigma}^{-1}(w)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p6.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_w )</annotation></semantics></math> consists of 648 elements):</p> </div> <div class="ltx_theorem ltx_theorem_example" id="S4.Thmthm3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmthm3.1.1.1">Example 4.3</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmthm3.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmthm3.p1"> <p class="ltx_p" id="S4.Thmthm3.p1.2">Let us consider <math alttext="\cal A=\{a,b,c\},\,\cal B=\{d,e\}" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.1.m1.7"><semantics id="S4.Thmthm3.p1.1.m1.7a"><mrow id="S4.Thmthm3.p1.1.m1.7.7.2" xref="S4.Thmthm3.p1.1.m1.7.7.3.cmml"><mrow id="S4.Thmthm3.p1.1.m1.6.6.1.1" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.1.m1.6.6.1.1.2" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.2.cmml">𝒜</mi><mo id="S4.Thmthm3.p1.1.m1.6.6.1.1.1" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.1.cmml">=</mo><mrow id="S4.Thmthm3.p1.1.m1.6.6.1.1.3.2" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.3.1.cmml"><mo id="S4.Thmthm3.p1.1.m1.6.6.1.1.3.2.1" stretchy="false" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.1.m1.1.1" xref="S4.Thmthm3.p1.1.m1.1.1.cmml">𝒶</mi><mo id="S4.Thmthm3.p1.1.m1.6.6.1.1.3.2.2" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.1.m1.2.2" xref="S4.Thmthm3.p1.1.m1.2.2.cmml">𝒷</mi><mo id="S4.Thmthm3.p1.1.m1.6.6.1.1.3.2.3" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.1.m1.3.3" xref="S4.Thmthm3.p1.1.m1.3.3.cmml">𝒸</mi><mo id="S4.Thmthm3.p1.1.m1.6.6.1.1.3.2.4" stretchy="false" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.3.1.cmml">}</mo></mrow></mrow><mo id="S4.Thmthm3.p1.1.m1.7.7.2.3" rspace="0.337em" xref="S4.Thmthm3.p1.1.m1.7.7.3a.cmml">,</mo><mrow id="S4.Thmthm3.p1.1.m1.7.7.2.2" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.1.m1.7.7.2.2.2" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.2.cmml">ℬ</mi><mo id="S4.Thmthm3.p1.1.m1.7.7.2.2.1" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.1.cmml">=</mo><mrow id="S4.Thmthm3.p1.1.m1.7.7.2.2.3.2" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.3.1.cmml"><mo id="S4.Thmthm3.p1.1.m1.7.7.2.2.3.2.1" stretchy="false" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.1.m1.4.4" xref="S4.Thmthm3.p1.1.m1.4.4.cmml">𝒹</mi><mo id="S4.Thmthm3.p1.1.m1.7.7.2.2.3.2.2" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.1.m1.5.5" xref="S4.Thmthm3.p1.1.m1.5.5.cmml">ℯ</mi><mo id="S4.Thmthm3.p1.1.m1.7.7.2.2.3.2.3" stretchy="false" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.1.m1.7b"><apply id="S4.Thmthm3.p1.1.m1.7.7.3.cmml" xref="S4.Thmthm3.p1.1.m1.7.7.2"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.1.m1.7.7.3a.cmml" xref="S4.Thmthm3.p1.1.m1.7.7.2.3">formulae-sequence</csymbol><apply id="S4.Thmthm3.p1.1.m1.6.6.1.1.cmml" xref="S4.Thmthm3.p1.1.m1.6.6.1.1"><eq id="S4.Thmthm3.p1.1.m1.6.6.1.1.1.cmml" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.1"></eq><ci id="S4.Thmthm3.p1.1.m1.6.6.1.1.2.cmml" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.2">𝒜</ci><set id="S4.Thmthm3.p1.1.m1.6.6.1.1.3.1.cmml" xref="S4.Thmthm3.p1.1.m1.6.6.1.1.3.2"><ci id="S4.Thmthm3.p1.1.m1.1.1.cmml" xref="S4.Thmthm3.p1.1.m1.1.1">𝒶</ci><ci id="S4.Thmthm3.p1.1.m1.2.2.cmml" xref="S4.Thmthm3.p1.1.m1.2.2">𝒷</ci><ci id="S4.Thmthm3.p1.1.m1.3.3.cmml" xref="S4.Thmthm3.p1.1.m1.3.3">𝒸</ci></set></apply><apply id="S4.Thmthm3.p1.1.m1.7.7.2.2.cmml" xref="S4.Thmthm3.p1.1.m1.7.7.2.2"><eq id="S4.Thmthm3.p1.1.m1.7.7.2.2.1.cmml" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.1"></eq><ci id="S4.Thmthm3.p1.1.m1.7.7.2.2.2.cmml" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.2">ℬ</ci><set id="S4.Thmthm3.p1.1.m1.7.7.2.2.3.1.cmml" xref="S4.Thmthm3.p1.1.m1.7.7.2.2.3.2"><ci id="S4.Thmthm3.p1.1.m1.4.4.cmml" xref="S4.Thmthm3.p1.1.m1.4.4">𝒹</ci><ci id="S4.Thmthm3.p1.1.m1.5.5.cmml" xref="S4.Thmthm3.p1.1.m1.5.5">ℯ</ci></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.1.m1.7c">\cal A=\{a,b,c\},\,\cal B=\{d,e\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.1.m1.7d">caligraphic_A = { caligraphic_a , caligraphic_b , caligraphic_c } , caligraphic_B = { caligraphic_d , caligraphic_e }</annotation></semantics></math> and <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.2.m2.1"><semantics id="S4.Thmthm3.p1.2.m2.1a"><mi id="S4.Thmthm3.p1.2.m2.1.1" xref="S4.Thmthm3.p1.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.2.m2.1b"><ci id="S4.Thmthm3.p1.2.m2.1.1.cmml" xref="S4.Thmthm3.p1.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.2.m2.1d">italic_σ</annotation></semantics></math> given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="a\mapsto ded,\,b\mapsto de,\,c\mapsto dedd\,." class="ltx_Math" display="block" id="S4.Ex12.m1.1"><semantics id="S4.Ex12.m1.1a"><mrow id="S4.Ex12.m1.1.1.1"><mrow id="S4.Ex12.m1.1.1.1.1.2" xref="S4.Ex12.m1.1.1.1.1.3.cmml"><mrow id="S4.Ex12.m1.1.1.1.1.1.1" xref="S4.Ex12.m1.1.1.1.1.1.1.cmml"><mi id="S4.Ex12.m1.1.1.1.1.1.1.2" xref="S4.Ex12.m1.1.1.1.1.1.1.2.cmml">a</mi><mo id="S4.Ex12.m1.1.1.1.1.1.1.1" stretchy="false" xref="S4.Ex12.m1.1.1.1.1.1.1.1.cmml">↦</mo><mrow id="S4.Ex12.m1.1.1.1.1.1.1.3" xref="S4.Ex12.m1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex12.m1.1.1.1.1.1.1.3.2" xref="S4.Ex12.m1.1.1.1.1.1.1.3.2.cmml">d</mi><mo id="S4.Ex12.m1.1.1.1.1.1.1.3.1" xref="S4.Ex12.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex12.m1.1.1.1.1.1.1.3.3" xref="S4.Ex12.m1.1.1.1.1.1.1.3.3.cmml">e</mi><mo id="S4.Ex12.m1.1.1.1.1.1.1.3.1a" xref="S4.Ex12.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex12.m1.1.1.1.1.1.1.3.4" xref="S4.Ex12.m1.1.1.1.1.1.1.3.4.cmml">d</mi></mrow></mrow><mo id="S4.Ex12.m1.1.1.1.1.2.3" rspace="0.337em" xref="S4.Ex12.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S4.Ex12.m1.1.1.1.1.2.2.2" xref="S4.Ex12.m1.1.1.1.1.2.2.3.cmml"><mrow id="S4.Ex12.m1.1.1.1.1.2.2.1.1" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.cmml"><mi id="S4.Ex12.m1.1.1.1.1.2.2.1.1.2" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.2.cmml">b</mi><mo id="S4.Ex12.m1.1.1.1.1.2.2.1.1.1" stretchy="false" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.1.cmml">↦</mo><mrow id="S4.Ex12.m1.1.1.1.1.2.2.1.1.3" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.cmml"><mi id="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.2" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.2.cmml">d</mi><mo id="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.1" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.3" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.3.cmml">e</mi></mrow></mrow><mo id="S4.Ex12.m1.1.1.1.1.2.2.2.3" rspace="0.337em" xref="S4.Ex12.m1.1.1.1.1.2.2.3a.cmml">,</mo><mrow id="S4.Ex12.m1.1.1.1.1.2.2.2.2" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.cmml"><mi id="S4.Ex12.m1.1.1.1.1.2.2.2.2.2" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.2.cmml">c</mi><mo id="S4.Ex12.m1.1.1.1.1.2.2.2.2.1" stretchy="false" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.1.cmml">↦</mo><mrow id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.cmml"><mi id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.2" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.2.cmml">d</mi><mo id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.1" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.1.cmml">⁢</mo><mi id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.3" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.3.cmml">e</mi><mo id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.1a" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.1.cmml">⁢</mo><mi id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.4" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.4.cmml">d</mi><mo id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.1b" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.1.cmml">⁢</mo><mi id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.5" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.5.cmml">d</mi></mrow></mrow></mrow></mrow><mo id="S4.Ex12.m1.1.1.1.2" lspace="0.170em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex12.m1.1b"><apply id="S4.Ex12.m1.1.1.1.1.3.cmml" xref="S4.Ex12.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.1.1.3a.cmml" xref="S4.Ex12.m1.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="S4.Ex12.m1.1.1.1.1.1.1.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1"><csymbol cd="latexml" id="S4.Ex12.m1.1.1.1.1.1.1.1.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.1">maps-to</csymbol><ci id="S4.Ex12.m1.1.1.1.1.1.1.2.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.2">𝑎</ci><apply id="S4.Ex12.m1.1.1.1.1.1.1.3.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3"><times id="S4.Ex12.m1.1.1.1.1.1.1.3.1.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3.1"></times><ci id="S4.Ex12.m1.1.1.1.1.1.1.3.2.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3.2">𝑑</ci><ci id="S4.Ex12.m1.1.1.1.1.1.1.3.3.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3.3">𝑒</ci><ci id="S4.Ex12.m1.1.1.1.1.1.1.3.4.cmml" xref="S4.Ex12.m1.1.1.1.1.1.1.3.4">𝑑</ci></apply></apply><apply id="S4.Ex12.m1.1.1.1.1.2.2.3.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.Ex12.m1.1.1.1.1.2.2.3a.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.3">formulae-sequence</csymbol><apply id="S4.Ex12.m1.1.1.1.1.2.2.1.1.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1"><csymbol cd="latexml" id="S4.Ex12.m1.1.1.1.1.2.2.1.1.1.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.1">maps-to</csymbol><ci id="S4.Ex12.m1.1.1.1.1.2.2.1.1.2.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.2">𝑏</ci><apply id="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.3"><times id="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.1.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.1"></times><ci id="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.2.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.2">𝑑</ci><ci id="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.3.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.1.1.3.3">𝑒</ci></apply></apply><apply id="S4.Ex12.m1.1.1.1.1.2.2.2.2.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2"><csymbol cd="latexml" id="S4.Ex12.m1.1.1.1.1.2.2.2.2.1.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.1">maps-to</csymbol><ci id="S4.Ex12.m1.1.1.1.1.2.2.2.2.2.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.2">𝑐</ci><apply id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3"><times id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.1.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.1"></times><ci id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.2.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.2">𝑑</ci><ci id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.3.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.3">𝑒</ci><ci id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.4.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.4">𝑑</ci><ci id="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.5.cmml" xref="S4.Ex12.m1.1.1.1.1.2.2.2.2.3.5">𝑑</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.1c">a\mapsto ded,\,b\mapsto de,\,c\mapsto dedd\,.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.1d">italic_a ↦ italic_d italic_e italic_d , italic_b ↦ italic_d italic_e , italic_c ↦ italic_d italic_e italic_d italic_d .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmthm3.p1.5">Let us compute <math alttext="\mu^{\sigma}(w)\,[=\sigma M(\mu)(w)]" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.3.m1.4"><semantics id="S4.Thmthm3.p1.3.m1.4a"><mrow id="S4.Thmthm3.p1.3.m1.4.4" xref="S4.Thmthm3.p1.3.m1.4.4.cmml"><mrow id="S4.Thmthm3.p1.3.m1.4.4.3" xref="S4.Thmthm3.p1.3.m1.4.4.3.cmml"><msup id="S4.Thmthm3.p1.3.m1.4.4.3.2" xref="S4.Thmthm3.p1.3.m1.4.4.3.2.cmml"><mi id="S4.Thmthm3.p1.3.m1.4.4.3.2.2" xref="S4.Thmthm3.p1.3.m1.4.4.3.2.2.cmml">μ</mi><mi id="S4.Thmthm3.p1.3.m1.4.4.3.2.3" xref="S4.Thmthm3.p1.3.m1.4.4.3.2.3.cmml">σ</mi></msup><mo id="S4.Thmthm3.p1.3.m1.4.4.3.1" xref="S4.Thmthm3.p1.3.m1.4.4.3.1.cmml">⁢</mo><mrow id="S4.Thmthm3.p1.3.m1.4.4.3.3.2" xref="S4.Thmthm3.p1.3.m1.4.4.3.cmml"><mo id="S4.Thmthm3.p1.3.m1.4.4.3.3.2.1" stretchy="false" xref="S4.Thmthm3.p1.3.m1.4.4.3.cmml">(</mo><mi id="S4.Thmthm3.p1.3.m1.1.1" xref="S4.Thmthm3.p1.3.m1.1.1.cmml">w</mi><mo id="S4.Thmthm3.p1.3.m1.4.4.3.3.2.2" stretchy="false" xref="S4.Thmthm3.p1.3.m1.4.4.3.cmml">)</mo></mrow></mrow><mspace id="S4.Thmthm3.p1.3.m1.4.4a" width="0.559em" xref="S4.Thmthm3.p1.3.m1.4.4.cmml"></mspace><mrow id="S4.Thmthm3.p1.3.m1.4.4.1.1" xref="S4.Thmthm3.p1.3.m1.4.4.1.2.cmml"><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.2" stretchy="false" xref="S4.Thmthm3.p1.3.m1.4.4.1.2.1.cmml">[</mo><mrow id="S4.Thmthm3.p1.3.m1.4.4.1.1.1" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.cmml"><mi id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.2" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.2.cmml"></mi><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.1" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.1.cmml">=</mo><mrow id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.cmml"><mi id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.2" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.2.cmml">σ</mi><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.1" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.3" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.3.cmml">M</mi><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.1a" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.4.2" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.cmml"><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.4.2.1" stretchy="false" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.cmml">(</mo><mi id="S4.Thmthm3.p1.3.m1.2.2" xref="S4.Thmthm3.p1.3.m1.2.2.cmml">μ</mi><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.4.2.2" stretchy="false" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.cmml">)</mo></mrow><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.1b" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.5.2" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.cmml"><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.5.2.1" stretchy="false" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.cmml">(</mo><mi id="S4.Thmthm3.p1.3.m1.3.3" xref="S4.Thmthm3.p1.3.m1.3.3.cmml">w</mi><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.5.2.2" stretchy="false" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Thmthm3.p1.3.m1.4.4.1.1.3" stretchy="false" xref="S4.Thmthm3.p1.3.m1.4.4.1.2.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.3.m1.4b"><apply id="S4.Thmthm3.p1.3.m1.4.4.cmml" xref="S4.Thmthm3.p1.3.m1.4.4"><csymbol cd="latexml" id="S4.Thmthm3.p1.3.m1.4.4.2.cmml" xref="S4.Thmthm3.p1.3.m1.4.4">annotated</csymbol><apply id="S4.Thmthm3.p1.3.m1.4.4.3.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.3"><times id="S4.Thmthm3.p1.3.m1.4.4.3.1.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.3.1"></times><apply id="S4.Thmthm3.p1.3.m1.4.4.3.2.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.3.2"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.3.m1.4.4.3.2.1.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.3.2">superscript</csymbol><ci id="S4.Thmthm3.p1.3.m1.4.4.3.2.2.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.3.2.2">𝜇</ci><ci id="S4.Thmthm3.p1.3.m1.4.4.3.2.3.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.3.2.3">𝜎</ci></apply><ci id="S4.Thmthm3.p1.3.m1.1.1.cmml" xref="S4.Thmthm3.p1.3.m1.1.1">𝑤</ci></apply><apply id="S4.Thmthm3.p1.3.m1.4.4.1.2.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1"><csymbol cd="latexml" id="S4.Thmthm3.p1.3.m1.4.4.1.2.1.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.2">delimited-[]</csymbol><apply id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1"><eq id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.1.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.1"></eq><csymbol cd="latexml" id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.2.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.2">absent</csymbol><apply id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3"><times id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.1.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.1"></times><ci id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.2.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.2">𝜎</ci><ci id="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.3.cmml" xref="S4.Thmthm3.p1.3.m1.4.4.1.1.1.3.3">𝑀</ci><ci id="S4.Thmthm3.p1.3.m1.2.2.cmml" xref="S4.Thmthm3.p1.3.m1.2.2">𝜇</ci><ci id="S4.Thmthm3.p1.3.m1.3.3.cmml" xref="S4.Thmthm3.p1.3.m1.3.3">𝑤</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.3.m1.4c">\mu^{\sigma}(w)\,[=\sigma M(\mu)(w)]</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.3.m1.4d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_w ) [ = italic_σ italic_M ( italic_μ ) ( italic_w ) ]</annotation></semantics></math> for any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.4.m2.1"><semantics id="S4.Thmthm3.p1.4.m2.1a"><mi id="S4.Thmthm3.p1.4.m2.1.1" xref="S4.Thmthm3.p1.4.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.4.m2.1b"><ci id="S4.Thmthm3.p1.4.m2.1.1.cmml" xref="S4.Thmthm3.p1.4.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.4.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.4.m2.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.5.m3.1"><semantics id="S4.Thmthm3.p1.5.m3.1a"><msup id="S4.Thmthm3.p1.5.m3.1.1" xref="S4.Thmthm3.p1.5.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.5.m3.1.1.2" xref="S4.Thmthm3.p1.5.m3.1.1.2.cmml">𝒜</mi><mi id="S4.Thmthm3.p1.5.m3.1.1.3" xref="S4.Thmthm3.p1.5.m3.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.5.m3.1b"><apply id="S4.Thmthm3.p1.5.m3.1.1.cmml" xref="S4.Thmthm3.p1.5.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.5.m3.1.1.1.cmml" xref="S4.Thmthm3.p1.5.m3.1.1">superscript</csymbol><ci id="S4.Thmthm3.p1.5.m3.1.1.2.cmml" xref="S4.Thmthm3.p1.5.m3.1.1.2">𝒜</ci><ci id="S4.Thmthm3.p1.5.m3.1.1.3.cmml" xref="S4.Thmthm3.p1.5.m3.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.5.m3.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.5.m3.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, for the randomly picked word</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w=dded\,." class="ltx_Math" display="block" id="S4.Ex13.m1.1"><semantics id="S4.Ex13.m1.1a"><mrow id="S4.Ex13.m1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.cmml"><mrow id="S4.Ex13.m1.1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.cmml"><mi id="S4.Ex13.m1.1.1.1.1.2" xref="S4.Ex13.m1.1.1.1.1.2.cmml">w</mi><mo id="S4.Ex13.m1.1.1.1.1.1" xref="S4.Ex13.m1.1.1.1.1.1.cmml">=</mo><mrow id="S4.Ex13.m1.1.1.1.1.3" xref="S4.Ex13.m1.1.1.1.1.3.cmml"><mi id="S4.Ex13.m1.1.1.1.1.3.2" xref="S4.Ex13.m1.1.1.1.1.3.2.cmml">d</mi><mo id="S4.Ex13.m1.1.1.1.1.3.1" xref="S4.Ex13.m1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex13.m1.1.1.1.1.3.3" xref="S4.Ex13.m1.1.1.1.1.3.3.cmml">d</mi><mo id="S4.Ex13.m1.1.1.1.1.3.1a" xref="S4.Ex13.m1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex13.m1.1.1.1.1.3.4" xref="S4.Ex13.m1.1.1.1.1.3.4.cmml">e</mi><mo id="S4.Ex13.m1.1.1.1.1.3.1b" xref="S4.Ex13.m1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S4.Ex13.m1.1.1.1.1.3.5" xref="S4.Ex13.m1.1.1.1.1.3.5.cmml">d</mi></mrow></mrow><mo id="S4.Ex13.m1.1.1.1.2" lspace="0.170em" xref="S4.Ex13.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex13.m1.1b"><apply id="S4.Ex13.m1.1.1.1.1.cmml" xref="S4.Ex13.m1.1.1.1"><eq id="S4.Ex13.m1.1.1.1.1.1.cmml" xref="S4.Ex13.m1.1.1.1.1.1"></eq><ci id="S4.Ex13.m1.1.1.1.1.2.cmml" xref="S4.Ex13.m1.1.1.1.1.2">𝑤</ci><apply id="S4.Ex13.m1.1.1.1.1.3.cmml" xref="S4.Ex13.m1.1.1.1.1.3"><times id="S4.Ex13.m1.1.1.1.1.3.1.cmml" xref="S4.Ex13.m1.1.1.1.1.3.1"></times><ci id="S4.Ex13.m1.1.1.1.1.3.2.cmml" xref="S4.Ex13.m1.1.1.1.1.3.2">𝑑</ci><ci id="S4.Ex13.m1.1.1.1.1.3.3.cmml" xref="S4.Ex13.m1.1.1.1.1.3.3">𝑑</ci><ci id="S4.Ex13.m1.1.1.1.1.3.4.cmml" xref="S4.Ex13.m1.1.1.1.1.3.4">𝑒</ci><ci id="S4.Ex13.m1.1.1.1.1.3.5.cmml" xref="S4.Ex13.m1.1.1.1.1.3.5">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m1.1c">w=dded\,.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.1d">italic_w = italic_d italic_d italic_e italic_d .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmthm3.p1.7">By (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.E1" title="In 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.1</span></a>) it suffices to consider any <math alttext="u\in\cal A^{*}" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.6.m1.1"><semantics id="S4.Thmthm3.p1.6.m1.1a"><mrow id="S4.Thmthm3.p1.6.m1.1.1" xref="S4.Thmthm3.p1.6.m1.1.1.cmml"><mi id="S4.Thmthm3.p1.6.m1.1.1.2" xref="S4.Thmthm3.p1.6.m1.1.1.2.cmml">u</mi><mo id="S4.Thmthm3.p1.6.m1.1.1.1" xref="S4.Thmthm3.p1.6.m1.1.1.1.cmml">∈</mo><msup id="S4.Thmthm3.p1.6.m1.1.1.3" xref="S4.Thmthm3.p1.6.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.6.m1.1.1.3.2" xref="S4.Thmthm3.p1.6.m1.1.1.3.2.cmml">𝒜</mi><mo id="S4.Thmthm3.p1.6.m1.1.1.3.3" xref="S4.Thmthm3.p1.6.m1.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.6.m1.1b"><apply id="S4.Thmthm3.p1.6.m1.1.1.cmml" xref="S4.Thmthm3.p1.6.m1.1.1"><in id="S4.Thmthm3.p1.6.m1.1.1.1.cmml" xref="S4.Thmthm3.p1.6.m1.1.1.1"></in><ci id="S4.Thmthm3.p1.6.m1.1.1.2.cmml" xref="S4.Thmthm3.p1.6.m1.1.1.2">𝑢</ci><apply id="S4.Thmthm3.p1.6.m1.1.1.3.cmml" xref="S4.Thmthm3.p1.6.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.6.m1.1.1.3.1.cmml" xref="S4.Thmthm3.p1.6.m1.1.1.3">superscript</csymbol><ci id="S4.Thmthm3.p1.6.m1.1.1.3.2.cmml" xref="S4.Thmthm3.p1.6.m1.1.1.3.2">𝒜</ci><times id="S4.Thmthm3.p1.6.m1.1.1.3.3.cmml" xref="S4.Thmthm3.p1.6.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.6.m1.1c">u\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.6.m1.1d">italic_u ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of length <math alttext="|u|\leq 3" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.7.m2.1"><semantics id="S4.Thmthm3.p1.7.m2.1a"><mrow id="S4.Thmthm3.p1.7.m2.1.2" xref="S4.Thmthm3.p1.7.m2.1.2.cmml"><mrow id="S4.Thmthm3.p1.7.m2.1.2.2.2" xref="S4.Thmthm3.p1.7.m2.1.2.2.1.cmml"><mo id="S4.Thmthm3.p1.7.m2.1.2.2.2.1" stretchy="false" xref="S4.Thmthm3.p1.7.m2.1.2.2.1.1.cmml">|</mo><mi id="S4.Thmthm3.p1.7.m2.1.1" xref="S4.Thmthm3.p1.7.m2.1.1.cmml">u</mi><mo id="S4.Thmthm3.p1.7.m2.1.2.2.2.2" stretchy="false" xref="S4.Thmthm3.p1.7.m2.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.Thmthm3.p1.7.m2.1.2.1" xref="S4.Thmthm3.p1.7.m2.1.2.1.cmml">≤</mo><mn id="S4.Thmthm3.p1.7.m2.1.2.3" xref="S4.Thmthm3.p1.7.m2.1.2.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.7.m2.1b"><apply id="S4.Thmthm3.p1.7.m2.1.2.cmml" xref="S4.Thmthm3.p1.7.m2.1.2"><leq id="S4.Thmthm3.p1.7.m2.1.2.1.cmml" xref="S4.Thmthm3.p1.7.m2.1.2.1"></leq><apply id="S4.Thmthm3.p1.7.m2.1.2.2.1.cmml" xref="S4.Thmthm3.p1.7.m2.1.2.2.2"><abs id="S4.Thmthm3.p1.7.m2.1.2.2.1.1.cmml" xref="S4.Thmthm3.p1.7.m2.1.2.2.2.1"></abs><ci id="S4.Thmthm3.p1.7.m2.1.1.cmml" xref="S4.Thmthm3.p1.7.m2.1.1">𝑢</ci></apply><cn id="S4.Thmthm3.p1.7.m2.1.2.3.cmml" type="integer" xref="S4.Thmthm3.p1.7.m2.1.2.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.7.m2.1c">|u|\leq 3</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.7.m2.1d">| italic_u | ≤ 3</annotation></semantics></math>. We quickly check that that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lfloor\sigma(u)\rfloor_{w}=0\quad\text{for}\quad u\in\{a,b,c,ab,ba,bb,bc,cb\}" class="ltx_Math" display="block" id="S4.Ex14.m1.8"><semantics id="S4.Ex14.m1.8a"><mrow id="S4.Ex14.m1.8.8.2" xref="S4.Ex14.m1.8.8.3.cmml"><mrow id="S4.Ex14.m1.7.7.1.1" xref="S4.Ex14.m1.7.7.1.1.cmml"><msub id="S4.Ex14.m1.7.7.1.1.1" xref="S4.Ex14.m1.7.7.1.1.1.cmml"><mrow id="S4.Ex14.m1.7.7.1.1.1.1.1" xref="S4.Ex14.m1.7.7.1.1.1.1.2.cmml"><mo id="S4.Ex14.m1.7.7.1.1.1.1.1.2" stretchy="false" xref="S4.Ex14.m1.7.7.1.1.1.1.2.1.cmml">⌊</mo><mrow id="S4.Ex14.m1.7.7.1.1.1.1.1.1" xref="S4.Ex14.m1.7.7.1.1.1.1.1.1.cmml"><mi id="S4.Ex14.m1.7.7.1.1.1.1.1.1.2" xref="S4.Ex14.m1.7.7.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S4.Ex14.m1.7.7.1.1.1.1.1.1.1" xref="S4.Ex14.m1.7.7.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex14.m1.7.7.1.1.1.1.1.1.3.2" xref="S4.Ex14.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S4.Ex14.m1.7.7.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex14.m1.7.7.1.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex14.m1.1.1" xref="S4.Ex14.m1.1.1.cmml">u</mi><mo id="S4.Ex14.m1.7.7.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex14.m1.7.7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex14.m1.7.7.1.1.1.1.1.3" stretchy="false" xref="S4.Ex14.m1.7.7.1.1.1.1.2.1.cmml">⌋</mo></mrow><mi id="S4.Ex14.m1.7.7.1.1.1.3" xref="S4.Ex14.m1.7.7.1.1.1.3.cmml">w</mi></msub><mo id="S4.Ex14.m1.7.7.1.1.2" xref="S4.Ex14.m1.7.7.1.1.2.cmml">=</mo><mrow id="S4.Ex14.m1.7.7.1.1.3.2" xref="S4.Ex14.m1.7.7.1.1.3.1.cmml"><mn id="S4.Ex14.m1.5.5" xref="S4.Ex14.m1.5.5.cmml">0</mn><mspace id="S4.Ex14.m1.7.7.1.1.3.2.1" width="1em" xref="S4.Ex14.m1.7.7.1.1.3.1.cmml"></mspace><mtext id="S4.Ex14.m1.6.6" xref="S4.Ex14.m1.6.6a.cmml">for</mtext></mrow></mrow><mspace id="S4.Ex14.m1.8.8.2.3" width="1em" xref="S4.Ex14.m1.8.8.3a.cmml"></mspace><mrow id="S4.Ex14.m1.8.8.2.2" xref="S4.Ex14.m1.8.8.2.2.cmml"><mi id="S4.Ex14.m1.8.8.2.2.7" xref="S4.Ex14.m1.8.8.2.2.7.cmml">u</mi><mo id="S4.Ex14.m1.8.8.2.2.6" xref="S4.Ex14.m1.8.8.2.2.6.cmml">∈</mo><mrow id="S4.Ex14.m1.8.8.2.2.5.5" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml"><mo id="S4.Ex14.m1.8.8.2.2.5.5.6" stretchy="false" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">{</mo><mi id="S4.Ex14.m1.2.2" xref="S4.Ex14.m1.2.2.cmml">a</mi><mo id="S4.Ex14.m1.8.8.2.2.5.5.7" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">,</mo><mi id="S4.Ex14.m1.3.3" xref="S4.Ex14.m1.3.3.cmml">b</mi><mo id="S4.Ex14.m1.8.8.2.2.5.5.8" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">,</mo><mi id="S4.Ex14.m1.4.4" xref="S4.Ex14.m1.4.4.cmml">c</mi><mo id="S4.Ex14.m1.8.8.2.2.5.5.9" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">,</mo><mrow id="S4.Ex14.m1.8.8.2.2.1.1.1" xref="S4.Ex14.m1.8.8.2.2.1.1.1.cmml"><mi id="S4.Ex14.m1.8.8.2.2.1.1.1.2" xref="S4.Ex14.m1.8.8.2.2.1.1.1.2.cmml">a</mi><mo id="S4.Ex14.m1.8.8.2.2.1.1.1.1" xref="S4.Ex14.m1.8.8.2.2.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex14.m1.8.8.2.2.1.1.1.3" xref="S4.Ex14.m1.8.8.2.2.1.1.1.3.cmml">b</mi></mrow><mo id="S4.Ex14.m1.8.8.2.2.5.5.10" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">,</mo><mrow id="S4.Ex14.m1.8.8.2.2.2.2.2" xref="S4.Ex14.m1.8.8.2.2.2.2.2.cmml"><mi id="S4.Ex14.m1.8.8.2.2.2.2.2.2" xref="S4.Ex14.m1.8.8.2.2.2.2.2.2.cmml">b</mi><mo id="S4.Ex14.m1.8.8.2.2.2.2.2.1" xref="S4.Ex14.m1.8.8.2.2.2.2.2.1.cmml">⁢</mo><mi id="S4.Ex14.m1.8.8.2.2.2.2.2.3" xref="S4.Ex14.m1.8.8.2.2.2.2.2.3.cmml">a</mi></mrow><mo id="S4.Ex14.m1.8.8.2.2.5.5.11" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">,</mo><mrow id="S4.Ex14.m1.8.8.2.2.3.3.3" xref="S4.Ex14.m1.8.8.2.2.3.3.3.cmml"><mi id="S4.Ex14.m1.8.8.2.2.3.3.3.2" xref="S4.Ex14.m1.8.8.2.2.3.3.3.2.cmml">b</mi><mo id="S4.Ex14.m1.8.8.2.2.3.3.3.1" xref="S4.Ex14.m1.8.8.2.2.3.3.3.1.cmml">⁢</mo><mi id="S4.Ex14.m1.8.8.2.2.3.3.3.3" xref="S4.Ex14.m1.8.8.2.2.3.3.3.3.cmml">b</mi></mrow><mo id="S4.Ex14.m1.8.8.2.2.5.5.12" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">,</mo><mrow id="S4.Ex14.m1.8.8.2.2.4.4.4" xref="S4.Ex14.m1.8.8.2.2.4.4.4.cmml"><mi id="S4.Ex14.m1.8.8.2.2.4.4.4.2" xref="S4.Ex14.m1.8.8.2.2.4.4.4.2.cmml">b</mi><mo id="S4.Ex14.m1.8.8.2.2.4.4.4.1" xref="S4.Ex14.m1.8.8.2.2.4.4.4.1.cmml">⁢</mo><mi id="S4.Ex14.m1.8.8.2.2.4.4.4.3" xref="S4.Ex14.m1.8.8.2.2.4.4.4.3.cmml">c</mi></mrow><mo id="S4.Ex14.m1.8.8.2.2.5.5.13" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">,</mo><mrow id="S4.Ex14.m1.8.8.2.2.5.5.5" xref="S4.Ex14.m1.8.8.2.2.5.5.5.cmml"><mi id="S4.Ex14.m1.8.8.2.2.5.5.5.2" xref="S4.Ex14.m1.8.8.2.2.5.5.5.2.cmml">c</mi><mo id="S4.Ex14.m1.8.8.2.2.5.5.5.1" xref="S4.Ex14.m1.8.8.2.2.5.5.5.1.cmml">⁢</mo><mi id="S4.Ex14.m1.8.8.2.2.5.5.5.3" xref="S4.Ex14.m1.8.8.2.2.5.5.5.3.cmml">b</mi></mrow><mo id="S4.Ex14.m1.8.8.2.2.5.5.14" stretchy="false" xref="S4.Ex14.m1.8.8.2.2.5.6.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex14.m1.8b"><apply id="S4.Ex14.m1.8.8.3.cmml" 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xref="S4.Ex14.m1.8.8.2.2.4.4.4.2">𝑏</ci><ci id="S4.Ex14.m1.8.8.2.2.4.4.4.3.cmml" xref="S4.Ex14.m1.8.8.2.2.4.4.4.3">𝑐</ci></apply><apply id="S4.Ex14.m1.8.8.2.2.5.5.5.cmml" xref="S4.Ex14.m1.8.8.2.2.5.5.5"><times id="S4.Ex14.m1.8.8.2.2.5.5.5.1.cmml" xref="S4.Ex14.m1.8.8.2.2.5.5.5.1"></times><ci id="S4.Ex14.m1.8.8.2.2.5.5.5.2.cmml" xref="S4.Ex14.m1.8.8.2.2.5.5.5.2">𝑐</ci><ci id="S4.Ex14.m1.8.8.2.2.5.5.5.3.cmml" xref="S4.Ex14.m1.8.8.2.2.5.5.5.3">𝑏</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex14.m1.8c">\lfloor\sigma(u)\rfloor_{w}=0\quad\text{for}\quad u\in\{a,b,c,ab,ba,bb,bc,cb\}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.8d">⌊ italic_σ ( italic_u ) ⌋ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = 0 for italic_u ∈ { italic_a , italic_b , italic_c , italic_a italic_b , italic_b italic_a , italic_b italic_b , italic_b italic_c , italic_c italic_b }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmthm3.p1.13">and</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lfloor\sigma(u)\rfloor_{w}=1\quad\text{for}\quad u\in\{aa,ac,ca,cc\}\,." class="ltx_Math" display="block" id="S4.Ex15.m1.4"><semantics id="S4.Ex15.m1.4a"><mrow id="S4.Ex15.m1.4.4.1"><mrow id="S4.Ex15.m1.4.4.1.1.2" xref="S4.Ex15.m1.4.4.1.1.3.cmml"><mrow id="S4.Ex15.m1.4.4.1.1.1.1" xref="S4.Ex15.m1.4.4.1.1.1.1.cmml"><msub id="S4.Ex15.m1.4.4.1.1.1.1.1" xref="S4.Ex15.m1.4.4.1.1.1.1.1.cmml"><mrow id="S4.Ex15.m1.4.4.1.1.1.1.1.1.1" xref="S4.Ex15.m1.4.4.1.1.1.1.1.1.2.cmml"><mo id="S4.Ex15.m1.4.4.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex15.m1.4.4.1.1.1.1.1.1.2.1.cmml">⌊</mo><mrow id="S4.Ex15.m1.4.4.1.1.1.1.1.1.1.1" 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xref="S4.Ex15.m1.2.2.cmml">1</mn><mspace id="S4.Ex15.m1.4.4.1.1.1.1.3.2.1" width="1em" xref="S4.Ex15.m1.4.4.1.1.1.1.3.1.cmml"></mspace><mtext id="S4.Ex15.m1.3.3" xref="S4.Ex15.m1.3.3a.cmml">for</mtext></mrow></mrow><mspace id="S4.Ex15.m1.4.4.1.1.2.3" width="1em" xref="S4.Ex15.m1.4.4.1.1.3a.cmml"></mspace><mrow id="S4.Ex15.m1.4.4.1.1.2.2" xref="S4.Ex15.m1.4.4.1.1.2.2.cmml"><mi id="S4.Ex15.m1.4.4.1.1.2.2.6" xref="S4.Ex15.m1.4.4.1.1.2.2.6.cmml">u</mi><mo id="S4.Ex15.m1.4.4.1.1.2.2.5" xref="S4.Ex15.m1.4.4.1.1.2.2.5.cmml">∈</mo><mrow id="S4.Ex15.m1.4.4.1.1.2.2.4.4" xref="S4.Ex15.m1.4.4.1.1.2.2.4.5.cmml"><mo id="S4.Ex15.m1.4.4.1.1.2.2.4.4.5" stretchy="false" xref="S4.Ex15.m1.4.4.1.1.2.2.4.5.cmml">{</mo><mrow id="S4.Ex15.m1.4.4.1.1.2.2.1.1.1" xref="S4.Ex15.m1.4.4.1.1.2.2.1.1.1.cmml"><mi id="S4.Ex15.m1.4.4.1.1.2.2.1.1.1.2" xref="S4.Ex15.m1.4.4.1.1.2.2.1.1.1.2.cmml">a</mi><mo id="S4.Ex15.m1.4.4.1.1.2.2.1.1.1.1" xref="S4.Ex15.m1.4.4.1.1.2.2.1.1.1.1.cmml">⁢</mo><mi 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id="S4.Ex15.m1.4.4.1.1.2.2.2.2.2.2.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.2.2.2.2">𝑎</ci><ci id="S4.Ex15.m1.4.4.1.1.2.2.2.2.2.3.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.2.2.2.3">𝑐</ci></apply><apply id="S4.Ex15.m1.4.4.1.1.2.2.3.3.3.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.3.3.3"><times id="S4.Ex15.m1.4.4.1.1.2.2.3.3.3.1.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.3.3.3.1"></times><ci id="S4.Ex15.m1.4.4.1.1.2.2.3.3.3.2.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.3.3.3.2">𝑐</ci><ci id="S4.Ex15.m1.4.4.1.1.2.2.3.3.3.3.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.3.3.3.3">𝑎</ci></apply><apply id="S4.Ex15.m1.4.4.1.1.2.2.4.4.4.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.4.4.4"><times id="S4.Ex15.m1.4.4.1.1.2.2.4.4.4.1.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.4.4.4.1"></times><ci id="S4.Ex15.m1.4.4.1.1.2.2.4.4.4.2.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.4.4.4.2">𝑐</ci><ci id="S4.Ex15.m1.4.4.1.1.2.2.4.4.4.3.cmml" xref="S4.Ex15.m1.4.4.1.1.2.2.4.4.4.3">𝑐</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex15.m1.4c">\lfloor\sigma(u)\rfloor_{w}=1\quad\text{for}\quad u\in\{aa,ac,ca,cc\}\,.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.4d">⌊ italic_σ ( italic_u ) ⌋ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = 1 for italic_u ∈ { italic_a italic_a , italic_a italic_c , italic_c italic_a , italic_c italic_c } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmthm3.p1.12">For any word <math alttext="u=x_{1}x_{2}x_{3}\in\cal A^{*}" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.8.m1.1"><semantics id="S4.Thmthm3.p1.8.m1.1a"><mrow id="S4.Thmthm3.p1.8.m1.1.1" xref="S4.Thmthm3.p1.8.m1.1.1.cmml"><mi id="S4.Thmthm3.p1.8.m1.1.1.2" xref="S4.Thmthm3.p1.8.m1.1.1.2.cmml">u</mi><mo id="S4.Thmthm3.p1.8.m1.1.1.3" xref="S4.Thmthm3.p1.8.m1.1.1.3.cmml">=</mo><mrow id="S4.Thmthm3.p1.8.m1.1.1.4" xref="S4.Thmthm3.p1.8.m1.1.1.4.cmml"><msub id="S4.Thmthm3.p1.8.m1.1.1.4.2" xref="S4.Thmthm3.p1.8.m1.1.1.4.2.cmml"><mi id="S4.Thmthm3.p1.8.m1.1.1.4.2.2" xref="S4.Thmthm3.p1.8.m1.1.1.4.2.2.cmml">x</mi><mn id="S4.Thmthm3.p1.8.m1.1.1.4.2.3" xref="S4.Thmthm3.p1.8.m1.1.1.4.2.3.cmml">1</mn></msub><mo id="S4.Thmthm3.p1.8.m1.1.1.4.1" xref="S4.Thmthm3.p1.8.m1.1.1.4.1.cmml">⁢</mo><msub id="S4.Thmthm3.p1.8.m1.1.1.4.3" xref="S4.Thmthm3.p1.8.m1.1.1.4.3.cmml"><mi id="S4.Thmthm3.p1.8.m1.1.1.4.3.2" xref="S4.Thmthm3.p1.8.m1.1.1.4.3.2.cmml">x</mi><mn id="S4.Thmthm3.p1.8.m1.1.1.4.3.3" xref="S4.Thmthm3.p1.8.m1.1.1.4.3.3.cmml">2</mn></msub><mo id="S4.Thmthm3.p1.8.m1.1.1.4.1a" xref="S4.Thmthm3.p1.8.m1.1.1.4.1.cmml">⁢</mo><msub id="S4.Thmthm3.p1.8.m1.1.1.4.4" xref="S4.Thmthm3.p1.8.m1.1.1.4.4.cmml"><mi id="S4.Thmthm3.p1.8.m1.1.1.4.4.2" xref="S4.Thmthm3.p1.8.m1.1.1.4.4.2.cmml">x</mi><mn id="S4.Thmthm3.p1.8.m1.1.1.4.4.3" xref="S4.Thmthm3.p1.8.m1.1.1.4.4.3.cmml">3</mn></msub></mrow><mo id="S4.Thmthm3.p1.8.m1.1.1.5" xref="S4.Thmthm3.p1.8.m1.1.1.5.cmml">∈</mo><msup id="S4.Thmthm3.p1.8.m1.1.1.6" xref="S4.Thmthm3.p1.8.m1.1.1.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.8.m1.1.1.6.2" xref="S4.Thmthm3.p1.8.m1.1.1.6.2.cmml">𝒜</mi><mo id="S4.Thmthm3.p1.8.m1.1.1.6.3" xref="S4.Thmthm3.p1.8.m1.1.1.6.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.8.m1.1b"><apply id="S4.Thmthm3.p1.8.m1.1.1.cmml" xref="S4.Thmthm3.p1.8.m1.1.1"><and id="S4.Thmthm3.p1.8.m1.1.1a.cmml" xref="S4.Thmthm3.p1.8.m1.1.1"></and><apply id="S4.Thmthm3.p1.8.m1.1.1b.cmml" xref="S4.Thmthm3.p1.8.m1.1.1"><eq id="S4.Thmthm3.p1.8.m1.1.1.3.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.3"></eq><ci id="S4.Thmthm3.p1.8.m1.1.1.2.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.2">𝑢</ci><apply id="S4.Thmthm3.p1.8.m1.1.1.4.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4"><times id="S4.Thmthm3.p1.8.m1.1.1.4.1.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.1"></times><apply id="S4.Thmthm3.p1.8.m1.1.1.4.2.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.2"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.8.m1.1.1.4.2.1.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.2">subscript</csymbol><ci id="S4.Thmthm3.p1.8.m1.1.1.4.2.2.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.2.2">𝑥</ci><cn id="S4.Thmthm3.p1.8.m1.1.1.4.2.3.cmml" type="integer" xref="S4.Thmthm3.p1.8.m1.1.1.4.2.3">1</cn></apply><apply id="S4.Thmthm3.p1.8.m1.1.1.4.3.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.3"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.8.m1.1.1.4.3.1.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.3">subscript</csymbol><ci id="S4.Thmthm3.p1.8.m1.1.1.4.3.2.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.3.2">𝑥</ci><cn id="S4.Thmthm3.p1.8.m1.1.1.4.3.3.cmml" type="integer" xref="S4.Thmthm3.p1.8.m1.1.1.4.3.3">2</cn></apply><apply id="S4.Thmthm3.p1.8.m1.1.1.4.4.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.4"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.8.m1.1.1.4.4.1.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.4">subscript</csymbol><ci id="S4.Thmthm3.p1.8.m1.1.1.4.4.2.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.4.4.2">𝑥</ci><cn id="S4.Thmthm3.p1.8.m1.1.1.4.4.3.cmml" type="integer" xref="S4.Thmthm3.p1.8.m1.1.1.4.4.3">3</cn></apply></apply></apply><apply id="S4.Thmthm3.p1.8.m1.1.1c.cmml" xref="S4.Thmthm3.p1.8.m1.1.1"><in id="S4.Thmthm3.p1.8.m1.1.1.5.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.5"></in><share href="https://arxiv.org/html/2211.11234v4#S4.Thmthm3.p1.8.m1.1.1.4.cmml" id="S4.Thmthm3.p1.8.m1.1.1d.cmml" xref="S4.Thmthm3.p1.8.m1.1.1"></share><apply id="S4.Thmthm3.p1.8.m1.1.1.6.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.6"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.8.m1.1.1.6.1.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.6">superscript</csymbol><ci id="S4.Thmthm3.p1.8.m1.1.1.6.2.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.6.2">𝒜</ci><times id="S4.Thmthm3.p1.8.m1.1.1.6.3.cmml" xref="S4.Thmthm3.p1.8.m1.1.1.6.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.8.m1.1c">u=x_{1}x_{2}x_{3}\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.8.m1.1d">italic_u = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="x_{1},x_{2},x_{3}\in\cal A" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.9.m2.3"><semantics id="S4.Thmthm3.p1.9.m2.3a"><mrow id="S4.Thmthm3.p1.9.m2.3.3" xref="S4.Thmthm3.p1.9.m2.3.3.cmml"><mrow id="S4.Thmthm3.p1.9.m2.3.3.3.3" xref="S4.Thmthm3.p1.9.m2.3.3.3.4.cmml"><msub id="S4.Thmthm3.p1.9.m2.1.1.1.1.1" xref="S4.Thmthm3.p1.9.m2.1.1.1.1.1.cmml"><mi id="S4.Thmthm3.p1.9.m2.1.1.1.1.1.2" xref="S4.Thmthm3.p1.9.m2.1.1.1.1.1.2.cmml">x</mi><mn id="S4.Thmthm3.p1.9.m2.1.1.1.1.1.3" xref="S4.Thmthm3.p1.9.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmthm3.p1.9.m2.3.3.3.3.4" xref="S4.Thmthm3.p1.9.m2.3.3.3.4.cmml">,</mo><msub id="S4.Thmthm3.p1.9.m2.2.2.2.2.2" xref="S4.Thmthm3.p1.9.m2.2.2.2.2.2.cmml"><mi id="S4.Thmthm3.p1.9.m2.2.2.2.2.2.2" xref="S4.Thmthm3.p1.9.m2.2.2.2.2.2.2.cmml">x</mi><mn id="S4.Thmthm3.p1.9.m2.2.2.2.2.2.3" xref="S4.Thmthm3.p1.9.m2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S4.Thmthm3.p1.9.m2.3.3.3.3.5" xref="S4.Thmthm3.p1.9.m2.3.3.3.4.cmml">,</mo><msub id="S4.Thmthm3.p1.9.m2.3.3.3.3.3" xref="S4.Thmthm3.p1.9.m2.3.3.3.3.3.cmml"><mi id="S4.Thmthm3.p1.9.m2.3.3.3.3.3.2" xref="S4.Thmthm3.p1.9.m2.3.3.3.3.3.2.cmml">x</mi><mn id="S4.Thmthm3.p1.9.m2.3.3.3.3.3.3" xref="S4.Thmthm3.p1.9.m2.3.3.3.3.3.3.cmml">3</mn></msub></mrow><mo id="S4.Thmthm3.p1.9.m2.3.3.4" xref="S4.Thmthm3.p1.9.m2.3.3.4.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm3.p1.9.m2.3.3.5" xref="S4.Thmthm3.p1.9.m2.3.3.5.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.9.m2.3b"><apply id="S4.Thmthm3.p1.9.m2.3.3.cmml" xref="S4.Thmthm3.p1.9.m2.3.3"><in id="S4.Thmthm3.p1.9.m2.3.3.4.cmml" xref="S4.Thmthm3.p1.9.m2.3.3.4"></in><list id="S4.Thmthm3.p1.9.m2.3.3.3.4.cmml" xref="S4.Thmthm3.p1.9.m2.3.3.3.3"><apply id="S4.Thmthm3.p1.9.m2.1.1.1.1.1.cmml" xref="S4.Thmthm3.p1.9.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.9.m2.1.1.1.1.1.1.cmml" xref="S4.Thmthm3.p1.9.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.Thmthm3.p1.9.m2.1.1.1.1.1.2.cmml" xref="S4.Thmthm3.p1.9.m2.1.1.1.1.1.2">𝑥</ci><cn id="S4.Thmthm3.p1.9.m2.1.1.1.1.1.3.cmml" type="integer" xref="S4.Thmthm3.p1.9.m2.1.1.1.1.1.3">1</cn></apply><apply id="S4.Thmthm3.p1.9.m2.2.2.2.2.2.cmml" xref="S4.Thmthm3.p1.9.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.9.m2.2.2.2.2.2.1.cmml" xref="S4.Thmthm3.p1.9.m2.2.2.2.2.2">subscript</csymbol><ci id="S4.Thmthm3.p1.9.m2.2.2.2.2.2.2.cmml" xref="S4.Thmthm3.p1.9.m2.2.2.2.2.2.2">𝑥</ci><cn id="S4.Thmthm3.p1.9.m2.2.2.2.2.2.3.cmml" type="integer" xref="S4.Thmthm3.p1.9.m2.2.2.2.2.2.3">2</cn></apply><apply id="S4.Thmthm3.p1.9.m2.3.3.3.3.3.cmml" xref="S4.Thmthm3.p1.9.m2.3.3.3.3.3"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.9.m2.3.3.3.3.3.1.cmml" xref="S4.Thmthm3.p1.9.m2.3.3.3.3.3">subscript</csymbol><ci id="S4.Thmthm3.p1.9.m2.3.3.3.3.3.2.cmml" xref="S4.Thmthm3.p1.9.m2.3.3.3.3.3.2">𝑥</ci><cn id="S4.Thmthm3.p1.9.m2.3.3.3.3.3.3.cmml" type="integer" xref="S4.Thmthm3.p1.9.m2.3.3.3.3.3.3">3</cn></apply></list><ci id="S4.Thmthm3.p1.9.m2.3.3.5.cmml" xref="S4.Thmthm3.p1.9.m2.3.3.5">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.9.m2.3c">x_{1},x_{2},x_{3}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.9.m2.3d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> and <math alttext="x_{2}\neq b" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.10.m3.1"><semantics id="S4.Thmthm3.p1.10.m3.1a"><mrow id="S4.Thmthm3.p1.10.m3.1.1" xref="S4.Thmthm3.p1.10.m3.1.1.cmml"><msub id="S4.Thmthm3.p1.10.m3.1.1.2" xref="S4.Thmthm3.p1.10.m3.1.1.2.cmml"><mi id="S4.Thmthm3.p1.10.m3.1.1.2.2" xref="S4.Thmthm3.p1.10.m3.1.1.2.2.cmml">x</mi><mn id="S4.Thmthm3.p1.10.m3.1.1.2.3" xref="S4.Thmthm3.p1.10.m3.1.1.2.3.cmml">2</mn></msub><mo id="S4.Thmthm3.p1.10.m3.1.1.1" xref="S4.Thmthm3.p1.10.m3.1.1.1.cmml">≠</mo><mi id="S4.Thmthm3.p1.10.m3.1.1.3" xref="S4.Thmthm3.p1.10.m3.1.1.3.cmml">b</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.10.m3.1b"><apply id="S4.Thmthm3.p1.10.m3.1.1.cmml" xref="S4.Thmthm3.p1.10.m3.1.1"><neq id="S4.Thmthm3.p1.10.m3.1.1.1.cmml" xref="S4.Thmthm3.p1.10.m3.1.1.1"></neq><apply id="S4.Thmthm3.p1.10.m3.1.1.2.cmml" xref="S4.Thmthm3.p1.10.m3.1.1.2"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.10.m3.1.1.2.1.cmml" xref="S4.Thmthm3.p1.10.m3.1.1.2">subscript</csymbol><ci id="S4.Thmthm3.p1.10.m3.1.1.2.2.cmml" xref="S4.Thmthm3.p1.10.m3.1.1.2.2">𝑥</ci><cn id="S4.Thmthm3.p1.10.m3.1.1.2.3.cmml" type="integer" xref="S4.Thmthm3.p1.10.m3.1.1.2.3">2</cn></apply><ci id="S4.Thmthm3.p1.10.m3.1.1.3.cmml" xref="S4.Thmthm3.p1.10.m3.1.1.3">𝑏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.10.m3.1c">x_{2}\neq b</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.10.m3.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≠ italic_b</annotation></semantics></math> the definition of <math alttext="\lfloor\sigma(u)\rfloor_{w}" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.11.m4.2"><semantics id="S4.Thmthm3.p1.11.m4.2a"><msub id="S4.Thmthm3.p1.11.m4.2.2" xref="S4.Thmthm3.p1.11.m4.2.2.cmml"><mrow id="S4.Thmthm3.p1.11.m4.2.2.1.1" xref="S4.Thmthm3.p1.11.m4.2.2.1.2.cmml"><mo id="S4.Thmthm3.p1.11.m4.2.2.1.1.2" stretchy="false" xref="S4.Thmthm3.p1.11.m4.2.2.1.2.1.cmml">⌊</mo><mrow id="S4.Thmthm3.p1.11.m4.2.2.1.1.1" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1.cmml"><mi id="S4.Thmthm3.p1.11.m4.2.2.1.1.1.2" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1.2.cmml">σ</mi><mo id="S4.Thmthm3.p1.11.m4.2.2.1.1.1.1" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmthm3.p1.11.m4.2.2.1.1.1.3.2" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1.cmml"><mo id="S4.Thmthm3.p1.11.m4.2.2.1.1.1.3.2.1" stretchy="false" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1.cmml">(</mo><mi id="S4.Thmthm3.p1.11.m4.1.1" xref="S4.Thmthm3.p1.11.m4.1.1.cmml">u</mi><mo id="S4.Thmthm3.p1.11.m4.2.2.1.1.1.3.2.2" stretchy="false" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmthm3.p1.11.m4.2.2.1.1.3" stretchy="false" xref="S4.Thmthm3.p1.11.m4.2.2.1.2.1.cmml">⌋</mo></mrow><mi id="S4.Thmthm3.p1.11.m4.2.2.3" xref="S4.Thmthm3.p1.11.m4.2.2.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.11.m4.2b"><apply id="S4.Thmthm3.p1.11.m4.2.2.cmml" xref="S4.Thmthm3.p1.11.m4.2.2"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.11.m4.2.2.2.cmml" xref="S4.Thmthm3.p1.11.m4.2.2">subscript</csymbol><apply id="S4.Thmthm3.p1.11.m4.2.2.1.2.cmml" xref="S4.Thmthm3.p1.11.m4.2.2.1.1"><floor id="S4.Thmthm3.p1.11.m4.2.2.1.2.1.cmml" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.2"></floor><apply id="S4.Thmthm3.p1.11.m4.2.2.1.1.1.cmml" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1"><times id="S4.Thmthm3.p1.11.m4.2.2.1.1.1.1.cmml" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1.1"></times><ci id="S4.Thmthm3.p1.11.m4.2.2.1.1.1.2.cmml" xref="S4.Thmthm3.p1.11.m4.2.2.1.1.1.2">𝜎</ci><ci id="S4.Thmthm3.p1.11.m4.1.1.cmml" xref="S4.Thmthm3.p1.11.m4.1.1">𝑢</ci></apply></apply><ci id="S4.Thmthm3.p1.11.m4.2.2.3.cmml" xref="S4.Thmthm3.p1.11.m4.2.2.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.11.m4.2c">\lfloor\sigma(u)\rfloor_{w}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.11.m4.2d">⌊ italic_σ ( italic_u ) ⌋ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> gives directly <math alttext="\lfloor\sigma(u)\rfloor_{w}=0" class="ltx_Math" display="inline" id="S4.Thmthm3.p1.12.m5.2"><semantics id="S4.Thmthm3.p1.12.m5.2a"><mrow id="S4.Thmthm3.p1.12.m5.2.2" xref="S4.Thmthm3.p1.12.m5.2.2.cmml"><msub id="S4.Thmthm3.p1.12.m5.2.2.1" xref="S4.Thmthm3.p1.12.m5.2.2.1.cmml"><mrow id="S4.Thmthm3.p1.12.m5.2.2.1.1.1" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.2.cmml"><mo id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.2" stretchy="false" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.2.1.cmml">⌊</mo><mrow id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.cmml"><mi id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.2" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.2.cmml">σ</mi><mo id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.1" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.3.2" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.cmml"><mo id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.cmml">(</mo><mi id="S4.Thmthm3.p1.12.m5.1.1" xref="S4.Thmthm3.p1.12.m5.1.1.cmml">u</mi><mo id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.3" stretchy="false" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.2.1.cmml">⌋</mo></mrow><mi id="S4.Thmthm3.p1.12.m5.2.2.1.3" xref="S4.Thmthm3.p1.12.m5.2.2.1.3.cmml">w</mi></msub><mo id="S4.Thmthm3.p1.12.m5.2.2.2" xref="S4.Thmthm3.p1.12.m5.2.2.2.cmml">=</mo><mn id="S4.Thmthm3.p1.12.m5.2.2.3" xref="S4.Thmthm3.p1.12.m5.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm3.p1.12.m5.2b"><apply id="S4.Thmthm3.p1.12.m5.2.2.cmml" xref="S4.Thmthm3.p1.12.m5.2.2"><eq id="S4.Thmthm3.p1.12.m5.2.2.2.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.2"></eq><apply id="S4.Thmthm3.p1.12.m5.2.2.1.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.1"><csymbol cd="ambiguous" id="S4.Thmthm3.p1.12.m5.2.2.1.2.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.1">subscript</csymbol><apply id="S4.Thmthm3.p1.12.m5.2.2.1.1.2.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1"><floor id="S4.Thmthm3.p1.12.m5.2.2.1.1.2.1.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.2"></floor><apply id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1"><times id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.1.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.1"></times><ci id="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.2.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.1.1.1.1.2">𝜎</ci><ci id="S4.Thmthm3.p1.12.m5.1.1.cmml" xref="S4.Thmthm3.p1.12.m5.1.1">𝑢</ci></apply></apply><ci id="S4.Thmthm3.p1.12.m5.2.2.1.3.cmml" xref="S4.Thmthm3.p1.12.m5.2.2.1.3">𝑤</ci></apply><cn id="S4.Thmthm3.p1.12.m5.2.2.3.cmml" type="integer" xref="S4.Thmthm3.p1.12.m5.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm3.p1.12.m5.2c">\lfloor\sigma(u)\rfloor_{w}=0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm3.p1.12.m5.2d">⌊ italic_σ ( italic_u ) ⌋ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = 0</annotation></semantics></math>. It remains to check that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex16"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lfloor\sigma(u)\rfloor_{w}=0\quad\text{for}\quad u\in\{bba,bbb,bbc\}\,," class="ltx_Math" display="block" id="S4.Ex16.m1.4"><semantics id="S4.Ex16.m1.4a"><mrow id="S4.Ex16.m1.4.4.1"><mrow id="S4.Ex16.m1.4.4.1.1.2" xref="S4.Ex16.m1.4.4.1.1.3.cmml"><mrow id="S4.Ex16.m1.4.4.1.1.1.1" xref="S4.Ex16.m1.4.4.1.1.1.1.cmml"><msub id="S4.Ex16.m1.4.4.1.1.1.1.1" xref="S4.Ex16.m1.4.4.1.1.1.1.1.cmml"><mrow id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.2.cmml"><mo id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.2.1.cmml">⌊</mo><mrow id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.2" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.1" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.3.2" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex16.m1.1.1" xref="S4.Ex16.m1.1.1.cmml">u</mi><mo id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex16.m1.4.4.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.Ex16.m1.4.4.1.1.1.1.1.1.2.1.cmml">⌋</mo></mrow><mi id="S4.Ex16.m1.4.4.1.1.1.1.1.3" xref="S4.Ex16.m1.4.4.1.1.1.1.1.3.cmml">w</mi></msub><mo id="S4.Ex16.m1.4.4.1.1.1.1.2" xref="S4.Ex16.m1.4.4.1.1.1.1.2.cmml">=</mo><mrow id="S4.Ex16.m1.4.4.1.1.1.1.3.2" xref="S4.Ex16.m1.4.4.1.1.1.1.3.1.cmml"><mn id="S4.Ex16.m1.2.2" xref="S4.Ex16.m1.2.2.cmml">0</mn><mspace id="S4.Ex16.m1.4.4.1.1.1.1.3.2.1" width="1em" xref="S4.Ex16.m1.4.4.1.1.1.1.3.1.cmml"></mspace><mtext id="S4.Ex16.m1.3.3" xref="S4.Ex16.m1.3.3a.cmml">for</mtext></mrow></mrow><mspace id="S4.Ex16.m1.4.4.1.1.2.3" width="1em" xref="S4.Ex16.m1.4.4.1.1.3a.cmml"></mspace><mrow id="S4.Ex16.m1.4.4.1.1.2.2" xref="S4.Ex16.m1.4.4.1.1.2.2.cmml"><mi id="S4.Ex16.m1.4.4.1.1.2.2.5" xref="S4.Ex16.m1.4.4.1.1.2.2.5.cmml">u</mi><mo id="S4.Ex16.m1.4.4.1.1.2.2.4" xref="S4.Ex16.m1.4.4.1.1.2.2.4.cmml">∈</mo><mrow id="S4.Ex16.m1.4.4.1.1.2.2.3.3" xref="S4.Ex16.m1.4.4.1.1.2.2.3.4.cmml"><mo id="S4.Ex16.m1.4.4.1.1.2.2.3.3.4" stretchy="false" xref="S4.Ex16.m1.4.4.1.1.2.2.3.4.cmml">{</mo><mrow id="S4.Ex16.m1.4.4.1.1.2.2.1.1.1" xref="S4.Ex16.m1.4.4.1.1.2.2.1.1.1.cmml"><mi id="S4.Ex16.m1.4.4.1.1.2.2.1.1.1.2" xref="S4.Ex16.m1.4.4.1.1.2.2.1.1.1.2.cmml">b</mi><mo id="S4.Ex16.m1.4.4.1.1.2.2.1.1.1.1" xref="S4.Ex16.m1.4.4.1.1.2.2.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex16.m1.4.4.1.1.2.2.1.1.1.3" xref="S4.Ex16.m1.4.4.1.1.2.2.1.1.1.3.cmml">b</mi><mo 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u\in\{bba,bbb,bbc\}\,,</annotation><annotation encoding="application/x-llamapun" id="S4.Ex16.m1.4d">⌊ italic_σ ( italic_u ) ⌋ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT = 0 for italic_u ∈ { italic_b italic_b italic_a , italic_b italic_b italic_b , italic_b italic_b italic_c } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmthm3.p1.14">and</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lfloor\sigma(u)\rfloor_{w}=1\quad\text{for}\quad u\in\{aba,abb,abc,cba,cbb,% cbc\}\,," class="ltx_Math" display="block" id="S4.Ex17.m1.4"><semantics id="S4.Ex17.m1.4a"><mrow id="S4.Ex17.m1.4.4.1"><mrow id="S4.Ex17.m1.4.4.1.1.2" xref="S4.Ex17.m1.4.4.1.1.3.cmml"><mrow id="S4.Ex17.m1.4.4.1.1.1.1" 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italic_b , italic_c italic_b italic_c } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmthm3.p1.15">in order to obtained the desired formula</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu^{\sigma}(w)=\mu(aa)+\mu(ac)+\mu(ca)+\mu(cc)+\mu(aba)+\mu(abb)+\mu(abc)+\mu% (cba)+\mu(cbb)+\mu(cbc)" class="ltx_Math" display="block" id="S4.Ex18.m1.11"><semantics id="S4.Ex18.m1.11a"><mrow id="S4.Ex18.m1.11.11" xref="S4.Ex18.m1.11.11.cmml"><mrow id="S4.Ex18.m1.11.11.12" xref="S4.Ex18.m1.11.11.12.cmml"><msup id="S4.Ex18.m1.11.11.12.2" xref="S4.Ex18.m1.11.11.12.2.cmml"><mi id="S4.Ex18.m1.11.11.12.2.2" xref="S4.Ex18.m1.11.11.12.2.2.cmml">μ</mi><mi id="S4.Ex18.m1.11.11.12.2.3" xref="S4.Ex18.m1.11.11.12.2.3.cmml">σ</mi></msup><mo id="S4.Ex18.m1.11.11.12.1" xref="S4.Ex18.m1.11.11.12.1.cmml">⁢</mo><mrow id="S4.Ex18.m1.11.11.12.3.2" xref="S4.Ex18.m1.11.11.12.cmml"><mo id="S4.Ex18.m1.11.11.12.3.2.1" stretchy="false" xref="S4.Ex18.m1.11.11.12.cmml">(</mo><mi id="S4.Ex18.m1.1.1" xref="S4.Ex18.m1.1.1.cmml">w</mi><mo id="S4.Ex18.m1.11.11.12.3.2.2" stretchy="false" xref="S4.Ex18.m1.11.11.12.cmml">)</mo></mrow></mrow><mo id="S4.Ex18.m1.11.11.11" xref="S4.Ex18.m1.11.11.11.cmml">=</mo><mrow id="S4.Ex18.m1.11.11.10" xref="S4.Ex18.m1.11.11.10.cmml"><mrow id="S4.Ex18.m1.2.2.1.1" xref="S4.Ex18.m1.2.2.1.1.cmml"><mi id="S4.Ex18.m1.2.2.1.1.3" xref="S4.Ex18.m1.2.2.1.1.3.cmml">μ</mi><mo id="S4.Ex18.m1.2.2.1.1.2" xref="S4.Ex18.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="S4.Ex18.m1.2.2.1.1.1.1" xref="S4.Ex18.m1.2.2.1.1.1.1.1.cmml"><mo id="S4.Ex18.m1.2.2.1.1.1.1.2" stretchy="false" xref="S4.Ex18.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex18.m1.2.2.1.1.1.1.1" 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id="S4.Ex18.m1.6.6.5.5.1.1.3" stretchy="false" xref="S4.Ex18.m1.6.6.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex18.m1.11.11.10.11d" xref="S4.Ex18.m1.11.11.10.11.cmml">+</mo><mrow id="S4.Ex18.m1.7.7.6.6" xref="S4.Ex18.m1.7.7.6.6.cmml"><mi id="S4.Ex18.m1.7.7.6.6.3" xref="S4.Ex18.m1.7.7.6.6.3.cmml">μ</mi><mo id="S4.Ex18.m1.7.7.6.6.2" xref="S4.Ex18.m1.7.7.6.6.2.cmml">⁢</mo><mrow id="S4.Ex18.m1.7.7.6.6.1.1" xref="S4.Ex18.m1.7.7.6.6.1.1.1.cmml"><mo id="S4.Ex18.m1.7.7.6.6.1.1.2" stretchy="false" xref="S4.Ex18.m1.7.7.6.6.1.1.1.cmml">(</mo><mrow id="S4.Ex18.m1.7.7.6.6.1.1.1" xref="S4.Ex18.m1.7.7.6.6.1.1.1.cmml"><mi id="S4.Ex18.m1.7.7.6.6.1.1.1.2" xref="S4.Ex18.m1.7.7.6.6.1.1.1.2.cmml">a</mi><mo id="S4.Ex18.m1.7.7.6.6.1.1.1.1" xref="S4.Ex18.m1.7.7.6.6.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex18.m1.7.7.6.6.1.1.1.3" xref="S4.Ex18.m1.7.7.6.6.1.1.1.3.cmml">b</mi><mo id="S4.Ex18.m1.7.7.6.6.1.1.1.1a" xref="S4.Ex18.m1.7.7.6.6.1.1.1.1.cmml">⁢</mo><mi id="S4.Ex18.m1.7.7.6.6.1.1.1.4" 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italic_b italic_b ) + italic_μ ( italic_a italic_b italic_c ) + italic_μ ( italic_c italic_b italic_a ) + italic_μ ( italic_c italic_b italic_b ) + italic_μ ( italic_c italic_b italic_c )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <table class="ltx_equation ltx_eqn_table" id="S4.Ex19"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="=\mu(aa)+\mu(ac)+\mu(ca)+\mu(cc)+\mu(ab)+\mu(cb)=\mu(a)+\mu(c)\,." class="ltx_Math" display="block" id="S4.Ex19.m1.3"><semantics id="S4.Ex19.m1.3a"><mrow id="S4.Ex19.m1.3.3.1" xref="S4.Ex19.m1.3.3.1.1.cmml"><mrow id="S4.Ex19.m1.3.3.1.1" xref="S4.Ex19.m1.3.3.1.1.cmml"><mi id="S4.Ex19.m1.3.3.1.1.8" xref="S4.Ex19.m1.3.3.1.1.8.cmml"></mi><mo id="S4.Ex19.m1.3.3.1.1.9" xref="S4.Ex19.m1.3.3.1.1.9.cmml">=</mo><mrow id="S4.Ex19.m1.3.3.1.1.6" 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id="S4.Ex19.m1.3.3.1.1.11.2.1.cmml" xref="S4.Ex19.m1.3.3.1.1.11.2.1"></times><ci id="S4.Ex19.m1.3.3.1.1.11.2.2.cmml" xref="S4.Ex19.m1.3.3.1.1.11.2.2">𝜇</ci><ci id="S4.Ex19.m1.1.1.cmml" xref="S4.Ex19.m1.1.1">𝑎</ci></apply><apply id="S4.Ex19.m1.3.3.1.1.11.3.cmml" xref="S4.Ex19.m1.3.3.1.1.11.3"><times id="S4.Ex19.m1.3.3.1.1.11.3.1.cmml" xref="S4.Ex19.m1.3.3.1.1.11.3.1"></times><ci id="S4.Ex19.m1.3.3.1.1.11.3.2.cmml" xref="S4.Ex19.m1.3.3.1.1.11.3.2">𝜇</ci><ci id="S4.Ex19.m1.2.2.cmml" xref="S4.Ex19.m1.2.2">𝑐</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex19.m1.3c">=\mu(aa)+\mu(ac)+\mu(ca)+\mu(cc)+\mu(ab)+\mu(cb)=\mu(a)+\mu(c)\,.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex19.m1.3d">= italic_μ ( italic_a italic_a ) + italic_μ ( italic_a italic_c ) + italic_μ ( italic_c italic_a ) + italic_μ ( italic_c italic_c ) + italic_μ ( italic_a italic_b ) + italic_μ ( italic_c italic_b ) = italic_μ ( italic_a ) + italic_μ ( italic_c ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S4.Thmthm4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmthm4.1.1.1">Remark 4.4</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmthm4.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmthm4.p1"> <p class="ltx_p" id="S4.Thmthm4.p1.9">Since a factor <math alttext="w^{\prime}\in\cal B^{*}" class="ltx_Math" display="inline" id="S4.Thmthm4.p1.1.m1.1"><semantics id="S4.Thmthm4.p1.1.m1.1a"><mrow id="S4.Thmthm4.p1.1.m1.1.1" xref="S4.Thmthm4.p1.1.m1.1.1.cmml"><msup id="S4.Thmthm4.p1.1.m1.1.1.2" xref="S4.Thmthm4.p1.1.m1.1.1.2.cmml"><mi id="S4.Thmthm4.p1.1.m1.1.1.2.2" xref="S4.Thmthm4.p1.1.m1.1.1.2.2.cmml">w</mi><mo id="S4.Thmthm4.p1.1.m1.1.1.2.3" xref="S4.Thmthm4.p1.1.m1.1.1.2.3.cmml">′</mo></msup><mo 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xref="S4.Thmthm4.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmthm4.p1.1.m1.1.1.3.1.cmml" xref="S4.Thmthm4.p1.1.m1.1.1.3">superscript</csymbol><ci id="S4.Thmthm4.p1.1.m1.1.1.3.2.cmml" xref="S4.Thmthm4.p1.1.m1.1.1.3.2">ℬ</ci><times id="S4.Thmthm4.p1.1.m1.1.1.3.3.cmml" xref="S4.Thmthm4.p1.1.m1.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm4.p1.1.m1.1c">w^{\prime}\in\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm4.p1.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> of length <math alttext="|w^{\prime}|=1" class="ltx_Math" display="inline" id="S4.Thmthm4.p1.2.m2.1"><semantics id="S4.Thmthm4.p1.2.m2.1a"><mrow id="S4.Thmthm4.p1.2.m2.1.1" xref="S4.Thmthm4.p1.2.m2.1.1.cmml"><mrow id="S4.Thmthm4.p1.2.m2.1.1.1.1" xref="S4.Thmthm4.p1.2.m2.1.1.1.2.cmml"><mo id="S4.Thmthm4.p1.2.m2.1.1.1.1.2" 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xref="S4.Thmthm4.p1.3.m3.2.2.1.3">𝜎</ci><apply id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1"><times id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.1"></times><apply id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.2.1.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.2">subscript</csymbol><ci id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.2.2.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.2.2">𝑥</ci><cn id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.2.3">1</cn></apply><ci id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.3.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.3">…</ci><apply id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.4.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.4.1.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.4">subscript</csymbol><ci id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.4.2.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.4.2">𝑥</ci><ci id="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.4.3.cmml" xref="S4.Thmthm4.p1.3.m3.2.2.1.1.1.1.4.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm4.p1.3.m3.2c">\sigma(w)=\sigma(x_{1}\ldots x_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm4.p1.3.m3.2d">italic_σ ( italic_w ) = italic_σ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> cannot “overlap” from some <math alttext="\sigma(x_{i})" class="ltx_Math" display="inline" id="S4.Thmthm4.p1.4.m4.1"><semantics id="S4.Thmthm4.p1.4.m4.1a"><mrow id="S4.Thmthm4.p1.4.m4.1.1" xref="S4.Thmthm4.p1.4.m4.1.1.cmml"><mi id="S4.Thmthm4.p1.4.m4.1.1.3" xref="S4.Thmthm4.p1.4.m4.1.1.3.cmml">σ</mi><mo id="S4.Thmthm4.p1.4.m4.1.1.2" xref="S4.Thmthm4.p1.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.Thmthm4.p1.4.m4.1.1.1.1" xref="S4.Thmthm4.p1.4.m4.1.1.1.1.1.cmml"><mo id="S4.Thmthm4.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S4.Thmthm4.p1.4.m4.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmthm4.p1.4.m4.1.1.1.1.1" xref="S4.Thmthm4.p1.4.m4.1.1.1.1.1.cmml"><mi id="S4.Thmthm4.p1.4.m4.1.1.1.1.1.2" xref="S4.Thmthm4.p1.4.m4.1.1.1.1.1.2.cmml">x</mi><mi id="S4.Thmthm4.p1.4.m4.1.1.1.1.1.3" xref="S4.Thmthm4.p1.4.m4.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.Thmthm4.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S4.Thmthm4.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm4.p1.4.m4.1b"><apply id="S4.Thmthm4.p1.4.m4.1.1.cmml" xref="S4.Thmthm4.p1.4.m4.1.1"><times id="S4.Thmthm4.p1.4.m4.1.1.2.cmml" xref="S4.Thmthm4.p1.4.m4.1.1.2"></times><ci id="S4.Thmthm4.p1.4.m4.1.1.3.cmml" xref="S4.Thmthm4.p1.4.m4.1.1.3">𝜎</ci><apply id="S4.Thmthm4.p1.4.m4.1.1.1.1.1.cmml" xref="S4.Thmthm4.p1.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmthm4.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.Thmthm4.p1.4.m4.1.1.1.1">subscript</csymbol><ci id="S4.Thmthm4.p1.4.m4.1.1.1.1.1.2.cmml" xref="S4.Thmthm4.p1.4.m4.1.1.1.1.1.2">𝑥</ci><ci id="S4.Thmthm4.p1.4.m4.1.1.1.1.1.3.cmml" xref="S4.Thmthm4.p1.4.m4.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm4.p1.4.m4.1c">\sigma(x_{i})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm4.p1.4.m4.1d">italic_σ ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> into the adjacent <math alttext="\sigma(x_{i+1})" class="ltx_Math" display="inline" id="S4.Thmthm4.p1.5.m5.1"><semantics id="S4.Thmthm4.p1.5.m5.1a"><mrow id="S4.Thmthm4.p1.5.m5.1.1" xref="S4.Thmthm4.p1.5.m5.1.1.cmml"><mi id="S4.Thmthm4.p1.5.m5.1.1.3" xref="S4.Thmthm4.p1.5.m5.1.1.3.cmml">σ</mi><mo id="S4.Thmthm4.p1.5.m5.1.1.2" xref="S4.Thmthm4.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S4.Thmthm4.p1.5.m5.1.1.1.1" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.cmml"><mo id="S4.Thmthm4.p1.5.m5.1.1.1.1.2" stretchy="false" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.cmml">(</mo><msub id="S4.Thmthm4.p1.5.m5.1.1.1.1.1" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.cmml"><mi id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.2" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.2.cmml">x</mi><mrow id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.cmml"><mi id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.2" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.2.cmml">i</mi><mo id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.1" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.1.cmml">+</mo><mn id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.3" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.Thmthm4.p1.5.m5.1.1.1.1.3" stretchy="false" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm4.p1.5.m5.1b"><apply id="S4.Thmthm4.p1.5.m5.1.1.cmml" xref="S4.Thmthm4.p1.5.m5.1.1"><times id="S4.Thmthm4.p1.5.m5.1.1.2.cmml" xref="S4.Thmthm4.p1.5.m5.1.1.2"></times><ci id="S4.Thmthm4.p1.5.m5.1.1.3.cmml" xref="S4.Thmthm4.p1.5.m5.1.1.3">𝜎</ci><apply id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.cmml" xref="S4.Thmthm4.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.1.cmml" xref="S4.Thmthm4.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.2.cmml" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.2">𝑥</ci><apply id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.cmml" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3"><plus id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.1.cmml" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.1"></plus><ci id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.2.cmml" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.2">𝑖</ci><cn id="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmthm4.p1.5.m5.1.1.1.1.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm4.p1.5.m5.1c">\sigma(x_{i+1})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm4.p1.5.m5.1d">italic_σ ( italic_x start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, the only essential occurrences of <math alttext="w^{\prime}" class="ltx_Math" display="inline" id="S4.Thmthm4.p1.6.m6.1"><semantics id="S4.Thmthm4.p1.6.m6.1a"><msup id="S4.Thmthm4.p1.6.m6.1.1" xref="S4.Thmthm4.p1.6.m6.1.1.cmml"><mi id="S4.Thmthm4.p1.6.m6.1.1.2" xref="S4.Thmthm4.p1.6.m6.1.1.2.cmml">w</mi><mo id="S4.Thmthm4.p1.6.m6.1.1.3" xref="S4.Thmthm4.p1.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmthm4.p1.6.m6.1b"><apply id="S4.Thmthm4.p1.6.m6.1.1.cmml" xref="S4.Thmthm4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmthm4.p1.6.m6.1.1.1.cmml" xref="S4.Thmthm4.p1.6.m6.1.1">superscript</csymbol><ci id="S4.Thmthm4.p1.6.m6.1.1.2.cmml" xref="S4.Thmthm4.p1.6.m6.1.1.2">𝑤</ci><ci id="S4.Thmthm4.p1.6.m6.1.1.3.cmml" xref="S4.Thmthm4.p1.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm4.p1.6.m6.1c">w^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm4.p1.6.m6.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in any <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S4.Thmthm4.p1.7.m7.1"><semantics id="S4.Thmthm4.p1.7.m7.1a"><mrow id="S4.Thmthm4.p1.7.m7.1.2" xref="S4.Thmthm4.p1.7.m7.1.2.cmml"><mi id="S4.Thmthm4.p1.7.m7.1.2.2" xref="S4.Thmthm4.p1.7.m7.1.2.2.cmml">σ</mi><mo id="S4.Thmthm4.p1.7.m7.1.2.1" xref="S4.Thmthm4.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S4.Thmthm4.p1.7.m7.1.2.3.2" xref="S4.Thmthm4.p1.7.m7.1.2.cmml"><mo id="S4.Thmthm4.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S4.Thmthm4.p1.7.m7.1.2.cmml">(</mo><mi id="S4.Thmthm4.p1.7.m7.1.1" xref="S4.Thmthm4.p1.7.m7.1.1.cmml">w</mi><mo id="S4.Thmthm4.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S4.Thmthm4.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm4.p1.7.m7.1b"><apply id="S4.Thmthm4.p1.7.m7.1.2.cmml" xref="S4.Thmthm4.p1.7.m7.1.2"><times id="S4.Thmthm4.p1.7.m7.1.2.1.cmml" xref="S4.Thmthm4.p1.7.m7.1.2.1"></times><ci id="S4.Thmthm4.p1.7.m7.1.2.2.cmml" xref="S4.Thmthm4.p1.7.m7.1.2.2">𝜎</ci><ci id="S4.Thmthm4.p1.7.m7.1.1.cmml" xref="S4.Thmthm4.p1.7.m7.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm4.p1.7.m7.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm4.p1.7.m7.1d">italic_σ ( italic_w )</annotation></semantics></math> can take place if one has <math alttext="|w|=1" class="ltx_Math" display="inline" id="S4.Thmthm4.p1.8.m8.1"><semantics id="S4.Thmthm4.p1.8.m8.1a"><mrow id="S4.Thmthm4.p1.8.m8.1.2" xref="S4.Thmthm4.p1.8.m8.1.2.cmml"><mrow id="S4.Thmthm4.p1.8.m8.1.2.2.2" xref="S4.Thmthm4.p1.8.m8.1.2.2.1.cmml"><mo id="S4.Thmthm4.p1.8.m8.1.2.2.2.1" stretchy="false" xref="S4.Thmthm4.p1.8.m8.1.2.2.1.1.cmml">|</mo><mi id="S4.Thmthm4.p1.8.m8.1.1" xref="S4.Thmthm4.p1.8.m8.1.1.cmml">w</mi><mo id="S4.Thmthm4.p1.8.m8.1.2.2.2.2" stretchy="false" xref="S4.Thmthm4.p1.8.m8.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.Thmthm4.p1.8.m8.1.2.1" xref="S4.Thmthm4.p1.8.m8.1.2.1.cmml">=</mo><mn id="S4.Thmthm4.p1.8.m8.1.2.3" xref="S4.Thmthm4.p1.8.m8.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm4.p1.8.m8.1b"><apply id="S4.Thmthm4.p1.8.m8.1.2.cmml" xref="S4.Thmthm4.p1.8.m8.1.2"><eq id="S4.Thmthm4.p1.8.m8.1.2.1.cmml" xref="S4.Thmthm4.p1.8.m8.1.2.1"></eq><apply id="S4.Thmthm4.p1.8.m8.1.2.2.1.cmml" xref="S4.Thmthm4.p1.8.m8.1.2.2.2"><abs id="S4.Thmthm4.p1.8.m8.1.2.2.1.1.cmml" xref="S4.Thmthm4.p1.8.m8.1.2.2.2.1"></abs><ci id="S4.Thmthm4.p1.8.m8.1.1.cmml" xref="S4.Thmthm4.p1.8.m8.1.1">𝑤</ci></apply><cn id="S4.Thmthm4.p1.8.m8.1.2.3.cmml" type="integer" xref="S4.Thmthm4.p1.8.m8.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm4.p1.8.m8.1c">|w|=1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm4.p1.8.m8.1d">| italic_w | = 1</annotation></semantics></math>. Hence we deduce from (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.E2" title="In Proposition 4.2. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.2</span></a>) for any <math alttext="b_{j}\in\cal B" class="ltx_Math" display="inline" id="S4.Thmthm4.p1.9.m9.1"><semantics id="S4.Thmthm4.p1.9.m9.1a"><mrow id="S4.Thmthm4.p1.9.m9.1.1" xref="S4.Thmthm4.p1.9.m9.1.1.cmml"><msub id="S4.Thmthm4.p1.9.m9.1.1.2" xref="S4.Thmthm4.p1.9.m9.1.1.2.cmml"><mi id="S4.Thmthm4.p1.9.m9.1.1.2.2" xref="S4.Thmthm4.p1.9.m9.1.1.2.2.cmml">b</mi><mi id="S4.Thmthm4.p1.9.m9.1.1.2.3" xref="S4.Thmthm4.p1.9.m9.1.1.2.3.cmml">j</mi></msub><mo id="S4.Thmthm4.p1.9.m9.1.1.1" xref="S4.Thmthm4.p1.9.m9.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm4.p1.9.m9.1.1.3" xref="S4.Thmthm4.p1.9.m9.1.1.3.cmml">ℬ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm4.p1.9.m9.1b"><apply id="S4.Thmthm4.p1.9.m9.1.1.cmml" xref="S4.Thmthm4.p1.9.m9.1.1"><in id="S4.Thmthm4.p1.9.m9.1.1.1.cmml" xref="S4.Thmthm4.p1.9.m9.1.1.1"></in><apply id="S4.Thmthm4.p1.9.m9.1.1.2.cmml" xref="S4.Thmthm4.p1.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.Thmthm4.p1.9.m9.1.1.2.1.cmml" xref="S4.Thmthm4.p1.9.m9.1.1.2">subscript</csymbol><ci id="S4.Thmthm4.p1.9.m9.1.1.2.2.cmml" xref="S4.Thmthm4.p1.9.m9.1.1.2.2">𝑏</ci><ci id="S4.Thmthm4.p1.9.m9.1.1.2.3.cmml" xref="S4.Thmthm4.p1.9.m9.1.1.2.3">𝑗</ci></apply><ci id="S4.Thmthm4.p1.9.m9.1.1.3.cmml" xref="S4.Thmthm4.p1.9.m9.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm4.p1.9.m9.1c">b_{j}\in\cal B</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm4.p1.9.m9.1d">italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ caligraphic_B</annotation></semantics></math> the formula</p> <table class="ltx_equation ltx_eqn_table" id="S4.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(4.4)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu^{\sigma}(b_{j})\,[=\sigma M(\mu)(b_{j})]=\sum_{a_{k}\in\cal A}{|\sigma(a_{% k})|}_{b_{j}}\cdot\mu(a_{k})\,." class="ltx_Math" display="block" id="S4.E4.m1.2"><semantics id="S4.E4.m1.2a"><mrow id="S4.E4.m1.2.2.1" xref="S4.E4.m1.2.2.1.1.cmml"><mrow id="S4.E4.m1.2.2.1.1" xref="S4.E4.m1.2.2.1.1.cmml"><mrow id="S4.E4.m1.2.2.1.1.2" xref="S4.E4.m1.2.2.1.1.2.cmml"><mrow id="S4.E4.m1.2.2.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.cmml"><msup id="S4.E4.m1.2.2.1.1.1.1.3" xref="S4.E4.m1.2.2.1.1.1.1.3.cmml"><mi id="S4.E4.m1.2.2.1.1.1.1.3.2" xref="S4.E4.m1.2.2.1.1.1.1.3.2.cmml">μ</mi><mi id="S4.E4.m1.2.2.1.1.1.1.3.3" xref="S4.E4.m1.2.2.1.1.1.1.3.3.cmml">σ</mi></msup><mo id="S4.E4.m1.2.2.1.1.1.1.2" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E4.m1.2.2.1.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.cmml"><mo 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id="S4.E4.m1.2c">\mu^{\sigma}(b_{j})\,[=\sigma M(\mu)(b_{j})]=\sum_{a_{k}\in\cal A}{|\sigma(a_{% k})|}_{b_{j}}\cdot\mu(a_{k})\,.</annotation><annotation encoding="application/x-llamapun" id="S4.E4.m1.2d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) [ = italic_σ italic_M ( italic_μ ) ( italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ] = ∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT | italic_σ ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⋅ italic_μ ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S4.SS2.p7"> <p class="ltx_p" id="S4.SS2.p7.3">Every shift-invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS2.p7.1.m1.1"><semantics id="S4.SS2.p7.1.m1.1a"><mi id="S4.SS2.p7.1.m1.1.1" xref="S4.SS2.p7.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.1.m1.1b"><ci id="S4.SS2.p7.1.m1.1.1.cmml" xref="S4.SS2.p7.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.1.m1.1d">italic_μ</annotation></semantics></math> on <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S4.SS2.p7.2.m2.1"><semantics id="S4.SS2.p7.2.m2.1a"><msup id="S4.SS2.p7.2.m2.1.1" xref="S4.SS2.p7.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p7.2.m2.1.1.2" xref="S4.SS2.p7.2.m2.1.1.2.cmml">𝒜</mi><mi id="S4.SS2.p7.2.m2.1.1.3" xref="S4.SS2.p7.2.m2.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p7.2.m2.1b"><apply id="S4.SS2.p7.2.m2.1.1.cmml" xref="S4.SS2.p7.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p7.2.m2.1.1.1.cmml" xref="S4.SS2.p7.2.m2.1.1">superscript</csymbol><ci id="S4.SS2.p7.2.m2.1.1.2.cmml" xref="S4.SS2.p7.2.m2.1.1.2">𝒜</ci><ci id="S4.SS2.p7.2.m2.1.1.3.cmml" xref="S4.SS2.p7.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.2.m2.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.2.m2.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> defines canonically a <span class="ltx_text ltx_font_italic" id="S4.SS2.p7.3.1">letter frequency vector</span> <math alttext="\vec{v}(\mu)=(\mu([a_{k}]))_{a_{k}\in\cal A}\in\mathbb{R}_{\geq 0}A" class="ltx_Math" display="inline" id="S4.SS2.p7.3.m3.2"><semantics id="S4.SS2.p7.3.m3.2a"><mrow id="S4.SS2.p7.3.m3.2.2" xref="S4.SS2.p7.3.m3.2.2.cmml"><mrow id="S4.SS2.p7.3.m3.2.2.3" xref="S4.SS2.p7.3.m3.2.2.3.cmml"><mover accent="true" 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xref="S4.SS2.p7.3.m3.2.2.1.3.2.3">𝑘</ci></apply><ci id="S4.SS2.p7.3.m3.2.2.1.3.3.cmml" xref="S4.SS2.p7.3.m3.2.2.1.3.3">𝒜</ci></apply></apply></apply><apply id="S4.SS2.p7.3.m3.2.2c.cmml" xref="S4.SS2.p7.3.m3.2.2"><in id="S4.SS2.p7.3.m3.2.2.5.cmml" xref="S4.SS2.p7.3.m3.2.2.5"></in><share href="https://arxiv.org/html/2211.11234v4#S4.SS2.p7.3.m3.2.2.1.cmml" id="S4.SS2.p7.3.m3.2.2d.cmml" xref="S4.SS2.p7.3.m3.2.2"></share><apply id="S4.SS2.p7.3.m3.2.2.6.cmml" xref="S4.SS2.p7.3.m3.2.2.6"><times id="S4.SS2.p7.3.m3.2.2.6.1.cmml" xref="S4.SS2.p7.3.m3.2.2.6.1"></times><apply id="S4.SS2.p7.3.m3.2.2.6.2.cmml" xref="S4.SS2.p7.3.m3.2.2.6.2"><csymbol cd="ambiguous" id="S4.SS2.p7.3.m3.2.2.6.2.1.cmml" xref="S4.SS2.p7.3.m3.2.2.6.2">subscript</csymbol><ci id="S4.SS2.p7.3.m3.2.2.6.2.2.cmml" xref="S4.SS2.p7.3.m3.2.2.6.2.2">ℝ</ci><apply id="S4.SS2.p7.3.m3.2.2.6.2.3.cmml" xref="S4.SS2.p7.3.m3.2.2.6.2.3"><geq id="S4.SS2.p7.3.m3.2.2.6.2.3.1.cmml" xref="S4.SS2.p7.3.m3.2.2.6.2.3.1"></geq><csymbol cd="latexml" id="S4.SS2.p7.3.m3.2.2.6.2.3.2.cmml" xref="S4.SS2.p7.3.m3.2.2.6.2.3.2">absent</csymbol><cn id="S4.SS2.p7.3.m3.2.2.6.2.3.3.cmml" type="integer" xref="S4.SS2.p7.3.m3.2.2.6.2.3.3">0</cn></apply></apply><ci id="S4.SS2.p7.3.m3.2.2.6.3.cmml" xref="S4.SS2.p7.3.m3.2.2.6.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p7.3.m3.2c">\vec{v}(\mu)=(\mu([a_{k}]))_{a_{k}\in\cal A}\in\mathbb{R}_{\geq 0}A</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p7.3.m3.2d">over→ start_ARG italic_v end_ARG ( italic_μ ) = ( italic_μ ( [ italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] ) ) start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_A end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT italic_A</annotation></semantics></math>, which plays an important role in many contexts (see for instance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite> or <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib4" title="">4</a>]</cite>).</p> </div> <div class="ltx_para" id="S4.SS2.p8"> <p class="ltx_p" id="S4.SS2.p8.1">We observe directly from (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S4.E4" title="In Remark 4.4. ‣ 4.2. An alternative evaluation method ‣ 4. Evaluation of the transferred measure 𝜎⁢𝑀⁢(𝜇) ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">4.4</span></a>):</p> </div> <div class="ltx_theorem ltx_theorem_prop" id="S4.Thmthm5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmthm5.1.1.1">Proposition 4.5</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmthm5.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmthm5.p1"> <p class="ltx_p" id="S4.Thmthm5.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmthm5.p1.6.6">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.1.1.m1.1"><semantics id="S4.Thmthm5.p1.1.1.m1.1a"><mrow id="S4.Thmthm5.p1.1.1.m1.1.1" xref="S4.Thmthm5.p1.1.1.m1.1.1.cmml"><mi id="S4.Thmthm5.p1.1.1.m1.1.1.2" xref="S4.Thmthm5.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S4.Thmthm5.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.Thmthm5.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S4.Thmthm5.p1.1.1.m1.1.1.3" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.cmml"><msup id="S4.Thmthm5.p1.1.1.m1.1.1.3.2" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm5.p1.1.1.m1.1.1.3.2.2" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S4.Thmthm5.p1.1.1.m1.1.1.3.2.3" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S4.Thmthm5.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S4.Thmthm5.p1.1.1.m1.1.1.3.3" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm5.p1.1.1.m1.1.1.3.3.2" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S4.Thmthm5.p1.1.1.m1.1.1.3.3.3" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.1.1.m1.1b"><apply id="S4.Thmthm5.p1.1.1.m1.1.1.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1"><ci id="S4.Thmthm5.p1.1.1.m1.1.1.1.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.1">:</ci><ci id="S4.Thmthm5.p1.1.1.m1.1.1.2.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S4.Thmthm5.p1.1.1.m1.1.1.3.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3"><ci id="S4.Thmthm5.p1.1.1.m1.1.1.3.1.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.1">→</ci><apply id="S4.Thmthm5.p1.1.1.m1.1.1.3.2.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmthm5.p1.1.1.m1.1.1.3.2.1.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S4.Thmthm5.p1.1.1.m1.1.1.3.2.2.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S4.Thmthm5.p1.1.1.m1.1.1.3.2.3.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S4.Thmthm5.p1.1.1.m1.1.1.3.3.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S4.Thmthm5.p1.1.1.m1.1.1.3.3.1.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S4.Thmthm5.p1.1.1.m1.1.1.3.3.2.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S4.Thmthm5.p1.1.1.m1.1.1.3.3.3.cmml" xref="S4.Thmthm5.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-erasing monoid morphism, and let <math alttext="\mu\in\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.2.2.m2.1"><semantics id="S4.Thmthm5.p1.2.2.m2.1a"><mrow id="S4.Thmthm5.p1.2.2.m2.1.1" xref="S4.Thmthm5.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmthm5.p1.2.2.m2.1.1.3" xref="S4.Thmthm5.p1.2.2.m2.1.1.3.cmml">μ</mi><mo id="S4.Thmthm5.p1.2.2.m2.1.1.2" xref="S4.Thmthm5.p1.2.2.m2.1.1.2.cmml">∈</mo><mrow id="S4.Thmthm5.p1.2.2.m2.1.1.1" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm5.p1.2.2.m2.1.1.1.3" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.3.cmml">ℳ</mi><mo id="S4.Thmthm5.p1.2.2.m2.1.1.1.2" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.2" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.3" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.2.2.m2.1b"><apply id="S4.Thmthm5.p1.2.2.m2.1.1.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1"><in id="S4.Thmthm5.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.2"></in><ci id="S4.Thmthm5.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.3">𝜇</ci><apply id="S4.Thmthm5.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.1"><times id="S4.Thmthm5.p1.2.2.m2.1.1.1.2.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.2"></times><ci id="S4.Thmthm5.p1.2.2.m2.1.1.1.3.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.3">ℳ</ci><apply id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.2">𝒜</ci><ci id="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S4.Thmthm5.p1.2.2.m2.1.1.1.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.2.2.m2.1c">\mu\in\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.2.2.m2.1d">italic_μ ∈ caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> be an invariant measure. Then the letter frequency vectors <math alttext="\vec{v}(\mu)" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.3.3.m3.1"><semantics id="S4.Thmthm5.p1.3.3.m3.1a"><mrow id="S4.Thmthm5.p1.3.3.m3.1.2" xref="S4.Thmthm5.p1.3.3.m3.1.2.cmml"><mover accent="true" id="S4.Thmthm5.p1.3.3.m3.1.2.2" xref="S4.Thmthm5.p1.3.3.m3.1.2.2.cmml"><mi id="S4.Thmthm5.p1.3.3.m3.1.2.2.2" xref="S4.Thmthm5.p1.3.3.m3.1.2.2.2.cmml">v</mi><mo id="S4.Thmthm5.p1.3.3.m3.1.2.2.1" stretchy="false" xref="S4.Thmthm5.p1.3.3.m3.1.2.2.1.cmml">→</mo></mover><mo id="S4.Thmthm5.p1.3.3.m3.1.2.1" xref="S4.Thmthm5.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.Thmthm5.p1.3.3.m3.1.2.3.2" xref="S4.Thmthm5.p1.3.3.m3.1.2.cmml"><mo id="S4.Thmthm5.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="S4.Thmthm5.p1.3.3.m3.1.2.cmml">(</mo><mi id="S4.Thmthm5.p1.3.3.m3.1.1" xref="S4.Thmthm5.p1.3.3.m3.1.1.cmml">μ</mi><mo id="S4.Thmthm5.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="S4.Thmthm5.p1.3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.3.3.m3.1b"><apply id="S4.Thmthm5.p1.3.3.m3.1.2.cmml" xref="S4.Thmthm5.p1.3.3.m3.1.2"><times id="S4.Thmthm5.p1.3.3.m3.1.2.1.cmml" xref="S4.Thmthm5.p1.3.3.m3.1.2.1"></times><apply id="S4.Thmthm5.p1.3.3.m3.1.2.2.cmml" xref="S4.Thmthm5.p1.3.3.m3.1.2.2"><ci id="S4.Thmthm5.p1.3.3.m3.1.2.2.1.cmml" xref="S4.Thmthm5.p1.3.3.m3.1.2.2.1">→</ci><ci id="S4.Thmthm5.p1.3.3.m3.1.2.2.2.cmml" xref="S4.Thmthm5.p1.3.3.m3.1.2.2.2">𝑣</ci></apply><ci id="S4.Thmthm5.p1.3.3.m3.1.1.cmml" xref="S4.Thmthm5.p1.3.3.m3.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.3.3.m3.1c">\vec{v}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.3.3.m3.1d">over→ start_ARG italic_v end_ARG ( italic_μ )</annotation></semantics></math> and <math alttext="\vec{v}(\mu^{\sigma})" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.4.4.m4.1"><semantics id="S4.Thmthm5.p1.4.4.m4.1a"><mrow id="S4.Thmthm5.p1.4.4.m4.1.1" xref="S4.Thmthm5.p1.4.4.m4.1.1.cmml"><mover accent="true" id="S4.Thmthm5.p1.4.4.m4.1.1.3" xref="S4.Thmthm5.p1.4.4.m4.1.1.3.cmml"><mi id="S4.Thmthm5.p1.4.4.m4.1.1.3.2" xref="S4.Thmthm5.p1.4.4.m4.1.1.3.2.cmml">v</mi><mo id="S4.Thmthm5.p1.4.4.m4.1.1.3.1" stretchy="false" xref="S4.Thmthm5.p1.4.4.m4.1.1.3.1.cmml">→</mo></mover><mo id="S4.Thmthm5.p1.4.4.m4.1.1.2" xref="S4.Thmthm5.p1.4.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.Thmthm5.p1.4.4.m4.1.1.1.1" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.cmml"><mo id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.2" stretchy="false" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.cmml">(</mo><msup id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.cmml"><mi id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.2" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.2.cmml">μ</mi><mi id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.3" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.3.cmml">σ</mi></msup><mo id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.3" stretchy="false" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.4.4.m4.1b"><apply id="S4.Thmthm5.p1.4.4.m4.1.1.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1"><times id="S4.Thmthm5.p1.4.4.m4.1.1.2.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1.2"></times><apply id="S4.Thmthm5.p1.4.4.m4.1.1.3.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1.3"><ci id="S4.Thmthm5.p1.4.4.m4.1.1.3.1.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1.3.1">→</ci><ci id="S4.Thmthm5.p1.4.4.m4.1.1.3.2.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1.3.2">𝑣</ci></apply><apply id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1">superscript</csymbol><ci id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.2">𝜇</ci><ci id="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.3.cmml" xref="S4.Thmthm5.p1.4.4.m4.1.1.1.1.1.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.4.4.m4.1c">\vec{v}(\mu^{\sigma})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.4.4.m4.1d">over→ start_ARG italic_v end_ARG ( italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT )</annotation></semantics></math>, associated to <math alttext="\mu" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.5.5.m5.1"><semantics id="S4.Thmthm5.p1.5.5.m5.1a"><mi id="S4.Thmthm5.p1.5.5.m5.1.1" xref="S4.Thmthm5.p1.5.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.5.5.m5.1b"><ci id="S4.Thmthm5.p1.5.5.m5.1.1.cmml" xref="S4.Thmthm5.p1.5.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.5.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.5.5.m5.1d">italic_μ</annotation></semantics></math> and to its transferred measure <math alttext="\mu^{\sigma}=\sigma M(\mu)\in\cal M(\cal B^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.6.6.m6.2"><semantics id="S4.Thmthm5.p1.6.6.m6.2a"><mrow id="S4.Thmthm5.p1.6.6.m6.2.2" xref="S4.Thmthm5.p1.6.6.m6.2.2.cmml"><msup id="S4.Thmthm5.p1.6.6.m6.2.2.3" xref="S4.Thmthm5.p1.6.6.m6.2.2.3.cmml"><mi id="S4.Thmthm5.p1.6.6.m6.2.2.3.2" xref="S4.Thmthm5.p1.6.6.m6.2.2.3.2.cmml">μ</mi><mi id="S4.Thmthm5.p1.6.6.m6.2.2.3.3" xref="S4.Thmthm5.p1.6.6.m6.2.2.3.3.cmml">σ</mi></msup><mo id="S4.Thmthm5.p1.6.6.m6.2.2.4" xref="S4.Thmthm5.p1.6.6.m6.2.2.4.cmml">=</mo><mrow id="S4.Thmthm5.p1.6.6.m6.2.2.5" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.cmml"><mi id="S4.Thmthm5.p1.6.6.m6.2.2.5.2" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.2.cmml">σ</mi><mo id="S4.Thmthm5.p1.6.6.m6.2.2.5.1" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.1.cmml">⁢</mo><mi id="S4.Thmthm5.p1.6.6.m6.2.2.5.3" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.3.cmml">M</mi><mo id="S4.Thmthm5.p1.6.6.m6.2.2.5.1a" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.1.cmml">⁢</mo><mrow id="S4.Thmthm5.p1.6.6.m6.2.2.5.4.2" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.cmml"><mo id="S4.Thmthm5.p1.6.6.m6.2.2.5.4.2.1" stretchy="false" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.cmml">(</mo><mi id="S4.Thmthm5.p1.6.6.m6.1.1" xref="S4.Thmthm5.p1.6.6.m6.1.1.cmml">μ</mi><mo id="S4.Thmthm5.p1.6.6.m6.2.2.5.4.2.2" stretchy="false" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.cmml">)</mo></mrow></mrow><mo id="S4.Thmthm5.p1.6.6.m6.2.2.6" xref="S4.Thmthm5.p1.6.6.m6.2.2.6.cmml">∈</mo><mrow id="S4.Thmthm5.p1.6.6.m6.2.2.1" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm5.p1.6.6.m6.2.2.1.3" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.3.cmml">ℳ</mi><mo id="S4.Thmthm5.p1.6.6.m6.2.2.1.2" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.cmml"><mo id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.2" stretchy="false" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.cmml">(</mo><msup id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.2" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.2.cmml">ℬ</mi><mi id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.3" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.3" stretchy="false" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.6.6.m6.2b"><apply id="S4.Thmthm5.p1.6.6.m6.2.2.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2"><and id="S4.Thmthm5.p1.6.6.m6.2.2a.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2"></and><apply id="S4.Thmthm5.p1.6.6.m6.2.2b.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2"><eq id="S4.Thmthm5.p1.6.6.m6.2.2.4.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.4"></eq><apply id="S4.Thmthm5.p1.6.6.m6.2.2.3.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.3"><csymbol cd="ambiguous" id="S4.Thmthm5.p1.6.6.m6.2.2.3.1.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.3">superscript</csymbol><ci id="S4.Thmthm5.p1.6.6.m6.2.2.3.2.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.3.2">𝜇</ci><ci id="S4.Thmthm5.p1.6.6.m6.2.2.3.3.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.3.3">𝜎</ci></apply><apply id="S4.Thmthm5.p1.6.6.m6.2.2.5.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.5"><times id="S4.Thmthm5.p1.6.6.m6.2.2.5.1.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.1"></times><ci id="S4.Thmthm5.p1.6.6.m6.2.2.5.2.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.2">𝜎</ci><ci id="S4.Thmthm5.p1.6.6.m6.2.2.5.3.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.5.3">𝑀</ci><ci id="S4.Thmthm5.p1.6.6.m6.1.1.cmml" xref="S4.Thmthm5.p1.6.6.m6.1.1">𝜇</ci></apply></apply><apply id="S4.Thmthm5.p1.6.6.m6.2.2c.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2"><in id="S4.Thmthm5.p1.6.6.m6.2.2.6.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.6"></in><share href="https://arxiv.org/html/2211.11234v4#S4.Thmthm5.p1.6.6.m6.2.2.5.cmml" id="S4.Thmthm5.p1.6.6.m6.2.2d.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2"></share><apply id="S4.Thmthm5.p1.6.6.m6.2.2.1.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.1"><times id="S4.Thmthm5.p1.6.6.m6.2.2.1.2.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.2"></times><ci id="S4.Thmthm5.p1.6.6.m6.2.2.1.3.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.3">ℳ</ci><apply id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.1.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.2.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.2">ℬ</ci><ci id="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.3.cmml" xref="S4.Thmthm5.p1.6.6.m6.2.2.1.1.1.1.3">ℤ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.6.6.m6.2c">\mu^{\sigma}=\sigma M(\mu)\in\cal M(\cal B^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.6.6.m6.2d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT = italic_σ italic_M ( italic_μ ) ∈ caligraphic_M ( caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> respectively, satisfy</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(4.5)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\vec{v}(\mu^{\sigma})=M(\sigma)\cdot\vec{v}(\mu)\,," class="ltx_Math" display="block" id="S4.E5.m1.3"><semantics id="S4.E5.m1.3a"><mrow id="S4.E5.m1.3.3.1" xref="S4.E5.m1.3.3.1.1.cmml"><mrow id="S4.E5.m1.3.3.1.1" xref="S4.E5.m1.3.3.1.1.cmml"><mrow id="S4.E5.m1.3.3.1.1.1" xref="S4.E5.m1.3.3.1.1.1.cmml"><mover accent="true" id="S4.E5.m1.3.3.1.1.1.3" xref="S4.E5.m1.3.3.1.1.1.3.cmml"><mi id="S4.E5.m1.3.3.1.1.1.3.2" xref="S4.E5.m1.3.3.1.1.1.3.2.cmml">v</mi><mo id="S4.E5.m1.3.3.1.1.1.3.1" stretchy="false" xref="S4.E5.m1.3.3.1.1.1.3.1.cmml">→</mo></mover><mo id="S4.E5.m1.3.3.1.1.1.2" xref="S4.E5.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S4.E5.m1.3.3.1.1.1.1.1" xref="S4.E5.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S4.E5.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S4.E5.m1.3.3.1.1.1.1.1.1.cmml">(</mo><msup id="S4.E5.m1.3.3.1.1.1.1.1.1" xref="S4.E5.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S4.E5.m1.3.3.1.1.1.1.1.1.2" xref="S4.E5.m1.3.3.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S4.E5.m1.3.3.1.1.1.1.1.1.3" xref="S4.E5.m1.3.3.1.1.1.1.1.1.3.cmml">σ</mi></msup><mo id="S4.E5.m1.3.3.1.1.1.1.1.3" stretchy="false" xref="S4.E5.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.3.3.1.1.2" xref="S4.E5.m1.3.3.1.1.2.cmml">=</mo><mrow id="S4.E5.m1.3.3.1.1.3" xref="S4.E5.m1.3.3.1.1.3.cmml"><mrow id="S4.E5.m1.3.3.1.1.3.2" xref="S4.E5.m1.3.3.1.1.3.2.cmml"><mrow id="S4.E5.m1.3.3.1.1.3.2.2" xref="S4.E5.m1.3.3.1.1.3.2.2.cmml"><mi id="S4.E5.m1.3.3.1.1.3.2.2.2" xref="S4.E5.m1.3.3.1.1.3.2.2.2.cmml">M</mi><mo id="S4.E5.m1.3.3.1.1.3.2.2.1" xref="S4.E5.m1.3.3.1.1.3.2.2.1.cmml">⁢</mo><mrow id="S4.E5.m1.3.3.1.1.3.2.2.3.2" xref="S4.E5.m1.3.3.1.1.3.2.2.cmml"><mo id="S4.E5.m1.3.3.1.1.3.2.2.3.2.1" stretchy="false" xref="S4.E5.m1.3.3.1.1.3.2.2.cmml">(</mo><mi id="S4.E5.m1.1.1" xref="S4.E5.m1.1.1.cmml">σ</mi><mo id="S4.E5.m1.3.3.1.1.3.2.2.3.2.2" rspace="0.055em" stretchy="false" xref="S4.E5.m1.3.3.1.1.3.2.2.cmml">)</mo></mrow></mrow><mo id="S4.E5.m1.3.3.1.1.3.2.1" rspace="0.222em" xref="S4.E5.m1.3.3.1.1.3.2.1.cmml">⋅</mo><mover accent="true" id="S4.E5.m1.3.3.1.1.3.2.3" xref="S4.E5.m1.3.3.1.1.3.2.3.cmml"><mi id="S4.E5.m1.3.3.1.1.3.2.3.2" xref="S4.E5.m1.3.3.1.1.3.2.3.2.cmml">v</mi><mo id="S4.E5.m1.3.3.1.1.3.2.3.1" stretchy="false" xref="S4.E5.m1.3.3.1.1.3.2.3.1.cmml">→</mo></mover></mrow><mo id="S4.E5.m1.3.3.1.1.3.1" xref="S4.E5.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S4.E5.m1.3.3.1.1.3.3.2" xref="S4.E5.m1.3.3.1.1.3.cmml"><mo id="S4.E5.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S4.E5.m1.3.3.1.1.3.cmml">(</mo><mi id="S4.E5.m1.2.2" xref="S4.E5.m1.2.2.cmml">μ</mi><mo id="S4.E5.m1.3.3.1.1.3.3.2.2" rspace="0.170em" stretchy="false" xref="S4.E5.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S4.E5.m1.3.3.1.2" xref="S4.E5.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E5.m1.3b"><apply id="S4.E5.m1.3.3.1.1.cmml" xref="S4.E5.m1.3.3.1"><eq id="S4.E5.m1.3.3.1.1.2.cmml" xref="S4.E5.m1.3.3.1.1.2"></eq><apply id="S4.E5.m1.3.3.1.1.1.cmml" xref="S4.E5.m1.3.3.1.1.1"><times id="S4.E5.m1.3.3.1.1.1.2.cmml" xref="S4.E5.m1.3.3.1.1.1.2"></times><apply id="S4.E5.m1.3.3.1.1.1.3.cmml" xref="S4.E5.m1.3.3.1.1.1.3"><ci id="S4.E5.m1.3.3.1.1.1.3.1.cmml" xref="S4.E5.m1.3.3.1.1.1.3.1">→</ci><ci id="S4.E5.m1.3.3.1.1.1.3.2.cmml" xref="S4.E5.m1.3.3.1.1.1.3.2">𝑣</ci></apply><apply id="S4.E5.m1.3.3.1.1.1.1.1.1.cmml" xref="S4.E5.m1.3.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.E5.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S4.E5.m1.3.3.1.1.1.1.1">superscript</csymbol><ci id="S4.E5.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S4.E5.m1.3.3.1.1.1.1.1.1.2">𝜇</ci><ci id="S4.E5.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S4.E5.m1.3.3.1.1.1.1.1.1.3">𝜎</ci></apply></apply><apply id="S4.E5.m1.3.3.1.1.3.cmml" xref="S4.E5.m1.3.3.1.1.3"><times id="S4.E5.m1.3.3.1.1.3.1.cmml" xref="S4.E5.m1.3.3.1.1.3.1"></times><apply id="S4.E5.m1.3.3.1.1.3.2.cmml" xref="S4.E5.m1.3.3.1.1.3.2"><ci id="S4.E5.m1.3.3.1.1.3.2.1.cmml" xref="S4.E5.m1.3.3.1.1.3.2.1">⋅</ci><apply id="S4.E5.m1.3.3.1.1.3.2.2.cmml" xref="S4.E5.m1.3.3.1.1.3.2.2"><times id="S4.E5.m1.3.3.1.1.3.2.2.1.cmml" xref="S4.E5.m1.3.3.1.1.3.2.2.1"></times><ci id="S4.E5.m1.3.3.1.1.3.2.2.2.cmml" xref="S4.E5.m1.3.3.1.1.3.2.2.2">𝑀</ci><ci id="S4.E5.m1.1.1.cmml" xref="S4.E5.m1.1.1">𝜎</ci></apply><apply id="S4.E5.m1.3.3.1.1.3.2.3.cmml" xref="S4.E5.m1.3.3.1.1.3.2.3"><ci id="S4.E5.m1.3.3.1.1.3.2.3.1.cmml" xref="S4.E5.m1.3.3.1.1.3.2.3.1">→</ci><ci id="S4.E5.m1.3.3.1.1.3.2.3.2.cmml" xref="S4.E5.m1.3.3.1.1.3.2.3.2">𝑣</ci></apply></apply><ci id="S4.E5.m1.2.2.cmml" xref="S4.E5.m1.2.2">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E5.m1.3c">\vec{v}(\mu^{\sigma})=M(\sigma)\cdot\vec{v}(\mu)\,,</annotation><annotation encoding="application/x-llamapun" id="S4.E5.m1.3d">over→ start_ARG italic_v end_ARG ( italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ) = italic_M ( italic_σ ) ⋅ over→ start_ARG italic_v end_ARG ( italic_μ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmthm5.p1.10"><span class="ltx_text ltx_font_italic" id="S4.Thmthm5.p1.10.4">where <math alttext="M(\sigma)" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.7.1.m1.1"><semantics id="S4.Thmthm5.p1.7.1.m1.1a"><mrow id="S4.Thmthm5.p1.7.1.m1.1.2" xref="S4.Thmthm5.p1.7.1.m1.1.2.cmml"><mi id="S4.Thmthm5.p1.7.1.m1.1.2.2" xref="S4.Thmthm5.p1.7.1.m1.1.2.2.cmml">M</mi><mo id="S4.Thmthm5.p1.7.1.m1.1.2.1" xref="S4.Thmthm5.p1.7.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.Thmthm5.p1.7.1.m1.1.2.3.2" xref="S4.Thmthm5.p1.7.1.m1.1.2.cmml"><mo id="S4.Thmthm5.p1.7.1.m1.1.2.3.2.1" stretchy="false" xref="S4.Thmthm5.p1.7.1.m1.1.2.cmml">(</mo><mi id="S4.Thmthm5.p1.7.1.m1.1.1" xref="S4.Thmthm5.p1.7.1.m1.1.1.cmml">σ</mi><mo id="S4.Thmthm5.p1.7.1.m1.1.2.3.2.2" stretchy="false" xref="S4.Thmthm5.p1.7.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.7.1.m1.1b"><apply id="S4.Thmthm5.p1.7.1.m1.1.2.cmml" xref="S4.Thmthm5.p1.7.1.m1.1.2"><times id="S4.Thmthm5.p1.7.1.m1.1.2.1.cmml" xref="S4.Thmthm5.p1.7.1.m1.1.2.1"></times><ci id="S4.Thmthm5.p1.7.1.m1.1.2.2.cmml" xref="S4.Thmthm5.p1.7.1.m1.1.2.2">𝑀</ci><ci id="S4.Thmthm5.p1.7.1.m1.1.1.cmml" xref="S4.Thmthm5.p1.7.1.m1.1.1">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.7.1.m1.1c">M(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.7.1.m1.1d">italic_M ( italic_σ )</annotation></semantics></math> denotes the incidence matrix of <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.8.2.m2.1"><semantics id="S4.Thmthm5.p1.8.2.m2.1a"><mi id="S4.Thmthm5.p1.8.2.m2.1.1" xref="S4.Thmthm5.p1.8.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.8.2.m2.1b"><ci id="S4.Thmthm5.p1.8.2.m2.1.1.cmml" xref="S4.Thmthm5.p1.8.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.8.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.8.2.m2.1d">italic_σ</annotation></semantics></math> - see (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E1" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.1</span></a>). <span class="ltx_text ltx_inline-block" id="S4.Thmthm5.p1.9.3.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.9.3.1.m1.1"><semantics id="S4.Thmthm5.p1.9.3.1.m1.1a"><mo id="S4.Thmthm5.p1.9.3.1.m1.1.1" xref="S4.Thmthm5.p1.9.3.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.9.3.1.m1.1b"><csymbol cd="latexml" id="S4.Thmthm5.p1.9.3.1.m1.1.1.cmml" xref="S4.Thmthm5.p1.9.3.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.9.3.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.9.3.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S4.Thmthm5.p1.10.4.m3.1"><semantics id="S4.Thmthm5.p1.10.4.m3.1a"><mo id="S4.Thmthm5.p1.10.4.m3.1.1" xref="S4.Thmthm5.p1.10.4.m3.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S4.Thmthm5.p1.10.4.m3.1b"><csymbol cd="latexml" id="S4.Thmthm5.p1.10.4.m3.1.1.cmml" xref="S4.Thmthm5.p1.10.4.m3.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmthm5.p1.10.4.m3.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S4.Thmthm5.p1.10.4.m3.1d">⊓</annotation></semantics></math></span></p> </div> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5. </span>Shift-orbit injectivity and related notions</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.2">In this section we will establish a natural criterion which guaranties that the measure transfer map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><mrow id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mi id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S5.p1.1.m1.1.1.1" xref="S5.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S5.p1.1.m1.1.1.3" xref="S5.p1.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><times id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1.1"></times><ci id="S5.p1.1.m1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.2">𝜎</ci><ci id="S5.p1.1.m1.1.1.3.cmml" xref="S5.p1.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">italic_σ italic_M</annotation></semantics></math> is <math alttext="1-1" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mrow id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml"><mn id="S5.p1.2.m2.1.1.2" xref="S5.p1.2.m2.1.1.2.cmml">1</mn><mo id="S5.p1.2.m2.1.1.1" xref="S5.p1.2.m2.1.1.1.cmml">−</mo><mn id="S5.p1.2.m2.1.1.3" xref="S5.p1.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><apply id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1"><minus id="S5.p1.2.m2.1.1.1.cmml" xref="S5.p1.2.m2.1.1.1"></minus><cn id="S5.p1.2.m2.1.1.2.cmml" type="integer" xref="S5.p1.2.m2.1.1.2">1</cn><cn id="S5.p1.2.m2.1.1.3.cmml" type="integer" xref="S5.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">1-1</annotation><annotation encoding="application/x-llamapun" id="S5.p1.2.m2.1d">1 - 1</annotation></semantics></math>, when restricted to measures which are supported by suitable subshifts. We first need to recall and specify the notation from Section <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS3" title="2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.3</span></a>:</p> </div> <div class="ltx_theorem ltx_theorem_defn" id="S5.Thmthm1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm1.1.1.1">Definition 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm1.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm1.p1"> <p class="ltx_p" id="S5.Thmthm1.p1.2">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.Thmthm1.p1.1.m1.1"><semantics id="S5.Thmthm1.p1.1.m1.1a"><mrow id="S5.Thmthm1.p1.1.m1.1.1" xref="S5.Thmthm1.p1.1.m1.1.1.cmml"><mi id="S5.Thmthm1.p1.1.m1.1.1.2" xref="S5.Thmthm1.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S5.Thmthm1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm1.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm1.p1.1.m1.1.1.3" xref="S5.Thmthm1.p1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm1.p1.1.m1.1.1.3.2" xref="S5.Thmthm1.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm1.p1.1.m1.1.1.3.2.2" xref="S5.Thmthm1.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm1.p1.1.m1.1.1.3.2.3" xref="S5.Thmthm1.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm1.p1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm1.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm1.p1.1.m1.1.1.3.3" xref="S5.Thmthm1.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm1.p1.1.m1.1.1.3.3.2" xref="S5.Thmthm1.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.Thmthm1.p1.1.m1.1.1.3.3.3" xref="S5.Thmthm1.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm1.p1.1.m1.1b"><apply id="S5.Thmthm1.p1.1.m1.1.1.cmml" xref="S5.Thmthm1.p1.1.m1.1.1"><ci id="S5.Thmthm1.p1.1.m1.1.1.1.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.1">:</ci><ci id="S5.Thmthm1.p1.1.m1.1.1.2.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.2">𝜎</ci><apply id="S5.Thmthm1.p1.1.m1.1.1.3.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3"><ci id="S5.Thmthm1.p1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm1.p1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm1.p1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm1.p1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm1.p1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm1.p1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm1.p1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm1.p1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S5.Thmthm1.p1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm1.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm1.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm1.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-erasing monoid morphism, and let <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm1.p1.2.m2.1"><semantics id="S5.Thmthm1.p1.2.m2.1a"><mrow id="S5.Thmthm1.p1.2.m2.1.1" xref="S5.Thmthm1.p1.2.m2.1.1.cmml"><mi id="S5.Thmthm1.p1.2.m2.1.1.2" xref="S5.Thmthm1.p1.2.m2.1.1.2.cmml">X</mi><mo id="S5.Thmthm1.p1.2.m2.1.1.1" xref="S5.Thmthm1.p1.2.m2.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm1.p1.2.m2.1.1.3" xref="S5.Thmthm1.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm1.p1.2.m2.1.1.3.2" xref="S5.Thmthm1.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm1.p1.2.m2.1.1.3.3" xref="S5.Thmthm1.p1.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm1.p1.2.m2.1b"><apply id="S5.Thmthm1.p1.2.m2.1.1.cmml" xref="S5.Thmthm1.p1.2.m2.1.1"><subset id="S5.Thmthm1.p1.2.m2.1.1.1.cmml" xref="S5.Thmthm1.p1.2.m2.1.1.1"></subset><ci id="S5.Thmthm1.p1.2.m2.1.1.2.cmml" xref="S5.Thmthm1.p1.2.m2.1.1.2">𝑋</ci><apply id="S5.Thmthm1.p1.2.m2.1.1.3.cmml" xref="S5.Thmthm1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm1.p1.2.m2.1.1.3.1.cmml" xref="S5.Thmthm1.p1.2.m2.1.1.3">superscript</csymbol><ci id="S5.Thmthm1.p1.2.m2.1.1.3.2.cmml" xref="S5.Thmthm1.p1.2.m2.1.1.3.2">𝒜</ci><ci id="S5.Thmthm1.p1.2.m2.1.1.3.3.cmml" xref="S5.Thmthm1.p1.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm1.p1.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm1.p1.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> be any subshift.</p> <ol class="ltx_enumerate" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.4">We say that <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.1"><semantics id="S5.I1.i1.p1.1.m1.1a"><mi id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.1b"><ci id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">italic_σ</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S5.I1.i1.p1.2.1">shift-orbit injective in <math alttext="X" class="ltx_Math" display="inline" id="S5.I1.i1.p1.2.1.m1.1"><semantics id="S5.I1.i1.p1.2.1.m1.1a"><mi id="S5.I1.i1.p1.2.1.m1.1.1" xref="S5.I1.i1.p1.2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.2.1.m1.1b"><ci id="S5.I1.i1.p1.2.1.m1.1.1.cmml" xref="S5.I1.i1.p1.2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.2.1.m1.1d">italic_X</annotation></semantics></math></span> if the map <math alttext="\sigma^{T}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.3.m2.1"><semantics id="S5.I1.i1.p1.3.m2.1a"><msup id="S5.I1.i1.p1.3.m2.1.1" xref="S5.I1.i1.p1.3.m2.1.1.cmml"><mi id="S5.I1.i1.p1.3.m2.1.1.2" xref="S5.I1.i1.p1.3.m2.1.1.2.cmml">σ</mi><mi id="S5.I1.i1.p1.3.m2.1.1.3" xref="S5.I1.i1.p1.3.m2.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.3.m2.1b"><apply id="S5.I1.i1.p1.3.m2.1.1.cmml" xref="S5.I1.i1.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.3.m2.1.1.1.cmml" xref="S5.I1.i1.p1.3.m2.1.1">superscript</csymbol><ci id="S5.I1.i1.p1.3.m2.1.1.2.cmml" xref="S5.I1.i1.p1.3.m2.1.1.2">𝜎</ci><ci id="S5.I1.i1.p1.3.m2.1.1.3.cmml" xref="S5.I1.i1.p1.3.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.3.m2.1c">\sigma^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.3.m2.1d">italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> restricted to the shift-orbits of <math alttext="X" class="ltx_Math" display="inline" id="S5.I1.i1.p1.4.m3.1"><semantics id="S5.I1.i1.p1.4.m3.1a"><mi id="S5.I1.i1.p1.4.m3.1.1" xref="S5.I1.i1.p1.4.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.4.m3.1b"><ci id="S5.I1.i1.p1.4.m3.1.1.cmml" xref="S5.I1.i1.p1.4.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.4.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.4.m3.1d">italic_X</annotation></semantics></math> is injective.</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.4">We say that <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.1"><semantics id="S5.I1.i2.p1.1.m1.1a"><mi id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.1b"><ci id="S5.I1.i2.p1.1.m1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.1.m1.1d">italic_σ</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S5.I1.i2.p1.2.1">shift-period preserving in <math alttext="X" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.1.m1.1"><semantics id="S5.I1.i2.p1.2.1.m1.1a"><mi id="S5.I1.i2.p1.2.1.m1.1.1" xref="S5.I1.i2.p1.2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.1.m1.1b"><ci id="S5.I1.i2.p1.2.1.m1.1.1.cmml" xref="S5.I1.i2.p1.2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.1.m1.1d">italic_X</annotation></semantics></math></span> if <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I1.i2.p1.3.m2.1"><semantics id="S5.I1.i2.p1.3.m2.1a"><mi id="S5.I1.i2.p1.3.m2.1.1" xref="S5.I1.i2.p1.3.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.3.m2.1b"><ci id="S5.I1.i2.p1.3.m2.1.1.cmml" xref="S5.I1.i2.p1.3.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.3.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.3.m2.1d">italic_σ</annotation></semantics></math> preserves the shift-period for every periodic orbit in <math alttext="X" class="ltx_Math" display="inline" id="S5.I1.i2.p1.4.m3.1"><semantics id="S5.I1.i2.p1.4.m3.1a"><mi id="S5.I1.i2.p1.4.m3.1.1" xref="S5.I1.i2.p1.4.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.4.m3.1b"><ci id="S5.I1.i2.p1.4.m3.1.1.cmml" xref="S5.I1.i2.p1.4.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.4.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.4.m3.1d">italic_X</annotation></semantics></math> (see Definition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm10" title="Definition 2.10. ‣ 2.3.2. Shift-period preservation ‣ 2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.10</span></a>).</p> </div> </li> </ol> </div> </div> <div class="ltx_theorem ltx_theorem_lem" id="S5.Thmthm2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm2.1.1.1">Lemma 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm2.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm2.p1"> <p class="ltx_p" id="S5.Thmthm2.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmthm2.p1.5.5">Let <math alttext="\sigma^{\prime}:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.Thmthm2.p1.1.1.m1.1"><semantics id="S5.Thmthm2.p1.1.1.m1.1a"><mrow id="S5.Thmthm2.p1.1.1.m1.1.1" xref="S5.Thmthm2.p1.1.1.m1.1.1.cmml"><msup id="S5.Thmthm2.p1.1.1.m1.1.1.2" xref="S5.Thmthm2.p1.1.1.m1.1.1.2.cmml"><mi id="S5.Thmthm2.p1.1.1.m1.1.1.2.2" xref="S5.Thmthm2.p1.1.1.m1.1.1.2.2.cmml">σ</mi><mo id="S5.Thmthm2.p1.1.1.m1.1.1.2.3" xref="S5.Thmthm2.p1.1.1.m1.1.1.2.3.cmml">′</mo></msup><mo id="S5.Thmthm2.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm2.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm2.p1.1.1.m1.1.1.3" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm2.p1.1.1.m1.1.1.3.2" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm2.p1.1.1.m1.1.1.3.2.2" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm2.p1.1.1.m1.1.1.3.2.3" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm2.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm2.p1.1.1.m1.1.1.3.3" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm2.p1.1.1.m1.1.1.3.3.2" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.Thmthm2.p1.1.1.m1.1.1.3.3.3" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm2.p1.1.1.m1.1b"><apply id="S5.Thmthm2.p1.1.1.m1.1.1.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1"><ci id="S5.Thmthm2.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.1">:</ci><apply id="S5.Thmthm2.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.1.1.m1.1.1.2.1.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.2">superscript</csymbol><ci id="S5.Thmthm2.p1.1.1.m1.1.1.2.2.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.2.2">𝜎</ci><ci id="S5.Thmthm2.p1.1.1.m1.1.1.2.3.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.2.3">′</ci></apply><apply id="S5.Thmthm2.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3"><ci id="S5.Thmthm2.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm2.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm2.p1.1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm2.p1.1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm2.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm2.p1.1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S5.Thmthm2.p1.1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm2.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm2.p1.1.1.m1.1c">\sigma^{\prime}:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm2.p1.1.1.m1.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\sigma^{\prime\prime}:\cal B^{*}\to\cal C^{*}" class="ltx_Math" display="inline" id="S5.Thmthm2.p1.2.2.m2.1"><semantics id="S5.Thmthm2.p1.2.2.m2.1a"><mrow id="S5.Thmthm2.p1.2.2.m2.1.1" xref="S5.Thmthm2.p1.2.2.m2.1.1.cmml"><msup id="S5.Thmthm2.p1.2.2.m2.1.1.2" xref="S5.Thmthm2.p1.2.2.m2.1.1.2.cmml"><mi id="S5.Thmthm2.p1.2.2.m2.1.1.2.2" xref="S5.Thmthm2.p1.2.2.m2.1.1.2.2.cmml">σ</mi><mo id="S5.Thmthm2.p1.2.2.m2.1.1.2.3" xref="S5.Thmthm2.p1.2.2.m2.1.1.2.3.cmml">′′</mo></msup><mo id="S5.Thmthm2.p1.2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm2.p1.2.2.m2.1.1.1.cmml">:</mo><mrow id="S5.Thmthm2.p1.2.2.m2.1.1.3" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.cmml"><msup id="S5.Thmthm2.p1.2.2.m2.1.1.3.2" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm2.p1.2.2.m2.1.1.3.2.2" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.2.2.cmml">ℬ</mi><mo id="S5.Thmthm2.p1.2.2.m2.1.1.3.2.3" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm2.p1.2.2.m2.1.1.3.1" stretchy="false" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm2.p1.2.2.m2.1.1.3.3" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm2.p1.2.2.m2.1.1.3.3.2" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.3.2.cmml">𝒞</mi><mo id="S5.Thmthm2.p1.2.2.m2.1.1.3.3.3" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm2.p1.2.2.m2.1b"><apply id="S5.Thmthm2.p1.2.2.m2.1.1.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1"><ci id="S5.Thmthm2.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.1">:</ci><apply id="S5.Thmthm2.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.2.2.m2.1.1.2.1.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.2">superscript</csymbol><ci id="S5.Thmthm2.p1.2.2.m2.1.1.2.2.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.2.2">𝜎</ci><ci id="S5.Thmthm2.p1.2.2.m2.1.1.2.3.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.2.3">′′</ci></apply><apply id="S5.Thmthm2.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3"><ci id="S5.Thmthm2.p1.2.2.m2.1.1.3.1.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.1">→</ci><apply id="S5.Thmthm2.p1.2.2.m2.1.1.3.2.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.2.2.m2.1.1.3.2.1.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm2.p1.2.2.m2.1.1.3.2.2.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.2.2">ℬ</ci><times id="S5.Thmthm2.p1.2.2.m2.1.1.3.2.3.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.2.3"></times></apply><apply id="S5.Thmthm2.p1.2.2.m2.1.1.3.3.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.2.2.m2.1.1.3.3.1.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm2.p1.2.2.m2.1.1.3.3.2.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.3.2">𝒞</ci><times id="S5.Thmthm2.p1.2.2.m2.1.1.3.3.3.cmml" xref="S5.Thmthm2.p1.2.2.m2.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm2.p1.2.2.m2.1c">\sigma^{\prime\prime}:\cal B^{*}\to\cal C^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm2.p1.2.2.m2.1d">italic_σ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT : caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be two non-erasing morphisms (so that the composition <math alttext="\sigma:=\sigma^{\prime\prime}\circ\sigma^{\prime}" class="ltx_Math" display="inline" id="S5.Thmthm2.p1.3.3.m3.1"><semantics id="S5.Thmthm2.p1.3.3.m3.1a"><mrow id="S5.Thmthm2.p1.3.3.m3.1.1" xref="S5.Thmthm2.p1.3.3.m3.1.1.cmml"><mi id="S5.Thmthm2.p1.3.3.m3.1.1.2" xref="S5.Thmthm2.p1.3.3.m3.1.1.2.cmml">σ</mi><mo id="S5.Thmthm2.p1.3.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm2.p1.3.3.m3.1.1.1.cmml">:=</mo><mrow id="S5.Thmthm2.p1.3.3.m3.1.1.3" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.cmml"><msup id="S5.Thmthm2.p1.3.3.m3.1.1.3.2" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.2.cmml"><mi id="S5.Thmthm2.p1.3.3.m3.1.1.3.2.2" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.2.2.cmml">σ</mi><mo id="S5.Thmthm2.p1.3.3.m3.1.1.3.2.3" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.2.3.cmml">′′</mo></msup><mo id="S5.Thmthm2.p1.3.3.m3.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.1.cmml">∘</mo><msup id="S5.Thmthm2.p1.3.3.m3.1.1.3.3" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.3.cmml"><mi id="S5.Thmthm2.p1.3.3.m3.1.1.3.3.2" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.3.2.cmml">σ</mi><mo id="S5.Thmthm2.p1.3.3.m3.1.1.3.3.3" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm2.p1.3.3.m3.1b"><apply id="S5.Thmthm2.p1.3.3.m3.1.1.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1"><csymbol cd="latexml" id="S5.Thmthm2.p1.3.3.m3.1.1.1.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.1">assign</csymbol><ci id="S5.Thmthm2.p1.3.3.m3.1.1.2.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.2">𝜎</ci><apply id="S5.Thmthm2.p1.3.3.m3.1.1.3.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3"><compose id="S5.Thmthm2.p1.3.3.m3.1.1.3.1.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.1"></compose><apply id="S5.Thmthm2.p1.3.3.m3.1.1.3.2.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.3.3.m3.1.1.3.2.1.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm2.p1.3.3.m3.1.1.3.2.2.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.2.2">𝜎</ci><ci id="S5.Thmthm2.p1.3.3.m3.1.1.3.2.3.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.2.3">′′</ci></apply><apply id="S5.Thmthm2.p1.3.3.m3.1.1.3.3.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.3.3.m3.1.1.3.3.1.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm2.p1.3.3.m3.1.1.3.3.2.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.3.2">𝜎</ci><ci id="S5.Thmthm2.p1.3.3.m3.1.1.3.3.3.cmml" xref="S5.Thmthm2.p1.3.3.m3.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm2.p1.3.3.m3.1c">\sigma:=\sigma^{\prime\prime}\circ\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm2.p1.3.3.m3.1d">italic_σ := italic_σ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ∘ italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is also non-erasing.) For any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm2.p1.4.4.m4.1"><semantics id="S5.Thmthm2.p1.4.4.m4.1a"><mrow id="S5.Thmthm2.p1.4.4.m4.1.1" xref="S5.Thmthm2.p1.4.4.m4.1.1.cmml"><mi id="S5.Thmthm2.p1.4.4.m4.1.1.2" xref="S5.Thmthm2.p1.4.4.m4.1.1.2.cmml">X</mi><mo id="S5.Thmthm2.p1.4.4.m4.1.1.1" xref="S5.Thmthm2.p1.4.4.m4.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm2.p1.4.4.m4.1.1.3" xref="S5.Thmthm2.p1.4.4.m4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm2.p1.4.4.m4.1.1.3.2" xref="S5.Thmthm2.p1.4.4.m4.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm2.p1.4.4.m4.1.1.3.3" xref="S5.Thmthm2.p1.4.4.m4.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm2.p1.4.4.m4.1b"><apply id="S5.Thmthm2.p1.4.4.m4.1.1.cmml" xref="S5.Thmthm2.p1.4.4.m4.1.1"><subset id="S5.Thmthm2.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmthm2.p1.4.4.m4.1.1.1"></subset><ci id="S5.Thmthm2.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmthm2.p1.4.4.m4.1.1.2">𝑋</ci><apply id="S5.Thmthm2.p1.4.4.m4.1.1.3.cmml" xref="S5.Thmthm2.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm2.p1.4.4.m4.1.1.3.1.cmml" xref="S5.Thmthm2.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="S5.Thmthm2.p1.4.4.m4.1.1.3.2.cmml" xref="S5.Thmthm2.p1.4.4.m4.1.1.3.2">𝒜</ci><ci id="S5.Thmthm2.p1.4.4.m4.1.1.3.3.cmml" xref="S5.Thmthm2.p1.4.4.m4.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm2.p1.4.4.m4.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm2.p1.4.4.m4.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> consider the image subshift <math alttext="Y=\sigma(X)" class="ltx_Math" display="inline" id="S5.Thmthm2.p1.5.5.m5.1"><semantics id="S5.Thmthm2.p1.5.5.m5.1a"><mrow id="S5.Thmthm2.p1.5.5.m5.1.2" xref="S5.Thmthm2.p1.5.5.m5.1.2.cmml"><mi id="S5.Thmthm2.p1.5.5.m5.1.2.2" xref="S5.Thmthm2.p1.5.5.m5.1.2.2.cmml">Y</mi><mo id="S5.Thmthm2.p1.5.5.m5.1.2.1" xref="S5.Thmthm2.p1.5.5.m5.1.2.1.cmml">=</mo><mrow id="S5.Thmthm2.p1.5.5.m5.1.2.3" xref="S5.Thmthm2.p1.5.5.m5.1.2.3.cmml"><mi id="S5.Thmthm2.p1.5.5.m5.1.2.3.2" xref="S5.Thmthm2.p1.5.5.m5.1.2.3.2.cmml">σ</mi><mo id="S5.Thmthm2.p1.5.5.m5.1.2.3.1" xref="S5.Thmthm2.p1.5.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S5.Thmthm2.p1.5.5.m5.1.2.3.3.2" xref="S5.Thmthm2.p1.5.5.m5.1.2.3.cmml"><mo id="S5.Thmthm2.p1.5.5.m5.1.2.3.3.2.1" stretchy="false" xref="S5.Thmthm2.p1.5.5.m5.1.2.3.cmml">(</mo><mi id="S5.Thmthm2.p1.5.5.m5.1.1" xref="S5.Thmthm2.p1.5.5.m5.1.1.cmml">X</mi><mo id="S5.Thmthm2.p1.5.5.m5.1.2.3.3.2.2" stretchy="false" xref="S5.Thmthm2.p1.5.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm2.p1.5.5.m5.1b"><apply id="S5.Thmthm2.p1.5.5.m5.1.2.cmml" xref="S5.Thmthm2.p1.5.5.m5.1.2"><eq id="S5.Thmthm2.p1.5.5.m5.1.2.1.cmml" xref="S5.Thmthm2.p1.5.5.m5.1.2.1"></eq><ci id="S5.Thmthm2.p1.5.5.m5.1.2.2.cmml" xref="S5.Thmthm2.p1.5.5.m5.1.2.2">𝑌</ci><apply id="S5.Thmthm2.p1.5.5.m5.1.2.3.cmml" xref="S5.Thmthm2.p1.5.5.m5.1.2.3"><times id="S5.Thmthm2.p1.5.5.m5.1.2.3.1.cmml" xref="S5.Thmthm2.p1.5.5.m5.1.2.3.1"></times><ci id="S5.Thmthm2.p1.5.5.m5.1.2.3.2.cmml" xref="S5.Thmthm2.p1.5.5.m5.1.2.3.2">𝜎</ci><ci id="S5.Thmthm2.p1.5.5.m5.1.1.cmml" xref="S5.Thmthm2.p1.5.5.m5.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm2.p1.5.5.m5.1c">Y=\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm2.p1.5.5.m5.1d">italic_Y = italic_σ ( italic_X )</annotation></semantics></math>. Then we have:</span></p> <ol class="ltx_enumerate" id="S5.I2"> <li class="ltx_item" id="S5.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S5.I2.i1.p1"> <p class="ltx_p" id="S5.I2.i1.p1.6"><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.6.1">The map </span><math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I2.i1.p1.1.m1.1"><semantics id="S5.I2.i1.p1.1.m1.1a"><mi id="S5.I2.i1.p1.1.m1.1.1" xref="S5.I2.i1.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.1.m1.1b"><ci id="S5.I2.i1.p1.1.m1.1.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.1.m1.1d">italic_σ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.6.2"> is shift-orbit injective in </span><math alttext="X" class="ltx_Math" display="inline" id="S5.I2.i1.p1.2.m2.1"><semantics id="S5.I2.i1.p1.2.m2.1a"><mi id="S5.I2.i1.p1.2.m2.1.1" xref="S5.I2.i1.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.2.m2.1b"><ci id="S5.I2.i1.p1.2.m2.1.1.cmml" xref="S5.I2.i1.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.2.m2.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.6.3"> if and only if </span><math alttext="\sigma^{\prime}" class="ltx_Math" display="inline" id="S5.I2.i1.p1.3.m3.1"><semantics id="S5.I2.i1.p1.3.m3.1a"><msup id="S5.I2.i1.p1.3.m3.1.1" xref="S5.I2.i1.p1.3.m3.1.1.cmml"><mi id="S5.I2.i1.p1.3.m3.1.1.2" xref="S5.I2.i1.p1.3.m3.1.1.2.cmml">σ</mi><mo id="S5.I2.i1.p1.3.m3.1.1.3" xref="S5.I2.i1.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.3.m3.1b"><apply id="S5.I2.i1.p1.3.m3.1.1.cmml" xref="S5.I2.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.I2.i1.p1.3.m3.1.1.1.cmml" xref="S5.I2.i1.p1.3.m3.1.1">superscript</csymbol><ci id="S5.I2.i1.p1.3.m3.1.1.2.cmml" xref="S5.I2.i1.p1.3.m3.1.1.2">𝜎</ci><ci id="S5.I2.i1.p1.3.m3.1.1.3.cmml" xref="S5.I2.i1.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.3.m3.1c">\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.3.m3.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.6.4"> is shift-orbit injective in </span><math alttext="X" class="ltx_Math" display="inline" id="S5.I2.i1.p1.4.m4.1"><semantics id="S5.I2.i1.p1.4.m4.1a"><mi id="S5.I2.i1.p1.4.m4.1.1" xref="S5.I2.i1.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.4.m4.1b"><ci id="S5.I2.i1.p1.4.m4.1.1.cmml" xref="S5.I2.i1.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.4.m4.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.6.5"> and </span><math alttext="\sigma^{\prime\prime}" class="ltx_Math" display="inline" id="S5.I2.i1.p1.5.m5.1"><semantics id="S5.I2.i1.p1.5.m5.1a"><msup id="S5.I2.i1.p1.5.m5.1.1" xref="S5.I2.i1.p1.5.m5.1.1.cmml"><mi id="S5.I2.i1.p1.5.m5.1.1.2" xref="S5.I2.i1.p1.5.m5.1.1.2.cmml">σ</mi><mo id="S5.I2.i1.p1.5.m5.1.1.3" xref="S5.I2.i1.p1.5.m5.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.5.m5.1b"><apply id="S5.I2.i1.p1.5.m5.1.1.cmml" xref="S5.I2.i1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.I2.i1.p1.5.m5.1.1.1.cmml" xref="S5.I2.i1.p1.5.m5.1.1">superscript</csymbol><ci id="S5.I2.i1.p1.5.m5.1.1.2.cmml" xref="S5.I2.i1.p1.5.m5.1.1.2">𝜎</ci><ci id="S5.I2.i1.p1.5.m5.1.1.3.cmml" xref="S5.I2.i1.p1.5.m5.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.5.m5.1c">\sigma^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.5.m5.1d">italic_σ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.6.6"> is shift-orbit injective in </span><math alttext="Y" class="ltx_Math" display="inline" id="S5.I2.i1.p1.6.m6.1"><semantics id="S5.I2.i1.p1.6.m6.1a"><mi id="S5.I2.i1.p1.6.m6.1.1" xref="S5.I2.i1.p1.6.m6.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.6.m6.1b"><ci id="S5.I2.i1.p1.6.m6.1.1.cmml" xref="S5.I2.i1.p1.6.m6.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.6.m6.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.6.m6.1d">italic_Y</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.6.7">.</span></p> </div> </li> <li class="ltx_item" id="S5.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S5.I2.i2.p1"> <p class="ltx_p" id="S5.I2.i2.p1.6"><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.6.1">The map </span><math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I2.i2.p1.1.m1.1"><semantics id="S5.I2.i2.p1.1.m1.1a"><mi id="S5.I2.i2.p1.1.m1.1.1" xref="S5.I2.i2.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.1.m1.1b"><ci id="S5.I2.i2.p1.1.m1.1.1.cmml" xref="S5.I2.i2.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.1.m1.1d">italic_σ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.6.2"> is shift-period preserving in </span><math alttext="X" class="ltx_Math" display="inline" id="S5.I2.i2.p1.2.m2.1"><semantics id="S5.I2.i2.p1.2.m2.1a"><mi id="S5.I2.i2.p1.2.m2.1.1" xref="S5.I2.i2.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.2.m2.1b"><ci id="S5.I2.i2.p1.2.m2.1.1.cmml" xref="S5.I2.i2.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.2.m2.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.6.3"> if and only if </span><math alttext="\sigma^{\prime}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.3.m3.1"><semantics id="S5.I2.i2.p1.3.m3.1a"><msup id="S5.I2.i2.p1.3.m3.1.1" xref="S5.I2.i2.p1.3.m3.1.1.cmml"><mi id="S5.I2.i2.p1.3.m3.1.1.2" xref="S5.I2.i2.p1.3.m3.1.1.2.cmml">σ</mi><mo id="S5.I2.i2.p1.3.m3.1.1.3" xref="S5.I2.i2.p1.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.3.m3.1b"><apply id="S5.I2.i2.p1.3.m3.1.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p1.3.m3.1.1.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1">superscript</csymbol><ci id="S5.I2.i2.p1.3.m3.1.1.2.cmml" xref="S5.I2.i2.p1.3.m3.1.1.2">𝜎</ci><ci id="S5.I2.i2.p1.3.m3.1.1.3.cmml" xref="S5.I2.i2.p1.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.3.m3.1c">\sigma^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.3.m3.1d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.6.4"> is shift-period preserving in </span><math alttext="X" class="ltx_Math" display="inline" id="S5.I2.i2.p1.4.m4.1"><semantics id="S5.I2.i2.p1.4.m4.1a"><mi id="S5.I2.i2.p1.4.m4.1.1" xref="S5.I2.i2.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.4.m4.1b"><ci id="S5.I2.i2.p1.4.m4.1.1.cmml" xref="S5.I2.i2.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.4.m4.1d">italic_X</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.6.5"> and </span><math alttext="\sigma^{\prime\prime}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.5.m5.1"><semantics id="S5.I2.i2.p1.5.m5.1a"><msup id="S5.I2.i2.p1.5.m5.1.1" xref="S5.I2.i2.p1.5.m5.1.1.cmml"><mi id="S5.I2.i2.p1.5.m5.1.1.2" xref="S5.I2.i2.p1.5.m5.1.1.2.cmml">σ</mi><mo id="S5.I2.i2.p1.5.m5.1.1.3" xref="S5.I2.i2.p1.5.m5.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.5.m5.1b"><apply id="S5.I2.i2.p1.5.m5.1.1.cmml" xref="S5.I2.i2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.I2.i2.p1.5.m5.1.1.1.cmml" xref="S5.I2.i2.p1.5.m5.1.1">superscript</csymbol><ci id="S5.I2.i2.p1.5.m5.1.1.2.cmml" xref="S5.I2.i2.p1.5.m5.1.1.2">𝜎</ci><ci id="S5.I2.i2.p1.5.m5.1.1.3.cmml" xref="S5.I2.i2.p1.5.m5.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.5.m5.1c">\sigma^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.5.m5.1d">italic_σ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.6.6"> is shift-period preserving in </span><math alttext="Y" class="ltx_Math" display="inline" id="S5.I2.i2.p1.6.m6.1"><semantics id="S5.I2.i2.p1.6.m6.1a"><mi id="S5.I2.i2.p1.6.m6.1.1" xref="S5.I2.i2.p1.6.m6.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.6.m6.1b"><ci id="S5.I2.i2.p1.6.m6.1.1.cmml" xref="S5.I2.i2.p1.6.m6.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.6.m6.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.6.m6.1d">italic_Y</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.6.7">.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S5.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.1.p1"> <p class="ltx_p" id="S5.1.p1.10">Using the surjectivity from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm4" title="Lemma 2.4. ‣ 2.2. “Not so standard” basic facts and terminology ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.4</span></a> (2), we obtain the first of these statements as direct application of the fact that for any two composable surjective maps <math alttext="f" class="ltx_Math" display="inline" id="S5.1.p1.1.m1.1"><semantics id="S5.1.p1.1.m1.1a"><mi id="S5.1.p1.1.m1.1.1" xref="S5.1.p1.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.1.m1.1b"><ci id="S5.1.p1.1.m1.1.1.cmml" xref="S5.1.p1.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.1d">italic_f</annotation></semantics></math> and <math alttext="g" class="ltx_Math" display="inline" id="S5.1.p1.2.m2.1"><semantics id="S5.1.p1.2.m2.1a"><mi id="S5.1.p1.2.m2.1.1" xref="S5.1.p1.2.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.2.m2.1b"><ci id="S5.1.p1.2.m2.1.1.cmml" xref="S5.1.p1.2.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m2.1c">g</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m2.1d">italic_g</annotation></semantics></math> the composition <math alttext="g\circ f" class="ltx_Math" display="inline" id="S5.1.p1.3.m3.1"><semantics id="S5.1.p1.3.m3.1a"><mrow id="S5.1.p1.3.m3.1.1" xref="S5.1.p1.3.m3.1.1.cmml"><mi id="S5.1.p1.3.m3.1.1.2" xref="S5.1.p1.3.m3.1.1.2.cmml">g</mi><mo id="S5.1.p1.3.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.1.p1.3.m3.1.1.1.cmml">∘</mo><mi id="S5.1.p1.3.m3.1.1.3" xref="S5.1.p1.3.m3.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.3.m3.1b"><apply id="S5.1.p1.3.m3.1.1.cmml" xref="S5.1.p1.3.m3.1.1"><compose id="S5.1.p1.3.m3.1.1.1.cmml" xref="S5.1.p1.3.m3.1.1.1"></compose><ci id="S5.1.p1.3.m3.1.1.2.cmml" xref="S5.1.p1.3.m3.1.1.2">𝑔</ci><ci id="S5.1.p1.3.m3.1.1.3.cmml" xref="S5.1.p1.3.m3.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.3.m3.1c">g\circ f</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.3.m3.1d">italic_g ∘ italic_f</annotation></semantics></math> is injective if and only if both maps <math alttext="f" class="ltx_Math" display="inline" id="S5.1.p1.4.m4.1"><semantics id="S5.1.p1.4.m4.1a"><mi id="S5.1.p1.4.m4.1.1" xref="S5.1.p1.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.4.m4.1b"><ci id="S5.1.p1.4.m4.1.1.cmml" xref="S5.1.p1.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.4.m4.1d">italic_f</annotation></semantics></math> and <math alttext="g" class="ltx_Math" display="inline" id="S5.1.p1.5.m5.1"><semantics id="S5.1.p1.5.m5.1a"><mi id="S5.1.p1.5.m5.1.1" xref="S5.1.p1.5.m5.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S5.1.p1.5.m5.1b"><ci id="S5.1.p1.5.m5.1.1.cmml" xref="S5.1.p1.5.m5.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.5.m5.1c">g</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.5.m5.1d">italic_g</annotation></semantics></math> are injective. Statement (2) follows directly from the trivial observation that <math alttext="\sigma(w)" class="ltx_Math" display="inline" id="S5.1.p1.6.m6.1"><semantics id="S5.1.p1.6.m6.1a"><mrow id="S5.1.p1.6.m6.1.2" xref="S5.1.p1.6.m6.1.2.cmml"><mi id="S5.1.p1.6.m6.1.2.2" xref="S5.1.p1.6.m6.1.2.2.cmml">σ</mi><mo id="S5.1.p1.6.m6.1.2.1" xref="S5.1.p1.6.m6.1.2.1.cmml">⁢</mo><mrow id="S5.1.p1.6.m6.1.2.3.2" xref="S5.1.p1.6.m6.1.2.cmml"><mo id="S5.1.p1.6.m6.1.2.3.2.1" stretchy="false" xref="S5.1.p1.6.m6.1.2.cmml">(</mo><mi id="S5.1.p1.6.m6.1.1" xref="S5.1.p1.6.m6.1.1.cmml">w</mi><mo id="S5.1.p1.6.m6.1.2.3.2.2" stretchy="false" xref="S5.1.p1.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.6.m6.1b"><apply id="S5.1.p1.6.m6.1.2.cmml" xref="S5.1.p1.6.m6.1.2"><times id="S5.1.p1.6.m6.1.2.1.cmml" xref="S5.1.p1.6.m6.1.2.1"></times><ci id="S5.1.p1.6.m6.1.2.2.cmml" xref="S5.1.p1.6.m6.1.2.2">𝜎</ci><ci id="S5.1.p1.6.m6.1.1.cmml" xref="S5.1.p1.6.m6.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.6.m6.1c">\sigma(w)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.6.m6.1d">italic_σ ( italic_w )</annotation></semantics></math> is a proper power (see (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E4" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.4</span></a>)) if and only if one of the three, <math alttext="w,\sigma^{\prime}(w)" class="ltx_Math" display="inline" id="S5.1.p1.7.m7.3"><semantics id="S5.1.p1.7.m7.3a"><mrow id="S5.1.p1.7.m7.3.3.1" xref="S5.1.p1.7.m7.3.3.2.cmml"><mi id="S5.1.p1.7.m7.2.2" xref="S5.1.p1.7.m7.2.2.cmml">w</mi><mo id="S5.1.p1.7.m7.3.3.1.2" xref="S5.1.p1.7.m7.3.3.2.cmml">,</mo><mrow id="S5.1.p1.7.m7.3.3.1.1" xref="S5.1.p1.7.m7.3.3.1.1.cmml"><msup id="S5.1.p1.7.m7.3.3.1.1.2" xref="S5.1.p1.7.m7.3.3.1.1.2.cmml"><mi id="S5.1.p1.7.m7.3.3.1.1.2.2" xref="S5.1.p1.7.m7.3.3.1.1.2.2.cmml">σ</mi><mo id="S5.1.p1.7.m7.3.3.1.1.2.3" xref="S5.1.p1.7.m7.3.3.1.1.2.3.cmml">′</mo></msup><mo id="S5.1.p1.7.m7.3.3.1.1.1" xref="S5.1.p1.7.m7.3.3.1.1.1.cmml">⁢</mo><mrow id="S5.1.p1.7.m7.3.3.1.1.3.2" xref="S5.1.p1.7.m7.3.3.1.1.cmml"><mo id="S5.1.p1.7.m7.3.3.1.1.3.2.1" stretchy="false" xref="S5.1.p1.7.m7.3.3.1.1.cmml">(</mo><mi id="S5.1.p1.7.m7.1.1" xref="S5.1.p1.7.m7.1.1.cmml">w</mi><mo id="S5.1.p1.7.m7.3.3.1.1.3.2.2" stretchy="false" xref="S5.1.p1.7.m7.3.3.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.7.m7.3b"><list id="S5.1.p1.7.m7.3.3.2.cmml" xref="S5.1.p1.7.m7.3.3.1"><ci id="S5.1.p1.7.m7.2.2.cmml" xref="S5.1.p1.7.m7.2.2">𝑤</ci><apply id="S5.1.p1.7.m7.3.3.1.1.cmml" xref="S5.1.p1.7.m7.3.3.1.1"><times id="S5.1.p1.7.m7.3.3.1.1.1.cmml" xref="S5.1.p1.7.m7.3.3.1.1.1"></times><apply id="S5.1.p1.7.m7.3.3.1.1.2.cmml" xref="S5.1.p1.7.m7.3.3.1.1.2"><csymbol cd="ambiguous" id="S5.1.p1.7.m7.3.3.1.1.2.1.cmml" xref="S5.1.p1.7.m7.3.3.1.1.2">superscript</csymbol><ci id="S5.1.p1.7.m7.3.3.1.1.2.2.cmml" xref="S5.1.p1.7.m7.3.3.1.1.2.2">𝜎</ci><ci id="S5.1.p1.7.m7.3.3.1.1.2.3.cmml" xref="S5.1.p1.7.m7.3.3.1.1.2.3">′</ci></apply><ci id="S5.1.p1.7.m7.1.1.cmml" xref="S5.1.p1.7.m7.1.1">𝑤</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.7.m7.3c">w,\sigma^{\prime}(w)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.7.m7.3d">italic_w , italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w )</annotation></semantics></math> or <math alttext="\sigma^{\prime\prime}(\sigma^{\prime}(w))" class="ltx_Math" display="inline" id="S5.1.p1.8.m8.2"><semantics id="S5.1.p1.8.m8.2a"><mrow id="S5.1.p1.8.m8.2.2" xref="S5.1.p1.8.m8.2.2.cmml"><msup id="S5.1.p1.8.m8.2.2.3" xref="S5.1.p1.8.m8.2.2.3.cmml"><mi id="S5.1.p1.8.m8.2.2.3.2" xref="S5.1.p1.8.m8.2.2.3.2.cmml">σ</mi><mo id="S5.1.p1.8.m8.2.2.3.3" xref="S5.1.p1.8.m8.2.2.3.3.cmml">′′</mo></msup><mo id="S5.1.p1.8.m8.2.2.2" xref="S5.1.p1.8.m8.2.2.2.cmml">⁢</mo><mrow id="S5.1.p1.8.m8.2.2.1.1" xref="S5.1.p1.8.m8.2.2.1.1.1.cmml"><mo id="S5.1.p1.8.m8.2.2.1.1.2" stretchy="false" xref="S5.1.p1.8.m8.2.2.1.1.1.cmml">(</mo><mrow id="S5.1.p1.8.m8.2.2.1.1.1" xref="S5.1.p1.8.m8.2.2.1.1.1.cmml"><msup id="S5.1.p1.8.m8.2.2.1.1.1.2" xref="S5.1.p1.8.m8.2.2.1.1.1.2.cmml"><mi id="S5.1.p1.8.m8.2.2.1.1.1.2.2" xref="S5.1.p1.8.m8.2.2.1.1.1.2.2.cmml">σ</mi><mo id="S5.1.p1.8.m8.2.2.1.1.1.2.3" xref="S5.1.p1.8.m8.2.2.1.1.1.2.3.cmml">′</mo></msup><mo id="S5.1.p1.8.m8.2.2.1.1.1.1" xref="S5.1.p1.8.m8.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.1.p1.8.m8.2.2.1.1.1.3.2" xref="S5.1.p1.8.m8.2.2.1.1.1.cmml"><mo id="S5.1.p1.8.m8.2.2.1.1.1.3.2.1" stretchy="false" xref="S5.1.p1.8.m8.2.2.1.1.1.cmml">(</mo><mi id="S5.1.p1.8.m8.1.1" xref="S5.1.p1.8.m8.1.1.cmml">w</mi><mo id="S5.1.p1.8.m8.2.2.1.1.1.3.2.2" stretchy="false" xref="S5.1.p1.8.m8.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.1.p1.8.m8.2.2.1.1.3" stretchy="false" xref="S5.1.p1.8.m8.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.8.m8.2b"><apply id="S5.1.p1.8.m8.2.2.cmml" xref="S5.1.p1.8.m8.2.2"><times id="S5.1.p1.8.m8.2.2.2.cmml" xref="S5.1.p1.8.m8.2.2.2"></times><apply id="S5.1.p1.8.m8.2.2.3.cmml" xref="S5.1.p1.8.m8.2.2.3"><csymbol cd="ambiguous" id="S5.1.p1.8.m8.2.2.3.1.cmml" xref="S5.1.p1.8.m8.2.2.3">superscript</csymbol><ci id="S5.1.p1.8.m8.2.2.3.2.cmml" xref="S5.1.p1.8.m8.2.2.3.2">𝜎</ci><ci id="S5.1.p1.8.m8.2.2.3.3.cmml" xref="S5.1.p1.8.m8.2.2.3.3">′′</ci></apply><apply id="S5.1.p1.8.m8.2.2.1.1.1.cmml" xref="S5.1.p1.8.m8.2.2.1.1"><times id="S5.1.p1.8.m8.2.2.1.1.1.1.cmml" xref="S5.1.p1.8.m8.2.2.1.1.1.1"></times><apply id="S5.1.p1.8.m8.2.2.1.1.1.2.cmml" xref="S5.1.p1.8.m8.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S5.1.p1.8.m8.2.2.1.1.1.2.1.cmml" xref="S5.1.p1.8.m8.2.2.1.1.1.2">superscript</csymbol><ci id="S5.1.p1.8.m8.2.2.1.1.1.2.2.cmml" xref="S5.1.p1.8.m8.2.2.1.1.1.2.2">𝜎</ci><ci id="S5.1.p1.8.m8.2.2.1.1.1.2.3.cmml" xref="S5.1.p1.8.m8.2.2.1.1.1.2.3">′</ci></apply><ci id="S5.1.p1.8.m8.1.1.cmml" xref="S5.1.p1.8.m8.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.8.m8.2c">\sigma^{\prime\prime}(\sigma^{\prime}(w))</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.8.m8.2d">italic_σ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w ) )</annotation></semantics></math> is a proper power. <span class="ltx_text ltx_inline-block" id="S5.1.p1.9.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S5.1.p1.9.1.m1.1"><semantics id="S5.1.p1.9.1.m1.1a"><mo id="S5.1.p1.9.1.m1.1.1" xref="S5.1.p1.9.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S5.1.p1.9.1.m1.1b"><csymbol cd="latexml" id="S5.1.p1.9.1.m1.1.1.cmml" xref="S5.1.p1.9.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.9.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.9.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S5.1.p1.10.m9.1"><semantics id="S5.1.p1.10.m9.1a"><mo id="S5.1.p1.10.m9.1.1" xref="S5.1.p1.10.m9.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S5.1.p1.10.m9.1b"><csymbol cd="latexml" id="S5.1.p1.10.m9.1.1.cmml" xref="S5.1.p1.10.m9.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.10.m9.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.10.m9.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.2">Next we consider a subdivision length function <math alttext="\ell" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mi id="S5.p2.1.m1.1.1" mathvariant="normal" xref="S5.p2.1.m1.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">roman_ℓ</annotation></semantics></math> and the associated subdivision morphism <math alttext="\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><mrow id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml"><msub id="S5.p2.2.m2.1.1.2" xref="S5.p2.2.m2.1.1.2.cmml"><mi id="S5.p2.2.m2.1.1.2.2" xref="S5.p2.2.m2.1.1.2.2.cmml">π</mi><mi id="S5.p2.2.m2.1.1.2.3" mathvariant="normal" xref="S5.p2.2.m2.1.1.2.3.cmml">ℓ</mi></msub><mo id="S5.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S5.p2.2.m2.1.1.3" xref="S5.p2.2.m2.1.1.3.cmml"><msup id="S5.p2.2.m2.1.1.3.2" xref="S5.p2.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p2.2.m2.1.1.3.2.2" xref="S5.p2.2.m2.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.p2.2.m2.1.1.3.2.3" xref="S5.p2.2.m2.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.p2.2.m2.1.1.3.1" stretchy="false" xref="S5.p2.2.m2.1.1.3.1.cmml">→</mo><msubsup id="S5.p2.2.m2.1.1.3.3" xref="S5.p2.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p2.2.m2.1.1.3.3.2.2" xref="S5.p2.2.m2.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S5.p2.2.m2.1.1.3.3.2.3" mathvariant="normal" xref="S5.p2.2.m2.1.1.3.3.2.3.cmml">ℓ</mi><mo id="S5.p2.2.m2.1.1.3.3.3" xref="S5.p2.2.m2.1.1.3.3.3.cmml">∗</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><apply id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1"><ci id="S5.p2.2.m2.1.1.1.cmml" xref="S5.p2.2.m2.1.1.1">:</ci><apply id="S5.p2.2.m2.1.1.2.cmml" xref="S5.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.p2.2.m2.1.1.2.1.cmml" xref="S5.p2.2.m2.1.1.2">subscript</csymbol><ci id="S5.p2.2.m2.1.1.2.2.cmml" xref="S5.p2.2.m2.1.1.2.2">𝜋</ci><ci id="S5.p2.2.m2.1.1.2.3.cmml" xref="S5.p2.2.m2.1.1.2.3">ℓ</ci></apply><apply id="S5.p2.2.m2.1.1.3.cmml" xref="S5.p2.2.m2.1.1.3"><ci id="S5.p2.2.m2.1.1.3.1.cmml" xref="S5.p2.2.m2.1.1.3.1">→</ci><apply id="S5.p2.2.m2.1.1.3.2.cmml" xref="S5.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.p2.2.m2.1.1.3.2.1.cmml" xref="S5.p2.2.m2.1.1.3.2">superscript</csymbol><ci id="S5.p2.2.m2.1.1.3.2.2.cmml" xref="S5.p2.2.m2.1.1.3.2.2">𝒜</ci><times id="S5.p2.2.m2.1.1.3.2.3.cmml" xref="S5.p2.2.m2.1.1.3.2.3"></times></apply><apply id="S5.p2.2.m2.1.1.3.3.cmml" xref="S5.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.p2.2.m2.1.1.3.3.1.cmml" xref="S5.p2.2.m2.1.1.3.3">superscript</csymbol><apply id="S5.p2.2.m2.1.1.3.3.2.cmml" xref="S5.p2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.p2.2.m2.1.1.3.3.2.1.cmml" xref="S5.p2.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.p2.2.m2.1.1.3.3.2.2.cmml" xref="S5.p2.2.m2.1.1.3.3.2.2">𝒜</ci><ci id="S5.p2.2.m2.1.1.3.3.2.3.cmml" xref="S5.p2.2.m2.1.1.3.3.2.3">ℓ</ci></apply><times id="S5.p2.2.m2.1.1.3.3.3.cmml" xref="S5.p2.2.m2.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> as in Subsection <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.SS1" title="3.1. Subdivision morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.1</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="S5.Thmthm3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm3.1.1.1">Lemma 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm3.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm3.p1"> <p class="ltx_p" id="S5.Thmthm3.p1.2"><span class="ltx_text ltx_font_italic" id="S5.Thmthm3.p1.2.2">Any subdivision morphism <math alttext="\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}" class="ltx_Math" display="inline" id="S5.Thmthm3.p1.1.1.m1.1"><semantics id="S5.Thmthm3.p1.1.1.m1.1a"><mrow id="S5.Thmthm3.p1.1.1.m1.1.1" xref="S5.Thmthm3.p1.1.1.m1.1.1.cmml"><msub id="S5.Thmthm3.p1.1.1.m1.1.1.2" xref="S5.Thmthm3.p1.1.1.m1.1.1.2.cmml"><mi id="S5.Thmthm3.p1.1.1.m1.1.1.2.2" xref="S5.Thmthm3.p1.1.1.m1.1.1.2.2.cmml">π</mi><mi id="S5.Thmthm3.p1.1.1.m1.1.1.2.3" mathvariant="normal" xref="S5.Thmthm3.p1.1.1.m1.1.1.2.3.cmml">ℓ</mi></msub><mo id="S5.Thmthm3.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm3.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm3.p1.1.1.m1.1.1.3" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm3.p1.1.1.m1.1.1.3.2" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm3.p1.1.1.m1.1.1.3.2.2" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm3.p1.1.1.m1.1.1.3.2.3" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm3.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.1.cmml">→</mo><msubsup id="S5.Thmthm3.p1.1.1.m1.1.1.3.3" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.2" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.3" mathvariant="normal" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.3.cmml">ℓ</mi><mo id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.3" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm3.p1.1.1.m1.1b"><apply id="S5.Thmthm3.p1.1.1.m1.1.1.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1"><ci id="S5.Thmthm3.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.1">:</ci><apply id="S5.Thmthm3.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.Thmthm3.p1.1.1.m1.1.1.2.1.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.2">subscript</csymbol><ci id="S5.Thmthm3.p1.1.1.m1.1.1.2.2.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.2.2">𝜋</ci><ci id="S5.Thmthm3.p1.1.1.m1.1.1.2.3.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.2.3">ℓ</ci></apply><apply id="S5.Thmthm3.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3"><ci id="S5.Thmthm3.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm3.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm3.p1.1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm3.p1.1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm3.p1.1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3">superscript</csymbol><apply id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.1.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.2.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.2">𝒜</ci><ci id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.3.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3.2.3">ℓ</ci></apply><times id="S5.Thmthm3.p1.1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm3.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm3.p1.1.1.m1.1c">\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm3.p1.1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is both, shift-orbit injective and shift-period preserving in the full shift <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm3.p1.2.2.m2.1"><semantics id="S5.Thmthm3.p1.2.2.m2.1a"><msup id="S5.Thmthm3.p1.2.2.m2.1.1" xref="S5.Thmthm3.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm3.p1.2.2.m2.1.1.2" xref="S5.Thmthm3.p1.2.2.m2.1.1.2.cmml">𝒜</mi><mi id="S5.Thmthm3.p1.2.2.m2.1.1.3" xref="S5.Thmthm3.p1.2.2.m2.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S5.Thmthm3.p1.2.2.m2.1b"><apply id="S5.Thmthm3.p1.2.2.m2.1.1.cmml" xref="S5.Thmthm3.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmthm3.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmthm3.p1.2.2.m2.1.1">superscript</csymbol><ci id="S5.Thmthm3.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmthm3.p1.2.2.m2.1.1.2">𝒜</ci><ci id="S5.Thmthm3.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmthm3.p1.2.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm3.p1.2.2.m2.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm3.p1.2.2.m2.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.2.p1"> <p class="ltx_p" id="S5.2.p1.4">In order to see that <math alttext="\pi_{\ell}" class="ltx_Math" display="inline" id="S5.2.p1.1.m1.1"><semantics id="S5.2.p1.1.m1.1a"><msub id="S5.2.p1.1.m1.1.1" xref="S5.2.p1.1.m1.1.1.cmml"><mi id="S5.2.p1.1.m1.1.1.2" xref="S5.2.p1.1.m1.1.1.2.cmml">π</mi><mi id="S5.2.p1.1.m1.1.1.3" mathvariant="normal" xref="S5.2.p1.1.m1.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.2.p1.1.m1.1b"><apply id="S5.2.p1.1.m1.1.1.cmml" xref="S5.2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.2.p1.1.m1.1.1.1.cmml" xref="S5.2.p1.1.m1.1.1">subscript</csymbol><ci id="S5.2.p1.1.m1.1.1.2.cmml" xref="S5.2.p1.1.m1.1.1.2">𝜋</ci><ci id="S5.2.p1.1.m1.1.1.3.cmml" xref="S5.2.p1.1.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p1.1.m1.1c">\pi_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> is shift-orbit injective in the full shift we only need to observe that <math alttext="\pi_{\ell}^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal A_{\ell}^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.2.p1.2.m2.1"><semantics id="S5.2.p1.2.m2.1a"><mrow id="S5.2.p1.2.m2.1.1" xref="S5.2.p1.2.m2.1.1.cmml"><msubsup id="S5.2.p1.2.m2.1.1.2" xref="S5.2.p1.2.m2.1.1.2.cmml"><mi id="S5.2.p1.2.m2.1.1.2.2.2" xref="S5.2.p1.2.m2.1.1.2.2.2.cmml">π</mi><mi id="S5.2.p1.2.m2.1.1.2.2.3" mathvariant="normal" xref="S5.2.p1.2.m2.1.1.2.2.3.cmml">ℓ</mi><mi id="S5.2.p1.2.m2.1.1.2.3" xref="S5.2.p1.2.m2.1.1.2.3.cmml">ℤ</mi></msubsup><mo id="S5.2.p1.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.2.p1.2.m2.1.1.1.cmml">:</mo><mrow id="S5.2.p1.2.m2.1.1.3" xref="S5.2.p1.2.m2.1.1.3.cmml"><msup id="S5.2.p1.2.m2.1.1.3.2" xref="S5.2.p1.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.2.p1.2.m2.1.1.3.2.2" xref="S5.2.p1.2.m2.1.1.3.2.2.cmml">𝒜</mi><mi id="S5.2.p1.2.m2.1.1.3.2.3" xref="S5.2.p1.2.m2.1.1.3.2.3.cmml">ℤ</mi></msup><mo id="S5.2.p1.2.m2.1.1.3.1" stretchy="false" xref="S5.2.p1.2.m2.1.1.3.1.cmml">→</mo><msubsup id="S5.2.p1.2.m2.1.1.3.3" xref="S5.2.p1.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.2.p1.2.m2.1.1.3.3.2.2" xref="S5.2.p1.2.m2.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S5.2.p1.2.m2.1.1.3.3.2.3" mathvariant="normal" xref="S5.2.p1.2.m2.1.1.3.3.2.3.cmml">ℓ</mi><mi id="S5.2.p1.2.m2.1.1.3.3.3" xref="S5.2.p1.2.m2.1.1.3.3.3.cmml">ℤ</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p1.2.m2.1b"><apply id="S5.2.p1.2.m2.1.1.cmml" xref="S5.2.p1.2.m2.1.1"><ci id="S5.2.p1.2.m2.1.1.1.cmml" xref="S5.2.p1.2.m2.1.1.1">:</ci><apply id="S5.2.p1.2.m2.1.1.2.cmml" xref="S5.2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.2.p1.2.m2.1.1.2.1.cmml" xref="S5.2.p1.2.m2.1.1.2">superscript</csymbol><apply id="S5.2.p1.2.m2.1.1.2.2.cmml" xref="S5.2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.2.p1.2.m2.1.1.2.2.1.cmml" xref="S5.2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S5.2.p1.2.m2.1.1.2.2.2.cmml" xref="S5.2.p1.2.m2.1.1.2.2.2">𝜋</ci><ci id="S5.2.p1.2.m2.1.1.2.2.3.cmml" xref="S5.2.p1.2.m2.1.1.2.2.3">ℓ</ci></apply><ci id="S5.2.p1.2.m2.1.1.2.3.cmml" xref="S5.2.p1.2.m2.1.1.2.3">ℤ</ci></apply><apply id="S5.2.p1.2.m2.1.1.3.cmml" xref="S5.2.p1.2.m2.1.1.3"><ci id="S5.2.p1.2.m2.1.1.3.1.cmml" xref="S5.2.p1.2.m2.1.1.3.1">→</ci><apply id="S5.2.p1.2.m2.1.1.3.2.cmml" xref="S5.2.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.2.p1.2.m2.1.1.3.2.1.cmml" xref="S5.2.p1.2.m2.1.1.3.2">superscript</csymbol><ci id="S5.2.p1.2.m2.1.1.3.2.2.cmml" xref="S5.2.p1.2.m2.1.1.3.2.2">𝒜</ci><ci id="S5.2.p1.2.m2.1.1.3.2.3.cmml" xref="S5.2.p1.2.m2.1.1.3.2.3">ℤ</ci></apply><apply id="S5.2.p1.2.m2.1.1.3.3.cmml" xref="S5.2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.2.p1.2.m2.1.1.3.3.1.cmml" xref="S5.2.p1.2.m2.1.1.3.3">superscript</csymbol><apply id="S5.2.p1.2.m2.1.1.3.3.2.cmml" xref="S5.2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.2.p1.2.m2.1.1.3.3.2.1.cmml" xref="S5.2.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.2.p1.2.m2.1.1.3.3.2.2.cmml" xref="S5.2.p1.2.m2.1.1.3.3.2.2">𝒜</ci><ci id="S5.2.p1.2.m2.1.1.3.3.2.3.cmml" xref="S5.2.p1.2.m2.1.1.3.3.2.3">ℓ</ci></apply><ci id="S5.2.p1.2.m2.1.1.3.3.3.cmml" xref="S5.2.p1.2.m2.1.1.3.3.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p1.2.m2.1c">\pi_{\ell}^{\mathbb{Z}}:\cal A^{\mathbb{Z}}\to\cal A_{\ell}^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p1.2.m2.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT : caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is injective, and that two biinfinite words from <math alttext="\pi_{\ell}^{\mathbb{Z}}(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S5.2.p1.3.m3.1"><semantics id="S5.2.p1.3.m3.1a"><mrow id="S5.2.p1.3.m3.1.1" xref="S5.2.p1.3.m3.1.1.cmml"><msubsup id="S5.2.p1.3.m3.1.1.3" xref="S5.2.p1.3.m3.1.1.3.cmml"><mi id="S5.2.p1.3.m3.1.1.3.2.2" xref="S5.2.p1.3.m3.1.1.3.2.2.cmml">π</mi><mi id="S5.2.p1.3.m3.1.1.3.2.3" mathvariant="normal" xref="S5.2.p1.3.m3.1.1.3.2.3.cmml">ℓ</mi><mi id="S5.2.p1.3.m3.1.1.3.3" xref="S5.2.p1.3.m3.1.1.3.3.cmml">ℤ</mi></msubsup><mo id="S5.2.p1.3.m3.1.1.2" xref="S5.2.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S5.2.p1.3.m3.1.1.1.1" xref="S5.2.p1.3.m3.1.1.1.1.1.cmml"><mo id="S5.2.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S5.2.p1.3.m3.1.1.1.1.1.cmml">(</mo><msup id="S5.2.p1.3.m3.1.1.1.1.1" xref="S5.2.p1.3.m3.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.2.p1.3.m3.1.1.1.1.1.2" xref="S5.2.p1.3.m3.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S5.2.p1.3.m3.1.1.1.1.1.3" xref="S5.2.p1.3.m3.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S5.2.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S5.2.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.2.p1.3.m3.1b"><apply id="S5.2.p1.3.m3.1.1.cmml" xref="S5.2.p1.3.m3.1.1"><times id="S5.2.p1.3.m3.1.1.2.cmml" xref="S5.2.p1.3.m3.1.1.2"></times><apply id="S5.2.p1.3.m3.1.1.3.cmml" xref="S5.2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.2.p1.3.m3.1.1.3.1.cmml" xref="S5.2.p1.3.m3.1.1.3">superscript</csymbol><apply id="S5.2.p1.3.m3.1.1.3.2.cmml" xref="S5.2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.2.p1.3.m3.1.1.3.2.1.cmml" xref="S5.2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.2.p1.3.m3.1.1.3.2.2.cmml" xref="S5.2.p1.3.m3.1.1.3.2.2">𝜋</ci><ci id="S5.2.p1.3.m3.1.1.3.2.3.cmml" xref="S5.2.p1.3.m3.1.1.3.2.3">ℓ</ci></apply><ci id="S5.2.p1.3.m3.1.1.3.3.cmml" xref="S5.2.p1.3.m3.1.1.3.3">ℤ</ci></apply><apply id="S5.2.p1.3.m3.1.1.1.1.1.cmml" xref="S5.2.p1.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S5.2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S5.2.p1.3.m3.1.1.1.1">superscript</csymbol><ci id="S5.2.p1.3.m3.1.1.1.1.1.2.cmml" xref="S5.2.p1.3.m3.1.1.1.1.1.2">𝒜</ci><ci id="S5.2.p1.3.m3.1.1.1.1.1.3.cmml" xref="S5.2.p1.3.m3.1.1.1.1.1.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p1.3.m3.1c">\pi_{\ell}^{\mathbb{Z}}(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S5.2.p1.3.m3.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> belong to the same shift-orbit if and only if their preimages in <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.2.p1.4.m4.1"><semantics id="S5.2.p1.4.m4.1a"><msup id="S5.2.p1.4.m4.1.1" xref="S5.2.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.2.p1.4.m4.1.1.2" xref="S5.2.p1.4.m4.1.1.2.cmml">𝒜</mi><mi id="S5.2.p1.4.m4.1.1.3" xref="S5.2.p1.4.m4.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S5.2.p1.4.m4.1b"><apply id="S5.2.p1.4.m4.1.1.cmml" xref="S5.2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.2.p1.4.m4.1.1.1.cmml" xref="S5.2.p1.4.m4.1.1">superscript</csymbol><ci id="S5.2.p1.4.m4.1.1.2.cmml" xref="S5.2.p1.4.m4.1.1.2">𝒜</ci><ci id="S5.2.p1.4.m4.1.1.3.cmml" xref="S5.2.p1.4.m4.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p1.4.m4.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p1.4.m4.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> belong to the same orbit.</p> </div> <div class="ltx_para" id="S5.3.p2"> <p class="ltx_p" id="S5.3.p2.6">Similarly, from the definition of <math alttext="\pi_{\ell}" class="ltx_Math" display="inline" id="S5.3.p2.1.m1.1"><semantics id="S5.3.p2.1.m1.1a"><msub id="S5.3.p2.1.m1.1.1" xref="S5.3.p2.1.m1.1.1.cmml"><mi id="S5.3.p2.1.m1.1.1.2" xref="S5.3.p2.1.m1.1.1.2.cmml">π</mi><mi id="S5.3.p2.1.m1.1.1.3" mathvariant="normal" xref="S5.3.p2.1.m1.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.3.p2.1.m1.1b"><apply id="S5.3.p2.1.m1.1.1.cmml" xref="S5.3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.3.p2.1.m1.1.1.1.cmml" xref="S5.3.p2.1.m1.1.1">subscript</csymbol><ci id="S5.3.p2.1.m1.1.1.2.cmml" xref="S5.3.p2.1.m1.1.1.2">𝜋</ci><ci id="S5.3.p2.1.m1.1.1.3.cmml" xref="S5.3.p2.1.m1.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p2.1.m1.1c">\pi_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p2.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> it follows directly that the image of any <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S5.3.p2.2.m2.1"><semantics id="S5.3.p2.2.m2.1a"><mrow id="S5.3.p2.2.m2.1.1" xref="S5.3.p2.2.m2.1.1.cmml"><mi id="S5.3.p2.2.m2.1.1.2" xref="S5.3.p2.2.m2.1.1.2.cmml">w</mi><mo id="S5.3.p2.2.m2.1.1.1" xref="S5.3.p2.2.m2.1.1.1.cmml">∈</mo><msup id="S5.3.p2.2.m2.1.1.3" xref="S5.3.p2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.3.p2.2.m2.1.1.3.2" xref="S5.3.p2.2.m2.1.1.3.2.cmml">𝒜</mi><mo id="S5.3.p2.2.m2.1.1.3.3" xref="S5.3.p2.2.m2.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.3.p2.2.m2.1b"><apply id="S5.3.p2.2.m2.1.1.cmml" xref="S5.3.p2.2.m2.1.1"><in id="S5.3.p2.2.m2.1.1.1.cmml" xref="S5.3.p2.2.m2.1.1.1"></in><ci id="S5.3.p2.2.m2.1.1.2.cmml" xref="S5.3.p2.2.m2.1.1.2">𝑤</ci><apply id="S5.3.p2.2.m2.1.1.3.cmml" xref="S5.3.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.3.p2.2.m2.1.1.3.1.cmml" xref="S5.3.p2.2.m2.1.1.3">superscript</csymbol><ci id="S5.3.p2.2.m2.1.1.3.2.cmml" xref="S5.3.p2.2.m2.1.1.3.2">𝒜</ci><times id="S5.3.p2.2.m2.1.1.3.3.cmml" xref="S5.3.p2.2.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p2.2.m2.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p2.2.m2.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is a proper power if and only <math alttext="w" class="ltx_Math" display="inline" id="S5.3.p2.3.m3.1"><semantics id="S5.3.p2.3.m3.1a"><mi id="S5.3.p2.3.m3.1.1" xref="S5.3.p2.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S5.3.p2.3.m3.1b"><ci id="S5.3.p2.3.m3.1.1.cmml" xref="S5.3.p2.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p2.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S5.3.p2.3.m3.1d">italic_w</annotation></semantics></math> itself is a proper power. This implies directly (see Definition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm10" title="Definition 2.10. ‣ 2.3.2. Shift-period preservation ‣ 2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.10</span></a>) that <math alttext="\pi_{\ell}" class="ltx_Math" display="inline" id="S5.3.p2.4.m4.1"><semantics id="S5.3.p2.4.m4.1a"><msub id="S5.3.p2.4.m4.1.1" xref="S5.3.p2.4.m4.1.1.cmml"><mi id="S5.3.p2.4.m4.1.1.2" xref="S5.3.p2.4.m4.1.1.2.cmml">π</mi><mi id="S5.3.p2.4.m4.1.1.3" mathvariant="normal" xref="S5.3.p2.4.m4.1.1.3.cmml">ℓ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.3.p2.4.m4.1b"><apply id="S5.3.p2.4.m4.1.1.cmml" xref="S5.3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S5.3.p2.4.m4.1.1.1.cmml" xref="S5.3.p2.4.m4.1.1">subscript</csymbol><ci id="S5.3.p2.4.m4.1.1.2.cmml" xref="S5.3.p2.4.m4.1.1.2">𝜋</ci><ci id="S5.3.p2.4.m4.1.1.3.cmml" xref="S5.3.p2.4.m4.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p2.4.m4.1c">\pi_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S5.3.p2.4.m4.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> is shift-period preserving in the full shift. <span class="ltx_text ltx_inline-block" id="S5.3.p2.5.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S5.3.p2.5.1.m1.1"><semantics id="S5.3.p2.5.1.m1.1a"><mo id="S5.3.p2.5.1.m1.1.1" xref="S5.3.p2.5.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S5.3.p2.5.1.m1.1b"><csymbol cd="latexml" id="S5.3.p2.5.1.m1.1.1.cmml" xref="S5.3.p2.5.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p2.5.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S5.3.p2.5.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S5.3.p2.6.m5.1"><semantics id="S5.3.p2.6.m5.1a"><mo id="S5.3.p2.6.m5.1.1" xref="S5.3.p2.6.m5.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S5.3.p2.6.m5.1b"><csymbol cd="latexml" id="S5.3.p2.6.m5.1.1.cmml" xref="S5.3.p2.6.m5.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.3.p2.6.m5.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S5.3.p2.6.m5.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_lem" id="S5.Thmthm4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm4.1.1.1">Lemma 5.4</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm4.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm4.p1"> <p class="ltx_p" id="S5.Thmthm4.p1.2"><span class="ltx_text ltx_font_italic" id="S5.Thmthm4.p1.2.2">For any subdivision morphism <math alttext="\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}" class="ltx_Math" display="inline" id="S5.Thmthm4.p1.1.1.m1.1"><semantics id="S5.Thmthm4.p1.1.1.m1.1a"><mrow id="S5.Thmthm4.p1.1.1.m1.1.1" xref="S5.Thmthm4.p1.1.1.m1.1.1.cmml"><msub id="S5.Thmthm4.p1.1.1.m1.1.1.2" xref="S5.Thmthm4.p1.1.1.m1.1.1.2.cmml"><mi id="S5.Thmthm4.p1.1.1.m1.1.1.2.2" xref="S5.Thmthm4.p1.1.1.m1.1.1.2.2.cmml">π</mi><mi id="S5.Thmthm4.p1.1.1.m1.1.1.2.3" mathvariant="normal" xref="S5.Thmthm4.p1.1.1.m1.1.1.2.3.cmml">ℓ</mi></msub><mo id="S5.Thmthm4.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm4.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm4.p1.1.1.m1.1.1.3" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm4.p1.1.1.m1.1.1.3.2" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm4.p1.1.1.m1.1.1.3.2.2" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm4.p1.1.1.m1.1.1.3.2.3" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm4.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.1.cmml">→</mo><msubsup id="S5.Thmthm4.p1.1.1.m1.1.1.3.3" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.2" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.2.cmml">𝒜</mi><mi id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.3" mathvariant="normal" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.3.cmml">ℓ</mi><mo id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.3" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm4.p1.1.1.m1.1b"><apply id="S5.Thmthm4.p1.1.1.m1.1.1.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1"><ci id="S5.Thmthm4.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.1">:</ci><apply id="S5.Thmthm4.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.1.1.m1.1.1.2.1.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.2">subscript</csymbol><ci id="S5.Thmthm4.p1.1.1.m1.1.1.2.2.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.2.2">𝜋</ci><ci id="S5.Thmthm4.p1.1.1.m1.1.1.2.3.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.2.3">ℓ</ci></apply><apply id="S5.Thmthm4.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3"><ci id="S5.Thmthm4.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm4.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm4.p1.1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm4.p1.1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3">superscript</csymbol><apply id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.1.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3">subscript</csymbol><ci id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.2.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.2">𝒜</ci><ci id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.3.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3.2.3">ℓ</ci></apply><times id="S5.Thmthm4.p1.1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm4.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm4.p1.1.1.m1.1c">\pi_{\ell}:\cal A^{*}\to\cal A_{\ell}^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm4.p1.1.1.m1.1d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the induced transfer map <math alttext="\pi_{\ell}M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal A_{\ell}^{\mathbb{Z}})\,% ,\,\,\mu\mapsto\mu_{\ell}" class="ltx_Math" display="inline" id="S5.Thmthm4.p1.2.2.m2.2"><semantics id="S5.Thmthm4.p1.2.2.m2.2a"><mrow id="S5.Thmthm4.p1.2.2.m2.2.2" xref="S5.Thmthm4.p1.2.2.m2.2.2.cmml"><mrow id="S5.Thmthm4.p1.2.2.m2.2.2.4" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.cmml"><msub id="S5.Thmthm4.p1.2.2.m2.2.2.4.2" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.2.cmml"><mi id="S5.Thmthm4.p1.2.2.m2.2.2.4.2.2" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.2.2.cmml">π</mi><mi id="S5.Thmthm4.p1.2.2.m2.2.2.4.2.3" mathvariant="normal" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.2.3.cmml">ℓ</mi></msub><mo id="S5.Thmthm4.p1.2.2.m2.2.2.4.1" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.1.cmml">⁢</mo><mi id="S5.Thmthm4.p1.2.2.m2.2.2.4.3" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.3.cmml">M</mi></mrow><mo id="S5.Thmthm4.p1.2.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm4.p1.2.2.m2.2.2.3.cmml">:</mo><mrow id="S5.Thmthm4.p1.2.2.m2.2.2.2.2" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.3.cmml"><mrow id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.cmml"><mrow id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.3" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.3.cmml">ℳ</mi><mo id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.2" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.2" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.2.cmml">𝒜</mi><mi id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.3" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.3.cmml">ℤ</mi></msup><mo id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.3.cmml">→</mo><mrow id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.3" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.3.cmml">ℳ</mi><mo id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.2" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.cmml"><mo id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.2" stretchy="false" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.cmml">(</mo><msubsup id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.2" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.2.cmml">𝒜</mi><mi id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.3" mathvariant="normal" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.3.cmml">ℓ</mi><mi id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.3" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.3.cmml">ℤ</mi></msubsup><mo id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.3" rspace="0.170em" stretchy="false" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.3" rspace="0.497em" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.3a.cmml">,</mo><mrow id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.cmml"><mi id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.2" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.2.cmml">μ</mi><mo id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.1" stretchy="false" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.1.cmml">↦</mo><msub id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.cmml"><mi id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.2" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.2.cmml">μ</mi><mi id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.3" mathvariant="normal" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.3.cmml">ℓ</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm4.p1.2.2.m2.2b"><apply id="S5.Thmthm4.p1.2.2.m2.2.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2"><ci id="S5.Thmthm4.p1.2.2.m2.2.2.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.3">:</ci><apply id="S5.Thmthm4.p1.2.2.m2.2.2.4.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.4"><times id="S5.Thmthm4.p1.2.2.m2.2.2.4.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.1"></times><apply id="S5.Thmthm4.p1.2.2.m2.2.2.4.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.2"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.2.2.m2.2.2.4.2.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.2">subscript</csymbol><ci id="S5.Thmthm4.p1.2.2.m2.2.2.4.2.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.2.2">𝜋</ci><ci id="S5.Thmthm4.p1.2.2.m2.2.2.4.2.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.2.3">ℓ</ci></apply><ci id="S5.Thmthm4.p1.2.2.m2.2.2.4.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.4.3">𝑀</ci></apply><apply id="S5.Thmthm4.p1.2.2.m2.2.2.2.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.2.2.m2.2.2.2.3a.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.3">formulae-sequence</csymbol><apply id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1"><ci id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.3">→</ci><apply id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1"><times id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.2"></times><ci id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.3">ℳ</ci><apply id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.2">𝒜</ci><ci id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.1.1.1.1.3">ℤ</ci></apply></apply><apply id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2"><times id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.2"></times><ci id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.3">ℳ</ci><apply id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1">superscript</csymbol><apply id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1">subscript</csymbol><ci id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.2">𝒜</ci><ci id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.2.3">ℓ</ci></apply><ci id="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.1.1.1.1.1.2.1.1.1.3">ℤ</ci></apply></apply></apply><apply id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2"><csymbol cd="latexml" id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.1">maps-to</csymbol><ci id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.2">𝜇</ci><apply id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.1.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3">subscript</csymbol><ci id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.2.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.2">𝜇</ci><ci id="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.3.cmml" xref="S5.Thmthm4.p1.2.2.m2.2.2.2.2.2.3.3">ℓ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm4.p1.2.2.m2.2c">\pi_{\ell}M:\cal M(\cal A^{\mathbb{Z}})\to\cal M(\cal A_{\ell}^{\mathbb{Z}})\,% ,\,\,\mu\mapsto\mu_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm4.p1.2.2.m2.2d">italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) → caligraphic_M ( caligraphic_A start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ) , italic_μ ↦ italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> on the measure cones is injective.</span></p> </div> </div> <div class="ltx_proof" id="S5.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.4.p1"> <p class="ltx_p" id="S5.4.p1.8">Two measures <math alttext="\mu,\mu^{\prime}\in\cal M(\cal A^{\mathbb{Z}})" class="ltx_Math" display="inline" id="S5.4.p1.1.m1.3"><semantics id="S5.4.p1.1.m1.3a"><mrow id="S5.4.p1.1.m1.3.3" xref="S5.4.p1.1.m1.3.3.cmml"><mrow id="S5.4.p1.1.m1.2.2.1.1" xref="S5.4.p1.1.m1.2.2.1.2.cmml"><mi id="S5.4.p1.1.m1.1.1" xref="S5.4.p1.1.m1.1.1.cmml">μ</mi><mo id="S5.4.p1.1.m1.2.2.1.1.2" xref="S5.4.p1.1.m1.2.2.1.2.cmml">,</mo><msup id="S5.4.p1.1.m1.2.2.1.1.1" xref="S5.4.p1.1.m1.2.2.1.1.1.cmml"><mi id="S5.4.p1.1.m1.2.2.1.1.1.2" xref="S5.4.p1.1.m1.2.2.1.1.1.2.cmml">μ</mi><mo id="S5.4.p1.1.m1.2.2.1.1.1.3" xref="S5.4.p1.1.m1.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S5.4.p1.1.m1.3.3.3" xref="S5.4.p1.1.m1.3.3.3.cmml">∈</mo><mrow id="S5.4.p1.1.m1.3.3.2" xref="S5.4.p1.1.m1.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p1.1.m1.3.3.2.3" xref="S5.4.p1.1.m1.3.3.2.3.cmml">ℳ</mi><mo id="S5.4.p1.1.m1.3.3.2.2" xref="S5.4.p1.1.m1.3.3.2.2.cmml">⁢</mo><mrow id="S5.4.p1.1.m1.3.3.2.1.1" xref="S5.4.p1.1.m1.3.3.2.1.1.1.cmml"><mo id="S5.4.p1.1.m1.3.3.2.1.1.2" stretchy="false" xref="S5.4.p1.1.m1.3.3.2.1.1.1.cmml">(</mo><msup id="S5.4.p1.1.m1.3.3.2.1.1.1" xref="S5.4.p1.1.m1.3.3.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p1.1.m1.3.3.2.1.1.1.2" xref="S5.4.p1.1.m1.3.3.2.1.1.1.2.cmml">𝒜</mi><mi id="S5.4.p1.1.m1.3.3.2.1.1.1.3" xref="S5.4.p1.1.m1.3.3.2.1.1.1.3.cmml">ℤ</mi></msup><mo id="S5.4.p1.1.m1.3.3.2.1.1.3" stretchy="false" xref="S5.4.p1.1.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p1.1.m1.3b"><apply id="S5.4.p1.1.m1.3.3.cmml" xref="S5.4.p1.1.m1.3.3"><in id="S5.4.p1.1.m1.3.3.3.cmml" xref="S5.4.p1.1.m1.3.3.3"></in><list id="S5.4.p1.1.m1.2.2.1.2.cmml" xref="S5.4.p1.1.m1.2.2.1.1"><ci id="S5.4.p1.1.m1.1.1.cmml" xref="S5.4.p1.1.m1.1.1">𝜇</ci><apply id="S5.4.p1.1.m1.2.2.1.1.1.cmml" xref="S5.4.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.4.p1.1.m1.2.2.1.1.1.1.cmml" xref="S5.4.p1.1.m1.2.2.1.1.1">superscript</csymbol><ci id="S5.4.p1.1.m1.2.2.1.1.1.2.cmml" xref="S5.4.p1.1.m1.2.2.1.1.1.2">𝜇</ci><ci id="S5.4.p1.1.m1.2.2.1.1.1.3.cmml" xref="S5.4.p1.1.m1.2.2.1.1.1.3">′</ci></apply></list><apply id="S5.4.p1.1.m1.3.3.2.cmml" xref="S5.4.p1.1.m1.3.3.2"><times id="S5.4.p1.1.m1.3.3.2.2.cmml" xref="S5.4.p1.1.m1.3.3.2.2"></times><ci id="S5.4.p1.1.m1.3.3.2.3.cmml" xref="S5.4.p1.1.m1.3.3.2.3">ℳ</ci><apply id="S5.4.p1.1.m1.3.3.2.1.1.1.cmml" xref="S5.4.p1.1.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="S5.4.p1.1.m1.3.3.2.1.1.1.1.cmml" xref="S5.4.p1.1.m1.3.3.2.1.1">superscript</csymbol><ci id="S5.4.p1.1.m1.3.3.2.1.1.1.2.cmml" xref="S5.4.p1.1.m1.3.3.2.1.1.1.2">𝒜</ci><ci id="S5.4.p1.1.m1.3.3.2.1.1.1.3.cmml" xref="S5.4.p1.1.m1.3.3.2.1.1.1.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p1.1.m1.3c">\mu,\mu^{\prime}\in\cal M(\cal A^{\mathbb{Z}})</annotation><annotation encoding="application/x-llamapun" id="S5.4.p1.1.m1.3d">italic_μ , italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_M ( caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT )</annotation></semantics></math> are distinct if and only there exists a word <math alttext="w\in\cal A^{*}" class="ltx_Math" display="inline" id="S5.4.p1.2.m2.1"><semantics id="S5.4.p1.2.m2.1a"><mrow id="S5.4.p1.2.m2.1.1" xref="S5.4.p1.2.m2.1.1.cmml"><mi id="S5.4.p1.2.m2.1.1.2" xref="S5.4.p1.2.m2.1.1.2.cmml">w</mi><mo id="S5.4.p1.2.m2.1.1.1" xref="S5.4.p1.2.m2.1.1.1.cmml">∈</mo><msup id="S5.4.p1.2.m2.1.1.3" xref="S5.4.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p1.2.m2.1.1.3.2" xref="S5.4.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mo id="S5.4.p1.2.m2.1.1.3.3" xref="S5.4.p1.2.m2.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p1.2.m2.1b"><apply id="S5.4.p1.2.m2.1.1.cmml" xref="S5.4.p1.2.m2.1.1"><in id="S5.4.p1.2.m2.1.1.1.cmml" xref="S5.4.p1.2.m2.1.1.1"></in><ci id="S5.4.p1.2.m2.1.1.2.cmml" xref="S5.4.p1.2.m2.1.1.2">𝑤</ci><apply id="S5.4.p1.2.m2.1.1.3.cmml" xref="S5.4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.4.p1.2.m2.1.1.3.1.cmml" xref="S5.4.p1.2.m2.1.1.3">superscript</csymbol><ci id="S5.4.p1.2.m2.1.1.3.2.cmml" xref="S5.4.p1.2.m2.1.1.3.2">𝒜</ci><times id="S5.4.p1.2.m2.1.1.3.3.cmml" xref="S5.4.p1.2.m2.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p1.2.m2.1c">w\in\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p1.2.m2.1d">italic_w ∈ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> where the associated weight functions satisfy <math alttext="\mu(w)\neq\mu^{\prime}(w)" class="ltx_Math" display="inline" id="S5.4.p1.3.m3.2"><semantics id="S5.4.p1.3.m3.2a"><mrow id="S5.4.p1.3.m3.2.3" xref="S5.4.p1.3.m3.2.3.cmml"><mrow id="S5.4.p1.3.m3.2.3.2" xref="S5.4.p1.3.m3.2.3.2.cmml"><mi id="S5.4.p1.3.m3.2.3.2.2" xref="S5.4.p1.3.m3.2.3.2.2.cmml">μ</mi><mo id="S5.4.p1.3.m3.2.3.2.1" xref="S5.4.p1.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S5.4.p1.3.m3.2.3.2.3.2" xref="S5.4.p1.3.m3.2.3.2.cmml"><mo id="S5.4.p1.3.m3.2.3.2.3.2.1" stretchy="false" xref="S5.4.p1.3.m3.2.3.2.cmml">(</mo><mi id="S5.4.p1.3.m3.1.1" xref="S5.4.p1.3.m3.1.1.cmml">w</mi><mo id="S5.4.p1.3.m3.2.3.2.3.2.2" stretchy="false" xref="S5.4.p1.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.4.p1.3.m3.2.3.1" xref="S5.4.p1.3.m3.2.3.1.cmml">≠</mo><mrow id="S5.4.p1.3.m3.2.3.3" xref="S5.4.p1.3.m3.2.3.3.cmml"><msup id="S5.4.p1.3.m3.2.3.3.2" xref="S5.4.p1.3.m3.2.3.3.2.cmml"><mi id="S5.4.p1.3.m3.2.3.3.2.2" xref="S5.4.p1.3.m3.2.3.3.2.2.cmml">μ</mi><mo id="S5.4.p1.3.m3.2.3.3.2.3" xref="S5.4.p1.3.m3.2.3.3.2.3.cmml">′</mo></msup><mo id="S5.4.p1.3.m3.2.3.3.1" xref="S5.4.p1.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S5.4.p1.3.m3.2.3.3.3.2" xref="S5.4.p1.3.m3.2.3.3.cmml"><mo id="S5.4.p1.3.m3.2.3.3.3.2.1" stretchy="false" xref="S5.4.p1.3.m3.2.3.3.cmml">(</mo><mi id="S5.4.p1.3.m3.2.2" xref="S5.4.p1.3.m3.2.2.cmml">w</mi><mo id="S5.4.p1.3.m3.2.3.3.3.2.2" stretchy="false" xref="S5.4.p1.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p1.3.m3.2b"><apply id="S5.4.p1.3.m3.2.3.cmml" xref="S5.4.p1.3.m3.2.3"><neq id="S5.4.p1.3.m3.2.3.1.cmml" xref="S5.4.p1.3.m3.2.3.1"></neq><apply id="S5.4.p1.3.m3.2.3.2.cmml" xref="S5.4.p1.3.m3.2.3.2"><times id="S5.4.p1.3.m3.2.3.2.1.cmml" xref="S5.4.p1.3.m3.2.3.2.1"></times><ci id="S5.4.p1.3.m3.2.3.2.2.cmml" xref="S5.4.p1.3.m3.2.3.2.2">𝜇</ci><ci id="S5.4.p1.3.m3.1.1.cmml" xref="S5.4.p1.3.m3.1.1">𝑤</ci></apply><apply id="S5.4.p1.3.m3.2.3.3.cmml" xref="S5.4.p1.3.m3.2.3.3"><times id="S5.4.p1.3.m3.2.3.3.1.cmml" xref="S5.4.p1.3.m3.2.3.3.1"></times><apply id="S5.4.p1.3.m3.2.3.3.2.cmml" xref="S5.4.p1.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S5.4.p1.3.m3.2.3.3.2.1.cmml" xref="S5.4.p1.3.m3.2.3.3.2">superscript</csymbol><ci id="S5.4.p1.3.m3.2.3.3.2.2.cmml" xref="S5.4.p1.3.m3.2.3.3.2.2">𝜇</ci><ci id="S5.4.p1.3.m3.2.3.3.2.3.cmml" xref="S5.4.p1.3.m3.2.3.3.2.3">′</ci></apply><ci id="S5.4.p1.3.m3.2.2.cmml" xref="S5.4.p1.3.m3.2.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p1.3.m3.2c">\mu(w)\neq\mu^{\prime}(w)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p1.3.m3.2d">italic_μ ( italic_w ) ≠ italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w )</annotation></semantics></math>. From the definition of the subdivision measure in Definition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm3" title="Definition 3.3. ‣ 3.1. Subdivision morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.3</span></a> we see directly that <math alttext="\mu_{\ell}(\pi_{\ell}(w))=\mu(w)" class="ltx_Math" display="inline" id="S5.4.p1.4.m4.3"><semantics id="S5.4.p1.4.m4.3a"><mrow id="S5.4.p1.4.m4.3.3" xref="S5.4.p1.4.m4.3.3.cmml"><mrow id="S5.4.p1.4.m4.3.3.1" xref="S5.4.p1.4.m4.3.3.1.cmml"><msub id="S5.4.p1.4.m4.3.3.1.3" xref="S5.4.p1.4.m4.3.3.1.3.cmml"><mi id="S5.4.p1.4.m4.3.3.1.3.2" xref="S5.4.p1.4.m4.3.3.1.3.2.cmml">μ</mi><mi id="S5.4.p1.4.m4.3.3.1.3.3" mathvariant="normal" xref="S5.4.p1.4.m4.3.3.1.3.3.cmml">ℓ</mi></msub><mo id="S5.4.p1.4.m4.3.3.1.2" xref="S5.4.p1.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S5.4.p1.4.m4.3.3.1.1.1" xref="S5.4.p1.4.m4.3.3.1.1.1.1.cmml"><mo id="S5.4.p1.4.m4.3.3.1.1.1.2" stretchy="false" xref="S5.4.p1.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.4.p1.4.m4.3.3.1.1.1.1" xref="S5.4.p1.4.m4.3.3.1.1.1.1.cmml"><msub id="S5.4.p1.4.m4.3.3.1.1.1.1.2" xref="S5.4.p1.4.m4.3.3.1.1.1.1.2.cmml"><mi id="S5.4.p1.4.m4.3.3.1.1.1.1.2.2" xref="S5.4.p1.4.m4.3.3.1.1.1.1.2.2.cmml">π</mi><mi id="S5.4.p1.4.m4.3.3.1.1.1.1.2.3" mathvariant="normal" xref="S5.4.p1.4.m4.3.3.1.1.1.1.2.3.cmml">ℓ</mi></msub><mo id="S5.4.p1.4.m4.3.3.1.1.1.1.1" xref="S5.4.p1.4.m4.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.4.p1.4.m4.3.3.1.1.1.1.3.2" xref="S5.4.p1.4.m4.3.3.1.1.1.1.cmml"><mo id="S5.4.p1.4.m4.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S5.4.p1.4.m4.3.3.1.1.1.1.cmml">(</mo><mi id="S5.4.p1.4.m4.1.1" xref="S5.4.p1.4.m4.1.1.cmml">w</mi><mo id="S5.4.p1.4.m4.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S5.4.p1.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.4.p1.4.m4.3.3.1.1.1.3" stretchy="false" xref="S5.4.p1.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.4.p1.4.m4.3.3.2" xref="S5.4.p1.4.m4.3.3.2.cmml">=</mo><mrow id="S5.4.p1.4.m4.3.3.3" xref="S5.4.p1.4.m4.3.3.3.cmml"><mi id="S5.4.p1.4.m4.3.3.3.2" xref="S5.4.p1.4.m4.3.3.3.2.cmml">μ</mi><mo id="S5.4.p1.4.m4.3.3.3.1" xref="S5.4.p1.4.m4.3.3.3.1.cmml">⁢</mo><mrow id="S5.4.p1.4.m4.3.3.3.3.2" xref="S5.4.p1.4.m4.3.3.3.cmml"><mo id="S5.4.p1.4.m4.3.3.3.3.2.1" stretchy="false" xref="S5.4.p1.4.m4.3.3.3.cmml">(</mo><mi id="S5.4.p1.4.m4.2.2" xref="S5.4.p1.4.m4.2.2.cmml">w</mi><mo id="S5.4.p1.4.m4.3.3.3.3.2.2" stretchy="false" xref="S5.4.p1.4.m4.3.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p1.4.m4.3b"><apply id="S5.4.p1.4.m4.3.3.cmml" xref="S5.4.p1.4.m4.3.3"><eq id="S5.4.p1.4.m4.3.3.2.cmml" xref="S5.4.p1.4.m4.3.3.2"></eq><apply id="S5.4.p1.4.m4.3.3.1.cmml" xref="S5.4.p1.4.m4.3.3.1"><times id="S5.4.p1.4.m4.3.3.1.2.cmml" xref="S5.4.p1.4.m4.3.3.1.2"></times><apply id="S5.4.p1.4.m4.3.3.1.3.cmml" xref="S5.4.p1.4.m4.3.3.1.3"><csymbol cd="ambiguous" id="S5.4.p1.4.m4.3.3.1.3.1.cmml" xref="S5.4.p1.4.m4.3.3.1.3">subscript</csymbol><ci id="S5.4.p1.4.m4.3.3.1.3.2.cmml" xref="S5.4.p1.4.m4.3.3.1.3.2">𝜇</ci><ci id="S5.4.p1.4.m4.3.3.1.3.3.cmml" xref="S5.4.p1.4.m4.3.3.1.3.3">ℓ</ci></apply><apply id="S5.4.p1.4.m4.3.3.1.1.1.1.cmml" xref="S5.4.p1.4.m4.3.3.1.1.1"><times id="S5.4.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="S5.4.p1.4.m4.3.3.1.1.1.1.1"></times><apply id="S5.4.p1.4.m4.3.3.1.1.1.1.2.cmml" xref="S5.4.p1.4.m4.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.4.p1.4.m4.3.3.1.1.1.1.2.1.cmml" xref="S5.4.p1.4.m4.3.3.1.1.1.1.2">subscript</csymbol><ci id="S5.4.p1.4.m4.3.3.1.1.1.1.2.2.cmml" xref="S5.4.p1.4.m4.3.3.1.1.1.1.2.2">𝜋</ci><ci id="S5.4.p1.4.m4.3.3.1.1.1.1.2.3.cmml" xref="S5.4.p1.4.m4.3.3.1.1.1.1.2.3">ℓ</ci></apply><ci id="S5.4.p1.4.m4.1.1.cmml" xref="S5.4.p1.4.m4.1.1">𝑤</ci></apply></apply><apply id="S5.4.p1.4.m4.3.3.3.cmml" xref="S5.4.p1.4.m4.3.3.3"><times id="S5.4.p1.4.m4.3.3.3.1.cmml" xref="S5.4.p1.4.m4.3.3.3.1"></times><ci id="S5.4.p1.4.m4.3.3.3.2.cmml" xref="S5.4.p1.4.m4.3.3.3.2">𝜇</ci><ci id="S5.4.p1.4.m4.2.2.cmml" xref="S5.4.p1.4.m4.2.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p1.4.m4.3c">\mu_{\ell}(\pi_{\ell}(w))=\mu(w)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p1.4.m4.3d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_w ) ) = italic_μ ( italic_w )</annotation></semantics></math> and <math alttext="\mu^{\prime}_{\ell}(\pi_{\ell}(w))=\mu^{\prime}(w)" class="ltx_Math" display="inline" id="S5.4.p1.5.m5.3"><semantics id="S5.4.p1.5.m5.3a"><mrow id="S5.4.p1.5.m5.3.3" xref="S5.4.p1.5.m5.3.3.cmml"><mrow id="S5.4.p1.5.m5.3.3.1" xref="S5.4.p1.5.m5.3.3.1.cmml"><msubsup id="S5.4.p1.5.m5.3.3.1.3" xref="S5.4.p1.5.m5.3.3.1.3.cmml"><mi id="S5.4.p1.5.m5.3.3.1.3.2.2" xref="S5.4.p1.5.m5.3.3.1.3.2.2.cmml">μ</mi><mi id="S5.4.p1.5.m5.3.3.1.3.3" mathvariant="normal" xref="S5.4.p1.5.m5.3.3.1.3.3.cmml">ℓ</mi><mo id="S5.4.p1.5.m5.3.3.1.3.2.3" xref="S5.4.p1.5.m5.3.3.1.3.2.3.cmml">′</mo></msubsup><mo id="S5.4.p1.5.m5.3.3.1.2" xref="S5.4.p1.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="S5.4.p1.5.m5.3.3.1.1.1" xref="S5.4.p1.5.m5.3.3.1.1.1.1.cmml"><mo id="S5.4.p1.5.m5.3.3.1.1.1.2" stretchy="false" xref="S5.4.p1.5.m5.3.3.1.1.1.1.cmml">(</mo><mrow id="S5.4.p1.5.m5.3.3.1.1.1.1" xref="S5.4.p1.5.m5.3.3.1.1.1.1.cmml"><msub id="S5.4.p1.5.m5.3.3.1.1.1.1.2" xref="S5.4.p1.5.m5.3.3.1.1.1.1.2.cmml"><mi id="S5.4.p1.5.m5.3.3.1.1.1.1.2.2" xref="S5.4.p1.5.m5.3.3.1.1.1.1.2.2.cmml">π</mi><mi id="S5.4.p1.5.m5.3.3.1.1.1.1.2.3" mathvariant="normal" xref="S5.4.p1.5.m5.3.3.1.1.1.1.2.3.cmml">ℓ</mi></msub><mo id="S5.4.p1.5.m5.3.3.1.1.1.1.1" xref="S5.4.p1.5.m5.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.4.p1.5.m5.3.3.1.1.1.1.3.2" xref="S5.4.p1.5.m5.3.3.1.1.1.1.cmml"><mo id="S5.4.p1.5.m5.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S5.4.p1.5.m5.3.3.1.1.1.1.cmml">(</mo><mi id="S5.4.p1.5.m5.1.1" xref="S5.4.p1.5.m5.1.1.cmml">w</mi><mo id="S5.4.p1.5.m5.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S5.4.p1.5.m5.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.4.p1.5.m5.3.3.1.1.1.3" stretchy="false" xref="S5.4.p1.5.m5.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.4.p1.5.m5.3.3.2" xref="S5.4.p1.5.m5.3.3.2.cmml">=</mo><mrow id="S5.4.p1.5.m5.3.3.3" xref="S5.4.p1.5.m5.3.3.3.cmml"><msup id="S5.4.p1.5.m5.3.3.3.2" xref="S5.4.p1.5.m5.3.3.3.2.cmml"><mi id="S5.4.p1.5.m5.3.3.3.2.2" xref="S5.4.p1.5.m5.3.3.3.2.2.cmml">μ</mi><mo id="S5.4.p1.5.m5.3.3.3.2.3" xref="S5.4.p1.5.m5.3.3.3.2.3.cmml">′</mo></msup><mo id="S5.4.p1.5.m5.3.3.3.1" xref="S5.4.p1.5.m5.3.3.3.1.cmml">⁢</mo><mrow id="S5.4.p1.5.m5.3.3.3.3.2" xref="S5.4.p1.5.m5.3.3.3.cmml"><mo id="S5.4.p1.5.m5.3.3.3.3.2.1" stretchy="false" xref="S5.4.p1.5.m5.3.3.3.cmml">(</mo><mi id="S5.4.p1.5.m5.2.2" xref="S5.4.p1.5.m5.2.2.cmml">w</mi><mo id="S5.4.p1.5.m5.3.3.3.3.2.2" stretchy="false" xref="S5.4.p1.5.m5.3.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p1.5.m5.3b"><apply id="S5.4.p1.5.m5.3.3.cmml" xref="S5.4.p1.5.m5.3.3"><eq id="S5.4.p1.5.m5.3.3.2.cmml" xref="S5.4.p1.5.m5.3.3.2"></eq><apply id="S5.4.p1.5.m5.3.3.1.cmml" xref="S5.4.p1.5.m5.3.3.1"><times id="S5.4.p1.5.m5.3.3.1.2.cmml" xref="S5.4.p1.5.m5.3.3.1.2"></times><apply id="S5.4.p1.5.m5.3.3.1.3.cmml" xref="S5.4.p1.5.m5.3.3.1.3"><csymbol cd="ambiguous" id="S5.4.p1.5.m5.3.3.1.3.1.cmml" xref="S5.4.p1.5.m5.3.3.1.3">subscript</csymbol><apply id="S5.4.p1.5.m5.3.3.1.3.2.cmml" xref="S5.4.p1.5.m5.3.3.1.3"><csymbol cd="ambiguous" id="S5.4.p1.5.m5.3.3.1.3.2.1.cmml" xref="S5.4.p1.5.m5.3.3.1.3">superscript</csymbol><ci id="S5.4.p1.5.m5.3.3.1.3.2.2.cmml" xref="S5.4.p1.5.m5.3.3.1.3.2.2">𝜇</ci><ci id="S5.4.p1.5.m5.3.3.1.3.2.3.cmml" xref="S5.4.p1.5.m5.3.3.1.3.2.3">′</ci></apply><ci id="S5.4.p1.5.m5.3.3.1.3.3.cmml" xref="S5.4.p1.5.m5.3.3.1.3.3">ℓ</ci></apply><apply id="S5.4.p1.5.m5.3.3.1.1.1.1.cmml" xref="S5.4.p1.5.m5.3.3.1.1.1"><times id="S5.4.p1.5.m5.3.3.1.1.1.1.1.cmml" xref="S5.4.p1.5.m5.3.3.1.1.1.1.1"></times><apply id="S5.4.p1.5.m5.3.3.1.1.1.1.2.cmml" xref="S5.4.p1.5.m5.3.3.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.4.p1.5.m5.3.3.1.1.1.1.2.1.cmml" xref="S5.4.p1.5.m5.3.3.1.1.1.1.2">subscript</csymbol><ci id="S5.4.p1.5.m5.3.3.1.1.1.1.2.2.cmml" xref="S5.4.p1.5.m5.3.3.1.1.1.1.2.2">𝜋</ci><ci id="S5.4.p1.5.m5.3.3.1.1.1.1.2.3.cmml" xref="S5.4.p1.5.m5.3.3.1.1.1.1.2.3">ℓ</ci></apply><ci id="S5.4.p1.5.m5.1.1.cmml" xref="S5.4.p1.5.m5.1.1">𝑤</ci></apply></apply><apply id="S5.4.p1.5.m5.3.3.3.cmml" xref="S5.4.p1.5.m5.3.3.3"><times id="S5.4.p1.5.m5.3.3.3.1.cmml" xref="S5.4.p1.5.m5.3.3.3.1"></times><apply id="S5.4.p1.5.m5.3.3.3.2.cmml" xref="S5.4.p1.5.m5.3.3.3.2"><csymbol cd="ambiguous" id="S5.4.p1.5.m5.3.3.3.2.1.cmml" xref="S5.4.p1.5.m5.3.3.3.2">superscript</csymbol><ci id="S5.4.p1.5.m5.3.3.3.2.2.cmml" xref="S5.4.p1.5.m5.3.3.3.2.2">𝜇</ci><ci id="S5.4.p1.5.m5.3.3.3.2.3.cmml" xref="S5.4.p1.5.m5.3.3.3.2.3">′</ci></apply><ci id="S5.4.p1.5.m5.2.2.cmml" xref="S5.4.p1.5.m5.2.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p1.5.m5.3c">\mu^{\prime}_{\ell}(\pi_{\ell}(w))=\mu^{\prime}(w)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p1.5.m5.3d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_π start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( italic_w ) ) = italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_w )</annotation></semantics></math>. It follows that <math alttext="\mu_{\ell}\neq\mu^{\prime}_{\ell}" class="ltx_Math" display="inline" id="S5.4.p1.6.m6.1"><semantics id="S5.4.p1.6.m6.1a"><mrow id="S5.4.p1.6.m6.1.1" xref="S5.4.p1.6.m6.1.1.cmml"><msub id="S5.4.p1.6.m6.1.1.2" xref="S5.4.p1.6.m6.1.1.2.cmml"><mi id="S5.4.p1.6.m6.1.1.2.2" xref="S5.4.p1.6.m6.1.1.2.2.cmml">μ</mi><mi id="S5.4.p1.6.m6.1.1.2.3" mathvariant="normal" xref="S5.4.p1.6.m6.1.1.2.3.cmml">ℓ</mi></msub><mo id="S5.4.p1.6.m6.1.1.1" xref="S5.4.p1.6.m6.1.1.1.cmml">≠</mo><msubsup id="S5.4.p1.6.m6.1.1.3" xref="S5.4.p1.6.m6.1.1.3.cmml"><mi id="S5.4.p1.6.m6.1.1.3.2.2" xref="S5.4.p1.6.m6.1.1.3.2.2.cmml">μ</mi><mi id="S5.4.p1.6.m6.1.1.3.3" mathvariant="normal" xref="S5.4.p1.6.m6.1.1.3.3.cmml">ℓ</mi><mo id="S5.4.p1.6.m6.1.1.3.2.3" xref="S5.4.p1.6.m6.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p1.6.m6.1b"><apply id="S5.4.p1.6.m6.1.1.cmml" xref="S5.4.p1.6.m6.1.1"><neq id="S5.4.p1.6.m6.1.1.1.cmml" xref="S5.4.p1.6.m6.1.1.1"></neq><apply id="S5.4.p1.6.m6.1.1.2.cmml" xref="S5.4.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S5.4.p1.6.m6.1.1.2.1.cmml" xref="S5.4.p1.6.m6.1.1.2">subscript</csymbol><ci id="S5.4.p1.6.m6.1.1.2.2.cmml" xref="S5.4.p1.6.m6.1.1.2.2">𝜇</ci><ci id="S5.4.p1.6.m6.1.1.2.3.cmml" xref="S5.4.p1.6.m6.1.1.2.3">ℓ</ci></apply><apply id="S5.4.p1.6.m6.1.1.3.cmml" xref="S5.4.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.4.p1.6.m6.1.1.3.1.cmml" xref="S5.4.p1.6.m6.1.1.3">subscript</csymbol><apply id="S5.4.p1.6.m6.1.1.3.2.cmml" xref="S5.4.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S5.4.p1.6.m6.1.1.3.2.1.cmml" xref="S5.4.p1.6.m6.1.1.3">superscript</csymbol><ci id="S5.4.p1.6.m6.1.1.3.2.2.cmml" xref="S5.4.p1.6.m6.1.1.3.2.2">𝜇</ci><ci id="S5.4.p1.6.m6.1.1.3.2.3.cmml" xref="S5.4.p1.6.m6.1.1.3.2.3">′</ci></apply><ci id="S5.4.p1.6.m6.1.1.3.3.cmml" xref="S5.4.p1.6.m6.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p1.6.m6.1c">\mu_{\ell}\neq\mu^{\prime}_{\ell}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p1.6.m6.1d">italic_μ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ≠ italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S5.4.p1.7.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S5.4.p1.7.1.m1.1"><semantics id="S5.4.p1.7.1.m1.1a"><mo id="S5.4.p1.7.1.m1.1.1" xref="S5.4.p1.7.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S5.4.p1.7.1.m1.1b"><csymbol cd="latexml" id="S5.4.p1.7.1.m1.1.1.cmml" xref="S5.4.p1.7.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p1.7.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S5.4.p1.7.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S5.4.p1.8.m7.1"><semantics id="S5.4.p1.8.m7.1a"><mo id="S5.4.p1.8.m7.1.1" xref="S5.4.p1.8.m7.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S5.4.p1.8.m7.1b"><csymbol cd="latexml" id="S5.4.p1.8.m7.1.1.cmml" xref="S5.4.p1.8.m7.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p1.8.m7.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S5.4.p1.8.m7.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1">We can now observe:</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="S5.Thmthm5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm5.1.1.1">Lemma 5.5</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm5.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm5.p1"> <p class="ltx_p" id="S5.Thmthm5.p1.5"><span class="ltx_text ltx_font_italic" id="S5.Thmthm5.p1.5.5">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.Thmthm5.p1.1.1.m1.1"><semantics id="S5.Thmthm5.p1.1.1.m1.1a"><mrow id="S5.Thmthm5.p1.1.1.m1.1.1" xref="S5.Thmthm5.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmthm5.p1.1.1.m1.1.1.2" xref="S5.Thmthm5.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S5.Thmthm5.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm5.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm5.p1.1.1.m1.1.1.3" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm5.p1.1.1.m1.1.1.3.2" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm5.p1.1.1.m1.1.1.3.2.2" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm5.p1.1.1.m1.1.1.3.2.3" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm5.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm5.p1.1.1.m1.1.1.3.3" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm5.p1.1.1.m1.1.1.3.3.2" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.Thmthm5.p1.1.1.m1.1.1.3.3.3" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm5.p1.1.1.m1.1b"><apply id="S5.Thmthm5.p1.1.1.m1.1.1.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1"><ci id="S5.Thmthm5.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.1">:</ci><ci id="S5.Thmthm5.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S5.Thmthm5.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3"><ci id="S5.Thmthm5.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm5.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm5.p1.1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm5.p1.1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm5.p1.1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm5.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm5.p1.1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm5.p1.1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S5.Thmthm5.p1.1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm5.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm5.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm5.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-erasing monoid morphism, let <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm5.p1.2.2.m2.1"><semantics id="S5.Thmthm5.p1.2.2.m2.1a"><mrow id="S5.Thmthm5.p1.2.2.m2.1.1" xref="S5.Thmthm5.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmthm5.p1.2.2.m2.1.1.2" xref="S5.Thmthm5.p1.2.2.m2.1.1.2.cmml">X</mi><mo id="S5.Thmthm5.p1.2.2.m2.1.1.1" xref="S5.Thmthm5.p1.2.2.m2.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm5.p1.2.2.m2.1.1.3" xref="S5.Thmthm5.p1.2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm5.p1.2.2.m2.1.1.3.2" xref="S5.Thmthm5.p1.2.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm5.p1.2.2.m2.1.1.3.3" xref="S5.Thmthm5.p1.2.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm5.p1.2.2.m2.1b"><apply id="S5.Thmthm5.p1.2.2.m2.1.1.cmml" xref="S5.Thmthm5.p1.2.2.m2.1.1"><subset id="S5.Thmthm5.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmthm5.p1.2.2.m2.1.1.1"></subset><ci id="S5.Thmthm5.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmthm5.p1.2.2.m2.1.1.2">𝑋</ci><apply id="S5.Thmthm5.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmthm5.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm5.p1.2.2.m2.1.1.3.1.cmml" xref="S5.Thmthm5.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S5.Thmthm5.p1.2.2.m2.1.1.3.2.cmml" xref="S5.Thmthm5.p1.2.2.m2.1.1.3.2">𝒜</ci><ci id="S5.Thmthm5.p1.2.2.m2.1.1.3.3.cmml" xref="S5.Thmthm5.p1.2.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm5.p1.2.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm5.p1.2.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> be any subshift, and let <math alttext="\mu\in\cal M(X)" class="ltx_Math" display="inline" id="S5.Thmthm5.p1.3.3.m3.1"><semantics id="S5.Thmthm5.p1.3.3.m3.1a"><mrow id="S5.Thmthm5.p1.3.3.m3.1.2" xref="S5.Thmthm5.p1.3.3.m3.1.2.cmml"><mi id="S5.Thmthm5.p1.3.3.m3.1.2.2" xref="S5.Thmthm5.p1.3.3.m3.1.2.2.cmml">μ</mi><mo id="S5.Thmthm5.p1.3.3.m3.1.2.1" xref="S5.Thmthm5.p1.3.3.m3.1.2.1.cmml">∈</mo><mrow id="S5.Thmthm5.p1.3.3.m3.1.2.3" xref="S5.Thmthm5.p1.3.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm5.p1.3.3.m3.1.2.3.2" xref="S5.Thmthm5.p1.3.3.m3.1.2.3.2.cmml">ℳ</mi><mo id="S5.Thmthm5.p1.3.3.m3.1.2.3.1" xref="S5.Thmthm5.p1.3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S5.Thmthm5.p1.3.3.m3.1.2.3.3.2" xref="S5.Thmthm5.p1.3.3.m3.1.2.3.cmml"><mo id="S5.Thmthm5.p1.3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S5.Thmthm5.p1.3.3.m3.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm5.p1.3.3.m3.1.1" xref="S5.Thmthm5.p1.3.3.m3.1.1.cmml">𝒳</mi><mo id="S5.Thmthm5.p1.3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S5.Thmthm5.p1.3.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm5.p1.3.3.m3.1b"><apply id="S5.Thmthm5.p1.3.3.m3.1.2.cmml" xref="S5.Thmthm5.p1.3.3.m3.1.2"><in id="S5.Thmthm5.p1.3.3.m3.1.2.1.cmml" xref="S5.Thmthm5.p1.3.3.m3.1.2.1"></in><ci id="S5.Thmthm5.p1.3.3.m3.1.2.2.cmml" xref="S5.Thmthm5.p1.3.3.m3.1.2.2">𝜇</ci><apply id="S5.Thmthm5.p1.3.3.m3.1.2.3.cmml" xref="S5.Thmthm5.p1.3.3.m3.1.2.3"><times id="S5.Thmthm5.p1.3.3.m3.1.2.3.1.cmml" xref="S5.Thmthm5.p1.3.3.m3.1.2.3.1"></times><ci id="S5.Thmthm5.p1.3.3.m3.1.2.3.2.cmml" xref="S5.Thmthm5.p1.3.3.m3.1.2.3.2">ℳ</ci><ci id="S5.Thmthm5.p1.3.3.m3.1.1.cmml" xref="S5.Thmthm5.p1.3.3.m3.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm5.p1.3.3.m3.1c">\mu\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm5.p1.3.3.m3.1d">italic_μ ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math> be an ergodic measure. Then <math alttext="\mu^{\sigma}" class="ltx_Math" display="inline" id="S5.Thmthm5.p1.4.4.m4.1"><semantics id="S5.Thmthm5.p1.4.4.m4.1a"><msup id="S5.Thmthm5.p1.4.4.m4.1.1" xref="S5.Thmthm5.p1.4.4.m4.1.1.cmml"><mi id="S5.Thmthm5.p1.4.4.m4.1.1.2" xref="S5.Thmthm5.p1.4.4.m4.1.1.2.cmml">μ</mi><mi id="S5.Thmthm5.p1.4.4.m4.1.1.3" xref="S5.Thmthm5.p1.4.4.m4.1.1.3.cmml">σ</mi></msup><annotation-xml encoding="MathML-Content" id="S5.Thmthm5.p1.4.4.m4.1b"><apply id="S5.Thmthm5.p1.4.4.m4.1.1.cmml" xref="S5.Thmthm5.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="S5.Thmthm5.p1.4.4.m4.1.1.1.cmml" xref="S5.Thmthm5.p1.4.4.m4.1.1">superscript</csymbol><ci id="S5.Thmthm5.p1.4.4.m4.1.1.2.cmml" xref="S5.Thmthm5.p1.4.4.m4.1.1.2">𝜇</ci><ci id="S5.Thmthm5.p1.4.4.m4.1.1.3.cmml" xref="S5.Thmthm5.p1.4.4.m4.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm5.p1.4.4.m4.1c">\mu^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm5.p1.4.4.m4.1d">italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math> is an ergodic measure on the image subshift <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S5.Thmthm5.p1.5.5.m5.1"><semantics id="S5.Thmthm5.p1.5.5.m5.1a"><mrow id="S5.Thmthm5.p1.5.5.m5.1.2" xref="S5.Thmthm5.p1.5.5.m5.1.2.cmml"><mi id="S5.Thmthm5.p1.5.5.m5.1.2.2" xref="S5.Thmthm5.p1.5.5.m5.1.2.2.cmml">σ</mi><mo id="S5.Thmthm5.p1.5.5.m5.1.2.1" xref="S5.Thmthm5.p1.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S5.Thmthm5.p1.5.5.m5.1.2.3.2" xref="S5.Thmthm5.p1.5.5.m5.1.2.cmml"><mo id="S5.Thmthm5.p1.5.5.m5.1.2.3.2.1" stretchy="false" xref="S5.Thmthm5.p1.5.5.m5.1.2.cmml">(</mo><mi id="S5.Thmthm5.p1.5.5.m5.1.1" xref="S5.Thmthm5.p1.5.5.m5.1.1.cmml">X</mi><mo id="S5.Thmthm5.p1.5.5.m5.1.2.3.2.2" stretchy="false" xref="S5.Thmthm5.p1.5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm5.p1.5.5.m5.1b"><apply id="S5.Thmthm5.p1.5.5.m5.1.2.cmml" xref="S5.Thmthm5.p1.5.5.m5.1.2"><times id="S5.Thmthm5.p1.5.5.m5.1.2.1.cmml" xref="S5.Thmthm5.p1.5.5.m5.1.2.1"></times><ci id="S5.Thmthm5.p1.5.5.m5.1.2.2.cmml" xref="S5.Thmthm5.p1.5.5.m5.1.2.2">𝜎</ci><ci id="S5.Thmthm5.p1.5.5.m5.1.1.cmml" xref="S5.Thmthm5.p1.5.5.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm5.p1.5.5.m5.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm5.p1.5.5.m5.1d">italic_σ ( italic_X )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.5.p1"> <p class="ltx_p" id="S5.5.p1.10">As before, we write <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.5.p1.1.m1.1"><semantics id="S5.5.p1.1.m1.1a"><mi id="S5.5.p1.1.m1.1.1" xref="S5.5.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.5.p1.1.m1.1b"><ci id="S5.5.p1.1.m1.1.1.cmml" xref="S5.5.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.1.m1.1d">italic_σ</annotation></semantics></math> as composition <math alttext="\sigma=\alpha_{\sigma}\circ\pi_{\sigma}" class="ltx_Math" display="inline" id="S5.5.p1.2.m2.1"><semantics id="S5.5.p1.2.m2.1a"><mrow id="S5.5.p1.2.m2.1.1" xref="S5.5.p1.2.m2.1.1.cmml"><mi id="S5.5.p1.2.m2.1.1.2" xref="S5.5.p1.2.m2.1.1.2.cmml">σ</mi><mo id="S5.5.p1.2.m2.1.1.1" xref="S5.5.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S5.5.p1.2.m2.1.1.3" xref="S5.5.p1.2.m2.1.1.3.cmml"><msub id="S5.5.p1.2.m2.1.1.3.2" xref="S5.5.p1.2.m2.1.1.3.2.cmml"><mi id="S5.5.p1.2.m2.1.1.3.2.2" xref="S5.5.p1.2.m2.1.1.3.2.2.cmml">α</mi><mi id="S5.5.p1.2.m2.1.1.3.2.3" xref="S5.5.p1.2.m2.1.1.3.2.3.cmml">σ</mi></msub><mo id="S5.5.p1.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S5.5.p1.2.m2.1.1.3.1.cmml">∘</mo><msub id="S5.5.p1.2.m2.1.1.3.3" xref="S5.5.p1.2.m2.1.1.3.3.cmml"><mi id="S5.5.p1.2.m2.1.1.3.3.2" xref="S5.5.p1.2.m2.1.1.3.3.2.cmml">π</mi><mi id="S5.5.p1.2.m2.1.1.3.3.3" xref="S5.5.p1.2.m2.1.1.3.3.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.2.m2.1b"><apply id="S5.5.p1.2.m2.1.1.cmml" xref="S5.5.p1.2.m2.1.1"><eq id="S5.5.p1.2.m2.1.1.1.cmml" xref="S5.5.p1.2.m2.1.1.1"></eq><ci id="S5.5.p1.2.m2.1.1.2.cmml" xref="S5.5.p1.2.m2.1.1.2">𝜎</ci><apply id="S5.5.p1.2.m2.1.1.3.cmml" xref="S5.5.p1.2.m2.1.1.3"><compose id="S5.5.p1.2.m2.1.1.3.1.cmml" xref="S5.5.p1.2.m2.1.1.3.1"></compose><apply id="S5.5.p1.2.m2.1.1.3.2.cmml" xref="S5.5.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.5.p1.2.m2.1.1.3.2.1.cmml" xref="S5.5.p1.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.5.p1.2.m2.1.1.3.2.2.cmml" xref="S5.5.p1.2.m2.1.1.3.2.2">𝛼</ci><ci id="S5.5.p1.2.m2.1.1.3.2.3.cmml" xref="S5.5.p1.2.m2.1.1.3.2.3">𝜎</ci></apply><apply id="S5.5.p1.2.m2.1.1.3.3.cmml" xref="S5.5.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.5.p1.2.m2.1.1.3.3.1.cmml" xref="S5.5.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.5.p1.2.m2.1.1.3.3.2.cmml" xref="S5.5.p1.2.m2.1.1.3.3.2">𝜋</ci><ci id="S5.5.p1.2.m2.1.1.3.3.3.cmml" xref="S5.5.p1.2.m2.1.1.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.2.m2.1c">\sigma=\alpha_{\sigma}\circ\pi_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.2.m2.1d">italic_σ = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∘ italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> of the subdivision morphism <math alttext="\pi_{\sigma}" class="ltx_Math" display="inline" id="S5.5.p1.3.m3.1"><semantics id="S5.5.p1.3.m3.1a"><msub id="S5.5.p1.3.m3.1.1" xref="S5.5.p1.3.m3.1.1.cmml"><mi id="S5.5.p1.3.m3.1.1.2" xref="S5.5.p1.3.m3.1.1.2.cmml">π</mi><mi id="S5.5.p1.3.m3.1.1.3" xref="S5.5.p1.3.m3.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.5.p1.3.m3.1b"><apply id="S5.5.p1.3.m3.1.1.cmml" xref="S5.5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.5.p1.3.m3.1.1.1.cmml" xref="S5.5.p1.3.m3.1.1">subscript</csymbol><ci id="S5.5.p1.3.m3.1.1.2.cmml" xref="S5.5.p1.3.m3.1.1.2">𝜋</ci><ci id="S5.5.p1.3.m3.1.1.3.cmml" xref="S5.5.p1.3.m3.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.3.m3.1c">\pi_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.3.m3.1d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> and the letter-to-letter morphism <math alttext="\alpha_{\sigma}\," class="ltx_Math" display="inline" id="S5.5.p1.4.m4.1"><semantics id="S5.5.p1.4.m4.1a"><msub id="S5.5.p1.4.m4.1.1" xref="S5.5.p1.4.m4.1.1.cmml"><mi id="S5.5.p1.4.m4.1.1.2" xref="S5.5.p1.4.m4.1.1.2.cmml">α</mi><mi id="S5.5.p1.4.m4.1.1.3" xref="S5.5.p1.4.m4.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.5.p1.4.m4.1b"><apply id="S5.5.p1.4.m4.1.1.cmml" xref="S5.5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.5.p1.4.m4.1.1.1.cmml" xref="S5.5.p1.4.m4.1.1">subscript</csymbol><ci id="S5.5.p1.4.m4.1.1.2.cmml" xref="S5.5.p1.4.m4.1.1.2">𝛼</ci><ci id="S5.5.p1.4.m4.1.1.3.cmml" xref="S5.5.p1.4.m4.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.4.m4.1c">\alpha_{\sigma}\,</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.4.m4.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. From Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm4" title="Lemma 5.4. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.4</span></a> we know that the induced map <math alttext="\pi_{\sigma}M:\cal M(X)\to\cal M(\pi_{\sigma}(X))" class="ltx_Math" display="inline" id="S5.5.p1.5.m5.3"><semantics id="S5.5.p1.5.m5.3a"><mrow id="S5.5.p1.5.m5.3.3" xref="S5.5.p1.5.m5.3.3.cmml"><mrow id="S5.5.p1.5.m5.3.3.3" xref="S5.5.p1.5.m5.3.3.3.cmml"><msub id="S5.5.p1.5.m5.3.3.3.2" xref="S5.5.p1.5.m5.3.3.3.2.cmml"><mi id="S5.5.p1.5.m5.3.3.3.2.2" xref="S5.5.p1.5.m5.3.3.3.2.2.cmml">π</mi><mi id="S5.5.p1.5.m5.3.3.3.2.3" xref="S5.5.p1.5.m5.3.3.3.2.3.cmml">σ</mi></msub><mo id="S5.5.p1.5.m5.3.3.3.1" xref="S5.5.p1.5.m5.3.3.3.1.cmml">⁢</mo><mi id="S5.5.p1.5.m5.3.3.3.3" xref="S5.5.p1.5.m5.3.3.3.3.cmml">M</mi></mrow><mo id="S5.5.p1.5.m5.3.3.2" lspace="0.278em" rspace="0.278em" xref="S5.5.p1.5.m5.3.3.2.cmml">:</mo><mrow id="S5.5.p1.5.m5.3.3.1" xref="S5.5.p1.5.m5.3.3.1.cmml"><mrow id="S5.5.p1.5.m5.3.3.1.3" xref="S5.5.p1.5.m5.3.3.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.5.m5.3.3.1.3.2" xref="S5.5.p1.5.m5.3.3.1.3.2.cmml">ℳ</mi><mo id="S5.5.p1.5.m5.3.3.1.3.1" xref="S5.5.p1.5.m5.3.3.1.3.1.cmml">⁢</mo><mrow id="S5.5.p1.5.m5.3.3.1.3.3.2" xref="S5.5.p1.5.m5.3.3.1.3.cmml"><mo id="S5.5.p1.5.m5.3.3.1.3.3.2.1" stretchy="false" xref="S5.5.p1.5.m5.3.3.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.5.m5.1.1" xref="S5.5.p1.5.m5.1.1.cmml">𝒳</mi><mo id="S5.5.p1.5.m5.3.3.1.3.3.2.2" stretchy="false" xref="S5.5.p1.5.m5.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="S5.5.p1.5.m5.3.3.1.2" stretchy="false" xref="S5.5.p1.5.m5.3.3.1.2.cmml">→</mo><mrow id="S5.5.p1.5.m5.3.3.1.1" xref="S5.5.p1.5.m5.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.5.m5.3.3.1.1.3" xref="S5.5.p1.5.m5.3.3.1.1.3.cmml">ℳ</mi><mo id="S5.5.p1.5.m5.3.3.1.1.2" xref="S5.5.p1.5.m5.3.3.1.1.2.cmml">⁢</mo><mrow id="S5.5.p1.5.m5.3.3.1.1.1.1" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.cmml"><mo id="S5.5.p1.5.m5.3.3.1.1.1.1.2" stretchy="false" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.5.p1.5.m5.3.3.1.1.1.1.1" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.cmml"><msub id="S5.5.p1.5.m5.3.3.1.1.1.1.1.2" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.cmml"><mi id="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.2" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.2.cmml">π</mi><mi id="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.3" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.3.cmml">σ</mi></msub><mo id="S5.5.p1.5.m5.3.3.1.1.1.1.1.1" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.5.p1.5.m5.3.3.1.1.1.1.1.3.2" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.cmml"><mo id="S5.5.p1.5.m5.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.5.m5.2.2" xref="S5.5.p1.5.m5.2.2.cmml">𝒳</mi><mo id="S5.5.p1.5.m5.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.5.p1.5.m5.3.3.1.1.1.1.3" stretchy="false" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.5.m5.3b"><apply id="S5.5.p1.5.m5.3.3.cmml" xref="S5.5.p1.5.m5.3.3"><ci id="S5.5.p1.5.m5.3.3.2.cmml" xref="S5.5.p1.5.m5.3.3.2">:</ci><apply id="S5.5.p1.5.m5.3.3.3.cmml" xref="S5.5.p1.5.m5.3.3.3"><times id="S5.5.p1.5.m5.3.3.3.1.cmml" xref="S5.5.p1.5.m5.3.3.3.1"></times><apply id="S5.5.p1.5.m5.3.3.3.2.cmml" xref="S5.5.p1.5.m5.3.3.3.2"><csymbol cd="ambiguous" id="S5.5.p1.5.m5.3.3.3.2.1.cmml" xref="S5.5.p1.5.m5.3.3.3.2">subscript</csymbol><ci id="S5.5.p1.5.m5.3.3.3.2.2.cmml" xref="S5.5.p1.5.m5.3.3.3.2.2">𝜋</ci><ci id="S5.5.p1.5.m5.3.3.3.2.3.cmml" xref="S5.5.p1.5.m5.3.3.3.2.3">𝜎</ci></apply><ci id="S5.5.p1.5.m5.3.3.3.3.cmml" xref="S5.5.p1.5.m5.3.3.3.3">𝑀</ci></apply><apply id="S5.5.p1.5.m5.3.3.1.cmml" xref="S5.5.p1.5.m5.3.3.1"><ci id="S5.5.p1.5.m5.3.3.1.2.cmml" xref="S5.5.p1.5.m5.3.3.1.2">→</ci><apply id="S5.5.p1.5.m5.3.3.1.3.cmml" xref="S5.5.p1.5.m5.3.3.1.3"><times id="S5.5.p1.5.m5.3.3.1.3.1.cmml" xref="S5.5.p1.5.m5.3.3.1.3.1"></times><ci id="S5.5.p1.5.m5.3.3.1.3.2.cmml" xref="S5.5.p1.5.m5.3.3.1.3.2">ℳ</ci><ci id="S5.5.p1.5.m5.1.1.cmml" xref="S5.5.p1.5.m5.1.1">𝒳</ci></apply><apply id="S5.5.p1.5.m5.3.3.1.1.cmml" xref="S5.5.p1.5.m5.3.3.1.1"><times id="S5.5.p1.5.m5.3.3.1.1.2.cmml" xref="S5.5.p1.5.m5.3.3.1.1.2"></times><ci id="S5.5.p1.5.m5.3.3.1.1.3.cmml" xref="S5.5.p1.5.m5.3.3.1.1.3">ℳ</ci><apply id="S5.5.p1.5.m5.3.3.1.1.1.1.1.cmml" xref="S5.5.p1.5.m5.3.3.1.1.1.1"><times id="S5.5.p1.5.m5.3.3.1.1.1.1.1.1.cmml" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.1"></times><apply id="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.cmml" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.1.cmml" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.2">subscript</csymbol><ci id="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.2.cmml" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.2">𝜋</ci><ci id="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.3.cmml" xref="S5.5.p1.5.m5.3.3.1.1.1.1.1.2.3">𝜎</ci></apply><ci id="S5.5.p1.5.m5.2.2.cmml" xref="S5.5.p1.5.m5.2.2">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.5.m5.3c">\pi_{\sigma}M:\cal M(X)\to\cal M(\pi_{\sigma}(X))</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.5.m5.3d">italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( caligraphic_X ) )</annotation></semantics></math> is injective and hence, by Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm7" title="Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.7</span></a> (a) and the surjectivity from Proposition 4.4 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>, an <math alttext="\mathbb{R}_{\geq 0}" class="ltx_Math" display="inline" id="S5.5.p1.6.m6.1"><semantics id="S5.5.p1.6.m6.1a"><msub id="S5.5.p1.6.m6.1.1" xref="S5.5.p1.6.m6.1.1.cmml"><mi id="S5.5.p1.6.m6.1.1.2" xref="S5.5.p1.6.m6.1.1.2.cmml">ℝ</mi><mrow id="S5.5.p1.6.m6.1.1.3" xref="S5.5.p1.6.m6.1.1.3.cmml"><mi id="S5.5.p1.6.m6.1.1.3.2" xref="S5.5.p1.6.m6.1.1.3.2.cmml"></mi><mo id="S5.5.p1.6.m6.1.1.3.1" xref="S5.5.p1.6.m6.1.1.3.1.cmml">≥</mo><mn id="S5.5.p1.6.m6.1.1.3.3" xref="S5.5.p1.6.m6.1.1.3.3.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.5.p1.6.m6.1b"><apply id="S5.5.p1.6.m6.1.1.cmml" xref="S5.5.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.5.p1.6.m6.1.1.1.cmml" xref="S5.5.p1.6.m6.1.1">subscript</csymbol><ci id="S5.5.p1.6.m6.1.1.2.cmml" xref="S5.5.p1.6.m6.1.1.2">ℝ</ci><apply id="S5.5.p1.6.m6.1.1.3.cmml" xref="S5.5.p1.6.m6.1.1.3"><geq id="S5.5.p1.6.m6.1.1.3.1.cmml" xref="S5.5.p1.6.m6.1.1.3.1"></geq><csymbol cd="latexml" id="S5.5.p1.6.m6.1.1.3.2.cmml" xref="S5.5.p1.6.m6.1.1.3.2">absent</csymbol><cn id="S5.5.p1.6.m6.1.1.3.3.cmml" type="integer" xref="S5.5.p1.6.m6.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.6.m6.1c">\mathbb{R}_{\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.6.m6.1d">blackboard_R start_POSTSUBSCRIPT ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math>-linear isomorphism of cones. It follows that any extremal point of the cone <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S5.5.p1.7.m7.1"><semantics id="S5.5.p1.7.m7.1a"><mrow id="S5.5.p1.7.m7.1.2" xref="S5.5.p1.7.m7.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.7.m7.1.2.2" xref="S5.5.p1.7.m7.1.2.2.cmml">ℳ</mi><mo id="S5.5.p1.7.m7.1.2.1" xref="S5.5.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S5.5.p1.7.m7.1.2.3.2" xref="S5.5.p1.7.m7.1.2.cmml"><mo id="S5.5.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S5.5.p1.7.m7.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.7.m7.1.1" xref="S5.5.p1.7.m7.1.1.cmml">𝒳</mi><mo id="S5.5.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S5.5.p1.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.7.m7.1b"><apply id="S5.5.p1.7.m7.1.2.cmml" xref="S5.5.p1.7.m7.1.2"><times id="S5.5.p1.7.m7.1.2.1.cmml" xref="S5.5.p1.7.m7.1.2.1"></times><ci id="S5.5.p1.7.m7.1.2.2.cmml" xref="S5.5.p1.7.m7.1.2.2">ℳ</ci><ci id="S5.5.p1.7.m7.1.1.cmml" xref="S5.5.p1.7.m7.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.7.m7.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.7.m7.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math> is mapped to an extremal point of the cone <math alttext="\cal M(\pi_{\sigma}(X))" class="ltx_Math" display="inline" id="S5.5.p1.8.m8.2"><semantics id="S5.5.p1.8.m8.2a"><mrow id="S5.5.p1.8.m8.2.2" xref="S5.5.p1.8.m8.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.8.m8.2.2.3" xref="S5.5.p1.8.m8.2.2.3.cmml">ℳ</mi><mo id="S5.5.p1.8.m8.2.2.2" xref="S5.5.p1.8.m8.2.2.2.cmml">⁢</mo><mrow id="S5.5.p1.8.m8.2.2.1.1" xref="S5.5.p1.8.m8.2.2.1.1.1.cmml"><mo id="S5.5.p1.8.m8.2.2.1.1.2" stretchy="false" xref="S5.5.p1.8.m8.2.2.1.1.1.cmml">(</mo><mrow id="S5.5.p1.8.m8.2.2.1.1.1" xref="S5.5.p1.8.m8.2.2.1.1.1.cmml"><msub id="S5.5.p1.8.m8.2.2.1.1.1.2" xref="S5.5.p1.8.m8.2.2.1.1.1.2.cmml"><mi id="S5.5.p1.8.m8.2.2.1.1.1.2.2" xref="S5.5.p1.8.m8.2.2.1.1.1.2.2.cmml">π</mi><mi id="S5.5.p1.8.m8.2.2.1.1.1.2.3" xref="S5.5.p1.8.m8.2.2.1.1.1.2.3.cmml">σ</mi></msub><mo id="S5.5.p1.8.m8.2.2.1.1.1.1" xref="S5.5.p1.8.m8.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S5.5.p1.8.m8.2.2.1.1.1.3.2" xref="S5.5.p1.8.m8.2.2.1.1.1.cmml"><mo id="S5.5.p1.8.m8.2.2.1.1.1.3.2.1" stretchy="false" xref="S5.5.p1.8.m8.2.2.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.8.m8.1.1" xref="S5.5.p1.8.m8.1.1.cmml">𝒳</mi><mo id="S5.5.p1.8.m8.2.2.1.1.1.3.2.2" stretchy="false" xref="S5.5.p1.8.m8.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.5.p1.8.m8.2.2.1.1.3" stretchy="false" xref="S5.5.p1.8.m8.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.8.m8.2b"><apply id="S5.5.p1.8.m8.2.2.cmml" xref="S5.5.p1.8.m8.2.2"><times id="S5.5.p1.8.m8.2.2.2.cmml" xref="S5.5.p1.8.m8.2.2.2"></times><ci id="S5.5.p1.8.m8.2.2.3.cmml" xref="S5.5.p1.8.m8.2.2.3">ℳ</ci><apply id="S5.5.p1.8.m8.2.2.1.1.1.cmml" xref="S5.5.p1.8.m8.2.2.1.1"><times id="S5.5.p1.8.m8.2.2.1.1.1.1.cmml" xref="S5.5.p1.8.m8.2.2.1.1.1.1"></times><apply id="S5.5.p1.8.m8.2.2.1.1.1.2.cmml" xref="S5.5.p1.8.m8.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S5.5.p1.8.m8.2.2.1.1.1.2.1.cmml" xref="S5.5.p1.8.m8.2.2.1.1.1.2">subscript</csymbol><ci id="S5.5.p1.8.m8.2.2.1.1.1.2.2.cmml" xref="S5.5.p1.8.m8.2.2.1.1.1.2.2">𝜋</ci><ci id="S5.5.p1.8.m8.2.2.1.1.1.2.3.cmml" xref="S5.5.p1.8.m8.2.2.1.1.1.2.3">𝜎</ci></apply><ci id="S5.5.p1.8.m8.1.1.cmml" xref="S5.5.p1.8.m8.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.8.m8.2c">\cal M(\pi_{\sigma}(X))</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.8.m8.2d">caligraphic_M ( italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( caligraphic_X ) )</annotation></semantics></math>. But a measure is ergodic if and only if it is an extremal point of the corresponding measure cone, so that, for any ergodic measure <math alttext="\mu\in\cal M(X)" class="ltx_Math" display="inline" id="S5.5.p1.9.m9.1"><semantics id="S5.5.p1.9.m9.1a"><mrow id="S5.5.p1.9.m9.1.2" xref="S5.5.p1.9.m9.1.2.cmml"><mi id="S5.5.p1.9.m9.1.2.2" xref="S5.5.p1.9.m9.1.2.2.cmml">μ</mi><mo id="S5.5.p1.9.m9.1.2.1" xref="S5.5.p1.9.m9.1.2.1.cmml">∈</mo><mrow id="S5.5.p1.9.m9.1.2.3" xref="S5.5.p1.9.m9.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.9.m9.1.2.3.2" xref="S5.5.p1.9.m9.1.2.3.2.cmml">ℳ</mi><mo id="S5.5.p1.9.m9.1.2.3.1" xref="S5.5.p1.9.m9.1.2.3.1.cmml">⁢</mo><mrow id="S5.5.p1.9.m9.1.2.3.3.2" xref="S5.5.p1.9.m9.1.2.3.cmml"><mo id="S5.5.p1.9.m9.1.2.3.3.2.1" stretchy="false" xref="S5.5.p1.9.m9.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.5.p1.9.m9.1.1" xref="S5.5.p1.9.m9.1.1.cmml">𝒳</mi><mo id="S5.5.p1.9.m9.1.2.3.3.2.2" stretchy="false" xref="S5.5.p1.9.m9.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p1.9.m9.1b"><apply id="S5.5.p1.9.m9.1.2.cmml" xref="S5.5.p1.9.m9.1.2"><in id="S5.5.p1.9.m9.1.2.1.cmml" xref="S5.5.p1.9.m9.1.2.1"></in><ci id="S5.5.p1.9.m9.1.2.2.cmml" xref="S5.5.p1.9.m9.1.2.2">𝜇</ci><apply id="S5.5.p1.9.m9.1.2.3.cmml" xref="S5.5.p1.9.m9.1.2.3"><times id="S5.5.p1.9.m9.1.2.3.1.cmml" xref="S5.5.p1.9.m9.1.2.3.1"></times><ci id="S5.5.p1.9.m9.1.2.3.2.cmml" xref="S5.5.p1.9.m9.1.2.3.2">ℳ</ci><ci id="S5.5.p1.9.m9.1.1.cmml" xref="S5.5.p1.9.m9.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.9.m9.1c">\mu\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.9.m9.1d">italic_μ ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math> the transferred measure <math alttext="\mu^{\pi_{\sigma}}" class="ltx_Math" display="inline" id="S5.5.p1.10.m10.1"><semantics id="S5.5.p1.10.m10.1a"><msup id="S5.5.p1.10.m10.1.1" xref="S5.5.p1.10.m10.1.1.cmml"><mi id="S5.5.p1.10.m10.1.1.2" xref="S5.5.p1.10.m10.1.1.2.cmml">μ</mi><msub id="S5.5.p1.10.m10.1.1.3" xref="S5.5.p1.10.m10.1.1.3.cmml"><mi id="S5.5.p1.10.m10.1.1.3.2" xref="S5.5.p1.10.m10.1.1.3.2.cmml">π</mi><mi id="S5.5.p1.10.m10.1.1.3.3" xref="S5.5.p1.10.m10.1.1.3.3.cmml">σ</mi></msub></msup><annotation-xml encoding="MathML-Content" id="S5.5.p1.10.m10.1b"><apply id="S5.5.p1.10.m10.1.1.cmml" xref="S5.5.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S5.5.p1.10.m10.1.1.1.cmml" xref="S5.5.p1.10.m10.1.1">superscript</csymbol><ci id="S5.5.p1.10.m10.1.1.2.cmml" xref="S5.5.p1.10.m10.1.1.2">𝜇</ci><apply id="S5.5.p1.10.m10.1.1.3.cmml" xref="S5.5.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.5.p1.10.m10.1.1.3.1.cmml" xref="S5.5.p1.10.m10.1.1.3">subscript</csymbol><ci id="S5.5.p1.10.m10.1.1.3.2.cmml" xref="S5.5.p1.10.m10.1.1.3.2">𝜋</ci><ci id="S5.5.p1.10.m10.1.1.3.3.cmml" xref="S5.5.p1.10.m10.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p1.10.m10.1c">\mu^{\pi_{\sigma}}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p1.10.m10.1d">italic_μ start_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is also ergodic.</p> </div> <div class="ltx_para" id="S5.6.p2"> <p class="ltx_p" id="S5.6.p2.4">We now use the well known - and easy to prove - fact (see Proposition 7.9 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib8" title="">8</a>]</cite>), that for any ergodic measure the push-forward measure with respect to any factor map (such as our letter-to-letter morphism <math alttext="\alpha_{\sigma}" class="ltx_Math" display="inline" id="S5.6.p2.1.m1.1"><semantics id="S5.6.p2.1.m1.1a"><msub id="S5.6.p2.1.m1.1.1" xref="S5.6.p2.1.m1.1.1.cmml"><mi id="S5.6.p2.1.m1.1.1.2" xref="S5.6.p2.1.m1.1.1.2.cmml">α</mi><mi id="S5.6.p2.1.m1.1.1.3" xref="S5.6.p2.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.6.p2.1.m1.1b"><apply id="S5.6.p2.1.m1.1.1.cmml" xref="S5.6.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.1.m1.1.1.1.cmml" xref="S5.6.p2.1.m1.1.1">subscript</csymbol><ci id="S5.6.p2.1.m1.1.1.2.cmml" xref="S5.6.p2.1.m1.1.1.2">𝛼</ci><ci id="S5.6.p2.1.m1.1.1.3.cmml" xref="S5.6.p2.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.1.m1.1c">\alpha_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>) is again ergodic, in order to conclude that the measure <math alttext="(\mu^{\pi_{\sigma}})^{\alpha_{\sigma}}=\mu^{\sigma}" class="ltx_Math" display="inline" id="S5.6.p2.2.m2.1"><semantics id="S5.6.p2.2.m2.1a"><mrow id="S5.6.p2.2.m2.1.1" xref="S5.6.p2.2.m2.1.1.cmml"><msup id="S5.6.p2.2.m2.1.1.1" xref="S5.6.p2.2.m2.1.1.1.cmml"><mrow id="S5.6.p2.2.m2.1.1.1.1.1" xref="S5.6.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="S5.6.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S5.6.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S5.6.p2.2.m2.1.1.1.1.1.1" xref="S5.6.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="S5.6.p2.2.m2.1.1.1.1.1.1.2" xref="S5.6.p2.2.m2.1.1.1.1.1.1.2.cmml">μ</mi><msub id="S5.6.p2.2.m2.1.1.1.1.1.1.3" xref="S5.6.p2.2.m2.1.1.1.1.1.1.3.cmml"><mi id="S5.6.p2.2.m2.1.1.1.1.1.1.3.2" xref="S5.6.p2.2.m2.1.1.1.1.1.1.3.2.cmml">π</mi><mi id="S5.6.p2.2.m2.1.1.1.1.1.1.3.3" xref="S5.6.p2.2.m2.1.1.1.1.1.1.3.3.cmml">σ</mi></msub></msup><mo id="S5.6.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.6.p2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><msub id="S5.6.p2.2.m2.1.1.1.3" xref="S5.6.p2.2.m2.1.1.1.3.cmml"><mi id="S5.6.p2.2.m2.1.1.1.3.2" xref="S5.6.p2.2.m2.1.1.1.3.2.cmml">α</mi><mi id="S5.6.p2.2.m2.1.1.1.3.3" xref="S5.6.p2.2.m2.1.1.1.3.3.cmml">σ</mi></msub></msup><mo id="S5.6.p2.2.m2.1.1.2" xref="S5.6.p2.2.m2.1.1.2.cmml">=</mo><msup id="S5.6.p2.2.m2.1.1.3" xref="S5.6.p2.2.m2.1.1.3.cmml"><mi id="S5.6.p2.2.m2.1.1.3.2" xref="S5.6.p2.2.m2.1.1.3.2.cmml">μ</mi><mi id="S5.6.p2.2.m2.1.1.3.3" xref="S5.6.p2.2.m2.1.1.3.3.cmml">σ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p2.2.m2.1b"><apply id="S5.6.p2.2.m2.1.1.cmml" xref="S5.6.p2.2.m2.1.1"><eq id="S5.6.p2.2.m2.1.1.2.cmml" xref="S5.6.p2.2.m2.1.1.2"></eq><apply id="S5.6.p2.2.m2.1.1.1.cmml" xref="S5.6.p2.2.m2.1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.2.m2.1.1.1.2.cmml" xref="S5.6.p2.2.m2.1.1.1">superscript</csymbol><apply id="S5.6.p2.2.m2.1.1.1.1.1.1.cmml" xref="S5.6.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.6.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.6.p2.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S5.6.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.6.p2.2.m2.1.1.1.1.1.1.2">𝜇</ci><apply id="S5.6.p2.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.6.p2.2.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S5.6.p2.2.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.6.p2.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S5.6.p2.2.m2.1.1.1.1.1.1.3.2">𝜋</ci><ci id="S5.6.p2.2.m2.1.1.1.1.1.1.3.3.cmml" xref="S5.6.p2.2.m2.1.1.1.1.1.1.3.3">𝜎</ci></apply></apply><apply id="S5.6.p2.2.m2.1.1.1.3.cmml" xref="S5.6.p2.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.2.m2.1.1.1.3.1.cmml" xref="S5.6.p2.2.m2.1.1.1.3">subscript</csymbol><ci id="S5.6.p2.2.m2.1.1.1.3.2.cmml" xref="S5.6.p2.2.m2.1.1.1.3.2">𝛼</ci><ci id="S5.6.p2.2.m2.1.1.1.3.3.cmml" xref="S5.6.p2.2.m2.1.1.1.3.3">𝜎</ci></apply></apply><apply id="S5.6.p2.2.m2.1.1.3.cmml" xref="S5.6.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.6.p2.2.m2.1.1.3.1.cmml" xref="S5.6.p2.2.m2.1.1.3">superscript</csymbol><ci id="S5.6.p2.2.m2.1.1.3.2.cmml" xref="S5.6.p2.2.m2.1.1.3.2">𝜇</ci><ci id="S5.6.p2.2.m2.1.1.3.3.cmml" xref="S5.6.p2.2.m2.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.2.m2.1c">(\mu^{\pi_{\sigma}})^{\alpha_{\sigma}}=\mu^{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.2.m2.1d">( italic_μ start_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = italic_μ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT</annotation></semantics></math> is ergodic. <span class="ltx_text ltx_inline-block" id="S5.6.p2.3.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S5.6.p2.3.1.m1.1"><semantics id="S5.6.p2.3.1.m1.1a"><mo id="S5.6.p2.3.1.m1.1.1" xref="S5.6.p2.3.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S5.6.p2.3.1.m1.1b"><csymbol cd="latexml" id="S5.6.p2.3.1.m1.1.1.cmml" xref="S5.6.p2.3.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.3.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.3.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S5.6.p2.4.m3.1"><semantics id="S5.6.p2.4.m3.1a"><mo id="S5.6.p2.4.m3.1.1" xref="S5.6.p2.4.m3.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S5.6.p2.4.m3.1b"><csymbol cd="latexml" id="S5.6.p2.4.m3.1.1.cmml" xref="S5.6.p2.4.m3.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p2.4.m3.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S5.6.p2.4.m3.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">We can now start the proof of the main result of this paper; its core (Theorem <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm7" title="Theorem 6.7. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.7</span></a>) will however be delayed until the next section.</p> </div> <div class="ltx_theorem ltx_theorem_thm" id="S5.Thmthm6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm6.1.1.1">Theorem 5.6</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm6.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm6.p1"> <p class="ltx_p" id="S5.Thmthm6.p1.3"><span class="ltx_text ltx_font_italic" id="S5.Thmthm6.p1.3.3">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.Thmthm6.p1.1.1.m1.1"><semantics id="S5.Thmthm6.p1.1.1.m1.1a"><mrow id="S5.Thmthm6.p1.1.1.m1.1.1" xref="S5.Thmthm6.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmthm6.p1.1.1.m1.1.1.2" xref="S5.Thmthm6.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S5.Thmthm6.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm6.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm6.p1.1.1.m1.1.1.3" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm6.p1.1.1.m1.1.1.3.2" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm6.p1.1.1.m1.1.1.3.2.2" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm6.p1.1.1.m1.1.1.3.2.3" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm6.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm6.p1.1.1.m1.1.1.3.3" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm6.p1.1.1.m1.1.1.3.3.2" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.Thmthm6.p1.1.1.m1.1.1.3.3.3" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm6.p1.1.1.m1.1b"><apply id="S5.Thmthm6.p1.1.1.m1.1.1.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1"><ci id="S5.Thmthm6.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.1">:</ci><ci id="S5.Thmthm6.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S5.Thmthm6.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3"><ci id="S5.Thmthm6.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm6.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm6.p1.1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm6.p1.1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm6.p1.1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm6.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm6.p1.1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm6.p1.1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S5.Thmthm6.p1.1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm6.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm6.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm6.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-erasing morphism of free monoids on finite alphabets, and let <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm6.p1.2.2.m2.1"><semantics id="S5.Thmthm6.p1.2.2.m2.1a"><mrow id="S5.Thmthm6.p1.2.2.m2.1.1" xref="S5.Thmthm6.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmthm6.p1.2.2.m2.1.1.2" xref="S5.Thmthm6.p1.2.2.m2.1.1.2.cmml">X</mi><mo id="S5.Thmthm6.p1.2.2.m2.1.1.1" xref="S5.Thmthm6.p1.2.2.m2.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm6.p1.2.2.m2.1.1.3" xref="S5.Thmthm6.p1.2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm6.p1.2.2.m2.1.1.3.2" xref="S5.Thmthm6.p1.2.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm6.p1.2.2.m2.1.1.3.3" xref="S5.Thmthm6.p1.2.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm6.p1.2.2.m2.1b"><apply id="S5.Thmthm6.p1.2.2.m2.1.1.cmml" xref="S5.Thmthm6.p1.2.2.m2.1.1"><subset id="S5.Thmthm6.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmthm6.p1.2.2.m2.1.1.1"></subset><ci id="S5.Thmthm6.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmthm6.p1.2.2.m2.1.1.2">𝑋</ci><apply id="S5.Thmthm6.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmthm6.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm6.p1.2.2.m2.1.1.3.1.cmml" xref="S5.Thmthm6.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S5.Thmthm6.p1.2.2.m2.1.1.3.2.cmml" xref="S5.Thmthm6.p1.2.2.m2.1.1.3.2">𝒜</ci><ci id="S5.Thmthm6.p1.2.2.m2.1.1.3.3.cmml" xref="S5.Thmthm6.p1.2.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm6.p1.2.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm6.p1.2.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> be a subshift, with image subshift <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S5.Thmthm6.p1.3.3.m3.1"><semantics id="S5.Thmthm6.p1.3.3.m3.1a"><mrow id="S5.Thmthm6.p1.3.3.m3.1.2" xref="S5.Thmthm6.p1.3.3.m3.1.2.cmml"><mi id="S5.Thmthm6.p1.3.3.m3.1.2.2" xref="S5.Thmthm6.p1.3.3.m3.1.2.2.cmml">σ</mi><mo id="S5.Thmthm6.p1.3.3.m3.1.2.1" xref="S5.Thmthm6.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S5.Thmthm6.p1.3.3.m3.1.2.3.2" xref="S5.Thmthm6.p1.3.3.m3.1.2.cmml"><mo id="S5.Thmthm6.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="S5.Thmthm6.p1.3.3.m3.1.2.cmml">(</mo><mi id="S5.Thmthm6.p1.3.3.m3.1.1" xref="S5.Thmthm6.p1.3.3.m3.1.1.cmml">X</mi><mo id="S5.Thmthm6.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="S5.Thmthm6.p1.3.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm6.p1.3.3.m3.1b"><apply id="S5.Thmthm6.p1.3.3.m3.1.2.cmml" xref="S5.Thmthm6.p1.3.3.m3.1.2"><times id="S5.Thmthm6.p1.3.3.m3.1.2.1.cmml" xref="S5.Thmthm6.p1.3.3.m3.1.2.1"></times><ci id="S5.Thmthm6.p1.3.3.m3.1.2.2.cmml" xref="S5.Thmthm6.p1.3.3.m3.1.2.2">𝜎</ci><ci id="S5.Thmthm6.p1.3.3.m3.1.1.cmml" xref="S5.Thmthm6.p1.3.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm6.p1.3.3.m3.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm6.p1.3.3.m3.1d">italic_σ ( italic_X )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S5.Thmthm6.p2"> <p class="ltx_p" id="S5.Thmthm6.p2.3"><span class="ltx_text ltx_font_italic" id="S5.Thmthm6.p2.3.3">If <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm6.p2.1.1.m1.1"><semantics id="S5.Thmthm6.p2.1.1.m1.1a"><mi id="S5.Thmthm6.p2.1.1.m1.1.1" xref="S5.Thmthm6.p2.1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm6.p2.1.1.m1.1b"><ci id="S5.Thmthm6.p2.1.1.m1.1.1.cmml" xref="S5.Thmthm6.p2.1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm6.p2.1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm6.p2.1.1.m1.1d">italic_σ</annotation></semantics></math> is shift-orbit injective in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm6.p2.2.2.m2.1"><semantics id="S5.Thmthm6.p2.2.2.m2.1a"><mi id="S5.Thmthm6.p2.2.2.m2.1.1" xref="S5.Thmthm6.p2.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm6.p2.2.2.m2.1b"><ci id="S5.Thmthm6.p2.2.2.m2.1.1.cmml" xref="S5.Thmthm6.p2.2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm6.p2.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm6.p2.2.2.m2.1d">italic_X</annotation></semantics></math>, then the measure transfer map <math alttext="\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))" class="ltx_Math" display="inline" id="S5.Thmthm6.p2.3.3.m3.3"><semantics id="S5.Thmthm6.p2.3.3.m3.3a"><mrow id="S5.Thmthm6.p2.3.3.m3.3.3" xref="S5.Thmthm6.p2.3.3.m3.3.3.cmml"><mrow id="S5.Thmthm6.p2.3.3.m3.3.3.3" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.cmml"><msub id="S5.Thmthm6.p2.3.3.m3.3.3.3.2" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.2.cmml"><mi id="S5.Thmthm6.p2.3.3.m3.3.3.3.2.2" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.2.2.cmml">σ</mi><mi id="S5.Thmthm6.p2.3.3.m3.3.3.3.2.3" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.2.3.cmml">X</mi></msub><mo id="S5.Thmthm6.p2.3.3.m3.3.3.3.1" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.1.cmml">⁢</mo><mi id="S5.Thmthm6.p2.3.3.m3.3.3.3.3" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.3.cmml">M</mi></mrow><mo id="S5.Thmthm6.p2.3.3.m3.3.3.2" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm6.p2.3.3.m3.3.3.2.cmml">:</mo><mrow id="S5.Thmthm6.p2.3.3.m3.3.3.1" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.cmml"><mrow id="S5.Thmthm6.p2.3.3.m3.3.3.1.3" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm6.p2.3.3.m3.3.3.1.3.2" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3.2.cmml">ℳ</mi><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.3.1" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3.1.cmml">⁢</mo><mrow id="S5.Thmthm6.p2.3.3.m3.3.3.1.3.3.2" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3.cmml"><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.3.3.2.1" stretchy="false" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm6.p2.3.3.m3.1.1" xref="S5.Thmthm6.p2.3.3.m3.1.1.cmml">𝒳</mi><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.3.3.2.2" stretchy="false" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.2" stretchy="false" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.2.cmml">→</mo><mrow id="S5.Thmthm6.p2.3.3.m3.3.3.1.1" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.3" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.3.cmml">ℳ</mi><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.2" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.2.cmml">⁢</mo><mrow id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.cmml"><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.2" stretchy="false" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.cmml"><mi id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.2" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.2.cmml">σ</mi><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.1" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.3.2" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.cmml"><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm6.p2.3.3.m3.2.2" xref="S5.Thmthm6.p2.3.3.m3.2.2.cmml">𝒳</mi><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.3" stretchy="false" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm6.p2.3.3.m3.3b"><apply id="S5.Thmthm6.p2.3.3.m3.3.3.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3"><ci id="S5.Thmthm6.p2.3.3.m3.3.3.2.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.2">:</ci><apply id="S5.Thmthm6.p2.3.3.m3.3.3.3.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.3"><times id="S5.Thmthm6.p2.3.3.m3.3.3.3.1.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.1"></times><apply id="S5.Thmthm6.p2.3.3.m3.3.3.3.2.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.2"><csymbol cd="ambiguous" id="S5.Thmthm6.p2.3.3.m3.3.3.3.2.1.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.2">subscript</csymbol><ci id="S5.Thmthm6.p2.3.3.m3.3.3.3.2.2.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.2.2">𝜎</ci><ci id="S5.Thmthm6.p2.3.3.m3.3.3.3.2.3.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.2.3">𝑋</ci></apply><ci id="S5.Thmthm6.p2.3.3.m3.3.3.3.3.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.3.3">𝑀</ci></apply><apply id="S5.Thmthm6.p2.3.3.m3.3.3.1.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1"><ci id="S5.Thmthm6.p2.3.3.m3.3.3.1.2.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.2">→</ci><apply id="S5.Thmthm6.p2.3.3.m3.3.3.1.3.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3"><times id="S5.Thmthm6.p2.3.3.m3.3.3.1.3.1.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3.1"></times><ci id="S5.Thmthm6.p2.3.3.m3.3.3.1.3.2.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.3.2">ℳ</ci><ci id="S5.Thmthm6.p2.3.3.m3.1.1.cmml" xref="S5.Thmthm6.p2.3.3.m3.1.1">𝒳</ci></apply><apply id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1"><times id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.2.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.2"></times><ci id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.3.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.3">ℳ</ci><apply id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1"><times id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.1.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.1"></times><ci id="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.2.cmml" xref="S5.Thmthm6.p2.3.3.m3.3.3.1.1.1.1.1.2">𝜎</ci><ci id="S5.Thmthm6.p2.3.3.m3.2.2.cmml" xref="S5.Thmthm6.p2.3.3.m3.2.2">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm6.p2.3.3.m3.3c">\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm6.p2.3.3.m3.3d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) )</annotation></semantics></math> is injective.</span></p> </div> </div> <div class="ltx_proof" id="S5.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.7.p1"> <p class="ltx_p" id="S5.7.p1.2">We consider the canonical decomposition <math alttext="\sigma=\alpha_{\sigma}\circ\pi_{\sigma}" class="ltx_Math" display="inline" id="S5.7.p1.1.m1.1"><semantics id="S5.7.p1.1.m1.1a"><mrow id="S5.7.p1.1.m1.1.1" xref="S5.7.p1.1.m1.1.1.cmml"><mi id="S5.7.p1.1.m1.1.1.2" xref="S5.7.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S5.7.p1.1.m1.1.1.1" xref="S5.7.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S5.7.p1.1.m1.1.1.3" xref="S5.7.p1.1.m1.1.1.3.cmml"><msub id="S5.7.p1.1.m1.1.1.3.2" xref="S5.7.p1.1.m1.1.1.3.2.cmml"><mi id="S5.7.p1.1.m1.1.1.3.2.2" xref="S5.7.p1.1.m1.1.1.3.2.2.cmml">α</mi><mi id="S5.7.p1.1.m1.1.1.3.2.3" xref="S5.7.p1.1.m1.1.1.3.2.3.cmml">σ</mi></msub><mo id="S5.7.p1.1.m1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S5.7.p1.1.m1.1.1.3.1.cmml">∘</mo><msub id="S5.7.p1.1.m1.1.1.3.3" xref="S5.7.p1.1.m1.1.1.3.3.cmml"><mi id="S5.7.p1.1.m1.1.1.3.3.2" xref="S5.7.p1.1.m1.1.1.3.3.2.cmml">π</mi><mi id="S5.7.p1.1.m1.1.1.3.3.3" xref="S5.7.p1.1.m1.1.1.3.3.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.7.p1.1.m1.1b"><apply id="S5.7.p1.1.m1.1.1.cmml" xref="S5.7.p1.1.m1.1.1"><eq id="S5.7.p1.1.m1.1.1.1.cmml" xref="S5.7.p1.1.m1.1.1.1"></eq><ci id="S5.7.p1.1.m1.1.1.2.cmml" xref="S5.7.p1.1.m1.1.1.2">𝜎</ci><apply id="S5.7.p1.1.m1.1.1.3.cmml" xref="S5.7.p1.1.m1.1.1.3"><compose id="S5.7.p1.1.m1.1.1.3.1.cmml" xref="S5.7.p1.1.m1.1.1.3.1"></compose><apply id="S5.7.p1.1.m1.1.1.3.2.cmml" xref="S5.7.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.7.p1.1.m1.1.1.3.2.1.cmml" xref="S5.7.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S5.7.p1.1.m1.1.1.3.2.2.cmml" xref="S5.7.p1.1.m1.1.1.3.2.2">𝛼</ci><ci id="S5.7.p1.1.m1.1.1.3.2.3.cmml" xref="S5.7.p1.1.m1.1.1.3.2.3">𝜎</ci></apply><apply id="S5.7.p1.1.m1.1.1.3.3.cmml" xref="S5.7.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.7.p1.1.m1.1.1.3.3.1.cmml" xref="S5.7.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S5.7.p1.1.m1.1.1.3.3.2.cmml" xref="S5.7.p1.1.m1.1.1.3.3.2">𝜋</ci><ci id="S5.7.p1.1.m1.1.1.3.3.3.cmml" xref="S5.7.p1.1.m1.1.1.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.1.m1.1c">\sigma=\alpha_{\sigma}\circ\pi_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.1.m1.1d">italic_σ = italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∘ italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> from equality (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.E4" title="In Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.4</span></a>) and obtain from the functoriality of the measure transfer (see Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm7" title="Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.7</span></a> (b)) the decomposition <math alttext="\sigma_{X}M=(\alpha_{\sigma})_{\sigma(X)}M\circ(\pi_{\sigma})_{X}M" class="ltx_Math" display="inline" id="S5.7.p1.2.m2.3"><semantics id="S5.7.p1.2.m2.3a"><mrow id="S5.7.p1.2.m2.3.3" xref="S5.7.p1.2.m2.3.3.cmml"><mrow id="S5.7.p1.2.m2.3.3.4" xref="S5.7.p1.2.m2.3.3.4.cmml"><msub id="S5.7.p1.2.m2.3.3.4.2" xref="S5.7.p1.2.m2.3.3.4.2.cmml"><mi id="S5.7.p1.2.m2.3.3.4.2.2" xref="S5.7.p1.2.m2.3.3.4.2.2.cmml">σ</mi><mi id="S5.7.p1.2.m2.3.3.4.2.3" xref="S5.7.p1.2.m2.3.3.4.2.3.cmml">X</mi></msub><mo id="S5.7.p1.2.m2.3.3.4.1" xref="S5.7.p1.2.m2.3.3.4.1.cmml">⁢</mo><mi id="S5.7.p1.2.m2.3.3.4.3" xref="S5.7.p1.2.m2.3.3.4.3.cmml">M</mi></mrow><mo id="S5.7.p1.2.m2.3.3.3" xref="S5.7.p1.2.m2.3.3.3.cmml">=</mo><mrow id="S5.7.p1.2.m2.3.3.2" xref="S5.7.p1.2.m2.3.3.2.cmml"><mrow id="S5.7.p1.2.m2.3.3.2.2" xref="S5.7.p1.2.m2.3.3.2.2.cmml"><mrow id="S5.7.p1.2.m2.2.2.1.1.1" xref="S5.7.p1.2.m2.2.2.1.1.1.cmml"><msub id="S5.7.p1.2.m2.2.2.1.1.1.1" xref="S5.7.p1.2.m2.2.2.1.1.1.1.cmml"><mrow id="S5.7.p1.2.m2.2.2.1.1.1.1.1.1" xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml"><mo id="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1" xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml"><mi id="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.2" xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.2.cmml">α</mi><mi id="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.3" xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.3.cmml">σ</mi></msub><mo id="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.7.p1.2.m2.1.1.1" xref="S5.7.p1.2.m2.1.1.1.cmml"><mi id="S5.7.p1.2.m2.1.1.1.3" xref="S5.7.p1.2.m2.1.1.1.3.cmml">σ</mi><mo id="S5.7.p1.2.m2.1.1.1.2" xref="S5.7.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S5.7.p1.2.m2.1.1.1.4.2" 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xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.2.cmml" xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.2">𝛼</ci><ci id="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.3.cmml" xref="S5.7.p1.2.m2.2.2.1.1.1.1.1.1.1.3">𝜎</ci></apply><apply id="S5.7.p1.2.m2.1.1.1.cmml" xref="S5.7.p1.2.m2.1.1.1"><times id="S5.7.p1.2.m2.1.1.1.2.cmml" xref="S5.7.p1.2.m2.1.1.1.2"></times><ci id="S5.7.p1.2.m2.1.1.1.3.cmml" xref="S5.7.p1.2.m2.1.1.1.3">𝜎</ci><ci id="S5.7.p1.2.m2.1.1.1.1.cmml" xref="S5.7.p1.2.m2.1.1.1.1">𝑋</ci></apply></apply><ci id="S5.7.p1.2.m2.2.2.1.1.1.3.cmml" xref="S5.7.p1.2.m2.2.2.1.1.1.3">𝑀</ci></apply><apply id="S5.7.p1.2.m2.3.3.2.2.2.cmml" xref="S5.7.p1.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.7.p1.2.m2.3.3.2.2.2.2.cmml" xref="S5.7.p1.2.m2.3.3.2.2.2">subscript</csymbol><apply id="S5.7.p1.2.m2.3.3.2.2.2.1.1.1.cmml" xref="S5.7.p1.2.m2.3.3.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.7.p1.2.m2.3.3.2.2.2.1.1.1.1.cmml" xref="S5.7.p1.2.m2.3.3.2.2.2.1.1">subscript</csymbol><ci id="S5.7.p1.2.m2.3.3.2.2.2.1.1.1.2.cmml" xref="S5.7.p1.2.m2.3.3.2.2.2.1.1.1.2">𝜋</ci><ci id="S5.7.p1.2.m2.3.3.2.2.2.1.1.1.3.cmml" xref="S5.7.p1.2.m2.3.3.2.2.2.1.1.1.3">𝜎</ci></apply><ci id="S5.7.p1.2.m2.3.3.2.2.2.3.cmml" xref="S5.7.p1.2.m2.3.3.2.2.2.3">𝑋</ci></apply></apply><ci id="S5.7.p1.2.m2.3.3.2.4.cmml" xref="S5.7.p1.2.m2.3.3.2.4">𝑀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.7.p1.2.m2.3c">\sigma_{X}M=(\alpha_{\sigma})_{\sigma(X)}M\circ(\pi_{\sigma})_{X}M</annotation><annotation encoding="application/x-llamapun" id="S5.7.p1.2.m2.3d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M = ( italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_σ ( italic_X ) end_POSTSUBSCRIPT italic_M ∘ ( italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.8.p2"> <p class="ltx_p" id="S5.8.p2.6">From Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm2" title="Lemma 5.2. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.2</span></a> (1) and Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm3" title="Lemma 5.3. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.3</span></a> we obtain directly that the morphism <math alttext="\alpha_{\sigma}" class="ltx_Math" display="inline" id="S5.8.p2.1.m1.1"><semantics id="S5.8.p2.1.m1.1a"><msub id="S5.8.p2.1.m1.1.1" xref="S5.8.p2.1.m1.1.1.cmml"><mi id="S5.8.p2.1.m1.1.1.2" xref="S5.8.p2.1.m1.1.1.2.cmml">α</mi><mi id="S5.8.p2.1.m1.1.1.3" xref="S5.8.p2.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.8.p2.1.m1.1b"><apply id="S5.8.p2.1.m1.1.1.cmml" xref="S5.8.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.8.p2.1.m1.1.1.1.cmml" xref="S5.8.p2.1.m1.1.1">subscript</csymbol><ci id="S5.8.p2.1.m1.1.1.2.cmml" xref="S5.8.p2.1.m1.1.1.2">𝛼</ci><ci id="S5.8.p2.1.m1.1.1.3.cmml" xref="S5.8.p2.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.8.p2.1.m1.1c">\alpha_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S5.8.p2.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is shift-orbit injective. We can thus apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm4" title="Lemma 5.4. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.4</span></a> to <math alttext="(\pi_{\sigma})_{X}M" class="ltx_Math" display="inline" id="S5.8.p2.2.m2.1"><semantics id="S5.8.p2.2.m2.1a"><mrow id="S5.8.p2.2.m2.1.1" xref="S5.8.p2.2.m2.1.1.cmml"><msub id="S5.8.p2.2.m2.1.1.1" xref="S5.8.p2.2.m2.1.1.1.cmml"><mrow id="S5.8.p2.2.m2.1.1.1.1.1" xref="S5.8.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="S5.8.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S5.8.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S5.8.p2.2.m2.1.1.1.1.1.1" xref="S5.8.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="S5.8.p2.2.m2.1.1.1.1.1.1.2" xref="S5.8.p2.2.m2.1.1.1.1.1.1.2.cmml">π</mi><mi id="S5.8.p2.2.m2.1.1.1.1.1.1.3" xref="S5.8.p2.2.m2.1.1.1.1.1.1.3.cmml">σ</mi></msub><mo id="S5.8.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.8.p2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.8.p2.2.m2.1.1.1.3" xref="S5.8.p2.2.m2.1.1.1.3.cmml">X</mi></msub><mo id="S5.8.p2.2.m2.1.1.2" xref="S5.8.p2.2.m2.1.1.2.cmml">⁢</mo><mi id="S5.8.p2.2.m2.1.1.3" xref="S5.8.p2.2.m2.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.8.p2.2.m2.1b"><apply id="S5.8.p2.2.m2.1.1.cmml" xref="S5.8.p2.2.m2.1.1"><times id="S5.8.p2.2.m2.1.1.2.cmml" xref="S5.8.p2.2.m2.1.1.2"></times><apply id="S5.8.p2.2.m2.1.1.1.cmml" xref="S5.8.p2.2.m2.1.1.1"><csymbol cd="ambiguous" id="S5.8.p2.2.m2.1.1.1.2.cmml" xref="S5.8.p2.2.m2.1.1.1">subscript</csymbol><apply id="S5.8.p2.2.m2.1.1.1.1.1.1.cmml" xref="S5.8.p2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.8.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.8.p2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S5.8.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.8.p2.2.m2.1.1.1.1.1.1.2">𝜋</ci><ci id="S5.8.p2.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.8.p2.2.m2.1.1.1.1.1.1.3">𝜎</ci></apply><ci id="S5.8.p2.2.m2.1.1.1.3.cmml" xref="S5.8.p2.2.m2.1.1.1.3">𝑋</ci></apply><ci id="S5.8.p2.2.m2.1.1.3.cmml" xref="S5.8.p2.2.m2.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.8.p2.2.m2.1c">(\pi_{\sigma})_{X}M</annotation><annotation encoding="application/x-llamapun" id="S5.8.p2.2.m2.1d">( italic_π start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm7" title="Theorem 6.7. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.7</span></a> to <math alttext="(\alpha_{\sigma})_{\sigma(X)}M" class="ltx_Math" display="inline" id="S5.8.p2.3.m3.2"><semantics id="S5.8.p2.3.m3.2a"><mrow id="S5.8.p2.3.m3.2.2" xref="S5.8.p2.3.m3.2.2.cmml"><msub id="S5.8.p2.3.m3.2.2.1" xref="S5.8.p2.3.m3.2.2.1.cmml"><mrow id="S5.8.p2.3.m3.2.2.1.1.1" xref="S5.8.p2.3.m3.2.2.1.1.1.1.cmml"><mo id="S5.8.p2.3.m3.2.2.1.1.1.2" stretchy="false" xref="S5.8.p2.3.m3.2.2.1.1.1.1.cmml">(</mo><msub id="S5.8.p2.3.m3.2.2.1.1.1.1" xref="S5.8.p2.3.m3.2.2.1.1.1.1.cmml"><mi id="S5.8.p2.3.m3.2.2.1.1.1.1.2" xref="S5.8.p2.3.m3.2.2.1.1.1.1.2.cmml">α</mi><mi id="S5.8.p2.3.m3.2.2.1.1.1.1.3" xref="S5.8.p2.3.m3.2.2.1.1.1.1.3.cmml">σ</mi></msub><mo id="S5.8.p2.3.m3.2.2.1.1.1.3" stretchy="false" xref="S5.8.p2.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.8.p2.3.m3.1.1.1" xref="S5.8.p2.3.m3.1.1.1.cmml"><mi id="S5.8.p2.3.m3.1.1.1.3" xref="S5.8.p2.3.m3.1.1.1.3.cmml">σ</mi><mo id="S5.8.p2.3.m3.1.1.1.2" xref="S5.8.p2.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S5.8.p2.3.m3.1.1.1.4.2" xref="S5.8.p2.3.m3.1.1.1.cmml"><mo id="S5.8.p2.3.m3.1.1.1.4.2.1" stretchy="false" xref="S5.8.p2.3.m3.1.1.1.cmml">(</mo><mi id="S5.8.p2.3.m3.1.1.1.1" xref="S5.8.p2.3.m3.1.1.1.1.cmml">X</mi><mo id="S5.8.p2.3.m3.1.1.1.4.2.2" stretchy="false" xref="S5.8.p2.3.m3.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S5.8.p2.3.m3.2.2.2" xref="S5.8.p2.3.m3.2.2.2.cmml">⁢</mo><mi id="S5.8.p2.3.m3.2.2.3" xref="S5.8.p2.3.m3.2.2.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.8.p2.3.m3.2b"><apply id="S5.8.p2.3.m3.2.2.cmml" xref="S5.8.p2.3.m3.2.2"><times id="S5.8.p2.3.m3.2.2.2.cmml" xref="S5.8.p2.3.m3.2.2.2"></times><apply id="S5.8.p2.3.m3.2.2.1.cmml" xref="S5.8.p2.3.m3.2.2.1"><csymbol cd="ambiguous" id="S5.8.p2.3.m3.2.2.1.2.cmml" xref="S5.8.p2.3.m3.2.2.1">subscript</csymbol><apply id="S5.8.p2.3.m3.2.2.1.1.1.1.cmml" xref="S5.8.p2.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.8.p2.3.m3.2.2.1.1.1.1.1.cmml" xref="S5.8.p2.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S5.8.p2.3.m3.2.2.1.1.1.1.2.cmml" xref="S5.8.p2.3.m3.2.2.1.1.1.1.2">𝛼</ci><ci id="S5.8.p2.3.m3.2.2.1.1.1.1.3.cmml" xref="S5.8.p2.3.m3.2.2.1.1.1.1.3">𝜎</ci></apply><apply id="S5.8.p2.3.m3.1.1.1.cmml" xref="S5.8.p2.3.m3.1.1.1"><times id="S5.8.p2.3.m3.1.1.1.2.cmml" xref="S5.8.p2.3.m3.1.1.1.2"></times><ci id="S5.8.p2.3.m3.1.1.1.3.cmml" xref="S5.8.p2.3.m3.1.1.1.3">𝜎</ci><ci id="S5.8.p2.3.m3.1.1.1.1.cmml" xref="S5.8.p2.3.m3.1.1.1.1">𝑋</ci></apply></apply><ci id="S5.8.p2.3.m3.2.2.3.cmml" xref="S5.8.p2.3.m3.2.2.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.8.p2.3.m3.2c">(\alpha_{\sigma})_{\sigma(X)}M</annotation><annotation encoding="application/x-llamapun" id="S5.8.p2.3.m3.2d">( italic_α start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_σ ( italic_X ) end_POSTSUBSCRIPT italic_M</annotation></semantics></math> to deduce that <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S5.8.p2.4.m4.1"><semantics id="S5.8.p2.4.m4.1a"><mrow id="S5.8.p2.4.m4.1.1" xref="S5.8.p2.4.m4.1.1.cmml"><msub id="S5.8.p2.4.m4.1.1.2" xref="S5.8.p2.4.m4.1.1.2.cmml"><mi id="S5.8.p2.4.m4.1.1.2.2" xref="S5.8.p2.4.m4.1.1.2.2.cmml">σ</mi><mi id="S5.8.p2.4.m4.1.1.2.3" xref="S5.8.p2.4.m4.1.1.2.3.cmml">X</mi></msub><mo id="S5.8.p2.4.m4.1.1.1" xref="S5.8.p2.4.m4.1.1.1.cmml">⁢</mo><mi id="S5.8.p2.4.m4.1.1.3" xref="S5.8.p2.4.m4.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.8.p2.4.m4.1b"><apply id="S5.8.p2.4.m4.1.1.cmml" xref="S5.8.p2.4.m4.1.1"><times id="S5.8.p2.4.m4.1.1.1.cmml" xref="S5.8.p2.4.m4.1.1.1"></times><apply id="S5.8.p2.4.m4.1.1.2.cmml" xref="S5.8.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.8.p2.4.m4.1.1.2.1.cmml" xref="S5.8.p2.4.m4.1.1.2">subscript</csymbol><ci id="S5.8.p2.4.m4.1.1.2.2.cmml" xref="S5.8.p2.4.m4.1.1.2.2">𝜎</ci><ci id="S5.8.p2.4.m4.1.1.2.3.cmml" xref="S5.8.p2.4.m4.1.1.2.3">𝑋</ci></apply><ci id="S5.8.p2.4.m4.1.1.3.cmml" xref="S5.8.p2.4.m4.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.8.p2.4.m4.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S5.8.p2.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> is injective. <span class="ltx_text ltx_inline-block" id="S5.8.p2.5.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S5.8.p2.5.1.m1.1"><semantics id="S5.8.p2.5.1.m1.1a"><mo id="S5.8.p2.5.1.m1.1.1" xref="S5.8.p2.5.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S5.8.p2.5.1.m1.1b"><csymbol cd="latexml" id="S5.8.p2.5.1.m1.1.1.cmml" xref="S5.8.p2.5.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.8.p2.5.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S5.8.p2.5.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S5.8.p2.6.m5.1"><semantics id="S5.8.p2.6.m5.1a"><mo id="S5.8.p2.6.m5.1.1" xref="S5.8.p2.6.m5.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S5.8.p2.6.m5.1b"><csymbol cd="latexml" id="S5.8.p2.6.m5.1.1.cmml" xref="S5.8.p2.6.m5.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.8.p2.6.m5.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S5.8.p2.6.m5.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.1">We will terminate this section with a discussion that compares the above introduced notions to the more frequently used notions of morphisms that are “recognizable” or a “recognizable for aperiodic points” (see Fig.1). For the convenience of the reader we briefly recall the definitions:</p> </div> <div class="ltx_theorem ltx_theorem_defn" id="S5.Thmthm7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm7.1.1.1">Definition 5.7</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm7.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm7.p1"> <p class="ltx_p" id="S5.Thmthm7.p1.3">Let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.Thmthm7.p1.1.m1.1"><semantics id="S5.Thmthm7.p1.1.m1.1a"><mrow id="S5.Thmthm7.p1.1.m1.1.1" xref="S5.Thmthm7.p1.1.m1.1.1.cmml"><mi id="S5.Thmthm7.p1.1.m1.1.1.2" xref="S5.Thmthm7.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S5.Thmthm7.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm7.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm7.p1.1.m1.1.1.3" xref="S5.Thmthm7.p1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm7.p1.1.m1.1.1.3.2" xref="S5.Thmthm7.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm7.p1.1.m1.1.1.3.2.2" xref="S5.Thmthm7.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm7.p1.1.m1.1.1.3.2.3" xref="S5.Thmthm7.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm7.p1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm7.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm7.p1.1.m1.1.1.3.3" xref="S5.Thmthm7.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm7.p1.1.m1.1.1.3.3.2" xref="S5.Thmthm7.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.Thmthm7.p1.1.m1.1.1.3.3.3" xref="S5.Thmthm7.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p1.1.m1.1b"><apply id="S5.Thmthm7.p1.1.m1.1.1.cmml" xref="S5.Thmthm7.p1.1.m1.1.1"><ci id="S5.Thmthm7.p1.1.m1.1.1.1.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.1">:</ci><ci id="S5.Thmthm7.p1.1.m1.1.1.2.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.2">𝜎</ci><apply id="S5.Thmthm7.p1.1.m1.1.1.3.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3"><ci id="S5.Thmthm7.p1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm7.p1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm7.p1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm7.p1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm7.p1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm7.p1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm7.p1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm7.p1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S5.Thmthm7.p1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm7.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be a non-erasing morphism, and let <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm7.p1.2.m2.1"><semantics id="S5.Thmthm7.p1.2.m2.1a"><mrow id="S5.Thmthm7.p1.2.m2.1.1" xref="S5.Thmthm7.p1.2.m2.1.1.cmml"><mi id="S5.Thmthm7.p1.2.m2.1.1.2" xref="S5.Thmthm7.p1.2.m2.1.1.2.cmml">X</mi><mo id="S5.Thmthm7.p1.2.m2.1.1.1" xref="S5.Thmthm7.p1.2.m2.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm7.p1.2.m2.1.1.3" xref="S5.Thmthm7.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm7.p1.2.m2.1.1.3.2" xref="S5.Thmthm7.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm7.p1.2.m2.1.1.3.3" xref="S5.Thmthm7.p1.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p1.2.m2.1b"><apply id="S5.Thmthm7.p1.2.m2.1.1.cmml" xref="S5.Thmthm7.p1.2.m2.1.1"><subset id="S5.Thmthm7.p1.2.m2.1.1.1.cmml" xref="S5.Thmthm7.p1.2.m2.1.1.1"></subset><ci id="S5.Thmthm7.p1.2.m2.1.1.2.cmml" xref="S5.Thmthm7.p1.2.m2.1.1.2">𝑋</ci><apply id="S5.Thmthm7.p1.2.m2.1.1.3.cmml" xref="S5.Thmthm7.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm7.p1.2.m2.1.1.3.1.cmml" xref="S5.Thmthm7.p1.2.m2.1.1.3">superscript</csymbol><ci id="S5.Thmthm7.p1.2.m2.1.1.3.2.cmml" xref="S5.Thmthm7.p1.2.m2.1.1.3.2">𝒜</ci><ci id="S5.Thmthm7.p1.2.m2.1.1.3.3.cmml" xref="S5.Thmthm7.p1.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p1.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p1.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> be a subshift over <math alttext="\cal A" class="ltx_Math" display="inline" id="S5.Thmthm7.p1.3.m3.1"><semantics id="S5.Thmthm7.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm7.p1.3.m3.1.1" xref="S5.Thmthm7.p1.3.m3.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p1.3.m3.1b"><ci id="S5.Thmthm7.p1.3.m3.1.1.cmml" xref="S5.Thmthm7.p1.3.m3.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p1.3.m3.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p1.3.m3.1d">caligraphic_A</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S5.Thmthm7.p2"> <p class="ltx_p" id="S5.Thmthm7.p2.2">(1) Then <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm7.p2.1.m1.1"><semantics id="S5.Thmthm7.p2.1.m1.1a"><mi id="S5.Thmthm7.p2.1.m1.1.1" xref="S5.Thmthm7.p2.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p2.1.m1.1b"><ci id="S5.Thmthm7.p2.1.m1.1.1.cmml" xref="S5.Thmthm7.p2.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p2.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p2.1.m1.1d">italic_σ</annotation></semantics></math> is said <span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span> Some authors say “recognizable on <math alttext="X" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">X</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">italic_X</annotation></semantics></math>”.</span></span></span> to be <span class="ltx_text ltx_font_italic" id="S5.Thmthm7.p2.2.1">recognizable in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm7.p2.2.1.m1.1"><semantics id="S5.Thmthm7.p2.2.1.m1.1a"><mi id="S5.Thmthm7.p2.2.1.m1.1.1" xref="S5.Thmthm7.p2.2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p2.2.1.m1.1b"><ci id="S5.Thmthm7.p2.2.1.m1.1.1.cmml" xref="S5.Thmthm7.p2.2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p2.2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p2.2.1.m1.1d">italic_X</annotation></semantics></math></span> if the following conclusion is true:</p> </div> <div class="ltx_para ltx_noindent" id="S5.Thmthm7.p3"> <p class="ltx_p" id="S5.Thmthm7.p3.2">Consider biinfinite words <math alttext="{\bf x},{\bf x^{\prime}}\in X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm7.p3.1.m1.2"><semantics id="S5.Thmthm7.p3.1.m1.2a"><mrow id="S5.Thmthm7.p3.1.m1.2.2" xref="S5.Thmthm7.p3.1.m1.2.2.cmml"><mrow id="S5.Thmthm7.p3.1.m1.2.2.1.1" xref="S5.Thmthm7.p3.1.m1.2.2.1.2.cmml"><mi id="S5.Thmthm7.p3.1.m1.1.1" xref="S5.Thmthm7.p3.1.m1.1.1.cmml">𝐱</mi><mo id="S5.Thmthm7.p3.1.m1.2.2.1.1.2" xref="S5.Thmthm7.p3.1.m1.2.2.1.2.cmml">,</mo><msup id="S5.Thmthm7.p3.1.m1.2.2.1.1.1" xref="S5.Thmthm7.p3.1.m1.2.2.1.1.1.cmml"><mi id="S5.Thmthm7.p3.1.m1.2.2.1.1.1.2" xref="S5.Thmthm7.p3.1.m1.2.2.1.1.1.2.cmml">𝐱</mi><mo id="S5.Thmthm7.p3.1.m1.2.2.1.1.1.3" xref="S5.Thmthm7.p3.1.m1.2.2.1.1.1.3.cmml">′</mo></msup></mrow><mo id="S5.Thmthm7.p3.1.m1.2.2.3" xref="S5.Thmthm7.p3.1.m1.2.2.3.cmml">∈</mo><mi id="S5.Thmthm7.p3.1.m1.2.2.4" xref="S5.Thmthm7.p3.1.m1.2.2.4.cmml">X</mi><mo id="S5.Thmthm7.p3.1.m1.2.2.5" xref="S5.Thmthm7.p3.1.m1.2.2.5.cmml">⊆</mo><msup id="S5.Thmthm7.p3.1.m1.2.2.6" xref="S5.Thmthm7.p3.1.m1.2.2.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm7.p3.1.m1.2.2.6.2" xref="S5.Thmthm7.p3.1.m1.2.2.6.2.cmml">𝒜</mi><mi id="S5.Thmthm7.p3.1.m1.2.2.6.3" xref="S5.Thmthm7.p3.1.m1.2.2.6.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p3.1.m1.2b"><apply id="S5.Thmthm7.p3.1.m1.2.2.cmml" xref="S5.Thmthm7.p3.1.m1.2.2"><and id="S5.Thmthm7.p3.1.m1.2.2a.cmml" xref="S5.Thmthm7.p3.1.m1.2.2"></and><apply id="S5.Thmthm7.p3.1.m1.2.2b.cmml" xref="S5.Thmthm7.p3.1.m1.2.2"><in id="S5.Thmthm7.p3.1.m1.2.2.3.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.3"></in><list id="S5.Thmthm7.p3.1.m1.2.2.1.2.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.1.1"><ci id="S5.Thmthm7.p3.1.m1.1.1.cmml" xref="S5.Thmthm7.p3.1.m1.1.1">𝐱</ci><apply id="S5.Thmthm7.p3.1.m1.2.2.1.1.1.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmthm7.p3.1.m1.2.2.1.1.1.1.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.1.1.1">superscript</csymbol><ci id="S5.Thmthm7.p3.1.m1.2.2.1.1.1.2.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.1.1.1.2">𝐱</ci><ci id="S5.Thmthm7.p3.1.m1.2.2.1.1.1.3.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.1.1.1.3">′</ci></apply></list><ci id="S5.Thmthm7.p3.1.m1.2.2.4.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.4">𝑋</ci></apply><apply id="S5.Thmthm7.p3.1.m1.2.2c.cmml" xref="S5.Thmthm7.p3.1.m1.2.2"><subset id="S5.Thmthm7.p3.1.m1.2.2.5.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.5"></subset><share href="https://arxiv.org/html/2211.11234v4#S5.Thmthm7.p3.1.m1.2.2.4.cmml" id="S5.Thmthm7.p3.1.m1.2.2d.cmml" xref="S5.Thmthm7.p3.1.m1.2.2"></share><apply id="S5.Thmthm7.p3.1.m1.2.2.6.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.6"><csymbol cd="ambiguous" id="S5.Thmthm7.p3.1.m1.2.2.6.1.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.6">superscript</csymbol><ci id="S5.Thmthm7.p3.1.m1.2.2.6.2.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.6.2">𝒜</ci><ci id="S5.Thmthm7.p3.1.m1.2.2.6.3.cmml" xref="S5.Thmthm7.p3.1.m1.2.2.6.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p3.1.m1.2c">{\bf x},{\bf x^{\prime}}\in X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p3.1.m1.2d">bold_x , bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="{\bf y}\in\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm7.p3.2.m2.1"><semantics id="S5.Thmthm7.p3.2.m2.1a"><mrow id="S5.Thmthm7.p3.2.m2.1.1" xref="S5.Thmthm7.p3.2.m2.1.1.cmml"><mi id="S5.Thmthm7.p3.2.m2.1.1.2" xref="S5.Thmthm7.p3.2.m2.1.1.2.cmml">𝐲</mi><mo id="S5.Thmthm7.p3.2.m2.1.1.1" xref="S5.Thmthm7.p3.2.m2.1.1.1.cmml">∈</mo><msup id="S5.Thmthm7.p3.2.m2.1.1.3" xref="S5.Thmthm7.p3.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm7.p3.2.m2.1.1.3.2" xref="S5.Thmthm7.p3.2.m2.1.1.3.2.cmml">ℬ</mi><mi id="S5.Thmthm7.p3.2.m2.1.1.3.3" xref="S5.Thmthm7.p3.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p3.2.m2.1b"><apply id="S5.Thmthm7.p3.2.m2.1.1.cmml" xref="S5.Thmthm7.p3.2.m2.1.1"><in id="S5.Thmthm7.p3.2.m2.1.1.1.cmml" xref="S5.Thmthm7.p3.2.m2.1.1.1"></in><ci id="S5.Thmthm7.p3.2.m2.1.1.2.cmml" xref="S5.Thmthm7.p3.2.m2.1.1.2">𝐲</ci><apply id="S5.Thmthm7.p3.2.m2.1.1.3.cmml" xref="S5.Thmthm7.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm7.p3.2.m2.1.1.3.1.cmml" xref="S5.Thmthm7.p3.2.m2.1.1.3">superscript</csymbol><ci id="S5.Thmthm7.p3.2.m2.1.1.3.2.cmml" xref="S5.Thmthm7.p3.2.m2.1.1.3.2">ℬ</ci><ci id="S5.Thmthm7.p3.2.m2.1.1.3.3.cmml" xref="S5.Thmthm7.p3.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p3.2.m2.1c">{\bf y}\in\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p3.2.m2.1d">bold_y ∈ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> which satisfy</p> <ol class="ltx_enumerate" id="S5.I3"> <li class="ltx_item" id="S5.I3.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(*)</span> <div class="ltx_para" id="S5.I3.ix1.p1"> <p class="ltx_p" id="S5.I3.ix1.p1.11"><math alttext="{\bf y}=T^{k}(\sigma^{\mathbb{Z}}({\bf x}))" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.1.m1.2"><semantics id="S5.I3.ix1.p1.1.m1.2a"><mrow id="S5.I3.ix1.p1.1.m1.2.2" xref="S5.I3.ix1.p1.1.m1.2.2.cmml"><mi id="S5.I3.ix1.p1.1.m1.2.2.3" xref="S5.I3.ix1.p1.1.m1.2.2.3.cmml">𝐲</mi><mo id="S5.I3.ix1.p1.1.m1.2.2.2" xref="S5.I3.ix1.p1.1.m1.2.2.2.cmml">=</mo><mrow id="S5.I3.ix1.p1.1.m1.2.2.1" xref="S5.I3.ix1.p1.1.m1.2.2.1.cmml"><msup id="S5.I3.ix1.p1.1.m1.2.2.1.3" xref="S5.I3.ix1.p1.1.m1.2.2.1.3.cmml"><mi id="S5.I3.ix1.p1.1.m1.2.2.1.3.2" xref="S5.I3.ix1.p1.1.m1.2.2.1.3.2.cmml">T</mi><mi id="S5.I3.ix1.p1.1.m1.2.2.1.3.3" xref="S5.I3.ix1.p1.1.m1.2.2.1.3.3.cmml">k</mi></msup><mo id="S5.I3.ix1.p1.1.m1.2.2.1.2" xref="S5.I3.ix1.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S5.I3.ix1.p1.1.m1.2.2.1.1.1" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.cmml"><msup id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.cmml"><mi id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.2" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.2.cmml">σ</mi><mi id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.3" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.3.cmml">ℤ</mi></msup><mo id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.1" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.3.2" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mi id="S5.I3.ix1.p1.1.m1.1.1" xref="S5.I3.ix1.p1.1.m1.1.1.cmml">𝐱</mi><mo id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo 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xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1"><times id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.1"></times><apply id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.1.cmml" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2">superscript</csymbol><ci id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.2.cmml" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.2">𝜎</ci><ci id="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.3.cmml" xref="S5.I3.ix1.p1.1.m1.2.2.1.1.1.1.2.3">ℤ</ci></apply><ci id="S5.I3.ix1.p1.1.m1.1.1.cmml" xref="S5.I3.ix1.p1.1.m1.1.1">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.1.m1.2c">{\bf y}=T^{k}(\sigma^{\mathbb{Z}}({\bf x}))</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.1.m1.2d">bold_y = italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x ) )</annotation></semantics></math> and <math alttext="{\bf y}=T^{\ell}(\sigma^{\mathbb{Z}}({\bf x^{\prime}}))" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.2.m2.1"><semantics id="S5.I3.ix1.p1.2.m2.1a"><mrow id="S5.I3.ix1.p1.2.m2.1.1" xref="S5.I3.ix1.p1.2.m2.1.1.cmml"><mi id="S5.I3.ix1.p1.2.m2.1.1.3" xref="S5.I3.ix1.p1.2.m2.1.1.3.cmml">𝐲</mi><mo id="S5.I3.ix1.p1.2.m2.1.1.2" xref="S5.I3.ix1.p1.2.m2.1.1.2.cmml">=</mo><mrow id="S5.I3.ix1.p1.2.m2.1.1.1" xref="S5.I3.ix1.p1.2.m2.1.1.1.cmml"><msup id="S5.I3.ix1.p1.2.m2.1.1.1.3" xref="S5.I3.ix1.p1.2.m2.1.1.1.3.cmml"><mi id="S5.I3.ix1.p1.2.m2.1.1.1.3.2" xref="S5.I3.ix1.p1.2.m2.1.1.1.3.2.cmml">T</mi><mi id="S5.I3.ix1.p1.2.m2.1.1.1.3.3" mathvariant="normal" xref="S5.I3.ix1.p1.2.m2.1.1.1.3.3.cmml">ℓ</mi></msup><mo id="S5.I3.ix1.p1.2.m2.1.1.1.2" xref="S5.I3.ix1.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S5.I3.ix1.p1.2.m2.1.1.1.1.1" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.cmml"><msup id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.cmml"><mi id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.2" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.2.cmml">σ</mi><mi id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.3" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.3.cmml">ℤ</mi></msup><mo id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.2" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.2" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.3" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.2.m2.1b"><apply id="S5.I3.ix1.p1.2.m2.1.1.cmml" xref="S5.I3.ix1.p1.2.m2.1.1"><eq id="S5.I3.ix1.p1.2.m2.1.1.2.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.2"></eq><ci id="S5.I3.ix1.p1.2.m2.1.1.3.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.3">𝐲</ci><apply id="S5.I3.ix1.p1.2.m2.1.1.1.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1"><times id="S5.I3.ix1.p1.2.m2.1.1.1.2.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.2"></times><apply id="S5.I3.ix1.p1.2.m2.1.1.1.3.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.2.m2.1.1.1.3.1.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.3">superscript</csymbol><ci id="S5.I3.ix1.p1.2.m2.1.1.1.3.2.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.3.2">𝑇</ci><ci id="S5.I3.ix1.p1.2.m2.1.1.1.3.3.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.3.3">ℓ</ci></apply><apply id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1"><times id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.2"></times><apply id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3">superscript</csymbol><ci id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.2">𝜎</ci><ci id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.3.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.3.3">ℤ</ci></apply><apply id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.I3.ix1.p1.2.m2.1.1.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.2.m2.1c">{\bf y}=T^{\ell}(\sigma^{\mathbb{Z}}({\bf x^{\prime}}))</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.2.m2.1d">bold_y = italic_T start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ( italic_σ start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT ( bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math> for some integers <math alttext="k,\ell" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.3.m3.2"><semantics id="S5.I3.ix1.p1.3.m3.2a"><mrow id="S5.I3.ix1.p1.3.m3.2.3.2" xref="S5.I3.ix1.p1.3.m3.2.3.1.cmml"><mi id="S5.I3.ix1.p1.3.m3.1.1" xref="S5.I3.ix1.p1.3.m3.1.1.cmml">k</mi><mo id="S5.I3.ix1.p1.3.m3.2.3.2.1" xref="S5.I3.ix1.p1.3.m3.2.3.1.cmml">,</mo><mi id="S5.I3.ix1.p1.3.m3.2.2" mathvariant="normal" xref="S5.I3.ix1.p1.3.m3.2.2.cmml">ℓ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.3.m3.2b"><list id="S5.I3.ix1.p1.3.m3.2.3.1.cmml" xref="S5.I3.ix1.p1.3.m3.2.3.2"><ci id="S5.I3.ix1.p1.3.m3.1.1.cmml" xref="S5.I3.ix1.p1.3.m3.1.1">𝑘</ci><ci id="S5.I3.ix1.p1.3.m3.2.2.cmml" xref="S5.I3.ix1.p1.3.m3.2.2">ℓ</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.3.m3.2c">k,\ell</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.3.m3.2d">italic_k , roman_ℓ</annotation></semantics></math> which satisfy <math alttext="0\leq k\leq|\sigma(x_{1})|-1" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.4.m4.1"><semantics id="S5.I3.ix1.p1.4.m4.1a"><mrow id="S5.I3.ix1.p1.4.m4.1.1" xref="S5.I3.ix1.p1.4.m4.1.1.cmml"><mn id="S5.I3.ix1.p1.4.m4.1.1.3" xref="S5.I3.ix1.p1.4.m4.1.1.3.cmml">0</mn><mo id="S5.I3.ix1.p1.4.m4.1.1.4" xref="S5.I3.ix1.p1.4.m4.1.1.4.cmml">≤</mo><mi id="S5.I3.ix1.p1.4.m4.1.1.5" xref="S5.I3.ix1.p1.4.m4.1.1.5.cmml">k</mi><mo id="S5.I3.ix1.p1.4.m4.1.1.6" xref="S5.I3.ix1.p1.4.m4.1.1.6.cmml">≤</mo><mrow id="S5.I3.ix1.p1.4.m4.1.1.1" xref="S5.I3.ix1.p1.4.m4.1.1.1.cmml"><mrow id="S5.I3.ix1.p1.4.m4.1.1.1.1.1" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.2.cmml"><mo id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.2.1.cmml">|</mo><mrow id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.3" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.3.cmml">σ</mi><mo id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.2" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.2" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mn id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.3" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.I3.ix1.p1.4.m4.1.1.1.2" xref="S5.I3.ix1.p1.4.m4.1.1.1.2.cmml">−</mo><mn id="S5.I3.ix1.p1.4.m4.1.1.1.3" xref="S5.I3.ix1.p1.4.m4.1.1.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.4.m4.1b"><apply id="S5.I3.ix1.p1.4.m4.1.1.cmml" xref="S5.I3.ix1.p1.4.m4.1.1"><and id="S5.I3.ix1.p1.4.m4.1.1a.cmml" xref="S5.I3.ix1.p1.4.m4.1.1"></and><apply id="S5.I3.ix1.p1.4.m4.1.1b.cmml" xref="S5.I3.ix1.p1.4.m4.1.1"><leq id="S5.I3.ix1.p1.4.m4.1.1.4.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.4"></leq><cn id="S5.I3.ix1.p1.4.m4.1.1.3.cmml" type="integer" xref="S5.I3.ix1.p1.4.m4.1.1.3">0</cn><ci id="S5.I3.ix1.p1.4.m4.1.1.5.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.5">𝑘</ci></apply><apply id="S5.I3.ix1.p1.4.m4.1.1c.cmml" xref="S5.I3.ix1.p1.4.m4.1.1"><leq id="S5.I3.ix1.p1.4.m4.1.1.6.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.6"></leq><share href="https://arxiv.org/html/2211.11234v4#S5.I3.ix1.p1.4.m4.1.1.5.cmml" id="S5.I3.ix1.p1.4.m4.1.1d.cmml" xref="S5.I3.ix1.p1.4.m4.1.1"></share><apply id="S5.I3.ix1.p1.4.m4.1.1.1.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1"><minus id="S5.I3.ix1.p1.4.m4.1.1.1.2.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.2"></minus><apply id="S5.I3.ix1.p1.4.m4.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1"><abs id="S5.I3.ix1.p1.4.m4.1.1.1.1.2.1.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.2"></abs><apply id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1"><times id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.2"></times><ci id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.3">𝜎</ci><apply id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.2">𝑥</ci><cn id="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.I3.ix1.p1.4.m4.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply></apply><cn id="S5.I3.ix1.p1.4.m4.1.1.1.3.cmml" type="integer" xref="S5.I3.ix1.p1.4.m4.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.4.m4.1c">0\leq k\leq|\sigma(x_{1})|-1</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.4.m4.1d">0 ≤ italic_k ≤ | italic_σ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) | - 1</annotation></semantics></math> and <math alttext="0\leq\ell\leq|\sigma(x^{\prime}_{1})|-1" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.5.m5.1"><semantics id="S5.I3.ix1.p1.5.m5.1a"><mrow id="S5.I3.ix1.p1.5.m5.1.1" xref="S5.I3.ix1.p1.5.m5.1.1.cmml"><mn id="S5.I3.ix1.p1.5.m5.1.1.3" xref="S5.I3.ix1.p1.5.m5.1.1.3.cmml">0</mn><mo id="S5.I3.ix1.p1.5.m5.1.1.4" xref="S5.I3.ix1.p1.5.m5.1.1.4.cmml">≤</mo><mi id="S5.I3.ix1.p1.5.m5.1.1.5" mathvariant="normal" xref="S5.I3.ix1.p1.5.m5.1.1.5.cmml">ℓ</mi><mo id="S5.I3.ix1.p1.5.m5.1.1.6" xref="S5.I3.ix1.p1.5.m5.1.1.6.cmml">≤</mo><mrow id="S5.I3.ix1.p1.5.m5.1.1.1" xref="S5.I3.ix1.p1.5.m5.1.1.1.cmml"><mrow id="S5.I3.ix1.p1.5.m5.1.1.1.1.1" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.2.cmml"><mo id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.2.1.cmml">|</mo><mrow id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.3" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.3.cmml">σ</mi><mo id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.2" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml">(</mo><msubsup id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.2" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.2.cmml">x</mi><mn id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.3" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.3.cmml">1</mn><mo id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.3" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.3.cmml">′</mo></msubsup><mo id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S5.I3.ix1.p1.5.m5.1.1.1.2" xref="S5.I3.ix1.p1.5.m5.1.1.1.2.cmml">−</mo><mn id="S5.I3.ix1.p1.5.m5.1.1.1.3" xref="S5.I3.ix1.p1.5.m5.1.1.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.5.m5.1b"><apply id="S5.I3.ix1.p1.5.m5.1.1.cmml" xref="S5.I3.ix1.p1.5.m5.1.1"><and id="S5.I3.ix1.p1.5.m5.1.1a.cmml" xref="S5.I3.ix1.p1.5.m5.1.1"></and><apply id="S5.I3.ix1.p1.5.m5.1.1b.cmml" xref="S5.I3.ix1.p1.5.m5.1.1"><leq id="S5.I3.ix1.p1.5.m5.1.1.4.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.4"></leq><cn id="S5.I3.ix1.p1.5.m5.1.1.3.cmml" type="integer" xref="S5.I3.ix1.p1.5.m5.1.1.3">0</cn><ci id="S5.I3.ix1.p1.5.m5.1.1.5.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.5">ℓ</ci></apply><apply id="S5.I3.ix1.p1.5.m5.1.1c.cmml" xref="S5.I3.ix1.p1.5.m5.1.1"><leq id="S5.I3.ix1.p1.5.m5.1.1.6.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.6"></leq><share href="https://arxiv.org/html/2211.11234v4#S5.I3.ix1.p1.5.m5.1.1.5.cmml" id="S5.I3.ix1.p1.5.m5.1.1d.cmml" xref="S5.I3.ix1.p1.5.m5.1.1"></share><apply id="S5.I3.ix1.p1.5.m5.1.1.1.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1"><minus id="S5.I3.ix1.p1.5.m5.1.1.1.2.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.2"></minus><apply id="S5.I3.ix1.p1.5.m5.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1"><abs id="S5.I3.ix1.p1.5.m5.1.1.1.1.2.1.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.2"></abs><apply id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1"><times id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.2"></times><ci id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.3">𝜎</ci><apply id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1">subscript</csymbol><apply id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.2">𝑥</ci><ci id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.3.cmml" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.2.3">′</ci></apply><cn id="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S5.I3.ix1.p1.5.m5.1.1.1.1.1.1.1.1.1.3">1</cn></apply></apply></apply><cn id="S5.I3.ix1.p1.5.m5.1.1.1.3.cmml" type="integer" xref="S5.I3.ix1.p1.5.m5.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.5.m5.1c">0\leq\ell\leq|\sigma(x^{\prime}_{1})|-1</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.5.m5.1d">0 ≤ roman_ℓ ≤ | italic_σ ( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) | - 1</annotation></semantics></math>, where <math alttext="x_{1}" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.6.m6.1"><semantics id="S5.I3.ix1.p1.6.m6.1a"><msub id="S5.I3.ix1.p1.6.m6.1.1" xref="S5.I3.ix1.p1.6.m6.1.1.cmml"><mi id="S5.I3.ix1.p1.6.m6.1.1.2" xref="S5.I3.ix1.p1.6.m6.1.1.2.cmml">x</mi><mn id="S5.I3.ix1.p1.6.m6.1.1.3" xref="S5.I3.ix1.p1.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.6.m6.1b"><apply id="S5.I3.ix1.p1.6.m6.1.1.cmml" xref="S5.I3.ix1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.6.m6.1.1.1.cmml" xref="S5.I3.ix1.p1.6.m6.1.1">subscript</csymbol><ci id="S5.I3.ix1.p1.6.m6.1.1.2.cmml" xref="S5.I3.ix1.p1.6.m6.1.1.2">𝑥</ci><cn id="S5.I3.ix1.p1.6.m6.1.1.3.cmml" type="integer" xref="S5.I3.ix1.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.6.m6.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.6.m6.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x^{\prime}_{1}" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.7.m7.1"><semantics id="S5.I3.ix1.p1.7.m7.1a"><msubsup id="S5.I3.ix1.p1.7.m7.1.1" xref="S5.I3.ix1.p1.7.m7.1.1.cmml"><mi id="S5.I3.ix1.p1.7.m7.1.1.2.2" xref="S5.I3.ix1.p1.7.m7.1.1.2.2.cmml">x</mi><mn id="S5.I3.ix1.p1.7.m7.1.1.3" xref="S5.I3.ix1.p1.7.m7.1.1.3.cmml">1</mn><mo id="S5.I3.ix1.p1.7.m7.1.1.2.3" xref="S5.I3.ix1.p1.7.m7.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.7.m7.1b"><apply id="S5.I3.ix1.p1.7.m7.1.1.cmml" xref="S5.I3.ix1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.7.m7.1.1.1.cmml" xref="S5.I3.ix1.p1.7.m7.1.1">subscript</csymbol><apply id="S5.I3.ix1.p1.7.m7.1.1.2.cmml" xref="S5.I3.ix1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.7.m7.1.1.2.1.cmml" xref="S5.I3.ix1.p1.7.m7.1.1">superscript</csymbol><ci id="S5.I3.ix1.p1.7.m7.1.1.2.2.cmml" xref="S5.I3.ix1.p1.7.m7.1.1.2.2">𝑥</ci><ci id="S5.I3.ix1.p1.7.m7.1.1.2.3.cmml" xref="S5.I3.ix1.p1.7.m7.1.1.2.3">′</ci></apply><cn id="S5.I3.ix1.p1.7.m7.1.1.3.cmml" type="integer" xref="S5.I3.ix1.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.7.m7.1c">x^{\prime}_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.7.m7.1d">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> are the first letters of the positive half-words <math alttext="{\bf x_{[1,\infty)}}=x_{1}x_{2}\ldots" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.8.m8.2"><semantics id="S5.I3.ix1.p1.8.m8.2a"><mrow id="S5.I3.ix1.p1.8.m8.2.3" xref="S5.I3.ix1.p1.8.m8.2.3.cmml"><msub id="S5.I3.ix1.p1.8.m8.2.3.2" xref="S5.I3.ix1.p1.8.m8.2.3.2.cmml"><mi id="S5.I3.ix1.p1.8.m8.2.3.2.2" xref="S5.I3.ix1.p1.8.m8.2.3.2.2.cmml">𝐱</mi><mrow id="S5.I3.ix1.p1.8.m8.2.2.2.4" xref="S5.I3.ix1.p1.8.m8.2.2.2.3.cmml"><mo id="S5.I3.ix1.p1.8.m8.2.2.2.4.1" stretchy="false" xref="S5.I3.ix1.p1.8.m8.2.2.2.3.cmml">[</mo><mn id="S5.I3.ix1.p1.8.m8.1.1.1.1" xref="S5.I3.ix1.p1.8.m8.1.1.1.1.cmml">𝟏</mn><mo id="S5.I3.ix1.p1.8.m8.2.2.2.4.2" xref="S5.I3.ix1.p1.8.m8.2.2.2.3.cmml">,</mo><mi id="S5.I3.ix1.p1.8.m8.2.2.2.2" mathvariant="normal" xref="S5.I3.ix1.p1.8.m8.2.2.2.2.cmml">∞</mi><mo id="S5.I3.ix1.p1.8.m8.2.2.2.4.3" stretchy="false" xref="S5.I3.ix1.p1.8.m8.2.2.2.3.cmml">)</mo></mrow></msub><mo id="S5.I3.ix1.p1.8.m8.2.3.1" xref="S5.I3.ix1.p1.8.m8.2.3.1.cmml">=</mo><mrow id="S5.I3.ix1.p1.8.m8.2.3.3" xref="S5.I3.ix1.p1.8.m8.2.3.3.cmml"><msub id="S5.I3.ix1.p1.8.m8.2.3.3.2" xref="S5.I3.ix1.p1.8.m8.2.3.3.2.cmml"><mi id="S5.I3.ix1.p1.8.m8.2.3.3.2.2" xref="S5.I3.ix1.p1.8.m8.2.3.3.2.2.cmml">x</mi><mn id="S5.I3.ix1.p1.8.m8.2.3.3.2.3" xref="S5.I3.ix1.p1.8.m8.2.3.3.2.3.cmml">1</mn></msub><mo id="S5.I3.ix1.p1.8.m8.2.3.3.1" xref="S5.I3.ix1.p1.8.m8.2.3.3.1.cmml">⁢</mo><msub id="S5.I3.ix1.p1.8.m8.2.3.3.3" xref="S5.I3.ix1.p1.8.m8.2.3.3.3.cmml"><mi id="S5.I3.ix1.p1.8.m8.2.3.3.3.2" xref="S5.I3.ix1.p1.8.m8.2.3.3.3.2.cmml">x</mi><mn id="S5.I3.ix1.p1.8.m8.2.3.3.3.3" xref="S5.I3.ix1.p1.8.m8.2.3.3.3.3.cmml">2</mn></msub><mo id="S5.I3.ix1.p1.8.m8.2.3.3.1a" xref="S5.I3.ix1.p1.8.m8.2.3.3.1.cmml">⁢</mo><mi id="S5.I3.ix1.p1.8.m8.2.3.3.4" mathvariant="normal" xref="S5.I3.ix1.p1.8.m8.2.3.3.4.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.8.m8.2b"><apply id="S5.I3.ix1.p1.8.m8.2.3.cmml" xref="S5.I3.ix1.p1.8.m8.2.3"><eq id="S5.I3.ix1.p1.8.m8.2.3.1.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.1"></eq><apply id="S5.I3.ix1.p1.8.m8.2.3.2.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.2"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.8.m8.2.3.2.1.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.2">subscript</csymbol><ci id="S5.I3.ix1.p1.8.m8.2.3.2.2.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.2.2">𝐱</ci><interval closure="closed-open" id="S5.I3.ix1.p1.8.m8.2.2.2.3.cmml" xref="S5.I3.ix1.p1.8.m8.2.2.2.4"><cn id="S5.I3.ix1.p1.8.m8.1.1.1.1.cmml" type="integer" xref="S5.I3.ix1.p1.8.m8.1.1.1.1">1</cn><infinity id="S5.I3.ix1.p1.8.m8.2.2.2.2.cmml" xref="S5.I3.ix1.p1.8.m8.2.2.2.2"></infinity></interval></apply><apply id="S5.I3.ix1.p1.8.m8.2.3.3.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3"><times id="S5.I3.ix1.p1.8.m8.2.3.3.1.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3.1"></times><apply id="S5.I3.ix1.p1.8.m8.2.3.3.2.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3.2"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.8.m8.2.3.3.2.1.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3.2">subscript</csymbol><ci id="S5.I3.ix1.p1.8.m8.2.3.3.2.2.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3.2.2">𝑥</ci><cn id="S5.I3.ix1.p1.8.m8.2.3.3.2.3.cmml" type="integer" xref="S5.I3.ix1.p1.8.m8.2.3.3.2.3">1</cn></apply><apply id="S5.I3.ix1.p1.8.m8.2.3.3.3.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3.3"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.8.m8.2.3.3.3.1.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3.3">subscript</csymbol><ci id="S5.I3.ix1.p1.8.m8.2.3.3.3.2.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3.3.2">𝑥</ci><cn id="S5.I3.ix1.p1.8.m8.2.3.3.3.3.cmml" type="integer" xref="S5.I3.ix1.p1.8.m8.2.3.3.3.3">2</cn></apply><ci id="S5.I3.ix1.p1.8.m8.2.3.3.4.cmml" xref="S5.I3.ix1.p1.8.m8.2.3.3.4">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.8.m8.2c">{\bf x_{[1,\infty)}}=x_{1}x_{2}\ldots</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.8.m8.2d">bold_x start_POSTSUBSCRIPT [ bold_1 , ∞ ) end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT …</annotation></semantics></math> of <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.9.m9.1"><semantics id="S5.I3.ix1.p1.9.m9.1a"><mi id="S5.I3.ix1.p1.9.m9.1.1" xref="S5.I3.ix1.p1.9.m9.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.9.m9.1b"><ci id="S5.I3.ix1.p1.9.m9.1.1.cmml" xref="S5.I3.ix1.p1.9.m9.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.9.m9.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.9.m9.1d">bold_x</annotation></semantics></math> and <math alttext="{\bf x^{\prime}_{[1,\infty)}}=x^{\prime}_{1}x^{\prime}_{2}\ldots" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.10.m10.2"><semantics id="S5.I3.ix1.p1.10.m10.2a"><mrow id="S5.I3.ix1.p1.10.m10.2.3" xref="S5.I3.ix1.p1.10.m10.2.3.cmml"><msubsup id="S5.I3.ix1.p1.10.m10.2.3.2" xref="S5.I3.ix1.p1.10.m10.2.3.2.cmml"><mi id="S5.I3.ix1.p1.10.m10.2.3.2.2.2" xref="S5.I3.ix1.p1.10.m10.2.3.2.2.2.cmml">𝐱</mi><mrow id="S5.I3.ix1.p1.10.m10.2.2.2.4" xref="S5.I3.ix1.p1.10.m10.2.2.2.3.cmml"><mo id="S5.I3.ix1.p1.10.m10.2.2.2.4.1" stretchy="false" xref="S5.I3.ix1.p1.10.m10.2.2.2.3.cmml">[</mo><mn id="S5.I3.ix1.p1.10.m10.1.1.1.1" xref="S5.I3.ix1.p1.10.m10.1.1.1.1.cmml">𝟏</mn><mo id="S5.I3.ix1.p1.10.m10.2.2.2.4.2" xref="S5.I3.ix1.p1.10.m10.2.2.2.3.cmml">,</mo><mi id="S5.I3.ix1.p1.10.m10.2.2.2.2" mathvariant="normal" xref="S5.I3.ix1.p1.10.m10.2.2.2.2.cmml">∞</mi><mo id="S5.I3.ix1.p1.10.m10.2.2.2.4.3" stretchy="false" xref="S5.I3.ix1.p1.10.m10.2.2.2.3.cmml">)</mo></mrow><mo id="S5.I3.ix1.p1.10.m10.2.3.2.2.3" xref="S5.I3.ix1.p1.10.m10.2.3.2.2.3.cmml">′</mo></msubsup><mo id="S5.I3.ix1.p1.10.m10.2.3.1" xref="S5.I3.ix1.p1.10.m10.2.3.1.cmml">=</mo><mrow id="S5.I3.ix1.p1.10.m10.2.3.3" xref="S5.I3.ix1.p1.10.m10.2.3.3.cmml"><msubsup id="S5.I3.ix1.p1.10.m10.2.3.3.2" xref="S5.I3.ix1.p1.10.m10.2.3.3.2.cmml"><mi id="S5.I3.ix1.p1.10.m10.2.3.3.2.2.2" xref="S5.I3.ix1.p1.10.m10.2.3.3.2.2.2.cmml">x</mi><mn id="S5.I3.ix1.p1.10.m10.2.3.3.2.3" xref="S5.I3.ix1.p1.10.m10.2.3.3.2.3.cmml">1</mn><mo id="S5.I3.ix1.p1.10.m10.2.3.3.2.2.3" xref="S5.I3.ix1.p1.10.m10.2.3.3.2.2.3.cmml">′</mo></msubsup><mo id="S5.I3.ix1.p1.10.m10.2.3.3.1" xref="S5.I3.ix1.p1.10.m10.2.3.3.1.cmml">⁢</mo><msubsup id="S5.I3.ix1.p1.10.m10.2.3.3.3" xref="S5.I3.ix1.p1.10.m10.2.3.3.3.cmml"><mi id="S5.I3.ix1.p1.10.m10.2.3.3.3.2.2" xref="S5.I3.ix1.p1.10.m10.2.3.3.3.2.2.cmml">x</mi><mn id="S5.I3.ix1.p1.10.m10.2.3.3.3.3" xref="S5.I3.ix1.p1.10.m10.2.3.3.3.3.cmml">2</mn><mo id="S5.I3.ix1.p1.10.m10.2.3.3.3.2.3" xref="S5.I3.ix1.p1.10.m10.2.3.3.3.2.3.cmml">′</mo></msubsup><mo id="S5.I3.ix1.p1.10.m10.2.3.3.1a" xref="S5.I3.ix1.p1.10.m10.2.3.3.1.cmml">⁢</mo><mi id="S5.I3.ix1.p1.10.m10.2.3.3.4" mathvariant="normal" xref="S5.I3.ix1.p1.10.m10.2.3.3.4.cmml">…</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.10.m10.2b"><apply id="S5.I3.ix1.p1.10.m10.2.3.cmml" xref="S5.I3.ix1.p1.10.m10.2.3"><eq id="S5.I3.ix1.p1.10.m10.2.3.1.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.1"></eq><apply id="S5.I3.ix1.p1.10.m10.2.3.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.2"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.10.m10.2.3.2.1.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.2">subscript</csymbol><apply id="S5.I3.ix1.p1.10.m10.2.3.2.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.2"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.10.m10.2.3.2.2.1.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.2">superscript</csymbol><ci id="S5.I3.ix1.p1.10.m10.2.3.2.2.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.2.2.2">𝐱</ci><ci id="S5.I3.ix1.p1.10.m10.2.3.2.2.3.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.2.2.3">′</ci></apply><interval closure="closed-open" id="S5.I3.ix1.p1.10.m10.2.2.2.3.cmml" xref="S5.I3.ix1.p1.10.m10.2.2.2.4"><cn id="S5.I3.ix1.p1.10.m10.1.1.1.1.cmml" type="integer" xref="S5.I3.ix1.p1.10.m10.1.1.1.1">1</cn><infinity id="S5.I3.ix1.p1.10.m10.2.2.2.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.2.2.2"></infinity></interval></apply><apply id="S5.I3.ix1.p1.10.m10.2.3.3.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3"><times id="S5.I3.ix1.p1.10.m10.2.3.3.1.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.1"></times><apply id="S5.I3.ix1.p1.10.m10.2.3.3.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.2"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.10.m10.2.3.3.2.1.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.2">subscript</csymbol><apply id="S5.I3.ix1.p1.10.m10.2.3.3.2.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.2"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.10.m10.2.3.3.2.2.1.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.2">superscript</csymbol><ci id="S5.I3.ix1.p1.10.m10.2.3.3.2.2.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.2.2.2">𝑥</ci><ci id="S5.I3.ix1.p1.10.m10.2.3.3.2.2.3.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.2.2.3">′</ci></apply><cn id="S5.I3.ix1.p1.10.m10.2.3.3.2.3.cmml" type="integer" xref="S5.I3.ix1.p1.10.m10.2.3.3.2.3">1</cn></apply><apply id="S5.I3.ix1.p1.10.m10.2.3.3.3.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.3"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.10.m10.2.3.3.3.1.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.3">subscript</csymbol><apply id="S5.I3.ix1.p1.10.m10.2.3.3.3.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.3"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.10.m10.2.3.3.3.2.1.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.3">superscript</csymbol><ci id="S5.I3.ix1.p1.10.m10.2.3.3.3.2.2.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.3.2.2">𝑥</ci><ci id="S5.I3.ix1.p1.10.m10.2.3.3.3.2.3.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.3.2.3">′</ci></apply><cn id="S5.I3.ix1.p1.10.m10.2.3.3.3.3.cmml" type="integer" xref="S5.I3.ix1.p1.10.m10.2.3.3.3.3">2</cn></apply><ci id="S5.I3.ix1.p1.10.m10.2.3.3.4.cmml" xref="S5.I3.ix1.p1.10.m10.2.3.3.4">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.10.m10.2c">{\bf x^{\prime}_{[1,\infty)}}=x^{\prime}_{1}x^{\prime}_{2}\ldots</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.10.m10.2d">bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT [ bold_1 , ∞ ) end_POSTSUBSCRIPT = italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT …</annotation></semantics></math> of <math alttext="{\bf x^{\prime}}" class="ltx_Math" display="inline" id="S5.I3.ix1.p1.11.m11.1"><semantics id="S5.I3.ix1.p1.11.m11.1a"><msup id="S5.I3.ix1.p1.11.m11.1.1" xref="S5.I3.ix1.p1.11.m11.1.1.cmml"><mi id="S5.I3.ix1.p1.11.m11.1.1.2" xref="S5.I3.ix1.p1.11.m11.1.1.2.cmml">𝐱</mi><mo id="S5.I3.ix1.p1.11.m11.1.1.3" xref="S5.I3.ix1.p1.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I3.ix1.p1.11.m11.1b"><apply id="S5.I3.ix1.p1.11.m11.1.1.cmml" xref="S5.I3.ix1.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S5.I3.ix1.p1.11.m11.1.1.1.cmml" xref="S5.I3.ix1.p1.11.m11.1.1">superscript</csymbol><ci id="S5.I3.ix1.p1.11.m11.1.1.2.cmml" xref="S5.I3.ix1.p1.11.m11.1.1.2">𝐱</ci><ci id="S5.I3.ix1.p1.11.m11.1.1.3.cmml" xref="S5.I3.ix1.p1.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.ix1.p1.11.m11.1c">{\bf x^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.ix1.p1.11.m11.1d">bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> respectively.</p> </div> </li> </ol> <p class="ltx_p" id="S5.Thmthm7.p3.4">Then one has <math alttext="{\bf x}=\bf x^{\prime}" class="ltx_Math" display="inline" id="S5.Thmthm7.p3.3.m1.1"><semantics id="S5.Thmthm7.p3.3.m1.1a"><mrow id="S5.Thmthm7.p3.3.m1.1.1" xref="S5.Thmthm7.p3.3.m1.1.1.cmml"><mi id="S5.Thmthm7.p3.3.m1.1.1.2" xref="S5.Thmthm7.p3.3.m1.1.1.2.cmml">𝐱</mi><mo id="S5.Thmthm7.p3.3.m1.1.1.1" xref="S5.Thmthm7.p3.3.m1.1.1.1.cmml">=</mo><msup id="S5.Thmthm7.p3.3.m1.1.1.3" xref="S5.Thmthm7.p3.3.m1.1.1.3.cmml"><mi id="S5.Thmthm7.p3.3.m1.1.1.3.2" xref="S5.Thmthm7.p3.3.m1.1.1.3.2.cmml">𝐱</mi><mo id="S5.Thmthm7.p3.3.m1.1.1.3.3" xref="S5.Thmthm7.p3.3.m1.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p3.3.m1.1b"><apply id="S5.Thmthm7.p3.3.m1.1.1.cmml" xref="S5.Thmthm7.p3.3.m1.1.1"><eq id="S5.Thmthm7.p3.3.m1.1.1.1.cmml" xref="S5.Thmthm7.p3.3.m1.1.1.1"></eq><ci id="S5.Thmthm7.p3.3.m1.1.1.2.cmml" xref="S5.Thmthm7.p3.3.m1.1.1.2">𝐱</ci><apply id="S5.Thmthm7.p3.3.m1.1.1.3.cmml" xref="S5.Thmthm7.p3.3.m1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm7.p3.3.m1.1.1.3.1.cmml" xref="S5.Thmthm7.p3.3.m1.1.1.3">superscript</csymbol><ci id="S5.Thmthm7.p3.3.m1.1.1.3.2.cmml" xref="S5.Thmthm7.p3.3.m1.1.1.3.2">𝐱</ci><ci id="S5.Thmthm7.p3.3.m1.1.1.3.3.cmml" xref="S5.Thmthm7.p3.3.m1.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p3.3.m1.1c">{\bf x}=\bf x^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p3.3.m1.1d">bold_x = bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="k=\ell" class="ltx_Math" display="inline" id="S5.Thmthm7.p3.4.m2.1"><semantics id="S5.Thmthm7.p3.4.m2.1a"><mrow id="S5.Thmthm7.p3.4.m2.1.1" xref="S5.Thmthm7.p3.4.m2.1.1.cmml"><mi id="S5.Thmthm7.p3.4.m2.1.1.2" xref="S5.Thmthm7.p3.4.m2.1.1.2.cmml">k</mi><mo id="S5.Thmthm7.p3.4.m2.1.1.1" xref="S5.Thmthm7.p3.4.m2.1.1.1.cmml">=</mo><mi id="S5.Thmthm7.p3.4.m2.1.1.3" mathvariant="normal" xref="S5.Thmthm7.p3.4.m2.1.1.3.cmml">ℓ</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p3.4.m2.1b"><apply id="S5.Thmthm7.p3.4.m2.1.1.cmml" xref="S5.Thmthm7.p3.4.m2.1.1"><eq id="S5.Thmthm7.p3.4.m2.1.1.1.cmml" xref="S5.Thmthm7.p3.4.m2.1.1.1"></eq><ci id="S5.Thmthm7.p3.4.m2.1.1.2.cmml" xref="S5.Thmthm7.p3.4.m2.1.1.2">𝑘</ci><ci id="S5.Thmthm7.p3.4.m2.1.1.3.cmml" xref="S5.Thmthm7.p3.4.m2.1.1.3">ℓ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p3.4.m2.1c">k=\ell</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p3.4.m2.1d">italic_k = roman_ℓ</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S5.Thmthm7.p4"> <p class="ltx_p" id="S5.Thmthm7.p4.3">(2) The morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm7.p4.1.m1.1"><semantics id="S5.Thmthm7.p4.1.m1.1a"><mi id="S5.Thmthm7.p4.1.m1.1.1" xref="S5.Thmthm7.p4.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p4.1.m1.1b"><ci id="S5.Thmthm7.p4.1.m1.1.1.cmml" xref="S5.Thmthm7.p4.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p4.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p4.1.m1.1d">italic_σ</annotation></semantics></math> is called <span class="ltx_text ltx_font_italic" id="S5.Thmthm7.p4.2.1">recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm7.p4.2.1.m1.1"><semantics id="S5.Thmthm7.p4.2.1.m1.1a"><mi id="S5.Thmthm7.p4.2.1.m1.1.1" xref="S5.Thmthm7.p4.2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p4.2.1.m1.1b"><ci id="S5.Thmthm7.p4.2.1.m1.1.1.cmml" xref="S5.Thmthm7.p4.2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p4.2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p4.2.1.m1.1d">italic_X</annotation></semantics></math></span> if the same conclusion as in (1) is true, but under the strengthened hypothesis that in addition <math alttext="\bf y" class="ltx_Math" display="inline" id="S5.Thmthm7.p4.3.m2.1"><semantics id="S5.Thmthm7.p4.3.m2.1a"><mi id="S5.Thmthm7.p4.3.m2.1.1" xref="S5.Thmthm7.p4.3.m2.1.1.cmml">𝐲</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm7.p4.3.m2.1b"><ci id="S5.Thmthm7.p4.3.m2.1.1.cmml" xref="S5.Thmthm7.p4.3.m2.1.1">𝐲</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm7.p4.3.m2.1c">\bf y</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm7.p4.3.m2.1d">bold_y</annotation></semantics></math> is assumed not to be a periodic word.</p> </div> </div> <div class="ltx_theorem ltx_theorem_warning" id="S5.Thmthm8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm8.1.1.1">Warning 5.8</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm8.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm8.p1"> <p class="ltx_p" id="S5.Thmthm8.p1.5">The terminology “recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm8.p1.1.m1.1"><semantics id="S5.Thmthm8.p1.1.m1.1a"><mi id="S5.Thmthm8.p1.1.m1.1.1" xref="S5.Thmthm8.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm8.p1.1.m1.1b"><ci id="S5.Thmthm8.p1.1.m1.1.1.cmml" xref="S5.Thmthm8.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm8.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm8.p1.1.m1.1d">italic_X</annotation></semantics></math>” should not be misunderstood as to be referring to aperiodic words in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm8.p1.2.m2.1"><semantics id="S5.Thmthm8.p1.2.m2.1a"><mi id="S5.Thmthm8.p1.2.m2.1.1" xref="S5.Thmthm8.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm8.p1.2.m2.1b"><ci id="S5.Thmthm8.p1.2.m2.1.1.cmml" xref="S5.Thmthm8.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm8.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm8.p1.2.m2.1d">italic_X</annotation></semantics></math>: it really does concern aperiodic words in the image subshift <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S5.Thmthm8.p1.3.m3.1"><semantics id="S5.Thmthm8.p1.3.m3.1a"><mrow id="S5.Thmthm8.p1.3.m3.1.2" xref="S5.Thmthm8.p1.3.m3.1.2.cmml"><mi id="S5.Thmthm8.p1.3.m3.1.2.2" xref="S5.Thmthm8.p1.3.m3.1.2.2.cmml">σ</mi><mo id="S5.Thmthm8.p1.3.m3.1.2.1" xref="S5.Thmthm8.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S5.Thmthm8.p1.3.m3.1.2.3.2" xref="S5.Thmthm8.p1.3.m3.1.2.cmml"><mo id="S5.Thmthm8.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S5.Thmthm8.p1.3.m3.1.2.cmml">(</mo><mi id="S5.Thmthm8.p1.3.m3.1.1" xref="S5.Thmthm8.p1.3.m3.1.1.cmml">X</mi><mo id="S5.Thmthm8.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S5.Thmthm8.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm8.p1.3.m3.1b"><apply id="S5.Thmthm8.p1.3.m3.1.2.cmml" xref="S5.Thmthm8.p1.3.m3.1.2"><times id="S5.Thmthm8.p1.3.m3.1.2.1.cmml" xref="S5.Thmthm8.p1.3.m3.1.2.1"></times><ci id="S5.Thmthm8.p1.3.m3.1.2.2.cmml" xref="S5.Thmthm8.p1.3.m3.1.2.2">𝜎</ci><ci id="S5.Thmthm8.p1.3.m3.1.1.cmml" xref="S5.Thmthm8.p1.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm8.p1.3.m3.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm8.p1.3.m3.1d">italic_σ ( italic_X )</annotation></semantics></math>. The alternative wording “recognizable in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm8.p1.4.m4.1"><semantics id="S5.Thmthm8.p1.4.m4.1a"><mi id="S5.Thmthm8.p1.4.m4.1.1" xref="S5.Thmthm8.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm8.p1.4.m4.1b"><ci id="S5.Thmthm8.p1.4.m4.1.1.cmml" xref="S5.Thmthm8.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm8.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm8.p1.4.m4.1d">italic_X</annotation></semantics></math> for aperiodic points of <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S5.Thmthm8.p1.5.m5.1"><semantics id="S5.Thmthm8.p1.5.m5.1a"><mrow id="S5.Thmthm8.p1.5.m5.1.2" xref="S5.Thmthm8.p1.5.m5.1.2.cmml"><mi id="S5.Thmthm8.p1.5.m5.1.2.2" xref="S5.Thmthm8.p1.5.m5.1.2.2.cmml">σ</mi><mo id="S5.Thmthm8.p1.5.m5.1.2.1" xref="S5.Thmthm8.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S5.Thmthm8.p1.5.m5.1.2.3.2" xref="S5.Thmthm8.p1.5.m5.1.2.cmml"><mo id="S5.Thmthm8.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S5.Thmthm8.p1.5.m5.1.2.cmml">(</mo><mi id="S5.Thmthm8.p1.5.m5.1.1" xref="S5.Thmthm8.p1.5.m5.1.1.cmml">X</mi><mo id="S5.Thmthm8.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S5.Thmthm8.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm8.p1.5.m5.1b"><apply id="S5.Thmthm8.p1.5.m5.1.2.cmml" xref="S5.Thmthm8.p1.5.m5.1.2"><times id="S5.Thmthm8.p1.5.m5.1.2.1.cmml" xref="S5.Thmthm8.p1.5.m5.1.2.1"></times><ci id="S5.Thmthm8.p1.5.m5.1.2.2.cmml" xref="S5.Thmthm8.p1.5.m5.1.2.2">𝜎</ci><ci id="S5.Thmthm8.p1.5.m5.1.1.cmml" xref="S5.Thmthm8.p1.5.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm8.p1.5.m5.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm8.p1.5.m5.1d">italic_σ ( italic_X )</annotation></semantics></math>” would be more accurate, but we prefer here to stick to the established terminology in the literature.</p> </div> </div> <figure class="ltx_figure" id="S5.F1"><svg class="ltx_picture ltx_centering" height="142.79" id="S5.F1.1.pic1" overflow="visible" version="1.1" width="633.65"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,142.79) matrix(1 0 0 -1 0 0) translate(99.38,0) translate(0,117.14)"><g fill="#F2F2F2"><path d="M 33.23 -87.66 L -33.23 -87.66 C -37.81 -87.66 -41.53 -91.38 -41.53 -95.97 L -41.53 -100.88 C -41.53 -105.47 -37.81 -109.19 -33.23 -109.19 L 33.23 -109.19 C 37.81 -109.19 41.53 -105.47 41.53 -100.88 L 41.53 -95.97 C 41.53 -91.38 37.81 -87.66 33.23 -87.66 Z M -41.53 -109.19" style="stroke:none"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 -36.92 -101.88)"><foreignobject height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="73.84"><span class="ltx_text" id="S5.F1.1.pic1.21.21.21.1.1">recognizable</span></foreignobject></g><g fill="#F2F2F2"><path d="M 298.53 -81.51 L 173.92 -81.51 C 169.33 -81.51 165.61 -85.23 165.61 -89.82 L 165.61 -107.03 C 165.61 -111.62 169.33 -115.34 173.92 -115.34 L 298.53 -115.34 C 303.11 -115.34 306.83 -111.62 306.83 -107.03 L 306.83 -89.82 C 306.83 -85.23 303.11 -81.51 298.53 -81.51 Z M 165.61 -115.34" style="stroke:none"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 170.23 -110.72)"><foreignobject height="24.6" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="131.99"><math alttext="\begin{array}[]{c}\text{recognizable}\\ \text{for aperiodic points}\end{array}" class="ltx_Math" display="inline" id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1a"><mtable id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1" rowspacing="0pt" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtr id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1a" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtd id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1b" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtext id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1a.cmml">recognizable</mtext></mtd></mtr><mtr id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1c" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtd id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1d" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtext id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1a.cmml">for aperiodic points</mtext></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1b"><matrix id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><matrixrow id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1a.cmml" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><ci id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1a.cmml" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1"><mtext id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1.cmml" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1">recognizable</mtext></ci></matrixrow><matrixrow id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1b.cmml" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1"><ci id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1a.cmml" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1"><mtext id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1.cmml" xref="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1">for aperiodic points</mtext></ci></matrixrow></matrix></annotation-xml><annotation encoding="application/x-tex" id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1c">\begin{array}[]{c}\text{recognizable}\\ \text{for aperiodic points}\end{array}</annotation><annotation encoding="application/x-llamapun" id="S5.F1.1.pic1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.m1.1d">start_ARRAY start_ROW start_CELL recognizable end_CELL end_ROW start_ROW start_CELL for aperiodic points end_CELL end_ROW end_ARRAY</annotation></semantics></math></foreignobject></g><g fill="#F2F2F2"><path d="M 51.71 25.65 L -91.08 25.65 C -95.67 25.65 -99.38 21.94 -99.38 17.35 L -99.38 -9.48 C -99.38 -14.06 -95.67 -17.78 -91.08 -17.78 L 51.71 -17.78 C 56.29 -17.78 60.01 -14.06 60.01 -9.48 L 60.01 17.35 C 60.01 21.94 56.29 25.65 51.71 25.65 Z M -99.38 -17.78" style="stroke:none"></path></g><g fill="#000000" stroke="#000000" transform="matrix(1.0 0.0 0.0 1.0 -94.77 -13.17)"><foreignobject height="34.21" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="150.17"><math alttext="\begin{array}[]{c}\text{shift-orbit injective}\\ \text{and}\\ \text{shift-period preserving}\end{array}" class="ltx_Math" display="inline" id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1a"><mtable id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1" rowspacing="0pt" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtr id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1a" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtd id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1b" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtext id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1a.cmml">shift-orbit injective</mtext></mtd></mtr><mtr id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1c" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtd id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1d" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtext id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1a.cmml">and</mtext></mtd></mtr><mtr id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1e" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtd id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1f" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mtext id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.1.1" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.3.1.1a.cmml">shift-period preserving</mtext></mtd></mtr></mtable><annotation-xml encoding="MathML-Content" id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1b"><matrix id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1"><matrixrow id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1a.cmml" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1"><ci id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1a.cmml" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1"><mtext id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1.cmml" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.1.1.1">shift-orbit injective</mtext></ci></matrixrow><matrixrow id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1b.cmml" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1"><ci id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1a.cmml" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1"><mtext id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1.cmml" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1.2.1.1">and</mtext></ci></matrixrow><matrixrow id="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1c.cmml" xref="S5.F1.1.pic1.2.2.2.2.2.2.2.2.2.2.2.2.1.1.1.1.1.1.1.1.1.1.m1.1.1"><ci 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id="S5.F1.1.pic1.15.15.15.15.15.15.15.15.15.15.15.15.1.1.1.1.1.1.1.1.1.m1.1d">⟹</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(-0.90631 0.42262 -0.42262 -0.90631 355.34 -26.05)"><foreignobject height="13.84" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="7.69"><math alttext="\!/" class="ltx_Math" display="inline" id="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1a"><mo id="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">/</mo><annotation-xml encoding="MathML-Content" id="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1b"><divide id="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1.1"></divide></annotation-xml><annotation encoding="application/x-tex" id="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1c">\!/</annotation><annotation encoding="application/x-llamapun" id="S5.F1.1.pic1.16.16.16.16.16.16.16.16.16.16.16.16.1.1.1.1.1.1.1.1.1.m1.1d">/</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(0.90631 0.42262 -0.42262 0.90631 351.09 -88.01)"><foreignobject height="12.45" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="10.38"><math alttext="\Longrightarrow" class="ltx_Math" display="inline" id="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1a"><mo id="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1.1" stretchy="false" xref="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">⟹</mo><annotation-xml encoding="MathML-Content" id="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1.1">⟹</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1c">\Longrightarrow</annotation><annotation encoding="application/x-llamapun" id="S5.F1.1.pic1.17.17.17.17.17.17.17.17.17.17.17.17.1.1.1.1.1.1.1.1.1.m1.1d">⟹</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(0.90631 0.42262 -0.42262 0.90631 352.31 -87.44)"><foreignobject height="13.84" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="7.69"><math alttext="\!/" class="ltx_Math" display="inline" id="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1a"><mo id="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">/</mo><annotation-xml encoding="MathML-Content" id="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1b"><divide id="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1.1"></divide></annotation-xml><annotation encoding="application/x-tex" id="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1c">\!/</annotation><annotation encoding="application/x-llamapun" id="S5.F1.1.pic1.18.18.18.18.18.18.18.18.18.18.18.18.1.1.1.1.1.1.1.1.1.m1.1d">/</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(-0.90631 -0.42262 0.42262 -0.90631 361.51 -89.16)"><foreignobject height="12.45" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="10.38"><math alttext="\Longrightarrow" class="ltx_Math" display="inline" id="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1a"><mo id="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1.1" stretchy="false" xref="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">⟹</mo><annotation-xml encoding="MathML-Content" id="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1b"><ci id="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1.1">⟹</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1c">\Longrightarrow</annotation><annotation encoding="application/x-llamapun" id="S5.F1.1.pic1.19.19.19.19.19.19.19.19.19.19.19.19.1.1.1.1.1.1.1.1.1.m1.1d">⟹</annotation></semantics></math></foreignobject></g><g fill="#000000" stroke="#000000" transform="matrix(-0.90631 -0.42262 0.42262 -0.90631 360.29 -89.73)"><foreignobject height="13.84" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="7.69"><math alttext="\!/" class="ltx_Math" display="inline" id="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1"><semantics id="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1a"><mo id="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1.1" xref="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1.1.cmml">/</mo><annotation-xml encoding="MathML-Content" id="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1b"><divide id="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1.1.cmml" xref="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1.1"></divide></annotation-xml><annotation encoding="application/x-tex" id="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1c">\!/</annotation><annotation encoding="application/x-llamapun" id="S5.F1.1.pic1.20.20.20.20.20.20.20.20.20.20.20.20.1.1.1.1.1.1.1.1.1.m1.1d">/</annotation></semantics></math></foreignobject></g></g></svg> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1. </span></figcaption> </figure> <div class="ltx_para" id="S5.p6"> <p class="ltx_p" id="S5.p6.5">It turns out (see Proposition 3.8 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>) that for any non-erasing morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.p6.1.m1.1"><semantics id="S5.p6.1.m1.1a"><mi id="S5.p6.1.m1.1.1" xref="S5.p6.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.p6.1.m1.1b"><ci id="S5.p6.1.m1.1.1.cmml" xref="S5.p6.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.p6.1.m1.1d">italic_σ</annotation></semantics></math> as above the condition “recognizable in a subshift <math alttext="X" class="ltx_Math" display="inline" id="S5.p6.2.m2.1"><semantics id="S5.p6.2.m2.1a"><mi id="S5.p6.2.m2.1.1" xref="S5.p6.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p6.2.m2.1b"><ci id="S5.p6.2.m2.1.1.cmml" xref="S5.p6.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p6.2.m2.1d">italic_X</annotation></semantics></math>” is equivalent to the condition “shift-orbit injective and shift-period preserving in <math alttext="X" class="ltx_Math" display="inline" id="S5.p6.3.m3.1"><semantics id="S5.p6.3.m3.1a"><mi id="S5.p6.3.m3.1.1" xref="S5.p6.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p6.3.m3.1b"><ci id="S5.p6.3.m3.1.1.cmml" xref="S5.p6.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p6.3.m3.1d">italic_X</annotation></semantics></math>”. Indeed, a quick proof for the injectivity of the measure transfer map <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S5.p6.4.m4.1"><semantics id="S5.p6.4.m4.1a"><mrow id="S5.p6.4.m4.1.1" xref="S5.p6.4.m4.1.1.cmml"><msub id="S5.p6.4.m4.1.1.2" xref="S5.p6.4.m4.1.1.2.cmml"><mi id="S5.p6.4.m4.1.1.2.2" xref="S5.p6.4.m4.1.1.2.2.cmml">σ</mi><mi id="S5.p6.4.m4.1.1.2.3" xref="S5.p6.4.m4.1.1.2.3.cmml">X</mi></msub><mo id="S5.p6.4.m4.1.1.1" xref="S5.p6.4.m4.1.1.1.cmml">⁢</mo><mi id="S5.p6.4.m4.1.1.3" xref="S5.p6.4.m4.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p6.4.m4.1b"><apply id="S5.p6.4.m4.1.1.cmml" xref="S5.p6.4.m4.1.1"><times id="S5.p6.4.m4.1.1.1.cmml" xref="S5.p6.4.m4.1.1.1"></times><apply id="S5.p6.4.m4.1.1.2.cmml" xref="S5.p6.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.p6.4.m4.1.1.2.1.cmml" xref="S5.p6.4.m4.1.1.2">subscript</csymbol><ci id="S5.p6.4.m4.1.1.2.2.cmml" xref="S5.p6.4.m4.1.1.2.2">𝜎</ci><ci id="S5.p6.4.m4.1.1.2.3.cmml" xref="S5.p6.4.m4.1.1.2.3">𝑋</ci></apply><ci id="S5.p6.4.m4.1.1.3.cmml" xref="S5.p6.4.m4.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.4.m4.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S5.p6.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> as in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm6" title="Theorem 5.6. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.6</span></a>, under the stronger hypothesis of “recognizability in <math alttext="X" class="ltx_Math" display="inline" id="S5.p6.5.m5.1"><semantics id="S5.p6.5.m5.1a"><mi id="S5.p6.5.m5.1.1" xref="S5.p6.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p6.5.m5.1b"><ci id="S5.p6.5.m5.1.1.cmml" xref="S5.p6.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p6.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p6.5.m5.1d">italic_X</annotation></semantics></math>” is given in section 3.3 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>.</p> </div> <div class="ltx_para" id="S5.p7"> <p class="ltx_p" id="S5.p7.4">However, since recognizability can only be achieved for an everywhere growing <math alttext="S" class="ltx_Math" display="inline" id="S5.p7.1.m1.1"><semantics id="S5.p7.1.m1.1a"><mi id="S5.p7.1.m1.1.1" xref="S5.p7.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.p7.1.m1.1b"><ci id="S5.p7.1.m1.1.1.cmml" xref="S5.p7.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p7.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.p7.1.m1.1d">italic_S</annotation></semantics></math>-adic development of a given subshift <math alttext="X" class="ltx_Math" display="inline" id="S5.p7.2.m2.1"><semantics id="S5.p7.2.m2.1a"><mi id="S5.p7.2.m2.1.1" xref="S5.p7.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p7.2.m2.1b"><ci id="S5.p7.2.m2.1.1.cmml" xref="S5.p7.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p7.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p7.2.m2.1d">italic_X</annotation></semantics></math> (for the terminology see for instance Section 2 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>) if <math alttext="X" class="ltx_Math" display="inline" id="S5.p7.3.m3.1"><semantics id="S5.p7.3.m3.1a"><mi id="S5.p7.3.m3.1.1" xref="S5.p7.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p7.3.m3.1b"><ci id="S5.p7.3.m3.1.1.cmml" xref="S5.p7.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p7.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p7.3.m3.1d">italic_X</annotation></semantics></math> is aperiodic, the much more popular hypothesis used by the <math alttext="S" class="ltx_Math" display="inline" id="S5.p7.4.m4.1"><semantics id="S5.p7.4.m4.1a"><mi id="S5.p7.4.m4.1.1" xref="S5.p7.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S5.p7.4.m4.1b"><ci id="S5.p7.4.m4.1.1.cmml" xref="S5.p7.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p7.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S5.p7.4.m4.1d">italic_S</annotation></semantics></math>-adic community is not “recognizable” but “recognizable for aperiodic points”, or “eventually recognizable for aperiodic points”. The relation of this notion to the concepts introduced in this section (see Fig. 1) is given by the following; a formal proof is given in Remark 3.10 (3) of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_prop" id="S5.Thmthm9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm9.1.1.1">Proposition 5.9</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm9.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm9.p1"> <p class="ltx_p" id="S5.Thmthm9.p1.6"><span class="ltx_text ltx_font_italic" id="S5.Thmthm9.p1.6.6">If a non-erasing morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.Thmthm9.p1.1.1.m1.1"><semantics id="S5.Thmthm9.p1.1.1.m1.1a"><mrow id="S5.Thmthm9.p1.1.1.m1.1.1" xref="S5.Thmthm9.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmthm9.p1.1.1.m1.1.1.2" xref="S5.Thmthm9.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S5.Thmthm9.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm9.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm9.p1.1.1.m1.1.1.3" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm9.p1.1.1.m1.1.1.3.2" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm9.p1.1.1.m1.1.1.3.2.2" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm9.p1.1.1.m1.1.1.3.2.3" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm9.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm9.p1.1.1.m1.1.1.3.3" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm9.p1.1.1.m1.1.1.3.3.2" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.Thmthm9.p1.1.1.m1.1.1.3.3.3" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm9.p1.1.1.m1.1b"><apply id="S5.Thmthm9.p1.1.1.m1.1.1.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1"><ci id="S5.Thmthm9.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.1">:</ci><ci id="S5.Thmthm9.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S5.Thmthm9.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3"><ci id="S5.Thmthm9.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm9.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm9.p1.1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm9.p1.1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm9.p1.1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm9.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm9.p1.1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm9.p1.1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S5.Thmthm9.p1.1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm9.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm9.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm9.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is shift-orbit injective in a subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm9.p1.2.2.m2.1"><semantics id="S5.Thmthm9.p1.2.2.m2.1a"><mrow id="S5.Thmthm9.p1.2.2.m2.1.1" xref="S5.Thmthm9.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmthm9.p1.2.2.m2.1.1.2" xref="S5.Thmthm9.p1.2.2.m2.1.1.2.cmml">X</mi><mo id="S5.Thmthm9.p1.2.2.m2.1.1.1" xref="S5.Thmthm9.p1.2.2.m2.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm9.p1.2.2.m2.1.1.3" xref="S5.Thmthm9.p1.2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm9.p1.2.2.m2.1.1.3.2" xref="S5.Thmthm9.p1.2.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm9.p1.2.2.m2.1.1.3.3" xref="S5.Thmthm9.p1.2.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm9.p1.2.2.m2.1b"><apply id="S5.Thmthm9.p1.2.2.m2.1.1.cmml" xref="S5.Thmthm9.p1.2.2.m2.1.1"><subset id="S5.Thmthm9.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmthm9.p1.2.2.m2.1.1.1"></subset><ci id="S5.Thmthm9.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmthm9.p1.2.2.m2.1.1.2">𝑋</ci><apply id="S5.Thmthm9.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmthm9.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm9.p1.2.2.m2.1.1.3.1.cmml" xref="S5.Thmthm9.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S5.Thmthm9.p1.2.2.m2.1.1.3.2.cmml" xref="S5.Thmthm9.p1.2.2.m2.1.1.3.2">𝒜</ci><ci id="S5.Thmthm9.p1.2.2.m2.1.1.3.3.cmml" xref="S5.Thmthm9.p1.2.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm9.p1.2.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm9.p1.2.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, then <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm9.p1.3.3.m3.1"><semantics id="S5.Thmthm9.p1.3.3.m3.1a"><mi id="S5.Thmthm9.p1.3.3.m3.1.1" xref="S5.Thmthm9.p1.3.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm9.p1.3.3.m3.1b"><ci id="S5.Thmthm9.p1.3.3.m3.1.1.cmml" xref="S5.Thmthm9.p1.3.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm9.p1.3.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm9.p1.3.3.m3.1d">italic_σ</annotation></semantics></math> is recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm9.p1.4.4.m4.1"><semantics id="S5.Thmthm9.p1.4.4.m4.1a"><mi id="S5.Thmthm9.p1.4.4.m4.1.1" xref="S5.Thmthm9.p1.4.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm9.p1.4.4.m4.1b"><ci id="S5.Thmthm9.p1.4.4.m4.1.1.cmml" xref="S5.Thmthm9.p1.4.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm9.p1.4.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm9.p1.4.4.m4.1d">italic_X</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S5.Thmthm9.p1.5.5.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S5.Thmthm9.p1.5.5.1.m1.1"><semantics id="S5.Thmthm9.p1.5.5.1.m1.1a"><mo id="S5.Thmthm9.p1.5.5.1.m1.1.1" xref="S5.Thmthm9.p1.5.5.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S5.Thmthm9.p1.5.5.1.m1.1b"><csymbol cd="latexml" id="S5.Thmthm9.p1.5.5.1.m1.1.1.cmml" xref="S5.Thmthm9.p1.5.5.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm9.p1.5.5.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm9.p1.5.5.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S5.Thmthm9.p1.6.6.m5.1"><semantics id="S5.Thmthm9.p1.6.6.m5.1a"><mo id="S5.Thmthm9.p1.6.6.m5.1.1" xref="S5.Thmthm9.p1.6.6.m5.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S5.Thmthm9.p1.6.6.m5.1b"><csymbol cd="latexml" id="S5.Thmthm9.p1.6.6.m5.1.1.cmml" xref="S5.Thmthm9.p1.6.6.m5.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm9.p1.6.6.m5.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm9.p1.6.6.m5.1d">⊓</annotation></semantics></math></span></p> </div> </div> <div class="ltx_para" id="S5.p8"> <p class="ltx_p" id="S5.p8.9">The converse implication of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm9" title="Proposition 5.9. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.9</span></a> turns however out to be wrong: A very simple counterexample is given by the letter-to-letter morphism <math alttext="\sigma_{1}:\{a,b\}^{*}\to\{c\}^{*}" class="ltx_Math" display="inline" id="S5.p8.1.m1.3"><semantics id="S5.p8.1.m1.3a"><mrow id="S5.p8.1.m1.3.4" xref="S5.p8.1.m1.3.4.cmml"><msub id="S5.p8.1.m1.3.4.2" xref="S5.p8.1.m1.3.4.2.cmml"><mi id="S5.p8.1.m1.3.4.2.2" xref="S5.p8.1.m1.3.4.2.2.cmml">σ</mi><mn id="S5.p8.1.m1.3.4.2.3" xref="S5.p8.1.m1.3.4.2.3.cmml">1</mn></msub><mo id="S5.p8.1.m1.3.4.1" lspace="0.278em" rspace="0.278em" xref="S5.p8.1.m1.3.4.1.cmml">:</mo><mrow id="S5.p8.1.m1.3.4.3" xref="S5.p8.1.m1.3.4.3.cmml"><msup id="S5.p8.1.m1.3.4.3.2" xref="S5.p8.1.m1.3.4.3.2.cmml"><mrow id="S5.p8.1.m1.3.4.3.2.2.2" xref="S5.p8.1.m1.3.4.3.2.2.1.cmml"><mo id="S5.p8.1.m1.3.4.3.2.2.2.1" stretchy="false" xref="S5.p8.1.m1.3.4.3.2.2.1.cmml">{</mo><mi id="S5.p8.1.m1.1.1" xref="S5.p8.1.m1.1.1.cmml">a</mi><mo id="S5.p8.1.m1.3.4.3.2.2.2.2" xref="S5.p8.1.m1.3.4.3.2.2.1.cmml">,</mo><mi id="S5.p8.1.m1.2.2" xref="S5.p8.1.m1.2.2.cmml">b</mi><mo id="S5.p8.1.m1.3.4.3.2.2.2.3" stretchy="false" xref="S5.p8.1.m1.3.4.3.2.2.1.cmml">}</mo></mrow><mo id="S5.p8.1.m1.3.4.3.2.3" xref="S5.p8.1.m1.3.4.3.2.3.cmml">∗</mo></msup><mo id="S5.p8.1.m1.3.4.3.1" stretchy="false" xref="S5.p8.1.m1.3.4.3.1.cmml">→</mo><msup id="S5.p8.1.m1.3.4.3.3" xref="S5.p8.1.m1.3.4.3.3.cmml"><mrow id="S5.p8.1.m1.3.4.3.3.2.2" xref="S5.p8.1.m1.3.4.3.3.2.1.cmml"><mo id="S5.p8.1.m1.3.4.3.3.2.2.1" stretchy="false" xref="S5.p8.1.m1.3.4.3.3.2.1.cmml">{</mo><mi id="S5.p8.1.m1.3.3" xref="S5.p8.1.m1.3.3.cmml">c</mi><mo id="S5.p8.1.m1.3.4.3.3.2.2.2" stretchy="false" xref="S5.p8.1.m1.3.4.3.3.2.1.cmml">}</mo></mrow><mo id="S5.p8.1.m1.3.4.3.3.3" xref="S5.p8.1.m1.3.4.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p8.1.m1.3b"><apply id="S5.p8.1.m1.3.4.cmml" xref="S5.p8.1.m1.3.4"><ci id="S5.p8.1.m1.3.4.1.cmml" xref="S5.p8.1.m1.3.4.1">:</ci><apply id="S5.p8.1.m1.3.4.2.cmml" xref="S5.p8.1.m1.3.4.2"><csymbol cd="ambiguous" id="S5.p8.1.m1.3.4.2.1.cmml" xref="S5.p8.1.m1.3.4.2">subscript</csymbol><ci id="S5.p8.1.m1.3.4.2.2.cmml" xref="S5.p8.1.m1.3.4.2.2">𝜎</ci><cn id="S5.p8.1.m1.3.4.2.3.cmml" type="integer" xref="S5.p8.1.m1.3.4.2.3">1</cn></apply><apply id="S5.p8.1.m1.3.4.3.cmml" xref="S5.p8.1.m1.3.4.3"><ci id="S5.p8.1.m1.3.4.3.1.cmml" xref="S5.p8.1.m1.3.4.3.1">→</ci><apply id="S5.p8.1.m1.3.4.3.2.cmml" xref="S5.p8.1.m1.3.4.3.2"><csymbol cd="ambiguous" id="S5.p8.1.m1.3.4.3.2.1.cmml" xref="S5.p8.1.m1.3.4.3.2">superscript</csymbol><set id="S5.p8.1.m1.3.4.3.2.2.1.cmml" xref="S5.p8.1.m1.3.4.3.2.2.2"><ci id="S5.p8.1.m1.1.1.cmml" xref="S5.p8.1.m1.1.1">𝑎</ci><ci id="S5.p8.1.m1.2.2.cmml" xref="S5.p8.1.m1.2.2">𝑏</ci></set><times id="S5.p8.1.m1.3.4.3.2.3.cmml" xref="S5.p8.1.m1.3.4.3.2.3"></times></apply><apply id="S5.p8.1.m1.3.4.3.3.cmml" xref="S5.p8.1.m1.3.4.3.3"><csymbol cd="ambiguous" id="S5.p8.1.m1.3.4.3.3.1.cmml" xref="S5.p8.1.m1.3.4.3.3">superscript</csymbol><set id="S5.p8.1.m1.3.4.3.3.2.1.cmml" xref="S5.p8.1.m1.3.4.3.3.2.2"><ci id="S5.p8.1.m1.3.3.cmml" xref="S5.p8.1.m1.3.3">𝑐</ci></set><times id="S5.p8.1.m1.3.4.3.3.3.cmml" xref="S5.p8.1.m1.3.4.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.1.m1.3c">\sigma_{1}:\{a,b\}^{*}\to\{c\}^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p8.1.m1.3d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT : { italic_a , italic_b } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → { italic_c } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> from Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm9" title="Remark 2.9. ‣ 2.3.1. Typical injectivity problems ‣ 2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.9</span></a> and the subshift <math alttext="X=\{a^{\pm\infty},b^{\pm\infty}\}" class="ltx_Math" display="inline" id="S5.p8.2.m2.2"><semantics id="S5.p8.2.m2.2a"><mrow id="S5.p8.2.m2.2.2" xref="S5.p8.2.m2.2.2.cmml"><mi id="S5.p8.2.m2.2.2.4" xref="S5.p8.2.m2.2.2.4.cmml">X</mi><mo id="S5.p8.2.m2.2.2.3" xref="S5.p8.2.m2.2.2.3.cmml">=</mo><mrow id="S5.p8.2.m2.2.2.2.2" xref="S5.p8.2.m2.2.2.2.3.cmml"><mo id="S5.p8.2.m2.2.2.2.2.3" stretchy="false" xref="S5.p8.2.m2.2.2.2.3.cmml">{</mo><msup id="S5.p8.2.m2.1.1.1.1.1" xref="S5.p8.2.m2.1.1.1.1.1.cmml"><mi id="S5.p8.2.m2.1.1.1.1.1.2" xref="S5.p8.2.m2.1.1.1.1.1.2.cmml">a</mi><mrow id="S5.p8.2.m2.1.1.1.1.1.3" xref="S5.p8.2.m2.1.1.1.1.1.3.cmml"><mo id="S5.p8.2.m2.1.1.1.1.1.3a" xref="S5.p8.2.m2.1.1.1.1.1.3.cmml">±</mo><mi id="S5.p8.2.m2.1.1.1.1.1.3.2" mathvariant="normal" xref="S5.p8.2.m2.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S5.p8.2.m2.2.2.2.2.4" xref="S5.p8.2.m2.2.2.2.3.cmml">,</mo><msup id="S5.p8.2.m2.2.2.2.2.2" xref="S5.p8.2.m2.2.2.2.2.2.cmml"><mi id="S5.p8.2.m2.2.2.2.2.2.2" xref="S5.p8.2.m2.2.2.2.2.2.2.cmml">b</mi><mrow id="S5.p8.2.m2.2.2.2.2.2.3" xref="S5.p8.2.m2.2.2.2.2.2.3.cmml"><mo id="S5.p8.2.m2.2.2.2.2.2.3a" xref="S5.p8.2.m2.2.2.2.2.2.3.cmml">±</mo><mi id="S5.p8.2.m2.2.2.2.2.2.3.2" mathvariant="normal" xref="S5.p8.2.m2.2.2.2.2.2.3.2.cmml">∞</mi></mrow></msup><mo id="S5.p8.2.m2.2.2.2.2.5" stretchy="false" xref="S5.p8.2.m2.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p8.2.m2.2b"><apply id="S5.p8.2.m2.2.2.cmml" xref="S5.p8.2.m2.2.2"><eq id="S5.p8.2.m2.2.2.3.cmml" xref="S5.p8.2.m2.2.2.3"></eq><ci id="S5.p8.2.m2.2.2.4.cmml" xref="S5.p8.2.m2.2.2.4">𝑋</ci><set id="S5.p8.2.m2.2.2.2.3.cmml" xref="S5.p8.2.m2.2.2.2.2"><apply id="S5.p8.2.m2.1.1.1.1.1.cmml" xref="S5.p8.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p8.2.m2.1.1.1.1.1.1.cmml" xref="S5.p8.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S5.p8.2.m2.1.1.1.1.1.2.cmml" xref="S5.p8.2.m2.1.1.1.1.1.2">𝑎</ci><apply id="S5.p8.2.m2.1.1.1.1.1.3.cmml" xref="S5.p8.2.m2.1.1.1.1.1.3"><csymbol cd="latexml" id="S5.p8.2.m2.1.1.1.1.1.3.1.cmml" xref="S5.p8.2.m2.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S5.p8.2.m2.1.1.1.1.1.3.2.cmml" xref="S5.p8.2.m2.1.1.1.1.1.3.2"></infinity></apply></apply><apply id="S5.p8.2.m2.2.2.2.2.2.cmml" xref="S5.p8.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.p8.2.m2.2.2.2.2.2.1.cmml" xref="S5.p8.2.m2.2.2.2.2.2">superscript</csymbol><ci id="S5.p8.2.m2.2.2.2.2.2.2.cmml" xref="S5.p8.2.m2.2.2.2.2.2.2">𝑏</ci><apply id="S5.p8.2.m2.2.2.2.2.2.3.cmml" xref="S5.p8.2.m2.2.2.2.2.2.3"><csymbol cd="latexml" id="S5.p8.2.m2.2.2.2.2.2.3.1.cmml" xref="S5.p8.2.m2.2.2.2.2.2.3">plus-or-minus</csymbol><infinity id="S5.p8.2.m2.2.2.2.2.2.3.2.cmml" xref="S5.p8.2.m2.2.2.2.2.2.3.2"></infinity></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.2.m2.2c">X=\{a^{\pm\infty},b^{\pm\infty}\}</annotation><annotation encoding="application/x-llamapun" id="S5.p8.2.m2.2d">italic_X = { italic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT , italic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT }</annotation></semantics></math>. Since <math alttext="\{c\}^{*}" class="ltx_Math" display="inline" id="S5.p8.3.m3.1"><semantics id="S5.p8.3.m3.1a"><msup id="S5.p8.3.m3.1.2" xref="S5.p8.3.m3.1.2.cmml"><mrow id="S5.p8.3.m3.1.2.2.2" xref="S5.p8.3.m3.1.2.2.1.cmml"><mo id="S5.p8.3.m3.1.2.2.2.1" stretchy="false" xref="S5.p8.3.m3.1.2.2.1.cmml">{</mo><mi id="S5.p8.3.m3.1.1" xref="S5.p8.3.m3.1.1.cmml">c</mi><mo id="S5.p8.3.m3.1.2.2.2.2" stretchy="false" xref="S5.p8.3.m3.1.2.2.1.cmml">}</mo></mrow><mo id="S5.p8.3.m3.1.2.3" xref="S5.p8.3.m3.1.2.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S5.p8.3.m3.1b"><apply id="S5.p8.3.m3.1.2.cmml" xref="S5.p8.3.m3.1.2"><csymbol cd="ambiguous" id="S5.p8.3.m3.1.2.1.cmml" xref="S5.p8.3.m3.1.2">superscript</csymbol><set id="S5.p8.3.m3.1.2.2.1.cmml" xref="S5.p8.3.m3.1.2.2.2"><ci id="S5.p8.3.m3.1.1.cmml" xref="S5.p8.3.m3.1.1">𝑐</ci></set><times id="S5.p8.3.m3.1.2.3.cmml" xref="S5.p8.3.m3.1.2.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.3.m3.1c">\{c\}^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p8.3.m3.1d">{ italic_c } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> consists only of periodic words, it follows (see Warning <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm8" title="Warning 5.8. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.8</span></a>) that <math alttext="\sigma_{1}" class="ltx_Math" display="inline" id="S5.p8.4.m4.1"><semantics id="S5.p8.4.m4.1a"><msub id="S5.p8.4.m4.1.1" xref="S5.p8.4.m4.1.1.cmml"><mi id="S5.p8.4.m4.1.1.2" xref="S5.p8.4.m4.1.1.2.cmml">σ</mi><mn id="S5.p8.4.m4.1.1.3" xref="S5.p8.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p8.4.m4.1b"><apply id="S5.p8.4.m4.1.1.cmml" xref="S5.p8.4.m4.1.1"><csymbol cd="ambiguous" id="S5.p8.4.m4.1.1.1.cmml" xref="S5.p8.4.m4.1.1">subscript</csymbol><ci id="S5.p8.4.m4.1.1.2.cmml" xref="S5.p8.4.m4.1.1.2">𝜎</ci><cn id="S5.p8.4.m4.1.1.3.cmml" type="integer" xref="S5.p8.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.4.m4.1c">\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p8.4.m4.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is automatically recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.p8.5.m5.1"><semantics id="S5.p8.5.m5.1a"><mi id="S5.p8.5.m5.1.1" xref="S5.p8.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p8.5.m5.1b"><ci id="S5.p8.5.m5.1.1.cmml" xref="S5.p8.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p8.5.m5.1d">italic_X</annotation></semantics></math>. However, since <math alttext="\sigma_{1}^{T}(\cal O(a^{\pm\infty}))=\cal O(c^{\pm\infty})" class="ltx_Math" display="inline" id="S5.p8.6.m6.2"><semantics id="S5.p8.6.m6.2a"><mrow id="S5.p8.6.m6.2.2" xref="S5.p8.6.m6.2.2.cmml"><mrow id="S5.p8.6.m6.1.1.1" xref="S5.p8.6.m6.1.1.1.cmml"><msubsup id="S5.p8.6.m6.1.1.1.3" xref="S5.p8.6.m6.1.1.1.3.cmml"><mi id="S5.p8.6.m6.1.1.1.3.2.2" xref="S5.p8.6.m6.1.1.1.3.2.2.cmml">σ</mi><mn id="S5.p8.6.m6.1.1.1.3.2.3" xref="S5.p8.6.m6.1.1.1.3.2.3.cmml">1</mn><mi id="S5.p8.6.m6.1.1.1.3.3" xref="S5.p8.6.m6.1.1.1.3.3.cmml">T</mi></msubsup><mo id="S5.p8.6.m6.1.1.1.2" xref="S5.p8.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S5.p8.6.m6.1.1.1.1.1" xref="S5.p8.6.m6.1.1.1.1.1.1.cmml"><mo id="S5.p8.6.m6.1.1.1.1.1.2" stretchy="false" xref="S5.p8.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.p8.6.m6.1.1.1.1.1.1" xref="S5.p8.6.m6.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p8.6.m6.1.1.1.1.1.1.3" xref="S5.p8.6.m6.1.1.1.1.1.1.3.cmml">𝒪</mi><mo id="S5.p8.6.m6.1.1.1.1.1.1.2" xref="S5.p8.6.m6.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.p8.6.m6.1.1.1.1.1.1.1.1" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S5.p8.6.m6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.2" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.2.cmml">𝒶</mi><mrow id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3a" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3.cmml">±</mo><mi id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S5.p8.6.m6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p8.6.m6.1.1.1.1.1.3" stretchy="false" xref="S5.p8.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p8.6.m6.2.2.3" xref="S5.p8.6.m6.2.2.3.cmml">=</mo><mrow id="S5.p8.6.m6.2.2.2" xref="S5.p8.6.m6.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p8.6.m6.2.2.2.3" xref="S5.p8.6.m6.2.2.2.3.cmml">𝒪</mi><mo id="S5.p8.6.m6.2.2.2.2" xref="S5.p8.6.m6.2.2.2.2.cmml">⁢</mo><mrow id="S5.p8.6.m6.2.2.2.1.1" xref="S5.p8.6.m6.2.2.2.1.1.1.cmml"><mo id="S5.p8.6.m6.2.2.2.1.1.2" stretchy="false" xref="S5.p8.6.m6.2.2.2.1.1.1.cmml">(</mo><msup id="S5.p8.6.m6.2.2.2.1.1.1" xref="S5.p8.6.m6.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p8.6.m6.2.2.2.1.1.1.2" xref="S5.p8.6.m6.2.2.2.1.1.1.2.cmml">𝒸</mi><mrow id="S5.p8.6.m6.2.2.2.1.1.1.3" xref="S5.p8.6.m6.2.2.2.1.1.1.3.cmml"><mo id="S5.p8.6.m6.2.2.2.1.1.1.3a" xref="S5.p8.6.m6.2.2.2.1.1.1.3.cmml">±</mo><mi id="S5.p8.6.m6.2.2.2.1.1.1.3.2" mathvariant="normal" xref="S5.p8.6.m6.2.2.2.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S5.p8.6.m6.2.2.2.1.1.3" stretchy="false" xref="S5.p8.6.m6.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p8.6.m6.2b"><apply id="S5.p8.6.m6.2.2.cmml" xref="S5.p8.6.m6.2.2"><eq id="S5.p8.6.m6.2.2.3.cmml" xref="S5.p8.6.m6.2.2.3"></eq><apply id="S5.p8.6.m6.1.1.1.cmml" xref="S5.p8.6.m6.1.1.1"><times id="S5.p8.6.m6.1.1.1.2.cmml" xref="S5.p8.6.m6.1.1.1.2"></times><apply id="S5.p8.6.m6.1.1.1.3.cmml" xref="S5.p8.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="S5.p8.6.m6.1.1.1.3.1.cmml" xref="S5.p8.6.m6.1.1.1.3">superscript</csymbol><apply id="S5.p8.6.m6.1.1.1.3.2.cmml" xref="S5.p8.6.m6.1.1.1.3"><csymbol cd="ambiguous" id="S5.p8.6.m6.1.1.1.3.2.1.cmml" xref="S5.p8.6.m6.1.1.1.3">subscript</csymbol><ci id="S5.p8.6.m6.1.1.1.3.2.2.cmml" xref="S5.p8.6.m6.1.1.1.3.2.2">𝜎</ci><cn id="S5.p8.6.m6.1.1.1.3.2.3.cmml" type="integer" xref="S5.p8.6.m6.1.1.1.3.2.3">1</cn></apply><ci id="S5.p8.6.m6.1.1.1.3.3.cmml" xref="S5.p8.6.m6.1.1.1.3.3">𝑇</ci></apply><apply id="S5.p8.6.m6.1.1.1.1.1.1.cmml" xref="S5.p8.6.m6.1.1.1.1.1"><times id="S5.p8.6.m6.1.1.1.1.1.1.2.cmml" xref="S5.p8.6.m6.1.1.1.1.1.1.2"></times><ci id="S5.p8.6.m6.1.1.1.1.1.1.3.cmml" xref="S5.p8.6.m6.1.1.1.1.1.1.3">𝒪</ci><apply id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.cmml" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.2">𝒶</ci><apply id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3"><csymbol cd="latexml" id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.p8.6.m6.1.1.1.1.1.1.1.1.1.3.2"></infinity></apply></apply></apply></apply><apply id="S5.p8.6.m6.2.2.2.cmml" xref="S5.p8.6.m6.2.2.2"><times id="S5.p8.6.m6.2.2.2.2.cmml" xref="S5.p8.6.m6.2.2.2.2"></times><ci id="S5.p8.6.m6.2.2.2.3.cmml" xref="S5.p8.6.m6.2.2.2.3">𝒪</ci><apply id="S5.p8.6.m6.2.2.2.1.1.1.cmml" xref="S5.p8.6.m6.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.p8.6.m6.2.2.2.1.1.1.1.cmml" xref="S5.p8.6.m6.2.2.2.1.1">superscript</csymbol><ci id="S5.p8.6.m6.2.2.2.1.1.1.2.cmml" xref="S5.p8.6.m6.2.2.2.1.1.1.2">𝒸</ci><apply id="S5.p8.6.m6.2.2.2.1.1.1.3.cmml" xref="S5.p8.6.m6.2.2.2.1.1.1.3"><csymbol cd="latexml" id="S5.p8.6.m6.2.2.2.1.1.1.3.1.cmml" xref="S5.p8.6.m6.2.2.2.1.1.1.3">plus-or-minus</csymbol><infinity id="S5.p8.6.m6.2.2.2.1.1.1.3.2.cmml" xref="S5.p8.6.m6.2.2.2.1.1.1.3.2"></infinity></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.6.m6.2c">\sigma_{1}^{T}(\cal O(a^{\pm\infty}))=\cal O(c^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S5.p8.6.m6.2d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( caligraphic_O ( caligraphic_a start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) ) = caligraphic_O ( caligraphic_c start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\sigma_{1}^{T}(\cal O(b^{\pm\infty}))=\cal O(c^{\pm\infty})" class="ltx_Math" display="inline" id="S5.p8.7.m7.2"><semantics id="S5.p8.7.m7.2a"><mrow id="S5.p8.7.m7.2.2" xref="S5.p8.7.m7.2.2.cmml"><mrow id="S5.p8.7.m7.1.1.1" xref="S5.p8.7.m7.1.1.1.cmml"><msubsup id="S5.p8.7.m7.1.1.1.3" xref="S5.p8.7.m7.1.1.1.3.cmml"><mi id="S5.p8.7.m7.1.1.1.3.2.2" xref="S5.p8.7.m7.1.1.1.3.2.2.cmml">σ</mi><mn id="S5.p8.7.m7.1.1.1.3.2.3" xref="S5.p8.7.m7.1.1.1.3.2.3.cmml">1</mn><mi id="S5.p8.7.m7.1.1.1.3.3" xref="S5.p8.7.m7.1.1.1.3.3.cmml">T</mi></msubsup><mo id="S5.p8.7.m7.1.1.1.2" xref="S5.p8.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S5.p8.7.m7.1.1.1.1.1" xref="S5.p8.7.m7.1.1.1.1.1.1.cmml"><mo id="S5.p8.7.m7.1.1.1.1.1.2" stretchy="false" xref="S5.p8.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.p8.7.m7.1.1.1.1.1.1" xref="S5.p8.7.m7.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" 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stretchy="false" xref="S5.p8.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p8.7.m7.2.2.3" xref="S5.p8.7.m7.2.2.3.cmml">=</mo><mrow id="S5.p8.7.m7.2.2.2" xref="S5.p8.7.m7.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p8.7.m7.2.2.2.3" xref="S5.p8.7.m7.2.2.2.3.cmml">𝒪</mi><mo id="S5.p8.7.m7.2.2.2.2" xref="S5.p8.7.m7.2.2.2.2.cmml">⁢</mo><mrow id="S5.p8.7.m7.2.2.2.1.1" xref="S5.p8.7.m7.2.2.2.1.1.1.cmml"><mo id="S5.p8.7.m7.2.2.2.1.1.2" stretchy="false" xref="S5.p8.7.m7.2.2.2.1.1.1.cmml">(</mo><msup id="S5.p8.7.m7.2.2.2.1.1.1" xref="S5.p8.7.m7.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p8.7.m7.2.2.2.1.1.1.2" xref="S5.p8.7.m7.2.2.2.1.1.1.2.cmml">𝒸</mi><mrow id="S5.p8.7.m7.2.2.2.1.1.1.3" xref="S5.p8.7.m7.2.2.2.1.1.1.3.cmml"><mo id="S5.p8.7.m7.2.2.2.1.1.1.3a" xref="S5.p8.7.m7.2.2.2.1.1.1.3.cmml">±</mo><mi id="S5.p8.7.m7.2.2.2.1.1.1.3.2" mathvariant="normal" xref="S5.p8.7.m7.2.2.2.1.1.1.3.2.cmml">∞</mi></mrow></msup><mo id="S5.p8.7.m7.2.2.2.1.1.3" stretchy="false" xref="S5.p8.7.m7.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p8.7.m7.2b"><apply id="S5.p8.7.m7.2.2.cmml" xref="S5.p8.7.m7.2.2"><eq id="S5.p8.7.m7.2.2.3.cmml" xref="S5.p8.7.m7.2.2.3"></eq><apply id="S5.p8.7.m7.1.1.1.cmml" xref="S5.p8.7.m7.1.1.1"><times id="S5.p8.7.m7.1.1.1.2.cmml" xref="S5.p8.7.m7.1.1.1.2"></times><apply id="S5.p8.7.m7.1.1.1.3.cmml" xref="S5.p8.7.m7.1.1.1.3"><csymbol cd="ambiguous" id="S5.p8.7.m7.1.1.1.3.1.cmml" xref="S5.p8.7.m7.1.1.1.3">superscript</csymbol><apply id="S5.p8.7.m7.1.1.1.3.2.cmml" xref="S5.p8.7.m7.1.1.1.3"><csymbol cd="ambiguous" id="S5.p8.7.m7.1.1.1.3.2.1.cmml" xref="S5.p8.7.m7.1.1.1.3">subscript</csymbol><ci id="S5.p8.7.m7.1.1.1.3.2.2.cmml" xref="S5.p8.7.m7.1.1.1.3.2.2">𝜎</ci><cn id="S5.p8.7.m7.1.1.1.3.2.3.cmml" type="integer" xref="S5.p8.7.m7.1.1.1.3.2.3">1</cn></apply><ci id="S5.p8.7.m7.1.1.1.3.3.cmml" xref="S5.p8.7.m7.1.1.1.3.3">𝑇</ci></apply><apply id="S5.p8.7.m7.1.1.1.1.1.1.cmml" xref="S5.p8.7.m7.1.1.1.1.1"><times id="S5.p8.7.m7.1.1.1.1.1.1.2.cmml" xref="S5.p8.7.m7.1.1.1.1.1.1.2"></times><ci id="S5.p8.7.m7.1.1.1.1.1.1.3.cmml" xref="S5.p8.7.m7.1.1.1.1.1.1.3">𝒪</ci><apply id="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.cmml" xref="S5.p8.7.m7.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.1.cmml" xref="S5.p8.7.m7.1.1.1.1.1.1.1.1">superscript</csymbol><ci id="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.2.cmml" xref="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.2">𝒷</ci><apply id="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.3.cmml" xref="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.3"><csymbol cd="latexml" id="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.3">plus-or-minus</csymbol><infinity id="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S5.p8.7.m7.1.1.1.1.1.1.1.1.1.3.2"></infinity></apply></apply></apply></apply><apply id="S5.p8.7.m7.2.2.2.cmml" xref="S5.p8.7.m7.2.2.2"><times id="S5.p8.7.m7.2.2.2.2.cmml" xref="S5.p8.7.m7.2.2.2.2"></times><ci id="S5.p8.7.m7.2.2.2.3.cmml" xref="S5.p8.7.m7.2.2.2.3">𝒪</ci><apply id="S5.p8.7.m7.2.2.2.1.1.1.cmml" xref="S5.p8.7.m7.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.p8.7.m7.2.2.2.1.1.1.1.cmml" xref="S5.p8.7.m7.2.2.2.1.1">superscript</csymbol><ci id="S5.p8.7.m7.2.2.2.1.1.1.2.cmml" xref="S5.p8.7.m7.2.2.2.1.1.1.2">𝒸</ci><apply id="S5.p8.7.m7.2.2.2.1.1.1.3.cmml" xref="S5.p8.7.m7.2.2.2.1.1.1.3"><csymbol cd="latexml" id="S5.p8.7.m7.2.2.2.1.1.1.3.1.cmml" xref="S5.p8.7.m7.2.2.2.1.1.1.3">plus-or-minus</csymbol><infinity id="S5.p8.7.m7.2.2.2.1.1.1.3.2.cmml" xref="S5.p8.7.m7.2.2.2.1.1.1.3.2"></infinity></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.7.m7.2c">\sigma_{1}^{T}(\cal O(b^{\pm\infty}))=\cal O(c^{\pm\infty})</annotation><annotation encoding="application/x-llamapun" id="S5.p8.7.m7.2d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( caligraphic_O ( caligraphic_b start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT ) ) = caligraphic_O ( caligraphic_c start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT )</annotation></semantics></math>, the map <math alttext="\sigma_{1}" class="ltx_Math" display="inline" id="S5.p8.8.m8.1"><semantics id="S5.p8.8.m8.1a"><msub id="S5.p8.8.m8.1.1" xref="S5.p8.8.m8.1.1.cmml"><mi id="S5.p8.8.m8.1.1.2" xref="S5.p8.8.m8.1.1.2.cmml">σ</mi><mn id="S5.p8.8.m8.1.1.3" xref="S5.p8.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p8.8.m8.1b"><apply id="S5.p8.8.m8.1.1.cmml" xref="S5.p8.8.m8.1.1"><csymbol cd="ambiguous" id="S5.p8.8.m8.1.1.1.cmml" xref="S5.p8.8.m8.1.1">subscript</csymbol><ci id="S5.p8.8.m8.1.1.2.cmml" xref="S5.p8.8.m8.1.1.2">𝜎</ci><cn id="S5.p8.8.m8.1.1.3.cmml" type="integer" xref="S5.p8.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.8.m8.1c">\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p8.8.m8.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is not shift-orbit injective in <math alttext="X" class="ltx_Math" display="inline" id="S5.p8.9.m9.1"><semantics id="S5.p8.9.m9.1a"><mi id="S5.p8.9.m9.1.1" xref="S5.p8.9.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p8.9.m9.1b"><ci id="S5.p8.9.m9.1.1.cmml" xref="S5.p8.9.m9.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p8.9.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p8.9.m9.1d">italic_X</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.p9"> <p class="ltx_p" id="S5.p9.5">This example shows also that “<math alttext="\sigma" class="ltx_Math" display="inline" id="S5.p9.1.m1.1"><semantics id="S5.p9.1.m1.1a"><mi id="S5.p9.1.m1.1.1" xref="S5.p9.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.p9.1.m1.1b"><ci id="S5.p9.1.m1.1.1.cmml" xref="S5.p9.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p9.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.p9.1.m1.1d">italic_σ</annotation></semantics></math> recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.p9.2.m2.1"><semantics id="S5.p9.2.m2.1a"><mi id="S5.p9.2.m2.1.1" xref="S5.p9.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p9.2.m2.1b"><ci id="S5.p9.2.m2.1.1.cmml" xref="S5.p9.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p9.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p9.2.m2.1d">italic_X</annotation></semantics></math>” is in general too weak to be able to deduce that the measure transfer map <math alttext="\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))" class="ltx_Math" display="inline" id="S5.p9.3.m3.3"><semantics id="S5.p9.3.m3.3a"><mrow id="S5.p9.3.m3.3.3" xref="S5.p9.3.m3.3.3.cmml"><mrow id="S5.p9.3.m3.3.3.3" xref="S5.p9.3.m3.3.3.3.cmml"><msub id="S5.p9.3.m3.3.3.3.2" xref="S5.p9.3.m3.3.3.3.2.cmml"><mi id="S5.p9.3.m3.3.3.3.2.2" xref="S5.p9.3.m3.3.3.3.2.2.cmml">σ</mi><mi id="S5.p9.3.m3.3.3.3.2.3" xref="S5.p9.3.m3.3.3.3.2.3.cmml">X</mi></msub><mo id="S5.p9.3.m3.3.3.3.1" xref="S5.p9.3.m3.3.3.3.1.cmml">⁢</mo><mi id="S5.p9.3.m3.3.3.3.3" xref="S5.p9.3.m3.3.3.3.3.cmml">M</mi></mrow><mo id="S5.p9.3.m3.3.3.2" lspace="0.278em" rspace="0.278em" xref="S5.p9.3.m3.3.3.2.cmml">:</mo><mrow id="S5.p9.3.m3.3.3.1" xref="S5.p9.3.m3.3.3.1.cmml"><mrow id="S5.p9.3.m3.3.3.1.3" xref="S5.p9.3.m3.3.3.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p9.3.m3.3.3.1.3.2" xref="S5.p9.3.m3.3.3.1.3.2.cmml">ℳ</mi><mo id="S5.p9.3.m3.3.3.1.3.1" xref="S5.p9.3.m3.3.3.1.3.1.cmml">⁢</mo><mrow id="S5.p9.3.m3.3.3.1.3.3.2" xref="S5.p9.3.m3.3.3.1.3.cmml"><mo id="S5.p9.3.m3.3.3.1.3.3.2.1" stretchy="false" xref="S5.p9.3.m3.3.3.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.p9.3.m3.1.1" xref="S5.p9.3.m3.1.1.cmml">𝒳</mi><mo id="S5.p9.3.m3.3.3.1.3.3.2.2" stretchy="false" xref="S5.p9.3.m3.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="S5.p9.3.m3.3.3.1.2" stretchy="false" xref="S5.p9.3.m3.3.3.1.2.cmml">→</mo><mrow id="S5.p9.3.m3.3.3.1.1" xref="S5.p9.3.m3.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p9.3.m3.3.3.1.1.3" xref="S5.p9.3.m3.3.3.1.1.3.cmml">ℳ</mi><mo id="S5.p9.3.m3.3.3.1.1.2" xref="S5.p9.3.m3.3.3.1.1.2.cmml">⁢</mo><mrow id="S5.p9.3.m3.3.3.1.1.1.1" xref="S5.p9.3.m3.3.3.1.1.1.1.1.cmml"><mo id="S5.p9.3.m3.3.3.1.1.1.1.2" stretchy="false" xref="S5.p9.3.m3.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.p9.3.m3.3.3.1.1.1.1.1" xref="S5.p9.3.m3.3.3.1.1.1.1.1.cmml"><mi id="S5.p9.3.m3.3.3.1.1.1.1.1.2" xref="S5.p9.3.m3.3.3.1.1.1.1.1.2.cmml">σ</mi><mo id="S5.p9.3.m3.3.3.1.1.1.1.1.1" xref="S5.p9.3.m3.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.p9.3.m3.3.3.1.1.1.1.1.3.2" xref="S5.p9.3.m3.3.3.1.1.1.1.1.cmml"><mo id="S5.p9.3.m3.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="S5.p9.3.m3.3.3.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.p9.3.m3.2.2" xref="S5.p9.3.m3.2.2.cmml">𝒳</mi><mo id="S5.p9.3.m3.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="S5.p9.3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p9.3.m3.3.3.1.1.1.1.3" stretchy="false" xref="S5.p9.3.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p9.3.m3.3b"><apply id="S5.p9.3.m3.3.3.cmml" xref="S5.p9.3.m3.3.3"><ci id="S5.p9.3.m3.3.3.2.cmml" xref="S5.p9.3.m3.3.3.2">:</ci><apply id="S5.p9.3.m3.3.3.3.cmml" xref="S5.p9.3.m3.3.3.3"><times id="S5.p9.3.m3.3.3.3.1.cmml" xref="S5.p9.3.m3.3.3.3.1"></times><apply id="S5.p9.3.m3.3.3.3.2.cmml" xref="S5.p9.3.m3.3.3.3.2"><csymbol cd="ambiguous" id="S5.p9.3.m3.3.3.3.2.1.cmml" xref="S5.p9.3.m3.3.3.3.2">subscript</csymbol><ci id="S5.p9.3.m3.3.3.3.2.2.cmml" xref="S5.p9.3.m3.3.3.3.2.2">𝜎</ci><ci id="S5.p9.3.m3.3.3.3.2.3.cmml" xref="S5.p9.3.m3.3.3.3.2.3">𝑋</ci></apply><ci id="S5.p9.3.m3.3.3.3.3.cmml" xref="S5.p9.3.m3.3.3.3.3">𝑀</ci></apply><apply id="S5.p9.3.m3.3.3.1.cmml" xref="S5.p9.3.m3.3.3.1"><ci id="S5.p9.3.m3.3.3.1.2.cmml" xref="S5.p9.3.m3.3.3.1.2">→</ci><apply id="S5.p9.3.m3.3.3.1.3.cmml" xref="S5.p9.3.m3.3.3.1.3"><times id="S5.p9.3.m3.3.3.1.3.1.cmml" xref="S5.p9.3.m3.3.3.1.3.1"></times><ci id="S5.p9.3.m3.3.3.1.3.2.cmml" xref="S5.p9.3.m3.3.3.1.3.2">ℳ</ci><ci id="S5.p9.3.m3.1.1.cmml" xref="S5.p9.3.m3.1.1">𝒳</ci></apply><apply id="S5.p9.3.m3.3.3.1.1.cmml" xref="S5.p9.3.m3.3.3.1.1"><times id="S5.p9.3.m3.3.3.1.1.2.cmml" xref="S5.p9.3.m3.3.3.1.1.2"></times><ci id="S5.p9.3.m3.3.3.1.1.3.cmml" xref="S5.p9.3.m3.3.3.1.1.3">ℳ</ci><apply id="S5.p9.3.m3.3.3.1.1.1.1.1.cmml" xref="S5.p9.3.m3.3.3.1.1.1.1"><times id="S5.p9.3.m3.3.3.1.1.1.1.1.1.cmml" xref="S5.p9.3.m3.3.3.1.1.1.1.1.1"></times><ci id="S5.p9.3.m3.3.3.1.1.1.1.1.2.cmml" xref="S5.p9.3.m3.3.3.1.1.1.1.1.2">𝜎</ci><ci id="S5.p9.3.m3.2.2.cmml" xref="S5.p9.3.m3.2.2">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p9.3.m3.3c">\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S5.p9.3.m3.3d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) )</annotation></semantics></math> is injective: For <math alttext="\sigma_{1}" class="ltx_Math" display="inline" id="S5.p9.4.m4.1"><semantics id="S5.p9.4.m4.1a"><msub id="S5.p9.4.m4.1.1" xref="S5.p9.4.m4.1.1.cmml"><mi id="S5.p9.4.m4.1.1.2" xref="S5.p9.4.m4.1.1.2.cmml">σ</mi><mn id="S5.p9.4.m4.1.1.3" xref="S5.p9.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p9.4.m4.1b"><apply id="S5.p9.4.m4.1.1.cmml" xref="S5.p9.4.m4.1.1"><csymbol cd="ambiguous" id="S5.p9.4.m4.1.1.1.cmml" xref="S5.p9.4.m4.1.1">subscript</csymbol><ci id="S5.p9.4.m4.1.1.2.cmml" xref="S5.p9.4.m4.1.1.2">𝜎</ci><cn id="S5.p9.4.m4.1.1.3.cmml" type="integer" xref="S5.p9.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p9.4.m4.1c">\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p9.4.m4.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> as above one observes directly from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm7" title="Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.7</span></a> (d) that <math alttext="\sigma_{1}M(\mu_{a})=\sigma_{1}M(\mu_{b})=\mu_{c}" class="ltx_Math" display="inline" id="S5.p9.5.m5.2"><semantics id="S5.p9.5.m5.2a"><mrow id="S5.p9.5.m5.2.2" xref="S5.p9.5.m5.2.2.cmml"><mrow id="S5.p9.5.m5.1.1.1" xref="S5.p9.5.m5.1.1.1.cmml"><msub id="S5.p9.5.m5.1.1.1.3" xref="S5.p9.5.m5.1.1.1.3.cmml"><mi id="S5.p9.5.m5.1.1.1.3.2" xref="S5.p9.5.m5.1.1.1.3.2.cmml">σ</mi><mn id="S5.p9.5.m5.1.1.1.3.3" xref="S5.p9.5.m5.1.1.1.3.3.cmml">1</mn></msub><mo id="S5.p9.5.m5.1.1.1.2" xref="S5.p9.5.m5.1.1.1.2.cmml">⁢</mo><mi id="S5.p9.5.m5.1.1.1.4" xref="S5.p9.5.m5.1.1.1.4.cmml">M</mi><mo id="S5.p9.5.m5.1.1.1.2a" xref="S5.p9.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S5.p9.5.m5.1.1.1.1.1" xref="S5.p9.5.m5.1.1.1.1.1.1.cmml"><mo id="S5.p9.5.m5.1.1.1.1.1.2" stretchy="false" xref="S5.p9.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S5.p9.5.m5.1.1.1.1.1.1" xref="S5.p9.5.m5.1.1.1.1.1.1.cmml"><mi id="S5.p9.5.m5.1.1.1.1.1.1.2" xref="S5.p9.5.m5.1.1.1.1.1.1.2.cmml">μ</mi><mi id="S5.p9.5.m5.1.1.1.1.1.1.3" xref="S5.p9.5.m5.1.1.1.1.1.1.3.cmml">a</mi></msub><mo id="S5.p9.5.m5.1.1.1.1.1.3" stretchy="false" xref="S5.p9.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p9.5.m5.2.2.4" xref="S5.p9.5.m5.2.2.4.cmml">=</mo><mrow id="S5.p9.5.m5.2.2.2" xref="S5.p9.5.m5.2.2.2.cmml"><msub id="S5.p9.5.m5.2.2.2.3" xref="S5.p9.5.m5.2.2.2.3.cmml"><mi id="S5.p9.5.m5.2.2.2.3.2" xref="S5.p9.5.m5.2.2.2.3.2.cmml">σ</mi><mn id="S5.p9.5.m5.2.2.2.3.3" xref="S5.p9.5.m5.2.2.2.3.3.cmml">1</mn></msub><mo id="S5.p9.5.m5.2.2.2.2" xref="S5.p9.5.m5.2.2.2.2.cmml">⁢</mo><mi id="S5.p9.5.m5.2.2.2.4" xref="S5.p9.5.m5.2.2.2.4.cmml">M</mi><mo id="S5.p9.5.m5.2.2.2.2a" xref="S5.p9.5.m5.2.2.2.2.cmml">⁢</mo><mrow id="S5.p9.5.m5.2.2.2.1.1" xref="S5.p9.5.m5.2.2.2.1.1.1.cmml"><mo id="S5.p9.5.m5.2.2.2.1.1.2" stretchy="false" xref="S5.p9.5.m5.2.2.2.1.1.1.cmml">(</mo><msub id="S5.p9.5.m5.2.2.2.1.1.1" xref="S5.p9.5.m5.2.2.2.1.1.1.cmml"><mi id="S5.p9.5.m5.2.2.2.1.1.1.2" xref="S5.p9.5.m5.2.2.2.1.1.1.2.cmml">μ</mi><mi id="S5.p9.5.m5.2.2.2.1.1.1.3" xref="S5.p9.5.m5.2.2.2.1.1.1.3.cmml">b</mi></msub><mo id="S5.p9.5.m5.2.2.2.1.1.3" stretchy="false" xref="S5.p9.5.m5.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.p9.5.m5.2.2.5" xref="S5.p9.5.m5.2.2.5.cmml">=</mo><msub id="S5.p9.5.m5.2.2.6" xref="S5.p9.5.m5.2.2.6.cmml"><mi id="S5.p9.5.m5.2.2.6.2" xref="S5.p9.5.m5.2.2.6.2.cmml">μ</mi><mi id="S5.p9.5.m5.2.2.6.3" xref="S5.p9.5.m5.2.2.6.3.cmml">c</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p9.5.m5.2b"><apply id="S5.p9.5.m5.2.2.cmml" xref="S5.p9.5.m5.2.2"><and id="S5.p9.5.m5.2.2a.cmml" xref="S5.p9.5.m5.2.2"></and><apply id="S5.p9.5.m5.2.2b.cmml" xref="S5.p9.5.m5.2.2"><eq id="S5.p9.5.m5.2.2.4.cmml" xref="S5.p9.5.m5.2.2.4"></eq><apply id="S5.p9.5.m5.1.1.1.cmml" xref="S5.p9.5.m5.1.1.1"><times id="S5.p9.5.m5.1.1.1.2.cmml" xref="S5.p9.5.m5.1.1.1.2"></times><apply id="S5.p9.5.m5.1.1.1.3.cmml" xref="S5.p9.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="S5.p9.5.m5.1.1.1.3.1.cmml" xref="S5.p9.5.m5.1.1.1.3">subscript</csymbol><ci id="S5.p9.5.m5.1.1.1.3.2.cmml" xref="S5.p9.5.m5.1.1.1.3.2">𝜎</ci><cn id="S5.p9.5.m5.1.1.1.3.3.cmml" type="integer" xref="S5.p9.5.m5.1.1.1.3.3">1</cn></apply><ci id="S5.p9.5.m5.1.1.1.4.cmml" xref="S5.p9.5.m5.1.1.1.4">𝑀</ci><apply id="S5.p9.5.m5.1.1.1.1.1.1.cmml" xref="S5.p9.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p9.5.m5.1.1.1.1.1.1.1.cmml" xref="S5.p9.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S5.p9.5.m5.1.1.1.1.1.1.2.cmml" xref="S5.p9.5.m5.1.1.1.1.1.1.2">𝜇</ci><ci id="S5.p9.5.m5.1.1.1.1.1.1.3.cmml" xref="S5.p9.5.m5.1.1.1.1.1.1.3">𝑎</ci></apply></apply><apply id="S5.p9.5.m5.2.2.2.cmml" xref="S5.p9.5.m5.2.2.2"><times id="S5.p9.5.m5.2.2.2.2.cmml" xref="S5.p9.5.m5.2.2.2.2"></times><apply id="S5.p9.5.m5.2.2.2.3.cmml" xref="S5.p9.5.m5.2.2.2.3"><csymbol cd="ambiguous" id="S5.p9.5.m5.2.2.2.3.1.cmml" xref="S5.p9.5.m5.2.2.2.3">subscript</csymbol><ci id="S5.p9.5.m5.2.2.2.3.2.cmml" xref="S5.p9.5.m5.2.2.2.3.2">𝜎</ci><cn id="S5.p9.5.m5.2.2.2.3.3.cmml" type="integer" xref="S5.p9.5.m5.2.2.2.3.3">1</cn></apply><ci id="S5.p9.5.m5.2.2.2.4.cmml" xref="S5.p9.5.m5.2.2.2.4">𝑀</ci><apply id="S5.p9.5.m5.2.2.2.1.1.1.cmml" xref="S5.p9.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.p9.5.m5.2.2.2.1.1.1.1.cmml" xref="S5.p9.5.m5.2.2.2.1.1">subscript</csymbol><ci id="S5.p9.5.m5.2.2.2.1.1.1.2.cmml" xref="S5.p9.5.m5.2.2.2.1.1.1.2">𝜇</ci><ci id="S5.p9.5.m5.2.2.2.1.1.1.3.cmml" xref="S5.p9.5.m5.2.2.2.1.1.1.3">𝑏</ci></apply></apply></apply><apply id="S5.p9.5.m5.2.2c.cmml" xref="S5.p9.5.m5.2.2"><eq id="S5.p9.5.m5.2.2.5.cmml" xref="S5.p9.5.m5.2.2.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S5.p9.5.m5.2.2.2.cmml" id="S5.p9.5.m5.2.2d.cmml" xref="S5.p9.5.m5.2.2"></share><apply id="S5.p9.5.m5.2.2.6.cmml" xref="S5.p9.5.m5.2.2.6"><csymbol cd="ambiguous" id="S5.p9.5.m5.2.2.6.1.cmml" xref="S5.p9.5.m5.2.2.6">subscript</csymbol><ci id="S5.p9.5.m5.2.2.6.2.cmml" xref="S5.p9.5.m5.2.2.6.2">𝜇</ci><ci id="S5.p9.5.m5.2.2.6.3.cmml" xref="S5.p9.5.m5.2.2.6.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p9.5.m5.2c">\sigma_{1}M(\mu_{a})=\sigma_{1}M(\mu_{b})=\mu_{c}</annotation><annotation encoding="application/x-llamapun" id="S5.p9.5.m5.2d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M ( italic_μ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) = italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_M ( italic_μ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) = italic_μ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.p10"> <p class="ltx_p" id="S5.p10.4">We also observe that the injectivity of the measure transfer map <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S5.p10.1.m1.1"><semantics id="S5.p10.1.m1.1a"><mrow id="S5.p10.1.m1.1.1" xref="S5.p10.1.m1.1.1.cmml"><msub id="S5.p10.1.m1.1.1.2" xref="S5.p10.1.m1.1.1.2.cmml"><mi id="S5.p10.1.m1.1.1.2.2" xref="S5.p10.1.m1.1.1.2.2.cmml">σ</mi><mi id="S5.p10.1.m1.1.1.2.3" xref="S5.p10.1.m1.1.1.2.3.cmml">X</mi></msub><mo id="S5.p10.1.m1.1.1.1" xref="S5.p10.1.m1.1.1.1.cmml">⁢</mo><mi id="S5.p10.1.m1.1.1.3" xref="S5.p10.1.m1.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p10.1.m1.1b"><apply id="S5.p10.1.m1.1.1.cmml" xref="S5.p10.1.m1.1.1"><times id="S5.p10.1.m1.1.1.1.cmml" xref="S5.p10.1.m1.1.1.1"></times><apply id="S5.p10.1.m1.1.1.2.cmml" xref="S5.p10.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.p10.1.m1.1.1.2.1.cmml" xref="S5.p10.1.m1.1.1.2">subscript</csymbol><ci id="S5.p10.1.m1.1.1.2.2.cmml" xref="S5.p10.1.m1.1.1.2.2">𝜎</ci><ci id="S5.p10.1.m1.1.1.2.3.cmml" xref="S5.p10.1.m1.1.1.2.3">𝑋</ci></apply><ci id="S5.p10.1.m1.1.1.3.cmml" xref="S5.p10.1.m1.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.1.m1.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S5.p10.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> does in general not imply the injectivity of the map <math alttext="\sigma_{X}^{T}" class="ltx_Math" display="inline" id="S5.p10.2.m2.1"><semantics id="S5.p10.2.m2.1a"><msubsup id="S5.p10.2.m2.1.1" xref="S5.p10.2.m2.1.1.cmml"><mi id="S5.p10.2.m2.1.1.2.2" xref="S5.p10.2.m2.1.1.2.2.cmml">σ</mi><mi id="S5.p10.2.m2.1.1.2.3" xref="S5.p10.2.m2.1.1.2.3.cmml">X</mi><mi id="S5.p10.2.m2.1.1.3" xref="S5.p10.2.m2.1.1.3.cmml">T</mi></msubsup><annotation-xml encoding="MathML-Content" id="S5.p10.2.m2.1b"><apply id="S5.p10.2.m2.1.1.cmml" xref="S5.p10.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p10.2.m2.1.1.1.cmml" xref="S5.p10.2.m2.1.1">superscript</csymbol><apply id="S5.p10.2.m2.1.1.2.cmml" xref="S5.p10.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p10.2.m2.1.1.2.1.cmml" xref="S5.p10.2.m2.1.1">subscript</csymbol><ci id="S5.p10.2.m2.1.1.2.2.cmml" xref="S5.p10.2.m2.1.1.2.2">𝜎</ci><ci id="S5.p10.2.m2.1.1.2.3.cmml" xref="S5.p10.2.m2.1.1.2.3">𝑋</ci></apply><ci id="S5.p10.2.m2.1.1.3.cmml" xref="S5.p10.2.m2.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.2.m2.1c">\sigma_{X}^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.p10.2.m2.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> induced by <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.p10.3.m3.1"><semantics id="S5.p10.3.m3.1a"><mi id="S5.p10.3.m3.1.1" xref="S5.p10.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.p10.3.m3.1b"><ci id="S5.p10.3.m3.1.1.cmml" xref="S5.p10.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.p10.3.m3.1d">italic_σ</annotation></semantics></math> on the orbits of <math alttext="X" class="ltx_Math" display="inline" id="S5.p10.4.m4.1"><semantics id="S5.p10.4.m4.1a"><mi id="S5.p10.4.m4.1.1" xref="S5.p10.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p10.4.m4.1b"><ci id="S5.p10.4.m4.1.1.cmml" xref="S5.p10.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p10.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p10.4.m4.1d">italic_X</annotation></semantics></math>, so that the converse of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm6" title="Theorem 5.6. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.6</span></a> does not hold in full generality. Indeed, we have:</p> </div> <div class="ltx_theorem ltx_theorem_rem" id="S5.Thmthm10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm10.1.1.1">Remark 5.10</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm10.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm10.p1"> <p class="ltx_p" id="S5.Thmthm10.p1.1">Let <math alttext="X\subseteq\{a,b\}^{*}" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.1.m1.2"><semantics id="S5.Thmthm10.p1.1.m1.2a"><mrow id="S5.Thmthm10.p1.1.m1.2.3" xref="S5.Thmthm10.p1.1.m1.2.3.cmml"><mi id="S5.Thmthm10.p1.1.m1.2.3.2" xref="S5.Thmthm10.p1.1.m1.2.3.2.cmml">X</mi><mo id="S5.Thmthm10.p1.1.m1.2.3.1" xref="S5.Thmthm10.p1.1.m1.2.3.1.cmml">⊆</mo><msup id="S5.Thmthm10.p1.1.m1.2.3.3" xref="S5.Thmthm10.p1.1.m1.2.3.3.cmml"><mrow id="S5.Thmthm10.p1.1.m1.2.3.3.2.2" xref="S5.Thmthm10.p1.1.m1.2.3.3.2.1.cmml"><mo id="S5.Thmthm10.p1.1.m1.2.3.3.2.2.1" stretchy="false" xref="S5.Thmthm10.p1.1.m1.2.3.3.2.1.cmml">{</mo><mi id="S5.Thmthm10.p1.1.m1.1.1" xref="S5.Thmthm10.p1.1.m1.1.1.cmml">a</mi><mo id="S5.Thmthm10.p1.1.m1.2.3.3.2.2.2" xref="S5.Thmthm10.p1.1.m1.2.3.3.2.1.cmml">,</mo><mi id="S5.Thmthm10.p1.1.m1.2.2" xref="S5.Thmthm10.p1.1.m1.2.2.cmml">b</mi><mo id="S5.Thmthm10.p1.1.m1.2.3.3.2.2.3" stretchy="false" xref="S5.Thmthm10.p1.1.m1.2.3.3.2.1.cmml">}</mo></mrow><mo id="S5.Thmthm10.p1.1.m1.2.3.3.3" xref="S5.Thmthm10.p1.1.m1.2.3.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.1.m1.2b"><apply id="S5.Thmthm10.p1.1.m1.2.3.cmml" xref="S5.Thmthm10.p1.1.m1.2.3"><subset id="S5.Thmthm10.p1.1.m1.2.3.1.cmml" xref="S5.Thmthm10.p1.1.m1.2.3.1"></subset><ci id="S5.Thmthm10.p1.1.m1.2.3.2.cmml" xref="S5.Thmthm10.p1.1.m1.2.3.2">𝑋</ci><apply id="S5.Thmthm10.p1.1.m1.2.3.3.cmml" xref="S5.Thmthm10.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="S5.Thmthm10.p1.1.m1.2.3.3.1.cmml" xref="S5.Thmthm10.p1.1.m1.2.3.3">superscript</csymbol><set id="S5.Thmthm10.p1.1.m1.2.3.3.2.1.cmml" xref="S5.Thmthm10.p1.1.m1.2.3.3.2.2"><ci id="S5.Thmthm10.p1.1.m1.1.1.cmml" xref="S5.Thmthm10.p1.1.m1.1.1">𝑎</ci><ci id="S5.Thmthm10.p1.1.m1.2.2.cmml" xref="S5.Thmthm10.p1.1.m1.2.2">𝑏</ci></set><times id="S5.Thmthm10.p1.1.m1.2.3.3.3.cmml" xref="S5.Thmthm10.p1.1.m1.2.3.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.1.m1.2c">X\subseteq\{a,b\}^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.1.m1.2d">italic_X ⊆ { italic_a , italic_b } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be the countable subshift which consists of the two orbits defined by</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ldots aaa\ldots\qquad\text{and}\qquad\ldots aaabaaa\ldots\,." class="ltx_Math" display="block" id="S5.Ex1.m1.2"><semantics id="S5.Ex1.m1.2a"><mrow id="S5.Ex1.m1.2.2.1"><mrow id="S5.Ex1.m1.2.2.1.1.2" xref="S5.Ex1.m1.2.2.1.1.3.cmml"><mrow id="S5.Ex1.m1.2.2.1.1.1.1" xref="S5.Ex1.m1.2.2.1.1.1.1.cmml"><mi id="S5.Ex1.m1.2.2.1.1.1.1.2" mathvariant="normal" xref="S5.Ex1.m1.2.2.1.1.1.1.2.cmml">…</mi><mo id="S5.Ex1.m1.2.2.1.1.1.1.1" xref="S5.Ex1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.1.1.3" xref="S5.Ex1.m1.2.2.1.1.1.1.3.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.1.1.1a" xref="S5.Ex1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.1.1.4" xref="S5.Ex1.m1.2.2.1.1.1.1.4.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.1.1.1b" xref="S5.Ex1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.1.1.5" xref="S5.Ex1.m1.2.2.1.1.1.1.5.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.1.1.1c" xref="S5.Ex1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.1.1.6" mathvariant="normal" xref="S5.Ex1.m1.2.2.1.1.1.1.6.cmml">…</mi></mrow><mspace id="S5.Ex1.m1.2.2.1.1.2.3" width="2em" xref="S5.Ex1.m1.2.2.1.1.3.cmml"></mspace><mtext id="S5.Ex1.m1.1.1" xref="S5.Ex1.m1.1.1a.cmml">and</mtext><mspace id="S5.Ex1.m1.2.2.1.1.2.4" width="2em" xref="S5.Ex1.m1.2.2.1.1.3.cmml"></mspace><mrow id="S5.Ex1.m1.2.2.1.1.2.2" xref="S5.Ex1.m1.2.2.1.1.2.2.cmml"><mi id="S5.Ex1.m1.2.2.1.1.2.2.2" mathvariant="normal" xref="S5.Ex1.m1.2.2.1.1.2.2.2.cmml">…</mi><mo id="S5.Ex1.m1.2.2.1.1.2.2.1" xref="S5.Ex1.m1.2.2.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.2.2.3" xref="S5.Ex1.m1.2.2.1.1.2.2.3.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.2.2.1a" xref="S5.Ex1.m1.2.2.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.2.2.4" xref="S5.Ex1.m1.2.2.1.1.2.2.4.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.2.2.1b" xref="S5.Ex1.m1.2.2.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.2.2.5" xref="S5.Ex1.m1.2.2.1.1.2.2.5.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.2.2.1c" xref="S5.Ex1.m1.2.2.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.2.2.6" xref="S5.Ex1.m1.2.2.1.1.2.2.6.cmml">b</mi><mo id="S5.Ex1.m1.2.2.1.1.2.2.1d" xref="S5.Ex1.m1.2.2.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.2.2.7" xref="S5.Ex1.m1.2.2.1.1.2.2.7.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.2.2.1e" xref="S5.Ex1.m1.2.2.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.2.2.8" xref="S5.Ex1.m1.2.2.1.1.2.2.8.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.2.2.1f" xref="S5.Ex1.m1.2.2.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.2.2.9" xref="S5.Ex1.m1.2.2.1.1.2.2.9.cmml">a</mi><mo id="S5.Ex1.m1.2.2.1.1.2.2.1g" xref="S5.Ex1.m1.2.2.1.1.2.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.2.2.1.1.2.2.10" mathvariant="normal" xref="S5.Ex1.m1.2.2.1.1.2.2.10.cmml">…</mi></mrow></mrow><mo id="S5.Ex1.m1.2.2.1.2" lspace="0.170em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex1.m1.2b"><list id="S5.Ex1.m1.2.2.1.1.3.cmml" xref="S5.Ex1.m1.2.2.1.1.2"><apply id="S5.Ex1.m1.2.2.1.1.1.1.cmml" xref="S5.Ex1.m1.2.2.1.1.1.1"><times id="S5.Ex1.m1.2.2.1.1.1.1.1.cmml" xref="S5.Ex1.m1.2.2.1.1.1.1.1"></times><ci id="S5.Ex1.m1.2.2.1.1.1.1.2.cmml" xref="S5.Ex1.m1.2.2.1.1.1.1.2">…</ci><ci id="S5.Ex1.m1.2.2.1.1.1.1.3.cmml" xref="S5.Ex1.m1.2.2.1.1.1.1.3">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.1.1.4.cmml" xref="S5.Ex1.m1.2.2.1.1.1.1.4">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.1.1.5.cmml" xref="S5.Ex1.m1.2.2.1.1.1.1.5">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.1.1.6.cmml" xref="S5.Ex1.m1.2.2.1.1.1.1.6">…</ci></apply><ci id="S5.Ex1.m1.1.1a.cmml" xref="S5.Ex1.m1.1.1"><mtext id="S5.Ex1.m1.1.1.cmml" xref="S5.Ex1.m1.1.1">and</mtext></ci><apply id="S5.Ex1.m1.2.2.1.1.2.2.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2"><times id="S5.Ex1.m1.2.2.1.1.2.2.1.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.1"></times><ci id="S5.Ex1.m1.2.2.1.1.2.2.2.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.2">…</ci><ci id="S5.Ex1.m1.2.2.1.1.2.2.3.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.3">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.2.2.4.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.4">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.2.2.5.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.5">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.2.2.6.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.6">𝑏</ci><ci id="S5.Ex1.m1.2.2.1.1.2.2.7.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.7">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.2.2.8.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.8">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.2.2.9.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.9">𝑎</ci><ci id="S5.Ex1.m1.2.2.1.1.2.2.10.cmml" xref="S5.Ex1.m1.2.2.1.1.2.2.10">…</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex1.m1.2c">\ldots aaa\ldots\qquad\text{and}\qquad\ldots aaabaaa\ldots\,.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex1.m1.2d">… italic_a italic_a italic_a … and … italic_a italic_a italic_a italic_b italic_a italic_a italic_a … .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmthm10.p1.12">Set <math alttext="Y=\{c\}^{*}" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.2.m1.1"><semantics id="S5.Thmthm10.p1.2.m1.1a"><mrow id="S5.Thmthm10.p1.2.m1.1.2" xref="S5.Thmthm10.p1.2.m1.1.2.cmml"><mi id="S5.Thmthm10.p1.2.m1.1.2.2" xref="S5.Thmthm10.p1.2.m1.1.2.2.cmml">Y</mi><mo id="S5.Thmthm10.p1.2.m1.1.2.1" xref="S5.Thmthm10.p1.2.m1.1.2.1.cmml">=</mo><msup id="S5.Thmthm10.p1.2.m1.1.2.3" xref="S5.Thmthm10.p1.2.m1.1.2.3.cmml"><mrow id="S5.Thmthm10.p1.2.m1.1.2.3.2.2" xref="S5.Thmthm10.p1.2.m1.1.2.3.2.1.cmml"><mo id="S5.Thmthm10.p1.2.m1.1.2.3.2.2.1" stretchy="false" xref="S5.Thmthm10.p1.2.m1.1.2.3.2.1.cmml">{</mo><mi id="S5.Thmthm10.p1.2.m1.1.1" xref="S5.Thmthm10.p1.2.m1.1.1.cmml">c</mi><mo id="S5.Thmthm10.p1.2.m1.1.2.3.2.2.2" stretchy="false" xref="S5.Thmthm10.p1.2.m1.1.2.3.2.1.cmml">}</mo></mrow><mo id="S5.Thmthm10.p1.2.m1.1.2.3.3" xref="S5.Thmthm10.p1.2.m1.1.2.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.2.m1.1b"><apply id="S5.Thmthm10.p1.2.m1.1.2.cmml" xref="S5.Thmthm10.p1.2.m1.1.2"><eq id="S5.Thmthm10.p1.2.m1.1.2.1.cmml" xref="S5.Thmthm10.p1.2.m1.1.2.1"></eq><ci id="S5.Thmthm10.p1.2.m1.1.2.2.cmml" xref="S5.Thmthm10.p1.2.m1.1.2.2">𝑌</ci><apply id="S5.Thmthm10.p1.2.m1.1.2.3.cmml" xref="S5.Thmthm10.p1.2.m1.1.2.3"><csymbol cd="ambiguous" id="S5.Thmthm10.p1.2.m1.1.2.3.1.cmml" xref="S5.Thmthm10.p1.2.m1.1.2.3">superscript</csymbol><set id="S5.Thmthm10.p1.2.m1.1.2.3.2.1.cmml" xref="S5.Thmthm10.p1.2.m1.1.2.3.2.2"><ci id="S5.Thmthm10.p1.2.m1.1.1.cmml" xref="S5.Thmthm10.p1.2.m1.1.1">𝑐</ci></set><times id="S5.Thmthm10.p1.2.m1.1.2.3.3.cmml" xref="S5.Thmthm10.p1.2.m1.1.2.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.2.m1.1c">Y=\{c\}^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.2.m1.1d">italic_Y = { italic_c } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, and consider again the morphism <math alttext="\sigma_{1}" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.3.m2.1"><semantics id="S5.Thmthm10.p1.3.m2.1a"><msub id="S5.Thmthm10.p1.3.m2.1.1" xref="S5.Thmthm10.p1.3.m2.1.1.cmml"><mi id="S5.Thmthm10.p1.3.m2.1.1.2" xref="S5.Thmthm10.p1.3.m2.1.1.2.cmml">σ</mi><mn id="S5.Thmthm10.p1.3.m2.1.1.3" xref="S5.Thmthm10.p1.3.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.3.m2.1b"><apply id="S5.Thmthm10.p1.3.m2.1.1.cmml" xref="S5.Thmthm10.p1.3.m2.1.1"><csymbol cd="ambiguous" id="S5.Thmthm10.p1.3.m2.1.1.1.cmml" xref="S5.Thmthm10.p1.3.m2.1.1">subscript</csymbol><ci id="S5.Thmthm10.p1.3.m2.1.1.2.cmml" xref="S5.Thmthm10.p1.3.m2.1.1.2">𝜎</ci><cn id="S5.Thmthm10.p1.3.m2.1.1.3.cmml" type="integer" xref="S5.Thmthm10.p1.3.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.3.m2.1c">\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.3.m2.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> from Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm9" title="Remark 2.9. ‣ 2.3.1. Typical injectivity problems ‣ 2.3. About injectivity ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.9</span></a>, given by <math alttext="a\mapsto c" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.4.m3.1"><semantics id="S5.Thmthm10.p1.4.m3.1a"><mrow id="S5.Thmthm10.p1.4.m3.1.1" xref="S5.Thmthm10.p1.4.m3.1.1.cmml"><mi id="S5.Thmthm10.p1.4.m3.1.1.2" xref="S5.Thmthm10.p1.4.m3.1.1.2.cmml">a</mi><mo id="S5.Thmthm10.p1.4.m3.1.1.1" stretchy="false" xref="S5.Thmthm10.p1.4.m3.1.1.1.cmml">↦</mo><mi id="S5.Thmthm10.p1.4.m3.1.1.3" xref="S5.Thmthm10.p1.4.m3.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.4.m3.1b"><apply id="S5.Thmthm10.p1.4.m3.1.1.cmml" xref="S5.Thmthm10.p1.4.m3.1.1"><csymbol cd="latexml" id="S5.Thmthm10.p1.4.m3.1.1.1.cmml" xref="S5.Thmthm10.p1.4.m3.1.1.1">maps-to</csymbol><ci id="S5.Thmthm10.p1.4.m3.1.1.2.cmml" xref="S5.Thmthm10.p1.4.m3.1.1.2">𝑎</ci><ci id="S5.Thmthm10.p1.4.m3.1.1.3.cmml" xref="S5.Thmthm10.p1.4.m3.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.4.m3.1c">a\mapsto c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.4.m3.1d">italic_a ↦ italic_c</annotation></semantics></math> and <math alttext="b\mapsto c" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.5.m4.1"><semantics id="S5.Thmthm10.p1.5.m4.1a"><mrow id="S5.Thmthm10.p1.5.m4.1.1" xref="S5.Thmthm10.p1.5.m4.1.1.cmml"><mi id="S5.Thmthm10.p1.5.m4.1.1.2" xref="S5.Thmthm10.p1.5.m4.1.1.2.cmml">b</mi><mo id="S5.Thmthm10.p1.5.m4.1.1.1" stretchy="false" xref="S5.Thmthm10.p1.5.m4.1.1.1.cmml">↦</mo><mi id="S5.Thmthm10.p1.5.m4.1.1.3" xref="S5.Thmthm10.p1.5.m4.1.1.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.5.m4.1b"><apply id="S5.Thmthm10.p1.5.m4.1.1.cmml" xref="S5.Thmthm10.p1.5.m4.1.1"><csymbol cd="latexml" id="S5.Thmthm10.p1.5.m4.1.1.1.cmml" xref="S5.Thmthm10.p1.5.m4.1.1.1">maps-to</csymbol><ci id="S5.Thmthm10.p1.5.m4.1.1.2.cmml" xref="S5.Thmthm10.p1.5.m4.1.1.2">𝑏</ci><ci id="S5.Thmthm10.p1.5.m4.1.1.3.cmml" xref="S5.Thmthm10.p1.5.m4.1.1.3">𝑐</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.5.m4.1c">b\mapsto c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.5.m4.1d">italic_b ↦ italic_c</annotation></semantics></math>. This map is not injective on the shift-orbits of <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.6.m5.1"><semantics id="S5.Thmthm10.p1.6.m5.1a"><mi id="S5.Thmthm10.p1.6.m5.1.1" xref="S5.Thmthm10.p1.6.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.6.m5.1b"><ci id="S5.Thmthm10.p1.6.m5.1.1.cmml" xref="S5.Thmthm10.p1.6.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.6.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.6.m5.1d">italic_X</annotation></semantics></math>, but the map <math alttext="(\sigma_{1})_{X}M:\cal M(X)\to\cal M(Y)" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.7.m6.3"><semantics id="S5.Thmthm10.p1.7.m6.3a"><mrow id="S5.Thmthm10.p1.7.m6.3.3" xref="S5.Thmthm10.p1.7.m6.3.3.cmml"><mrow id="S5.Thmthm10.p1.7.m6.3.3.1" xref="S5.Thmthm10.p1.7.m6.3.3.1.cmml"><msub id="S5.Thmthm10.p1.7.m6.3.3.1.1" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.cmml"><mrow id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.cmml"><mo id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.2" stretchy="false" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.cmml">(</mo><msub id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.cmml"><mi id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.2" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.2.cmml">σ</mi><mn id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.3" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.3" stretchy="false" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.cmml">)</mo></mrow><mi id="S5.Thmthm10.p1.7.m6.3.3.1.1.3" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.3.cmml">X</mi></msub><mo id="S5.Thmthm10.p1.7.m6.3.3.1.2" xref="S5.Thmthm10.p1.7.m6.3.3.1.2.cmml">⁢</mo><mi id="S5.Thmthm10.p1.7.m6.3.3.1.3" xref="S5.Thmthm10.p1.7.m6.3.3.1.3.cmml">M</mi></mrow><mo id="S5.Thmthm10.p1.7.m6.3.3.2" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm10.p1.7.m6.3.3.2.cmml">:</mo><mrow id="S5.Thmthm10.p1.7.m6.3.3.3" xref="S5.Thmthm10.p1.7.m6.3.3.3.cmml"><mrow id="S5.Thmthm10.p1.7.m6.3.3.3.2" xref="S5.Thmthm10.p1.7.m6.3.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm10.p1.7.m6.3.3.3.2.2" xref="S5.Thmthm10.p1.7.m6.3.3.3.2.2.cmml">ℳ</mi><mo id="S5.Thmthm10.p1.7.m6.3.3.3.2.1" xref="S5.Thmthm10.p1.7.m6.3.3.3.2.1.cmml">⁢</mo><mrow id="S5.Thmthm10.p1.7.m6.3.3.3.2.3.2" xref="S5.Thmthm10.p1.7.m6.3.3.3.2.cmml"><mo id="S5.Thmthm10.p1.7.m6.3.3.3.2.3.2.1" stretchy="false" xref="S5.Thmthm10.p1.7.m6.3.3.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm10.p1.7.m6.1.1" xref="S5.Thmthm10.p1.7.m6.1.1.cmml">𝒳</mi><mo id="S5.Thmthm10.p1.7.m6.3.3.3.2.3.2.2" stretchy="false" xref="S5.Thmthm10.p1.7.m6.3.3.3.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm10.p1.7.m6.3.3.3.1" stretchy="false" xref="S5.Thmthm10.p1.7.m6.3.3.3.1.cmml">→</mo><mrow id="S5.Thmthm10.p1.7.m6.3.3.3.3" xref="S5.Thmthm10.p1.7.m6.3.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm10.p1.7.m6.3.3.3.3.2" xref="S5.Thmthm10.p1.7.m6.3.3.3.3.2.cmml">ℳ</mi><mo id="S5.Thmthm10.p1.7.m6.3.3.3.3.1" xref="S5.Thmthm10.p1.7.m6.3.3.3.3.1.cmml">⁢</mo><mrow id="S5.Thmthm10.p1.7.m6.3.3.3.3.3.2" xref="S5.Thmthm10.p1.7.m6.3.3.3.3.cmml"><mo id="S5.Thmthm10.p1.7.m6.3.3.3.3.3.2.1" stretchy="false" xref="S5.Thmthm10.p1.7.m6.3.3.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm10.p1.7.m6.2.2" xref="S5.Thmthm10.p1.7.m6.2.2.cmml">𝒴</mi><mo id="S5.Thmthm10.p1.7.m6.3.3.3.3.3.2.2" stretchy="false" xref="S5.Thmthm10.p1.7.m6.3.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.7.m6.3b"><apply id="S5.Thmthm10.p1.7.m6.3.3.cmml" xref="S5.Thmthm10.p1.7.m6.3.3"><ci id="S5.Thmthm10.p1.7.m6.3.3.2.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.2">:</ci><apply id="S5.Thmthm10.p1.7.m6.3.3.1.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1"><times id="S5.Thmthm10.p1.7.m6.3.3.1.2.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1.2"></times><apply id="S5.Thmthm10.p1.7.m6.3.3.1.1.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1.1"><csymbol cd="ambiguous" id="S5.Thmthm10.p1.7.m6.3.3.1.1.2.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1.1">subscript</csymbol><apply id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.1.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1">subscript</csymbol><ci id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.2.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.2">𝜎</ci><cn id="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.3.cmml" type="integer" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.1.1.1.3">1</cn></apply><ci id="S5.Thmthm10.p1.7.m6.3.3.1.1.3.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1.1.3">𝑋</ci></apply><ci id="S5.Thmthm10.p1.7.m6.3.3.1.3.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.1.3">𝑀</ci></apply><apply id="S5.Thmthm10.p1.7.m6.3.3.3.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.3"><ci id="S5.Thmthm10.p1.7.m6.3.3.3.1.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.3.1">→</ci><apply id="S5.Thmthm10.p1.7.m6.3.3.3.2.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.3.2"><times id="S5.Thmthm10.p1.7.m6.3.3.3.2.1.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.3.2.1"></times><ci id="S5.Thmthm10.p1.7.m6.3.3.3.2.2.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.3.2.2">ℳ</ci><ci id="S5.Thmthm10.p1.7.m6.1.1.cmml" xref="S5.Thmthm10.p1.7.m6.1.1">𝒳</ci></apply><apply id="S5.Thmthm10.p1.7.m6.3.3.3.3.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.3.3"><times id="S5.Thmthm10.p1.7.m6.3.3.3.3.1.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.3.3.1"></times><ci id="S5.Thmthm10.p1.7.m6.3.3.3.3.2.cmml" xref="S5.Thmthm10.p1.7.m6.3.3.3.3.2">ℳ</ci><ci id="S5.Thmthm10.p1.7.m6.2.2.cmml" xref="S5.Thmthm10.p1.7.m6.2.2">𝒴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.7.m6.3c">(\sigma_{1})_{X}M:\cal M(X)\to\cal M(Y)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.7.m6.3d">( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( caligraphic_Y )</annotation></semantics></math> is injective, since the only non-trivial invariant probability measure on <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.8.m7.1"><semantics id="S5.Thmthm10.p1.8.m7.1a"><mi id="S5.Thmthm10.p1.8.m7.1.1" xref="S5.Thmthm10.p1.8.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.8.m7.1b"><ci id="S5.Thmthm10.p1.8.m7.1.1.cmml" xref="S5.Thmthm10.p1.8.m7.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.8.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.8.m7.1d">italic_X</annotation></semantics></math> is the atomic measure <math alttext="\mu_{a}" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.9.m8.1"><semantics id="S5.Thmthm10.p1.9.m8.1a"><msub id="S5.Thmthm10.p1.9.m8.1.1" xref="S5.Thmthm10.p1.9.m8.1.1.cmml"><mi id="S5.Thmthm10.p1.9.m8.1.1.2" xref="S5.Thmthm10.p1.9.m8.1.1.2.cmml">μ</mi><mi id="S5.Thmthm10.p1.9.m8.1.1.3" xref="S5.Thmthm10.p1.9.m8.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.9.m8.1b"><apply id="S5.Thmthm10.p1.9.m8.1.1.cmml" xref="S5.Thmthm10.p1.9.m8.1.1"><csymbol cd="ambiguous" id="S5.Thmthm10.p1.9.m8.1.1.1.cmml" xref="S5.Thmthm10.p1.9.m8.1.1">subscript</csymbol><ci id="S5.Thmthm10.p1.9.m8.1.1.2.cmml" xref="S5.Thmthm10.p1.9.m8.1.1.2">𝜇</ci><ci id="S5.Thmthm10.p1.9.m8.1.1.3.cmml" xref="S5.Thmthm10.p1.9.m8.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.9.m8.1c">\mu_{a}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.9.m8.1d">italic_μ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> concentrated on the element <math alttext="\ldots aaa\ldots" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.10.m9.1"><semantics id="S5.Thmthm10.p1.10.m9.1a"><mrow id="S5.Thmthm10.p1.10.m9.1.1" xref="S5.Thmthm10.p1.10.m9.1.1.cmml"><mi id="S5.Thmthm10.p1.10.m9.1.1.2" mathvariant="normal" xref="S5.Thmthm10.p1.10.m9.1.1.2.cmml">…</mi><mo id="S5.Thmthm10.p1.10.m9.1.1.1" xref="S5.Thmthm10.p1.10.m9.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm10.p1.10.m9.1.1.3" xref="S5.Thmthm10.p1.10.m9.1.1.3.cmml">a</mi><mo id="S5.Thmthm10.p1.10.m9.1.1.1a" xref="S5.Thmthm10.p1.10.m9.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm10.p1.10.m9.1.1.4" xref="S5.Thmthm10.p1.10.m9.1.1.4.cmml">a</mi><mo id="S5.Thmthm10.p1.10.m9.1.1.1b" xref="S5.Thmthm10.p1.10.m9.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm10.p1.10.m9.1.1.5" xref="S5.Thmthm10.p1.10.m9.1.1.5.cmml">a</mi><mo id="S5.Thmthm10.p1.10.m9.1.1.1c" xref="S5.Thmthm10.p1.10.m9.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm10.p1.10.m9.1.1.6" mathvariant="normal" xref="S5.Thmthm10.p1.10.m9.1.1.6.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.10.m9.1b"><apply id="S5.Thmthm10.p1.10.m9.1.1.cmml" xref="S5.Thmthm10.p1.10.m9.1.1"><times id="S5.Thmthm10.p1.10.m9.1.1.1.cmml" xref="S5.Thmthm10.p1.10.m9.1.1.1"></times><ci id="S5.Thmthm10.p1.10.m9.1.1.2.cmml" xref="S5.Thmthm10.p1.10.m9.1.1.2">…</ci><ci id="S5.Thmthm10.p1.10.m9.1.1.3.cmml" xref="S5.Thmthm10.p1.10.m9.1.1.3">𝑎</ci><ci id="S5.Thmthm10.p1.10.m9.1.1.4.cmml" xref="S5.Thmthm10.p1.10.m9.1.1.4">𝑎</ci><ci id="S5.Thmthm10.p1.10.m9.1.1.5.cmml" xref="S5.Thmthm10.p1.10.m9.1.1.5">𝑎</ci><ci id="S5.Thmthm10.p1.10.m9.1.1.6.cmml" xref="S5.Thmthm10.p1.10.m9.1.1.6">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.10.m9.1c">\ldots aaa\ldots</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.10.m9.1d">… italic_a italic_a italic_a …</annotation></semantics></math>. This can be seen from a direct application of the Kirchhoff rules (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E2" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.2</span></a>). Alternatively one can use the fact that <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.11.m10.1"><semantics id="S5.Thmthm10.p1.11.m10.1a"><mi id="S5.Thmthm10.p1.11.m10.1.1" xref="S5.Thmthm10.p1.11.m10.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.11.m10.1b"><ci id="S5.Thmthm10.p1.11.m10.1.1.cmml" xref="S5.Thmthm10.p1.11.m10.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.11.m10.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.11.m10.1d">italic_X</annotation></semantics></math> is the substitutive subshift associated to the substitution given by <math alttext="a\mapsto a^{2}\,,\,\,b\mapsto aba" class="ltx_Math" display="inline" id="S5.Thmthm10.p1.12.m11.2"><semantics id="S5.Thmthm10.p1.12.m11.2a"><mrow id="S5.Thmthm10.p1.12.m11.2.2.2" xref="S5.Thmthm10.p1.12.m11.2.2.3.cmml"><mrow id="S5.Thmthm10.p1.12.m11.1.1.1.1" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.cmml"><mi id="S5.Thmthm10.p1.12.m11.1.1.1.1.2" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.2.cmml">a</mi><mo id="S5.Thmthm10.p1.12.m11.1.1.1.1.1" stretchy="false" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.1.cmml">↦</mo><msup id="S5.Thmthm10.p1.12.m11.1.1.1.1.3" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.3.cmml"><mi id="S5.Thmthm10.p1.12.m11.1.1.1.1.3.2" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.3.2.cmml">a</mi><mn id="S5.Thmthm10.p1.12.m11.1.1.1.1.3.3" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="S5.Thmthm10.p1.12.m11.2.2.2.3" rspace="0.497em" xref="S5.Thmthm10.p1.12.m11.2.2.3a.cmml">,</mo><mrow id="S5.Thmthm10.p1.12.m11.2.2.2.2" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.cmml"><mi id="S5.Thmthm10.p1.12.m11.2.2.2.2.2" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.2.cmml">b</mi><mo id="S5.Thmthm10.p1.12.m11.2.2.2.2.1" stretchy="false" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.1.cmml">↦</mo><mrow id="S5.Thmthm10.p1.12.m11.2.2.2.2.3" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.cmml"><mi id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.2" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.2.cmml">a</mi><mo id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.1" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.1.cmml">⁢</mo><mi id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.3" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.3.cmml">b</mi><mo id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.1a" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.1.cmml">⁢</mo><mi id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.4" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.4.cmml">a</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm10.p1.12.m11.2b"><apply id="S5.Thmthm10.p1.12.m11.2.2.3.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2"><csymbol cd="ambiguous" id="S5.Thmthm10.p1.12.m11.2.2.3a.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.3">formulae-sequence</csymbol><apply id="S5.Thmthm10.p1.12.m11.1.1.1.1.cmml" xref="S5.Thmthm10.p1.12.m11.1.1.1.1"><csymbol cd="latexml" id="S5.Thmthm10.p1.12.m11.1.1.1.1.1.cmml" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.1">maps-to</csymbol><ci id="S5.Thmthm10.p1.12.m11.1.1.1.1.2.cmml" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.2">𝑎</ci><apply id="S5.Thmthm10.p1.12.m11.1.1.1.1.3.cmml" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm10.p1.12.m11.1.1.1.1.3.1.cmml" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.3">superscript</csymbol><ci id="S5.Thmthm10.p1.12.m11.1.1.1.1.3.2.cmml" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.3.2">𝑎</ci><cn id="S5.Thmthm10.p1.12.m11.1.1.1.1.3.3.cmml" type="integer" xref="S5.Thmthm10.p1.12.m11.1.1.1.1.3.3">2</cn></apply></apply><apply id="S5.Thmthm10.p1.12.m11.2.2.2.2.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.2"><csymbol cd="latexml" id="S5.Thmthm10.p1.12.m11.2.2.2.2.1.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.1">maps-to</csymbol><ci id="S5.Thmthm10.p1.12.m11.2.2.2.2.2.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.2">𝑏</ci><apply id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3"><times id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.1.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.1"></times><ci id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.2.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.2">𝑎</ci><ci id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.3.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.3">𝑏</ci><ci id="S5.Thmthm10.p1.12.m11.2.2.2.2.3.4.cmml" xref="S5.Thmthm10.p1.12.m11.2.2.2.2.3.4">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm10.p1.12.m11.2c">a\mapsto a^{2}\,,\,\,b\mapsto aba</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm10.p1.12.m11.2d">italic_a ↦ italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_b ↦ italic_a italic_b italic_a</annotation></semantics></math> and apply Corollary 3.5 (1) of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib2" title="">2</a>]</cite>.</p> </div> </div> <div class="ltx_para" id="S5.p11"> <p class="ltx_p" id="S5.p11.9">More generally, if a subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.p11.1.m1.1"><semantics id="S5.p11.1.m1.1a"><mrow id="S5.p11.1.m1.1.1" xref="S5.p11.1.m1.1.1.cmml"><mi id="S5.p11.1.m1.1.1.2" xref="S5.p11.1.m1.1.1.2.cmml">X</mi><mo id="S5.p11.1.m1.1.1.1" xref="S5.p11.1.m1.1.1.1.cmml">⊆</mo><msup id="S5.p11.1.m1.1.1.3" xref="S5.p11.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p11.1.m1.1.1.3.2" xref="S5.p11.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S5.p11.1.m1.1.1.3.3" xref="S5.p11.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.p11.1.m1.1b"><apply id="S5.p11.1.m1.1.1.cmml" xref="S5.p11.1.m1.1.1"><subset id="S5.p11.1.m1.1.1.1.cmml" xref="S5.p11.1.m1.1.1.1"></subset><ci id="S5.p11.1.m1.1.1.2.cmml" xref="S5.p11.1.m1.1.1.2">𝑋</ci><apply id="S5.p11.1.m1.1.1.3.cmml" xref="S5.p11.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.p11.1.m1.1.1.3.1.cmml" xref="S5.p11.1.m1.1.1.3">superscript</csymbol><ci id="S5.p11.1.m1.1.1.3.2.cmml" xref="S5.p11.1.m1.1.1.3.2">𝒜</ci><ci id="S5.p11.1.m1.1.1.3.3.cmml" xref="S5.p11.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.1.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.p11.1.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> contains a non-empty subset <math alttext="X_{0}\subseteq X" class="ltx_Math" display="inline" id="S5.p11.2.m2.1"><semantics id="S5.p11.2.m2.1a"><mrow id="S5.p11.2.m2.1.1" xref="S5.p11.2.m2.1.1.cmml"><msub id="S5.p11.2.m2.1.1.2" xref="S5.p11.2.m2.1.1.2.cmml"><mi id="S5.p11.2.m2.1.1.2.2" xref="S5.p11.2.m2.1.1.2.2.cmml">X</mi><mn id="S5.p11.2.m2.1.1.2.3" xref="S5.p11.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S5.p11.2.m2.1.1.1" xref="S5.p11.2.m2.1.1.1.cmml">⊆</mo><mi id="S5.p11.2.m2.1.1.3" xref="S5.p11.2.m2.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p11.2.m2.1b"><apply id="S5.p11.2.m2.1.1.cmml" xref="S5.p11.2.m2.1.1"><subset id="S5.p11.2.m2.1.1.1.cmml" xref="S5.p11.2.m2.1.1.1"></subset><apply id="S5.p11.2.m2.1.1.2.cmml" xref="S5.p11.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.p11.2.m2.1.1.2.1.cmml" xref="S5.p11.2.m2.1.1.2">subscript</csymbol><ci id="S5.p11.2.m2.1.1.2.2.cmml" xref="S5.p11.2.m2.1.1.2.2">𝑋</ci><cn id="S5.p11.2.m2.1.1.2.3.cmml" type="integer" xref="S5.p11.2.m2.1.1.2.3">0</cn></apply><ci id="S5.p11.2.m2.1.1.3.cmml" xref="S5.p11.2.m2.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.2.m2.1c">X_{0}\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S5.p11.2.m2.1d">italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⊆ italic_X</annotation></semantics></math> which consists entirely of biinfinite words that lie outside the support of any invariant measure on <math alttext="X" class="ltx_Math" display="inline" id="S5.p11.3.m3.1"><semantics id="S5.p11.3.m3.1a"><mi id="S5.p11.3.m3.1.1" xref="S5.p11.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.p11.3.m3.1b"><ci id="S5.p11.3.m3.1.1.cmml" xref="S5.p11.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.p11.3.m3.1d">italic_X</annotation></semantics></math>, then the violation of the injectivity of <math alttext="\sigma^{T}" class="ltx_Math" display="inline" id="S5.p11.4.m4.1"><semantics id="S5.p11.4.m4.1a"><msup id="S5.p11.4.m4.1.1" xref="S5.p11.4.m4.1.1.cmml"><mi id="S5.p11.4.m4.1.1.2" xref="S5.p11.4.m4.1.1.2.cmml">σ</mi><mi id="S5.p11.4.m4.1.1.3" xref="S5.p11.4.m4.1.1.3.cmml">T</mi></msup><annotation-xml encoding="MathML-Content" id="S5.p11.4.m4.1b"><apply id="S5.p11.4.m4.1.1.cmml" xref="S5.p11.4.m4.1.1"><csymbol cd="ambiguous" id="S5.p11.4.m4.1.1.1.cmml" xref="S5.p11.4.m4.1.1">superscript</csymbol><ci id="S5.p11.4.m4.1.1.2.cmml" xref="S5.p11.4.m4.1.1.2">𝜎</ci><ci id="S5.p11.4.m4.1.1.3.cmml" xref="S5.p11.4.m4.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.4.m4.1c">\sigma^{T}</annotation><annotation encoding="application/x-llamapun" id="S5.p11.4.m4.1d">italic_σ start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT</annotation></semantics></math> on <math alttext="X_{0}" class="ltx_Math" display="inline" id="S5.p11.5.m5.1"><semantics id="S5.p11.5.m5.1a"><msub id="S5.p11.5.m5.1.1" xref="S5.p11.5.m5.1.1.cmml"><mi id="S5.p11.5.m5.1.1.2" xref="S5.p11.5.m5.1.1.2.cmml">X</mi><mn id="S5.p11.5.m5.1.1.3" xref="S5.p11.5.m5.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p11.5.m5.1b"><apply id="S5.p11.5.m5.1.1.cmml" xref="S5.p11.5.m5.1.1"><csymbol cd="ambiguous" id="S5.p11.5.m5.1.1.1.cmml" xref="S5.p11.5.m5.1.1">subscript</csymbol><ci id="S5.p11.5.m5.1.1.2.cmml" xref="S5.p11.5.m5.1.1.2">𝑋</ci><cn id="S5.p11.5.m5.1.1.3.cmml" type="integer" xref="S5.p11.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.5.m5.1c">X_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.p11.5.m5.1d">italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> will not have any impact on the injectivity of the measure transfer map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S5.p11.6.m6.1"><semantics id="S5.p11.6.m6.1a"><mrow id="S5.p11.6.m6.1.1" xref="S5.p11.6.m6.1.1.cmml"><mi id="S5.p11.6.m6.1.1.2" xref="S5.p11.6.m6.1.1.2.cmml">σ</mi><mo id="S5.p11.6.m6.1.1.1" xref="S5.p11.6.m6.1.1.1.cmml">⁢</mo><mi id="S5.p11.6.m6.1.1.3" xref="S5.p11.6.m6.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p11.6.m6.1b"><apply id="S5.p11.6.m6.1.1.cmml" xref="S5.p11.6.m6.1.1"><times id="S5.p11.6.m6.1.1.1.cmml" xref="S5.p11.6.m6.1.1.1"></times><ci id="S5.p11.6.m6.1.1.2.cmml" xref="S5.p11.6.m6.1.1.2">𝜎</ci><ci id="S5.p11.6.m6.1.1.3.cmml" xref="S5.p11.6.m6.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.6.m6.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S5.p11.6.m6.1d">italic_σ italic_M</annotation></semantics></math>, for any morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.p11.7.m7.1"><semantics id="S5.p11.7.m7.1a"><mrow id="S5.p11.7.m7.1.1" xref="S5.p11.7.m7.1.1.cmml"><mi id="S5.p11.7.m7.1.1.2" xref="S5.p11.7.m7.1.1.2.cmml">σ</mi><mo id="S5.p11.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.p11.7.m7.1.1.1.cmml">:</mo><mrow id="S5.p11.7.m7.1.1.3" xref="S5.p11.7.m7.1.1.3.cmml"><msup id="S5.p11.7.m7.1.1.3.2" xref="S5.p11.7.m7.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p11.7.m7.1.1.3.2.2" xref="S5.p11.7.m7.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.p11.7.m7.1.1.3.2.3" xref="S5.p11.7.m7.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.p11.7.m7.1.1.3.1" stretchy="false" xref="S5.p11.7.m7.1.1.3.1.cmml">→</mo><msup id="S5.p11.7.m7.1.1.3.3" xref="S5.p11.7.m7.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p11.7.m7.1.1.3.3.2" xref="S5.p11.7.m7.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.p11.7.m7.1.1.3.3.3" xref="S5.p11.7.m7.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p11.7.m7.1b"><apply id="S5.p11.7.m7.1.1.cmml" xref="S5.p11.7.m7.1.1"><ci id="S5.p11.7.m7.1.1.1.cmml" xref="S5.p11.7.m7.1.1.1">:</ci><ci id="S5.p11.7.m7.1.1.2.cmml" xref="S5.p11.7.m7.1.1.2">𝜎</ci><apply id="S5.p11.7.m7.1.1.3.cmml" xref="S5.p11.7.m7.1.1.3"><ci id="S5.p11.7.m7.1.1.3.1.cmml" xref="S5.p11.7.m7.1.1.3.1">→</ci><apply id="S5.p11.7.m7.1.1.3.2.cmml" xref="S5.p11.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S5.p11.7.m7.1.1.3.2.1.cmml" xref="S5.p11.7.m7.1.1.3.2">superscript</csymbol><ci id="S5.p11.7.m7.1.1.3.2.2.cmml" xref="S5.p11.7.m7.1.1.3.2.2">𝒜</ci><times id="S5.p11.7.m7.1.1.3.2.3.cmml" xref="S5.p11.7.m7.1.1.3.2.3"></times></apply><apply id="S5.p11.7.m7.1.1.3.3.cmml" xref="S5.p11.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S5.p11.7.m7.1.1.3.3.1.cmml" xref="S5.p11.7.m7.1.1.3.3">superscript</csymbol><ci id="S5.p11.7.m7.1.1.3.3.2.cmml" xref="S5.p11.7.m7.1.1.3.3.2">ℬ</ci><times id="S5.p11.7.m7.1.1.3.3.3.cmml" xref="S5.p11.7.m7.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.7.m7.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.p11.7.m7.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. This is the basic principle which is now used in the proof of the following proposition, which shows that the injectivity of a map <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S5.p11.8.m8.1"><semantics id="S5.p11.8.m8.1a"><mrow id="S5.p11.8.m8.1.1" xref="S5.p11.8.m8.1.1.cmml"><msub id="S5.p11.8.m8.1.1.2" xref="S5.p11.8.m8.1.1.2.cmml"><mi id="S5.p11.8.m8.1.1.2.2" xref="S5.p11.8.m8.1.1.2.2.cmml">σ</mi><mi id="S5.p11.8.m8.1.1.2.3" xref="S5.p11.8.m8.1.1.2.3.cmml">X</mi></msub><mo id="S5.p11.8.m8.1.1.1" xref="S5.p11.8.m8.1.1.1.cmml">⁢</mo><mi id="S5.p11.8.m8.1.1.3" xref="S5.p11.8.m8.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.p11.8.m8.1b"><apply id="S5.p11.8.m8.1.1.cmml" xref="S5.p11.8.m8.1.1"><times id="S5.p11.8.m8.1.1.1.cmml" xref="S5.p11.8.m8.1.1.1"></times><apply id="S5.p11.8.m8.1.1.2.cmml" xref="S5.p11.8.m8.1.1.2"><csymbol cd="ambiguous" id="S5.p11.8.m8.1.1.2.1.cmml" xref="S5.p11.8.m8.1.1.2">subscript</csymbol><ci id="S5.p11.8.m8.1.1.2.2.cmml" xref="S5.p11.8.m8.1.1.2.2">𝜎</ci><ci id="S5.p11.8.m8.1.1.2.3.cmml" xref="S5.p11.8.m8.1.1.2.3">𝑋</ci></apply><ci id="S5.p11.8.m8.1.1.3.cmml" xref="S5.p11.8.m8.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.8.m8.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S5.p11.8.m8.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> as above is in general too weak to deduce that <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.p11.9.m9.1"><semantics id="S5.p11.9.m9.1a"><mi id="S5.p11.9.m9.1.1" xref="S5.p11.9.m9.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.p11.9.m9.1b"><ci id="S5.p11.9.m9.1.1.cmml" xref="S5.p11.9.m9.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p11.9.m9.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.p11.9.m9.1d">italic_σ</annotation></semantics></math> is recognizable for aperiodic points. It completes the set of implications and their refusals between the properties discussed here, as summarized in Fig. 1.</p> </div> <div class="ltx_theorem ltx_theorem_prop" id="S5.Thmthm11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm11.1.1.1">Proposition 5.11</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm11.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm11.p1"> <p class="ltx_p" id="S5.Thmthm11.p1.4"><span class="ltx_text ltx_font_italic" id="S5.Thmthm11.p1.4.4">There exist a subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm11.p1.1.1.m1.1"><semantics id="S5.Thmthm11.p1.1.1.m1.1a"><mrow id="S5.Thmthm11.p1.1.1.m1.1.1" xref="S5.Thmthm11.p1.1.1.m1.1.1.cmml"><mi id="S5.Thmthm11.p1.1.1.m1.1.1.2" xref="S5.Thmthm11.p1.1.1.m1.1.1.2.cmml">X</mi><mo id="S5.Thmthm11.p1.1.1.m1.1.1.1" xref="S5.Thmthm11.p1.1.1.m1.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm11.p1.1.1.m1.1.1.3" xref="S5.Thmthm11.p1.1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm11.p1.1.1.m1.1.1.3.2" xref="S5.Thmthm11.p1.1.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm11.p1.1.1.m1.1.1.3.3" xref="S5.Thmthm11.p1.1.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm11.p1.1.1.m1.1b"><apply id="S5.Thmthm11.p1.1.1.m1.1.1.cmml" xref="S5.Thmthm11.p1.1.1.m1.1.1"><subset id="S5.Thmthm11.p1.1.1.m1.1.1.1.cmml" xref="S5.Thmthm11.p1.1.1.m1.1.1.1"></subset><ci id="S5.Thmthm11.p1.1.1.m1.1.1.2.cmml" xref="S5.Thmthm11.p1.1.1.m1.1.1.2">𝑋</ci><apply id="S5.Thmthm11.p1.1.1.m1.1.1.3.cmml" xref="S5.Thmthm11.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm11.p1.1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm11.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S5.Thmthm11.p1.1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm11.p1.1.1.m1.1.1.3.2">𝒜</ci><ci id="S5.Thmthm11.p1.1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm11.p1.1.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm11.p1.1.1.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm11.p1.1.1.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and a morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.Thmthm11.p1.2.2.m2.1"><semantics id="S5.Thmthm11.p1.2.2.m2.1a"><mrow id="S5.Thmthm11.p1.2.2.m2.1.1" xref="S5.Thmthm11.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmthm11.p1.2.2.m2.1.1.2" xref="S5.Thmthm11.p1.2.2.m2.1.1.2.cmml">σ</mi><mo id="S5.Thmthm11.p1.2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm11.p1.2.2.m2.1.1.1.cmml">:</mo><mrow id="S5.Thmthm11.p1.2.2.m2.1.1.3" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.cmml"><msup id="S5.Thmthm11.p1.2.2.m2.1.1.3.2" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm11.p1.2.2.m2.1.1.3.2.2" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm11.p1.2.2.m2.1.1.3.2.3" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm11.p1.2.2.m2.1.1.3.1" stretchy="false" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm11.p1.2.2.m2.1.1.3.3" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm11.p1.2.2.m2.1.1.3.3.2" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.Thmthm11.p1.2.2.m2.1.1.3.3.3" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm11.p1.2.2.m2.1b"><apply id="S5.Thmthm11.p1.2.2.m2.1.1.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1"><ci id="S5.Thmthm11.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.1">:</ci><ci id="S5.Thmthm11.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.2">𝜎</ci><apply id="S5.Thmthm11.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3"><ci id="S5.Thmthm11.p1.2.2.m2.1.1.3.1.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.1">→</ci><apply id="S5.Thmthm11.p1.2.2.m2.1.1.3.2.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm11.p1.2.2.m2.1.1.3.2.1.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm11.p1.2.2.m2.1.1.3.2.2.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm11.p1.2.2.m2.1.1.3.2.3.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.2.3"></times></apply><apply id="S5.Thmthm11.p1.2.2.m2.1.1.3.3.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm11.p1.2.2.m2.1.1.3.3.1.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm11.p1.2.2.m2.1.1.3.3.2.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.3.2">ℬ</ci><times id="S5.Thmthm11.p1.2.2.m2.1.1.3.3.3.cmml" xref="S5.Thmthm11.p1.2.2.m2.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm11.p1.2.2.m2.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm11.p1.2.2.m2.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> which is not recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm11.p1.3.3.m3.1"><semantics id="S5.Thmthm11.p1.3.3.m3.1a"><mi id="S5.Thmthm11.p1.3.3.m3.1.1" xref="S5.Thmthm11.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm11.p1.3.3.m3.1b"><ci id="S5.Thmthm11.p1.3.3.m3.1.1.cmml" xref="S5.Thmthm11.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm11.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm11.p1.3.3.m3.1d">italic_X</annotation></semantics></math>, while the induced measure transfer map <math alttext="\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))" class="ltx_Math" display="inline" id="S5.Thmthm11.p1.4.4.m4.3"><semantics id="S5.Thmthm11.p1.4.4.m4.3a"><mrow id="S5.Thmthm11.p1.4.4.m4.3.3" xref="S5.Thmthm11.p1.4.4.m4.3.3.cmml"><mrow id="S5.Thmthm11.p1.4.4.m4.3.3.3" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.cmml"><msub id="S5.Thmthm11.p1.4.4.m4.3.3.3.2" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.2.cmml"><mi id="S5.Thmthm11.p1.4.4.m4.3.3.3.2.2" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.2.2.cmml">σ</mi><mi id="S5.Thmthm11.p1.4.4.m4.3.3.3.2.3" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.2.3.cmml">X</mi></msub><mo id="S5.Thmthm11.p1.4.4.m4.3.3.3.1" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.1.cmml">⁢</mo><mi id="S5.Thmthm11.p1.4.4.m4.3.3.3.3" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.3.cmml">M</mi></mrow><mo id="S5.Thmthm11.p1.4.4.m4.3.3.2" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm11.p1.4.4.m4.3.3.2.cmml">:</mo><mrow id="S5.Thmthm11.p1.4.4.m4.3.3.1" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.cmml"><mrow id="S5.Thmthm11.p1.4.4.m4.3.3.1.3" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm11.p1.4.4.m4.3.3.1.3.2" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3.2.cmml">ℳ</mi><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.3.1" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3.1.cmml">⁢</mo><mrow id="S5.Thmthm11.p1.4.4.m4.3.3.1.3.3.2" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3.cmml"><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.3.3.2.1" stretchy="false" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm11.p1.4.4.m4.1.1" xref="S5.Thmthm11.p1.4.4.m4.1.1.cmml">𝒳</mi><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.3.3.2.2" stretchy="false" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.2" stretchy="false" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.2.cmml">→</mo><mrow id="S5.Thmthm11.p1.4.4.m4.3.3.1.1" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.3" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.3.cmml">ℳ</mi><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.2" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.2.cmml">⁢</mo><mrow id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.cmml"><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.2" stretchy="false" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.cmml"><mi id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.2" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.2.cmml">σ</mi><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.1" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.3.2" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.cmml"><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.3.2.1" stretchy="false" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm11.p1.4.4.m4.2.2" xref="S5.Thmthm11.p1.4.4.m4.2.2.cmml">𝒳</mi><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.3.2.2" stretchy="false" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.3" stretchy="false" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm11.p1.4.4.m4.3b"><apply id="S5.Thmthm11.p1.4.4.m4.3.3.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3"><ci id="S5.Thmthm11.p1.4.4.m4.3.3.2.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.2">:</ci><apply id="S5.Thmthm11.p1.4.4.m4.3.3.3.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.3"><times id="S5.Thmthm11.p1.4.4.m4.3.3.3.1.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.1"></times><apply id="S5.Thmthm11.p1.4.4.m4.3.3.3.2.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.2"><csymbol cd="ambiguous" id="S5.Thmthm11.p1.4.4.m4.3.3.3.2.1.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.2">subscript</csymbol><ci id="S5.Thmthm11.p1.4.4.m4.3.3.3.2.2.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.2.2">𝜎</ci><ci id="S5.Thmthm11.p1.4.4.m4.3.3.3.2.3.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.2.3">𝑋</ci></apply><ci id="S5.Thmthm11.p1.4.4.m4.3.3.3.3.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.3.3">𝑀</ci></apply><apply id="S5.Thmthm11.p1.4.4.m4.3.3.1.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1"><ci id="S5.Thmthm11.p1.4.4.m4.3.3.1.2.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.2">→</ci><apply id="S5.Thmthm11.p1.4.4.m4.3.3.1.3.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3"><times id="S5.Thmthm11.p1.4.4.m4.3.3.1.3.1.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3.1"></times><ci id="S5.Thmthm11.p1.4.4.m4.3.3.1.3.2.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.3.2">ℳ</ci><ci id="S5.Thmthm11.p1.4.4.m4.1.1.cmml" xref="S5.Thmthm11.p1.4.4.m4.1.1">𝒳</ci></apply><apply id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1"><times id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.2.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.2"></times><ci id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.3.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.3">ℳ</ci><apply id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1"><times id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.1.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.1"></times><ci id="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.2.cmml" xref="S5.Thmthm11.p1.4.4.m4.3.3.1.1.1.1.1.2">𝜎</ci><ci id="S5.Thmthm11.p1.4.4.m4.2.2.cmml" xref="S5.Thmthm11.p1.4.4.m4.2.2">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm11.p1.4.4.m4.3c">\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm11.p1.4.4.m4.3d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) )</annotation></semantics></math> is injective.</span></p> </div> </div> <div class="ltx_proof" id="S5.9"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.9.p1"> <p class="ltx_p" id="S5.9.p1.8">We will give a concrete example of <math alttext="X" class="ltx_Math" display="inline" id="S5.9.p1.1.m1.1"><semantics id="S5.9.p1.1.m1.1a"><mi id="S5.9.p1.1.m1.1.1" xref="S5.9.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.9.p1.1.m1.1b"><ci id="S5.9.p1.1.m1.1.1.cmml" xref="S5.9.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.1.m1.1d">italic_X</annotation></semantics></math> and <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.9.p1.2.m2.1"><semantics id="S5.9.p1.2.m2.1a"><mi id="S5.9.p1.2.m2.1.1" xref="S5.9.p1.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.9.p1.2.m2.1b"><ci id="S5.9.p1.2.m2.1.1.cmml" xref="S5.9.p1.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.2.m2.1d">italic_σ</annotation></semantics></math> as claimed: Set <math alttext="\cal A=\{a,b,c\}" class="ltx_Math" display="inline" id="S5.9.p1.3.m3.3"><semantics id="S5.9.p1.3.m3.3a"><mrow id="S5.9.p1.3.m3.3.4" xref="S5.9.p1.3.m3.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.3.m3.3.4.2" xref="S5.9.p1.3.m3.3.4.2.cmml">𝒜</mi><mo id="S5.9.p1.3.m3.3.4.1" xref="S5.9.p1.3.m3.3.4.1.cmml">=</mo><mrow id="S5.9.p1.3.m3.3.4.3.2" xref="S5.9.p1.3.m3.3.4.3.1.cmml"><mo id="S5.9.p1.3.m3.3.4.3.2.1" stretchy="false" xref="S5.9.p1.3.m3.3.4.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.3.m3.1.1" xref="S5.9.p1.3.m3.1.1.cmml">𝒶</mi><mo id="S5.9.p1.3.m3.3.4.3.2.2" xref="S5.9.p1.3.m3.3.4.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.3.m3.2.2" xref="S5.9.p1.3.m3.2.2.cmml">𝒷</mi><mo id="S5.9.p1.3.m3.3.4.3.2.3" xref="S5.9.p1.3.m3.3.4.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.3.m3.3.3" xref="S5.9.p1.3.m3.3.3.cmml">𝒸</mi><mo id="S5.9.p1.3.m3.3.4.3.2.4" stretchy="false" xref="S5.9.p1.3.m3.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.3.m3.3b"><apply id="S5.9.p1.3.m3.3.4.cmml" xref="S5.9.p1.3.m3.3.4"><eq id="S5.9.p1.3.m3.3.4.1.cmml" xref="S5.9.p1.3.m3.3.4.1"></eq><ci id="S5.9.p1.3.m3.3.4.2.cmml" xref="S5.9.p1.3.m3.3.4.2">𝒜</ci><set id="S5.9.p1.3.m3.3.4.3.1.cmml" xref="S5.9.p1.3.m3.3.4.3.2"><ci id="S5.9.p1.3.m3.1.1.cmml" xref="S5.9.p1.3.m3.1.1">𝒶</ci><ci id="S5.9.p1.3.m3.2.2.cmml" xref="S5.9.p1.3.m3.2.2">𝒷</ci><ci id="S5.9.p1.3.m3.3.3.cmml" xref="S5.9.p1.3.m3.3.3">𝒸</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.3.m3.3c">\cal A=\{a,b,c\}</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.3.m3.3d">caligraphic_A = { caligraphic_a , caligraphic_b , caligraphic_c }</annotation></semantics></math> and <math alttext="\cal B=\{d,e\}" class="ltx_Math" display="inline" id="S5.9.p1.4.m4.2"><semantics id="S5.9.p1.4.m4.2a"><mrow id="S5.9.p1.4.m4.2.3" xref="S5.9.p1.4.m4.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.4.m4.2.3.2" xref="S5.9.p1.4.m4.2.3.2.cmml">ℬ</mi><mo id="S5.9.p1.4.m4.2.3.1" xref="S5.9.p1.4.m4.2.3.1.cmml">=</mo><mrow id="S5.9.p1.4.m4.2.3.3.2" xref="S5.9.p1.4.m4.2.3.3.1.cmml"><mo id="S5.9.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S5.9.p1.4.m4.2.3.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.4.m4.1.1" xref="S5.9.p1.4.m4.1.1.cmml">𝒹</mi><mo id="S5.9.p1.4.m4.2.3.3.2.2" xref="S5.9.p1.4.m4.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.4.m4.2.2" xref="S5.9.p1.4.m4.2.2.cmml">ℯ</mi><mo id="S5.9.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S5.9.p1.4.m4.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.4.m4.2b"><apply id="S5.9.p1.4.m4.2.3.cmml" xref="S5.9.p1.4.m4.2.3"><eq id="S5.9.p1.4.m4.2.3.1.cmml" xref="S5.9.p1.4.m4.2.3.1"></eq><ci id="S5.9.p1.4.m4.2.3.2.cmml" xref="S5.9.p1.4.m4.2.3.2">ℬ</ci><set id="S5.9.p1.4.m4.2.3.3.1.cmml" xref="S5.9.p1.4.m4.2.3.3.2"><ci id="S5.9.p1.4.m4.1.1.cmml" xref="S5.9.p1.4.m4.1.1">𝒹</ci><ci id="S5.9.p1.4.m4.2.2.cmml" xref="S5.9.p1.4.m4.2.2">ℯ</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.4.m4.2c">\cal B=\{d,e\}</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.4.m4.2d">caligraphic_B = { caligraphic_d , caligraphic_e }</annotation></semantics></math>, and let <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.9.p1.5.m5.1"><semantics id="S5.9.p1.5.m5.1a"><mrow id="S5.9.p1.5.m5.1.1" xref="S5.9.p1.5.m5.1.1.cmml"><mi id="S5.9.p1.5.m5.1.1.2" xref="S5.9.p1.5.m5.1.1.2.cmml">σ</mi><mo id="S5.9.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.9.p1.5.m5.1.1.1.cmml">:</mo><mrow id="S5.9.p1.5.m5.1.1.3" xref="S5.9.p1.5.m5.1.1.3.cmml"><msup id="S5.9.p1.5.m5.1.1.3.2" xref="S5.9.p1.5.m5.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.5.m5.1.1.3.2.2" xref="S5.9.p1.5.m5.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.9.p1.5.m5.1.1.3.2.3" xref="S5.9.p1.5.m5.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.9.p1.5.m5.1.1.3.1" stretchy="false" xref="S5.9.p1.5.m5.1.1.3.1.cmml">→</mo><msup id="S5.9.p1.5.m5.1.1.3.3" xref="S5.9.p1.5.m5.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.5.m5.1.1.3.3.2" xref="S5.9.p1.5.m5.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.9.p1.5.m5.1.1.3.3.3" xref="S5.9.p1.5.m5.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.5.m5.1b"><apply id="S5.9.p1.5.m5.1.1.cmml" xref="S5.9.p1.5.m5.1.1"><ci id="S5.9.p1.5.m5.1.1.1.cmml" xref="S5.9.p1.5.m5.1.1.1">:</ci><ci id="S5.9.p1.5.m5.1.1.2.cmml" xref="S5.9.p1.5.m5.1.1.2">𝜎</ci><apply id="S5.9.p1.5.m5.1.1.3.cmml" xref="S5.9.p1.5.m5.1.1.3"><ci id="S5.9.p1.5.m5.1.1.3.1.cmml" xref="S5.9.p1.5.m5.1.1.3.1">→</ci><apply id="S5.9.p1.5.m5.1.1.3.2.cmml" xref="S5.9.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S5.9.p1.5.m5.1.1.3.2.1.cmml" xref="S5.9.p1.5.m5.1.1.3.2">superscript</csymbol><ci id="S5.9.p1.5.m5.1.1.3.2.2.cmml" xref="S5.9.p1.5.m5.1.1.3.2.2">𝒜</ci><times id="S5.9.p1.5.m5.1.1.3.2.3.cmml" xref="S5.9.p1.5.m5.1.1.3.2.3"></times></apply><apply id="S5.9.p1.5.m5.1.1.3.3.cmml" xref="S5.9.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S5.9.p1.5.m5.1.1.3.3.1.cmml" xref="S5.9.p1.5.m5.1.1.3.3">superscript</csymbol><ci id="S5.9.p1.5.m5.1.1.3.3.2.cmml" xref="S5.9.p1.5.m5.1.1.3.3.2">ℬ</ci><times id="S5.9.p1.5.m5.1.1.3.3.3.cmml" xref="S5.9.p1.5.m5.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.5.m5.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.5.m5.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be the letter-to-letter morphism given by <math alttext="\sigma(a)=d" class="ltx_Math" display="inline" id="S5.9.p1.6.m6.1"><semantics id="S5.9.p1.6.m6.1a"><mrow id="S5.9.p1.6.m6.1.2" xref="S5.9.p1.6.m6.1.2.cmml"><mrow id="S5.9.p1.6.m6.1.2.2" xref="S5.9.p1.6.m6.1.2.2.cmml"><mi id="S5.9.p1.6.m6.1.2.2.2" xref="S5.9.p1.6.m6.1.2.2.2.cmml">σ</mi><mo id="S5.9.p1.6.m6.1.2.2.1" xref="S5.9.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S5.9.p1.6.m6.1.2.2.3.2" xref="S5.9.p1.6.m6.1.2.2.cmml"><mo id="S5.9.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S5.9.p1.6.m6.1.2.2.cmml">(</mo><mi id="S5.9.p1.6.m6.1.1" xref="S5.9.p1.6.m6.1.1.cmml">a</mi><mo id="S5.9.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S5.9.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.9.p1.6.m6.1.2.1" xref="S5.9.p1.6.m6.1.2.1.cmml">=</mo><mi id="S5.9.p1.6.m6.1.2.3" xref="S5.9.p1.6.m6.1.2.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.6.m6.1b"><apply id="S5.9.p1.6.m6.1.2.cmml" xref="S5.9.p1.6.m6.1.2"><eq id="S5.9.p1.6.m6.1.2.1.cmml" xref="S5.9.p1.6.m6.1.2.1"></eq><apply id="S5.9.p1.6.m6.1.2.2.cmml" xref="S5.9.p1.6.m6.1.2.2"><times id="S5.9.p1.6.m6.1.2.2.1.cmml" xref="S5.9.p1.6.m6.1.2.2.1"></times><ci id="S5.9.p1.6.m6.1.2.2.2.cmml" xref="S5.9.p1.6.m6.1.2.2.2">𝜎</ci><ci id="S5.9.p1.6.m6.1.1.cmml" xref="S5.9.p1.6.m6.1.1">𝑎</ci></apply><ci id="S5.9.p1.6.m6.1.2.3.cmml" xref="S5.9.p1.6.m6.1.2.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.6.m6.1c">\sigma(a)=d</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.6.m6.1d">italic_σ ( italic_a ) = italic_d</annotation></semantics></math> and <math alttext="\sigma(b)=\sigma(c)=e" class="ltx_Math" display="inline" id="S5.9.p1.7.m7.2"><semantics id="S5.9.p1.7.m7.2a"><mrow id="S5.9.p1.7.m7.2.3" xref="S5.9.p1.7.m7.2.3.cmml"><mrow id="S5.9.p1.7.m7.2.3.2" xref="S5.9.p1.7.m7.2.3.2.cmml"><mi id="S5.9.p1.7.m7.2.3.2.2" xref="S5.9.p1.7.m7.2.3.2.2.cmml">σ</mi><mo id="S5.9.p1.7.m7.2.3.2.1" xref="S5.9.p1.7.m7.2.3.2.1.cmml">⁢</mo><mrow id="S5.9.p1.7.m7.2.3.2.3.2" xref="S5.9.p1.7.m7.2.3.2.cmml"><mo id="S5.9.p1.7.m7.2.3.2.3.2.1" stretchy="false" xref="S5.9.p1.7.m7.2.3.2.cmml">(</mo><mi id="S5.9.p1.7.m7.1.1" xref="S5.9.p1.7.m7.1.1.cmml">b</mi><mo id="S5.9.p1.7.m7.2.3.2.3.2.2" stretchy="false" xref="S5.9.p1.7.m7.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.9.p1.7.m7.2.3.3" xref="S5.9.p1.7.m7.2.3.3.cmml">=</mo><mrow id="S5.9.p1.7.m7.2.3.4" xref="S5.9.p1.7.m7.2.3.4.cmml"><mi id="S5.9.p1.7.m7.2.3.4.2" xref="S5.9.p1.7.m7.2.3.4.2.cmml">σ</mi><mo id="S5.9.p1.7.m7.2.3.4.1" xref="S5.9.p1.7.m7.2.3.4.1.cmml">⁢</mo><mrow id="S5.9.p1.7.m7.2.3.4.3.2" xref="S5.9.p1.7.m7.2.3.4.cmml"><mo id="S5.9.p1.7.m7.2.3.4.3.2.1" stretchy="false" xref="S5.9.p1.7.m7.2.3.4.cmml">(</mo><mi id="S5.9.p1.7.m7.2.2" xref="S5.9.p1.7.m7.2.2.cmml">c</mi><mo id="S5.9.p1.7.m7.2.3.4.3.2.2" stretchy="false" xref="S5.9.p1.7.m7.2.3.4.cmml">)</mo></mrow></mrow><mo id="S5.9.p1.7.m7.2.3.5" xref="S5.9.p1.7.m7.2.3.5.cmml">=</mo><mi id="S5.9.p1.7.m7.2.3.6" xref="S5.9.p1.7.m7.2.3.6.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.7.m7.2b"><apply id="S5.9.p1.7.m7.2.3.cmml" xref="S5.9.p1.7.m7.2.3"><and id="S5.9.p1.7.m7.2.3a.cmml" xref="S5.9.p1.7.m7.2.3"></and><apply id="S5.9.p1.7.m7.2.3b.cmml" xref="S5.9.p1.7.m7.2.3"><eq id="S5.9.p1.7.m7.2.3.3.cmml" xref="S5.9.p1.7.m7.2.3.3"></eq><apply id="S5.9.p1.7.m7.2.3.2.cmml" xref="S5.9.p1.7.m7.2.3.2"><times id="S5.9.p1.7.m7.2.3.2.1.cmml" xref="S5.9.p1.7.m7.2.3.2.1"></times><ci id="S5.9.p1.7.m7.2.3.2.2.cmml" xref="S5.9.p1.7.m7.2.3.2.2">𝜎</ci><ci id="S5.9.p1.7.m7.1.1.cmml" xref="S5.9.p1.7.m7.1.1">𝑏</ci></apply><apply id="S5.9.p1.7.m7.2.3.4.cmml" xref="S5.9.p1.7.m7.2.3.4"><times id="S5.9.p1.7.m7.2.3.4.1.cmml" xref="S5.9.p1.7.m7.2.3.4.1"></times><ci id="S5.9.p1.7.m7.2.3.4.2.cmml" xref="S5.9.p1.7.m7.2.3.4.2">𝜎</ci><ci id="S5.9.p1.7.m7.2.2.cmml" xref="S5.9.p1.7.m7.2.2">𝑐</ci></apply></apply><apply id="S5.9.p1.7.m7.2.3c.cmml" xref="S5.9.p1.7.m7.2.3"><eq id="S5.9.p1.7.m7.2.3.5.cmml" xref="S5.9.p1.7.m7.2.3.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S5.9.p1.7.m7.2.3.4.cmml" id="S5.9.p1.7.m7.2.3d.cmml" xref="S5.9.p1.7.m7.2.3"></share><ci id="S5.9.p1.7.m7.2.3.6.cmml" xref="S5.9.p1.7.m7.2.3.6">𝑒</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.7.m7.2c">\sigma(b)=\sigma(c)=e</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.7.m7.2d">italic_σ ( italic_b ) = italic_σ ( italic_c ) = italic_e</annotation></semantics></math>. Let now <math alttext="X\subseteq\cal A^{*}" class="ltx_Math" display="inline" id="S5.9.p1.8.m8.1"><semantics id="S5.9.p1.8.m8.1a"><mrow id="S5.9.p1.8.m8.1.1" xref="S5.9.p1.8.m8.1.1.cmml"><mi id="S5.9.p1.8.m8.1.1.2" xref="S5.9.p1.8.m8.1.1.2.cmml">X</mi><mo id="S5.9.p1.8.m8.1.1.1" xref="S5.9.p1.8.m8.1.1.1.cmml">⊆</mo><msup id="S5.9.p1.8.m8.1.1.3" xref="S5.9.p1.8.m8.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.9.p1.8.m8.1.1.3.2" xref="S5.9.p1.8.m8.1.1.3.2.cmml">𝒜</mi><mo id="S5.9.p1.8.m8.1.1.3.3" xref="S5.9.p1.8.m8.1.1.3.3.cmml">∗</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.8.m8.1b"><apply id="S5.9.p1.8.m8.1.1.cmml" xref="S5.9.p1.8.m8.1.1"><subset id="S5.9.p1.8.m8.1.1.1.cmml" xref="S5.9.p1.8.m8.1.1.1"></subset><ci id="S5.9.p1.8.m8.1.1.2.cmml" xref="S5.9.p1.8.m8.1.1.2">𝑋</ci><apply id="S5.9.p1.8.m8.1.1.3.cmml" xref="S5.9.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S5.9.p1.8.m8.1.1.3.1.cmml" xref="S5.9.p1.8.m8.1.1.3">superscript</csymbol><ci id="S5.9.p1.8.m8.1.1.3.2.cmml" xref="S5.9.p1.8.m8.1.1.3.2">𝒜</ci><times id="S5.9.p1.8.m8.1.1.3.3.cmml" xref="S5.9.p1.8.m8.1.1.3.3"></times></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.8.m8.1c">X\subseteq\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.8.m8.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> be the countable subshift which consists of the three orbits defined by</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\ldots aaa\ldots\,\,\,\,\,,\,\,\,\,\,\ldots aaabaaa\ldots\qquad\text{and}% \qquad\ldots aaacaaa\ldots\,." class="ltx_Math" display="block" id="S5.Ex2.m1.2"><semantics id="S5.Ex2.m1.2a"><mrow id="S5.Ex2.m1.2.2.1"><mrow id="S5.Ex2.m1.2.2.1.1.3" xref="S5.Ex2.m1.2.2.1.1.4.cmml"><mrow id="S5.Ex2.m1.2.2.1.1.1.1" xref="S5.Ex2.m1.2.2.1.1.1.1.cmml"><mi id="S5.Ex2.m1.2.2.1.1.1.1.2" mathvariant="normal" xref="S5.Ex2.m1.2.2.1.1.1.1.2.cmml">…</mi><mo id="S5.Ex2.m1.2.2.1.1.1.1.1" xref="S5.Ex2.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex2.m1.2.2.1.1.1.1.3" xref="S5.Ex2.m1.2.2.1.1.1.1.3.cmml">a</mi><mo id="S5.Ex2.m1.2.2.1.1.1.1.1a" xref="S5.Ex2.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex2.m1.2.2.1.1.1.1.4" xref="S5.Ex2.m1.2.2.1.1.1.1.4.cmml">a</mi><mo id="S5.Ex2.m1.2.2.1.1.1.1.1b" xref="S5.Ex2.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex2.m1.2.2.1.1.1.1.5" xref="S5.Ex2.m1.2.2.1.1.1.1.5.cmml">a</mi><mo id="S5.Ex2.m1.2.2.1.1.1.1.1c" xref="S5.Ex2.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Ex2.m1.2.2.1.1.1.1.6" mathvariant="normal" xref="S5.Ex2.m1.2.2.1.1.1.1.6.cmml">…</mi></mrow><mo id="S5.Ex2.m1.2.2.1.1.3.4" lspace="0.830em" rspace="0.997em" xref="S5.Ex2.m1.2.2.1.1.4.cmml">,</mo><mrow id="S5.Ex2.m1.2.2.1.1.2.2" xref="S5.Ex2.m1.2.2.1.1.2.2.cmml"><mi 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.9.p1.18">As in Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm10" title="Remark 5.10. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.10</span></a> the only invariant probability measure on <math alttext="X" class="ltx_Math" display="inline" id="S5.9.p1.9.m1.1"><semantics id="S5.9.p1.9.m1.1a"><mi id="S5.9.p1.9.m1.1.1" xref="S5.9.p1.9.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.9.p1.9.m1.1b"><ci id="S5.9.p1.9.m1.1.1.cmml" xref="S5.9.p1.9.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.9.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.9.m1.1d">italic_X</annotation></semantics></math> is the characteristic measure <math alttext="\mu_{a}\," class="ltx_Math" display="inline" id="S5.9.p1.10.m2.1"><semantics id="S5.9.p1.10.m2.1a"><msub id="S5.9.p1.10.m2.1.1" xref="S5.9.p1.10.m2.1.1.cmml"><mi id="S5.9.p1.10.m2.1.1.2" xref="S5.9.p1.10.m2.1.1.2.cmml">μ</mi><mi 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xref="S5.9.p1.11.m3.3.3.3.2">subscript</csymbol><ci id="S5.9.p1.11.m3.3.3.3.2.2.cmml" xref="S5.9.p1.11.m3.3.3.3.2.2">𝜎</ci><ci id="S5.9.p1.11.m3.3.3.3.2.3.cmml" xref="S5.9.p1.11.m3.3.3.3.2.3">𝑋</ci></apply><ci id="S5.9.p1.11.m3.3.3.3.3.cmml" xref="S5.9.p1.11.m3.3.3.3.3">𝑀</ci></apply><apply id="S5.9.p1.11.m3.3.3.1.cmml" xref="S5.9.p1.11.m3.3.3.1"><ci id="S5.9.p1.11.m3.3.3.1.2.cmml" xref="S5.9.p1.11.m3.3.3.1.2">→</ci><apply id="S5.9.p1.11.m3.3.3.1.3.cmml" xref="S5.9.p1.11.m3.3.3.1.3"><times id="S5.9.p1.11.m3.3.3.1.3.1.cmml" xref="S5.9.p1.11.m3.3.3.1.3.1"></times><ci id="S5.9.p1.11.m3.3.3.1.3.2.cmml" xref="S5.9.p1.11.m3.3.3.1.3.2">ℳ</ci><ci id="S5.9.p1.11.m3.1.1.cmml" xref="S5.9.p1.11.m3.1.1">𝒳</ci></apply><apply id="S5.9.p1.11.m3.3.3.1.1.cmml" xref="S5.9.p1.11.m3.3.3.1.1"><times id="S5.9.p1.11.m3.3.3.1.1.2.cmml" xref="S5.9.p1.11.m3.3.3.1.1.2"></times><ci id="S5.9.p1.11.m3.3.3.1.1.3.cmml" xref="S5.9.p1.11.m3.3.3.1.1.3">ℳ</ci><apply id="S5.9.p1.11.m3.3.3.1.1.1.1.1.cmml" xref="S5.9.p1.11.m3.3.3.1.1.1.1"><times id="S5.9.p1.11.m3.3.3.1.1.1.1.1.1.cmml" xref="S5.9.p1.11.m3.3.3.1.1.1.1.1.1"></times><ci id="S5.9.p1.11.m3.3.3.1.1.1.1.1.2.cmml" xref="S5.9.p1.11.m3.3.3.1.1.1.1.1.2">𝜎</ci><ci id="S5.9.p1.11.m3.2.2.cmml" xref="S5.9.p1.11.m3.2.2">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.11.m3.3c">\sigma_{X}M:\cal M(X)\to\cal M(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.11.m3.3d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) )</annotation></semantics></math> is automatically injective. On the other hand, the two biinfninite indexed words <math alttext="\ldots aaa\cdot baaa\ldots" class="ltx_Math" display="inline" id="S5.9.p1.12.m4.1"><semantics id="S5.9.p1.12.m4.1a"><mrow id="S5.9.p1.12.m4.1.1" xref="S5.9.p1.12.m4.1.1.cmml"><mrow id="S5.9.p1.12.m4.1.1.2" xref="S5.9.p1.12.m4.1.1.2.cmml"><mrow id="S5.9.p1.12.m4.1.1.2.2" xref="S5.9.p1.12.m4.1.1.2.2.cmml"><mi id="S5.9.p1.12.m4.1.1.2.2.2" mathvariant="normal" xref="S5.9.p1.12.m4.1.1.2.2.2.cmml">…</mi><mo id="S5.9.p1.12.m4.1.1.2.2.1" xref="S5.9.p1.12.m4.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.12.m4.1.1.2.2.3" xref="S5.9.p1.12.m4.1.1.2.2.3.cmml">a</mi><mo id="S5.9.p1.12.m4.1.1.2.2.1a" xref="S5.9.p1.12.m4.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.12.m4.1.1.2.2.4" xref="S5.9.p1.12.m4.1.1.2.2.4.cmml">a</mi><mo id="S5.9.p1.12.m4.1.1.2.2.1b" xref="S5.9.p1.12.m4.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.12.m4.1.1.2.2.5" xref="S5.9.p1.12.m4.1.1.2.2.5.cmml">a</mi></mrow><mo id="S5.9.p1.12.m4.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.9.p1.12.m4.1.1.2.1.cmml">⋅</mo><mi id="S5.9.p1.12.m4.1.1.2.3" xref="S5.9.p1.12.m4.1.1.2.3.cmml">b</mi></mrow><mo id="S5.9.p1.12.m4.1.1.1" xref="S5.9.p1.12.m4.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.12.m4.1.1.3" xref="S5.9.p1.12.m4.1.1.3.cmml">a</mi><mo id="S5.9.p1.12.m4.1.1.1a" xref="S5.9.p1.12.m4.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.12.m4.1.1.4" xref="S5.9.p1.12.m4.1.1.4.cmml">a</mi><mo id="S5.9.p1.12.m4.1.1.1b" xref="S5.9.p1.12.m4.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.12.m4.1.1.5" xref="S5.9.p1.12.m4.1.1.5.cmml">a</mi><mo id="S5.9.p1.12.m4.1.1.1c" xref="S5.9.p1.12.m4.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.12.m4.1.1.6" mathvariant="normal" xref="S5.9.p1.12.m4.1.1.6.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.12.m4.1b"><apply id="S5.9.p1.12.m4.1.1.cmml" xref="S5.9.p1.12.m4.1.1"><times id="S5.9.p1.12.m4.1.1.1.cmml" xref="S5.9.p1.12.m4.1.1.1"></times><apply id="S5.9.p1.12.m4.1.1.2.cmml" xref="S5.9.p1.12.m4.1.1.2"><ci id="S5.9.p1.12.m4.1.1.2.1.cmml" xref="S5.9.p1.12.m4.1.1.2.1">⋅</ci><apply id="S5.9.p1.12.m4.1.1.2.2.cmml" xref="S5.9.p1.12.m4.1.1.2.2"><times id="S5.9.p1.12.m4.1.1.2.2.1.cmml" xref="S5.9.p1.12.m4.1.1.2.2.1"></times><ci id="S5.9.p1.12.m4.1.1.2.2.2.cmml" xref="S5.9.p1.12.m4.1.1.2.2.2">…</ci><ci id="S5.9.p1.12.m4.1.1.2.2.3.cmml" xref="S5.9.p1.12.m4.1.1.2.2.3">𝑎</ci><ci id="S5.9.p1.12.m4.1.1.2.2.4.cmml" xref="S5.9.p1.12.m4.1.1.2.2.4">𝑎</ci><ci id="S5.9.p1.12.m4.1.1.2.2.5.cmml" xref="S5.9.p1.12.m4.1.1.2.2.5">𝑎</ci></apply><ci id="S5.9.p1.12.m4.1.1.2.3.cmml" xref="S5.9.p1.12.m4.1.1.2.3">𝑏</ci></apply><ci id="S5.9.p1.12.m4.1.1.3.cmml" xref="S5.9.p1.12.m4.1.1.3">𝑎</ci><ci id="S5.9.p1.12.m4.1.1.4.cmml" xref="S5.9.p1.12.m4.1.1.4">𝑎</ci><ci id="S5.9.p1.12.m4.1.1.5.cmml" xref="S5.9.p1.12.m4.1.1.5">𝑎</ci><ci id="S5.9.p1.12.m4.1.1.6.cmml" xref="S5.9.p1.12.m4.1.1.6">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.12.m4.1c">\ldots aaa\cdot baaa\ldots</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.12.m4.1d">… italic_a italic_a italic_a ⋅ italic_b italic_a italic_a italic_a …</annotation></semantics></math> and <math alttext="\ldots aaa\cdot caaa\ldots" class="ltx_Math" display="inline" id="S5.9.p1.13.m5.1"><semantics id="S5.9.p1.13.m5.1a"><mrow id="S5.9.p1.13.m5.1.1" xref="S5.9.p1.13.m5.1.1.cmml"><mrow id="S5.9.p1.13.m5.1.1.2" xref="S5.9.p1.13.m5.1.1.2.cmml"><mrow id="S5.9.p1.13.m5.1.1.2.2" xref="S5.9.p1.13.m5.1.1.2.2.cmml"><mi id="S5.9.p1.13.m5.1.1.2.2.2" mathvariant="normal" xref="S5.9.p1.13.m5.1.1.2.2.2.cmml">…</mi><mo id="S5.9.p1.13.m5.1.1.2.2.1" xref="S5.9.p1.13.m5.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.13.m5.1.1.2.2.3" xref="S5.9.p1.13.m5.1.1.2.2.3.cmml">a</mi><mo id="S5.9.p1.13.m5.1.1.2.2.1a" xref="S5.9.p1.13.m5.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.13.m5.1.1.2.2.4" xref="S5.9.p1.13.m5.1.1.2.2.4.cmml">a</mi><mo id="S5.9.p1.13.m5.1.1.2.2.1b" xref="S5.9.p1.13.m5.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.13.m5.1.1.2.2.5" xref="S5.9.p1.13.m5.1.1.2.2.5.cmml">a</mi></mrow><mo id="S5.9.p1.13.m5.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.9.p1.13.m5.1.1.2.1.cmml">⋅</mo><mi id="S5.9.p1.13.m5.1.1.2.3" xref="S5.9.p1.13.m5.1.1.2.3.cmml">c</mi></mrow><mo id="S5.9.p1.13.m5.1.1.1" xref="S5.9.p1.13.m5.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.13.m5.1.1.3" xref="S5.9.p1.13.m5.1.1.3.cmml">a</mi><mo id="S5.9.p1.13.m5.1.1.1a" xref="S5.9.p1.13.m5.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.13.m5.1.1.4" xref="S5.9.p1.13.m5.1.1.4.cmml">a</mi><mo id="S5.9.p1.13.m5.1.1.1b" xref="S5.9.p1.13.m5.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.13.m5.1.1.5" xref="S5.9.p1.13.m5.1.1.5.cmml">a</mi><mo id="S5.9.p1.13.m5.1.1.1c" xref="S5.9.p1.13.m5.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.13.m5.1.1.6" mathvariant="normal" xref="S5.9.p1.13.m5.1.1.6.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.13.m5.1b"><apply id="S5.9.p1.13.m5.1.1.cmml" xref="S5.9.p1.13.m5.1.1"><times id="S5.9.p1.13.m5.1.1.1.cmml" xref="S5.9.p1.13.m5.1.1.1"></times><apply id="S5.9.p1.13.m5.1.1.2.cmml" xref="S5.9.p1.13.m5.1.1.2"><ci id="S5.9.p1.13.m5.1.1.2.1.cmml" xref="S5.9.p1.13.m5.1.1.2.1">⋅</ci><apply id="S5.9.p1.13.m5.1.1.2.2.cmml" xref="S5.9.p1.13.m5.1.1.2.2"><times id="S5.9.p1.13.m5.1.1.2.2.1.cmml" xref="S5.9.p1.13.m5.1.1.2.2.1"></times><ci id="S5.9.p1.13.m5.1.1.2.2.2.cmml" xref="S5.9.p1.13.m5.1.1.2.2.2">…</ci><ci id="S5.9.p1.13.m5.1.1.2.2.3.cmml" xref="S5.9.p1.13.m5.1.1.2.2.3">𝑎</ci><ci id="S5.9.p1.13.m5.1.1.2.2.4.cmml" xref="S5.9.p1.13.m5.1.1.2.2.4">𝑎</ci><ci id="S5.9.p1.13.m5.1.1.2.2.5.cmml" xref="S5.9.p1.13.m5.1.1.2.2.5">𝑎</ci></apply><ci id="S5.9.p1.13.m5.1.1.2.3.cmml" xref="S5.9.p1.13.m5.1.1.2.3">𝑐</ci></apply><ci id="S5.9.p1.13.m5.1.1.3.cmml" xref="S5.9.p1.13.m5.1.1.3">𝑎</ci><ci id="S5.9.p1.13.m5.1.1.4.cmml" xref="S5.9.p1.13.m5.1.1.4">𝑎</ci><ci id="S5.9.p1.13.m5.1.1.5.cmml" xref="S5.9.p1.13.m5.1.1.5">𝑎</ci><ci id="S5.9.p1.13.m5.1.1.6.cmml" xref="S5.9.p1.13.m5.1.1.6">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.13.m5.1c">\ldots aaa\cdot caaa\ldots</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.13.m5.1d">… italic_a italic_a italic_a ⋅ italic_c italic_a italic_a italic_a …</annotation></semantics></math> lie in distinct shift-orbits and have the same image <math alttext="\ldots ddd\cdot eddd\ldots\," class="ltx_Math" display="inline" id="S5.9.p1.14.m6.1"><semantics id="S5.9.p1.14.m6.1a"><mrow id="S5.9.p1.14.m6.1.1" xref="S5.9.p1.14.m6.1.1.cmml"><mrow id="S5.9.p1.14.m6.1.1.2" xref="S5.9.p1.14.m6.1.1.2.cmml"><mrow id="S5.9.p1.14.m6.1.1.2.2" xref="S5.9.p1.14.m6.1.1.2.2.cmml"><mi id="S5.9.p1.14.m6.1.1.2.2.2" mathvariant="normal" xref="S5.9.p1.14.m6.1.1.2.2.2.cmml">…</mi><mo id="S5.9.p1.14.m6.1.1.2.2.1" xref="S5.9.p1.14.m6.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.14.m6.1.1.2.2.3" xref="S5.9.p1.14.m6.1.1.2.2.3.cmml">d</mi><mo id="S5.9.p1.14.m6.1.1.2.2.1a" xref="S5.9.p1.14.m6.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.14.m6.1.1.2.2.4" xref="S5.9.p1.14.m6.1.1.2.2.4.cmml">d</mi><mo id="S5.9.p1.14.m6.1.1.2.2.1b" xref="S5.9.p1.14.m6.1.1.2.2.1.cmml">⁢</mo><mi id="S5.9.p1.14.m6.1.1.2.2.5" xref="S5.9.p1.14.m6.1.1.2.2.5.cmml">d</mi></mrow><mo id="S5.9.p1.14.m6.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.9.p1.14.m6.1.1.2.1.cmml">⋅</mo><mi id="S5.9.p1.14.m6.1.1.2.3" xref="S5.9.p1.14.m6.1.1.2.3.cmml">e</mi></mrow><mo id="S5.9.p1.14.m6.1.1.1" xref="S5.9.p1.14.m6.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.14.m6.1.1.3" xref="S5.9.p1.14.m6.1.1.3.cmml">d</mi><mo id="S5.9.p1.14.m6.1.1.1a" xref="S5.9.p1.14.m6.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.14.m6.1.1.4" xref="S5.9.p1.14.m6.1.1.4.cmml">d</mi><mo id="S5.9.p1.14.m6.1.1.1b" xref="S5.9.p1.14.m6.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.14.m6.1.1.5" xref="S5.9.p1.14.m6.1.1.5.cmml">d</mi><mo id="S5.9.p1.14.m6.1.1.1c" xref="S5.9.p1.14.m6.1.1.1.cmml">⁢</mo><mi id="S5.9.p1.14.m6.1.1.6" mathvariant="normal" xref="S5.9.p1.14.m6.1.1.6.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.9.p1.14.m6.1b"><apply id="S5.9.p1.14.m6.1.1.cmml" xref="S5.9.p1.14.m6.1.1"><times id="S5.9.p1.14.m6.1.1.1.cmml" xref="S5.9.p1.14.m6.1.1.1"></times><apply id="S5.9.p1.14.m6.1.1.2.cmml" xref="S5.9.p1.14.m6.1.1.2"><ci id="S5.9.p1.14.m6.1.1.2.1.cmml" xref="S5.9.p1.14.m6.1.1.2.1">⋅</ci><apply id="S5.9.p1.14.m6.1.1.2.2.cmml" xref="S5.9.p1.14.m6.1.1.2.2"><times id="S5.9.p1.14.m6.1.1.2.2.1.cmml" xref="S5.9.p1.14.m6.1.1.2.2.1"></times><ci id="S5.9.p1.14.m6.1.1.2.2.2.cmml" xref="S5.9.p1.14.m6.1.1.2.2.2">…</ci><ci id="S5.9.p1.14.m6.1.1.2.2.3.cmml" xref="S5.9.p1.14.m6.1.1.2.2.3">𝑑</ci><ci id="S5.9.p1.14.m6.1.1.2.2.4.cmml" xref="S5.9.p1.14.m6.1.1.2.2.4">𝑑</ci><ci id="S5.9.p1.14.m6.1.1.2.2.5.cmml" xref="S5.9.p1.14.m6.1.1.2.2.5">𝑑</ci></apply><ci id="S5.9.p1.14.m6.1.1.2.3.cmml" xref="S5.9.p1.14.m6.1.1.2.3">𝑒</ci></apply><ci id="S5.9.p1.14.m6.1.1.3.cmml" xref="S5.9.p1.14.m6.1.1.3">𝑑</ci><ci id="S5.9.p1.14.m6.1.1.4.cmml" xref="S5.9.p1.14.m6.1.1.4">𝑑</ci><ci id="S5.9.p1.14.m6.1.1.5.cmml" xref="S5.9.p1.14.m6.1.1.5">𝑑</ci><ci id="S5.9.p1.14.m6.1.1.6.cmml" xref="S5.9.p1.14.m6.1.1.6">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.14.m6.1c">\ldots ddd\cdot eddd\ldots\,</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.14.m6.1d">… italic_d italic_d italic_d ⋅ italic_e italic_d italic_d italic_d …</annotation></semantics></math>, which is aperiodic. It follows that <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.9.p1.15.m7.1"><semantics id="S5.9.p1.15.m7.1a"><mi id="S5.9.p1.15.m7.1.1" xref="S5.9.p1.15.m7.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.9.p1.15.m7.1b"><ci id="S5.9.p1.15.m7.1.1.cmml" xref="S5.9.p1.15.m7.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.15.m7.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.15.m7.1d">italic_σ</annotation></semantics></math> is not recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.9.p1.16.m8.1"><semantics id="S5.9.p1.16.m8.1a"><mi id="S5.9.p1.16.m8.1.1" xref="S5.9.p1.16.m8.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.9.p1.16.m8.1b"><ci id="S5.9.p1.16.m8.1.1.cmml" xref="S5.9.p1.16.m8.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.16.m8.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.16.m8.1d">italic_X</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S5.9.p1.17.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S5.9.p1.17.1.m1.1"><semantics id="S5.9.p1.17.1.m1.1a"><mo id="S5.9.p1.17.1.m1.1.1" xref="S5.9.p1.17.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S5.9.p1.17.1.m1.1b"><csymbol cd="latexml" id="S5.9.p1.17.1.m1.1.1.cmml" xref="S5.9.p1.17.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.17.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.17.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S5.9.p1.18.m9.1"><semantics id="S5.9.p1.18.m9.1a"><mo id="S5.9.p1.18.m9.1.1" xref="S5.9.p1.18.m9.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S5.9.p1.18.m9.1b"><csymbol cd="latexml" id="S5.9.p1.18.m9.1.1.cmml" xref="S5.9.p1.18.m9.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S5.9.p1.18.m9.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S5.9.p1.18.m9.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S5.Thmthm12"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm12.1.1.1">Remark 5.12</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm12.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm12.p1"> <p class="ltx_p" id="S5.Thmthm12.p1.10">There exist also stronger but slightly more intricate examples as given in the last proof, where the image subshift <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.1.m1.1"><semantics id="S5.Thmthm12.p1.1.m1.1a"><mrow id="S5.Thmthm12.p1.1.m1.1.2" xref="S5.Thmthm12.p1.1.m1.1.2.cmml"><mi id="S5.Thmthm12.p1.1.m1.1.2.2" xref="S5.Thmthm12.p1.1.m1.1.2.2.cmml">σ</mi><mo id="S5.Thmthm12.p1.1.m1.1.2.1" xref="S5.Thmthm12.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.Thmthm12.p1.1.m1.1.2.3.2" xref="S5.Thmthm12.p1.1.m1.1.2.cmml"><mo id="S5.Thmthm12.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.Thmthm12.p1.1.m1.1.2.cmml">(</mo><mi id="S5.Thmthm12.p1.1.m1.1.1" xref="S5.Thmthm12.p1.1.m1.1.1.cmml">X</mi><mo id="S5.Thmthm12.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.Thmthm12.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p1.1.m1.1b"><apply id="S5.Thmthm12.p1.1.m1.1.2.cmml" xref="S5.Thmthm12.p1.1.m1.1.2"><times id="S5.Thmthm12.p1.1.m1.1.2.1.cmml" xref="S5.Thmthm12.p1.1.m1.1.2.1"></times><ci id="S5.Thmthm12.p1.1.m1.1.2.2.cmml" xref="S5.Thmthm12.p1.1.m1.1.2.2">𝜎</ci><ci id="S5.Thmthm12.p1.1.m1.1.1.cmml" xref="S5.Thmthm12.p1.1.m1.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.1.m1.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.1.m1.1d">italic_σ ( italic_X )</annotation></semantics></math> is completely aperiodic. They are based on the same construction principle, but the periodic orbit of <math alttext="\ldots aaa\ldots" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.2.m2.1"><semantics id="S5.Thmthm12.p1.2.m2.1a"><mrow id="S5.Thmthm12.p1.2.m2.1.1" xref="S5.Thmthm12.p1.2.m2.1.1.cmml"><mi id="S5.Thmthm12.p1.2.m2.1.1.2" mathvariant="normal" xref="S5.Thmthm12.p1.2.m2.1.1.2.cmml">…</mi><mo id="S5.Thmthm12.p1.2.m2.1.1.1" xref="S5.Thmthm12.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p1.2.m2.1.1.3" xref="S5.Thmthm12.p1.2.m2.1.1.3.cmml">a</mi><mo id="S5.Thmthm12.p1.2.m2.1.1.1a" xref="S5.Thmthm12.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p1.2.m2.1.1.4" xref="S5.Thmthm12.p1.2.m2.1.1.4.cmml">a</mi><mo id="S5.Thmthm12.p1.2.m2.1.1.1b" xref="S5.Thmthm12.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p1.2.m2.1.1.5" xref="S5.Thmthm12.p1.2.m2.1.1.5.cmml">a</mi><mo id="S5.Thmthm12.p1.2.m2.1.1.1c" xref="S5.Thmthm12.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p1.2.m2.1.1.6" mathvariant="normal" xref="S5.Thmthm12.p1.2.m2.1.1.6.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p1.2.m2.1b"><apply id="S5.Thmthm12.p1.2.m2.1.1.cmml" xref="S5.Thmthm12.p1.2.m2.1.1"><times id="S5.Thmthm12.p1.2.m2.1.1.1.cmml" xref="S5.Thmthm12.p1.2.m2.1.1.1"></times><ci id="S5.Thmthm12.p1.2.m2.1.1.2.cmml" xref="S5.Thmthm12.p1.2.m2.1.1.2">…</ci><ci id="S5.Thmthm12.p1.2.m2.1.1.3.cmml" xref="S5.Thmthm12.p1.2.m2.1.1.3">𝑎</ci><ci id="S5.Thmthm12.p1.2.m2.1.1.4.cmml" xref="S5.Thmthm12.p1.2.m2.1.1.4">𝑎</ci><ci id="S5.Thmthm12.p1.2.m2.1.1.5.cmml" xref="S5.Thmthm12.p1.2.m2.1.1.5">𝑎</ci><ci id="S5.Thmthm12.p1.2.m2.1.1.6.cmml" xref="S5.Thmthm12.p1.2.m2.1.1.6">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.2.m2.1c">\ldots aaa\ldots</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.2.m2.1d">… italic_a italic_a italic_a …</annotation></semantics></math> is replaced by the minimal and uniquely ergodic subshift <math alttext="X_{\tau}" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.3.m3.1"><semantics id="S5.Thmthm12.p1.3.m3.1a"><msub id="S5.Thmthm12.p1.3.m3.1.1" xref="S5.Thmthm12.p1.3.m3.1.1.cmml"><mi id="S5.Thmthm12.p1.3.m3.1.1.2" xref="S5.Thmthm12.p1.3.m3.1.1.2.cmml">X</mi><mi id="S5.Thmthm12.p1.3.m3.1.1.3" xref="S5.Thmthm12.p1.3.m3.1.1.3.cmml">τ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p1.3.m3.1b"><apply id="S5.Thmthm12.p1.3.m3.1.1.cmml" xref="S5.Thmthm12.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.Thmthm12.p1.3.m3.1.1.1.cmml" xref="S5.Thmthm12.p1.3.m3.1.1">subscript</csymbol><ci id="S5.Thmthm12.p1.3.m3.1.1.2.cmml" xref="S5.Thmthm12.p1.3.m3.1.1.2">𝑋</ci><ci id="S5.Thmthm12.p1.3.m3.1.1.3.cmml" xref="S5.Thmthm12.p1.3.m3.1.1.3">𝜏</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.3.m3.1c">X_{\tau}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.3.m3.1d">italic_X start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT</annotation></semantics></math> defined by any primitive substitution <math alttext="\tau:\cal A^{*}\to\cal A^{*}" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.4.m4.1"><semantics id="S5.Thmthm12.p1.4.m4.1a"><mrow id="S5.Thmthm12.p1.4.m4.1.1" xref="S5.Thmthm12.p1.4.m4.1.1.cmml"><mi id="S5.Thmthm12.p1.4.m4.1.1.2" xref="S5.Thmthm12.p1.4.m4.1.1.2.cmml">τ</mi><mo id="S5.Thmthm12.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm12.p1.4.m4.1.1.1.cmml">:</mo><mrow id="S5.Thmthm12.p1.4.m4.1.1.3" xref="S5.Thmthm12.p1.4.m4.1.1.3.cmml"><msup id="S5.Thmthm12.p1.4.m4.1.1.3.2" xref="S5.Thmthm12.p1.4.m4.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p1.4.m4.1.1.3.2.2" xref="S5.Thmthm12.p1.4.m4.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm12.p1.4.m4.1.1.3.2.3" xref="S5.Thmthm12.p1.4.m4.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm12.p1.4.m4.1.1.3.1" stretchy="false" xref="S5.Thmthm12.p1.4.m4.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm12.p1.4.m4.1.1.3.3" xref="S5.Thmthm12.p1.4.m4.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p1.4.m4.1.1.3.3.2" xref="S5.Thmthm12.p1.4.m4.1.1.3.3.2.cmml">𝒜</mi><mo id="S5.Thmthm12.p1.4.m4.1.1.3.3.3" xref="S5.Thmthm12.p1.4.m4.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p1.4.m4.1b"><apply id="S5.Thmthm12.p1.4.m4.1.1.cmml" xref="S5.Thmthm12.p1.4.m4.1.1"><ci id="S5.Thmthm12.p1.4.m4.1.1.1.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.1">:</ci><ci id="S5.Thmthm12.p1.4.m4.1.1.2.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.2">𝜏</ci><apply id="S5.Thmthm12.p1.4.m4.1.1.3.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3"><ci id="S5.Thmthm12.p1.4.m4.1.1.3.1.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.1">→</ci><apply id="S5.Thmthm12.p1.4.m4.1.1.3.2.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm12.p1.4.m4.1.1.3.2.1.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm12.p1.4.m4.1.1.3.2.2.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm12.p1.4.m4.1.1.3.2.3.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.2.3"></times></apply><apply id="S5.Thmthm12.p1.4.m4.1.1.3.3.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm12.p1.4.m4.1.1.3.3.1.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm12.p1.4.m4.1.1.3.3.2.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.3.2">𝒜</ci><times id="S5.Thmthm12.p1.4.m4.1.1.3.3.3.cmml" xref="S5.Thmthm12.p1.4.m4.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.4.m4.1c">\tau:\cal A^{*}\to\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.4.m4.1d">italic_τ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, which in addition we assume to possess a left-infinite word <math alttext="V" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.5.m5.1"><semantics id="S5.Thmthm12.p1.5.m5.1a"><mi id="S5.Thmthm12.p1.5.m5.1.1" xref="S5.Thmthm12.p1.5.m5.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p1.5.m5.1b"><ci id="S5.Thmthm12.p1.5.m5.1.1.cmml" xref="S5.Thmthm12.p1.5.m5.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.5.m5.1c">V</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.5.m5.1d">italic_V</annotation></semantics></math> and a right-infinite word <math alttext="W" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.6.m6.1"><semantics id="S5.Thmthm12.p1.6.m6.1a"><mi id="S5.Thmthm12.p1.6.m6.1.1" xref="S5.Thmthm12.p1.6.m6.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p1.6.m6.1b"><ci id="S5.Thmthm12.p1.6.m6.1.1.cmml" xref="S5.Thmthm12.p1.6.m6.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.6.m6.1c">W</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.6.m6.1d">italic_W</annotation></semantics></math> that are both fixed by <math alttext="\tau" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.7.m7.1"><semantics id="S5.Thmthm12.p1.7.m7.1a"><mi id="S5.Thmthm12.p1.7.m7.1.1" xref="S5.Thmthm12.p1.7.m7.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p1.7.m7.1b"><ci id="S5.Thmthm12.p1.7.m7.1.1.cmml" xref="S5.Thmthm12.p1.7.m7.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.7.m7.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.7.m7.1d">italic_τ</annotation></semantics></math>. Such a substitution is given for example by the square of the Fibonacci substitution <math alttext="\tau(x)=xyx\,,\,\,\tau(y)=xy" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.8.m8.4"><semantics id="S5.Thmthm12.p1.8.m8.4a"><mrow id="S5.Thmthm12.p1.8.m8.4.4.2" xref="S5.Thmthm12.p1.8.m8.4.4.3.cmml"><mrow id="S5.Thmthm12.p1.8.m8.3.3.1.1" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.cmml"><mrow id="S5.Thmthm12.p1.8.m8.3.3.1.1.2" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2.cmml"><mi id="S5.Thmthm12.p1.8.m8.3.3.1.1.2.2" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2.2.cmml">τ</mi><mo id="S5.Thmthm12.p1.8.m8.3.3.1.1.2.1" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S5.Thmthm12.p1.8.m8.3.3.1.1.2.3.2" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2.cmml"><mo id="S5.Thmthm12.p1.8.m8.3.3.1.1.2.3.2.1" stretchy="false" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2.cmml">(</mo><mi id="S5.Thmthm12.p1.8.m8.1.1" xref="S5.Thmthm12.p1.8.m8.1.1.cmml">x</mi><mo id="S5.Thmthm12.p1.8.m8.3.3.1.1.2.3.2.2" stretchy="false" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm12.p1.8.m8.3.3.1.1.1" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.1.cmml">=</mo><mrow id="S5.Thmthm12.p1.8.m8.3.3.1.1.3" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.cmml"><mi id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.2" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.2.cmml">x</mi><mo id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.1" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.1.cmml">⁢</mo><mi id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.3" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.3.cmml">y</mi><mo id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.1a" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.1.cmml">⁢</mo><mi id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.4" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.4.cmml">x</mi></mrow></mrow><mo id="S5.Thmthm12.p1.8.m8.4.4.2.3" lspace="0.170em" rspace="0.497em" xref="S5.Thmthm12.p1.8.m8.4.4.3a.cmml">,</mo><mrow id="S5.Thmthm12.p1.8.m8.4.4.2.2" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.cmml"><mrow id="S5.Thmthm12.p1.8.m8.4.4.2.2.2" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2.cmml"><mi id="S5.Thmthm12.p1.8.m8.4.4.2.2.2.2" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2.2.cmml">τ</mi><mo id="S5.Thmthm12.p1.8.m8.4.4.2.2.2.1" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2.1.cmml">⁢</mo><mrow id="S5.Thmthm12.p1.8.m8.4.4.2.2.2.3.2" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2.cmml"><mo id="S5.Thmthm12.p1.8.m8.4.4.2.2.2.3.2.1" stretchy="false" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2.cmml">(</mo><mi id="S5.Thmthm12.p1.8.m8.2.2" xref="S5.Thmthm12.p1.8.m8.2.2.cmml">y</mi><mo id="S5.Thmthm12.p1.8.m8.4.4.2.2.2.3.2.2" stretchy="false" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm12.p1.8.m8.4.4.2.2.1" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.1.cmml">=</mo><mrow id="S5.Thmthm12.p1.8.m8.4.4.2.2.3" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.3.cmml"><mi id="S5.Thmthm12.p1.8.m8.4.4.2.2.3.2" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.3.2.cmml">x</mi><mo id="S5.Thmthm12.p1.8.m8.4.4.2.2.3.1" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.3.1.cmml">⁢</mo><mi id="S5.Thmthm12.p1.8.m8.4.4.2.2.3.3" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.3.3.cmml">y</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p1.8.m8.4b"><apply id="S5.Thmthm12.p1.8.m8.4.4.3.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2"><csymbol cd="ambiguous" id="S5.Thmthm12.p1.8.m8.4.4.3a.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.3">formulae-sequence</csymbol><apply id="S5.Thmthm12.p1.8.m8.3.3.1.1.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1"><eq id="S5.Thmthm12.p1.8.m8.3.3.1.1.1.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.1"></eq><apply id="S5.Thmthm12.p1.8.m8.3.3.1.1.2.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2"><times id="S5.Thmthm12.p1.8.m8.3.3.1.1.2.1.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2.1"></times><ci id="S5.Thmthm12.p1.8.m8.3.3.1.1.2.2.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.2.2">𝜏</ci><ci id="S5.Thmthm12.p1.8.m8.1.1.cmml" xref="S5.Thmthm12.p1.8.m8.1.1">𝑥</ci></apply><apply id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3"><times id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.1.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.1"></times><ci id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.2.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.2">𝑥</ci><ci id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.3.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.3">𝑦</ci><ci id="S5.Thmthm12.p1.8.m8.3.3.1.1.3.4.cmml" xref="S5.Thmthm12.p1.8.m8.3.3.1.1.3.4">𝑥</ci></apply></apply><apply id="S5.Thmthm12.p1.8.m8.4.4.2.2.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2"><eq id="S5.Thmthm12.p1.8.m8.4.4.2.2.1.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.1"></eq><apply id="S5.Thmthm12.p1.8.m8.4.4.2.2.2.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2"><times id="S5.Thmthm12.p1.8.m8.4.4.2.2.2.1.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2.1"></times><ci id="S5.Thmthm12.p1.8.m8.4.4.2.2.2.2.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.2.2">𝜏</ci><ci id="S5.Thmthm12.p1.8.m8.2.2.cmml" xref="S5.Thmthm12.p1.8.m8.2.2">𝑦</ci></apply><apply id="S5.Thmthm12.p1.8.m8.4.4.2.2.3.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.3"><times id="S5.Thmthm12.p1.8.m8.4.4.2.2.3.1.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.3.1"></times><ci id="S5.Thmthm12.p1.8.m8.4.4.2.2.3.2.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.3.2">𝑥</ci><ci id="S5.Thmthm12.p1.8.m8.4.4.2.2.3.3.cmml" xref="S5.Thmthm12.p1.8.m8.4.4.2.2.3.3">𝑦</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.8.m8.4c">\tau(x)=xyx\,,\,\,\tau(y)=xy</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.8.m8.4d">italic_τ ( italic_x ) = italic_x italic_y italic_x , italic_τ ( italic_y ) = italic_x italic_y</annotation></semantics></math> with <math alttext="V=\ldots\tau^{2}(xy)\,\tau(xy)\,xy\,x" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.9.m9.2"><semantics id="S5.Thmthm12.p1.9.m9.2a"><mrow id="S5.Thmthm12.p1.9.m9.2.2" xref="S5.Thmthm12.p1.9.m9.2.2.cmml"><mi id="S5.Thmthm12.p1.9.m9.2.2.4" xref="S5.Thmthm12.p1.9.m9.2.2.4.cmml">V</mi><mo id="S5.Thmthm12.p1.9.m9.2.2.3" xref="S5.Thmthm12.p1.9.m9.2.2.3.cmml">=</mo><mrow id="S5.Thmthm12.p1.9.m9.2.2.2" xref="S5.Thmthm12.p1.9.m9.2.2.2.cmml"><mi id="S5.Thmthm12.p1.9.m9.2.2.2.4" mathvariant="normal" xref="S5.Thmthm12.p1.9.m9.2.2.2.4.cmml">…</mi><mo id="S5.Thmthm12.p1.9.m9.2.2.2.3" xref="S5.Thmthm12.p1.9.m9.2.2.2.3.cmml">⁢</mo><msup id="S5.Thmthm12.p1.9.m9.2.2.2.5" xref="S5.Thmthm12.p1.9.m9.2.2.2.5.cmml"><mi id="S5.Thmthm12.p1.9.m9.2.2.2.5.2" xref="S5.Thmthm12.p1.9.m9.2.2.2.5.2.cmml">τ</mi><mn id="S5.Thmthm12.p1.9.m9.2.2.2.5.3" xref="S5.Thmthm12.p1.9.m9.2.2.2.5.3.cmml">2</mn></msup><mo id="S5.Thmthm12.p1.9.m9.2.2.2.3a" xref="S5.Thmthm12.p1.9.m9.2.2.2.3.cmml">⁢</mo><mrow id="S5.Thmthm12.p1.9.m9.1.1.1.1.1" xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.cmml"><mo id="S5.Thmthm12.p1.9.m9.1.1.1.1.1.2" stretchy="false" xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1" xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.cmml"><mi id="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.2" xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.2.cmml">x</mi><mo id="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.1" xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.3" xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.3.cmml">y</mi></mrow><mo id="S5.Thmthm12.p1.9.m9.1.1.1.1.1.3" stretchy="false" xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmthm12.p1.9.m9.2.2.2.3b" lspace="0.170em" xref="S5.Thmthm12.p1.9.m9.2.2.2.3.cmml">⁢</mo><mi id="S5.Thmthm12.p1.9.m9.2.2.2.6" xref="S5.Thmthm12.p1.9.m9.2.2.2.6.cmml">τ</mi><mo id="S5.Thmthm12.p1.9.m9.2.2.2.3c" xref="S5.Thmthm12.p1.9.m9.2.2.2.3.cmml">⁢</mo><mrow id="S5.Thmthm12.p1.9.m9.2.2.2.2.1" xref="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.cmml"><mo id="S5.Thmthm12.p1.9.m9.2.2.2.2.1.2" stretchy="false" xref="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.cmml">(</mo><mrow id="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1" xref="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.cmml"><mi id="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.2" xref="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.2.cmml">x</mi><mo id="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.1" 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xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.2">𝑥</ci><ci id="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.3.cmml" xref="S5.Thmthm12.p1.9.m9.1.1.1.1.1.1.3">𝑦</ci></apply><ci id="S5.Thmthm12.p1.9.m9.2.2.2.6.cmml" xref="S5.Thmthm12.p1.9.m9.2.2.2.6">𝜏</ci><apply id="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.cmml" xref="S5.Thmthm12.p1.9.m9.2.2.2.2.1"><times id="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.1.cmml" xref="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.1"></times><ci id="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.2.cmml" xref="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.2">𝑥</ci><ci id="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.3.cmml" xref="S5.Thmthm12.p1.9.m9.2.2.2.2.1.1.3">𝑦</ci></apply><ci id="S5.Thmthm12.p1.9.m9.2.2.2.7.cmml" xref="S5.Thmthm12.p1.9.m9.2.2.2.7">𝑥</ci><ci id="S5.Thmthm12.p1.9.m9.2.2.2.8.cmml" xref="S5.Thmthm12.p1.9.m9.2.2.2.8">𝑦</ci><ci id="S5.Thmthm12.p1.9.m9.2.2.2.9.cmml" xref="S5.Thmthm12.p1.9.m9.2.2.2.9">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.9.m9.2c">V=\ldots\tau^{2}(xy)\,\tau(xy)\,xy\,x</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.9.m9.2d">italic_V = … italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_x italic_y ) italic_τ ( italic_x italic_y ) italic_x italic_y italic_x</annotation></semantics></math> and <math alttext="W=x\,yx\,\tau(yx)\,\tau^{2}(yx)\ldots" class="ltx_Math" display="inline" id="S5.Thmthm12.p1.10.m10.2"><semantics id="S5.Thmthm12.p1.10.m10.2a"><mrow id="S5.Thmthm12.p1.10.m10.2.2" xref="S5.Thmthm12.p1.10.m10.2.2.cmml"><mi id="S5.Thmthm12.p1.10.m10.2.2.4" xref="S5.Thmthm12.p1.10.m10.2.2.4.cmml">W</mi><mo id="S5.Thmthm12.p1.10.m10.2.2.3" xref="S5.Thmthm12.p1.10.m10.2.2.3.cmml">=</mo><mrow id="S5.Thmthm12.p1.10.m10.2.2.2" xref="S5.Thmthm12.p1.10.m10.2.2.2.cmml"><mi id="S5.Thmthm12.p1.10.m10.2.2.2.4" xref="S5.Thmthm12.p1.10.m10.2.2.2.4.cmml">x</mi><mo id="S5.Thmthm12.p1.10.m10.2.2.2.3" lspace="0.170em" 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xref="S5.Thmthm12.p1.10.m10.2.2.4">𝑊</ci><apply id="S5.Thmthm12.p1.10.m10.2.2.2.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2"><times id="S5.Thmthm12.p1.10.m10.2.2.2.3.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.3"></times><ci id="S5.Thmthm12.p1.10.m10.2.2.2.4.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.4">𝑥</ci><ci id="S5.Thmthm12.p1.10.m10.2.2.2.5.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.5">𝑦</ci><ci id="S5.Thmthm12.p1.10.m10.2.2.2.6.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.6">𝑥</ci><ci id="S5.Thmthm12.p1.10.m10.2.2.2.7.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.7">𝜏</ci><apply id="S5.Thmthm12.p1.10.m10.1.1.1.1.1.1.cmml" xref="S5.Thmthm12.p1.10.m10.1.1.1.1.1"><times id="S5.Thmthm12.p1.10.m10.1.1.1.1.1.1.1.cmml" xref="S5.Thmthm12.p1.10.m10.1.1.1.1.1.1.1"></times><ci id="S5.Thmthm12.p1.10.m10.1.1.1.1.1.1.2.cmml" xref="S5.Thmthm12.p1.10.m10.1.1.1.1.1.1.2">𝑦</ci><ci id="S5.Thmthm12.p1.10.m10.1.1.1.1.1.1.3.cmml" xref="S5.Thmthm12.p1.10.m10.1.1.1.1.1.1.3">𝑥</ci></apply><apply id="S5.Thmthm12.p1.10.m10.2.2.2.8.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.8"><csymbol cd="ambiguous" id="S5.Thmthm12.p1.10.m10.2.2.2.8.1.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.8">superscript</csymbol><ci id="S5.Thmthm12.p1.10.m10.2.2.2.8.2.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.8.2">𝜏</ci><cn id="S5.Thmthm12.p1.10.m10.2.2.2.8.3.cmml" type="integer" xref="S5.Thmthm12.p1.10.m10.2.2.2.8.3">2</cn></apply><apply id="S5.Thmthm12.p1.10.m10.2.2.2.2.1.1.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.2.1"><times id="S5.Thmthm12.p1.10.m10.2.2.2.2.1.1.1.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.2.1.1.1"></times><ci id="S5.Thmthm12.p1.10.m10.2.2.2.2.1.1.2.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.2.1.1.2">𝑦</ci><ci id="S5.Thmthm12.p1.10.m10.2.2.2.2.1.1.3.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.2.1.1.3">𝑥</ci></apply><ci id="S5.Thmthm12.p1.10.m10.2.2.2.9.cmml" xref="S5.Thmthm12.p1.10.m10.2.2.2.9">…</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p1.10.m10.2c">W=x\,yx\,\tau(yx)\,\tau^{2}(yx)\ldots</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p1.10.m10.2d">italic_W = italic_x italic_y italic_x italic_τ ( italic_y italic_x ) italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_y italic_x ) …</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.Thmthm12.p2"> <p class="ltx_p" id="S5.Thmthm12.p2.28">One now considers three additional letters <math alttext="a,b,c" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.1.m1.3"><semantics id="S5.Thmthm12.p2.1.m1.3a"><mrow id="S5.Thmthm12.p2.1.m1.3.4.2" xref="S5.Thmthm12.p2.1.m1.3.4.1.cmml"><mi id="S5.Thmthm12.p2.1.m1.1.1" xref="S5.Thmthm12.p2.1.m1.1.1.cmml">a</mi><mo id="S5.Thmthm12.p2.1.m1.3.4.2.1" xref="S5.Thmthm12.p2.1.m1.3.4.1.cmml">,</mo><mi id="S5.Thmthm12.p2.1.m1.2.2" xref="S5.Thmthm12.p2.1.m1.2.2.cmml">b</mi><mo id="S5.Thmthm12.p2.1.m1.3.4.2.2" xref="S5.Thmthm12.p2.1.m1.3.4.1.cmml">,</mo><mi id="S5.Thmthm12.p2.1.m1.3.3" xref="S5.Thmthm12.p2.1.m1.3.3.cmml">c</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.1.m1.3b"><list id="S5.Thmthm12.p2.1.m1.3.4.1.cmml" xref="S5.Thmthm12.p2.1.m1.3.4.2"><ci id="S5.Thmthm12.p2.1.m1.1.1.cmml" xref="S5.Thmthm12.p2.1.m1.1.1">𝑎</ci><ci id="S5.Thmthm12.p2.1.m1.2.2.cmml" xref="S5.Thmthm12.p2.1.m1.2.2">𝑏</ci><ci id="S5.Thmthm12.p2.1.m1.3.3.cmml" xref="S5.Thmthm12.p2.1.m1.3.3">𝑐</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.1.m1.3c">a,b,c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.1.m1.3d">italic_a , italic_b , italic_c</annotation></semantics></math> not contained in <math alttext="\cal A" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.2.m2.1"><semantics id="S5.Thmthm12.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.2.m2.1.1" xref="S5.Thmthm12.p2.2.m2.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.2.m2.1b"><ci id="S5.Thmthm12.p2.2.m2.1.1.cmml" xref="S5.Thmthm12.p2.2.m2.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.2.m2.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.2.m2.1d">caligraphic_A</annotation></semantics></math> and defines the letter-to-letter morphism <math alttext="\sigma:(\{b,c\}\cup\cal A)^{*}\to(\{a\}\cup\cal A)^{*}" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.3.m3.5"><semantics id="S5.Thmthm12.p2.3.m3.5a"><mrow id="S5.Thmthm12.p2.3.m3.5.5" xref="S5.Thmthm12.p2.3.m3.5.5.cmml"><mi id="S5.Thmthm12.p2.3.m3.5.5.4" xref="S5.Thmthm12.p2.3.m3.5.5.4.cmml">σ</mi><mo id="S5.Thmthm12.p2.3.m3.5.5.3" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm12.p2.3.m3.5.5.3.cmml">:</mo><mrow id="S5.Thmthm12.p2.3.m3.5.5.2" xref="S5.Thmthm12.p2.3.m3.5.5.2.cmml"><msup id="S5.Thmthm12.p2.3.m3.4.4.1.1" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.cmml"><mrow id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.cmml"><mo id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.2" stretchy="false" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.cmml"><mrow id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.2" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.1.cmml"><mo id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.2.1" stretchy="false" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.1.cmml">{</mo><mi id="S5.Thmthm12.p2.3.m3.1.1" xref="S5.Thmthm12.p2.3.m3.1.1.cmml">b</mi><mo id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.2.2" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.1.cmml">,</mo><mi id="S5.Thmthm12.p2.3.m3.2.2" xref="S5.Thmthm12.p2.3.m3.2.2.cmml">c</mi><mo id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.2.3" stretchy="false" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.1.cmml">}</mo></mrow><mo id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.1" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.1.cmml">∪</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.3" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.3.cmml">𝒜</mi></mrow><mo id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.3" stretchy="false" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmthm12.p2.3.m3.4.4.1.1.3" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.3.cmml">∗</mo></msup><mo id="S5.Thmthm12.p2.3.m3.5.5.2.3" stretchy="false" xref="S5.Thmthm12.p2.3.m3.5.5.2.3.cmml">→</mo><msup id="S5.Thmthm12.p2.3.m3.5.5.2.2" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.cmml"><mrow id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.cmml"><mo id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.2" stretchy="false" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.cmml">(</mo><mrow id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.cmml"><mrow id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.2.2" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.2.1.cmml"><mo id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.2.2.1" stretchy="false" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.2.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.3.m3.3.3" xref="S5.Thmthm12.p2.3.m3.3.3.cmml">𝒶</mi><mo id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.2.2.2" stretchy="false" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.2.1.cmml">}</mo></mrow><mo id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.1" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.1.cmml">∪</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.3" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.3.cmml">𝒜</mi></mrow><mo id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.3" stretchy="false" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmthm12.p2.3.m3.5.5.2.2.3" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.3.m3.5b"><apply id="S5.Thmthm12.p2.3.m3.5.5.cmml" xref="S5.Thmthm12.p2.3.m3.5.5"><ci id="S5.Thmthm12.p2.3.m3.5.5.3.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.3">:</ci><ci id="S5.Thmthm12.p2.3.m3.5.5.4.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.4">𝜎</ci><apply id="S5.Thmthm12.p2.3.m3.5.5.2.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2"><ci id="S5.Thmthm12.p2.3.m3.5.5.2.3.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2.3">→</ci><apply id="S5.Thmthm12.p2.3.m3.4.4.1.1.cmml" xref="S5.Thmthm12.p2.3.m3.4.4.1.1"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.3.m3.4.4.1.1.2.cmml" xref="S5.Thmthm12.p2.3.m3.4.4.1.1">superscript</csymbol><apply id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.cmml" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1"><union id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.1.cmml" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.1"></union><set id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.1.cmml" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.2.2"><ci id="S5.Thmthm12.p2.3.m3.1.1.cmml" xref="S5.Thmthm12.p2.3.m3.1.1">𝑏</ci><ci id="S5.Thmthm12.p2.3.m3.2.2.cmml" xref="S5.Thmthm12.p2.3.m3.2.2">𝑐</ci></set><ci id="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.3.cmml" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.1.1.1.3">𝒜</ci></apply><times id="S5.Thmthm12.p2.3.m3.4.4.1.1.3.cmml" xref="S5.Thmthm12.p2.3.m3.4.4.1.1.3"></times></apply><apply id="S5.Thmthm12.p2.3.m3.5.5.2.2.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2.2"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.3.m3.5.5.2.2.2.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2.2">superscript</csymbol><apply id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1"><union id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.1.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.1"></union><set id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.2.1.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.2.2"><ci id="S5.Thmthm12.p2.3.m3.3.3.cmml" xref="S5.Thmthm12.p2.3.m3.3.3">𝒶</ci></set><ci id="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.3.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.1.1.1.3">𝒜</ci></apply><times id="S5.Thmthm12.p2.3.m3.5.5.2.2.3.cmml" xref="S5.Thmthm12.p2.3.m3.5.5.2.2.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.3.m3.5c">\sigma:(\{b,c\}\cup\cal A)^{*}\to(\{a\}\cup\cal A)^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.3.m3.5d">italic_σ : ( { italic_b , italic_c } ∪ caligraphic_A ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → ( { caligraphic_a } ∪ caligraphic_A ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> by setting <math alttext="\sigma(a_{i})=a_{i}" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.4.m4.1"><semantics id="S5.Thmthm12.p2.4.m4.1a"><mrow id="S5.Thmthm12.p2.4.m4.1.1" xref="S5.Thmthm12.p2.4.m4.1.1.cmml"><mrow id="S5.Thmthm12.p2.4.m4.1.1.1" xref="S5.Thmthm12.p2.4.m4.1.1.1.cmml"><mi id="S5.Thmthm12.p2.4.m4.1.1.1.3" xref="S5.Thmthm12.p2.4.m4.1.1.1.3.cmml">σ</mi><mo id="S5.Thmthm12.p2.4.m4.1.1.1.2" xref="S5.Thmthm12.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmthm12.p2.4.m4.1.1.1.1.1" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.2" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.2.cmml">a</mi><mi id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.3" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm12.p2.4.m4.1.1.2" xref="S5.Thmthm12.p2.4.m4.1.1.2.cmml">=</mo><msub id="S5.Thmthm12.p2.4.m4.1.1.3" xref="S5.Thmthm12.p2.4.m4.1.1.3.cmml"><mi id="S5.Thmthm12.p2.4.m4.1.1.3.2" xref="S5.Thmthm12.p2.4.m4.1.1.3.2.cmml">a</mi><mi id="S5.Thmthm12.p2.4.m4.1.1.3.3" xref="S5.Thmthm12.p2.4.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.4.m4.1b"><apply id="S5.Thmthm12.p2.4.m4.1.1.cmml" xref="S5.Thmthm12.p2.4.m4.1.1"><eq id="S5.Thmthm12.p2.4.m4.1.1.2.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.2"></eq><apply id="S5.Thmthm12.p2.4.m4.1.1.1.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.1"><times id="S5.Thmthm12.p2.4.m4.1.1.1.2.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.1.2"></times><ci id="S5.Thmthm12.p2.4.m4.1.1.1.3.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.1.3">𝜎</ci><apply id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.2">𝑎</ci><ci id="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="S5.Thmthm12.p2.4.m4.1.1.3.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.4.m4.1.1.3.1.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.3">subscript</csymbol><ci id="S5.Thmthm12.p2.4.m4.1.1.3.2.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.3.2">𝑎</ci><ci id="S5.Thmthm12.p2.4.m4.1.1.3.3.cmml" xref="S5.Thmthm12.p2.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.4.m4.1c">\sigma(a_{i})=a_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.4.m4.1d">italic_σ ( italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for any <math alttext="a_{i}\in\cal A" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.5.m5.1"><semantics id="S5.Thmthm12.p2.5.m5.1a"><mrow id="S5.Thmthm12.p2.5.m5.1.1" xref="S5.Thmthm12.p2.5.m5.1.1.cmml"><msub id="S5.Thmthm12.p2.5.m5.1.1.2" xref="S5.Thmthm12.p2.5.m5.1.1.2.cmml"><mi id="S5.Thmthm12.p2.5.m5.1.1.2.2" xref="S5.Thmthm12.p2.5.m5.1.1.2.2.cmml">a</mi><mi id="S5.Thmthm12.p2.5.m5.1.1.2.3" xref="S5.Thmthm12.p2.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S5.Thmthm12.p2.5.m5.1.1.1" xref="S5.Thmthm12.p2.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.5.m5.1.1.3" xref="S5.Thmthm12.p2.5.m5.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.5.m5.1b"><apply id="S5.Thmthm12.p2.5.m5.1.1.cmml" xref="S5.Thmthm12.p2.5.m5.1.1"><in id="S5.Thmthm12.p2.5.m5.1.1.1.cmml" xref="S5.Thmthm12.p2.5.m5.1.1.1"></in><apply id="S5.Thmthm12.p2.5.m5.1.1.2.cmml" xref="S5.Thmthm12.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.5.m5.1.1.2.1.cmml" xref="S5.Thmthm12.p2.5.m5.1.1.2">subscript</csymbol><ci id="S5.Thmthm12.p2.5.m5.1.1.2.2.cmml" xref="S5.Thmthm12.p2.5.m5.1.1.2.2">𝑎</ci><ci id="S5.Thmthm12.p2.5.m5.1.1.2.3.cmml" xref="S5.Thmthm12.p2.5.m5.1.1.2.3">𝑖</ci></apply><ci id="S5.Thmthm12.p2.5.m5.1.1.3.cmml" xref="S5.Thmthm12.p2.5.m5.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.5.m5.1c">a_{i}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math> and <math alttext="\sigma(b)=\sigma(c)=a" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.6.m6.2"><semantics id="S5.Thmthm12.p2.6.m6.2a"><mrow id="S5.Thmthm12.p2.6.m6.2.3" xref="S5.Thmthm12.p2.6.m6.2.3.cmml"><mrow id="S5.Thmthm12.p2.6.m6.2.3.2" xref="S5.Thmthm12.p2.6.m6.2.3.2.cmml"><mi id="S5.Thmthm12.p2.6.m6.2.3.2.2" xref="S5.Thmthm12.p2.6.m6.2.3.2.2.cmml">σ</mi><mo id="S5.Thmthm12.p2.6.m6.2.3.2.1" xref="S5.Thmthm12.p2.6.m6.2.3.2.1.cmml">⁢</mo><mrow id="S5.Thmthm12.p2.6.m6.2.3.2.3.2" xref="S5.Thmthm12.p2.6.m6.2.3.2.cmml"><mo id="S5.Thmthm12.p2.6.m6.2.3.2.3.2.1" stretchy="false" xref="S5.Thmthm12.p2.6.m6.2.3.2.cmml">(</mo><mi id="S5.Thmthm12.p2.6.m6.1.1" xref="S5.Thmthm12.p2.6.m6.1.1.cmml">b</mi><mo id="S5.Thmthm12.p2.6.m6.2.3.2.3.2.2" stretchy="false" xref="S5.Thmthm12.p2.6.m6.2.3.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm12.p2.6.m6.2.3.3" xref="S5.Thmthm12.p2.6.m6.2.3.3.cmml">=</mo><mrow id="S5.Thmthm12.p2.6.m6.2.3.4" xref="S5.Thmthm12.p2.6.m6.2.3.4.cmml"><mi id="S5.Thmthm12.p2.6.m6.2.3.4.2" xref="S5.Thmthm12.p2.6.m6.2.3.4.2.cmml">σ</mi><mo id="S5.Thmthm12.p2.6.m6.2.3.4.1" xref="S5.Thmthm12.p2.6.m6.2.3.4.1.cmml">⁢</mo><mrow id="S5.Thmthm12.p2.6.m6.2.3.4.3.2" xref="S5.Thmthm12.p2.6.m6.2.3.4.cmml"><mo id="S5.Thmthm12.p2.6.m6.2.3.4.3.2.1" stretchy="false" xref="S5.Thmthm12.p2.6.m6.2.3.4.cmml">(</mo><mi id="S5.Thmthm12.p2.6.m6.2.2" xref="S5.Thmthm12.p2.6.m6.2.2.cmml">c</mi><mo id="S5.Thmthm12.p2.6.m6.2.3.4.3.2.2" stretchy="false" xref="S5.Thmthm12.p2.6.m6.2.3.4.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm12.p2.6.m6.2.3.5" xref="S5.Thmthm12.p2.6.m6.2.3.5.cmml">=</mo><mi id="S5.Thmthm12.p2.6.m6.2.3.6" xref="S5.Thmthm12.p2.6.m6.2.3.6.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.6.m6.2b"><apply id="S5.Thmthm12.p2.6.m6.2.3.cmml" xref="S5.Thmthm12.p2.6.m6.2.3"><and id="S5.Thmthm12.p2.6.m6.2.3a.cmml" xref="S5.Thmthm12.p2.6.m6.2.3"></and><apply id="S5.Thmthm12.p2.6.m6.2.3b.cmml" xref="S5.Thmthm12.p2.6.m6.2.3"><eq id="S5.Thmthm12.p2.6.m6.2.3.3.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.3"></eq><apply id="S5.Thmthm12.p2.6.m6.2.3.2.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.2"><times id="S5.Thmthm12.p2.6.m6.2.3.2.1.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.2.1"></times><ci id="S5.Thmthm12.p2.6.m6.2.3.2.2.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.2.2">𝜎</ci><ci id="S5.Thmthm12.p2.6.m6.1.1.cmml" xref="S5.Thmthm12.p2.6.m6.1.1">𝑏</ci></apply><apply id="S5.Thmthm12.p2.6.m6.2.3.4.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.4"><times id="S5.Thmthm12.p2.6.m6.2.3.4.1.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.4.1"></times><ci id="S5.Thmthm12.p2.6.m6.2.3.4.2.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.4.2">𝜎</ci><ci id="S5.Thmthm12.p2.6.m6.2.2.cmml" xref="S5.Thmthm12.p2.6.m6.2.2">𝑐</ci></apply></apply><apply id="S5.Thmthm12.p2.6.m6.2.3c.cmml" xref="S5.Thmthm12.p2.6.m6.2.3"><eq id="S5.Thmthm12.p2.6.m6.2.3.5.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S5.Thmthm12.p2.6.m6.2.3.4.cmml" id="S5.Thmthm12.p2.6.m6.2.3d.cmml" xref="S5.Thmthm12.p2.6.m6.2.3"></share><ci id="S5.Thmthm12.p2.6.m6.2.3.6.cmml" xref="S5.Thmthm12.p2.6.m6.2.3.6">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.6.m6.2c">\sigma(b)=\sigma(c)=a</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.6.m6.2d">italic_σ ( italic_b ) = italic_σ ( italic_c ) = italic_a</annotation></semantics></math>. We define <math alttext="X=X_{\tau}\cup\cal O(VbW)\cup\cal O(VcW)" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.7.m7.2"><semantics id="S5.Thmthm12.p2.7.m7.2a"><mrow id="S5.Thmthm12.p2.7.m7.2.2" xref="S5.Thmthm12.p2.7.m7.2.2.cmml"><mi id="S5.Thmthm12.p2.7.m7.2.2.4" xref="S5.Thmthm12.p2.7.m7.2.2.4.cmml">X</mi><mo id="S5.Thmthm12.p2.7.m7.2.2.3" xref="S5.Thmthm12.p2.7.m7.2.2.3.cmml">=</mo><mrow id="S5.Thmthm12.p2.7.m7.2.2.2" xref="S5.Thmthm12.p2.7.m7.2.2.2.cmml"><msub id="S5.Thmthm12.p2.7.m7.2.2.2.4" xref="S5.Thmthm12.p2.7.m7.2.2.2.4.cmml"><mi id="S5.Thmthm12.p2.7.m7.2.2.2.4.2" xref="S5.Thmthm12.p2.7.m7.2.2.2.4.2.cmml">X</mi><mi id="S5.Thmthm12.p2.7.m7.2.2.2.4.3" xref="S5.Thmthm12.p2.7.m7.2.2.2.4.3.cmml">τ</mi></msub><mo id="S5.Thmthm12.p2.7.m7.2.2.2.3" xref="S5.Thmthm12.p2.7.m7.2.2.2.3.cmml">∪</mo><mrow id="S5.Thmthm12.p2.7.m7.1.1.1.1" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.7.m7.1.1.1.1.3" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.3.cmml">𝒪</mi><mo id="S5.Thmthm12.p2.7.m7.1.1.1.1.2" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.2.cmml">⁢</mo><mrow id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.cmml"><mo id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.2" stretchy="false" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.2" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.2.cmml">𝒱</mi><mo id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.1" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.3" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.3.cmml">𝒷</mi><mo id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.1a" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.4" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.4.cmml">𝒲</mi></mrow><mo id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.3" stretchy="false" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm12.p2.7.m7.2.2.2.3a" xref="S5.Thmthm12.p2.7.m7.2.2.2.3.cmml">∪</mo><mrow id="S5.Thmthm12.p2.7.m7.2.2.2.2" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.7.m7.2.2.2.2.3" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.3.cmml">𝒪</mi><mo id="S5.Thmthm12.p2.7.m7.2.2.2.2.2" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.2.cmml">⁢</mo><mrow id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.cmml"><mo id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.2" stretchy="false" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.cmml">(</mo><mrow id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.2" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.2.cmml">𝒱</mi><mo id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.1" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.3" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.3.cmml">𝒸</mi><mo id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.1a" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.4" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.4.cmml">𝒲</mi></mrow><mo id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.3" stretchy="false" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.7.m7.2b"><apply id="S5.Thmthm12.p2.7.m7.2.2.cmml" xref="S5.Thmthm12.p2.7.m7.2.2"><eq id="S5.Thmthm12.p2.7.m7.2.2.3.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.3"></eq><ci id="S5.Thmthm12.p2.7.m7.2.2.4.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.4">𝑋</ci><apply id="S5.Thmthm12.p2.7.m7.2.2.2.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2"><union id="S5.Thmthm12.p2.7.m7.2.2.2.3.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.3"></union><apply id="S5.Thmthm12.p2.7.m7.2.2.2.4.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.4"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.7.m7.2.2.2.4.1.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.4">subscript</csymbol><ci id="S5.Thmthm12.p2.7.m7.2.2.2.4.2.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.4.2">𝑋</ci><ci id="S5.Thmthm12.p2.7.m7.2.2.2.4.3.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.4.3">𝜏</ci></apply><apply id="S5.Thmthm12.p2.7.m7.1.1.1.1.cmml" xref="S5.Thmthm12.p2.7.m7.1.1.1.1"><times id="S5.Thmthm12.p2.7.m7.1.1.1.1.2.cmml" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.2"></times><ci id="S5.Thmthm12.p2.7.m7.1.1.1.1.3.cmml" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.3">𝒪</ci><apply id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.cmml" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1"><times id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.1.cmml" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.1"></times><ci id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.2.cmml" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.2">𝒱</ci><ci id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.3.cmml" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.3">𝒷</ci><ci id="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.4.cmml" xref="S5.Thmthm12.p2.7.m7.1.1.1.1.1.1.1.4">𝒲</ci></apply></apply><apply id="S5.Thmthm12.p2.7.m7.2.2.2.2.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.2"><times id="S5.Thmthm12.p2.7.m7.2.2.2.2.2.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.2"></times><ci id="S5.Thmthm12.p2.7.m7.2.2.2.2.3.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.3">𝒪</ci><apply id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1"><times id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.1.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.1"></times><ci id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.2.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.2">𝒱</ci><ci id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.3.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.3">𝒸</ci><ci id="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.4.cmml" xref="S5.Thmthm12.p2.7.m7.2.2.2.2.1.1.1.4">𝒲</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.7.m7.2c">X=X_{\tau}\cup\cal O(VbW)\cup\cal O(VcW)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.7.m7.2d">italic_X = italic_X start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT ∪ caligraphic_O ( caligraphic_V caligraphic_b caligraphic_W ) ∪ caligraphic_O ( caligraphic_V caligraphic_c caligraphic_W )</annotation></semantics></math> and observe that the biinfinite indexed words <math alttext="V\cdot bW" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.8.m8.1"><semantics id="S5.Thmthm12.p2.8.m8.1a"><mrow id="S5.Thmthm12.p2.8.m8.1.1" xref="S5.Thmthm12.p2.8.m8.1.1.cmml"><mrow id="S5.Thmthm12.p2.8.m8.1.1.2" xref="S5.Thmthm12.p2.8.m8.1.1.2.cmml"><mi id="S5.Thmthm12.p2.8.m8.1.1.2.2" xref="S5.Thmthm12.p2.8.m8.1.1.2.2.cmml">V</mi><mo id="S5.Thmthm12.p2.8.m8.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmthm12.p2.8.m8.1.1.2.1.cmml">⋅</mo><mi id="S5.Thmthm12.p2.8.m8.1.1.2.3" xref="S5.Thmthm12.p2.8.m8.1.1.2.3.cmml">b</mi></mrow><mo id="S5.Thmthm12.p2.8.m8.1.1.1" xref="S5.Thmthm12.p2.8.m8.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p2.8.m8.1.1.3" xref="S5.Thmthm12.p2.8.m8.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.8.m8.1b"><apply id="S5.Thmthm12.p2.8.m8.1.1.cmml" xref="S5.Thmthm12.p2.8.m8.1.1"><times id="S5.Thmthm12.p2.8.m8.1.1.1.cmml" xref="S5.Thmthm12.p2.8.m8.1.1.1"></times><apply id="S5.Thmthm12.p2.8.m8.1.1.2.cmml" xref="S5.Thmthm12.p2.8.m8.1.1.2"><ci id="S5.Thmthm12.p2.8.m8.1.1.2.1.cmml" xref="S5.Thmthm12.p2.8.m8.1.1.2.1">⋅</ci><ci id="S5.Thmthm12.p2.8.m8.1.1.2.2.cmml" xref="S5.Thmthm12.p2.8.m8.1.1.2.2">𝑉</ci><ci id="S5.Thmthm12.p2.8.m8.1.1.2.3.cmml" xref="S5.Thmthm12.p2.8.m8.1.1.2.3">𝑏</ci></apply><ci id="S5.Thmthm12.p2.8.m8.1.1.3.cmml" xref="S5.Thmthm12.p2.8.m8.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.8.m8.1c">V\cdot bW</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.8.m8.1d">italic_V ⋅ italic_b italic_W</annotation></semantics></math> and <math alttext="V\cdot cW" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.9.m9.1"><semantics id="S5.Thmthm12.p2.9.m9.1a"><mrow id="S5.Thmthm12.p2.9.m9.1.1" xref="S5.Thmthm12.p2.9.m9.1.1.cmml"><mrow id="S5.Thmthm12.p2.9.m9.1.1.2" xref="S5.Thmthm12.p2.9.m9.1.1.2.cmml"><mi id="S5.Thmthm12.p2.9.m9.1.1.2.2" xref="S5.Thmthm12.p2.9.m9.1.1.2.2.cmml">V</mi><mo id="S5.Thmthm12.p2.9.m9.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmthm12.p2.9.m9.1.1.2.1.cmml">⋅</mo><mi id="S5.Thmthm12.p2.9.m9.1.1.2.3" xref="S5.Thmthm12.p2.9.m9.1.1.2.3.cmml">c</mi></mrow><mo id="S5.Thmthm12.p2.9.m9.1.1.1" xref="S5.Thmthm12.p2.9.m9.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p2.9.m9.1.1.3" xref="S5.Thmthm12.p2.9.m9.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.9.m9.1b"><apply id="S5.Thmthm12.p2.9.m9.1.1.cmml" xref="S5.Thmthm12.p2.9.m9.1.1"><times id="S5.Thmthm12.p2.9.m9.1.1.1.cmml" xref="S5.Thmthm12.p2.9.m9.1.1.1"></times><apply id="S5.Thmthm12.p2.9.m9.1.1.2.cmml" xref="S5.Thmthm12.p2.9.m9.1.1.2"><ci id="S5.Thmthm12.p2.9.m9.1.1.2.1.cmml" xref="S5.Thmthm12.p2.9.m9.1.1.2.1">⋅</ci><ci id="S5.Thmthm12.p2.9.m9.1.1.2.2.cmml" xref="S5.Thmthm12.p2.9.m9.1.1.2.2">𝑉</ci><ci id="S5.Thmthm12.p2.9.m9.1.1.2.3.cmml" xref="S5.Thmthm12.p2.9.m9.1.1.2.3">𝑐</ci></apply><ci id="S5.Thmthm12.p2.9.m9.1.1.3.cmml" xref="S5.Thmthm12.p2.9.m9.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.9.m9.1c">V\cdot cW</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.9.m9.1d">italic_V ⋅ italic_c italic_W</annotation></semantics></math> are sent by <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.10.m10.1"><semantics id="S5.Thmthm12.p2.10.m10.1a"><mi id="S5.Thmthm12.p2.10.m10.1.1" xref="S5.Thmthm12.p2.10.m10.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.10.m10.1b"><ci id="S5.Thmthm12.p2.10.m10.1.1.cmml" xref="S5.Thmthm12.p2.10.m10.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.10.m10.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.10.m10.1d">italic_σ</annotation></semantics></math> both to the same image word <math alttext="V\cdot aW" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.11.m11.1"><semantics id="S5.Thmthm12.p2.11.m11.1a"><mrow id="S5.Thmthm12.p2.11.m11.1.1" xref="S5.Thmthm12.p2.11.m11.1.1.cmml"><mrow id="S5.Thmthm12.p2.11.m11.1.1.2" xref="S5.Thmthm12.p2.11.m11.1.1.2.cmml"><mi id="S5.Thmthm12.p2.11.m11.1.1.2.2" xref="S5.Thmthm12.p2.11.m11.1.1.2.2.cmml">V</mi><mo id="S5.Thmthm12.p2.11.m11.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmthm12.p2.11.m11.1.1.2.1.cmml">⋅</mo><mi id="S5.Thmthm12.p2.11.m11.1.1.2.3" xref="S5.Thmthm12.p2.11.m11.1.1.2.3.cmml">a</mi></mrow><mo id="S5.Thmthm12.p2.11.m11.1.1.1" xref="S5.Thmthm12.p2.11.m11.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p2.11.m11.1.1.3" xref="S5.Thmthm12.p2.11.m11.1.1.3.cmml">W</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.11.m11.1b"><apply id="S5.Thmthm12.p2.11.m11.1.1.cmml" xref="S5.Thmthm12.p2.11.m11.1.1"><times id="S5.Thmthm12.p2.11.m11.1.1.1.cmml" xref="S5.Thmthm12.p2.11.m11.1.1.1"></times><apply id="S5.Thmthm12.p2.11.m11.1.1.2.cmml" xref="S5.Thmthm12.p2.11.m11.1.1.2"><ci id="S5.Thmthm12.p2.11.m11.1.1.2.1.cmml" xref="S5.Thmthm12.p2.11.m11.1.1.2.1">⋅</ci><ci id="S5.Thmthm12.p2.11.m11.1.1.2.2.cmml" xref="S5.Thmthm12.p2.11.m11.1.1.2.2">𝑉</ci><ci id="S5.Thmthm12.p2.11.m11.1.1.2.3.cmml" xref="S5.Thmthm12.p2.11.m11.1.1.2.3">𝑎</ci></apply><ci id="S5.Thmthm12.p2.11.m11.1.1.3.cmml" xref="S5.Thmthm12.p2.11.m11.1.1.3">𝑊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.11.m11.1c">V\cdot aW</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.11.m11.1d">italic_V ⋅ italic_a italic_W</annotation></semantics></math> which is not periodic, so that <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.12.m12.1"><semantics id="S5.Thmthm12.p2.12.m12.1a"><mi id="S5.Thmthm12.p2.12.m12.1.1" xref="S5.Thmthm12.p2.12.m12.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.12.m12.1b"><ci id="S5.Thmthm12.p2.12.m12.1.1.cmml" xref="S5.Thmthm12.p2.12.m12.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.12.m12.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.12.m12.1d">italic_σ</annotation></semantics></math> is not recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.13.m13.1"><semantics id="S5.Thmthm12.p2.13.m13.1a"><mi id="S5.Thmthm12.p2.13.m13.1.1" xref="S5.Thmthm12.p2.13.m13.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.13.m13.1b"><ci id="S5.Thmthm12.p2.13.m13.1.1.cmml" xref="S5.Thmthm12.p2.13.m13.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.13.m13.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.13.m13.1d">italic_X</annotation></semantics></math>. On the other hand, <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.14.m14.1"><semantics id="S5.Thmthm12.p2.14.m14.1a"><mi id="S5.Thmthm12.p2.14.m14.1.1" xref="S5.Thmthm12.p2.14.m14.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.14.m14.1b"><ci id="S5.Thmthm12.p2.14.m14.1.1.cmml" xref="S5.Thmthm12.p2.14.m14.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.14.m14.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.14.m14.1d">italic_X</annotation></semantics></math> is easily seen to be the subshift <math alttext="X=X_{\tau^{\prime}}" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.15.m15.1"><semantics id="S5.Thmthm12.p2.15.m15.1a"><mrow id="S5.Thmthm12.p2.15.m15.1.1" xref="S5.Thmthm12.p2.15.m15.1.1.cmml"><mi id="S5.Thmthm12.p2.15.m15.1.1.2" xref="S5.Thmthm12.p2.15.m15.1.1.2.cmml">X</mi><mo id="S5.Thmthm12.p2.15.m15.1.1.1" xref="S5.Thmthm12.p2.15.m15.1.1.1.cmml">=</mo><msub id="S5.Thmthm12.p2.15.m15.1.1.3" xref="S5.Thmthm12.p2.15.m15.1.1.3.cmml"><mi id="S5.Thmthm12.p2.15.m15.1.1.3.2" xref="S5.Thmthm12.p2.15.m15.1.1.3.2.cmml">X</mi><msup id="S5.Thmthm12.p2.15.m15.1.1.3.3" xref="S5.Thmthm12.p2.15.m15.1.1.3.3.cmml"><mi id="S5.Thmthm12.p2.15.m15.1.1.3.3.2" xref="S5.Thmthm12.p2.15.m15.1.1.3.3.2.cmml">τ</mi><mo id="S5.Thmthm12.p2.15.m15.1.1.3.3.3" xref="S5.Thmthm12.p2.15.m15.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.15.m15.1b"><apply id="S5.Thmthm12.p2.15.m15.1.1.cmml" xref="S5.Thmthm12.p2.15.m15.1.1"><eq id="S5.Thmthm12.p2.15.m15.1.1.1.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.1"></eq><ci id="S5.Thmthm12.p2.15.m15.1.1.2.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.2">𝑋</ci><apply id="S5.Thmthm12.p2.15.m15.1.1.3.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.15.m15.1.1.3.1.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.3">subscript</csymbol><ci id="S5.Thmthm12.p2.15.m15.1.1.3.2.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.3.2">𝑋</ci><apply id="S5.Thmthm12.p2.15.m15.1.1.3.3.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.15.m15.1.1.3.3.1.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm12.p2.15.m15.1.1.3.3.2.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.3.3.2">𝜏</ci><ci id="S5.Thmthm12.p2.15.m15.1.1.3.3.3.cmml" xref="S5.Thmthm12.p2.15.m15.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.15.m15.1c">X=X_{\tau^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.15.m15.1d">italic_X = italic_X start_POSTSUBSCRIPT italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> associated to the substitution <math alttext="\tau^{\prime}:(\{b,c\}\cup\cal A)^{*}\to(\{b,c\}\cup\cal A)^{*}" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.16.m16.6"><semantics id="S5.Thmthm12.p2.16.m16.6a"><mrow id="S5.Thmthm12.p2.16.m16.6.6" xref="S5.Thmthm12.p2.16.m16.6.6.cmml"><msup id="S5.Thmthm12.p2.16.m16.6.6.4" xref="S5.Thmthm12.p2.16.m16.6.6.4.cmml"><mi id="S5.Thmthm12.p2.16.m16.6.6.4.2" xref="S5.Thmthm12.p2.16.m16.6.6.4.2.cmml">τ</mi><mo id="S5.Thmthm12.p2.16.m16.6.6.4.3" xref="S5.Thmthm12.p2.16.m16.6.6.4.3.cmml">′</mo></msup><mo id="S5.Thmthm12.p2.16.m16.6.6.3" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm12.p2.16.m16.6.6.3.cmml">:</mo><mrow id="S5.Thmthm12.p2.16.m16.6.6.2" xref="S5.Thmthm12.p2.16.m16.6.6.2.cmml"><msup id="S5.Thmthm12.p2.16.m16.5.5.1.1" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.cmml"><mrow id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.cmml"><mo id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.2" stretchy="false" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.cmml">(</mo><mrow id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.cmml"><mrow id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.2" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.1.cmml"><mo id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.2.1" stretchy="false" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.1.cmml">{</mo><mi id="S5.Thmthm12.p2.16.m16.1.1" xref="S5.Thmthm12.p2.16.m16.1.1.cmml">b</mi><mo id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.2.2" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.1.cmml">,</mo><mi id="S5.Thmthm12.p2.16.m16.2.2" xref="S5.Thmthm12.p2.16.m16.2.2.cmml">c</mi><mo id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.2.3" stretchy="false" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.1.cmml">}</mo></mrow><mo id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.1" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.1.cmml">∪</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.3" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.3.cmml">𝒜</mi></mrow><mo id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.3" stretchy="false" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmthm12.p2.16.m16.5.5.1.1.3" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.3.cmml">∗</mo></msup><mo id="S5.Thmthm12.p2.16.m16.6.6.2.3" stretchy="false" xref="S5.Thmthm12.p2.16.m16.6.6.2.3.cmml">→</mo><msup id="S5.Thmthm12.p2.16.m16.6.6.2.2" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.cmml"><mrow id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.cmml"><mo id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.2" stretchy="false" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.cmml">(</mo><mrow id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.cmml"><mrow id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.2" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.1.cmml"><mo id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.2.1" stretchy="false" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.16.m16.3.3" xref="S5.Thmthm12.p2.16.m16.3.3.cmml">𝒷</mi><mo id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.2.2" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.16.m16.4.4" xref="S5.Thmthm12.p2.16.m16.4.4.cmml">𝒸</mi><mo id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.2.3" stretchy="false" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.1.cmml">}</mo></mrow><mo id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.1" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.1.cmml">∪</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.3" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.3.cmml">𝒜</mi></mrow><mo id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.3" stretchy="false" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.cmml">)</mo></mrow><mo id="S5.Thmthm12.p2.16.m16.6.6.2.2.3" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.16.m16.6b"><apply id="S5.Thmthm12.p2.16.m16.6.6.cmml" xref="S5.Thmthm12.p2.16.m16.6.6"><ci id="S5.Thmthm12.p2.16.m16.6.6.3.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.3">:</ci><apply id="S5.Thmthm12.p2.16.m16.6.6.4.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.4"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.16.m16.6.6.4.1.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.4">superscript</csymbol><ci id="S5.Thmthm12.p2.16.m16.6.6.4.2.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.4.2">𝜏</ci><ci id="S5.Thmthm12.p2.16.m16.6.6.4.3.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.4.3">′</ci></apply><apply id="S5.Thmthm12.p2.16.m16.6.6.2.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2"><ci id="S5.Thmthm12.p2.16.m16.6.6.2.3.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2.3">→</ci><apply id="S5.Thmthm12.p2.16.m16.5.5.1.1.cmml" xref="S5.Thmthm12.p2.16.m16.5.5.1.1"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.16.m16.5.5.1.1.2.cmml" xref="S5.Thmthm12.p2.16.m16.5.5.1.1">superscript</csymbol><apply id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.cmml" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1"><union id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.1.cmml" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.1"></union><set id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.1.cmml" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.2.2"><ci id="S5.Thmthm12.p2.16.m16.1.1.cmml" xref="S5.Thmthm12.p2.16.m16.1.1">𝑏</ci><ci id="S5.Thmthm12.p2.16.m16.2.2.cmml" xref="S5.Thmthm12.p2.16.m16.2.2">𝑐</ci></set><ci id="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.3.cmml" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.1.1.1.3">𝒜</ci></apply><times id="S5.Thmthm12.p2.16.m16.5.5.1.1.3.cmml" xref="S5.Thmthm12.p2.16.m16.5.5.1.1.3"></times></apply><apply id="S5.Thmthm12.p2.16.m16.6.6.2.2.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2.2"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.16.m16.6.6.2.2.2.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2.2">superscript</csymbol><apply id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1"><union id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.1.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.1"></union><set id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.1.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.2.2"><ci id="S5.Thmthm12.p2.16.m16.3.3.cmml" xref="S5.Thmthm12.p2.16.m16.3.3">𝒷</ci><ci id="S5.Thmthm12.p2.16.m16.4.4.cmml" xref="S5.Thmthm12.p2.16.m16.4.4">𝒸</ci></set><ci id="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.3.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.1.1.1.3">𝒜</ci></apply><times id="S5.Thmthm12.p2.16.m16.6.6.2.2.3.cmml" xref="S5.Thmthm12.p2.16.m16.6.6.2.2.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.16.m16.6c">\tau^{\prime}:(\{b,c\}\cup\cal A)^{*}\to(\{b,c\}\cup\cal A)^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.16.m16.6d">italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT : ( { italic_b , italic_c } ∪ caligraphic_A ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → ( { caligraphic_b , caligraphic_c } ∪ caligraphic_A ) start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> which coincides with <math alttext="\tau" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.17.m17.1"><semantics id="S5.Thmthm12.p2.17.m17.1a"><mi id="S5.Thmthm12.p2.17.m17.1.1" xref="S5.Thmthm12.p2.17.m17.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.17.m17.1b"><ci id="S5.Thmthm12.p2.17.m17.1.1.cmml" xref="S5.Thmthm12.p2.17.m17.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.17.m17.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.17.m17.1d">italic_τ</annotation></semantics></math> on <math alttext="\cal A" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.18.m18.1"><semantics id="S5.Thmthm12.p2.18.m18.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm12.p2.18.m18.1.1" xref="S5.Thmthm12.p2.18.m18.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.18.m18.1b"><ci id="S5.Thmthm12.p2.18.m18.1.1.cmml" xref="S5.Thmthm12.p2.18.m18.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.18.m18.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.18.m18.1d">caligraphic_A</annotation></semantics></math> and sends <math alttext="b" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.19.m19.1"><semantics id="S5.Thmthm12.p2.19.m19.1a"><mi id="S5.Thmthm12.p2.19.m19.1.1" xref="S5.Thmthm12.p2.19.m19.1.1.cmml">b</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.19.m19.1b"><ci id="S5.Thmthm12.p2.19.m19.1.1.cmml" xref="S5.Thmthm12.p2.19.m19.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.19.m19.1c">b</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.19.m19.1d">italic_b</annotation></semantics></math> and <math alttext="c" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.20.m20.1"><semantics id="S5.Thmthm12.p2.20.m20.1a"><mi id="S5.Thmthm12.p2.20.m20.1.1" xref="S5.Thmthm12.p2.20.m20.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.20.m20.1b"><ci id="S5.Thmthm12.p2.20.m20.1.1.cmml" xref="S5.Thmthm12.p2.20.m20.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.20.m20.1c">c</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.20.m20.1d">italic_c</annotation></semantics></math> to <math alttext="vbw" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.21.m21.1"><semantics id="S5.Thmthm12.p2.21.m21.1a"><mrow id="S5.Thmthm12.p2.21.m21.1.1" xref="S5.Thmthm12.p2.21.m21.1.1.cmml"><mi id="S5.Thmthm12.p2.21.m21.1.1.2" xref="S5.Thmthm12.p2.21.m21.1.1.2.cmml">v</mi><mo id="S5.Thmthm12.p2.21.m21.1.1.1" xref="S5.Thmthm12.p2.21.m21.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p2.21.m21.1.1.3" xref="S5.Thmthm12.p2.21.m21.1.1.3.cmml">b</mi><mo id="S5.Thmthm12.p2.21.m21.1.1.1a" xref="S5.Thmthm12.p2.21.m21.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p2.21.m21.1.1.4" xref="S5.Thmthm12.p2.21.m21.1.1.4.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.21.m21.1b"><apply id="S5.Thmthm12.p2.21.m21.1.1.cmml" xref="S5.Thmthm12.p2.21.m21.1.1"><times id="S5.Thmthm12.p2.21.m21.1.1.1.cmml" xref="S5.Thmthm12.p2.21.m21.1.1.1"></times><ci id="S5.Thmthm12.p2.21.m21.1.1.2.cmml" xref="S5.Thmthm12.p2.21.m21.1.1.2">𝑣</ci><ci id="S5.Thmthm12.p2.21.m21.1.1.3.cmml" xref="S5.Thmthm12.p2.21.m21.1.1.3">𝑏</ci><ci id="S5.Thmthm12.p2.21.m21.1.1.4.cmml" xref="S5.Thmthm12.p2.21.m21.1.1.4">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.21.m21.1c">vbw</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.21.m21.1d">italic_v italic_b italic_w</annotation></semantics></math> and <math alttext="vcw" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.22.m22.1"><semantics id="S5.Thmthm12.p2.22.m22.1a"><mrow id="S5.Thmthm12.p2.22.m22.1.1" xref="S5.Thmthm12.p2.22.m22.1.1.cmml"><mi id="S5.Thmthm12.p2.22.m22.1.1.2" xref="S5.Thmthm12.p2.22.m22.1.1.2.cmml">v</mi><mo id="S5.Thmthm12.p2.22.m22.1.1.1" xref="S5.Thmthm12.p2.22.m22.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p2.22.m22.1.1.3" xref="S5.Thmthm12.p2.22.m22.1.1.3.cmml">c</mi><mo id="S5.Thmthm12.p2.22.m22.1.1.1a" xref="S5.Thmthm12.p2.22.m22.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p2.22.m22.1.1.4" xref="S5.Thmthm12.p2.22.m22.1.1.4.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.22.m22.1b"><apply id="S5.Thmthm12.p2.22.m22.1.1.cmml" xref="S5.Thmthm12.p2.22.m22.1.1"><times id="S5.Thmthm12.p2.22.m22.1.1.1.cmml" xref="S5.Thmthm12.p2.22.m22.1.1.1"></times><ci id="S5.Thmthm12.p2.22.m22.1.1.2.cmml" xref="S5.Thmthm12.p2.22.m22.1.1.2">𝑣</ci><ci id="S5.Thmthm12.p2.22.m22.1.1.3.cmml" xref="S5.Thmthm12.p2.22.m22.1.1.3">𝑐</ci><ci id="S5.Thmthm12.p2.22.m22.1.1.4.cmml" xref="S5.Thmthm12.p2.22.m22.1.1.4">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.22.m22.1c">vcw</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.22.m22.1d">italic_v italic_c italic_w</annotation></semantics></math> respectively, for an arbitrary non-empty finite suffix <math alttext="v" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.23.m23.1"><semantics id="S5.Thmthm12.p2.23.m23.1a"><mi id="S5.Thmthm12.p2.23.m23.1.1" xref="S5.Thmthm12.p2.23.m23.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.23.m23.1b"><ci id="S5.Thmthm12.p2.23.m23.1.1.cmml" xref="S5.Thmthm12.p2.23.m23.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.23.m23.1c">v</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.23.m23.1d">italic_v</annotation></semantics></math> of <math alttext="V" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.24.m24.1"><semantics id="S5.Thmthm12.p2.24.m24.1a"><mi id="S5.Thmthm12.p2.24.m24.1.1" xref="S5.Thmthm12.p2.24.m24.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.24.m24.1b"><ci id="S5.Thmthm12.p2.24.m24.1.1.cmml" xref="S5.Thmthm12.p2.24.m24.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.24.m24.1c">V</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.24.m24.1d">italic_V</annotation></semantics></math> and prefix <math alttext="w" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.25.m25.1"><semantics id="S5.Thmthm12.p2.25.m25.1a"><mi id="S5.Thmthm12.p2.25.m25.1.1" xref="S5.Thmthm12.p2.25.m25.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.25.m25.1b"><ci id="S5.Thmthm12.p2.25.m25.1.1.cmml" xref="S5.Thmthm12.p2.25.m25.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.25.m25.1c">w</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.25.m25.1d">italic_w</annotation></semantics></math> of <math alttext="W" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.26.m26.1"><semantics id="S5.Thmthm12.p2.26.m26.1a"><mi id="S5.Thmthm12.p2.26.m26.1.1" xref="S5.Thmthm12.p2.26.m26.1.1.cmml">W</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.26.m26.1b"><ci id="S5.Thmthm12.p2.26.m26.1.1.cmml" xref="S5.Thmthm12.p2.26.m26.1.1">𝑊</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.26.m26.1c">W</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.26.m26.1d">italic_W</annotation></semantics></math>. It then follows from Corollary 3.5 (1) of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib2" title="">2</a>]</cite> that <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.27.m27.1"><semantics id="S5.Thmthm12.p2.27.m27.1a"><mi id="S5.Thmthm12.p2.27.m27.1.1" xref="S5.Thmthm12.p2.27.m27.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.27.m27.1b"><ci id="S5.Thmthm12.p2.27.m27.1.1.cmml" xref="S5.Thmthm12.p2.27.m27.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.27.m27.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.27.m27.1d">italic_X</annotation></semantics></math> is uniquely ergodic, so that <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S5.Thmthm12.p2.28.m28.1"><semantics id="S5.Thmthm12.p2.28.m28.1a"><mrow id="S5.Thmthm12.p2.28.m28.1.1" xref="S5.Thmthm12.p2.28.m28.1.1.cmml"><msub id="S5.Thmthm12.p2.28.m28.1.1.2" xref="S5.Thmthm12.p2.28.m28.1.1.2.cmml"><mi id="S5.Thmthm12.p2.28.m28.1.1.2.2" xref="S5.Thmthm12.p2.28.m28.1.1.2.2.cmml">σ</mi><mi id="S5.Thmthm12.p2.28.m28.1.1.2.3" xref="S5.Thmthm12.p2.28.m28.1.1.2.3.cmml">X</mi></msub><mo id="S5.Thmthm12.p2.28.m28.1.1.1" xref="S5.Thmthm12.p2.28.m28.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm12.p2.28.m28.1.1.3" xref="S5.Thmthm12.p2.28.m28.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm12.p2.28.m28.1b"><apply id="S5.Thmthm12.p2.28.m28.1.1.cmml" xref="S5.Thmthm12.p2.28.m28.1.1"><times id="S5.Thmthm12.p2.28.m28.1.1.1.cmml" xref="S5.Thmthm12.p2.28.m28.1.1.1"></times><apply id="S5.Thmthm12.p2.28.m28.1.1.2.cmml" xref="S5.Thmthm12.p2.28.m28.1.1.2"><csymbol cd="ambiguous" id="S5.Thmthm12.p2.28.m28.1.1.2.1.cmml" xref="S5.Thmthm12.p2.28.m28.1.1.2">subscript</csymbol><ci id="S5.Thmthm12.p2.28.m28.1.1.2.2.cmml" xref="S5.Thmthm12.p2.28.m28.1.1.2.2">𝜎</ci><ci id="S5.Thmthm12.p2.28.m28.1.1.2.3.cmml" xref="S5.Thmthm12.p2.28.m28.1.1.2.3">𝑋</ci></apply><ci id="S5.Thmthm12.p2.28.m28.1.1.3.cmml" xref="S5.Thmthm12.p2.28.m28.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm12.p2.28.m28.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm12.p2.28.m28.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math> is automatically injective.</p> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S5.Thmthm13"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm13.1.1.1">Remark 5.13</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm13.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm13.p1"> <p class="ltx_p" id="S5.Thmthm13.p1.10">(1) From the examples in the proof of Proposition 5.10 and from Remark 5.11 one may get the impression that the only possibility, for a subshift <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.1.m1.1"><semantics id="S5.Thmthm13.p1.1.m1.1a"><mi id="S5.Thmthm13.p1.1.m1.1.1" xref="S5.Thmthm13.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.1.m1.1b"><ci id="S5.Thmthm13.p1.1.m1.1.1.cmml" xref="S5.Thmthm13.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.1.m1.1d">italic_X</annotation></semantics></math> and a morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.2.m2.1"><semantics id="S5.Thmthm13.p1.2.m2.1a"><mi id="S5.Thmthm13.p1.2.m2.1.1" xref="S5.Thmthm13.p1.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.2.m2.1b"><ci id="S5.Thmthm13.p1.2.m2.1.1.cmml" xref="S5.Thmthm13.p1.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.2.m2.1d">italic_σ</annotation></semantics></math> which is not recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.3.m3.1"><semantics id="S5.Thmthm13.p1.3.m3.1a"><mi id="S5.Thmthm13.p1.3.m3.1.1" xref="S5.Thmthm13.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.3.m3.1b"><ci id="S5.Thmthm13.p1.3.m3.1.1.cmml" xref="S5.Thmthm13.p1.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.3.m3.1d">italic_X</annotation></semantics></math> while its associated measure transfer map <math alttext="\sigma M_{X}" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.4.m4.1"><semantics id="S5.Thmthm13.p1.4.m4.1a"><mrow id="S5.Thmthm13.p1.4.m4.1.1" xref="S5.Thmthm13.p1.4.m4.1.1.cmml"><mi id="S5.Thmthm13.p1.4.m4.1.1.2" xref="S5.Thmthm13.p1.4.m4.1.1.2.cmml">σ</mi><mo id="S5.Thmthm13.p1.4.m4.1.1.1" xref="S5.Thmthm13.p1.4.m4.1.1.1.cmml">⁢</mo><msub id="S5.Thmthm13.p1.4.m4.1.1.3" xref="S5.Thmthm13.p1.4.m4.1.1.3.cmml"><mi id="S5.Thmthm13.p1.4.m4.1.1.3.2" xref="S5.Thmthm13.p1.4.m4.1.1.3.2.cmml">M</mi><mi id="S5.Thmthm13.p1.4.m4.1.1.3.3" xref="S5.Thmthm13.p1.4.m4.1.1.3.3.cmml">X</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.4.m4.1b"><apply id="S5.Thmthm13.p1.4.m4.1.1.cmml" xref="S5.Thmthm13.p1.4.m4.1.1"><times id="S5.Thmthm13.p1.4.m4.1.1.1.cmml" xref="S5.Thmthm13.p1.4.m4.1.1.1"></times><ci id="S5.Thmthm13.p1.4.m4.1.1.2.cmml" xref="S5.Thmthm13.p1.4.m4.1.1.2">𝜎</ci><apply id="S5.Thmthm13.p1.4.m4.1.1.3.cmml" xref="S5.Thmthm13.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm13.p1.4.m4.1.1.3.1.cmml" xref="S5.Thmthm13.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.Thmthm13.p1.4.m4.1.1.3.2.cmml" xref="S5.Thmthm13.p1.4.m4.1.1.3.2">𝑀</ci><ci id="S5.Thmthm13.p1.4.m4.1.1.3.3.cmml" xref="S5.Thmthm13.p1.4.m4.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.4.m4.1c">\sigma M_{X}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.4.m4.1d">italic_σ italic_M start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> is injective, can occur if there is no invariant measure <math alttext="\mu\in\cal M(X)" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.5.m5.1"><semantics id="S5.Thmthm13.p1.5.m5.1a"><mrow id="S5.Thmthm13.p1.5.m5.1.2" xref="S5.Thmthm13.p1.5.m5.1.2.cmml"><mi id="S5.Thmthm13.p1.5.m5.1.2.2" xref="S5.Thmthm13.p1.5.m5.1.2.2.cmml">μ</mi><mo id="S5.Thmthm13.p1.5.m5.1.2.1" xref="S5.Thmthm13.p1.5.m5.1.2.1.cmml">∈</mo><mrow id="S5.Thmthm13.p1.5.m5.1.2.3" xref="S5.Thmthm13.p1.5.m5.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm13.p1.5.m5.1.2.3.2" xref="S5.Thmthm13.p1.5.m5.1.2.3.2.cmml">ℳ</mi><mo id="S5.Thmthm13.p1.5.m5.1.2.3.1" xref="S5.Thmthm13.p1.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S5.Thmthm13.p1.5.m5.1.2.3.3.2" xref="S5.Thmthm13.p1.5.m5.1.2.3.cmml"><mo id="S5.Thmthm13.p1.5.m5.1.2.3.3.2.1" stretchy="false" xref="S5.Thmthm13.p1.5.m5.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm13.p1.5.m5.1.1" xref="S5.Thmthm13.p1.5.m5.1.1.cmml">𝒳</mi><mo id="S5.Thmthm13.p1.5.m5.1.2.3.3.2.2" stretchy="false" xref="S5.Thmthm13.p1.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.5.m5.1b"><apply id="S5.Thmthm13.p1.5.m5.1.2.cmml" xref="S5.Thmthm13.p1.5.m5.1.2"><in id="S5.Thmthm13.p1.5.m5.1.2.1.cmml" xref="S5.Thmthm13.p1.5.m5.1.2.1"></in><ci id="S5.Thmthm13.p1.5.m5.1.2.2.cmml" xref="S5.Thmthm13.p1.5.m5.1.2.2">𝜇</ci><apply id="S5.Thmthm13.p1.5.m5.1.2.3.cmml" xref="S5.Thmthm13.p1.5.m5.1.2.3"><times id="S5.Thmthm13.p1.5.m5.1.2.3.1.cmml" xref="S5.Thmthm13.p1.5.m5.1.2.3.1"></times><ci id="S5.Thmthm13.p1.5.m5.1.2.3.2.cmml" xref="S5.Thmthm13.p1.5.m5.1.2.3.2">ℳ</ci><ci id="S5.Thmthm13.p1.5.m5.1.1.cmml" xref="S5.Thmthm13.p1.5.m5.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.5.m5.1c">\mu\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.5.m5.1d">italic_μ ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math> with <math alttext="\mbox{Supp}(\mu)=X" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.6.m6.1"><semantics id="S5.Thmthm13.p1.6.m6.1a"><mrow id="S5.Thmthm13.p1.6.m6.1.2" xref="S5.Thmthm13.p1.6.m6.1.2.cmml"><mrow id="S5.Thmthm13.p1.6.m6.1.2.2" xref="S5.Thmthm13.p1.6.m6.1.2.2.cmml"><mtext id="S5.Thmthm13.p1.6.m6.1.2.2.2" xref="S5.Thmthm13.p1.6.m6.1.2.2.2a.cmml">Supp</mtext><mo id="S5.Thmthm13.p1.6.m6.1.2.2.1" xref="S5.Thmthm13.p1.6.m6.1.2.2.1.cmml">⁢</mo><mrow id="S5.Thmthm13.p1.6.m6.1.2.2.3.2" xref="S5.Thmthm13.p1.6.m6.1.2.2.cmml"><mo id="S5.Thmthm13.p1.6.m6.1.2.2.3.2.1" stretchy="false" xref="S5.Thmthm13.p1.6.m6.1.2.2.cmml">(</mo><mi id="S5.Thmthm13.p1.6.m6.1.1" xref="S5.Thmthm13.p1.6.m6.1.1.cmml">μ</mi><mo id="S5.Thmthm13.p1.6.m6.1.2.2.3.2.2" stretchy="false" xref="S5.Thmthm13.p1.6.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm13.p1.6.m6.1.2.1" xref="S5.Thmthm13.p1.6.m6.1.2.1.cmml">=</mo><mi id="S5.Thmthm13.p1.6.m6.1.2.3" xref="S5.Thmthm13.p1.6.m6.1.2.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.6.m6.1b"><apply id="S5.Thmthm13.p1.6.m6.1.2.cmml" xref="S5.Thmthm13.p1.6.m6.1.2"><eq id="S5.Thmthm13.p1.6.m6.1.2.1.cmml" xref="S5.Thmthm13.p1.6.m6.1.2.1"></eq><apply id="S5.Thmthm13.p1.6.m6.1.2.2.cmml" xref="S5.Thmthm13.p1.6.m6.1.2.2"><times id="S5.Thmthm13.p1.6.m6.1.2.2.1.cmml" xref="S5.Thmthm13.p1.6.m6.1.2.2.1"></times><ci id="S5.Thmthm13.p1.6.m6.1.2.2.2a.cmml" xref="S5.Thmthm13.p1.6.m6.1.2.2.2"><mtext id="S5.Thmthm13.p1.6.m6.1.2.2.2.cmml" xref="S5.Thmthm13.p1.6.m6.1.2.2.2">Supp</mtext></ci><ci id="S5.Thmthm13.p1.6.m6.1.1.cmml" xref="S5.Thmthm13.p1.6.m6.1.1">𝜇</ci></apply><ci id="S5.Thmthm13.p1.6.m6.1.2.3.cmml" xref="S5.Thmthm13.p1.6.m6.1.2.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.6.m6.1c">\mbox{Supp}(\mu)=X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.6.m6.1d">Supp ( italic_μ ) = italic_X</annotation></semantics></math>. This impression, however, is treacherous: In Example 4.3 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib5" title="">5</a>]</cite> a uniquely ergodic infinite minimal subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.7.m7.1"><semantics id="S5.Thmthm13.p1.7.m7.1a"><mrow id="S5.Thmthm13.p1.7.m7.1.1" xref="S5.Thmthm13.p1.7.m7.1.1.cmml"><mi id="S5.Thmthm13.p1.7.m7.1.1.2" xref="S5.Thmthm13.p1.7.m7.1.1.2.cmml">X</mi><mo id="S5.Thmthm13.p1.7.m7.1.1.1" xref="S5.Thmthm13.p1.7.m7.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm13.p1.7.m7.1.1.3" xref="S5.Thmthm13.p1.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm13.p1.7.m7.1.1.3.2" xref="S5.Thmthm13.p1.7.m7.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm13.p1.7.m7.1.1.3.3" xref="S5.Thmthm13.p1.7.m7.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.7.m7.1b"><apply id="S5.Thmthm13.p1.7.m7.1.1.cmml" xref="S5.Thmthm13.p1.7.m7.1.1"><subset id="S5.Thmthm13.p1.7.m7.1.1.1.cmml" xref="S5.Thmthm13.p1.7.m7.1.1.1"></subset><ci id="S5.Thmthm13.p1.7.m7.1.1.2.cmml" xref="S5.Thmthm13.p1.7.m7.1.1.2">𝑋</ci><apply id="S5.Thmthm13.p1.7.m7.1.1.3.cmml" xref="S5.Thmthm13.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm13.p1.7.m7.1.1.3.1.cmml" xref="S5.Thmthm13.p1.7.m7.1.1.3">superscript</csymbol><ci id="S5.Thmthm13.p1.7.m7.1.1.3.2.cmml" xref="S5.Thmthm13.p1.7.m7.1.1.3.2">𝒜</ci><ci id="S5.Thmthm13.p1.7.m7.1.1.3.3.cmml" xref="S5.Thmthm13.p1.7.m7.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.7.m7.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.7.m7.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> (indeed the substitution subshift of a primitive substitution) is exhibited as well as a morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.8.m8.1"><semantics id="S5.Thmthm13.p1.8.m8.1a"><mi id="S5.Thmthm13.p1.8.m8.1.1" xref="S5.Thmthm13.p1.8.m8.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.8.m8.1b"><ci id="S5.Thmthm13.p1.8.m8.1.1.cmml" xref="S5.Thmthm13.p1.8.m8.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.8.m8.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.8.m8.1d">italic_σ</annotation></semantics></math> which is <math alttext="2:1" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.9.m9.1"><semantics id="S5.Thmthm13.p1.9.m9.1a"><mrow id="S5.Thmthm13.p1.9.m9.1.1" xref="S5.Thmthm13.p1.9.m9.1.1.cmml"><mn id="S5.Thmthm13.p1.9.m9.1.1.2" xref="S5.Thmthm13.p1.9.m9.1.1.2.cmml">2</mn><mo id="S5.Thmthm13.p1.9.m9.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm13.p1.9.m9.1.1.1.cmml">:</mo><mn id="S5.Thmthm13.p1.9.m9.1.1.3" xref="S5.Thmthm13.p1.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.9.m9.1b"><apply id="S5.Thmthm13.p1.9.m9.1.1.cmml" xref="S5.Thmthm13.p1.9.m9.1.1"><ci id="S5.Thmthm13.p1.9.m9.1.1.1.cmml" xref="S5.Thmthm13.p1.9.m9.1.1.1">:</ci><cn id="S5.Thmthm13.p1.9.m9.1.1.2.cmml" type="integer" xref="S5.Thmthm13.p1.9.m9.1.1.2">2</cn><cn id="S5.Thmthm13.p1.9.m9.1.1.3.cmml" type="integer" xref="S5.Thmthm13.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.9.m9.1c">2:1</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.9.m9.1d">2 : 1</annotation></semantics></math> on shift-orbits throughout all of <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm13.p1.10.m10.1"><semantics id="S5.Thmthm13.p1.10.m10.1a"><mi id="S5.Thmthm13.p1.10.m10.1.1" xref="S5.Thmthm13.p1.10.m10.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p1.10.m10.1b"><ci id="S5.Thmthm13.p1.10.m10.1.1.cmml" xref="S5.Thmthm13.p1.10.m10.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p1.10.m10.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p1.10.m10.1d">italic_X</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S5.Thmthm13.p2"> <p class="ltx_p" id="S5.Thmthm13.p2.3">(2) If one attempts to find additional conditions on <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm13.p2.1.m1.1"><semantics id="S5.Thmthm13.p2.1.m1.1a"><mi id="S5.Thmthm13.p2.1.m1.1.1" xref="S5.Thmthm13.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p2.1.m1.1b"><ci id="S5.Thmthm13.p2.1.m1.1.1.cmml" xref="S5.Thmthm13.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p2.1.m1.1d">italic_X</annotation></semantics></math> and <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm13.p2.2.m2.1"><semantics id="S5.Thmthm13.p2.2.m2.1a"><mi id="S5.Thmthm13.p2.2.m2.1.1" xref="S5.Thmthm13.p2.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p2.2.m2.1b"><ci id="S5.Thmthm13.p2.2.m2.1.1.cmml" xref="S5.Thmthm13.p2.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p2.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p2.2.m2.1d">italic_σ</annotation></semantics></math> which ensure that the two properties “recognizable for aperiodic points” and “injectivity of the measure transfer map <math alttext="\sigma_{X}M\," class="ltx_Math" display="inline" id="S5.Thmthm13.p2.3.m3.1"><semantics id="S5.Thmthm13.p2.3.m3.1a"><mrow id="S5.Thmthm13.p2.3.m3.1.1" xref="S5.Thmthm13.p2.3.m3.1.1.cmml"><msub id="S5.Thmthm13.p2.3.m3.1.1.2" xref="S5.Thmthm13.p2.3.m3.1.1.2.cmml"><mi id="S5.Thmthm13.p2.3.m3.1.1.2.2" xref="S5.Thmthm13.p2.3.m3.1.1.2.2.cmml">σ</mi><mi id="S5.Thmthm13.p2.3.m3.1.1.2.3" xref="S5.Thmthm13.p2.3.m3.1.1.2.3.cmml">X</mi></msub><mo id="S5.Thmthm13.p2.3.m3.1.1.1" xref="S5.Thmthm13.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S5.Thmthm13.p2.3.m3.1.1.3" xref="S5.Thmthm13.p2.3.m3.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p2.3.m3.1b"><apply id="S5.Thmthm13.p2.3.m3.1.1.cmml" xref="S5.Thmthm13.p2.3.m3.1.1"><times id="S5.Thmthm13.p2.3.m3.1.1.1.cmml" xref="S5.Thmthm13.p2.3.m3.1.1.1"></times><apply id="S5.Thmthm13.p2.3.m3.1.1.2.cmml" xref="S5.Thmthm13.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.Thmthm13.p2.3.m3.1.1.2.1.cmml" xref="S5.Thmthm13.p2.3.m3.1.1.2">subscript</csymbol><ci id="S5.Thmthm13.p2.3.m3.1.1.2.2.cmml" xref="S5.Thmthm13.p2.3.m3.1.1.2.2">𝜎</ci><ci id="S5.Thmthm13.p2.3.m3.1.1.2.3.cmml" xref="S5.Thmthm13.p2.3.m3.1.1.2.3">𝑋</ci></apply><ci id="S5.Thmthm13.p2.3.m3.1.1.3.cmml" xref="S5.Thmthm13.p2.3.m3.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p2.3.m3.1c">\sigma_{X}M\,</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p2.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math>” become equivalent, one should be aware of the following:</p> </div> <div class="ltx_para" id="S5.Thmthm13.p3"> <p class="ltx_p" id="S5.Thmthm13.p3.5">If <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm13.p3.1.m1.1"><semantics id="S5.Thmthm13.p3.1.m1.1a"><mi id="S5.Thmthm13.p3.1.m1.1.1" xref="S5.Thmthm13.p3.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p3.1.m1.1b"><ci id="S5.Thmthm13.p3.1.m1.1.1.cmml" xref="S5.Thmthm13.p3.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p3.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p3.1.m1.1d">italic_σ</annotation></semantics></math> is shift-orbit injective on <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm13.p3.2.m2.1"><semantics id="S5.Thmthm13.p3.2.m2.1a"><mi id="S5.Thmthm13.p3.2.m2.1.1" xref="S5.Thmthm13.p3.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p3.2.m2.1b"><ci id="S5.Thmthm13.p3.2.m2.1.1.cmml" xref="S5.Thmthm13.p3.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p3.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p3.2.m2.1d">italic_X</annotation></semantics></math>, then we have shown above that both properties are satisfied. If, on the other hand, two distinct shift-orbits <math alttext="\cal O({\bf x})\neq\cal O({\bf x^{\prime}})" class="ltx_Math" display="inline" id="S5.Thmthm13.p3.3.m3.2"><semantics id="S5.Thmthm13.p3.3.m3.2a"><mrow id="S5.Thmthm13.p3.3.m3.2.2" xref="S5.Thmthm13.p3.3.m3.2.2.cmml"><mrow id="S5.Thmthm13.p3.3.m3.2.2.3" xref="S5.Thmthm13.p3.3.m3.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm13.p3.3.m3.2.2.3.2" xref="S5.Thmthm13.p3.3.m3.2.2.3.2.cmml">𝒪</mi><mo id="S5.Thmthm13.p3.3.m3.2.2.3.1" xref="S5.Thmthm13.p3.3.m3.2.2.3.1.cmml">⁢</mo><mrow id="S5.Thmthm13.p3.3.m3.2.2.3.3.2" xref="S5.Thmthm13.p3.3.m3.2.2.3.cmml"><mo id="S5.Thmthm13.p3.3.m3.2.2.3.3.2.1" stretchy="false" xref="S5.Thmthm13.p3.3.m3.2.2.3.cmml">(</mo><mi id="S5.Thmthm13.p3.3.m3.1.1" xref="S5.Thmthm13.p3.3.m3.1.1.cmml">𝐱</mi><mo id="S5.Thmthm13.p3.3.m3.2.2.3.3.2.2" stretchy="false" xref="S5.Thmthm13.p3.3.m3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm13.p3.3.m3.2.2.2" 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id="S5.Thmthm13.p3.3.m3.2.2.cmml" xref="S5.Thmthm13.p3.3.m3.2.2"><neq id="S5.Thmthm13.p3.3.m3.2.2.2.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.2"></neq><apply id="S5.Thmthm13.p3.3.m3.2.2.3.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.3"><times id="S5.Thmthm13.p3.3.m3.2.2.3.1.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.3.1"></times><ci id="S5.Thmthm13.p3.3.m3.2.2.3.2.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.3.2">𝒪</ci><ci id="S5.Thmthm13.p3.3.m3.1.1.cmml" xref="S5.Thmthm13.p3.3.m3.1.1">𝐱</ci></apply><apply id="S5.Thmthm13.p3.3.m3.2.2.1.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.1"><times id="S5.Thmthm13.p3.3.m3.2.2.1.2.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.1.2"></times><ci id="S5.Thmthm13.p3.3.m3.2.2.1.3.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.1.3">𝒪</ci><apply id="S5.Thmthm13.p3.3.m3.2.2.1.1.1.1.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmthm13.p3.3.m3.2.2.1.1.1.1.1.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.1.1.1">superscript</csymbol><ci id="S5.Thmthm13.p3.3.m3.2.2.1.1.1.1.2.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.1.1.1.1.2">𝐱</ci><ci id="S5.Thmthm13.p3.3.m3.2.2.1.1.1.1.3.cmml" xref="S5.Thmthm13.p3.3.m3.2.2.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p3.3.m3.2c">\cal O({\bf x})\neq\cal O({\bf x^{\prime}})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p3.3.m3.2d">caligraphic_O ( bold_x ) ≠ caligraphic_O ( bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> of <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmthm13.p3.4.m4.1"><semantics id="S5.Thmthm13.p3.4.m4.1a"><mi id="S5.Thmthm13.p3.4.m4.1.1" xref="S5.Thmthm13.p3.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p3.4.m4.1b"><ci id="S5.Thmthm13.p3.4.m4.1.1.cmml" xref="S5.Thmthm13.p3.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p3.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p3.4.m4.1d">italic_X</annotation></semantics></math> have the same <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.Thmthm13.p3.5.m5.1"><semantics id="S5.Thmthm13.p3.5.m5.1a"><mi id="S5.Thmthm13.p3.5.m5.1.1" xref="S5.Thmthm13.p3.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmthm13.p3.5.m5.1b"><ci id="S5.Thmthm13.p3.5.m5.1.1.cmml" xref="S5.Thmthm13.p3.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm13.p3.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm13.p3.5.m5.1d">italic_σ</annotation></semantics></math>-image, then we observe:</p> <ol class="ltx_enumerate" id="S5.I4"> <li class="ltx_item" id="S5.I4.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S5.I4.ix1.p1"> <p class="ltx_p" id="S5.I4.ix1.p1.1">The first property is violated if and only if the image orbit <math alttext="\cal O({\sigma(\bf x}))=\cal O(\sigma({\bf x^{\prime}}))" class="ltx_Math" display="inline" id="S5.I4.ix1.p1.1.m1.3"><semantics id="S5.I4.ix1.p1.1.m1.3a"><mrow id="S5.I4.ix1.p1.1.m1.3.3" xref="S5.I4.ix1.p1.1.m1.3.3.cmml"><mrow id="S5.I4.ix1.p1.1.m1.2.2.1" xref="S5.I4.ix1.p1.1.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix1.p1.1.m1.2.2.1.3" xref="S5.I4.ix1.p1.1.m1.2.2.1.3.cmml">𝒪</mi><mo id="S5.I4.ix1.p1.1.m1.2.2.1.2" xref="S5.I4.ix1.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S5.I4.ix1.p1.1.m1.2.2.1.1.1" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.cmml"><mi id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.2" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.2.cmml">σ</mi><mo id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.1" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.3.2" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mi id="S5.I4.ix1.p1.1.m1.1.1" xref="S5.I4.ix1.p1.1.m1.1.1.cmml">𝐱</mi><mo id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.3" stretchy="false" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I4.ix1.p1.1.m1.3.3.3" xref="S5.I4.ix1.p1.1.m1.3.3.3.cmml">=</mo><mrow id="S5.I4.ix1.p1.1.m1.3.3.2" xref="S5.I4.ix1.p1.1.m1.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix1.p1.1.m1.3.3.2.3" xref="S5.I4.ix1.p1.1.m1.3.3.2.3.cmml">𝒪</mi><mo id="S5.I4.ix1.p1.1.m1.3.3.2.2" xref="S5.I4.ix1.p1.1.m1.3.3.2.2.cmml">⁢</mo><mrow id="S5.I4.ix1.p1.1.m1.3.3.2.1.1" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.cmml"><mo id="S5.I4.ix1.p1.1.m1.3.3.2.1.1.2" stretchy="false" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.cmml">(</mo><mrow 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xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.ix1.p1.1.m1.3b"><apply id="S5.I4.ix1.p1.1.m1.3.3.cmml" xref="S5.I4.ix1.p1.1.m1.3.3"><eq id="S5.I4.ix1.p1.1.m1.3.3.3.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.3"></eq><apply id="S5.I4.ix1.p1.1.m1.2.2.1.cmml" xref="S5.I4.ix1.p1.1.m1.2.2.1"><times id="S5.I4.ix1.p1.1.m1.2.2.1.2.cmml" xref="S5.I4.ix1.p1.1.m1.2.2.1.2"></times><ci id="S5.I4.ix1.p1.1.m1.2.2.1.3.cmml" xref="S5.I4.ix1.p1.1.m1.2.2.1.3">𝒪</ci><apply id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.cmml" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1"><times id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.1"></times><ci id="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="S5.I4.ix1.p1.1.m1.2.2.1.1.1.1.2">𝜎</ci><ci id="S5.I4.ix1.p1.1.m1.1.1.cmml" xref="S5.I4.ix1.p1.1.m1.1.1">𝐱</ci></apply></apply><apply id="S5.I4.ix1.p1.1.m1.3.3.2.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2"><times id="S5.I4.ix1.p1.1.m1.3.3.2.2.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.2"></times><ci id="S5.I4.ix1.p1.1.m1.3.3.2.3.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.3">𝒪</ci><apply id="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1"><times id="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.2.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.2"></times><ci id="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.3.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.3">𝜎</ci><apply id="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.1.1.1.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.1.1.1.1.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.1.1">superscript</csymbol><ci id="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.1.1.1.2.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.1.1.1.2">𝐱</ci><ci id="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.1.1.1.3.cmml" xref="S5.I4.ix1.p1.1.m1.3.3.2.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix1.p1.1.m1.3c">\cal O({\sigma(\bf x}))=\cal O(\sigma({\bf x^{\prime}}))</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix1.p1.1.m1.3d">caligraphic_O ( italic_σ ( bold_x ) ) = caligraphic_O ( italic_σ ( bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) )</annotation></semantics></math> is not periodic.</p> </div> </li> <li class="ltx_item" id="S5.I4.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S5.I4.ix2.p1"> <p class="ltx_p" id="S5.I4.ix2.p1.9">For the second property we consider the subshifts <math alttext="\overline{\cal O({\bf x})}" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.1.m1.1"><semantics id="S5.I4.ix2.p1.1.m1.1a"><mover accent="true" id="S5.I4.ix2.p1.1.m1.1.1" xref="S5.I4.ix2.p1.1.m1.1.1.cmml"><mrow id="S5.I4.ix2.p1.1.m1.1.1.1" xref="S5.I4.ix2.p1.1.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.1.m1.1.1.1.3" xref="S5.I4.ix2.p1.1.m1.1.1.1.3.cmml">𝒪</mi><mo id="S5.I4.ix2.p1.1.m1.1.1.1.2" xref="S5.I4.ix2.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.1.m1.1.1.1.4.2" xref="S5.I4.ix2.p1.1.m1.1.1.1.cmml"><mo id="S5.I4.ix2.p1.1.m1.1.1.1.4.2.1" stretchy="false" xref="S5.I4.ix2.p1.1.m1.1.1.1.cmml">(</mo><mi id="S5.I4.ix2.p1.1.m1.1.1.1.1" xref="S5.I4.ix2.p1.1.m1.1.1.1.1.cmml">𝐱</mi><mo id="S5.I4.ix2.p1.1.m1.1.1.1.4.2.2" stretchy="false" xref="S5.I4.ix2.p1.1.m1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I4.ix2.p1.1.m1.1.1.2" xref="S5.I4.ix2.p1.1.m1.1.1.2.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.1.m1.1b"><apply id="S5.I4.ix2.p1.1.m1.1.1.cmml" xref="S5.I4.ix2.p1.1.m1.1.1"><ci id="S5.I4.ix2.p1.1.m1.1.1.2.cmml" xref="S5.I4.ix2.p1.1.m1.1.1.2">¯</ci><apply id="S5.I4.ix2.p1.1.m1.1.1.1.cmml" xref="S5.I4.ix2.p1.1.m1.1.1.1"><times id="S5.I4.ix2.p1.1.m1.1.1.1.2.cmml" xref="S5.I4.ix2.p1.1.m1.1.1.1.2"></times><ci id="S5.I4.ix2.p1.1.m1.1.1.1.3.cmml" xref="S5.I4.ix2.p1.1.m1.1.1.1.3">𝒪</ci><ci id="S5.I4.ix2.p1.1.m1.1.1.1.1.cmml" xref="S5.I4.ix2.p1.1.m1.1.1.1.1">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.1.m1.1c">\overline{\cal O({\bf x})}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.1.m1.1d">over¯ start_ARG caligraphic_O ( bold_x ) end_ARG</annotation></semantics></math> and <math alttext="\overline{\cal O({\bf x^{\prime}})}" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.2.m2.1"><semantics id="S5.I4.ix2.p1.2.m2.1a"><mover accent="true" id="S5.I4.ix2.p1.2.m2.1.1" xref="S5.I4.ix2.p1.2.m2.1.1.cmml"><mrow id="S5.I4.ix2.p1.2.m2.1.1.1" xref="S5.I4.ix2.p1.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.2.m2.1.1.1.3" xref="S5.I4.ix2.p1.2.m2.1.1.1.3.cmml">𝒪</mi><mo id="S5.I4.ix2.p1.2.m2.1.1.1.2" xref="S5.I4.ix2.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.2.m2.1.1.1.1.1" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.2" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.3" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I4.ix2.p1.2.m2.1.1.2" xref="S5.I4.ix2.p1.2.m2.1.1.2.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.2.m2.1b"><apply id="S5.I4.ix2.p1.2.m2.1.1.cmml" xref="S5.I4.ix2.p1.2.m2.1.1"><ci id="S5.I4.ix2.p1.2.m2.1.1.2.cmml" xref="S5.I4.ix2.p1.2.m2.1.1.2">¯</ci><apply id="S5.I4.ix2.p1.2.m2.1.1.1.cmml" xref="S5.I4.ix2.p1.2.m2.1.1.1"><times id="S5.I4.ix2.p1.2.m2.1.1.1.2.cmml" xref="S5.I4.ix2.p1.2.m2.1.1.1.2"></times><ci id="S5.I4.ix2.p1.2.m2.1.1.1.3.cmml" xref="S5.I4.ix2.p1.2.m2.1.1.1.3">𝒪</ci><apply id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.2">𝐱</ci><ci id="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S5.I4.ix2.p1.2.m2.1.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.2.m2.1c">\overline{\cal O({\bf x^{\prime}})}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.2.m2.1d">over¯ start_ARG caligraphic_O ( bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG</annotation></semantics></math> in <math alttext="X" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.3.m3.1"><semantics id="S5.I4.ix2.p1.3.m3.1a"><mi id="S5.I4.ix2.p1.3.m3.1.1" xref="S5.I4.ix2.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.3.m3.1b"><ci id="S5.I4.ix2.p1.3.m3.1.1.cmml" xref="S5.I4.ix2.p1.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.3.m3.1d">italic_X</annotation></semantics></math> which are generated by <math alttext="\bf x" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.4.m4.1"><semantics id="S5.I4.ix2.p1.4.m4.1a"><mi id="S5.I4.ix2.p1.4.m4.1.1" xref="S5.I4.ix2.p1.4.m4.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.4.m4.1b"><ci id="S5.I4.ix2.p1.4.m4.1.1.cmml" xref="S5.I4.ix2.p1.4.m4.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.4.m4.1c">\bf x</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.4.m4.1d">bold_x</annotation></semantics></math> and <math alttext="\bf x^{\prime}" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.5.m5.1"><semantics id="S5.I4.ix2.p1.5.m5.1a"><msup id="S5.I4.ix2.p1.5.m5.1.1" xref="S5.I4.ix2.p1.5.m5.1.1.cmml"><mi id="S5.I4.ix2.p1.5.m5.1.1.2" xref="S5.I4.ix2.p1.5.m5.1.1.2.cmml">𝐱</mi><mo id="S5.I4.ix2.p1.5.m5.1.1.3" xref="S5.I4.ix2.p1.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.5.m5.1b"><apply id="S5.I4.ix2.p1.5.m5.1.1.cmml" xref="S5.I4.ix2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.I4.ix2.p1.5.m5.1.1.1.cmml" xref="S5.I4.ix2.p1.5.m5.1.1">superscript</csymbol><ci id="S5.I4.ix2.p1.5.m5.1.1.2.cmml" xref="S5.I4.ix2.p1.5.m5.1.1.2">𝐱</ci><ci id="S5.I4.ix2.p1.5.m5.1.1.3.cmml" xref="S5.I4.ix2.p1.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.5.m5.1c">\bf x^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.5.m5.1d">bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> respectively. In order to produce a violation to this second property, we need that <math alttext="\overline{\cal O({\bf x})}\neq\overline{\cal O({\bf x^{\prime}})}" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.6.m6.2"><semantics id="S5.I4.ix2.p1.6.m6.2a"><mrow id="S5.I4.ix2.p1.6.m6.2.3" xref="S5.I4.ix2.p1.6.m6.2.3.cmml"><mover accent="true" id="S5.I4.ix2.p1.6.m6.1.1" xref="S5.I4.ix2.p1.6.m6.1.1.cmml"><mrow id="S5.I4.ix2.p1.6.m6.1.1.1" xref="S5.I4.ix2.p1.6.m6.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.6.m6.1.1.1.3" xref="S5.I4.ix2.p1.6.m6.1.1.1.3.cmml">𝒪</mi><mo id="S5.I4.ix2.p1.6.m6.1.1.1.2" xref="S5.I4.ix2.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.6.m6.1.1.1.4.2" xref="S5.I4.ix2.p1.6.m6.1.1.1.cmml"><mo id="S5.I4.ix2.p1.6.m6.1.1.1.4.2.1" stretchy="false" xref="S5.I4.ix2.p1.6.m6.1.1.1.cmml">(</mo><mi id="S5.I4.ix2.p1.6.m6.1.1.1.1" xref="S5.I4.ix2.p1.6.m6.1.1.1.1.cmml">𝐱</mi><mo id="S5.I4.ix2.p1.6.m6.1.1.1.4.2.2" stretchy="false" xref="S5.I4.ix2.p1.6.m6.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I4.ix2.p1.6.m6.1.1.2" xref="S5.I4.ix2.p1.6.m6.1.1.2.cmml">¯</mo></mover><mo id="S5.I4.ix2.p1.6.m6.2.3.1" xref="S5.I4.ix2.p1.6.m6.2.3.1.cmml">≠</mo><mover accent="true" id="S5.I4.ix2.p1.6.m6.2.2" xref="S5.I4.ix2.p1.6.m6.2.2.cmml"><mrow id="S5.I4.ix2.p1.6.m6.2.2.1" xref="S5.I4.ix2.p1.6.m6.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.6.m6.2.2.1.3" xref="S5.I4.ix2.p1.6.m6.2.2.1.3.cmml">𝒪</mi><mo id="S5.I4.ix2.p1.6.m6.2.2.1.2" xref="S5.I4.ix2.p1.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.6.m6.2.2.1.1.1" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.cmml"><mo id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.cmml">(</mo><msup id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.2" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.2.cmml">𝐱</mi><mo id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.3" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.3" stretchy="false" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I4.ix2.p1.6.m6.2.2.2" xref="S5.I4.ix2.p1.6.m6.2.2.2.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.6.m6.2b"><apply id="S5.I4.ix2.p1.6.m6.2.3.cmml" xref="S5.I4.ix2.p1.6.m6.2.3"><neq id="S5.I4.ix2.p1.6.m6.2.3.1.cmml" xref="S5.I4.ix2.p1.6.m6.2.3.1"></neq><apply id="S5.I4.ix2.p1.6.m6.1.1.cmml" xref="S5.I4.ix2.p1.6.m6.1.1"><ci id="S5.I4.ix2.p1.6.m6.1.1.2.cmml" xref="S5.I4.ix2.p1.6.m6.1.1.2">¯</ci><apply id="S5.I4.ix2.p1.6.m6.1.1.1.cmml" xref="S5.I4.ix2.p1.6.m6.1.1.1"><times id="S5.I4.ix2.p1.6.m6.1.1.1.2.cmml" xref="S5.I4.ix2.p1.6.m6.1.1.1.2"></times><ci id="S5.I4.ix2.p1.6.m6.1.1.1.3.cmml" xref="S5.I4.ix2.p1.6.m6.1.1.1.3">𝒪</ci><ci id="S5.I4.ix2.p1.6.m6.1.1.1.1.cmml" xref="S5.I4.ix2.p1.6.m6.1.1.1.1">𝐱</ci></apply></apply><apply id="S5.I4.ix2.p1.6.m6.2.2.cmml" xref="S5.I4.ix2.p1.6.m6.2.2"><ci id="S5.I4.ix2.p1.6.m6.2.2.2.cmml" xref="S5.I4.ix2.p1.6.m6.2.2.2">¯</ci><apply id="S5.I4.ix2.p1.6.m6.2.2.1.cmml" xref="S5.I4.ix2.p1.6.m6.2.2.1"><times id="S5.I4.ix2.p1.6.m6.2.2.1.2.cmml" xref="S5.I4.ix2.p1.6.m6.2.2.1.2"></times><ci id="S5.I4.ix2.p1.6.m6.2.2.1.3.cmml" xref="S5.I4.ix2.p1.6.m6.2.2.1.3">𝒪</ci><apply id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.cmml" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1">superscript</csymbol><ci id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.2">𝐱</ci><ci id="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.3.cmml" xref="S5.I4.ix2.p1.6.m6.2.2.1.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.6.m6.2c">\overline{\cal O({\bf x})}\neq\overline{\cal O({\bf x^{\prime}})}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.6.m6.2d">over¯ start_ARG caligraphic_O ( bold_x ) end_ARG ≠ over¯ start_ARG caligraphic_O ( bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG</annotation></semantics></math>. If one has furthermore that their measure cones <math alttext="\cal M(\overline{\cal O({\bf x})})" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.7.m7.1"><semantics id="S5.I4.ix2.p1.7.m7.1a"><mrow id="S5.I4.ix2.p1.7.m7.1.2" xref="S5.I4.ix2.p1.7.m7.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.7.m7.1.2.2" xref="S5.I4.ix2.p1.7.m7.1.2.2.cmml">ℳ</mi><mo id="S5.I4.ix2.p1.7.m7.1.2.1" xref="S5.I4.ix2.p1.7.m7.1.2.1.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.7.m7.1.2.3.2" xref="S5.I4.ix2.p1.7.m7.1.1.cmml"><mo id="S5.I4.ix2.p1.7.m7.1.2.3.2.1" stretchy="false" xref="S5.I4.ix2.p1.7.m7.1.1.cmml">(</mo><mover accent="true" id="S5.I4.ix2.p1.7.m7.1.1" xref="S5.I4.ix2.p1.7.m7.1.1.cmml"><mrow id="S5.I4.ix2.p1.7.m7.1.1.1" xref="S5.I4.ix2.p1.7.m7.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.7.m7.1.1.1.3" xref="S5.I4.ix2.p1.7.m7.1.1.1.3.cmml">𝒪</mi><mo id="S5.I4.ix2.p1.7.m7.1.1.1.2" xref="S5.I4.ix2.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.7.m7.1.1.1.4.2" xref="S5.I4.ix2.p1.7.m7.1.1.1.cmml"><mo id="S5.I4.ix2.p1.7.m7.1.1.1.4.2.1" stretchy="false" xref="S5.I4.ix2.p1.7.m7.1.1.1.cmml">(</mo><mi id="S5.I4.ix2.p1.7.m7.1.1.1.1" xref="S5.I4.ix2.p1.7.m7.1.1.1.1.cmml">𝐱</mi><mo id="S5.I4.ix2.p1.7.m7.1.1.1.4.2.2" stretchy="false" xref="S5.I4.ix2.p1.7.m7.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I4.ix2.p1.7.m7.1.1.2" xref="S5.I4.ix2.p1.7.m7.1.1.2.cmml">¯</mo></mover><mo id="S5.I4.ix2.p1.7.m7.1.2.3.2.2" stretchy="false" xref="S5.I4.ix2.p1.7.m7.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.7.m7.1b"><apply id="S5.I4.ix2.p1.7.m7.1.2.cmml" xref="S5.I4.ix2.p1.7.m7.1.2"><times id="S5.I4.ix2.p1.7.m7.1.2.1.cmml" xref="S5.I4.ix2.p1.7.m7.1.2.1"></times><ci id="S5.I4.ix2.p1.7.m7.1.2.2.cmml" xref="S5.I4.ix2.p1.7.m7.1.2.2">ℳ</ci><apply id="S5.I4.ix2.p1.7.m7.1.1.cmml" xref="S5.I4.ix2.p1.7.m7.1.2.3.2"><ci id="S5.I4.ix2.p1.7.m7.1.1.2.cmml" xref="S5.I4.ix2.p1.7.m7.1.1.2">¯</ci><apply id="S5.I4.ix2.p1.7.m7.1.1.1.cmml" xref="S5.I4.ix2.p1.7.m7.1.1.1"><times id="S5.I4.ix2.p1.7.m7.1.1.1.2.cmml" xref="S5.I4.ix2.p1.7.m7.1.1.1.2"></times><ci id="S5.I4.ix2.p1.7.m7.1.1.1.3.cmml" xref="S5.I4.ix2.p1.7.m7.1.1.1.3">𝒪</ci><ci id="S5.I4.ix2.p1.7.m7.1.1.1.1.cmml" xref="S5.I4.ix2.p1.7.m7.1.1.1.1">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.7.m7.1c">\cal M(\overline{\cal O({\bf x})})</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.7.m7.1d">caligraphic_M ( over¯ start_ARG caligraphic_O ( bold_x ) end_ARG )</annotation></semantics></math> and <math alttext="\cal M(\overline{\cal O({\bf x^{\prime}})})" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.8.m8.1"><semantics id="S5.I4.ix2.p1.8.m8.1a"><mrow id="S5.I4.ix2.p1.8.m8.1.2" xref="S5.I4.ix2.p1.8.m8.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.8.m8.1.2.2" xref="S5.I4.ix2.p1.8.m8.1.2.2.cmml">ℳ</mi><mo id="S5.I4.ix2.p1.8.m8.1.2.1" xref="S5.I4.ix2.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.8.m8.1.2.3.2" xref="S5.I4.ix2.p1.8.m8.1.1.cmml"><mo id="S5.I4.ix2.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S5.I4.ix2.p1.8.m8.1.1.cmml">(</mo><mover accent="true" id="S5.I4.ix2.p1.8.m8.1.1" xref="S5.I4.ix2.p1.8.m8.1.1.cmml"><mrow id="S5.I4.ix2.p1.8.m8.1.1.1" xref="S5.I4.ix2.p1.8.m8.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.8.m8.1.1.1.3" xref="S5.I4.ix2.p1.8.m8.1.1.1.3.cmml">𝒪</mi><mo id="S5.I4.ix2.p1.8.m8.1.1.1.2" xref="S5.I4.ix2.p1.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.8.m8.1.1.1.1.1" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.cmml"><mo id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.2" stretchy="false" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.cmml"><mi id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.2" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.3" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.3" stretchy="false" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I4.ix2.p1.8.m8.1.1.2" xref="S5.I4.ix2.p1.8.m8.1.1.2.cmml">¯</mo></mover><mo id="S5.I4.ix2.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S5.I4.ix2.p1.8.m8.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.8.m8.1b"><apply id="S5.I4.ix2.p1.8.m8.1.2.cmml" xref="S5.I4.ix2.p1.8.m8.1.2"><times id="S5.I4.ix2.p1.8.m8.1.2.1.cmml" xref="S5.I4.ix2.p1.8.m8.1.2.1"></times><ci id="S5.I4.ix2.p1.8.m8.1.2.2.cmml" xref="S5.I4.ix2.p1.8.m8.1.2.2">ℳ</ci><apply id="S5.I4.ix2.p1.8.m8.1.1.cmml" xref="S5.I4.ix2.p1.8.m8.1.2.3.2"><ci id="S5.I4.ix2.p1.8.m8.1.1.2.cmml" xref="S5.I4.ix2.p1.8.m8.1.1.2">¯</ci><apply id="S5.I4.ix2.p1.8.m8.1.1.1.cmml" xref="S5.I4.ix2.p1.8.m8.1.1.1"><times id="S5.I4.ix2.p1.8.m8.1.1.1.2.cmml" xref="S5.I4.ix2.p1.8.m8.1.1.1.2"></times><ci id="S5.I4.ix2.p1.8.m8.1.1.1.3.cmml" xref="S5.I4.ix2.p1.8.m8.1.1.1.3">𝒪</ci><apply id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.cmml" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1">superscript</csymbol><ci id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.2">𝐱</ci><ci id="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="S5.I4.ix2.p1.8.m8.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.8.m8.1c">\cal M(\overline{\cal O({\bf x^{\prime}})})</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.8.m8.1d">caligraphic_M ( over¯ start_ARG caligraphic_O ( bold_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG )</annotation></semantics></math> are distinct subcones of <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S5.I4.ix2.p1.9.m9.1"><semantics id="S5.I4.ix2.p1.9.m9.1a"><mrow id="S5.I4.ix2.p1.9.m9.1.2" xref="S5.I4.ix2.p1.9.m9.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.9.m9.1.2.2" xref="S5.I4.ix2.p1.9.m9.1.2.2.cmml">ℳ</mi><mo id="S5.I4.ix2.p1.9.m9.1.2.1" xref="S5.I4.ix2.p1.9.m9.1.2.1.cmml">⁢</mo><mrow id="S5.I4.ix2.p1.9.m9.1.2.3.2" xref="S5.I4.ix2.p1.9.m9.1.2.cmml"><mo id="S5.I4.ix2.p1.9.m9.1.2.3.2.1" stretchy="false" xref="S5.I4.ix2.p1.9.m9.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S5.I4.ix2.p1.9.m9.1.1" xref="S5.I4.ix2.p1.9.m9.1.1.cmml">𝒳</mi><mo id="S5.I4.ix2.p1.9.m9.1.2.3.2.2" stretchy="false" xref="S5.I4.ix2.p1.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.ix2.p1.9.m9.1b"><apply id="S5.I4.ix2.p1.9.m9.1.2.cmml" xref="S5.I4.ix2.p1.9.m9.1.2"><times id="S5.I4.ix2.p1.9.m9.1.2.1.cmml" xref="S5.I4.ix2.p1.9.m9.1.2.1"></times><ci id="S5.I4.ix2.p1.9.m9.1.2.2.cmml" xref="S5.I4.ix2.p1.9.m9.1.2.2">ℳ</ci><ci id="S5.I4.ix2.p1.9.m9.1.1.cmml" xref="S5.I4.ix2.p1.9.m9.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.ix2.p1.9.m9.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S5.I4.ix2.p1.9.m9.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math>, then indeed the second property is violated.</p> </div> </li> </ol> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="S5.Thmthm14"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmthm14.1.1.1">Remark 5.14</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmthm14.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmthm14.p1"> <p class="ltx_p" id="S5.Thmthm14.p1.3">To finish this section, let us again consider the totality of the implications and their refusals as presented in Fig.1, for subshifts on which we impose additional assumptions. As before, we always consider a morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S5.Thmthm14.p1.1.m1.1"><semantics id="S5.Thmthm14.p1.1.m1.1a"><mrow id="S5.Thmthm14.p1.1.m1.1.1" xref="S5.Thmthm14.p1.1.m1.1.1.cmml"><mi id="S5.Thmthm14.p1.1.m1.1.1.2" xref="S5.Thmthm14.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S5.Thmthm14.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.Thmthm14.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S5.Thmthm14.p1.1.m1.1.1.3" xref="S5.Thmthm14.p1.1.m1.1.1.3.cmml"><msup id="S5.Thmthm14.p1.1.m1.1.1.3.2" xref="S5.Thmthm14.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm14.p1.1.m1.1.1.3.2.2" xref="S5.Thmthm14.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.Thmthm14.p1.1.m1.1.1.3.2.3" xref="S5.Thmthm14.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.Thmthm14.p1.1.m1.1.1.3.1" stretchy="false" xref="S5.Thmthm14.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S5.Thmthm14.p1.1.m1.1.1.3.3" xref="S5.Thmthm14.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm14.p1.1.m1.1.1.3.3.2" xref="S5.Thmthm14.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S5.Thmthm14.p1.1.m1.1.1.3.3.3" xref="S5.Thmthm14.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm14.p1.1.m1.1b"><apply id="S5.Thmthm14.p1.1.m1.1.1.cmml" xref="S5.Thmthm14.p1.1.m1.1.1"><ci id="S5.Thmthm14.p1.1.m1.1.1.1.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.1">:</ci><ci id="S5.Thmthm14.p1.1.m1.1.1.2.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.2">𝜎</ci><apply id="S5.Thmthm14.p1.1.m1.1.1.3.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3"><ci id="S5.Thmthm14.p1.1.m1.1.1.3.1.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.1">→</ci><apply id="S5.Thmthm14.p1.1.m1.1.1.3.2.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.Thmthm14.p1.1.m1.1.1.3.2.1.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.Thmthm14.p1.1.m1.1.1.3.2.2.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.Thmthm14.p1.1.m1.1.1.3.2.3.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S5.Thmthm14.p1.1.m1.1.1.3.3.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.Thmthm14.p1.1.m1.1.1.3.3.1.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.Thmthm14.p1.1.m1.1.1.3.3.2.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S5.Thmthm14.p1.1.m1.1.1.3.3.3.cmml" xref="S5.Thmthm14.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm14.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm14.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> as well as subshifts <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm14.p1.2.m2.1"><semantics id="S5.Thmthm14.p1.2.m2.1a"><mrow id="S5.Thmthm14.p1.2.m2.1.1" xref="S5.Thmthm14.p1.2.m2.1.1.cmml"><mi id="S5.Thmthm14.p1.2.m2.1.1.2" xref="S5.Thmthm14.p1.2.m2.1.1.2.cmml">X</mi><mo id="S5.Thmthm14.p1.2.m2.1.1.1" xref="S5.Thmthm14.p1.2.m2.1.1.1.cmml">⊆</mo><msup id="S5.Thmthm14.p1.2.m2.1.1.3" xref="S5.Thmthm14.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm14.p1.2.m2.1.1.3.2" xref="S5.Thmthm14.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S5.Thmthm14.p1.2.m2.1.1.3.3" xref="S5.Thmthm14.p1.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm14.p1.2.m2.1b"><apply id="S5.Thmthm14.p1.2.m2.1.1.cmml" xref="S5.Thmthm14.p1.2.m2.1.1"><subset id="S5.Thmthm14.p1.2.m2.1.1.1.cmml" xref="S5.Thmthm14.p1.2.m2.1.1.1"></subset><ci id="S5.Thmthm14.p1.2.m2.1.1.2.cmml" xref="S5.Thmthm14.p1.2.m2.1.1.2">𝑋</ci><apply id="S5.Thmthm14.p1.2.m2.1.1.3.cmml" xref="S5.Thmthm14.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmthm14.p1.2.m2.1.1.3.1.cmml" xref="S5.Thmthm14.p1.2.m2.1.1.3">superscript</csymbol><ci id="S5.Thmthm14.p1.2.m2.1.1.3.2.cmml" xref="S5.Thmthm14.p1.2.m2.1.1.3.2">𝒜</ci><ci id="S5.Thmthm14.p1.2.m2.1.1.3.3.cmml" xref="S5.Thmthm14.p1.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm14.p1.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm14.p1.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="Y=\sigma(X)\subseteq\cal B^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.Thmthm14.p1.3.m3.1"><semantics id="S5.Thmthm14.p1.3.m3.1a"><mrow id="S5.Thmthm14.p1.3.m3.1.2" xref="S5.Thmthm14.p1.3.m3.1.2.cmml"><mi id="S5.Thmthm14.p1.3.m3.1.2.2" xref="S5.Thmthm14.p1.3.m3.1.2.2.cmml">Y</mi><mo id="S5.Thmthm14.p1.3.m3.1.2.3" xref="S5.Thmthm14.p1.3.m3.1.2.3.cmml">=</mo><mrow id="S5.Thmthm14.p1.3.m3.1.2.4" xref="S5.Thmthm14.p1.3.m3.1.2.4.cmml"><mi id="S5.Thmthm14.p1.3.m3.1.2.4.2" xref="S5.Thmthm14.p1.3.m3.1.2.4.2.cmml">σ</mi><mo id="S5.Thmthm14.p1.3.m3.1.2.4.1" xref="S5.Thmthm14.p1.3.m3.1.2.4.1.cmml">⁢</mo><mrow id="S5.Thmthm14.p1.3.m3.1.2.4.3.2" xref="S5.Thmthm14.p1.3.m3.1.2.4.cmml"><mo id="S5.Thmthm14.p1.3.m3.1.2.4.3.2.1" stretchy="false" xref="S5.Thmthm14.p1.3.m3.1.2.4.cmml">(</mo><mi id="S5.Thmthm14.p1.3.m3.1.1" xref="S5.Thmthm14.p1.3.m3.1.1.cmml">X</mi><mo id="S5.Thmthm14.p1.3.m3.1.2.4.3.2.2" stretchy="false" xref="S5.Thmthm14.p1.3.m3.1.2.4.cmml">)</mo></mrow></mrow><mo id="S5.Thmthm14.p1.3.m3.1.2.5" xref="S5.Thmthm14.p1.3.m3.1.2.5.cmml">⊆</mo><msup id="S5.Thmthm14.p1.3.m3.1.2.6" xref="S5.Thmthm14.p1.3.m3.1.2.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmthm14.p1.3.m3.1.2.6.2" xref="S5.Thmthm14.p1.3.m3.1.2.6.2.cmml">ℬ</mi><mi id="S5.Thmthm14.p1.3.m3.1.2.6.3" xref="S5.Thmthm14.p1.3.m3.1.2.6.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmthm14.p1.3.m3.1b"><apply id="S5.Thmthm14.p1.3.m3.1.2.cmml" xref="S5.Thmthm14.p1.3.m3.1.2"><and id="S5.Thmthm14.p1.3.m3.1.2a.cmml" xref="S5.Thmthm14.p1.3.m3.1.2"></and><apply id="S5.Thmthm14.p1.3.m3.1.2b.cmml" xref="S5.Thmthm14.p1.3.m3.1.2"><eq id="S5.Thmthm14.p1.3.m3.1.2.3.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.3"></eq><ci id="S5.Thmthm14.p1.3.m3.1.2.2.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.2">𝑌</ci><apply id="S5.Thmthm14.p1.3.m3.1.2.4.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.4"><times id="S5.Thmthm14.p1.3.m3.1.2.4.1.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.4.1"></times><ci id="S5.Thmthm14.p1.3.m3.1.2.4.2.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.4.2">𝜎</ci><ci id="S5.Thmthm14.p1.3.m3.1.1.cmml" xref="S5.Thmthm14.p1.3.m3.1.1">𝑋</ci></apply></apply><apply id="S5.Thmthm14.p1.3.m3.1.2c.cmml" xref="S5.Thmthm14.p1.3.m3.1.2"><subset id="S5.Thmthm14.p1.3.m3.1.2.5.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.5"></subset><share href="https://arxiv.org/html/2211.11234v4#S5.Thmthm14.p1.3.m3.1.2.4.cmml" id="S5.Thmthm14.p1.3.m3.1.2d.cmml" xref="S5.Thmthm14.p1.3.m3.1.2"></share><apply id="S5.Thmthm14.p1.3.m3.1.2.6.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.6"><csymbol cd="ambiguous" id="S5.Thmthm14.p1.3.m3.1.2.6.1.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.6">superscript</csymbol><ci id="S5.Thmthm14.p1.3.m3.1.2.6.2.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.6.2">ℬ</ci><ci id="S5.Thmthm14.p1.3.m3.1.2.6.3.cmml" xref="S5.Thmthm14.p1.3.m3.1.2.6.3">ℤ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmthm14.p1.3.m3.1c">Y=\sigma(X)\subseteq\cal B^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmthm14.p1.3.m3.1d">italic_Y = italic_σ ( italic_X ) ⊆ caligraphic_B start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> <ol class="ltx_enumerate" id="S5.I5"> <li class="ltx_item" id="S5.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S5.I5.i1.p1"> <p class="ltx_p" id="S5.I5.i1.p1.3">If <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i1.p1.1.m1.1"><semantics id="S5.I5.i1.p1.1.m1.1a"><mi id="S5.I5.i1.p1.1.m1.1.1" xref="S5.I5.i1.p1.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.1.m1.1b"><ci id="S5.I5.i1.p1.1.m1.1.1.cmml" xref="S5.I5.i1.p1.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.1.m1.1d">italic_X</annotation></semantics></math> is minimal, then the four left-most properties in Fig.1 for <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I5.i1.p1.2.m2.1"><semantics id="S5.I5.i1.p1.2.m2.1a"><mi id="S5.I5.i1.p1.2.m2.1.1" xref="S5.I5.i1.p1.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.2.m2.1b"><ci id="S5.I5.i1.p1.2.m2.1.1.cmml" xref="S5.I5.i1.p1.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.2.m2.1d">italic_σ</annotation></semantics></math> on <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i1.p1.3.m3.1"><semantics id="S5.I5.i1.p1.3.m3.1a"><mi id="S5.I5.i1.p1.3.m3.1.1" xref="S5.I5.i1.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.3.m3.1b"><ci id="S5.I5.i1.p1.3.m3.1.1.cmml" xref="S5.I5.i1.p1.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.3.m3.1d">italic_X</annotation></semantics></math> are equivalent to each other, while the right-most property is a consequence, but the converse implication fails (as shown above in Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm13" title="Remark 5.13. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.13</span></a> (1)).</p> </div> </li> <li class="ltx_item" id="S5.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S5.I5.i2.p1"> <p class="ltx_p" id="S5.I5.i2.p1.6">The same is true if the image subshift <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S5.I5.i2.p1.1.m1.1"><semantics id="S5.I5.i2.p1.1.m1.1a"><mrow id="S5.I5.i2.p1.1.m1.1.2" xref="S5.I5.i2.p1.1.m1.1.2.cmml"><mi id="S5.I5.i2.p1.1.m1.1.2.2" xref="S5.I5.i2.p1.1.m1.1.2.2.cmml">σ</mi><mo id="S5.I5.i2.p1.1.m1.1.2.1" xref="S5.I5.i2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.I5.i2.p1.1.m1.1.2.3.2" xref="S5.I5.i2.p1.1.m1.1.2.cmml"><mo id="S5.I5.i2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.I5.i2.p1.1.m1.1.2.cmml">(</mo><mi id="S5.I5.i2.p1.1.m1.1.1" xref="S5.I5.i2.p1.1.m1.1.1.cmml">X</mi><mo id="S5.I5.i2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.I5.i2.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.1.m1.1b"><apply id="S5.I5.i2.p1.1.m1.1.2.cmml" xref="S5.I5.i2.p1.1.m1.1.2"><times id="S5.I5.i2.p1.1.m1.1.2.1.cmml" xref="S5.I5.i2.p1.1.m1.1.2.1"></times><ci id="S5.I5.i2.p1.1.m1.1.2.2.cmml" xref="S5.I5.i2.p1.1.m1.1.2.2">𝜎</ci><ci id="S5.I5.i2.p1.1.m1.1.1.cmml" xref="S5.I5.i2.p1.1.m1.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.1.m1.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.1.m1.1d">italic_σ ( italic_X )</annotation></semantics></math> (and hence also <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i2.p1.2.m2.1"><semantics id="S5.I5.i2.p1.2.m2.1a"><mi id="S5.I5.i2.p1.2.m2.1.1" xref="S5.I5.i2.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.2.m2.1b"><ci id="S5.I5.i2.p1.2.m2.1.1.cmml" xref="S5.I5.i2.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.2.m2.1d">italic_X</annotation></semantics></math> itself) is aperiodic, since in that case the equivalence between “<math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I5.i2.p1.3.m3.1"><semantics id="S5.I5.i2.p1.3.m3.1a"><mi id="S5.I5.i2.p1.3.m3.1.1" xref="S5.I5.i2.p1.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.3.m3.1b"><ci id="S5.I5.i2.p1.3.m3.1.1.cmml" xref="S5.I5.i2.p1.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.3.m3.1d">italic_σ</annotation></semantics></math> recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i2.p1.4.m4.1"><semantics id="S5.I5.i2.p1.4.m4.1a"><mi id="S5.I5.i2.p1.4.m4.1.1" xref="S5.I5.i2.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.4.m4.1b"><ci id="S5.I5.i2.p1.4.m4.1.1.cmml" xref="S5.I5.i2.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.4.m4.1d">italic_X</annotation></semantics></math>” and “<math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I5.i2.p1.5.m5.1"><semantics id="S5.I5.i2.p1.5.m5.1a"><mi id="S5.I5.i2.p1.5.m5.1.1" xref="S5.I5.i2.p1.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.5.m5.1b"><ci id="S5.I5.i2.p1.5.m5.1.1.cmml" xref="S5.I5.i2.p1.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.5.m5.1d">italic_σ</annotation></semantics></math> recognizable in <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i2.p1.6.m6.1"><semantics id="S5.I5.i2.p1.6.m6.1a"><mi id="S5.I5.i2.p1.6.m6.1.1" xref="S5.I5.i2.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.6.m6.1b"><ci id="S5.I5.i2.p1.6.m6.1.1.cmml" xref="S5.I5.i2.p1.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.6.m6.1d">italic_X</annotation></semantics></math>” comes for free.</p> </div> </li> <li class="ltx_item" id="S5.I5.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S5.I5.i3.p1"> <p class="ltx_p" id="S5.I5.i3.p1.2">If <math alttext="\sigma(X)" class="ltx_Math" display="inline" id="S5.I5.i3.p1.1.m1.1"><semantics id="S5.I5.i3.p1.1.m1.1a"><mrow id="S5.I5.i3.p1.1.m1.1.2" xref="S5.I5.i3.p1.1.m1.1.2.cmml"><mi id="S5.I5.i3.p1.1.m1.1.2.2" xref="S5.I5.i3.p1.1.m1.1.2.2.cmml">σ</mi><mo id="S5.I5.i3.p1.1.m1.1.2.1" xref="S5.I5.i3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.I5.i3.p1.1.m1.1.2.3.2" xref="S5.I5.i3.p1.1.m1.1.2.cmml"><mo id="S5.I5.i3.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.I5.i3.p1.1.m1.1.2.cmml">(</mo><mi id="S5.I5.i3.p1.1.m1.1.1" xref="S5.I5.i3.p1.1.m1.1.1.cmml">X</mi><mo id="S5.I5.i3.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.I5.i3.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.1.m1.1b"><apply id="S5.I5.i3.p1.1.m1.1.2.cmml" xref="S5.I5.i3.p1.1.m1.1.2"><times id="S5.I5.i3.p1.1.m1.1.2.1.cmml" xref="S5.I5.i3.p1.1.m1.1.2.1"></times><ci id="S5.I5.i3.p1.1.m1.1.2.2.cmml" xref="S5.I5.i3.p1.1.m1.1.2.2">𝜎</ci><ci id="S5.I5.i3.p1.1.m1.1.1.cmml" xref="S5.I5.i3.p1.1.m1.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.1.m1.1c">\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.1.m1.1d">italic_σ ( italic_X )</annotation></semantics></math> is not aperiodic, one may add the assumption that <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i3.p1.2.m2.1"><semantics id="S5.I5.i3.p1.2.m2.1a"><mi id="S5.I5.i3.p1.2.m2.1.1" xref="S5.I5.i3.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.p1.2.m2.1b"><ci id="S5.I5.i3.p1.2.m2.1.1.cmml" xref="S5.I5.i3.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.p1.2.m2.1d">italic_X</annotation></semantics></math> has a dense orbit, or a dense positive half-orbit. However, neither of these assumptions suffices to turn any of the 6 refused implications in Fig.1 into an implication. To be specific, we point out examples to the following three implication-refusals; the other three follow by using the implications from Fig.1.</p> <ol class="ltx_enumerate" id="S5.I5.i3.I1"> <li class="ltx_item" id="S5.I5.i3.I1.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S5.I5.i3.I1.ix1.p1"> <p class="ltx_p" id="S5.I5.i3.I1.ix1.p1.1">“shift-orbit injective” <math alttext="\,\,\Longrightarrow\!\!\!\!\!\!\!\!\!/\quad" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p1.1.m1.1"><semantics id="S5.I5.i3.I1.ix1.p1.1.m1.1a"><mrow id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.cmml"><mrow id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.2" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.2.cmml"></mi><mpadded width="0em"><mo id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.1" lspace="0.608em" stretchy="false" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.1.cmml">⟹</mo></mpadded><mo id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.3" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.3.cmml">/</mo></mrow><mspace id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.2" width="1em" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.cmml"></mspace></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p1.1.m1.1b"><apply id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1"><ci id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.1">⟹</ci><csymbol cd="latexml" id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.2">absent</csymbol><divide id="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p1.1.m1.1.1.1.1.3"></divide></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p1.1.m1.1c">\,\,\Longrightarrow\!\!\!\!\!\!\!\!\!/\quad</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p1.1.m1.1d">⟹ /</annotation></semantics></math> “shift-orbit injective and shift-period preserving”:</p> </div> <div class="ltx_para" id="S5.I5.i3.I1.ix1.p2"> <p class="ltx_p" id="S5.I5.i3.I1.ix1.p2.4">The simplest example with dense positive half-orbit in <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p2.1.m1.1"><semantics id="S5.I5.i3.I1.ix1.p2.1.m1.1a"><mi id="S5.I5.i3.I1.ix1.p2.1.m1.1.1" xref="S5.I5.i3.I1.ix1.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p2.1.m1.1b"><ci id="S5.I5.i3.I1.ix1.p2.1.m1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p2.1.m1.1d">italic_X</annotation></semantics></math> is given by <math alttext="X=\{a\}^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p2.2.m2.1"><semantics id="S5.I5.i3.I1.ix1.p2.2.m2.1a"><mrow id="S5.I5.i3.I1.ix1.p2.2.m2.1.2" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.cmml"><mi id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.2" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.2.cmml">X</mi><mo id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.1" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.1.cmml">=</mo><msup id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.cmml"><mrow id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.2.2" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.2.1.cmml"><mo id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.2.2.1" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.2.1.cmml">{</mo><mi id="S5.I5.i3.I1.ix1.p2.2.m2.1.1" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.1.cmml">a</mi><mo id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.2.2.2" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.2.1.cmml">}</mo></mrow><mi id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.3" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p2.2.m2.1b"><apply id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.cmml" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2"><eq id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.1.cmml" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.1"></eq><ci id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.2.cmml" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.2">𝑋</ci><apply id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.cmml" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.1.cmml" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3">superscript</csymbol><set id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.2.1.cmml" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.2.2"><ci id="S5.I5.i3.I1.ix1.p2.2.m2.1.1.cmml" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.1">𝑎</ci></set><ci id="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.3.cmml" xref="S5.I5.i3.I1.ix1.p2.2.m2.1.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p2.2.m2.1c">X=\{a\}^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p2.2.m2.1d">italic_X = { italic_a } start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\sigma:\{a\}^{*}\to\{b\}^{*}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p2.3.m3.2"><semantics id="S5.I5.i3.I1.ix1.p2.3.m3.2a"><mrow id="S5.I5.i3.I1.ix1.p2.3.m3.2.3" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.cmml"><mi id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.2" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.2.cmml">σ</mi><mo id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.1" lspace="0.278em" rspace="0.278em" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.1.cmml">:</mo><mrow id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.cmml"><msup id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.cmml"><mrow id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.2.2" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.2.1.cmml"><mo id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.2.2.1" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.2.1.cmml">{</mo><mi id="S5.I5.i3.I1.ix1.p2.3.m3.1.1" xref="S5.I5.i3.I1.ix1.p2.3.m3.1.1.cmml">a</mi><mo id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.2.2.2" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.2.1.cmml">}</mo></mrow><mo id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.3" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.3.cmml">∗</mo></msup><mo id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.1" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.1.cmml">→</mo><msup id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.cmml"><mrow id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.2.2" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.2.1.cmml"><mo id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.2.2.1" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.2.1.cmml">{</mo><mi id="S5.I5.i3.I1.ix1.p2.3.m3.2.2" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.2.cmml">b</mi><mo id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.2.2.2" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.2.1.cmml">}</mo></mrow><mo id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.3" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p2.3.m3.2b"><apply id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3"><ci id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.1.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.1">:</ci><ci id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.2.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.2">𝜎</ci><apply id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3"><ci id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.1.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.1">→</ci><apply id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.1.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2">superscript</csymbol><set id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.2.1.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.2.2"><ci id="S5.I5.i3.I1.ix1.p2.3.m3.1.1.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.1.1">𝑎</ci></set><times id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.3.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.2.3"></times></apply><apply id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.1.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3">superscript</csymbol><set id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.2.1.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.2.2"><ci id="S5.I5.i3.I1.ix1.p2.3.m3.2.2.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.2">𝑏</ci></set><times id="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.3.cmml" xref="S5.I5.i3.I1.ix1.p2.3.m3.2.3.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p2.3.m3.2c">\sigma:\{a\}^{*}\to\{b\}^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p2.3.m3.2d">italic_σ : { italic_a } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → { italic_b } start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\sigma(a)=b^{2}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p2.4.m4.1"><semantics id="S5.I5.i3.I1.ix1.p2.4.m4.1a"><mrow id="S5.I5.i3.I1.ix1.p2.4.m4.1.2" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.cmml"><mrow id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.cmml"><mi id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.2" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.2.cmml">σ</mi><mo id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.1" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.3.2" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.cmml"><mo id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.3.2.1" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.cmml">(</mo><mi id="S5.I5.i3.I1.ix1.p2.4.m4.1.1" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.1.cmml">a</mi><mo id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.3.2.2" stretchy="false" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.1" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.1.cmml">=</mo><msup id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.cmml"><mi id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.2" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.2.cmml">b</mi><mn id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.3" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p2.4.m4.1b"><apply id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2"><eq id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.1.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.1"></eq><apply id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2"><times id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.1.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.1"></times><ci id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.2.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.2.2">𝜎</ci><ci id="S5.I5.i3.I1.ix1.p2.4.m4.1.1.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.1">𝑎</ci></apply><apply id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.1.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.2.cmml" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.2">𝑏</ci><cn id="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.3.cmml" type="integer" xref="S5.I5.i3.I1.ix1.p2.4.m4.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p2.4.m4.1c">\sigma(a)=b^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p2.4.m4.1d">italic_σ ( italic_a ) = italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.I5.i3.I1.ix1.p3"> <p class="ltx_p" id="S5.I5.i3.I1.ix1.p3.20">More interesting examples are given by taking <math alttext="\tau:\cal A^{*}\to\cal A^{*}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.1.m1.1"><semantics id="S5.I5.i3.I1.ix1.p3.1.m1.1a"><mrow id="S5.I5.i3.I1.ix1.p3.1.m1.1.1" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.2" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.1.cmml">:</mo><mrow id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.cmml"><msup id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.2" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.3" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.1" stretchy="false" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.1.cmml">→</mo><msup id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.2" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.2.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.3" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.1.m1.1b"><apply id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1"><ci id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.1">:</ci><ci id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.2">𝜏</ci><apply id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3"><ci id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.1">→</ci><apply id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.1.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.2">𝒜</ci><times id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.2.3"></times></apply><apply id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.2">𝒜</ci><times id="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.3.cmml" xref="S5.I5.i3.I1.ix1.p3.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.1.m1.1c">\tau:\cal A^{*}\to\cal A^{*}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.1.m1.1d">italic_τ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> to be any primitive substitution on two or more letters, and set <math alttext="X^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.2.m2.1"><semantics id="S5.I5.i3.I1.ix1.p3.2.m2.1a"><msup id="S5.I5.i3.I1.ix1.p3.2.m2.1.1" xref="S5.I5.i3.I1.ix1.p3.2.m2.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.2.m2.1.1.2" xref="S5.I5.i3.I1.ix1.p3.2.m2.1.1.2.cmml">X</mi><mo id="S5.I5.i3.I1.ix1.p3.2.m2.1.1.3" xref="S5.I5.i3.I1.ix1.p3.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.2.m2.1b"><apply id="S5.I5.i3.I1.ix1.p3.2.m2.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.2.m2.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.2.m2.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.2.m2.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.2.m2.1.1.2">𝑋</ci><ci id="S5.I5.i3.I1.ix1.p3.2.m2.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.2.m2.1c">X^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.2.m2.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to be the substitution subshift <math alttext="X^{\prime}=X_{\tau}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.3.m3.1"><semantics id="S5.I5.i3.I1.ix1.p3.3.m3.1a"><mrow id="S5.I5.i3.I1.ix1.p3.3.m3.1.1" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.cmml"><msup id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.cmml"><mi id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.2" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.2.cmml">X</mi><mo id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.3" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.1" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.1.cmml">=</mo><msub id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.cmml"><mi id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.2" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.2.cmml">X</mi><mi id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.3" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.3.cmml">τ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.3.m3.1b"><apply id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1"><eq id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.1"></eq><apply id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.1.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.2">𝑋</ci><ci id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.2.3">′</ci></apply><apply id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3">subscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.2">𝑋</ci><ci id="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix1.p3.3.m3.1.1.3.3">𝜏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.3.m3.1c">X^{\prime}=X_{\tau}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.3.m3.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_X start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT</annotation></semantics></math>. One then “perturbes” <math alttext="\tau" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.4.m4.1"><semantics id="S5.I5.i3.I1.ix1.p3.4.m4.1a"><mi id="S5.I5.i3.I1.ix1.p3.4.m4.1.1" xref="S5.I5.i3.I1.ix1.p3.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.4.m4.1b"><ci id="S5.I5.i3.I1.ix1.p3.4.m4.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.4.m4.1d">italic_τ</annotation></semantics></math> into <math alttext="\tau^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.5.m5.1"><semantics id="S5.I5.i3.I1.ix1.p3.5.m5.1a"><msup id="S5.I5.i3.I1.ix1.p3.5.m5.1.1" xref="S5.I5.i3.I1.ix1.p3.5.m5.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.5.m5.1.1.2" xref="S5.I5.i3.I1.ix1.p3.5.m5.1.1.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix1.p3.5.m5.1.1.3" xref="S5.I5.i3.I1.ix1.p3.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.5.m5.1b"><apply id="S5.I5.i3.I1.ix1.p3.5.m5.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.5.m5.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.5.m5.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.5.m5.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.5.m5.1.1.2">𝜏</ci><ci id="S5.I5.i3.I1.ix1.p3.5.m5.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.5.m5.1c">\tau^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.5.m5.1d">italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> by adding a new generator <math alttext="a^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.6.m6.1"><semantics id="S5.I5.i3.I1.ix1.p3.6.m6.1a"><msup id="S5.I5.i3.I1.ix1.p3.6.m6.1.1" xref="S5.I5.i3.I1.ix1.p3.6.m6.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.6.m6.1.1.2" xref="S5.I5.i3.I1.ix1.p3.6.m6.1.1.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix1.p3.6.m6.1.1.3" xref="S5.I5.i3.I1.ix1.p3.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.6.m6.1b"><apply id="S5.I5.i3.I1.ix1.p3.6.m6.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.6.m6.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.6.m6.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.6.m6.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.6.m6.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix1.p3.6.m6.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.6.m6.1c">a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.6.m6.1d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\cal A" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.7.m7.1"><semantics id="S5.I5.i3.I1.ix1.p3.7.m7.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix1.p3.7.m7.1.1" xref="S5.I5.i3.I1.ix1.p3.7.m7.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.7.m7.1b"><ci id="S5.I5.i3.I1.ix1.p3.7.m7.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.7.m7.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.7.m7.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.7.m7.1d">caligraphic_A</annotation></semantics></math> to obtain <math alttext="\cal A^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.8.m8.1"><semantics id="S5.I5.i3.I1.ix1.p3.8.m8.1a"><msup id="S5.I5.i3.I1.ix1.p3.8.m8.1.1" xref="S5.I5.i3.I1.ix1.p3.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix1.p3.8.m8.1.1.2" xref="S5.I5.i3.I1.ix1.p3.8.m8.1.1.2.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix1.p3.8.m8.1.1.3" xref="S5.I5.i3.I1.ix1.p3.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.8.m8.1b"><apply id="S5.I5.i3.I1.ix1.p3.8.m8.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.8.m8.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.8.m8.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.8.m8.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.8.m8.1.1.2">𝒜</ci><ci id="S5.I5.i3.I1.ix1.p3.8.m8.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.8.m8.1c">\cal A^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.8.m8.1d">caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and defines <math alttext="\tau^{\prime}(a_{k})=\tau(a_{k})a^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.9.m9.2"><semantics id="S5.I5.i3.I1.ix1.p3.9.m9.2a"><mrow id="S5.I5.i3.I1.ix1.p3.9.m9.2.2" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.cmml"><mrow id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.cmml"><msup id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.3" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.3.cmml"><mi id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.3.2" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.3.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.3.3" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.3.3.cmml">′</mo></msup><mo id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.2" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1.cmml"><mo id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1.cmml">(</mo><msub id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1.2" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1.2.cmml">a</mi><mi id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1.3" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.3" stretchy="false" xref="S5.I5.i3.I1.ix1.p3.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.3" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.3.cmml">=</mo><mrow id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.cmml"><mi id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.3" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.3.cmml">τ</mi><mo id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.2" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.2.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.cmml"><mo id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.cmml">(</mo><msub id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.2" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.2.cmml">a</mi><mi id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.3" 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id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.2"></times><ci id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.3">𝜏</ci><apply id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1">subscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.1.1.1.3">𝑘</ci></apply><apply id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.4.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.4"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.4.1.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.4">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.4.2.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.4.2">𝑎</ci><ci id="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.4.3.cmml" xref="S5.I5.i3.I1.ix1.p3.9.m9.2.2.2.4.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.9.m9.2c">\tau^{\prime}(a_{k})=\tau(a_{k})a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.9.m9.2d">italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_τ ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> for any <math alttext="a_{k}\in\cal A" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.10.m10.1"><semantics id="S5.I5.i3.I1.ix1.p3.10.m10.1a"><mrow id="S5.I5.i3.I1.ix1.p3.10.m10.1.1" xref="S5.I5.i3.I1.ix1.p3.10.m10.1.1.cmml"><msub id="S5.I5.i3.I1.ix1.p3.10.m10.1.1.2" xref="S5.I5.i3.I1.ix1.p3.10.m10.1.1.2.cmml"><mi id="S5.I5.i3.I1.ix1.p3.10.m10.1.1.2.2" xref="S5.I5.i3.I1.ix1.p3.10.m10.1.1.2.2.cmml">a</mi><mi 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xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3.cmml"><mi id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3.2" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.2" xref="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.2.1" mathsize="142%" xref="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.2.1.cmml">′</mo><mo id="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.2.2" lspace="0em" xref="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.3.cmml">⁣</mo><mn id="S5.I5.i3.I1.ix1.p3.11.m11.1.1.1.1" xref="S5.I5.i3.I1.ix1.p3.11.m11.1.1.1.1.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.11.m11.3b"><apply id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3"><eq id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.2"></eq><apply id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1"><times id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.2"></times><apply id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.3.2">𝜏</ci><ci id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.3.3.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.3.3">′</ci></apply><apply id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.1.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.1.1.1.1.3">′</ci></apply></apply><apply id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.3.3.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.2"><ci id="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix1.p3.11.m11.2.2.2.2.1">′</ci><cn id="S5.I5.i3.I1.ix1.p3.11.m11.1.1.1.1.cmml" type="integer" xref="S5.I5.i3.I1.ix1.p3.11.m11.1.1.1.1">2</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.11.m11.3c">\tau^{\prime}(a^{\prime})=a^{\prime 2}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.11.m11.3d">italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_a start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Then the periodic orbit <math alttext="a^{\prime\pm\infty}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.12.m12.2"><semantics id="S5.I5.i3.I1.ix1.p3.12.m12.2a"><msup id="S5.I5.i3.I1.ix1.p3.12.m12.2.3" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.3.cmml"><mi id="S5.I5.i3.I1.ix1.p3.12.m12.2.3.2" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix1.p3.12.m12.1.1.1.1.1" mathsize="142%" xref="S5.I5.i3.I1.ix1.p3.12.m12.1.1.1.1.1.cmml">′</mo><mo id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.3" lspace="0em" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.3.cmml">⁣</mo><mrow id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2.cmml"><mo id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2a" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2.cmml">±</mo><mi id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2.2" mathvariant="normal" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2.2.cmml">∞</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.12.m12.2b"><apply id="S5.I5.i3.I1.ix1.p3.12.m12.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.12.m12.2.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.12.m12.2.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2"><ci id="S5.I5.i3.I1.ix1.p3.12.m12.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.12.m12.1.1.1.1.1">′</ci><apply id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2"><csymbol cd="latexml" id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2">plus-or-minus</csymbol><infinity id="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.12.m12.2.2.2.2.2.2"></infinity></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.12.m12.2c">a^{\prime\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.12.m12.2d">italic_a start_POSTSUPERSCRIPT ′ ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> belongs to the substitution subshift <math alttext="X:=X_{\tau^{\prime}}\," class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.13.m13.1"><semantics id="S5.I5.i3.I1.ix1.p3.13.m13.1a"><mrow id="S5.I5.i3.I1.ix1.p3.13.m13.1.1" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.2" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.2.cmml">X</mi><mo id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.1.cmml">:=</mo><msub id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.cmml"><mi id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.2" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.2.cmml">X</mi><msup id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.cmml"><mi id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.2" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.3" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.3.cmml">′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.13.m13.1b"><apply id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1"><csymbol cd="latexml" id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.1">assign</csymbol><ci id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.2">𝑋</ci><apply id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3">subscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.2">𝑋</ci><apply id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.2">𝜏</ci><ci id="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.3.cmml" xref="S5.I5.i3.I1.ix1.p3.13.m13.1.1.3.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.13.m13.1c">X:=X_{\tau^{\prime}}\,</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.13.m13.1d">italic_X := italic_X start_POSTSUBSCRIPT italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, and any <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.14.m14.1"><semantics id="S5.I5.i3.I1.ix1.p3.14.m14.1a"><mrow id="S5.I5.i3.I1.ix1.p3.14.m14.1.1" xref="S5.I5.i3.I1.ix1.p3.14.m14.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.14.m14.1.1.2" xref="S5.I5.i3.I1.ix1.p3.14.m14.1.1.2.cmml">𝐱</mi><mo id="S5.I5.i3.I1.ix1.p3.14.m14.1.1.1" xref="S5.I5.i3.I1.ix1.p3.14.m14.1.1.1.cmml">∈</mo><mi id="S5.I5.i3.I1.ix1.p3.14.m14.1.1.3" xref="S5.I5.i3.I1.ix1.p3.14.m14.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.14.m14.1b"><apply id="S5.I5.i3.I1.ix1.p3.14.m14.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.14.m14.1.1"><in id="S5.I5.i3.I1.ix1.p3.14.m14.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.14.m14.1.1.1"></in><ci id="S5.I5.i3.I1.ix1.p3.14.m14.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.14.m14.1.1.2">𝐱</ci><ci id="S5.I5.i3.I1.ix1.p3.14.m14.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.14.m14.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.14.m14.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.14.m14.1d">bold_x ∈ italic_X</annotation></semantics></math> which is different from <math alttext="a^{\prime\pm\infty}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.15.m15.2"><semantics id="S5.I5.i3.I1.ix1.p3.15.m15.2a"><msup id="S5.I5.i3.I1.ix1.p3.15.m15.2.3" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.3.cmml"><mi id="S5.I5.i3.I1.ix1.p3.15.m15.2.3.2" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix1.p3.15.m15.1.1.1.1.1" mathsize="142%" xref="S5.I5.i3.I1.ix1.p3.15.m15.1.1.1.1.1.cmml">′</mo><mo id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.3" lspace="0em" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.3.cmml">⁣</mo><mrow id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2.cmml"><mo id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2a" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2.cmml">±</mo><mi id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2.2" mathvariant="normal" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2.2.cmml">∞</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.15.m15.2b"><apply id="S5.I5.i3.I1.ix1.p3.15.m15.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.15.m15.2.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.15.m15.2.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2"><ci id="S5.I5.i3.I1.ix1.p3.15.m15.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.15.m15.1.1.1.1.1">′</ci><apply id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2"><csymbol cd="latexml" id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2">plus-or-minus</csymbol><infinity id="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.15.m15.2.2.2.2.2.2"></infinity></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.15.m15.2c">a^{\prime\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.15.m15.2d">italic_a start_POSTSUPERSCRIPT ′ ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> has positive half-orbit which is dense in <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.16.m16.1"><semantics id="S5.I5.i3.I1.ix1.p3.16.m16.1a"><mi id="S5.I5.i3.I1.ix1.p3.16.m16.1.1" xref="S5.I5.i3.I1.ix1.p3.16.m16.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.16.m16.1b"><ci id="S5.I5.i3.I1.ix1.p3.16.m16.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.16.m16.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.16.m16.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.16.m16.1d">italic_X</annotation></semantics></math>. Furthermore, <math alttext="\tau^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.17.m17.1"><semantics id="S5.I5.i3.I1.ix1.p3.17.m17.1a"><msup id="S5.I5.i3.I1.ix1.p3.17.m17.1.1" xref="S5.I5.i3.I1.ix1.p3.17.m17.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.17.m17.1.1.2" xref="S5.I5.i3.I1.ix1.p3.17.m17.1.1.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix1.p3.17.m17.1.1.3" xref="S5.I5.i3.I1.ix1.p3.17.m17.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.17.m17.1b"><apply id="S5.I5.i3.I1.ix1.p3.17.m17.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.17.m17.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.17.m17.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.17.m17.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.17.m17.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.17.m17.1.1.2">𝜏</ci><ci id="S5.I5.i3.I1.ix1.p3.17.m17.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.17.m17.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.17.m17.1c">\tau^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.17.m17.1d">italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is shift-orbit injective on <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.18.m18.1"><semantics id="S5.I5.i3.I1.ix1.p3.18.m18.1a"><mi id="S5.I5.i3.I1.ix1.p3.18.m18.1.1" xref="S5.I5.i3.I1.ix1.p3.18.m18.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.18.m18.1b"><ci id="S5.I5.i3.I1.ix1.p3.18.m18.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.18.m18.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.18.m18.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.18.m18.1d">italic_X</annotation></semantics></math>, while the shift-period of <math alttext="a^{\prime\pm\infty}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.19.m19.2"><semantics id="S5.I5.i3.I1.ix1.p3.19.m19.2a"><msup id="S5.I5.i3.I1.ix1.p3.19.m19.2.3" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.3.cmml"><mi id="S5.I5.i3.I1.ix1.p3.19.m19.2.3.2" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix1.p3.19.m19.1.1.1.1.1" mathsize="142%" xref="S5.I5.i3.I1.ix1.p3.19.m19.1.1.1.1.1.cmml">′</mo><mo id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.3" lspace="0em" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.3.cmml">⁣</mo><mrow id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2.cmml"><mo id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2a" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2.cmml">±</mo><mi id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2.2" mathvariant="normal" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2.2.cmml">∞</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.19.m19.2b"><apply id="S5.I5.i3.I1.ix1.p3.19.m19.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.19.m19.2.3.1.cmml" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.19.m19.2.3.2.cmml" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2"><ci id="S5.I5.i3.I1.ix1.p3.19.m19.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.19.m19.1.1.1.1.1">′</ci><apply id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2"><csymbol cd="latexml" id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2">plus-or-minus</csymbol><infinity id="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix1.p3.19.m19.2.2.2.2.2.2"></infinity></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.19.m19.2c">a^{\prime\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.19.m19.2d">italic_a start_POSTSUPERSCRIPT ′ ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> is not preserved by <math alttext="\tau^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix1.p3.20.m20.1"><semantics id="S5.I5.i3.I1.ix1.p3.20.m20.1a"><msup id="S5.I5.i3.I1.ix1.p3.20.m20.1.1" xref="S5.I5.i3.I1.ix1.p3.20.m20.1.1.cmml"><mi id="S5.I5.i3.I1.ix1.p3.20.m20.1.1.2" xref="S5.I5.i3.I1.ix1.p3.20.m20.1.1.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix1.p3.20.m20.1.1.3" xref="S5.I5.i3.I1.ix1.p3.20.m20.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix1.p3.20.m20.1b"><apply id="S5.I5.i3.I1.ix1.p3.20.m20.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.20.m20.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix1.p3.20.m20.1.1.1.cmml" xref="S5.I5.i3.I1.ix1.p3.20.m20.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix1.p3.20.m20.1.1.2.cmml" xref="S5.I5.i3.I1.ix1.p3.20.m20.1.1.2">𝜏</ci><ci id="S5.I5.i3.I1.ix1.p3.20.m20.1.1.3.cmml" xref="S5.I5.i3.I1.ix1.p3.20.m20.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix1.p3.20.m20.1c">\tau^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix1.p3.20.m20.1d">italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S5.I5.i3.I1.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S5.I5.i3.I1.ix2.p1"> <p class="ltx_p" id="S5.I5.i3.I1.ix2.p1.1">“recognizable for aperiodic points” <math alttext="\,\,\Longrightarrow\!\!\!\!\!\!\!\!\!/\quad" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p1.1.m1.1"><semantics id="S5.I5.i3.I1.ix2.p1.1.m1.1a"><mrow id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.cmml"><mrow id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.2" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.2.cmml"></mi><mpadded width="0em"><mo id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.1" lspace="0.608em" stretchy="false" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.1.cmml">⟹</mo></mpadded><mo id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.3" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.3.cmml">/</mo></mrow><mspace id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.2" width="1em" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.cmml"></mspace></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p1.1.m1.1b"><apply id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1"><ci id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.1">⟹</ci><csymbol cd="latexml" id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.2">absent</csymbol><divide id="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p1.1.m1.1.1.1.1.3"></divide></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p1.1.m1.1c">\,\,\Longrightarrow\!\!\!\!\!\!\!\!\!/\quad</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p1.1.m1.1d">⟹ /</annotation></semantics></math> “measure transfer injective”:</p> </div> <div class="ltx_para" id="S5.I5.i3.I1.ix2.p2"> <p class="ltx_p" id="S5.I5.i3.I1.ix2.p2.28">We start with <math alttext="\tau" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.1.m1.1"><semantics id="S5.I5.i3.I1.ix2.p2.1.m1.1a"><mi id="S5.I5.i3.I1.ix2.p2.1.m1.1.1" xref="S5.I5.i3.I1.ix2.p2.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.1.m1.1b"><ci id="S5.I5.i3.I1.ix2.p2.1.m1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.1.m1.1d">italic_τ</annotation></semantics></math> and <math alttext="X^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.2.m2.1"><semantics id="S5.I5.i3.I1.ix2.p2.2.m2.1a"><msup id="S5.I5.i3.I1.ix2.p2.2.m2.1.1" xref="S5.I5.i3.I1.ix2.p2.2.m2.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.2.m2.1.1.2" xref="S5.I5.i3.I1.ix2.p2.2.m2.1.1.2.cmml">X</mi><mo id="S5.I5.i3.I1.ix2.p2.2.m2.1.1.3" xref="S5.I5.i3.I1.ix2.p2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.2.m2.1b"><apply id="S5.I5.i3.I1.ix2.p2.2.m2.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.2.m2.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.2.m2.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.2.m2.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.2.m2.1.1.2">𝑋</ci><ci id="S5.I5.i3.I1.ix2.p2.2.m2.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.2.m2.1c">X^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.2.m2.1d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> as above, but add now two letters <math alttext="a^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.3.m3.1"><semantics id="S5.I5.i3.I1.ix2.p2.3.m3.1a"><msup id="S5.I5.i3.I1.ix2.p2.3.m3.1.1" xref="S5.I5.i3.I1.ix2.p2.3.m3.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.3.m3.1.1.2" xref="S5.I5.i3.I1.ix2.p2.3.m3.1.1.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.3.m3.1.1.3" xref="S5.I5.i3.I1.ix2.p2.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.3.m3.1b"><apply id="S5.I5.i3.I1.ix2.p2.3.m3.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.3.m3.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.3.m3.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.3.m3.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.3.m3.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.3.m3.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.3.m3.1c">a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.3.m3.1d">italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="a^{\prime\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.4.m4.1"><semantics id="S5.I5.i3.I1.ix2.p2.4.m4.1a"><msup id="S5.I5.i3.I1.ix2.p2.4.m4.1.1" xref="S5.I5.i3.I1.ix2.p2.4.m4.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.4.m4.1.1.2" xref="S5.I5.i3.I1.ix2.p2.4.m4.1.1.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.4.m4.1.1.3" xref="S5.I5.i3.I1.ix2.p2.4.m4.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.4.m4.1b"><apply id="S5.I5.i3.I1.ix2.p2.4.m4.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.4.m4.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.4.m4.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.4.m4.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.4.m4.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.4.m4.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.4.m4.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.4.m4.1c">a^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.4.m4.1d">italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\cal A" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.5.m5.1"><semantics id="S5.I5.i3.I1.ix2.p2.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.5.m5.1.1" xref="S5.I5.i3.I1.ix2.p2.5.m5.1.1.cmml">𝒜</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.5.m5.1b"><ci id="S5.I5.i3.I1.ix2.p2.5.m5.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.5.m5.1.1">𝒜</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.5.m5.1c">\cal A</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.5.m5.1d">caligraphic_A</annotation></semantics></math> to get the alphabet <math alttext="\cal A^{\prime\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.6.m6.1"><semantics id="S5.I5.i3.I1.ix2.p2.6.m6.1a"><msup id="S5.I5.i3.I1.ix2.p2.6.m6.1.1" xref="S5.I5.i3.I1.ix2.p2.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.6.m6.1.1.2" xref="S5.I5.i3.I1.ix2.p2.6.m6.1.1.2.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix2.p2.6.m6.1.1.3" xref="S5.I5.i3.I1.ix2.p2.6.m6.1.1.3.cmml">′′</mo></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.6.m6.1b"><apply id="S5.I5.i3.I1.ix2.p2.6.m6.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.6.m6.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.6.m6.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.6.m6.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.6.m6.1.1.2">𝒜</ci><ci id="S5.I5.i3.I1.ix2.p2.6.m6.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.6.m6.1.1.3">′′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.6.m6.1c">\cal A^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.6.m6.1d">caligraphic_A start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We then define the substitution <math alttext="\tau^{\prime\prime}:\cal A^{\prime\prime}\to\cal A^{\prime\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.7.m7.1"><semantics id="S5.I5.i3.I1.ix2.p2.7.m7.1a"><mrow id="S5.I5.i3.I1.ix2.p2.7.m7.1.1" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.cmml"><msup id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.cmml"><mi id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.2" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.3" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.3.cmml">′′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.1.cmml">:</mo><mrow id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.cmml"><msup id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.2" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.2.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.3" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.3.cmml">′′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.1" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.1.cmml">→</mo><msup id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.2" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.2.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.3" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.3.cmml">′′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.7.m7.1b"><apply id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1"><ci id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.1">:</ci><apply id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.2">𝜏</ci><ci id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.2.3">′′</ci></apply><apply id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3"><ci id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.1">→</ci><apply id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.2">𝒜</ci><ci id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.2.3">′′</ci></apply><apply id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.2">𝒜</ci><ci id="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.7.m7.1.1.3.3.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.7.m7.1c">\tau^{\prime\prime}:\cal A^{\prime\prime}\to\cal A^{\prime\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.7.m7.1d">italic_τ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT</annotation></semantics></math> by setting <math alttext="\tau^{\prime\prime}(a_{k})=a^{\prime\prime}\tau(a_{k})a^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.8.m8.2"><semantics id="S5.I5.i3.I1.ix2.p2.8.m8.2a"><mrow id="S5.I5.i3.I1.ix2.p2.8.m8.2.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.cmml"><mrow id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.cmml"><msup id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.3" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.3.cmml">′′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.cmml"><mo id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.cmml">(</mo><msub id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.2.cmml">a</mi><mi id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.3" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.3" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.3.cmml">=</mo><mrow id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.cmml"><msup id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.3" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.3.cmml">′′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.2.cmml">⁢</mo><mi id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.4" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.4.cmml">τ</mi><mo id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.2a" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.2.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.cmml"><mo id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.cmml">(</mo><msub id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.2.cmml">a</mi><mi id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.3.cmml">k</mi></msub><mo id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.3" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.2b" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.2.cmml">⁢</mo><msup id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.cmml"><mi id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.2" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.3" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.8.m8.2b"><apply id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2"><eq id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.3"></eq><apply id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1"><times id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.2"></times><apply id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.2">𝜏</ci><ci id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.3.3">′′</ci></apply><apply id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2"><times id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.2"></times><apply id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.3.3">′′</ci></apply><ci id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.4.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.4">𝜏</ci><apply id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1">subscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.1.1.1.3">𝑘</ci></apply><apply id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.1.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.2.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.3.cmml" xref="S5.I5.i3.I1.ix2.p2.8.m8.2.2.2.5.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.8.m8.2c">\tau^{\prime\prime}(a_{k})=a^{\prime\prime}\tau(a_{k})a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.8.m8.2d">italic_τ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_τ ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> for any <math alttext="a_{k}\in\cal A" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.9.m9.1"><semantics id="S5.I5.i3.I1.ix2.p2.9.m9.1a"><mrow id="S5.I5.i3.I1.ix2.p2.9.m9.1.1" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.cmml"><msub id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.cmml"><mi id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.2" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.2.cmml">a</mi><mi id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.3" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.3.cmml">k</mi></msub><mo id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.1" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.3" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.3.cmml">𝒜</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.9.m9.1b"><apply id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1"><in id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.1"></in><apply id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2">subscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.2.3">𝑘</ci></apply><ci id="S5.I5.i3.I1.ix2.p2.9.m9.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.9.m9.1.1.3">𝒜</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.9.m9.1c">a_{k}\in\cal A</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.9.m9.1d">italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ caligraphic_A</annotation></semantics></math>, as well as <math alttext="\tau^{\prime\prime}(a^{\prime})=a^{\prime 2}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.10.m10.3"><semantics id="S5.I5.i3.I1.ix2.p2.10.m10.3a"><mrow id="S5.I5.i3.I1.ix2.p2.10.m10.3.3" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.cmml"><mrow id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.cmml"><msup id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.2" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.3" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.3.cmml">′′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.2" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.2.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.cmml"><mo id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.cmml">(</mo><msup id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.3" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.2" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.2.cmml">=</mo><msup id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3.2" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.2.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.2.1.cmml">′</mo><mo id="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.2.2" lspace="0em" xref="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.3.cmml">⁣</mo><mn id="S5.I5.i3.I1.ix2.p2.10.m10.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.10.m10.1.1.1.1.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.10.m10.3b"><apply id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3"><eq id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.2"></eq><apply id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1"><times id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.2"></times><apply id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.2">𝜏</ci><ci id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.3.3">′′</ci></apply><apply id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.1.1.1.1.3">′</ci></apply></apply><apply id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.3.3.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.2"><ci id="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.10.m10.2.2.2.2.1">′</ci><cn id="S5.I5.i3.I1.ix2.p2.10.m10.1.1.1.1.cmml" type="integer" xref="S5.I5.i3.I1.ix2.p2.10.m10.1.1.1.1">2</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.10.m10.3c">\tau^{\prime\prime}(a^{\prime})=a^{\prime 2}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.10.m10.3d">italic_τ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_a start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\tau^{\prime\prime}(a^{\prime\prime})=a^{\prime\prime 2}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.11.m11.3"><semantics id="S5.I5.i3.I1.ix2.p2.11.m11.3a"><mrow id="S5.I5.i3.I1.ix2.p2.11.m11.3.3" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.cmml"><mrow id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.cmml"><msup id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.2" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.3" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.3.cmml">′′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.2" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.2.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.cmml"><mo id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.cmml">(</mo><msup id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.3.cmml">′′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.3" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.2" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.2.cmml">=</mo><msup id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3.2" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.2.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.2.1.cmml">′′</mo><mo id="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.2.2" lspace="0em" xref="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.3.cmml">⁣</mo><mn id="S5.I5.i3.I1.ix2.p2.11.m11.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.11.m11.1.1.1.1.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.11.m11.3b"><apply id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3"><eq id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.2"></eq><apply id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1"><times id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.2"></times><apply id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.2">𝜏</ci><ci id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.3.3">′′</ci></apply><apply id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.1.1.1.1.3">′′</ci></apply></apply><apply id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.3.3.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.2"><ci id="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.11.m11.2.2.2.2.1">′′</ci><cn id="S5.I5.i3.I1.ix2.p2.11.m11.1.1.1.1.cmml" type="integer" xref="S5.I5.i3.I1.ix2.p2.11.m11.1.1.1.1">2</cn></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.11.m11.3c">\tau^{\prime\prime}(a^{\prime\prime})=a^{\prime\prime 2}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.11.m11.3d">italic_τ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_a start_POSTSUPERSCRIPT ′ ′ 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Again, the periodic orbits <math alttext="a^{\prime\pm\infty}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.12.m12.2"><semantics id="S5.I5.i3.I1.ix2.p2.12.m12.2a"><msup id="S5.I5.i3.I1.ix2.p2.12.m12.2.3" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.12.m12.2.3.2" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.12.m12.1.1.1.1.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.12.m12.1.1.1.1.1.cmml">′</mo><mo id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.3" lspace="0em" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.3.cmml">⁣</mo><mrow id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2.cmml"><mo id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2a" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2.cmml">±</mo><mi id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2.2" mathvariant="normal" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2.2.cmml">∞</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.12.m12.2b"><apply id="S5.I5.i3.I1.ix2.p2.12.m12.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.12.m12.2.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.12.m12.2.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2"><ci id="S5.I5.i3.I1.ix2.p2.12.m12.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.12.m12.1.1.1.1.1">′</ci><apply id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2"><csymbol cd="latexml" id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2">plus-or-minus</csymbol><infinity id="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.12.m12.2.2.2.2.2.2"></infinity></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.12.m12.2c">a^{\prime\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.12.m12.2d">italic_a start_POSTSUPERSCRIPT ′ ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="a^{\prime\prime\pm\infty}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.13.m13.2"><semantics id="S5.I5.i3.I1.ix2.p2.13.m13.2a"><msup id="S5.I5.i3.I1.ix2.p2.13.m13.2.3" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.13.m13.2.3.2" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.13.m13.1.1.1.1.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.13.m13.1.1.1.1.1.cmml">′′</mo><mo id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.3" lspace="0em" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.3.cmml">⁣</mo><mrow id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2.cmml"><mo id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2a" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2.cmml">±</mo><mi id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2.2" mathvariant="normal" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2.2.cmml">∞</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.13.m13.2b"><apply id="S5.I5.i3.I1.ix2.p2.13.m13.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.13.m13.2.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.13.m13.2.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2"><ci id="S5.I5.i3.I1.ix2.p2.13.m13.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.13.m13.1.1.1.1.1">′′</ci><apply id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2"><csymbol cd="latexml" id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2">plus-or-minus</csymbol><infinity id="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.13.m13.2.2.2.2.2.2"></infinity></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.13.m13.2c">a^{\prime\prime\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.13.m13.2d">italic_a start_POSTSUPERSCRIPT ′ ′ ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> belong to the substitution subshift <math alttext="X:=X_{\tau^{\prime\prime}}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.14.m14.1"><semantics id="S5.I5.i3.I1.ix2.p2.14.m14.1a"><mrow id="S5.I5.i3.I1.ix2.p2.14.m14.1.1" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.2" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.2.cmml">X</mi><mo id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.1.cmml">:=</mo><msub id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.2" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.2.cmml">X</mi><msup id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.2" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.2.cmml">τ</mi><mo id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.3" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.3.cmml">′′</mo></msup></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.14.m14.1b"><apply id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1"><csymbol cd="latexml" id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.1">assign</csymbol><ci id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.2">𝑋</ci><apply id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3">subscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.2">𝑋</ci><apply id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.2">𝜏</ci><ci id="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.14.m14.1.1.3.3.3">′′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.14.m14.1c">X:=X_{\tau^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.14.m14.1d">italic_X := italic_X start_POSTSUBSCRIPT italic_τ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, and any <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.15.m15.1"><semantics id="S5.I5.i3.I1.ix2.p2.15.m15.1a"><mrow id="S5.I5.i3.I1.ix2.p2.15.m15.1.1" xref="S5.I5.i3.I1.ix2.p2.15.m15.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.15.m15.1.1.2" xref="S5.I5.i3.I1.ix2.p2.15.m15.1.1.2.cmml">𝐱</mi><mo id="S5.I5.i3.I1.ix2.p2.15.m15.1.1.1" xref="S5.I5.i3.I1.ix2.p2.15.m15.1.1.1.cmml">∈</mo><mi id="S5.I5.i3.I1.ix2.p2.15.m15.1.1.3" xref="S5.I5.i3.I1.ix2.p2.15.m15.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.15.m15.1b"><apply id="S5.I5.i3.I1.ix2.p2.15.m15.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.15.m15.1.1"><in id="S5.I5.i3.I1.ix2.p2.15.m15.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.15.m15.1.1.1"></in><ci id="S5.I5.i3.I1.ix2.p2.15.m15.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.15.m15.1.1.2">𝐱</ci><ci id="S5.I5.i3.I1.ix2.p2.15.m15.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.15.m15.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.15.m15.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.15.m15.1d">bold_x ∈ italic_X</annotation></semantics></math> which is different from <math alttext="a^{\prime\pm\infty}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.16.m16.2"><semantics id="S5.I5.i3.I1.ix2.p2.16.m16.2a"><msup id="S5.I5.i3.I1.ix2.p2.16.m16.2.3" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.16.m16.2.3.2" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.16.m16.1.1.1.1.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.16.m16.1.1.1.1.1.cmml">′</mo><mo id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.3" lspace="0em" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.3.cmml">⁣</mo><mrow id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2.cmml"><mo id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2a" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2.cmml">±</mo><mi id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2.2" mathvariant="normal" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2.2.cmml">∞</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.16.m16.2b"><apply id="S5.I5.i3.I1.ix2.p2.16.m16.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.16.m16.2.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.16.m16.2.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2"><ci id="S5.I5.i3.I1.ix2.p2.16.m16.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.16.m16.1.1.1.1.1">′</ci><apply id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2"><csymbol cd="latexml" id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2">plus-or-minus</csymbol><infinity id="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.16.m16.2.2.2.2.2.2"></infinity></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.16.m16.2c">a^{\prime\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.16.m16.2d">italic_a start_POSTSUPERSCRIPT ′ ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="a^{\prime\prime\pm\infty}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.17.m17.2"><semantics id="S5.I5.i3.I1.ix2.p2.17.m17.2a"><msup id="S5.I5.i3.I1.ix2.p2.17.m17.2.3" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.17.m17.2.3.2" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.3.2.cmml">a</mi><mrow id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.17.m17.1.1.1.1.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.17.m17.1.1.1.1.1.cmml">′′</mo><mo id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.3" lspace="0em" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.3.cmml">⁣</mo><mrow id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2.cmml"><mo id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2a" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2.cmml">±</mo><mi id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2.2" mathvariant="normal" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2.2.cmml">∞</mi></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.17.m17.2b"><apply id="S5.I5.i3.I1.ix2.p2.17.m17.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.17.m17.2.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.17.m17.2.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.3.2">𝑎</ci><list id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2"><ci id="S5.I5.i3.I1.ix2.p2.17.m17.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.17.m17.1.1.1.1.1">′′</ci><apply id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2"><csymbol cd="latexml" id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2">plus-or-minus</csymbol><infinity id="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.17.m17.2.2.2.2.2.2"></infinity></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.17.m17.2c">a^{\prime\prime\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.17.m17.2d">italic_a start_POSTSUPERSCRIPT ′ ′ ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> has positive half-orbit which is dense in <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.18.m18.1"><semantics id="S5.I5.i3.I1.ix2.p2.18.m18.1a"><mi id="S5.I5.i3.I1.ix2.p2.18.m18.1.1" xref="S5.I5.i3.I1.ix2.p2.18.m18.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.18.m18.1b"><ci id="S5.I5.i3.I1.ix2.p2.18.m18.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.18.m18.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.18.m18.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.18.m18.1d">italic_X</annotation></semantics></math>. For <math alttext="\cal A^{\prime}=\cal A\cup\{a^{\prime}\}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.19.m19.1"><semantics id="S5.I5.i3.I1.ix2.p2.19.m19.1a"><mrow id="S5.I5.i3.I1.ix2.p2.19.m19.1.1" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.cmml"><msup id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.2" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.2.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.3" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.3.cmml">′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.2" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.2.cmml">=</mo><mrow id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.3.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.2.cmml">∪</mo><mrow id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.2.cmml"><mo id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.2.cmml">{</mo><msup id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.2.cmml">𝒶</mi><mo id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.3" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.19.m19.1b"><apply id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1"><eq id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.2"></eq><apply id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.2">𝒜</ci><ci id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.3.3">′</ci></apply><apply id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1"><union id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.2"></union><ci id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.3">𝒜</ci><set id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1"><apply id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.2">𝒶</ci><ci id="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.19.m19.1.1.1.1.1.1.3">′</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.19.m19.1c">\cal A^{\prime}=\cal A\cup\{a^{\prime}\}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.19.m19.1d">caligraphic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = caligraphic_A ∪ { caligraphic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT }</annotation></semantics></math> we now define <math alttext="\sigma:\cal A^{\prime\prime*}\to\cal A^{\prime*}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.20.m20.4"><semantics id="S5.I5.i3.I1.ix2.p2.20.m20.4a"><mrow id="S5.I5.i3.I1.ix2.p2.20.m20.4.5" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.cmml"><mi id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.2" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.2.cmml">σ</mi><mo id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.1" lspace="0.278em" rspace="0.278em" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.1.cmml">:</mo><mrow id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.cmml"><msup id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2.2" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2.2.cmml">𝒜</mi><mrow id="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.2.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.2.1.cmml">′′</mo><mo id="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.2.2" lspace="0.222em" xref="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.3.cmml">⁣</mo><mo id="S5.I5.i3.I1.ix2.p2.20.m20.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.20.m20.1.1.1.1.cmml">∗</mo></mrow></msup><mo id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.1" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.1.cmml">→</mo><msup id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3.2" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3.2.cmml">𝒜</mi><mrow id="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.2" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.2.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.2.1.cmml">′</mo><mo id="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.2.2" lspace="0.222em" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.3.cmml">⁣</mo><mo id="S5.I5.i3.I1.ix2.p2.20.m20.3.3.1.1" xref="S5.I5.i3.I1.ix2.p2.20.m20.3.3.1.1.cmml">∗</mo></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.20.m20.4b"><apply id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5"><ci id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.1.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.1">:</ci><ci id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.2.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.2">𝜎</ci><apply id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3"><ci id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.1">→</ci><apply id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.2.2">𝒜</ci><list id="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.2"><ci id="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.2.2.2.2.1">′′</ci><times id="S5.I5.i3.I1.ix2.p2.20.m20.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.1.1.1.1"></times></list></apply><apply id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.5.3.3.2">𝒜</ci><list id="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.2"><ci id="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.4.4.2.2.1">′</ci><times id="S5.I5.i3.I1.ix2.p2.20.m20.3.3.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.20.m20.3.3.1.1"></times></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.20.m20.4c">\sigma:\cal A^{\prime\prime*}\to\cal A^{\prime*}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.20.m20.4d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ′ ′ ∗ end_POSTSUPERSCRIPT → caligraphic_A start_POSTSUPERSCRIPT ′ ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> to be the identity on <math alttext="\cal A^{*}\subseteq\cal A^{\prime\prime*}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.21.m21.2"><semantics id="S5.I5.i3.I1.ix2.p2.21.m21.2a"><mrow id="S5.I5.i3.I1.ix2.p2.21.m21.2.3" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.cmml"><msup id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.2" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.2.cmml">𝒜</mi><mo id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.3" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.3.cmml">∗</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.1" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.1.cmml">⊆</mo><msup id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3.2" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3.2.cmml">𝒜</mi><mrow id="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.3.cmml"><mo id="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.2.1" mathsize="142%" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.2.1.cmml">′′</mo><mo id="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.2.2" lspace="0.222em" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.3.cmml">⁣</mo><mo id="S5.I5.i3.I1.ix2.p2.21.m21.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.21.m21.1.1.1.1.cmml">∗</mo></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.21.m21.2b"><apply id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3"><subset id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.1"></subset><apply id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.2">𝒜</ci><times id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.2.3"></times></apply><apply id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.3.3.2">𝒜</ci><list id="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.2"><ci id="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.2.2.2.2.1">′′</ci><times id="S5.I5.i3.I1.ix2.p2.21.m21.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.21.m21.1.1.1.1"></times></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.21.m21.2c">\cal A^{*}\subseteq\cal A^{\prime\prime*}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.21.m21.2d">caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⊆ caligraphic_A start_POSTSUPERSCRIPT ′ ′ ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and by setting <math alttext="\sigma(a^{\prime})=\sigma(a^{\prime\prime})=a^{\prime}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.22.m22.2"><semantics id="S5.I5.i3.I1.ix2.p2.22.m22.2a"><mrow id="S5.I5.i3.I1.ix2.p2.22.m22.2.2" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.cmml"><mrow id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.3.cmml">σ</mi><mo id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.2.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.cmml"><mo id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.cmml">(</mo><msup id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.3" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.4" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.4.cmml">=</mo><mrow id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.cmml"><mi id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.3" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.3.cmml">σ</mi><mo id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.2" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.2.cmml">⁢</mo><mrow id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.cmml"><mo id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.2" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.cmml">(</mo><msup id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.2" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.3" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.3.cmml">′′</mo></msup><mo id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.3" stretchy="false" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.5" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.5.cmml">=</mo><msup id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.cmml"><mi id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.2" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.3" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.22.m22.2b"><apply id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2"><and id="S5.I5.i3.I1.ix2.p2.22.m22.2.2a.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2"></and><apply id="S5.I5.i3.I1.ix2.p2.22.m22.2.2b.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2"><eq id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.4.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.4"></eq><apply id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1"><times id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.2"></times><ci id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.3">𝜎</ci><apply id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2"><times id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.2"></times><ci id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.3">𝜎</ci><apply id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.1.1.1.3">′′</ci></apply></apply></apply><apply id="S5.I5.i3.I1.ix2.p2.22.m22.2.2c.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2"><eq id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.5.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S5.I5.i3.I1.ix2.p2.22.m22.2.2.2.cmml" id="S5.I5.i3.I1.ix2.p2.22.m22.2.2d.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2"></share><apply id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.1.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.2.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.3.cmml" xref="S5.I5.i3.I1.ix2.p2.22.m22.2.2.6.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.22.m22.2c">\sigma(a^{\prime})=\sigma(a^{\prime\prime})=a^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.22.m22.2d">italic_σ ( italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_σ ( italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) = italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Then <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.23.m23.1"><semantics id="S5.I5.i3.I1.ix2.p2.23.m23.1a"><mi id="S5.I5.i3.I1.ix2.p2.23.m23.1.1" xref="S5.I5.i3.I1.ix2.p2.23.m23.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.23.m23.1b"><ci id="S5.I5.i3.I1.ix2.p2.23.m23.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.23.m23.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.23.m23.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.23.m23.1d">italic_σ</annotation></semantics></math> is recognizable for aperiodic points in <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.24.m24.1"><semantics id="S5.I5.i3.I1.ix2.p2.24.m24.1a"><mi id="S5.I5.i3.I1.ix2.p2.24.m24.1.1" xref="S5.I5.i3.I1.ix2.p2.24.m24.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.24.m24.1b"><ci id="S5.I5.i3.I1.ix2.p2.24.m24.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.24.m24.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.24.m24.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.24.m24.1d">italic_X</annotation></semantics></math>, and the two characteristic measures <math alttext="\mu_{a^{\prime}}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.25.m25.1"><semantics id="S5.I5.i3.I1.ix2.p2.25.m25.1a"><msub id="S5.I5.i3.I1.ix2.p2.25.m25.1.1" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.2" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.2.cmml">μ</mi><msup id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.2" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.3" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.25.m25.1b"><apply id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1">subscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.2">𝜇</ci><apply id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.25.m25.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.25.m25.1c">\mu_{a^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.25.m25.1d">italic_μ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mu_{a^{\prime\prime}}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.26.m26.1"><semantics id="S5.I5.i3.I1.ix2.p2.26.m26.1a"><msub id="S5.I5.i3.I1.ix2.p2.26.m26.1.1" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.2" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.2.cmml">μ</mi><msup id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.2" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.3" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.3.cmml">′′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.26.m26.1b"><apply id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1">subscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.2">𝜇</ci><apply id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.26.m26.1.1.3.3">′′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.26.m26.1c">\mu_{a^{\prime\prime}}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.26.m26.1d">italic_μ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> are distinct, but have the same image <math alttext="\mu_{a^{\prime}}" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.27.m27.1"><semantics id="S5.I5.i3.I1.ix2.p2.27.m27.1a"><msub id="S5.I5.i3.I1.ix2.p2.27.m27.1.1" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.cmml"><mi id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.2" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.2.cmml">μ</mi><msup id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.cmml"><mi id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.2" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.2.cmml">a</mi><mo id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.3" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.3.cmml">′</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.27.m27.1b"><apply id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1">subscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.2">𝜇</ci><apply id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.1.cmml" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3">superscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.2.cmml" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.2">𝑎</ci><ci id="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.3.cmml" xref="S5.I5.i3.I1.ix2.p2.27.m27.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.27.m27.1c">\mu_{a^{\prime}}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.27.m27.1d">italic_μ start_POSTSUBSCRIPT italic_a start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> under the measure transfer map <math alttext="\sigma_{X}M" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix2.p2.28.m28.1"><semantics id="S5.I5.i3.I1.ix2.p2.28.m28.1a"><mrow id="S5.I5.i3.I1.ix2.p2.28.m28.1.1" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.cmml"><msub id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.cmml"><mi id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.2" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.2.cmml">σ</mi><mi id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.3" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.3.cmml">X</mi></msub><mo id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.1" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.1.cmml">⁢</mo><mi id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.3" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix2.p2.28.m28.1b"><apply id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1"><times id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.1.cmml" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.1"></times><apply id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.cmml" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2"><csymbol cd="ambiguous" id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.1.cmml" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2">subscript</csymbol><ci id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.2.cmml" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.2">𝜎</ci><ci id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.3.cmml" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.2.3">𝑋</ci></apply><ci id="S5.I5.i3.I1.ix2.p2.28.m28.1.1.3.cmml" xref="S5.I5.i3.I1.ix2.p2.28.m28.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix2.p2.28.m28.1c">\sigma_{X}M</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix2.p2.28.m28.1d">italic_σ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_M</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S5.I5.i3.I1.ix3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(c)</span> <div class="ltx_para" id="S5.I5.i3.I1.ix3.p1"> <p class="ltx_p" id="S5.I5.i3.I1.ix3.p1.1">“measure transfer injective” <math alttext="\,\,\Longrightarrow\!\!\!\!\!\!\!\!\!/\quad" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix3.p1.1.m1.1"><semantics id="S5.I5.i3.I1.ix3.p1.1.m1.1a"><mrow id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.cmml"><mrow id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.cmml"><mi id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.2" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.2.cmml"></mi><mpadded width="0em"><mo id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.1" lspace="0.608em" stretchy="false" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.1.cmml">⟹</mo></mpadded><mo id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.3" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.3.cmml">/</mo></mrow><mspace id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.2" width="1em" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.cmml"></mspace></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix3.p1.1.m1.1b"><apply id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1"><ci id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.1.cmml" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.1">⟹</ci><csymbol cd="latexml" id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.2.cmml" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.2">absent</csymbol><divide id="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.3.cmml" xref="S5.I5.i3.I1.ix3.p1.1.m1.1.1.1.1.3"></divide></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix3.p1.1.m1.1c">\,\,\Longrightarrow\!\!\!\!\!\!\!\!\!/\quad</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix3.p1.1.m1.1d">⟹ /</annotation></semantics></math> “recognizable for aperiodic points”:</p> </div> <div class="ltx_para" id="S5.I5.i3.I1.ix3.p2"> <p class="ltx_p" id="S5.I5.i3.I1.ix3.p2.2">Here we can point to the infinite minimal substitution subshift <math alttext="X" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix3.p2.1.m1.1"><semantics id="S5.I5.i3.I1.ix3.p2.1.m1.1a"><mi id="S5.I5.i3.I1.ix3.p2.1.m1.1.1" xref="S5.I5.i3.I1.ix3.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix3.p2.1.m1.1b"><ci id="S5.I5.i3.I1.ix3.p2.1.m1.1.1.cmml" xref="S5.I5.i3.I1.ix3.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix3.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix3.p2.1.m1.1d">italic_X</annotation></semantics></math> and the 2 : 1 morphism <math alttext="\sigma" class="ltx_Math" display="inline" id="S5.I5.i3.I1.ix3.p2.2.m2.1"><semantics id="S5.I5.i3.I1.ix3.p2.2.m2.1a"><mi id="S5.I5.i3.I1.ix3.p2.2.m2.1.1" xref="S5.I5.i3.I1.ix3.p2.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i3.I1.ix3.p2.2.m2.1b"><ci id="S5.I5.i3.I1.ix3.p2.2.m2.1.1.cmml" xref="S5.I5.i3.I1.ix3.p2.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i3.I1.ix3.p2.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i3.I1.ix3.p2.2.m2.1d">italic_σ</annotation></semantics></math> from Example 4.3 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib5" title="">5</a>]</cite> as cited above in Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm13" title="Remark 5.13. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.13</span></a> (1), which has all the required properties.</p> </div> </li> </ol> </div> </li> </ol> </div> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6. </span>The injectivity of the measure transfer for letter-to-letter morphisms</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.4">Throughout this section we assume that <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S6.p1.1.m1.1"><semantics id="S6.p1.1.m1.1a"><mrow id="S6.p1.1.m1.1.1" xref="S6.p1.1.m1.1.1.cmml"><mi id="S6.p1.1.m1.1.1.2" xref="S6.p1.1.m1.1.1.2.cmml">σ</mi><mo id="S6.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.p1.1.m1.1.1.1.cmml">:</mo><mrow id="S6.p1.1.m1.1.1.3" xref="S6.p1.1.m1.1.1.3.cmml"><msup id="S6.p1.1.m1.1.1.3.2" xref="S6.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p1.1.m1.1.1.3.2.2" xref="S6.p1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S6.p1.1.m1.1.1.3.2.3" xref="S6.p1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S6.p1.1.m1.1.1.3.1" stretchy="false" xref="S6.p1.1.m1.1.1.3.1.cmml">→</mo><msup id="S6.p1.1.m1.1.1.3.3" xref="S6.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p1.1.m1.1.1.3.3.2" xref="S6.p1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S6.p1.1.m1.1.1.3.3.3" xref="S6.p1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.1.m1.1b"><apply id="S6.p1.1.m1.1.1.cmml" xref="S6.p1.1.m1.1.1"><ci id="S6.p1.1.m1.1.1.1.cmml" xref="S6.p1.1.m1.1.1.1">:</ci><ci id="S6.p1.1.m1.1.1.2.cmml" xref="S6.p1.1.m1.1.1.2">𝜎</ci><apply id="S6.p1.1.m1.1.1.3.cmml" xref="S6.p1.1.m1.1.1.3"><ci id="S6.p1.1.m1.1.1.3.1.cmml" xref="S6.p1.1.m1.1.1.3.1">→</ci><apply id="S6.p1.1.m1.1.1.3.2.cmml" xref="S6.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.p1.1.m1.1.1.3.2.1.cmml" xref="S6.p1.1.m1.1.1.3.2">superscript</csymbol><ci id="S6.p1.1.m1.1.1.3.2.2.cmml" xref="S6.p1.1.m1.1.1.3.2.2">𝒜</ci><times id="S6.p1.1.m1.1.1.3.2.3.cmml" xref="S6.p1.1.m1.1.1.3.2.3"></times></apply><apply id="S6.p1.1.m1.1.1.3.3.cmml" xref="S6.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.p1.1.m1.1.1.3.3.1.cmml" xref="S6.p1.1.m1.1.1.3.3">superscript</csymbol><ci id="S6.p1.1.m1.1.1.3.3.2.cmml" xref="S6.p1.1.m1.1.1.3.3.2">ℬ</ci><times id="S6.p1.1.m1.1.1.3.3.3.cmml" xref="S6.p1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is a letter-to-letter morphism of free monoids, and that <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.p1.2.m2.1"><semantics id="S6.p1.2.m2.1a"><mrow id="S6.p1.2.m2.1.1" xref="S6.p1.2.m2.1.1.cmml"><mi id="S6.p1.2.m2.1.1.2" xref="S6.p1.2.m2.1.1.2.cmml">X</mi><mo id="S6.p1.2.m2.1.1.1" xref="S6.p1.2.m2.1.1.1.cmml">⊆</mo><msup id="S6.p1.2.m2.1.1.3" xref="S6.p1.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p1.2.m2.1.1.3.2" xref="S6.p1.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S6.p1.2.m2.1.1.3.3" xref="S6.p1.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.2.m2.1b"><apply id="S6.p1.2.m2.1.1.cmml" xref="S6.p1.2.m2.1.1"><subset id="S6.p1.2.m2.1.1.1.cmml" xref="S6.p1.2.m2.1.1.1"></subset><ci id="S6.p1.2.m2.1.1.2.cmml" xref="S6.p1.2.m2.1.1.2">𝑋</ci><apply id="S6.p1.2.m2.1.1.3.cmml" xref="S6.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.p1.2.m2.1.1.3.1.cmml" xref="S6.p1.2.m2.1.1.3">superscript</csymbol><ci id="S6.p1.2.m2.1.1.3.2.cmml" xref="S6.p1.2.m2.1.1.3.2">𝒜</ci><ci id="S6.p1.2.m2.1.1.3.3.cmml" xref="S6.p1.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.p1.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> is a given subshift. From Definition-Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm6" title="Definition-Remark 3.6. ‣ 3.3. The induced measure morphisms ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.6</span></a> we observe directly that in this special case the induced measure transfer map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S6.p1.3.m3.1"><semantics id="S6.p1.3.m3.1a"><mrow id="S6.p1.3.m3.1.1" xref="S6.p1.3.m3.1.1.cmml"><mi id="S6.p1.3.m3.1.1.2" xref="S6.p1.3.m3.1.1.2.cmml">σ</mi><mo id="S6.p1.3.m3.1.1.1" xref="S6.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="S6.p1.3.m3.1.1.3" xref="S6.p1.3.m3.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.3.m3.1b"><apply id="S6.p1.3.m3.1.1.cmml" xref="S6.p1.3.m3.1.1"><times id="S6.p1.3.m3.1.1.1.cmml" xref="S6.p1.3.m3.1.1.1"></times><ci id="S6.p1.3.m3.1.1.2.cmml" xref="S6.p1.3.m3.1.1.2">𝜎</ci><ci id="S6.p1.3.m3.1.1.3.cmml" xref="S6.p1.3.m3.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.3.m3.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S6.p1.3.m3.1d">italic_σ italic_M</annotation></semantics></math> is given by the classical push-forward map <math alttext="\mu\mapsto\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.p1.4.m4.1"><semantics id="S6.p1.4.m4.1a"><mrow id="S6.p1.4.m4.1.2" xref="S6.p1.4.m4.1.2.cmml"><mi id="S6.p1.4.m4.1.2.2" xref="S6.p1.4.m4.1.2.2.cmml">μ</mi><mo id="S6.p1.4.m4.1.2.1" stretchy="false" xref="S6.p1.4.m4.1.2.1.cmml">↦</mo><mrow id="S6.p1.4.m4.1.2.3" xref="S6.p1.4.m4.1.2.3.cmml"><msub id="S6.p1.4.m4.1.2.3.2" xref="S6.p1.4.m4.1.2.3.2.cmml"><mi id="S6.p1.4.m4.1.2.3.2.2" xref="S6.p1.4.m4.1.2.3.2.2.cmml">σ</mi><mo id="S6.p1.4.m4.1.2.3.2.3" xref="S6.p1.4.m4.1.2.3.2.3.cmml">∗</mo></msub><mo id="S6.p1.4.m4.1.2.3.1" xref="S6.p1.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S6.p1.4.m4.1.2.3.3.2" xref="S6.p1.4.m4.1.2.3.cmml"><mo id="S6.p1.4.m4.1.2.3.3.2.1" stretchy="false" xref="S6.p1.4.m4.1.2.3.cmml">(</mo><mi id="S6.p1.4.m4.1.1" xref="S6.p1.4.m4.1.1.cmml">μ</mi><mo id="S6.p1.4.m4.1.2.3.3.2.2" stretchy="false" xref="S6.p1.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p1.4.m4.1b"><apply id="S6.p1.4.m4.1.2.cmml" xref="S6.p1.4.m4.1.2"><csymbol cd="latexml" id="S6.p1.4.m4.1.2.1.cmml" xref="S6.p1.4.m4.1.2.1">maps-to</csymbol><ci id="S6.p1.4.m4.1.2.2.cmml" xref="S6.p1.4.m4.1.2.2">𝜇</ci><apply id="S6.p1.4.m4.1.2.3.cmml" xref="S6.p1.4.m4.1.2.3"><times id="S6.p1.4.m4.1.2.3.1.cmml" xref="S6.p1.4.m4.1.2.3.1"></times><apply id="S6.p1.4.m4.1.2.3.2.cmml" xref="S6.p1.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S6.p1.4.m4.1.2.3.2.1.cmml" xref="S6.p1.4.m4.1.2.3.2">subscript</csymbol><ci id="S6.p1.4.m4.1.2.3.2.2.cmml" xref="S6.p1.4.m4.1.2.3.2.2">𝜎</ci><times id="S6.p1.4.m4.1.2.3.2.3.cmml" xref="S6.p1.4.m4.1.2.3.2.3"></times></apply><ci id="S6.p1.4.m4.1.1.cmml" xref="S6.p1.4.m4.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p1.4.m4.1c">\mu\mapsto\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.p1.4.m4.1d">italic_μ ↦ italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="S6.Thmthm1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm1.1.1.1">Lemma 6.1</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm1.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm1.p1"> <p class="ltx_p" id="S6.Thmthm1.p1.5"><span class="ltx_text ltx_font_italic" id="S6.Thmthm1.p1.5.5">If <math alttext="\sigma" class="ltx_Math" display="inline" id="S6.Thmthm1.p1.1.1.m1.1"><semantics id="S6.Thmthm1.p1.1.1.m1.1a"><mi id="S6.Thmthm1.p1.1.1.m1.1.1" xref="S6.Thmthm1.p1.1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm1.p1.1.1.m1.1b"><ci id="S6.Thmthm1.p1.1.1.m1.1.1.cmml" xref="S6.Thmthm1.p1.1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm1.p1.1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm1.p1.1.1.m1.1d">italic_σ</annotation></semantics></math> is shift-orbit injective in <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm1.p1.2.2.m2.1"><semantics id="S6.Thmthm1.p1.2.2.m2.1a"><mi id="S6.Thmthm1.p1.2.2.m2.1.1" xref="S6.Thmthm1.p1.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm1.p1.2.2.m2.1b"><ci id="S6.Thmthm1.p1.2.2.m2.1.1.cmml" xref="S6.Thmthm1.p1.2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm1.p1.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm1.p1.2.2.m2.1d">italic_X</annotation></semantics></math>, and if for some <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S6.Thmthm1.p1.3.3.m3.1"><semantics id="S6.Thmthm1.p1.3.3.m3.1a"><mrow id="S6.Thmthm1.p1.3.3.m3.1.1" xref="S6.Thmthm1.p1.3.3.m3.1.1.cmml"><mi id="S6.Thmthm1.p1.3.3.m3.1.1.2" xref="S6.Thmthm1.p1.3.3.m3.1.1.2.cmml">𝐱</mi><mo id="S6.Thmthm1.p1.3.3.m3.1.1.1" xref="S6.Thmthm1.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S6.Thmthm1.p1.3.3.m3.1.1.3" xref="S6.Thmthm1.p1.3.3.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm1.p1.3.3.m3.1b"><apply id="S6.Thmthm1.p1.3.3.m3.1.1.cmml" xref="S6.Thmthm1.p1.3.3.m3.1.1"><in id="S6.Thmthm1.p1.3.3.m3.1.1.1.cmml" xref="S6.Thmthm1.p1.3.3.m3.1.1.1"></in><ci id="S6.Thmthm1.p1.3.3.m3.1.1.2.cmml" xref="S6.Thmthm1.p1.3.3.m3.1.1.2">𝐱</ci><ci id="S6.Thmthm1.p1.3.3.m3.1.1.3.cmml" xref="S6.Thmthm1.p1.3.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm1.p1.3.3.m3.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm1.p1.3.3.m3.1d">bold_x ∈ italic_X</annotation></semantics></math> the image word <math alttext="\sigma({\bf x})" class="ltx_Math" display="inline" id="S6.Thmthm1.p1.4.4.m4.1"><semantics id="S6.Thmthm1.p1.4.4.m4.1a"><mrow id="S6.Thmthm1.p1.4.4.m4.1.2" xref="S6.Thmthm1.p1.4.4.m4.1.2.cmml"><mi id="S6.Thmthm1.p1.4.4.m4.1.2.2" xref="S6.Thmthm1.p1.4.4.m4.1.2.2.cmml">σ</mi><mo id="S6.Thmthm1.p1.4.4.m4.1.2.1" xref="S6.Thmthm1.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S6.Thmthm1.p1.4.4.m4.1.2.3.2" xref="S6.Thmthm1.p1.4.4.m4.1.2.cmml"><mo id="S6.Thmthm1.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S6.Thmthm1.p1.4.4.m4.1.2.cmml">(</mo><mi id="S6.Thmthm1.p1.4.4.m4.1.1" xref="S6.Thmthm1.p1.4.4.m4.1.1.cmml">𝐱</mi><mo id="S6.Thmthm1.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S6.Thmthm1.p1.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm1.p1.4.4.m4.1b"><apply id="S6.Thmthm1.p1.4.4.m4.1.2.cmml" xref="S6.Thmthm1.p1.4.4.m4.1.2"><times id="S6.Thmthm1.p1.4.4.m4.1.2.1.cmml" xref="S6.Thmthm1.p1.4.4.m4.1.2.1"></times><ci id="S6.Thmthm1.p1.4.4.m4.1.2.2.cmml" xref="S6.Thmthm1.p1.4.4.m4.1.2.2">𝜎</ci><ci id="S6.Thmthm1.p1.4.4.m4.1.1.cmml" xref="S6.Thmthm1.p1.4.4.m4.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm1.p1.4.4.m4.1c">\sigma({\bf x})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm1.p1.4.4.m4.1d">italic_σ ( bold_x )</annotation></semantics></math> is periodic, then <math alttext="\bf x" class="ltx_Math" display="inline" id="S6.Thmthm1.p1.5.5.m5.1"><semantics id="S6.Thmthm1.p1.5.5.m5.1a"><mi id="S6.Thmthm1.p1.5.5.m5.1.1" xref="S6.Thmthm1.p1.5.5.m5.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm1.p1.5.5.m5.1b"><ci id="S6.Thmthm1.p1.5.5.m5.1.1.cmml" xref="S6.Thmthm1.p1.5.5.m5.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm1.p1.5.5.m5.1c">\bf x</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm1.p1.5.5.m5.1d">bold_x</annotation></semantics></math> is periodic too.</span></p> </div> </div> <div class="ltx_proof" id="S6.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.1.p1"> <p class="ltx_p" id="S6.1.p1.3">From the definition of the image subhift (see Definition-Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm2" title="Definition-Remark 2.2. ‣ 2.2. “Not so standard” basic facts and terminology ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.2</span></a> and Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.Thmthm4" title="Lemma 2.4. ‣ 2.2. “Not so standard” basic facts and terminology ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.4</span></a>) it follows directly that for any biinfinite word <math alttext="{\bf x}\in X" class="ltx_Math" display="inline" id="S6.1.p1.1.m1.1"><semantics id="S6.1.p1.1.m1.1a"><mrow id="S6.1.p1.1.m1.1.1" xref="S6.1.p1.1.m1.1.1.cmml"><mi id="S6.1.p1.1.m1.1.1.2" xref="S6.1.p1.1.m1.1.1.2.cmml">𝐱</mi><mo id="S6.1.p1.1.m1.1.1.1" xref="S6.1.p1.1.m1.1.1.1.cmml">∈</mo><mi id="S6.1.p1.1.m1.1.1.3" xref="S6.1.p1.1.m1.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.1.m1.1b"><apply id="S6.1.p1.1.m1.1.1.cmml" xref="S6.1.p1.1.m1.1.1"><in id="S6.1.p1.1.m1.1.1.1.cmml" xref="S6.1.p1.1.m1.1.1.1"></in><ci id="S6.1.p1.1.m1.1.1.2.cmml" xref="S6.1.p1.1.m1.1.1.2">𝐱</ci><ci id="S6.1.p1.1.m1.1.1.3.cmml" xref="S6.1.p1.1.m1.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.1.m1.1c">{\bf x}\in X</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.1.m1.1d">bold_x ∈ italic_X</annotation></semantics></math> the closure <math alttext="\overline{\cal O({\bf x})}" class="ltx_Math" display="inline" id="S6.1.p1.2.m2.1"><semantics id="S6.1.p1.2.m2.1a"><mover accent="true" id="S6.1.p1.2.m2.1.1" xref="S6.1.p1.2.m2.1.1.cmml"><mrow id="S6.1.p1.2.m2.1.1.1" xref="S6.1.p1.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.1.p1.2.m2.1.1.1.3" xref="S6.1.p1.2.m2.1.1.1.3.cmml">𝒪</mi><mo id="S6.1.p1.2.m2.1.1.1.2" xref="S6.1.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S6.1.p1.2.m2.1.1.1.4.2" xref="S6.1.p1.2.m2.1.1.1.cmml"><mo id="S6.1.p1.2.m2.1.1.1.4.2.1" stretchy="false" xref="S6.1.p1.2.m2.1.1.1.cmml">(</mo><mi id="S6.1.p1.2.m2.1.1.1.1" xref="S6.1.p1.2.m2.1.1.1.1.cmml">𝐱</mi><mo id="S6.1.p1.2.m2.1.1.1.4.2.2" stretchy="false" xref="S6.1.p1.2.m2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.1.p1.2.m2.1.1.2" xref="S6.1.p1.2.m2.1.1.2.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S6.1.p1.2.m2.1b"><apply id="S6.1.p1.2.m2.1.1.cmml" xref="S6.1.p1.2.m2.1.1"><ci id="S6.1.p1.2.m2.1.1.2.cmml" xref="S6.1.p1.2.m2.1.1.2">¯</ci><apply id="S6.1.p1.2.m2.1.1.1.cmml" xref="S6.1.p1.2.m2.1.1.1"><times id="S6.1.p1.2.m2.1.1.1.2.cmml" xref="S6.1.p1.2.m2.1.1.1.2"></times><ci id="S6.1.p1.2.m2.1.1.1.3.cmml" xref="S6.1.p1.2.m2.1.1.1.3">𝒪</ci><ci id="S6.1.p1.2.m2.1.1.1.1.cmml" xref="S6.1.p1.2.m2.1.1.1.1">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.2.m2.1c">\overline{\cal O({\bf x})}</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.2.m2.1d">over¯ start_ARG caligraphic_O ( bold_x ) end_ARG</annotation></semantics></math> of its orbit <math alttext="\cal O({\bf x})" class="ltx_Math" display="inline" id="S6.1.p1.3.m3.1"><semantics id="S6.1.p1.3.m3.1a"><mrow id="S6.1.p1.3.m3.1.2" xref="S6.1.p1.3.m3.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.1.p1.3.m3.1.2.2" xref="S6.1.p1.3.m3.1.2.2.cmml">𝒪</mi><mo id="S6.1.p1.3.m3.1.2.1" xref="S6.1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.1.p1.3.m3.1.2.3.2" xref="S6.1.p1.3.m3.1.2.cmml"><mo id="S6.1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S6.1.p1.3.m3.1.2.cmml">(</mo><mi id="S6.1.p1.3.m3.1.1" xref="S6.1.p1.3.m3.1.1.cmml">𝐱</mi><mo id="S6.1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S6.1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.1.p1.3.m3.1b"><apply id="S6.1.p1.3.m3.1.2.cmml" xref="S6.1.p1.3.m3.1.2"><times id="S6.1.p1.3.m3.1.2.1.cmml" xref="S6.1.p1.3.m3.1.2.1"></times><ci id="S6.1.p1.3.m3.1.2.2.cmml" xref="S6.1.p1.3.m3.1.2.2">𝒪</ci><ci id="S6.1.p1.3.m3.1.1.cmml" xref="S6.1.p1.3.m3.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.3.m3.1c">\cal O({\bf x})</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.3.m3.1d">caligraphic_O ( bold_x )</annotation></semantics></math> is a subshift which satisfies</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma(\overline{\cal O({\bf x})})=\overline{\cal O(\sigma({\bf x}))}\,." class="ltx_Math" display="block" id="S6.Ex1.m1.4"><semantics id="S6.Ex1.m1.4a"><mrow id="S6.Ex1.m1.4.4.1" xref="S6.Ex1.m1.4.4.1.1.cmml"><mrow id="S6.Ex1.m1.4.4.1.1" xref="S6.Ex1.m1.4.4.1.1.cmml"><mrow id="S6.Ex1.m1.4.4.1.1.2" xref="S6.Ex1.m1.4.4.1.1.2.cmml"><mi id="S6.Ex1.m1.4.4.1.1.2.2" xref="S6.Ex1.m1.4.4.1.1.2.2.cmml">σ</mi><mo id="S6.Ex1.m1.4.4.1.1.2.1" xref="S6.Ex1.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S6.Ex1.m1.4.4.1.1.2.3.2" xref="S6.Ex1.m1.1.1.cmml"><mo id="S6.Ex1.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="S6.Ex1.m1.1.1.cmml">(</mo><mover accent="true" id="S6.Ex1.m1.1.1" xref="S6.Ex1.m1.1.1.cmml"><mrow id="S6.Ex1.m1.1.1.1" xref="S6.Ex1.m1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex1.m1.1.1.1.3" xref="S6.Ex1.m1.1.1.1.3.cmml">𝒪</mi><mo id="S6.Ex1.m1.1.1.1.2" xref="S6.Ex1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex1.m1.1.1.1.4.2" xref="S6.Ex1.m1.1.1.1.cmml"><mo id="S6.Ex1.m1.1.1.1.4.2.1" stretchy="false" xref="S6.Ex1.m1.1.1.1.cmml">(</mo><mi id="S6.Ex1.m1.1.1.1.1" xref="S6.Ex1.m1.1.1.1.1.cmml">𝐱</mi><mo id="S6.Ex1.m1.1.1.1.4.2.2" stretchy="false" xref="S6.Ex1.m1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex1.m1.1.1.2" xref="S6.Ex1.m1.1.1.2.cmml">¯</mo></mover><mo id="S6.Ex1.m1.4.4.1.1.2.3.2.2" stretchy="false" xref="S6.Ex1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex1.m1.4.4.1.1.1" xref="S6.Ex1.m1.4.4.1.1.1.cmml">=</mo><mover accent="true" id="S6.Ex1.m1.3.3" xref="S6.Ex1.m1.3.3.cmml"><mrow id="S6.Ex1.m1.3.3.2" xref="S6.Ex1.m1.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex1.m1.3.3.2.4" xref="S6.Ex1.m1.3.3.2.4.cmml">𝒪</mi><mo id="S6.Ex1.m1.3.3.2.3" xref="S6.Ex1.m1.3.3.2.3.cmml">⁢</mo><mrow id="S6.Ex1.m1.3.3.2.2.1" xref="S6.Ex1.m1.3.3.2.2.1.1.cmml"><mo id="S6.Ex1.m1.3.3.2.2.1.2" stretchy="false" xref="S6.Ex1.m1.3.3.2.2.1.1.cmml">(</mo><mrow id="S6.Ex1.m1.3.3.2.2.1.1" xref="S6.Ex1.m1.3.3.2.2.1.1.cmml"><mi id="S6.Ex1.m1.3.3.2.2.1.1.2" xref="S6.Ex1.m1.3.3.2.2.1.1.2.cmml">σ</mi><mo id="S6.Ex1.m1.3.3.2.2.1.1.1" xref="S6.Ex1.m1.3.3.2.2.1.1.1.cmml">⁢</mo><mrow id="S6.Ex1.m1.3.3.2.2.1.1.3.2" xref="S6.Ex1.m1.3.3.2.2.1.1.cmml"><mo id="S6.Ex1.m1.3.3.2.2.1.1.3.2.1" stretchy="false" xref="S6.Ex1.m1.3.3.2.2.1.1.cmml">(</mo><mi id="S6.Ex1.m1.2.2.1.1" xref="S6.Ex1.m1.2.2.1.1.cmml">𝐱</mi><mo id="S6.Ex1.m1.3.3.2.2.1.1.3.2.2" stretchy="false" xref="S6.Ex1.m1.3.3.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex1.m1.3.3.2.2.1.3" stretchy="false" xref="S6.Ex1.m1.3.3.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex1.m1.3.3.3" xref="S6.Ex1.m1.3.3.3.cmml">¯</mo></mover></mrow><mo id="S6.Ex1.m1.4.4.1.2" lspace="0em" xref="S6.Ex1.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex1.m1.4b"><apply id="S6.Ex1.m1.4.4.1.1.cmml" xref="S6.Ex1.m1.4.4.1"><eq id="S6.Ex1.m1.4.4.1.1.1.cmml" xref="S6.Ex1.m1.4.4.1.1.1"></eq><apply id="S6.Ex1.m1.4.4.1.1.2.cmml" xref="S6.Ex1.m1.4.4.1.1.2"><times id="S6.Ex1.m1.4.4.1.1.2.1.cmml" xref="S6.Ex1.m1.4.4.1.1.2.1"></times><ci id="S6.Ex1.m1.4.4.1.1.2.2.cmml" xref="S6.Ex1.m1.4.4.1.1.2.2">𝜎</ci><apply id="S6.Ex1.m1.1.1.cmml" xref="S6.Ex1.m1.4.4.1.1.2.3.2"><ci id="S6.Ex1.m1.1.1.2.cmml" xref="S6.Ex1.m1.1.1.2">¯</ci><apply id="S6.Ex1.m1.1.1.1.cmml" xref="S6.Ex1.m1.1.1.1"><times id="S6.Ex1.m1.1.1.1.2.cmml" xref="S6.Ex1.m1.1.1.1.2"></times><ci id="S6.Ex1.m1.1.1.1.3.cmml" xref="S6.Ex1.m1.1.1.1.3">𝒪</ci><ci id="S6.Ex1.m1.1.1.1.1.cmml" xref="S6.Ex1.m1.1.1.1.1">𝐱</ci></apply></apply></apply><apply id="S6.Ex1.m1.3.3.cmml" xref="S6.Ex1.m1.3.3"><ci id="S6.Ex1.m1.3.3.3.cmml" xref="S6.Ex1.m1.3.3.3">¯</ci><apply id="S6.Ex1.m1.3.3.2.cmml" xref="S6.Ex1.m1.3.3.2"><times id="S6.Ex1.m1.3.3.2.3.cmml" xref="S6.Ex1.m1.3.3.2.3"></times><ci id="S6.Ex1.m1.3.3.2.4.cmml" xref="S6.Ex1.m1.3.3.2.4">𝒪</ci><apply id="S6.Ex1.m1.3.3.2.2.1.1.cmml" xref="S6.Ex1.m1.3.3.2.2.1"><times id="S6.Ex1.m1.3.3.2.2.1.1.1.cmml" xref="S6.Ex1.m1.3.3.2.2.1.1.1"></times><ci id="S6.Ex1.m1.3.3.2.2.1.1.2.cmml" xref="S6.Ex1.m1.3.3.2.2.1.1.2">𝜎</ci><ci id="S6.Ex1.m1.2.2.1.1.cmml" xref="S6.Ex1.m1.2.2.1.1">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex1.m1.4c">\sigma(\overline{\cal O({\bf x})})=\overline{\cal O(\sigma({\bf x}))}\,.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex1.m1.4d">italic_σ ( over¯ start_ARG caligraphic_O ( bold_x ) end_ARG ) = over¯ start_ARG caligraphic_O ( italic_σ ( bold_x ) ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.1.p1.5">The claim is hence a direct consequence of the well known and easy to prove fact that the shift-orbit of any biinfinite word is closed if and only if the word is periodic. <span class="ltx_text ltx_inline-block" id="S6.1.p1.4.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S6.1.p1.4.1.m1.1"><semantics id="S6.1.p1.4.1.m1.1a"><mo id="S6.1.p1.4.1.m1.1.1" xref="S6.1.p1.4.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S6.1.p1.4.1.m1.1b"><csymbol cd="latexml" id="S6.1.p1.4.1.m1.1.1.cmml" xref="S6.1.p1.4.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.4.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.4.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S6.1.p1.5.m1.1"><semantics id="S6.1.p1.5.m1.1a"><mo id="S6.1.p1.5.m1.1.1" xref="S6.1.p1.5.m1.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S6.1.p1.5.m1.1b"><csymbol cd="latexml" id="S6.1.p1.5.m1.1.1.cmml" xref="S6.1.p1.5.m1.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.1.p1.5.m1.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S6.1.p1.5.m1.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.6">For any word <math alttext="w\in\cal L(X)" class="ltx_Math" display="inline" id="S6.p2.1.m1.1"><semantics id="S6.p2.1.m1.1a"><mrow id="S6.p2.1.m1.1.2" xref="S6.p2.1.m1.1.2.cmml"><mi id="S6.p2.1.m1.1.2.2" xref="S6.p2.1.m1.1.2.2.cmml">w</mi><mo id="S6.p2.1.m1.1.2.1" xref="S6.p2.1.m1.1.2.1.cmml">∈</mo><mrow id="S6.p2.1.m1.1.2.3" xref="S6.p2.1.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p2.1.m1.1.2.3.2" xref="S6.p2.1.m1.1.2.3.2.cmml">ℒ</mi><mo id="S6.p2.1.m1.1.2.3.1" xref="S6.p2.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S6.p2.1.m1.1.2.3.3.2" xref="S6.p2.1.m1.1.2.3.cmml"><mo id="S6.p2.1.m1.1.2.3.3.2.1" stretchy="false" xref="S6.p2.1.m1.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p2.1.m1.1.1" xref="S6.p2.1.m1.1.1.cmml">𝒳</mi><mo id="S6.p2.1.m1.1.2.3.3.2.2" stretchy="false" xref="S6.p2.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.1.m1.1b"><apply id="S6.p2.1.m1.1.2.cmml" xref="S6.p2.1.m1.1.2"><in id="S6.p2.1.m1.1.2.1.cmml" xref="S6.p2.1.m1.1.2.1"></in><ci id="S6.p2.1.m1.1.2.2.cmml" xref="S6.p2.1.m1.1.2.2">𝑤</ci><apply id="S6.p2.1.m1.1.2.3.cmml" xref="S6.p2.1.m1.1.2.3"><times id="S6.p2.1.m1.1.2.3.1.cmml" xref="S6.p2.1.m1.1.2.3.1"></times><ci id="S6.p2.1.m1.1.2.3.2.cmml" xref="S6.p2.1.m1.1.2.3.2">ℒ</ci><ci id="S6.p2.1.m1.1.1.cmml" xref="S6.p2.1.m1.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.1.m1.1c">w\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.1.m1.1d">italic_w ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> and any integer <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S6.p2.2.m2.1"><semantics id="S6.p2.2.m2.1a"><mrow id="S6.p2.2.m2.1.1" xref="S6.p2.2.m2.1.1.cmml"><mi id="S6.p2.2.m2.1.1.2" xref="S6.p2.2.m2.1.1.2.cmml">n</mi><mo id="S6.p2.2.m2.1.1.1" xref="S6.p2.2.m2.1.1.1.cmml">≥</mo><mn id="S6.p2.2.m2.1.1.3" xref="S6.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.2.m2.1b"><apply id="S6.p2.2.m2.1.1.cmml" xref="S6.p2.2.m2.1.1"><geq id="S6.p2.2.m2.1.1.1.cmml" xref="S6.p2.2.m2.1.1.1"></geq><ci id="S6.p2.2.m2.1.1.2.cmml" xref="S6.p2.2.m2.1.1.2">𝑛</ci><cn id="S6.p2.2.m2.1.1.3.cmml" type="integer" xref="S6.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.2.m2.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.p2.2.m2.1d">italic_n ≥ 0</annotation></semantics></math> let us consider the set <math alttext="W_{n}(w)" class="ltx_Math" display="inline" id="S6.p2.3.m3.1"><semantics id="S6.p2.3.m3.1a"><mrow id="S6.p2.3.m3.1.2" xref="S6.p2.3.m3.1.2.cmml"><msub id="S6.p2.3.m3.1.2.2" xref="S6.p2.3.m3.1.2.2.cmml"><mi id="S6.p2.3.m3.1.2.2.2" xref="S6.p2.3.m3.1.2.2.2.cmml">W</mi><mi id="S6.p2.3.m3.1.2.2.3" xref="S6.p2.3.m3.1.2.2.3.cmml">n</mi></msub><mo id="S6.p2.3.m3.1.2.1" xref="S6.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.p2.3.m3.1.2.3.2" xref="S6.p2.3.m3.1.2.cmml"><mo id="S6.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S6.p2.3.m3.1.2.cmml">(</mo><mi id="S6.p2.3.m3.1.1" xref="S6.p2.3.m3.1.1.cmml">w</mi><mo id="S6.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S6.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.3.m3.1b"><apply id="S6.p2.3.m3.1.2.cmml" xref="S6.p2.3.m3.1.2"><times id="S6.p2.3.m3.1.2.1.cmml" xref="S6.p2.3.m3.1.2.1"></times><apply id="S6.p2.3.m3.1.2.2.cmml" xref="S6.p2.3.m3.1.2.2"><csymbol cd="ambiguous" id="S6.p2.3.m3.1.2.2.1.cmml" xref="S6.p2.3.m3.1.2.2">subscript</csymbol><ci id="S6.p2.3.m3.1.2.2.2.cmml" xref="S6.p2.3.m3.1.2.2.2">𝑊</ci><ci id="S6.p2.3.m3.1.2.2.3.cmml" xref="S6.p2.3.m3.1.2.2.3">𝑛</ci></apply><ci id="S6.p2.3.m3.1.1.cmml" xref="S6.p2.3.m3.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.3.m3.1c">W_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.3.m3.1d">italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> of both-sided prolongations of <math alttext="w" class="ltx_Math" display="inline" id="S6.p2.4.m4.1"><semantics id="S6.p2.4.m4.1a"><mi id="S6.p2.4.m4.1.1" xref="S6.p2.4.m4.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S6.p2.4.m4.1b"><ci id="S6.p2.4.m4.1.1.cmml" xref="S6.p2.4.m4.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.4.m4.1c">w</annotation><annotation encoding="application/x-llamapun" id="S6.p2.4.m4.1d">italic_w</annotation></semantics></math> in <math alttext="\cal L(X)" class="ltx_Math" display="inline" id="S6.p2.5.m5.1"><semantics id="S6.p2.5.m5.1a"><mrow id="S6.p2.5.m5.1.2" xref="S6.p2.5.m5.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p2.5.m5.1.2.2" xref="S6.p2.5.m5.1.2.2.cmml">ℒ</mi><mo id="S6.p2.5.m5.1.2.1" xref="S6.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S6.p2.5.m5.1.2.3.2" xref="S6.p2.5.m5.1.2.cmml"><mo id="S6.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S6.p2.5.m5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p2.5.m5.1.1" xref="S6.p2.5.m5.1.1.cmml">𝒳</mi><mo id="S6.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S6.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.5.m5.1b"><apply id="S6.p2.5.m5.1.2.cmml" xref="S6.p2.5.m5.1.2"><times id="S6.p2.5.m5.1.2.1.cmml" xref="S6.p2.5.m5.1.2.1"></times><ci id="S6.p2.5.m5.1.2.2.cmml" xref="S6.p2.5.m5.1.2.2">ℒ</ci><ci id="S6.p2.5.m5.1.1.cmml" xref="S6.p2.5.m5.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.5.m5.1c">\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.5.m5.1d">caligraphic_L ( caligraphic_X )</annotation></semantics></math> by <math alttext="n" class="ltx_Math" display="inline" id="S6.p2.6.m6.1"><semantics id="S6.p2.6.m6.1a"><mi id="S6.p2.6.m6.1.1" xref="S6.p2.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.p2.6.m6.1b"><ci id="S6.p2.6.m6.1.1.cmml" xref="S6.p2.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.p2.6.m6.1d">italic_n</annotation></semantics></math> letters on each side, i.e.</p> <table class="ltx_equation ltx_eqn_table" id="S6.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.1)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="W_{n}(w)\,\,:=\,\,\{uwv\in\cal L(X)\mid|u|=|v|=n\}\,." class="ltx_Math" display="block" id="S6.E1.m1.5"><semantics id="S6.E1.m1.5a"><mrow id="S6.E1.m1.5.5.1" xref="S6.E1.m1.5.5.1.1.cmml"><mrow id="S6.E1.m1.5.5.1.1" xref="S6.E1.m1.5.5.1.1.cmml"><mrow id="S6.E1.m1.5.5.1.1.4" xref="S6.E1.m1.5.5.1.1.4.cmml"><msub id="S6.E1.m1.5.5.1.1.4.2" xref="S6.E1.m1.5.5.1.1.4.2.cmml"><mi id="S6.E1.m1.5.5.1.1.4.2.2" xref="S6.E1.m1.5.5.1.1.4.2.2.cmml">W</mi><mi id="S6.E1.m1.5.5.1.1.4.2.3" xref="S6.E1.m1.5.5.1.1.4.2.3.cmml">n</mi></msub><mo id="S6.E1.m1.5.5.1.1.4.1" xref="S6.E1.m1.5.5.1.1.4.1.cmml">⁢</mo><mrow id="S6.E1.m1.5.5.1.1.4.3.2" xref="S6.E1.m1.5.5.1.1.4.cmml"><mo id="S6.E1.m1.5.5.1.1.4.3.2.1" stretchy="false" xref="S6.E1.m1.5.5.1.1.4.cmml">(</mo><mi id="S6.E1.m1.1.1" xref="S6.E1.m1.1.1.cmml">w</mi><mo id="S6.E1.m1.5.5.1.1.4.3.2.2" rspace="0.608em" stretchy="false" xref="S6.E1.m1.5.5.1.1.4.cmml">)</mo></mrow></mrow><mo id="S6.E1.m1.5.5.1.1.3" rspace="0.608em" xref="S6.E1.m1.5.5.1.1.3.cmml">:=</mo><mrow id="S6.E1.m1.5.5.1.1.2.2" xref="S6.E1.m1.5.5.1.1.2.3.cmml"><mo id="S6.E1.m1.5.5.1.1.2.2.3" stretchy="false" xref="S6.E1.m1.5.5.1.1.2.3.1.cmml">{</mo><mrow id="S6.E1.m1.5.5.1.1.1.1.1" xref="S6.E1.m1.5.5.1.1.1.1.1.cmml"><mrow id="S6.E1.m1.5.5.1.1.1.1.1.2" xref="S6.E1.m1.5.5.1.1.1.1.1.2.cmml"><mi id="S6.E1.m1.5.5.1.1.1.1.1.2.2" xref="S6.E1.m1.5.5.1.1.1.1.1.2.2.cmml">u</mi><mo id="S6.E1.m1.5.5.1.1.1.1.1.2.1" xref="S6.E1.m1.5.5.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S6.E1.m1.5.5.1.1.1.1.1.2.3" xref="S6.E1.m1.5.5.1.1.1.1.1.2.3.cmml">w</mi><mo id="S6.E1.m1.5.5.1.1.1.1.1.2.1a" xref="S6.E1.m1.5.5.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S6.E1.m1.5.5.1.1.1.1.1.2.4" xref="S6.E1.m1.5.5.1.1.1.1.1.2.4.cmml">v</mi></mrow><mo id="S6.E1.m1.5.5.1.1.1.1.1.1" xref="S6.E1.m1.5.5.1.1.1.1.1.1.cmml">∈</mo><mrow id="S6.E1.m1.5.5.1.1.1.1.1.3" xref="S6.E1.m1.5.5.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.E1.m1.5.5.1.1.1.1.1.3.2" xref="S6.E1.m1.5.5.1.1.1.1.1.3.2.cmml">ℒ</mi><mo id="S6.E1.m1.5.5.1.1.1.1.1.3.1" xref="S6.E1.m1.5.5.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S6.E1.m1.5.5.1.1.1.1.1.3.3.2" xref="S6.E1.m1.5.5.1.1.1.1.1.3.cmml"><mo id="S6.E1.m1.5.5.1.1.1.1.1.3.3.2.1" stretchy="false" xref="S6.E1.m1.5.5.1.1.1.1.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.E1.m1.2.2" xref="S6.E1.m1.2.2.cmml">𝒳</mi><mo id="S6.E1.m1.5.5.1.1.1.1.1.3.3.2.2" stretchy="false" xref="S6.E1.m1.5.5.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo fence="true" id="S6.E1.m1.5.5.1.1.2.2.4" lspace="0em" rspace="0em" xref="S6.E1.m1.5.5.1.1.2.3.1.cmml">∣</mo><mrow id="S6.E1.m1.5.5.1.1.2.2.2" xref="S6.E1.m1.5.5.1.1.2.2.2.cmml"><mrow id="S6.E1.m1.5.5.1.1.2.2.2.2.2" xref="S6.E1.m1.5.5.1.1.2.2.2.2.1.cmml"><mo id="S6.E1.m1.5.5.1.1.2.2.2.2.2.1" stretchy="false" xref="S6.E1.m1.5.5.1.1.2.2.2.2.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S6.E1.m1.3.3" xref="S6.E1.m1.3.3.cmml">𝓊</mi><mo id="S6.E1.m1.5.5.1.1.2.2.2.2.2.2" stretchy="false" xref="S6.E1.m1.5.5.1.1.2.2.2.2.1.1.cmml">|</mo></mrow><mo id="S6.E1.m1.5.5.1.1.2.2.2.3" xref="S6.E1.m1.5.5.1.1.2.2.2.3.cmml">=</mo><mrow id="S6.E1.m1.5.5.1.1.2.2.2.4.2" xref="S6.E1.m1.5.5.1.1.2.2.2.4.1.cmml"><mo id="S6.E1.m1.5.5.1.1.2.2.2.4.2.1" stretchy="false" xref="S6.E1.m1.5.5.1.1.2.2.2.4.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S6.E1.m1.4.4" xref="S6.E1.m1.4.4.cmml">𝓋</mi><mo id="S6.E1.m1.5.5.1.1.2.2.2.4.2.2" stretchy="false" xref="S6.E1.m1.5.5.1.1.2.2.2.4.1.1.cmml">|</mo></mrow><mo id="S6.E1.m1.5.5.1.1.2.2.2.5" xref="S6.E1.m1.5.5.1.1.2.2.2.5.cmml">=</mo><mi class="ltx_font_mathcaligraphic" id="S6.E1.m1.5.5.1.1.2.2.2.6" xref="S6.E1.m1.5.5.1.1.2.2.2.6.cmml">𝓃</mi></mrow><mo id="S6.E1.m1.5.5.1.1.2.2.5" stretchy="false" xref="S6.E1.m1.5.5.1.1.2.3.1.cmml">}</mo></mrow></mrow><mo id="S6.E1.m1.5.5.1.2" lspace="0.170em" xref="S6.E1.m1.5.5.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.E1.m1.5b"><apply id="S6.E1.m1.5.5.1.1.cmml" xref="S6.E1.m1.5.5.1"><csymbol cd="latexml" id="S6.E1.m1.5.5.1.1.3.cmml" xref="S6.E1.m1.5.5.1.1.3">assign</csymbol><apply id="S6.E1.m1.5.5.1.1.4.cmml" xref="S6.E1.m1.5.5.1.1.4"><times id="S6.E1.m1.5.5.1.1.4.1.cmml" xref="S6.E1.m1.5.5.1.1.4.1"></times><apply id="S6.E1.m1.5.5.1.1.4.2.cmml" xref="S6.E1.m1.5.5.1.1.4.2"><csymbol cd="ambiguous" id="S6.E1.m1.5.5.1.1.4.2.1.cmml" xref="S6.E1.m1.5.5.1.1.4.2">subscript</csymbol><ci id="S6.E1.m1.5.5.1.1.4.2.2.cmml" xref="S6.E1.m1.5.5.1.1.4.2.2">𝑊</ci><ci id="S6.E1.m1.5.5.1.1.4.2.3.cmml" xref="S6.E1.m1.5.5.1.1.4.2.3">𝑛</ci></apply><ci id="S6.E1.m1.1.1.cmml" xref="S6.E1.m1.1.1">𝑤</ci></apply><apply id="S6.E1.m1.5.5.1.1.2.3.cmml" xref="S6.E1.m1.5.5.1.1.2.2"><csymbol cd="latexml" id="S6.E1.m1.5.5.1.1.2.3.1.cmml" xref="S6.E1.m1.5.5.1.1.2.2.3">conditional-set</csymbol><apply id="S6.E1.m1.5.5.1.1.1.1.1.cmml" xref="S6.E1.m1.5.5.1.1.1.1.1"><in id="S6.E1.m1.5.5.1.1.1.1.1.1.cmml" xref="S6.E1.m1.5.5.1.1.1.1.1.1"></in><apply id="S6.E1.m1.5.5.1.1.1.1.1.2.cmml" 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id="S6.E1.m1.5.5.1.1.2.2.2.2.1.cmml" xref="S6.E1.m1.5.5.1.1.2.2.2.2.2"><abs id="S6.E1.m1.5.5.1.1.2.2.2.2.1.1.cmml" xref="S6.E1.m1.5.5.1.1.2.2.2.2.2.1"></abs><ci id="S6.E1.m1.3.3.cmml" xref="S6.E1.m1.3.3">𝓊</ci></apply><apply id="S6.E1.m1.5.5.1.1.2.2.2.4.1.cmml" xref="S6.E1.m1.5.5.1.1.2.2.2.4.2"><abs id="S6.E1.m1.5.5.1.1.2.2.2.4.1.1.cmml" xref="S6.E1.m1.5.5.1.1.2.2.2.4.2.1"></abs><ci id="S6.E1.m1.4.4.cmml" xref="S6.E1.m1.4.4">𝓋</ci></apply></apply><apply id="S6.E1.m1.5.5.1.1.2.2.2c.cmml" xref="S6.E1.m1.5.5.1.1.2.2.2"><eq id="S6.E1.m1.5.5.1.1.2.2.2.5.cmml" xref="S6.E1.m1.5.5.1.1.2.2.2.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S6.E1.m1.5.5.1.1.2.2.2.4.cmml" id="S6.E1.m1.5.5.1.1.2.2.2d.cmml" xref="S6.E1.m1.5.5.1.1.2.2.2"></share><ci id="S6.E1.m1.5.5.1.1.2.2.2.6.cmml" xref="S6.E1.m1.5.5.1.1.2.2.2.6">𝓃</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E1.m1.5c">W_{n}(w)\,\,:=\,\,\{uwv\in\cal L(X)\mid|u|=|v|=n\}\,.</annotation><annotation encoding="application/x-llamapun" id="S6.E1.m1.5d">italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) := { italic_u italic_w italic_v ∈ caligraphic_L ( caligraphic_X ) ∣ | caligraphic_u | = | caligraphic_v | = caligraphic_n } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.p2.10">For any <math alttext="w^{\prime}=uwv\in W_{n}(w)" class="ltx_Math" display="inline" id="S6.p2.7.m1.1"><semantics id="S6.p2.7.m1.1a"><mrow id="S6.p2.7.m1.1.2" xref="S6.p2.7.m1.1.2.cmml"><msup id="S6.p2.7.m1.1.2.2" xref="S6.p2.7.m1.1.2.2.cmml"><mi id="S6.p2.7.m1.1.2.2.2" xref="S6.p2.7.m1.1.2.2.2.cmml">w</mi><mo id="S6.p2.7.m1.1.2.2.3" xref="S6.p2.7.m1.1.2.2.3.cmml">′</mo></msup><mo id="S6.p2.7.m1.1.2.3" xref="S6.p2.7.m1.1.2.3.cmml">=</mo><mrow id="S6.p2.7.m1.1.2.4" xref="S6.p2.7.m1.1.2.4.cmml"><mi id="S6.p2.7.m1.1.2.4.2" xref="S6.p2.7.m1.1.2.4.2.cmml">u</mi><mo id="S6.p2.7.m1.1.2.4.1" xref="S6.p2.7.m1.1.2.4.1.cmml">⁢</mo><mi id="S6.p2.7.m1.1.2.4.3" xref="S6.p2.7.m1.1.2.4.3.cmml">w</mi><mo id="S6.p2.7.m1.1.2.4.1a" xref="S6.p2.7.m1.1.2.4.1.cmml">⁢</mo><mi id="S6.p2.7.m1.1.2.4.4" xref="S6.p2.7.m1.1.2.4.4.cmml">v</mi></mrow><mo id="S6.p2.7.m1.1.2.5" xref="S6.p2.7.m1.1.2.5.cmml">∈</mo><mrow id="S6.p2.7.m1.1.2.6" xref="S6.p2.7.m1.1.2.6.cmml"><msub id="S6.p2.7.m1.1.2.6.2" xref="S6.p2.7.m1.1.2.6.2.cmml"><mi id="S6.p2.7.m1.1.2.6.2.2" xref="S6.p2.7.m1.1.2.6.2.2.cmml">W</mi><mi id="S6.p2.7.m1.1.2.6.2.3" xref="S6.p2.7.m1.1.2.6.2.3.cmml">n</mi></msub><mo id="S6.p2.7.m1.1.2.6.1" xref="S6.p2.7.m1.1.2.6.1.cmml">⁢</mo><mrow id="S6.p2.7.m1.1.2.6.3.2" xref="S6.p2.7.m1.1.2.6.cmml"><mo id="S6.p2.7.m1.1.2.6.3.2.1" stretchy="false" xref="S6.p2.7.m1.1.2.6.cmml">(</mo><mi id="S6.p2.7.m1.1.1" xref="S6.p2.7.m1.1.1.cmml">w</mi><mo id="S6.p2.7.m1.1.2.6.3.2.2" stretchy="false" xref="S6.p2.7.m1.1.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.7.m1.1b"><apply id="S6.p2.7.m1.1.2.cmml" xref="S6.p2.7.m1.1.2"><and id="S6.p2.7.m1.1.2a.cmml" xref="S6.p2.7.m1.1.2"></and><apply id="S6.p2.7.m1.1.2b.cmml" xref="S6.p2.7.m1.1.2"><eq id="S6.p2.7.m1.1.2.3.cmml" xref="S6.p2.7.m1.1.2.3"></eq><apply id="S6.p2.7.m1.1.2.2.cmml" xref="S6.p2.7.m1.1.2.2"><csymbol cd="ambiguous" id="S6.p2.7.m1.1.2.2.1.cmml" xref="S6.p2.7.m1.1.2.2">superscript</csymbol><ci id="S6.p2.7.m1.1.2.2.2.cmml" xref="S6.p2.7.m1.1.2.2.2">𝑤</ci><ci id="S6.p2.7.m1.1.2.2.3.cmml" xref="S6.p2.7.m1.1.2.2.3">′</ci></apply><apply id="S6.p2.7.m1.1.2.4.cmml" xref="S6.p2.7.m1.1.2.4"><times id="S6.p2.7.m1.1.2.4.1.cmml" xref="S6.p2.7.m1.1.2.4.1"></times><ci id="S6.p2.7.m1.1.2.4.2.cmml" xref="S6.p2.7.m1.1.2.4.2">𝑢</ci><ci id="S6.p2.7.m1.1.2.4.3.cmml" xref="S6.p2.7.m1.1.2.4.3">𝑤</ci><ci id="S6.p2.7.m1.1.2.4.4.cmml" xref="S6.p2.7.m1.1.2.4.4">𝑣</ci></apply></apply><apply id="S6.p2.7.m1.1.2c.cmml" xref="S6.p2.7.m1.1.2"><in id="S6.p2.7.m1.1.2.5.cmml" xref="S6.p2.7.m1.1.2.5"></in><share href="https://arxiv.org/html/2211.11234v4#S6.p2.7.m1.1.2.4.cmml" id="S6.p2.7.m1.1.2d.cmml" xref="S6.p2.7.m1.1.2"></share><apply id="S6.p2.7.m1.1.2.6.cmml" xref="S6.p2.7.m1.1.2.6"><times id="S6.p2.7.m1.1.2.6.1.cmml" xref="S6.p2.7.m1.1.2.6.1"></times><apply id="S6.p2.7.m1.1.2.6.2.cmml" xref="S6.p2.7.m1.1.2.6.2"><csymbol cd="ambiguous" id="S6.p2.7.m1.1.2.6.2.1.cmml" xref="S6.p2.7.m1.1.2.6.2">subscript</csymbol><ci id="S6.p2.7.m1.1.2.6.2.2.cmml" xref="S6.p2.7.m1.1.2.6.2.2">𝑊</ci><ci id="S6.p2.7.m1.1.2.6.2.3.cmml" xref="S6.p2.7.m1.1.2.6.2.3">𝑛</ci></apply><ci id="S6.p2.7.m1.1.1.cmml" xref="S6.p2.7.m1.1.1">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.7.m1.1c">w^{\prime}=uwv\in W_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.7.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_u italic_w italic_v ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> we now consider the image <math alttext="\sigma(w^{\prime})" class="ltx_Math" display="inline" id="S6.p2.8.m2.1"><semantics id="S6.p2.8.m2.1a"><mrow id="S6.p2.8.m2.1.1" xref="S6.p2.8.m2.1.1.cmml"><mi id="S6.p2.8.m2.1.1.3" xref="S6.p2.8.m2.1.1.3.cmml">σ</mi><mo id="S6.p2.8.m2.1.1.2" xref="S6.p2.8.m2.1.1.2.cmml">⁢</mo><mrow id="S6.p2.8.m2.1.1.1.1" xref="S6.p2.8.m2.1.1.1.1.1.cmml"><mo id="S6.p2.8.m2.1.1.1.1.2" stretchy="false" xref="S6.p2.8.m2.1.1.1.1.1.cmml">(</mo><msup id="S6.p2.8.m2.1.1.1.1.1" xref="S6.p2.8.m2.1.1.1.1.1.cmml"><mi id="S6.p2.8.m2.1.1.1.1.1.2" xref="S6.p2.8.m2.1.1.1.1.1.2.cmml">w</mi><mo id="S6.p2.8.m2.1.1.1.1.1.3" xref="S6.p2.8.m2.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.p2.8.m2.1.1.1.1.3" stretchy="false" xref="S6.p2.8.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.8.m2.1b"><apply id="S6.p2.8.m2.1.1.cmml" xref="S6.p2.8.m2.1.1"><times id="S6.p2.8.m2.1.1.2.cmml" xref="S6.p2.8.m2.1.1.2"></times><ci id="S6.p2.8.m2.1.1.3.cmml" xref="S6.p2.8.m2.1.1.3">𝜎</ci><apply id="S6.p2.8.m2.1.1.1.1.1.cmml" xref="S6.p2.8.m2.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.8.m2.1.1.1.1.1.1.cmml" xref="S6.p2.8.m2.1.1.1.1">superscript</csymbol><ci id="S6.p2.8.m2.1.1.1.1.1.2.cmml" xref="S6.p2.8.m2.1.1.1.1.1.2">𝑤</ci><ci id="S6.p2.8.m2.1.1.1.1.1.3.cmml" xref="S6.p2.8.m2.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.8.m2.1c">\sigma(w^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.p2.8.m2.1d">italic_σ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and its preimage set <math alttext="\sigma^{-1}(\sigma(w^{\prime}))\cap\cal L(X)" class="ltx_Math" display="inline" id="S6.p2.9.m3.2"><semantics id="S6.p2.9.m3.2a"><mrow id="S6.p2.9.m3.2.2" xref="S6.p2.9.m3.2.2.cmml"><mrow id="S6.p2.9.m3.2.2.1" xref="S6.p2.9.m3.2.2.1.cmml"><msup id="S6.p2.9.m3.2.2.1.3" xref="S6.p2.9.m3.2.2.1.3.cmml"><mi id="S6.p2.9.m3.2.2.1.3.2" xref="S6.p2.9.m3.2.2.1.3.2.cmml">σ</mi><mrow id="S6.p2.9.m3.2.2.1.3.3" xref="S6.p2.9.m3.2.2.1.3.3.cmml"><mo id="S6.p2.9.m3.2.2.1.3.3a" xref="S6.p2.9.m3.2.2.1.3.3.cmml">−</mo><mn id="S6.p2.9.m3.2.2.1.3.3.2" xref="S6.p2.9.m3.2.2.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S6.p2.9.m3.2.2.1.2" xref="S6.p2.9.m3.2.2.1.2.cmml">⁢</mo><mrow id="S6.p2.9.m3.2.2.1.1.1" xref="S6.p2.9.m3.2.2.1.1.1.1.cmml"><mo id="S6.p2.9.m3.2.2.1.1.1.2" stretchy="false" xref="S6.p2.9.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.p2.9.m3.2.2.1.1.1.1" xref="S6.p2.9.m3.2.2.1.1.1.1.cmml"><mi id="S6.p2.9.m3.2.2.1.1.1.1.3" xref="S6.p2.9.m3.2.2.1.1.1.1.3.cmml">σ</mi><mo id="S6.p2.9.m3.2.2.1.1.1.1.2" xref="S6.p2.9.m3.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.p2.9.m3.2.2.1.1.1.1.1.1" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.cmml"><mo id="S6.p2.9.m3.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.cmml">(</mo><msup id="S6.p2.9.m3.2.2.1.1.1.1.1.1.1" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.cmml"><mi id="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.2" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.2.cmml">w</mi><mo id="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.3" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.p2.9.m3.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.p2.9.m3.2.2.1.1.1.3" stretchy="false" xref="S6.p2.9.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.p2.9.m3.2.2.2" xref="S6.p2.9.m3.2.2.2.cmml">∩</mo><mrow id="S6.p2.9.m3.2.2.3" xref="S6.p2.9.m3.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p2.9.m3.2.2.3.2" xref="S6.p2.9.m3.2.2.3.2.cmml">ℒ</mi><mo id="S6.p2.9.m3.2.2.3.1" xref="S6.p2.9.m3.2.2.3.1.cmml">⁢</mo><mrow id="S6.p2.9.m3.2.2.3.3.2" xref="S6.p2.9.m3.2.2.3.cmml"><mo id="S6.p2.9.m3.2.2.3.3.2.1" stretchy="false" xref="S6.p2.9.m3.2.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p2.9.m3.1.1" xref="S6.p2.9.m3.1.1.cmml">𝒳</mi><mo id="S6.p2.9.m3.2.2.3.3.2.2" stretchy="false" xref="S6.p2.9.m3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.9.m3.2b"><apply id="S6.p2.9.m3.2.2.cmml" xref="S6.p2.9.m3.2.2"><intersect id="S6.p2.9.m3.2.2.2.cmml" xref="S6.p2.9.m3.2.2.2"></intersect><apply id="S6.p2.9.m3.2.2.1.cmml" xref="S6.p2.9.m3.2.2.1"><times id="S6.p2.9.m3.2.2.1.2.cmml" xref="S6.p2.9.m3.2.2.1.2"></times><apply id="S6.p2.9.m3.2.2.1.3.cmml" xref="S6.p2.9.m3.2.2.1.3"><csymbol cd="ambiguous" id="S6.p2.9.m3.2.2.1.3.1.cmml" xref="S6.p2.9.m3.2.2.1.3">superscript</csymbol><ci id="S6.p2.9.m3.2.2.1.3.2.cmml" xref="S6.p2.9.m3.2.2.1.3.2">𝜎</ci><apply id="S6.p2.9.m3.2.2.1.3.3.cmml" xref="S6.p2.9.m3.2.2.1.3.3"><minus id="S6.p2.9.m3.2.2.1.3.3.1.cmml" xref="S6.p2.9.m3.2.2.1.3.3"></minus><cn id="S6.p2.9.m3.2.2.1.3.3.2.cmml" type="integer" xref="S6.p2.9.m3.2.2.1.3.3.2">1</cn></apply></apply><apply id="S6.p2.9.m3.2.2.1.1.1.1.cmml" xref="S6.p2.9.m3.2.2.1.1.1"><times id="S6.p2.9.m3.2.2.1.1.1.1.2.cmml" xref="S6.p2.9.m3.2.2.1.1.1.1.2"></times><ci id="S6.p2.9.m3.2.2.1.1.1.1.3.cmml" xref="S6.p2.9.m3.2.2.1.1.1.1.3">𝜎</ci><apply id="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.cmml" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1">superscript</csymbol><ci id="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.2.cmml" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.2">𝑤</ci><ci id="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.3.cmml" xref="S6.p2.9.m3.2.2.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply><apply id="S6.p2.9.m3.2.2.3.cmml" xref="S6.p2.9.m3.2.2.3"><times id="S6.p2.9.m3.2.2.3.1.cmml" xref="S6.p2.9.m3.2.2.3.1"></times><ci id="S6.p2.9.m3.2.2.3.2.cmml" xref="S6.p2.9.m3.2.2.3.2">ℒ</ci><ci id="S6.p2.9.m3.1.1.cmml" xref="S6.p2.9.m3.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.9.m3.2c">\sigma^{-1}(\sigma(w^{\prime}))\cap\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.9.m3.2d">italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_σ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) ∩ caligraphic_L ( caligraphic_X )</annotation></semantics></math>. We split the set <math alttext="W_{n}(w)" class="ltx_Math" display="inline" id="S6.p2.10.m4.1"><semantics id="S6.p2.10.m4.1a"><mrow id="S6.p2.10.m4.1.2" xref="S6.p2.10.m4.1.2.cmml"><msub id="S6.p2.10.m4.1.2.2" xref="S6.p2.10.m4.1.2.2.cmml"><mi id="S6.p2.10.m4.1.2.2.2" xref="S6.p2.10.m4.1.2.2.2.cmml">W</mi><mi id="S6.p2.10.m4.1.2.2.3" xref="S6.p2.10.m4.1.2.2.3.cmml">n</mi></msub><mo id="S6.p2.10.m4.1.2.1" xref="S6.p2.10.m4.1.2.1.cmml">⁢</mo><mrow id="S6.p2.10.m4.1.2.3.2" xref="S6.p2.10.m4.1.2.cmml"><mo id="S6.p2.10.m4.1.2.3.2.1" stretchy="false" xref="S6.p2.10.m4.1.2.cmml">(</mo><mi id="S6.p2.10.m4.1.1" xref="S6.p2.10.m4.1.1.cmml">w</mi><mo id="S6.p2.10.m4.1.2.3.2.2" stretchy="false" xref="S6.p2.10.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.10.m4.1b"><apply id="S6.p2.10.m4.1.2.cmml" xref="S6.p2.10.m4.1.2"><times id="S6.p2.10.m4.1.2.1.cmml" xref="S6.p2.10.m4.1.2.1"></times><apply id="S6.p2.10.m4.1.2.2.cmml" xref="S6.p2.10.m4.1.2.2"><csymbol cd="ambiguous" id="S6.p2.10.m4.1.2.2.1.cmml" xref="S6.p2.10.m4.1.2.2">subscript</csymbol><ci id="S6.p2.10.m4.1.2.2.2.cmml" xref="S6.p2.10.m4.1.2.2.2">𝑊</ci><ci id="S6.p2.10.m4.1.2.2.3.cmml" xref="S6.p2.10.m4.1.2.2.3">𝑛</ci></apply><ci id="S6.p2.10.m4.1.1.cmml" xref="S6.p2.10.m4.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.10.m4.1c">W_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.10.m4.1d">italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> into a disjoint union,</p> <table class="ltx_equation ltx_eqn_table" id="S6.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.2)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="W_{n}(w)=U_{n}(w)\sqcup A_{n}(w)\,," class="ltx_Math" display="block" id="S6.E2.m1.4"><semantics id="S6.E2.m1.4a"><mrow id="S6.E2.m1.4.4.1" xref="S6.E2.m1.4.4.1.1.cmml"><mrow id="S6.E2.m1.4.4.1.1" xref="S6.E2.m1.4.4.1.1.cmml"><mrow id="S6.E2.m1.4.4.1.1.2" xref="S6.E2.m1.4.4.1.1.2.cmml"><msub id="S6.E2.m1.4.4.1.1.2.2" xref="S6.E2.m1.4.4.1.1.2.2.cmml"><mi id="S6.E2.m1.4.4.1.1.2.2.2" xref="S6.E2.m1.4.4.1.1.2.2.2.cmml">W</mi><mi id="S6.E2.m1.4.4.1.1.2.2.3" xref="S6.E2.m1.4.4.1.1.2.2.3.cmml">n</mi></msub><mo id="S6.E2.m1.4.4.1.1.2.1" xref="S6.E2.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S6.E2.m1.4.4.1.1.2.3.2" xref="S6.E2.m1.4.4.1.1.2.cmml"><mo id="S6.E2.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="S6.E2.m1.4.4.1.1.2.cmml">(</mo><mi id="S6.E2.m1.1.1" xref="S6.E2.m1.1.1.cmml">w</mi><mo id="S6.E2.m1.4.4.1.1.2.3.2.2" stretchy="false" xref="S6.E2.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.E2.m1.4.4.1.1.1" xref="S6.E2.m1.4.4.1.1.1.cmml">=</mo><mrow id="S6.E2.m1.4.4.1.1.3" xref="S6.E2.m1.4.4.1.1.3.cmml"><mrow id="S6.E2.m1.4.4.1.1.3.2" xref="S6.E2.m1.4.4.1.1.3.2.cmml"><msub id="S6.E2.m1.4.4.1.1.3.2.2" xref="S6.E2.m1.4.4.1.1.3.2.2.cmml"><mi id="S6.E2.m1.4.4.1.1.3.2.2.2" xref="S6.E2.m1.4.4.1.1.3.2.2.2.cmml">U</mi><mi id="S6.E2.m1.4.4.1.1.3.2.2.3" xref="S6.E2.m1.4.4.1.1.3.2.2.3.cmml">n</mi></msub><mo id="S6.E2.m1.4.4.1.1.3.2.1" xref="S6.E2.m1.4.4.1.1.3.2.1.cmml">⁢</mo><mrow id="S6.E2.m1.4.4.1.1.3.2.3.2" xref="S6.E2.m1.4.4.1.1.3.2.cmml"><mo id="S6.E2.m1.4.4.1.1.3.2.3.2.1" stretchy="false" xref="S6.E2.m1.4.4.1.1.3.2.cmml">(</mo><mi id="S6.E2.m1.2.2" xref="S6.E2.m1.2.2.cmml">w</mi><mo id="S6.E2.m1.4.4.1.1.3.2.3.2.2" stretchy="false" xref="S6.E2.m1.4.4.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="S6.E2.m1.4.4.1.1.3.1" xref="S6.E2.m1.4.4.1.1.3.1.cmml">⊔</mo><mrow id="S6.E2.m1.4.4.1.1.3.3" xref="S6.E2.m1.4.4.1.1.3.3.cmml"><msub id="S6.E2.m1.4.4.1.1.3.3.2" xref="S6.E2.m1.4.4.1.1.3.3.2.cmml"><mi id="S6.E2.m1.4.4.1.1.3.3.2.2" xref="S6.E2.m1.4.4.1.1.3.3.2.2.cmml">A</mi><mi id="S6.E2.m1.4.4.1.1.3.3.2.3" xref="S6.E2.m1.4.4.1.1.3.3.2.3.cmml">n</mi></msub><mo id="S6.E2.m1.4.4.1.1.3.3.1" xref="S6.E2.m1.4.4.1.1.3.3.1.cmml">⁢</mo><mrow id="S6.E2.m1.4.4.1.1.3.3.3.2" xref="S6.E2.m1.4.4.1.1.3.3.cmml"><mo id="S6.E2.m1.4.4.1.1.3.3.3.2.1" stretchy="false" xref="S6.E2.m1.4.4.1.1.3.3.cmml">(</mo><mi id="S6.E2.m1.3.3" xref="S6.E2.m1.3.3.cmml">w</mi><mo id="S6.E2.m1.4.4.1.1.3.3.3.2.2" rspace="0.170em" stretchy="false" xref="S6.E2.m1.4.4.1.1.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.E2.m1.4.4.1.2" xref="S6.E2.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.E2.m1.4b"><apply id="S6.E2.m1.4.4.1.1.cmml" xref="S6.E2.m1.4.4.1"><eq id="S6.E2.m1.4.4.1.1.1.cmml" xref="S6.E2.m1.4.4.1.1.1"></eq><apply id="S6.E2.m1.4.4.1.1.2.cmml" xref="S6.E2.m1.4.4.1.1.2"><times id="S6.E2.m1.4.4.1.1.2.1.cmml" xref="S6.E2.m1.4.4.1.1.2.1"></times><apply id="S6.E2.m1.4.4.1.1.2.2.cmml" xref="S6.E2.m1.4.4.1.1.2.2"><csymbol cd="ambiguous" id="S6.E2.m1.4.4.1.1.2.2.1.cmml" xref="S6.E2.m1.4.4.1.1.2.2">subscript</csymbol><ci id="S6.E2.m1.4.4.1.1.2.2.2.cmml" xref="S6.E2.m1.4.4.1.1.2.2.2">𝑊</ci><ci id="S6.E2.m1.4.4.1.1.2.2.3.cmml" xref="S6.E2.m1.4.4.1.1.2.2.3">𝑛</ci></apply><ci id="S6.E2.m1.1.1.cmml" xref="S6.E2.m1.1.1">𝑤</ci></apply><apply id="S6.E2.m1.4.4.1.1.3.cmml" xref="S6.E2.m1.4.4.1.1.3"><csymbol cd="latexml" id="S6.E2.m1.4.4.1.1.3.1.cmml" xref="S6.E2.m1.4.4.1.1.3.1">square-union</csymbol><apply id="S6.E2.m1.4.4.1.1.3.2.cmml" xref="S6.E2.m1.4.4.1.1.3.2"><times id="S6.E2.m1.4.4.1.1.3.2.1.cmml" xref="S6.E2.m1.4.4.1.1.3.2.1"></times><apply id="S6.E2.m1.4.4.1.1.3.2.2.cmml" xref="S6.E2.m1.4.4.1.1.3.2.2"><csymbol cd="ambiguous" id="S6.E2.m1.4.4.1.1.3.2.2.1.cmml" xref="S6.E2.m1.4.4.1.1.3.2.2">subscript</csymbol><ci id="S6.E2.m1.4.4.1.1.3.2.2.2.cmml" xref="S6.E2.m1.4.4.1.1.3.2.2.2">𝑈</ci><ci id="S6.E2.m1.4.4.1.1.3.2.2.3.cmml" xref="S6.E2.m1.4.4.1.1.3.2.2.3">𝑛</ci></apply><ci id="S6.E2.m1.2.2.cmml" xref="S6.E2.m1.2.2">𝑤</ci></apply><apply id="S6.E2.m1.4.4.1.1.3.3.cmml" xref="S6.E2.m1.4.4.1.1.3.3"><times id="S6.E2.m1.4.4.1.1.3.3.1.cmml" xref="S6.E2.m1.4.4.1.1.3.3.1"></times><apply id="S6.E2.m1.4.4.1.1.3.3.2.cmml" xref="S6.E2.m1.4.4.1.1.3.3.2"><csymbol cd="ambiguous" id="S6.E2.m1.4.4.1.1.3.3.2.1.cmml" xref="S6.E2.m1.4.4.1.1.3.3.2">subscript</csymbol><ci id="S6.E2.m1.4.4.1.1.3.3.2.2.cmml" xref="S6.E2.m1.4.4.1.1.3.3.2.2">𝐴</ci><ci id="S6.E2.m1.4.4.1.1.3.3.2.3.cmml" xref="S6.E2.m1.4.4.1.1.3.3.2.3">𝑛</ci></apply><ci id="S6.E2.m1.3.3.cmml" xref="S6.E2.m1.3.3">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E2.m1.4c">W_{n}(w)=U_{n}(w)\sqcup A_{n}(w)\,,</annotation><annotation encoding="application/x-llamapun" id="S6.E2.m1.4d">italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) = italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ⊔ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.p2.23">by setting</p> <table class="ltx_equation ltx_eqn_table" id="S6.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.3)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w^{\prime}\in U_{n}(w)\,\,\Longleftrightarrow\,\,\sigma^{-1}(\sigma(w^{\prime}% ))\cap\cal L(X)\subseteq W_{n}(w)" class="ltx_Math" display="block" id="S6.E3.m1.4"><semantics id="S6.E3.m1.4a"><mrow id="S6.E3.m1.4.4" xref="S6.E3.m1.4.4.cmml"><mrow id="S6.E3.m1.4.4.3" xref="S6.E3.m1.4.4.3.cmml"><msup id="S6.E3.m1.4.4.3.2" xref="S6.E3.m1.4.4.3.2.cmml"><mi id="S6.E3.m1.4.4.3.2.2" xref="S6.E3.m1.4.4.3.2.2.cmml">w</mi><mo id="S6.E3.m1.4.4.3.2.3" xref="S6.E3.m1.4.4.3.2.3.cmml">′</mo></msup><mo id="S6.E3.m1.4.4.3.1" xref="S6.E3.m1.4.4.3.1.cmml">∈</mo><mrow id="S6.E3.m1.4.4.3.3" xref="S6.E3.m1.4.4.3.3.cmml"><msub id="S6.E3.m1.4.4.3.3.2" xref="S6.E3.m1.4.4.3.3.2.cmml"><mi id="S6.E3.m1.4.4.3.3.2.2" xref="S6.E3.m1.4.4.3.3.2.2.cmml">U</mi><mi id="S6.E3.m1.4.4.3.3.2.3" xref="S6.E3.m1.4.4.3.3.2.3.cmml">n</mi></msub><mo id="S6.E3.m1.4.4.3.3.1" xref="S6.E3.m1.4.4.3.3.1.cmml">⁢</mo><mrow id="S6.E3.m1.4.4.3.3.3.2" xref="S6.E3.m1.4.4.3.3.cmml"><mo id="S6.E3.m1.4.4.3.3.3.2.1" stretchy="false" xref="S6.E3.m1.4.4.3.3.cmml">(</mo><mi id="S6.E3.m1.1.1" xref="S6.E3.m1.1.1.cmml">w</mi><mo id="S6.E3.m1.4.4.3.3.3.2.2" rspace="0.330em" stretchy="false" xref="S6.E3.m1.4.4.3.3.cmml">)</mo></mrow></mrow></mrow><mo id="S6.E3.m1.4.4.2" rspace="0.608em" stretchy="false" xref="S6.E3.m1.4.4.2.cmml">⟺</mo><mrow id="S6.E3.m1.4.4.1" xref="S6.E3.m1.4.4.1.cmml"><mrow id="S6.E3.m1.4.4.1.1" xref="S6.E3.m1.4.4.1.1.cmml"><mrow id="S6.E3.m1.4.4.1.1.1" xref="S6.E3.m1.4.4.1.1.1.cmml"><msup id="S6.E3.m1.4.4.1.1.1.3" xref="S6.E3.m1.4.4.1.1.1.3.cmml"><mi id="S6.E3.m1.4.4.1.1.1.3.2" xref="S6.E3.m1.4.4.1.1.1.3.2.cmml">σ</mi><mrow id="S6.E3.m1.4.4.1.1.1.3.3" xref="S6.E3.m1.4.4.1.1.1.3.3.cmml"><mo id="S6.E3.m1.4.4.1.1.1.3.3a" xref="S6.E3.m1.4.4.1.1.1.3.3.cmml">−</mo><mn id="S6.E3.m1.4.4.1.1.1.3.3.2" xref="S6.E3.m1.4.4.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S6.E3.m1.4.4.1.1.1.2" xref="S6.E3.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S6.E3.m1.4.4.1.1.1.1.1" xref="S6.E3.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S6.E3.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S6.E3.m1.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.E3.m1.4.4.1.1.1.1.1.1" xref="S6.E3.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S6.E3.m1.4.4.1.1.1.1.1.1.3" xref="S6.E3.m1.4.4.1.1.1.1.1.1.3.cmml">σ</mi><mo id="S6.E3.m1.4.4.1.1.1.1.1.1.2" xref="S6.E3.m1.4.4.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.E3.m1.4.4.1.1.1.1.1.1.1.1" xref="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1.cmml"><mo id="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1" xref="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1.2" xref="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1.2.cmml">w</mi><mo id="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1.3" xref="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.E3.m1.4.4.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.E3.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="S6.E3.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.E3.m1.4.4.1.1.2" xref="S6.E3.m1.4.4.1.1.2.cmml">∩</mo><mrow id="S6.E3.m1.4.4.1.1.3" xref="S6.E3.m1.4.4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.E3.m1.4.4.1.1.3.2" xref="S6.E3.m1.4.4.1.1.3.2.cmml">ℒ</mi><mo id="S6.E3.m1.4.4.1.1.3.1" xref="S6.E3.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S6.E3.m1.4.4.1.1.3.3.2" xref="S6.E3.m1.4.4.1.1.3.cmml"><mo id="S6.E3.m1.4.4.1.1.3.3.2.1" stretchy="false" xref="S6.E3.m1.4.4.1.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.E3.m1.2.2" xref="S6.E3.m1.2.2.cmml">𝒳</mi><mo id="S6.E3.m1.4.4.1.1.3.3.2.2" stretchy="false" xref="S6.E3.m1.4.4.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S6.E3.m1.4.4.1.2" xref="S6.E3.m1.4.4.1.2.cmml">⊆</mo><mrow id="S6.E3.m1.4.4.1.3" xref="S6.E3.m1.4.4.1.3.cmml"><msub id="S6.E3.m1.4.4.1.3.2" xref="S6.E3.m1.4.4.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.E3.m1.4.4.1.3.2.2" xref="S6.E3.m1.4.4.1.3.2.2.cmml">𝒲</mi><mi class="ltx_font_mathcaligraphic" id="S6.E3.m1.4.4.1.3.2.3" xref="S6.E3.m1.4.4.1.3.2.3.cmml">𝓃</mi></msub><mo id="S6.E3.m1.4.4.1.3.1" xref="S6.E3.m1.4.4.1.3.1.cmml">⁢</mo><mrow id="S6.E3.m1.4.4.1.3.3.2" xref="S6.E3.m1.4.4.1.3.cmml"><mo id="S6.E3.m1.4.4.1.3.3.2.1" stretchy="false" xref="S6.E3.m1.4.4.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.E3.m1.3.3" xref="S6.E3.m1.3.3.cmml">𝓌</mi><mo id="S6.E3.m1.4.4.1.3.3.2.2" stretchy="false" xref="S6.E3.m1.4.4.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.E3.m1.4b"><apply id="S6.E3.m1.4.4.cmml" xref="S6.E3.m1.4.4"><ci id="S6.E3.m1.4.4.2.cmml" xref="S6.E3.m1.4.4.2">⟺</ci><apply id="S6.E3.m1.4.4.3.cmml" xref="S6.E3.m1.4.4.3"><in id="S6.E3.m1.4.4.3.1.cmml" xref="S6.E3.m1.4.4.3.1"></in><apply id="S6.E3.m1.4.4.3.2.cmml" xref="S6.E3.m1.4.4.3.2"><csymbol cd="ambiguous" id="S6.E3.m1.4.4.3.2.1.cmml" xref="S6.E3.m1.4.4.3.2">superscript</csymbol><ci id="S6.E3.m1.4.4.3.2.2.cmml" xref="S6.E3.m1.4.4.3.2.2">𝑤</ci><ci id="S6.E3.m1.4.4.3.2.3.cmml" xref="S6.E3.m1.4.4.3.2.3">′</ci></apply><apply id="S6.E3.m1.4.4.3.3.cmml" xref="S6.E3.m1.4.4.3.3"><times id="S6.E3.m1.4.4.3.3.1.cmml" xref="S6.E3.m1.4.4.3.3.1"></times><apply id="S6.E3.m1.4.4.3.3.2.cmml" xref="S6.E3.m1.4.4.3.3.2"><csymbol cd="ambiguous" id="S6.E3.m1.4.4.3.3.2.1.cmml" xref="S6.E3.m1.4.4.3.3.2">subscript</csymbol><ci id="S6.E3.m1.4.4.3.3.2.2.cmml" xref="S6.E3.m1.4.4.3.3.2.2">𝑈</ci><ci id="S6.E3.m1.4.4.3.3.2.3.cmml" xref="S6.E3.m1.4.4.3.3.2.3">𝑛</ci></apply><ci id="S6.E3.m1.1.1.cmml" xref="S6.E3.m1.1.1">𝑤</ci></apply></apply><apply id="S6.E3.m1.4.4.1.cmml" xref="S6.E3.m1.4.4.1"><subset id="S6.E3.m1.4.4.1.2.cmml" xref="S6.E3.m1.4.4.1.2"></subset><apply id="S6.E3.m1.4.4.1.1.cmml" xref="S6.E3.m1.4.4.1.1"><intersect id="S6.E3.m1.4.4.1.1.2.cmml" xref="S6.E3.m1.4.4.1.1.2"></intersect><apply id="S6.E3.m1.4.4.1.1.1.cmml" xref="S6.E3.m1.4.4.1.1.1"><times id="S6.E3.m1.4.4.1.1.1.2.cmml" xref="S6.E3.m1.4.4.1.1.1.2"></times><apply id="S6.E3.m1.4.4.1.1.1.3.cmml" xref="S6.E3.m1.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S6.E3.m1.4.4.1.1.1.3.1.cmml" xref="S6.E3.m1.4.4.1.1.1.3">superscript</csymbol><ci id="S6.E3.m1.4.4.1.1.1.3.2.cmml" xref="S6.E3.m1.4.4.1.1.1.3.2">𝜎</ci><apply id="S6.E3.m1.4.4.1.1.1.3.3.cmml" xref="S6.E3.m1.4.4.1.1.1.3.3"><minus id="S6.E3.m1.4.4.1.1.1.3.3.1.cmml" xref="S6.E3.m1.4.4.1.1.1.3.3"></minus><cn id="S6.E3.m1.4.4.1.1.1.3.3.2.cmml" type="integer" 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id="S6.E3.m1.4.4.1.3.cmml" xref="S6.E3.m1.4.4.1.3"><times id="S6.E3.m1.4.4.1.3.1.cmml" xref="S6.E3.m1.4.4.1.3.1"></times><apply id="S6.E3.m1.4.4.1.3.2.cmml" xref="S6.E3.m1.4.4.1.3.2"><csymbol cd="ambiguous" id="S6.E3.m1.4.4.1.3.2.1.cmml" xref="S6.E3.m1.4.4.1.3.2">subscript</csymbol><ci id="S6.E3.m1.4.4.1.3.2.2.cmml" xref="S6.E3.m1.4.4.1.3.2.2">𝒲</ci><ci id="S6.E3.m1.4.4.1.3.2.3.cmml" xref="S6.E3.m1.4.4.1.3.2.3">𝓃</ci></apply><ci id="S6.E3.m1.3.3.cmml" xref="S6.E3.m1.3.3">𝓌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E3.m1.4c">w^{\prime}\in U_{n}(w)\,\,\Longleftrightarrow\,\,\sigma^{-1}(\sigma(w^{\prime}% ))\cap\cal L(X)\subseteq W_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.E3.m1.4d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ⟺ italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_σ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) ∩ caligraphic_L ( caligraphic_X ) ⊆ caligraphic_W start_POSTSUBSCRIPT caligraphic_n end_POSTSUBSCRIPT ( caligraphic_w )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.p2.24">and</p> <table class="ltx_equation ltx_eqn_table" id="S6.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.4)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="w^{\prime}\in A_{n}(w)\,\,\Longleftrightarrow\,\,\sigma^{-1}(\sigma(w^{\prime}% ))\cap\cal L(X)\nsubseteq W_{n}(w)\,." class="ltx_Math" display="block" id="S6.E4.m1.4"><semantics id="S6.E4.m1.4a"><mrow id="S6.E4.m1.4.4.1" xref="S6.E4.m1.4.4.1.1.cmml"><mrow id="S6.E4.m1.4.4.1.1" xref="S6.E4.m1.4.4.1.1.cmml"><mrow id="S6.E4.m1.4.4.1.1.3" xref="S6.E4.m1.4.4.1.1.3.cmml"><msup id="S6.E4.m1.4.4.1.1.3.2" xref="S6.E4.m1.4.4.1.1.3.2.cmml"><mi id="S6.E4.m1.4.4.1.1.3.2.2" xref="S6.E4.m1.4.4.1.1.3.2.2.cmml">w</mi><mo id="S6.E4.m1.4.4.1.1.3.2.3" xref="S6.E4.m1.4.4.1.1.3.2.3.cmml">′</mo></msup><mo id="S6.E4.m1.4.4.1.1.3.1" xref="S6.E4.m1.4.4.1.1.3.1.cmml">∈</mo><mrow id="S6.E4.m1.4.4.1.1.3.3" xref="S6.E4.m1.4.4.1.1.3.3.cmml"><msub id="S6.E4.m1.4.4.1.1.3.3.2" xref="S6.E4.m1.4.4.1.1.3.3.2.cmml"><mi id="S6.E4.m1.4.4.1.1.3.3.2.2" xref="S6.E4.m1.4.4.1.1.3.3.2.2.cmml">A</mi><mi id="S6.E4.m1.4.4.1.1.3.3.2.3" xref="S6.E4.m1.4.4.1.1.3.3.2.3.cmml">n</mi></msub><mo id="S6.E4.m1.4.4.1.1.3.3.1" xref="S6.E4.m1.4.4.1.1.3.3.1.cmml">⁢</mo><mrow id="S6.E4.m1.4.4.1.1.3.3.3.2" xref="S6.E4.m1.4.4.1.1.3.3.cmml"><mo id="S6.E4.m1.4.4.1.1.3.3.3.2.1" stretchy="false" xref="S6.E4.m1.4.4.1.1.3.3.cmml">(</mo><mi id="S6.E4.m1.1.1" xref="S6.E4.m1.1.1.cmml">w</mi><mo id="S6.E4.m1.4.4.1.1.3.3.3.2.2" rspace="0.330em" stretchy="false" 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xref="S6.E4.m1.4.4.1.1.1.1.3.2">ℒ</ci><ci id="S6.E4.m1.2.2.cmml" xref="S6.E4.m1.2.2">𝒳</ci></apply></apply><apply id="S6.E4.m1.4.4.1.1.1.3.cmml" xref="S6.E4.m1.4.4.1.1.1.3"><times id="S6.E4.m1.4.4.1.1.1.3.1.cmml" xref="S6.E4.m1.4.4.1.1.1.3.1"></times><apply id="S6.E4.m1.4.4.1.1.1.3.2.cmml" xref="S6.E4.m1.4.4.1.1.1.3.2"><csymbol cd="ambiguous" id="S6.E4.m1.4.4.1.1.1.3.2.1.cmml" xref="S6.E4.m1.4.4.1.1.1.3.2">subscript</csymbol><ci id="S6.E4.m1.4.4.1.1.1.3.2.2.cmml" xref="S6.E4.m1.4.4.1.1.1.3.2.2">𝒲</ci><ci id="S6.E4.m1.4.4.1.1.1.3.2.3.cmml" xref="S6.E4.m1.4.4.1.1.1.3.2.3">𝓃</ci></apply><ci id="S6.E4.m1.3.3.cmml" xref="S6.E4.m1.3.3">𝓌</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E4.m1.4c">w^{\prime}\in A_{n}(w)\,\,\Longleftrightarrow\,\,\sigma^{-1}(\sigma(w^{\prime}% ))\cap\cal L(X)\nsubseteq W_{n}(w)\,.</annotation><annotation encoding="application/x-llamapun" id="S6.E4.m1.4d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ⟺ italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_σ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) ∩ caligraphic_L ( caligraphic_X ) ⊈ caligraphic_W start_POSTSUBSCRIPT caligraphic_n end_POSTSUBSCRIPT ( caligraphic_w ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.p2.22">In other words, for any <math alttext="w^{\prime}=uwv\in A_{n}(w)" class="ltx_Math" display="inline" id="S6.p2.11.m1.1"><semantics id="S6.p2.11.m1.1a"><mrow id="S6.p2.11.m1.1.2" xref="S6.p2.11.m1.1.2.cmml"><msup id="S6.p2.11.m1.1.2.2" xref="S6.p2.11.m1.1.2.2.cmml"><mi id="S6.p2.11.m1.1.2.2.2" xref="S6.p2.11.m1.1.2.2.2.cmml">w</mi><mo id="S6.p2.11.m1.1.2.2.3" xref="S6.p2.11.m1.1.2.2.3.cmml">′</mo></msup><mo id="S6.p2.11.m1.1.2.3" xref="S6.p2.11.m1.1.2.3.cmml">=</mo><mrow id="S6.p2.11.m1.1.2.4" xref="S6.p2.11.m1.1.2.4.cmml"><mi id="S6.p2.11.m1.1.2.4.2" xref="S6.p2.11.m1.1.2.4.2.cmml">u</mi><mo id="S6.p2.11.m1.1.2.4.1" xref="S6.p2.11.m1.1.2.4.1.cmml">⁢</mo><mi id="S6.p2.11.m1.1.2.4.3" xref="S6.p2.11.m1.1.2.4.3.cmml">w</mi><mo id="S6.p2.11.m1.1.2.4.1a" xref="S6.p2.11.m1.1.2.4.1.cmml">⁢</mo><mi id="S6.p2.11.m1.1.2.4.4" xref="S6.p2.11.m1.1.2.4.4.cmml">v</mi></mrow><mo id="S6.p2.11.m1.1.2.5" xref="S6.p2.11.m1.1.2.5.cmml">∈</mo><mrow id="S6.p2.11.m1.1.2.6" xref="S6.p2.11.m1.1.2.6.cmml"><msub id="S6.p2.11.m1.1.2.6.2" xref="S6.p2.11.m1.1.2.6.2.cmml"><mi id="S6.p2.11.m1.1.2.6.2.2" xref="S6.p2.11.m1.1.2.6.2.2.cmml">A</mi><mi id="S6.p2.11.m1.1.2.6.2.3" xref="S6.p2.11.m1.1.2.6.2.3.cmml">n</mi></msub><mo id="S6.p2.11.m1.1.2.6.1" xref="S6.p2.11.m1.1.2.6.1.cmml">⁢</mo><mrow id="S6.p2.11.m1.1.2.6.3.2" xref="S6.p2.11.m1.1.2.6.cmml"><mo id="S6.p2.11.m1.1.2.6.3.2.1" stretchy="false" xref="S6.p2.11.m1.1.2.6.cmml">(</mo><mi id="S6.p2.11.m1.1.1" xref="S6.p2.11.m1.1.1.cmml">w</mi><mo id="S6.p2.11.m1.1.2.6.3.2.2" stretchy="false" xref="S6.p2.11.m1.1.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.11.m1.1b"><apply id="S6.p2.11.m1.1.2.cmml" xref="S6.p2.11.m1.1.2"><and id="S6.p2.11.m1.1.2a.cmml" xref="S6.p2.11.m1.1.2"></and><apply id="S6.p2.11.m1.1.2b.cmml" xref="S6.p2.11.m1.1.2"><eq id="S6.p2.11.m1.1.2.3.cmml" xref="S6.p2.11.m1.1.2.3"></eq><apply id="S6.p2.11.m1.1.2.2.cmml" xref="S6.p2.11.m1.1.2.2"><csymbol cd="ambiguous" id="S6.p2.11.m1.1.2.2.1.cmml" xref="S6.p2.11.m1.1.2.2">superscript</csymbol><ci id="S6.p2.11.m1.1.2.2.2.cmml" xref="S6.p2.11.m1.1.2.2.2">𝑤</ci><ci id="S6.p2.11.m1.1.2.2.3.cmml" xref="S6.p2.11.m1.1.2.2.3">′</ci></apply><apply id="S6.p2.11.m1.1.2.4.cmml" xref="S6.p2.11.m1.1.2.4"><times id="S6.p2.11.m1.1.2.4.1.cmml" xref="S6.p2.11.m1.1.2.4.1"></times><ci id="S6.p2.11.m1.1.2.4.2.cmml" xref="S6.p2.11.m1.1.2.4.2">𝑢</ci><ci id="S6.p2.11.m1.1.2.4.3.cmml" xref="S6.p2.11.m1.1.2.4.3">𝑤</ci><ci id="S6.p2.11.m1.1.2.4.4.cmml" xref="S6.p2.11.m1.1.2.4.4">𝑣</ci></apply></apply><apply id="S6.p2.11.m1.1.2c.cmml" xref="S6.p2.11.m1.1.2"><in id="S6.p2.11.m1.1.2.5.cmml" xref="S6.p2.11.m1.1.2.5"></in><share href="https://arxiv.org/html/2211.11234v4#S6.p2.11.m1.1.2.4.cmml" id="S6.p2.11.m1.1.2d.cmml" xref="S6.p2.11.m1.1.2"></share><apply id="S6.p2.11.m1.1.2.6.cmml" xref="S6.p2.11.m1.1.2.6"><times id="S6.p2.11.m1.1.2.6.1.cmml" xref="S6.p2.11.m1.1.2.6.1"></times><apply id="S6.p2.11.m1.1.2.6.2.cmml" xref="S6.p2.11.m1.1.2.6.2"><csymbol cd="ambiguous" id="S6.p2.11.m1.1.2.6.2.1.cmml" xref="S6.p2.11.m1.1.2.6.2">subscript</csymbol><ci id="S6.p2.11.m1.1.2.6.2.2.cmml" xref="S6.p2.11.m1.1.2.6.2.2">𝐴</ci><ci id="S6.p2.11.m1.1.2.6.2.3.cmml" xref="S6.p2.11.m1.1.2.6.2.3">𝑛</ci></apply><ci id="S6.p2.11.m1.1.1.cmml" xref="S6.p2.11.m1.1.1">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.11.m1.1c">w^{\prime}=uwv\in A_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.11.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_u italic_w italic_v ∈ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> there exists a word <math alttext="u^{\prime}w^{\prime\prime}v^{\prime}\in\cal L(X)" class="ltx_Math" display="inline" id="S6.p2.12.m2.1"><semantics id="S6.p2.12.m2.1a"><mrow id="S6.p2.12.m2.1.2" xref="S6.p2.12.m2.1.2.cmml"><mrow id="S6.p2.12.m2.1.2.2" xref="S6.p2.12.m2.1.2.2.cmml"><msup id="S6.p2.12.m2.1.2.2.2" xref="S6.p2.12.m2.1.2.2.2.cmml"><mi id="S6.p2.12.m2.1.2.2.2.2" xref="S6.p2.12.m2.1.2.2.2.2.cmml">u</mi><mo id="S6.p2.12.m2.1.2.2.2.3" xref="S6.p2.12.m2.1.2.2.2.3.cmml">′</mo></msup><mo id="S6.p2.12.m2.1.2.2.1" xref="S6.p2.12.m2.1.2.2.1.cmml">⁢</mo><msup id="S6.p2.12.m2.1.2.2.3" xref="S6.p2.12.m2.1.2.2.3.cmml"><mi id="S6.p2.12.m2.1.2.2.3.2" xref="S6.p2.12.m2.1.2.2.3.2.cmml">w</mi><mo id="S6.p2.12.m2.1.2.2.3.3" xref="S6.p2.12.m2.1.2.2.3.3.cmml">′′</mo></msup><mo id="S6.p2.12.m2.1.2.2.1a" xref="S6.p2.12.m2.1.2.2.1.cmml">⁢</mo><msup id="S6.p2.12.m2.1.2.2.4" xref="S6.p2.12.m2.1.2.2.4.cmml"><mi id="S6.p2.12.m2.1.2.2.4.2" xref="S6.p2.12.m2.1.2.2.4.2.cmml">v</mi><mo id="S6.p2.12.m2.1.2.2.4.3" xref="S6.p2.12.m2.1.2.2.4.3.cmml">′</mo></msup></mrow><mo id="S6.p2.12.m2.1.2.1" xref="S6.p2.12.m2.1.2.1.cmml">∈</mo><mrow id="S6.p2.12.m2.1.2.3" xref="S6.p2.12.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p2.12.m2.1.2.3.2" xref="S6.p2.12.m2.1.2.3.2.cmml">ℒ</mi><mo id="S6.p2.12.m2.1.2.3.1" xref="S6.p2.12.m2.1.2.3.1.cmml">⁢</mo><mrow id="S6.p2.12.m2.1.2.3.3.2" xref="S6.p2.12.m2.1.2.3.cmml"><mo id="S6.p2.12.m2.1.2.3.3.2.1" stretchy="false" xref="S6.p2.12.m2.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p2.12.m2.1.1" xref="S6.p2.12.m2.1.1.cmml">𝒳</mi><mo id="S6.p2.12.m2.1.2.3.3.2.2" stretchy="false" xref="S6.p2.12.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.12.m2.1b"><apply id="S6.p2.12.m2.1.2.cmml" xref="S6.p2.12.m2.1.2"><in id="S6.p2.12.m2.1.2.1.cmml" xref="S6.p2.12.m2.1.2.1"></in><apply id="S6.p2.12.m2.1.2.2.cmml" xref="S6.p2.12.m2.1.2.2"><times id="S6.p2.12.m2.1.2.2.1.cmml" xref="S6.p2.12.m2.1.2.2.1"></times><apply id="S6.p2.12.m2.1.2.2.2.cmml" xref="S6.p2.12.m2.1.2.2.2"><csymbol cd="ambiguous" id="S6.p2.12.m2.1.2.2.2.1.cmml" xref="S6.p2.12.m2.1.2.2.2">superscript</csymbol><ci id="S6.p2.12.m2.1.2.2.2.2.cmml" xref="S6.p2.12.m2.1.2.2.2.2">𝑢</ci><ci id="S6.p2.12.m2.1.2.2.2.3.cmml" xref="S6.p2.12.m2.1.2.2.2.3">′</ci></apply><apply id="S6.p2.12.m2.1.2.2.3.cmml" xref="S6.p2.12.m2.1.2.2.3"><csymbol cd="ambiguous" id="S6.p2.12.m2.1.2.2.3.1.cmml" xref="S6.p2.12.m2.1.2.2.3">superscript</csymbol><ci id="S6.p2.12.m2.1.2.2.3.2.cmml" xref="S6.p2.12.m2.1.2.2.3.2">𝑤</ci><ci id="S6.p2.12.m2.1.2.2.3.3.cmml" xref="S6.p2.12.m2.1.2.2.3.3">′′</ci></apply><apply id="S6.p2.12.m2.1.2.2.4.cmml" xref="S6.p2.12.m2.1.2.2.4"><csymbol cd="ambiguous" id="S6.p2.12.m2.1.2.2.4.1.cmml" xref="S6.p2.12.m2.1.2.2.4">superscript</csymbol><ci id="S6.p2.12.m2.1.2.2.4.2.cmml" xref="S6.p2.12.m2.1.2.2.4.2">𝑣</ci><ci id="S6.p2.12.m2.1.2.2.4.3.cmml" xref="S6.p2.12.m2.1.2.2.4.3">′</ci></apply></apply><apply id="S6.p2.12.m2.1.2.3.cmml" xref="S6.p2.12.m2.1.2.3"><times id="S6.p2.12.m2.1.2.3.1.cmml" xref="S6.p2.12.m2.1.2.3.1"></times><ci id="S6.p2.12.m2.1.2.3.2.cmml" xref="S6.p2.12.m2.1.2.3.2">ℒ</ci><ci id="S6.p2.12.m2.1.1.cmml" xref="S6.p2.12.m2.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.12.m2.1c">u^{\prime}w^{\prime\prime}v^{\prime}\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.12.m2.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_w start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> with <math alttext="|u^{\prime}|=|v^{\prime}|=n" class="ltx_Math" display="inline" id="S6.p2.13.m3.2"><semantics id="S6.p2.13.m3.2a"><mrow id="S6.p2.13.m3.2.2" xref="S6.p2.13.m3.2.2.cmml"><mrow id="S6.p2.13.m3.1.1.1.1" xref="S6.p2.13.m3.1.1.1.2.cmml"><mo id="S6.p2.13.m3.1.1.1.1.2" stretchy="false" xref="S6.p2.13.m3.1.1.1.2.1.cmml">|</mo><msup id="S6.p2.13.m3.1.1.1.1.1" xref="S6.p2.13.m3.1.1.1.1.1.cmml"><mi id="S6.p2.13.m3.1.1.1.1.1.2" xref="S6.p2.13.m3.1.1.1.1.1.2.cmml">u</mi><mo id="S6.p2.13.m3.1.1.1.1.1.3" xref="S6.p2.13.m3.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.p2.13.m3.1.1.1.1.3" stretchy="false" xref="S6.p2.13.m3.1.1.1.2.1.cmml">|</mo></mrow><mo id="S6.p2.13.m3.2.2.4" xref="S6.p2.13.m3.2.2.4.cmml">=</mo><mrow id="S6.p2.13.m3.2.2.2.1" xref="S6.p2.13.m3.2.2.2.2.cmml"><mo id="S6.p2.13.m3.2.2.2.1.2" stretchy="false" xref="S6.p2.13.m3.2.2.2.2.1.cmml">|</mo><msup id="S6.p2.13.m3.2.2.2.1.1" xref="S6.p2.13.m3.2.2.2.1.1.cmml"><mi id="S6.p2.13.m3.2.2.2.1.1.2" xref="S6.p2.13.m3.2.2.2.1.1.2.cmml">v</mi><mo id="S6.p2.13.m3.2.2.2.1.1.3" xref="S6.p2.13.m3.2.2.2.1.1.3.cmml">′</mo></msup><mo id="S6.p2.13.m3.2.2.2.1.3" stretchy="false" xref="S6.p2.13.m3.2.2.2.2.1.cmml">|</mo></mrow><mo id="S6.p2.13.m3.2.2.5" xref="S6.p2.13.m3.2.2.5.cmml">=</mo><mi id="S6.p2.13.m3.2.2.6" xref="S6.p2.13.m3.2.2.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.13.m3.2b"><apply id="S6.p2.13.m3.2.2.cmml" xref="S6.p2.13.m3.2.2"><and id="S6.p2.13.m3.2.2a.cmml" xref="S6.p2.13.m3.2.2"></and><apply id="S6.p2.13.m3.2.2b.cmml" xref="S6.p2.13.m3.2.2"><eq id="S6.p2.13.m3.2.2.4.cmml" xref="S6.p2.13.m3.2.2.4"></eq><apply id="S6.p2.13.m3.1.1.1.2.cmml" xref="S6.p2.13.m3.1.1.1.1"><abs id="S6.p2.13.m3.1.1.1.2.1.cmml" xref="S6.p2.13.m3.1.1.1.1.2"></abs><apply id="S6.p2.13.m3.1.1.1.1.1.cmml" xref="S6.p2.13.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.13.m3.1.1.1.1.1.1.cmml" xref="S6.p2.13.m3.1.1.1.1.1">superscript</csymbol><ci id="S6.p2.13.m3.1.1.1.1.1.2.cmml" xref="S6.p2.13.m3.1.1.1.1.1.2">𝑢</ci><ci id="S6.p2.13.m3.1.1.1.1.1.3.cmml" xref="S6.p2.13.m3.1.1.1.1.1.3">′</ci></apply></apply><apply id="S6.p2.13.m3.2.2.2.2.cmml" xref="S6.p2.13.m3.2.2.2.1"><abs id="S6.p2.13.m3.2.2.2.2.1.cmml" xref="S6.p2.13.m3.2.2.2.1.2"></abs><apply id="S6.p2.13.m3.2.2.2.1.1.cmml" xref="S6.p2.13.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.p2.13.m3.2.2.2.1.1.1.cmml" xref="S6.p2.13.m3.2.2.2.1.1">superscript</csymbol><ci id="S6.p2.13.m3.2.2.2.1.1.2.cmml" xref="S6.p2.13.m3.2.2.2.1.1.2">𝑣</ci><ci id="S6.p2.13.m3.2.2.2.1.1.3.cmml" xref="S6.p2.13.m3.2.2.2.1.1.3">′</ci></apply></apply></apply><apply id="S6.p2.13.m3.2.2c.cmml" xref="S6.p2.13.m3.2.2"><eq id="S6.p2.13.m3.2.2.5.cmml" xref="S6.p2.13.m3.2.2.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S6.p2.13.m3.2.2.2.cmml" id="S6.p2.13.m3.2.2d.cmml" xref="S6.p2.13.m3.2.2"></share><ci id="S6.p2.13.m3.2.2.6.cmml" xref="S6.p2.13.m3.2.2.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.13.m3.2c">|u^{\prime}|=|v^{\prime}|=n</annotation><annotation encoding="application/x-llamapun" id="S6.p2.13.m3.2d">| italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | = | italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | = italic_n</annotation></semantics></math> and <math alttext="|w^{\prime\prime}|=|w|" class="ltx_Math" display="inline" id="S6.p2.14.m4.2"><semantics id="S6.p2.14.m4.2a"><mrow id="S6.p2.14.m4.2.2" xref="S6.p2.14.m4.2.2.cmml"><mrow id="S6.p2.14.m4.2.2.1.1" xref="S6.p2.14.m4.2.2.1.2.cmml"><mo id="S6.p2.14.m4.2.2.1.1.2" stretchy="false" xref="S6.p2.14.m4.2.2.1.2.1.cmml">|</mo><msup id="S6.p2.14.m4.2.2.1.1.1" xref="S6.p2.14.m4.2.2.1.1.1.cmml"><mi id="S6.p2.14.m4.2.2.1.1.1.2" xref="S6.p2.14.m4.2.2.1.1.1.2.cmml">w</mi><mo id="S6.p2.14.m4.2.2.1.1.1.3" xref="S6.p2.14.m4.2.2.1.1.1.3.cmml">′′</mo></msup><mo id="S6.p2.14.m4.2.2.1.1.3" stretchy="false" xref="S6.p2.14.m4.2.2.1.2.1.cmml">|</mo></mrow><mo id="S6.p2.14.m4.2.2.2" xref="S6.p2.14.m4.2.2.2.cmml">=</mo><mrow id="S6.p2.14.m4.2.2.3.2" xref="S6.p2.14.m4.2.2.3.1.cmml"><mo id="S6.p2.14.m4.2.2.3.2.1" stretchy="false" xref="S6.p2.14.m4.2.2.3.1.1.cmml">|</mo><mi id="S6.p2.14.m4.1.1" xref="S6.p2.14.m4.1.1.cmml">w</mi><mo id="S6.p2.14.m4.2.2.3.2.2" stretchy="false" xref="S6.p2.14.m4.2.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.14.m4.2b"><apply id="S6.p2.14.m4.2.2.cmml" xref="S6.p2.14.m4.2.2"><eq id="S6.p2.14.m4.2.2.2.cmml" xref="S6.p2.14.m4.2.2.2"></eq><apply id="S6.p2.14.m4.2.2.1.2.cmml" xref="S6.p2.14.m4.2.2.1.1"><abs id="S6.p2.14.m4.2.2.1.2.1.cmml" xref="S6.p2.14.m4.2.2.1.1.2"></abs><apply id="S6.p2.14.m4.2.2.1.1.1.cmml" xref="S6.p2.14.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.p2.14.m4.2.2.1.1.1.1.cmml" xref="S6.p2.14.m4.2.2.1.1.1">superscript</csymbol><ci id="S6.p2.14.m4.2.2.1.1.1.2.cmml" xref="S6.p2.14.m4.2.2.1.1.1.2">𝑤</ci><ci id="S6.p2.14.m4.2.2.1.1.1.3.cmml" xref="S6.p2.14.m4.2.2.1.1.1.3">′′</ci></apply></apply><apply id="S6.p2.14.m4.2.2.3.1.cmml" xref="S6.p2.14.m4.2.2.3.2"><abs id="S6.p2.14.m4.2.2.3.1.1.cmml" xref="S6.p2.14.m4.2.2.3.2.1"></abs><ci id="S6.p2.14.m4.1.1.cmml" xref="S6.p2.14.m4.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.14.m4.2c">|w^{\prime\prime}|=|w|</annotation><annotation encoding="application/x-llamapun" id="S6.p2.14.m4.2d">| italic_w start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT | = | italic_w |</annotation></semantics></math> such that <math alttext="w^{\prime\prime}\neq w" class="ltx_Math" display="inline" id="S6.p2.15.m5.1"><semantics id="S6.p2.15.m5.1a"><mrow id="S6.p2.15.m5.1.1" xref="S6.p2.15.m5.1.1.cmml"><msup id="S6.p2.15.m5.1.1.2" xref="S6.p2.15.m5.1.1.2.cmml"><mi id="S6.p2.15.m5.1.1.2.2" xref="S6.p2.15.m5.1.1.2.2.cmml">w</mi><mo id="S6.p2.15.m5.1.1.2.3" xref="S6.p2.15.m5.1.1.2.3.cmml">′′</mo></msup><mo id="S6.p2.15.m5.1.1.1" xref="S6.p2.15.m5.1.1.1.cmml">≠</mo><mi id="S6.p2.15.m5.1.1.3" xref="S6.p2.15.m5.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.15.m5.1b"><apply id="S6.p2.15.m5.1.1.cmml" xref="S6.p2.15.m5.1.1"><neq id="S6.p2.15.m5.1.1.1.cmml" xref="S6.p2.15.m5.1.1.1"></neq><apply id="S6.p2.15.m5.1.1.2.cmml" xref="S6.p2.15.m5.1.1.2"><csymbol cd="ambiguous" id="S6.p2.15.m5.1.1.2.1.cmml" xref="S6.p2.15.m5.1.1.2">superscript</csymbol><ci id="S6.p2.15.m5.1.1.2.2.cmml" xref="S6.p2.15.m5.1.1.2.2">𝑤</ci><ci id="S6.p2.15.m5.1.1.2.3.cmml" xref="S6.p2.15.m5.1.1.2.3">′′</ci></apply><ci id="S6.p2.15.m5.1.1.3.cmml" xref="S6.p2.15.m5.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.15.m5.1c">w^{\prime\prime}\neq w</annotation><annotation encoding="application/x-llamapun" id="S6.p2.15.m5.1d">italic_w start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ≠ italic_w</annotation></semantics></math> and <math alttext="\sigma(u^{\prime}w^{\prime\prime}v^{\prime})=\sigma(uwv)" class="ltx_Math" display="inline" id="S6.p2.16.m6.2"><semantics id="S6.p2.16.m6.2a"><mrow id="S6.p2.16.m6.2.2" xref="S6.p2.16.m6.2.2.cmml"><mrow id="S6.p2.16.m6.1.1.1" xref="S6.p2.16.m6.1.1.1.cmml"><mi id="S6.p2.16.m6.1.1.1.3" xref="S6.p2.16.m6.1.1.1.3.cmml">σ</mi><mo id="S6.p2.16.m6.1.1.1.2" xref="S6.p2.16.m6.1.1.1.2.cmml">⁢</mo><mrow id="S6.p2.16.m6.1.1.1.1.1" xref="S6.p2.16.m6.1.1.1.1.1.1.cmml"><mo id="S6.p2.16.m6.1.1.1.1.1.2" stretchy="false" xref="S6.p2.16.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.p2.16.m6.1.1.1.1.1.1" xref="S6.p2.16.m6.1.1.1.1.1.1.cmml"><msup id="S6.p2.16.m6.1.1.1.1.1.1.2" xref="S6.p2.16.m6.1.1.1.1.1.1.2.cmml"><mi id="S6.p2.16.m6.1.1.1.1.1.1.2.2" xref="S6.p2.16.m6.1.1.1.1.1.1.2.2.cmml">u</mi><mo id="S6.p2.16.m6.1.1.1.1.1.1.2.3" xref="S6.p2.16.m6.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S6.p2.16.m6.1.1.1.1.1.1.1" xref="S6.p2.16.m6.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S6.p2.16.m6.1.1.1.1.1.1.3" xref="S6.p2.16.m6.1.1.1.1.1.1.3.cmml"><mi id="S6.p2.16.m6.1.1.1.1.1.1.3.2" xref="S6.p2.16.m6.1.1.1.1.1.1.3.2.cmml">w</mi><mo id="S6.p2.16.m6.1.1.1.1.1.1.3.3" xref="S6.p2.16.m6.1.1.1.1.1.1.3.3.cmml">′′</mo></msup><mo id="S6.p2.16.m6.1.1.1.1.1.1.1a" xref="S6.p2.16.m6.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S6.p2.16.m6.1.1.1.1.1.1.4" xref="S6.p2.16.m6.1.1.1.1.1.1.4.cmml"><mi id="S6.p2.16.m6.1.1.1.1.1.1.4.2" xref="S6.p2.16.m6.1.1.1.1.1.1.4.2.cmml">v</mi><mo id="S6.p2.16.m6.1.1.1.1.1.1.4.3" xref="S6.p2.16.m6.1.1.1.1.1.1.4.3.cmml">′</mo></msup></mrow><mo id="S6.p2.16.m6.1.1.1.1.1.3" stretchy="false" xref="S6.p2.16.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.p2.16.m6.2.2.3" xref="S6.p2.16.m6.2.2.3.cmml">=</mo><mrow id="S6.p2.16.m6.2.2.2" xref="S6.p2.16.m6.2.2.2.cmml"><mi id="S6.p2.16.m6.2.2.2.3" xref="S6.p2.16.m6.2.2.2.3.cmml">σ</mi><mo id="S6.p2.16.m6.2.2.2.2" xref="S6.p2.16.m6.2.2.2.2.cmml">⁢</mo><mrow id="S6.p2.16.m6.2.2.2.1.1" xref="S6.p2.16.m6.2.2.2.1.1.1.cmml"><mo id="S6.p2.16.m6.2.2.2.1.1.2" stretchy="false" xref="S6.p2.16.m6.2.2.2.1.1.1.cmml">(</mo><mrow id="S6.p2.16.m6.2.2.2.1.1.1" xref="S6.p2.16.m6.2.2.2.1.1.1.cmml"><mi id="S6.p2.16.m6.2.2.2.1.1.1.2" xref="S6.p2.16.m6.2.2.2.1.1.1.2.cmml">u</mi><mo id="S6.p2.16.m6.2.2.2.1.1.1.1" xref="S6.p2.16.m6.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S6.p2.16.m6.2.2.2.1.1.1.3" xref="S6.p2.16.m6.2.2.2.1.1.1.3.cmml">w</mi><mo id="S6.p2.16.m6.2.2.2.1.1.1.1a" xref="S6.p2.16.m6.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S6.p2.16.m6.2.2.2.1.1.1.4" xref="S6.p2.16.m6.2.2.2.1.1.1.4.cmml">v</mi></mrow><mo id="S6.p2.16.m6.2.2.2.1.1.3" stretchy="false" xref="S6.p2.16.m6.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.16.m6.2b"><apply id="S6.p2.16.m6.2.2.cmml" xref="S6.p2.16.m6.2.2"><eq id="S6.p2.16.m6.2.2.3.cmml" xref="S6.p2.16.m6.2.2.3"></eq><apply id="S6.p2.16.m6.1.1.1.cmml" xref="S6.p2.16.m6.1.1.1"><times id="S6.p2.16.m6.1.1.1.2.cmml" xref="S6.p2.16.m6.1.1.1.2"></times><ci id="S6.p2.16.m6.1.1.1.3.cmml" xref="S6.p2.16.m6.1.1.1.3">𝜎</ci><apply id="S6.p2.16.m6.1.1.1.1.1.1.cmml" xref="S6.p2.16.m6.1.1.1.1.1"><times id="S6.p2.16.m6.1.1.1.1.1.1.1.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.1"></times><apply id="S6.p2.16.m6.1.1.1.1.1.1.2.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.p2.16.m6.1.1.1.1.1.1.2.1.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.2">superscript</csymbol><ci id="S6.p2.16.m6.1.1.1.1.1.1.2.2.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.2.2">𝑢</ci><ci id="S6.p2.16.m6.1.1.1.1.1.1.2.3.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.2.3">′</ci></apply><apply id="S6.p2.16.m6.1.1.1.1.1.1.3.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.p2.16.m6.1.1.1.1.1.1.3.1.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.3">superscript</csymbol><ci id="S6.p2.16.m6.1.1.1.1.1.1.3.2.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.3.2">𝑤</ci><ci id="S6.p2.16.m6.1.1.1.1.1.1.3.3.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.3.3">′′</ci></apply><apply id="S6.p2.16.m6.1.1.1.1.1.1.4.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S6.p2.16.m6.1.1.1.1.1.1.4.1.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.4">superscript</csymbol><ci id="S6.p2.16.m6.1.1.1.1.1.1.4.2.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.4.2">𝑣</ci><ci id="S6.p2.16.m6.1.1.1.1.1.1.4.3.cmml" xref="S6.p2.16.m6.1.1.1.1.1.1.4.3">′</ci></apply></apply></apply><apply id="S6.p2.16.m6.2.2.2.cmml" xref="S6.p2.16.m6.2.2.2"><times id="S6.p2.16.m6.2.2.2.2.cmml" xref="S6.p2.16.m6.2.2.2.2"></times><ci id="S6.p2.16.m6.2.2.2.3.cmml" xref="S6.p2.16.m6.2.2.2.3">𝜎</ci><apply id="S6.p2.16.m6.2.2.2.1.1.1.cmml" xref="S6.p2.16.m6.2.2.2.1.1"><times id="S6.p2.16.m6.2.2.2.1.1.1.1.cmml" xref="S6.p2.16.m6.2.2.2.1.1.1.1"></times><ci id="S6.p2.16.m6.2.2.2.1.1.1.2.cmml" xref="S6.p2.16.m6.2.2.2.1.1.1.2">𝑢</ci><ci id="S6.p2.16.m6.2.2.2.1.1.1.3.cmml" xref="S6.p2.16.m6.2.2.2.1.1.1.3">𝑤</ci><ci id="S6.p2.16.m6.2.2.2.1.1.1.4.cmml" xref="S6.p2.16.m6.2.2.2.1.1.1.4">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.16.m6.2c">\sigma(u^{\prime}w^{\prime\prime}v^{\prime})=\sigma(uwv)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.16.m6.2d">italic_σ ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_w start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_σ ( italic_u italic_w italic_v )</annotation></semantics></math>, while for any <math alttext="w^{\prime}=uwv\in U_{n}(w)" class="ltx_Math" display="inline" id="S6.p2.17.m7.1"><semantics id="S6.p2.17.m7.1a"><mrow id="S6.p2.17.m7.1.2" xref="S6.p2.17.m7.1.2.cmml"><msup id="S6.p2.17.m7.1.2.2" xref="S6.p2.17.m7.1.2.2.cmml"><mi id="S6.p2.17.m7.1.2.2.2" xref="S6.p2.17.m7.1.2.2.2.cmml">w</mi><mo id="S6.p2.17.m7.1.2.2.3" xref="S6.p2.17.m7.1.2.2.3.cmml">′</mo></msup><mo id="S6.p2.17.m7.1.2.3" xref="S6.p2.17.m7.1.2.3.cmml">=</mo><mrow id="S6.p2.17.m7.1.2.4" xref="S6.p2.17.m7.1.2.4.cmml"><mi id="S6.p2.17.m7.1.2.4.2" xref="S6.p2.17.m7.1.2.4.2.cmml">u</mi><mo id="S6.p2.17.m7.1.2.4.1" xref="S6.p2.17.m7.1.2.4.1.cmml">⁢</mo><mi id="S6.p2.17.m7.1.2.4.3" xref="S6.p2.17.m7.1.2.4.3.cmml">w</mi><mo id="S6.p2.17.m7.1.2.4.1a" xref="S6.p2.17.m7.1.2.4.1.cmml">⁢</mo><mi id="S6.p2.17.m7.1.2.4.4" xref="S6.p2.17.m7.1.2.4.4.cmml">v</mi></mrow><mo id="S6.p2.17.m7.1.2.5" xref="S6.p2.17.m7.1.2.5.cmml">∈</mo><mrow id="S6.p2.17.m7.1.2.6" xref="S6.p2.17.m7.1.2.6.cmml"><msub id="S6.p2.17.m7.1.2.6.2" xref="S6.p2.17.m7.1.2.6.2.cmml"><mi id="S6.p2.17.m7.1.2.6.2.2" xref="S6.p2.17.m7.1.2.6.2.2.cmml">U</mi><mi id="S6.p2.17.m7.1.2.6.2.3" xref="S6.p2.17.m7.1.2.6.2.3.cmml">n</mi></msub><mo id="S6.p2.17.m7.1.2.6.1" xref="S6.p2.17.m7.1.2.6.1.cmml">⁢</mo><mrow id="S6.p2.17.m7.1.2.6.3.2" xref="S6.p2.17.m7.1.2.6.cmml"><mo id="S6.p2.17.m7.1.2.6.3.2.1" stretchy="false" xref="S6.p2.17.m7.1.2.6.cmml">(</mo><mi id="S6.p2.17.m7.1.1" xref="S6.p2.17.m7.1.1.cmml">w</mi><mo id="S6.p2.17.m7.1.2.6.3.2.2" stretchy="false" xref="S6.p2.17.m7.1.2.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.17.m7.1b"><apply id="S6.p2.17.m7.1.2.cmml" xref="S6.p2.17.m7.1.2"><and id="S6.p2.17.m7.1.2a.cmml" xref="S6.p2.17.m7.1.2"></and><apply id="S6.p2.17.m7.1.2b.cmml" xref="S6.p2.17.m7.1.2"><eq id="S6.p2.17.m7.1.2.3.cmml" xref="S6.p2.17.m7.1.2.3"></eq><apply id="S6.p2.17.m7.1.2.2.cmml" xref="S6.p2.17.m7.1.2.2"><csymbol cd="ambiguous" id="S6.p2.17.m7.1.2.2.1.cmml" xref="S6.p2.17.m7.1.2.2">superscript</csymbol><ci id="S6.p2.17.m7.1.2.2.2.cmml" xref="S6.p2.17.m7.1.2.2.2">𝑤</ci><ci id="S6.p2.17.m7.1.2.2.3.cmml" xref="S6.p2.17.m7.1.2.2.3">′</ci></apply><apply id="S6.p2.17.m7.1.2.4.cmml" xref="S6.p2.17.m7.1.2.4"><times id="S6.p2.17.m7.1.2.4.1.cmml" xref="S6.p2.17.m7.1.2.4.1"></times><ci id="S6.p2.17.m7.1.2.4.2.cmml" xref="S6.p2.17.m7.1.2.4.2">𝑢</ci><ci id="S6.p2.17.m7.1.2.4.3.cmml" xref="S6.p2.17.m7.1.2.4.3">𝑤</ci><ci id="S6.p2.17.m7.1.2.4.4.cmml" xref="S6.p2.17.m7.1.2.4.4">𝑣</ci></apply></apply><apply id="S6.p2.17.m7.1.2c.cmml" xref="S6.p2.17.m7.1.2"><in id="S6.p2.17.m7.1.2.5.cmml" xref="S6.p2.17.m7.1.2.5"></in><share href="https://arxiv.org/html/2211.11234v4#S6.p2.17.m7.1.2.4.cmml" id="S6.p2.17.m7.1.2d.cmml" xref="S6.p2.17.m7.1.2"></share><apply id="S6.p2.17.m7.1.2.6.cmml" xref="S6.p2.17.m7.1.2.6"><times id="S6.p2.17.m7.1.2.6.1.cmml" xref="S6.p2.17.m7.1.2.6.1"></times><apply id="S6.p2.17.m7.1.2.6.2.cmml" xref="S6.p2.17.m7.1.2.6.2"><csymbol cd="ambiguous" id="S6.p2.17.m7.1.2.6.2.1.cmml" xref="S6.p2.17.m7.1.2.6.2">subscript</csymbol><ci id="S6.p2.17.m7.1.2.6.2.2.cmml" xref="S6.p2.17.m7.1.2.6.2.2">𝑈</ci><ci id="S6.p2.17.m7.1.2.6.2.3.cmml" xref="S6.p2.17.m7.1.2.6.2.3">𝑛</ci></apply><ci id="S6.p2.17.m7.1.1.cmml" xref="S6.p2.17.m7.1.1">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.17.m7.1c">w^{\prime}=uwv\in U_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.17.m7.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = italic_u italic_w italic_v ∈ italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> any word <math alttext="u^{\prime}w^{\prime\prime}v^{\prime}\in\cal L(X)" class="ltx_Math" display="inline" id="S6.p2.18.m8.1"><semantics id="S6.p2.18.m8.1a"><mrow id="S6.p2.18.m8.1.2" xref="S6.p2.18.m8.1.2.cmml"><mrow id="S6.p2.18.m8.1.2.2" xref="S6.p2.18.m8.1.2.2.cmml"><msup id="S6.p2.18.m8.1.2.2.2" xref="S6.p2.18.m8.1.2.2.2.cmml"><mi id="S6.p2.18.m8.1.2.2.2.2" xref="S6.p2.18.m8.1.2.2.2.2.cmml">u</mi><mo id="S6.p2.18.m8.1.2.2.2.3" xref="S6.p2.18.m8.1.2.2.2.3.cmml">′</mo></msup><mo id="S6.p2.18.m8.1.2.2.1" xref="S6.p2.18.m8.1.2.2.1.cmml">⁢</mo><msup id="S6.p2.18.m8.1.2.2.3" xref="S6.p2.18.m8.1.2.2.3.cmml"><mi id="S6.p2.18.m8.1.2.2.3.2" xref="S6.p2.18.m8.1.2.2.3.2.cmml">w</mi><mo id="S6.p2.18.m8.1.2.2.3.3" xref="S6.p2.18.m8.1.2.2.3.3.cmml">′′</mo></msup><mo id="S6.p2.18.m8.1.2.2.1a" xref="S6.p2.18.m8.1.2.2.1.cmml">⁢</mo><msup id="S6.p2.18.m8.1.2.2.4" xref="S6.p2.18.m8.1.2.2.4.cmml"><mi id="S6.p2.18.m8.1.2.2.4.2" xref="S6.p2.18.m8.1.2.2.4.2.cmml">v</mi><mo id="S6.p2.18.m8.1.2.2.4.3" xref="S6.p2.18.m8.1.2.2.4.3.cmml">′</mo></msup></mrow><mo id="S6.p2.18.m8.1.2.1" xref="S6.p2.18.m8.1.2.1.cmml">∈</mo><mrow id="S6.p2.18.m8.1.2.3" xref="S6.p2.18.m8.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p2.18.m8.1.2.3.2" xref="S6.p2.18.m8.1.2.3.2.cmml">ℒ</mi><mo id="S6.p2.18.m8.1.2.3.1" xref="S6.p2.18.m8.1.2.3.1.cmml">⁢</mo><mrow id="S6.p2.18.m8.1.2.3.3.2" xref="S6.p2.18.m8.1.2.3.cmml"><mo id="S6.p2.18.m8.1.2.3.3.2.1" stretchy="false" xref="S6.p2.18.m8.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p2.18.m8.1.1" xref="S6.p2.18.m8.1.1.cmml">𝒳</mi><mo id="S6.p2.18.m8.1.2.3.3.2.2" stretchy="false" xref="S6.p2.18.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.18.m8.1b"><apply id="S6.p2.18.m8.1.2.cmml" xref="S6.p2.18.m8.1.2"><in id="S6.p2.18.m8.1.2.1.cmml" xref="S6.p2.18.m8.1.2.1"></in><apply id="S6.p2.18.m8.1.2.2.cmml" xref="S6.p2.18.m8.1.2.2"><times id="S6.p2.18.m8.1.2.2.1.cmml" xref="S6.p2.18.m8.1.2.2.1"></times><apply id="S6.p2.18.m8.1.2.2.2.cmml" xref="S6.p2.18.m8.1.2.2.2"><csymbol cd="ambiguous" id="S6.p2.18.m8.1.2.2.2.1.cmml" xref="S6.p2.18.m8.1.2.2.2">superscript</csymbol><ci id="S6.p2.18.m8.1.2.2.2.2.cmml" xref="S6.p2.18.m8.1.2.2.2.2">𝑢</ci><ci id="S6.p2.18.m8.1.2.2.2.3.cmml" xref="S6.p2.18.m8.1.2.2.2.3">′</ci></apply><apply id="S6.p2.18.m8.1.2.2.3.cmml" xref="S6.p2.18.m8.1.2.2.3"><csymbol cd="ambiguous" id="S6.p2.18.m8.1.2.2.3.1.cmml" xref="S6.p2.18.m8.1.2.2.3">superscript</csymbol><ci id="S6.p2.18.m8.1.2.2.3.2.cmml" xref="S6.p2.18.m8.1.2.2.3.2">𝑤</ci><ci id="S6.p2.18.m8.1.2.2.3.3.cmml" xref="S6.p2.18.m8.1.2.2.3.3">′′</ci></apply><apply id="S6.p2.18.m8.1.2.2.4.cmml" xref="S6.p2.18.m8.1.2.2.4"><csymbol cd="ambiguous" id="S6.p2.18.m8.1.2.2.4.1.cmml" xref="S6.p2.18.m8.1.2.2.4">superscript</csymbol><ci id="S6.p2.18.m8.1.2.2.4.2.cmml" xref="S6.p2.18.m8.1.2.2.4.2">𝑣</ci><ci id="S6.p2.18.m8.1.2.2.4.3.cmml" xref="S6.p2.18.m8.1.2.2.4.3">′</ci></apply></apply><apply id="S6.p2.18.m8.1.2.3.cmml" xref="S6.p2.18.m8.1.2.3"><times id="S6.p2.18.m8.1.2.3.1.cmml" xref="S6.p2.18.m8.1.2.3.1"></times><ci id="S6.p2.18.m8.1.2.3.2.cmml" xref="S6.p2.18.m8.1.2.3.2">ℒ</ci><ci id="S6.p2.18.m8.1.1.cmml" xref="S6.p2.18.m8.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.18.m8.1c">u^{\prime}w^{\prime\prime}v^{\prime}\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.18.m8.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_w start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> with <math alttext="|u^{\prime}|=|v^{\prime}|=n" class="ltx_Math" display="inline" id="S6.p2.19.m9.2"><semantics id="S6.p2.19.m9.2a"><mrow id="S6.p2.19.m9.2.2" xref="S6.p2.19.m9.2.2.cmml"><mrow id="S6.p2.19.m9.1.1.1.1" xref="S6.p2.19.m9.1.1.1.2.cmml"><mo id="S6.p2.19.m9.1.1.1.1.2" stretchy="false" xref="S6.p2.19.m9.1.1.1.2.1.cmml">|</mo><msup id="S6.p2.19.m9.1.1.1.1.1" xref="S6.p2.19.m9.1.1.1.1.1.cmml"><mi id="S6.p2.19.m9.1.1.1.1.1.2" xref="S6.p2.19.m9.1.1.1.1.1.2.cmml">u</mi><mo id="S6.p2.19.m9.1.1.1.1.1.3" xref="S6.p2.19.m9.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S6.p2.19.m9.1.1.1.1.3" stretchy="false" xref="S6.p2.19.m9.1.1.1.2.1.cmml">|</mo></mrow><mo id="S6.p2.19.m9.2.2.4" xref="S6.p2.19.m9.2.2.4.cmml">=</mo><mrow id="S6.p2.19.m9.2.2.2.1" xref="S6.p2.19.m9.2.2.2.2.cmml"><mo id="S6.p2.19.m9.2.2.2.1.2" stretchy="false" xref="S6.p2.19.m9.2.2.2.2.1.cmml">|</mo><msup id="S6.p2.19.m9.2.2.2.1.1" xref="S6.p2.19.m9.2.2.2.1.1.cmml"><mi id="S6.p2.19.m9.2.2.2.1.1.2" xref="S6.p2.19.m9.2.2.2.1.1.2.cmml">v</mi><mo id="S6.p2.19.m9.2.2.2.1.1.3" xref="S6.p2.19.m9.2.2.2.1.1.3.cmml">′</mo></msup><mo id="S6.p2.19.m9.2.2.2.1.3" stretchy="false" xref="S6.p2.19.m9.2.2.2.2.1.cmml">|</mo></mrow><mo id="S6.p2.19.m9.2.2.5" xref="S6.p2.19.m9.2.2.5.cmml">=</mo><mi id="S6.p2.19.m9.2.2.6" xref="S6.p2.19.m9.2.2.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.19.m9.2b"><apply id="S6.p2.19.m9.2.2.cmml" xref="S6.p2.19.m9.2.2"><and id="S6.p2.19.m9.2.2a.cmml" xref="S6.p2.19.m9.2.2"></and><apply id="S6.p2.19.m9.2.2b.cmml" xref="S6.p2.19.m9.2.2"><eq id="S6.p2.19.m9.2.2.4.cmml" xref="S6.p2.19.m9.2.2.4"></eq><apply id="S6.p2.19.m9.1.1.1.2.cmml" xref="S6.p2.19.m9.1.1.1.1"><abs id="S6.p2.19.m9.1.1.1.2.1.cmml" xref="S6.p2.19.m9.1.1.1.1.2"></abs><apply id="S6.p2.19.m9.1.1.1.1.1.cmml" xref="S6.p2.19.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.p2.19.m9.1.1.1.1.1.1.cmml" xref="S6.p2.19.m9.1.1.1.1.1">superscript</csymbol><ci id="S6.p2.19.m9.1.1.1.1.1.2.cmml" xref="S6.p2.19.m9.1.1.1.1.1.2">𝑢</ci><ci id="S6.p2.19.m9.1.1.1.1.1.3.cmml" xref="S6.p2.19.m9.1.1.1.1.1.3">′</ci></apply></apply><apply id="S6.p2.19.m9.2.2.2.2.cmml" xref="S6.p2.19.m9.2.2.2.1"><abs id="S6.p2.19.m9.2.2.2.2.1.cmml" xref="S6.p2.19.m9.2.2.2.1.2"></abs><apply id="S6.p2.19.m9.2.2.2.1.1.cmml" xref="S6.p2.19.m9.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.p2.19.m9.2.2.2.1.1.1.cmml" xref="S6.p2.19.m9.2.2.2.1.1">superscript</csymbol><ci id="S6.p2.19.m9.2.2.2.1.1.2.cmml" xref="S6.p2.19.m9.2.2.2.1.1.2">𝑣</ci><ci id="S6.p2.19.m9.2.2.2.1.1.3.cmml" xref="S6.p2.19.m9.2.2.2.1.1.3">′</ci></apply></apply></apply><apply id="S6.p2.19.m9.2.2c.cmml" xref="S6.p2.19.m9.2.2"><eq id="S6.p2.19.m9.2.2.5.cmml" xref="S6.p2.19.m9.2.2.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S6.p2.19.m9.2.2.2.cmml" id="S6.p2.19.m9.2.2d.cmml" xref="S6.p2.19.m9.2.2"></share><ci id="S6.p2.19.m9.2.2.6.cmml" xref="S6.p2.19.m9.2.2.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.19.m9.2c">|u^{\prime}|=|v^{\prime}|=n</annotation><annotation encoding="application/x-llamapun" id="S6.p2.19.m9.2d">| italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | = | italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | = italic_n</annotation></semantics></math> and <math alttext="\sigma(u^{\prime}w^{\prime\prime}v^{\prime})=\sigma(uwv)" class="ltx_Math" display="inline" id="S6.p2.20.m10.2"><semantics id="S6.p2.20.m10.2a"><mrow id="S6.p2.20.m10.2.2" xref="S6.p2.20.m10.2.2.cmml"><mrow id="S6.p2.20.m10.1.1.1" xref="S6.p2.20.m10.1.1.1.cmml"><mi id="S6.p2.20.m10.1.1.1.3" xref="S6.p2.20.m10.1.1.1.3.cmml">σ</mi><mo id="S6.p2.20.m10.1.1.1.2" xref="S6.p2.20.m10.1.1.1.2.cmml">⁢</mo><mrow id="S6.p2.20.m10.1.1.1.1.1" xref="S6.p2.20.m10.1.1.1.1.1.1.cmml"><mo id="S6.p2.20.m10.1.1.1.1.1.2" stretchy="false" xref="S6.p2.20.m10.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.p2.20.m10.1.1.1.1.1.1" xref="S6.p2.20.m10.1.1.1.1.1.1.cmml"><msup id="S6.p2.20.m10.1.1.1.1.1.1.2" xref="S6.p2.20.m10.1.1.1.1.1.1.2.cmml"><mi id="S6.p2.20.m10.1.1.1.1.1.1.2.2" xref="S6.p2.20.m10.1.1.1.1.1.1.2.2.cmml">u</mi><mo id="S6.p2.20.m10.1.1.1.1.1.1.2.3" xref="S6.p2.20.m10.1.1.1.1.1.1.2.3.cmml">′</mo></msup><mo id="S6.p2.20.m10.1.1.1.1.1.1.1" xref="S6.p2.20.m10.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S6.p2.20.m10.1.1.1.1.1.1.3" xref="S6.p2.20.m10.1.1.1.1.1.1.3.cmml"><mi id="S6.p2.20.m10.1.1.1.1.1.1.3.2" xref="S6.p2.20.m10.1.1.1.1.1.1.3.2.cmml">w</mi><mo id="S6.p2.20.m10.1.1.1.1.1.1.3.3" xref="S6.p2.20.m10.1.1.1.1.1.1.3.3.cmml">′′</mo></msup><mo id="S6.p2.20.m10.1.1.1.1.1.1.1a" xref="S6.p2.20.m10.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S6.p2.20.m10.1.1.1.1.1.1.4" xref="S6.p2.20.m10.1.1.1.1.1.1.4.cmml"><mi id="S6.p2.20.m10.1.1.1.1.1.1.4.2" xref="S6.p2.20.m10.1.1.1.1.1.1.4.2.cmml">v</mi><mo id="S6.p2.20.m10.1.1.1.1.1.1.4.3" xref="S6.p2.20.m10.1.1.1.1.1.1.4.3.cmml">′</mo></msup></mrow><mo id="S6.p2.20.m10.1.1.1.1.1.3" stretchy="false" xref="S6.p2.20.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.p2.20.m10.2.2.3" xref="S6.p2.20.m10.2.2.3.cmml">=</mo><mrow id="S6.p2.20.m10.2.2.2" xref="S6.p2.20.m10.2.2.2.cmml"><mi id="S6.p2.20.m10.2.2.2.3" xref="S6.p2.20.m10.2.2.2.3.cmml">σ</mi><mo id="S6.p2.20.m10.2.2.2.2" xref="S6.p2.20.m10.2.2.2.2.cmml">⁢</mo><mrow id="S6.p2.20.m10.2.2.2.1.1" xref="S6.p2.20.m10.2.2.2.1.1.1.cmml"><mo id="S6.p2.20.m10.2.2.2.1.1.2" stretchy="false" xref="S6.p2.20.m10.2.2.2.1.1.1.cmml">(</mo><mrow id="S6.p2.20.m10.2.2.2.1.1.1" xref="S6.p2.20.m10.2.2.2.1.1.1.cmml"><mi id="S6.p2.20.m10.2.2.2.1.1.1.2" xref="S6.p2.20.m10.2.2.2.1.1.1.2.cmml">u</mi><mo id="S6.p2.20.m10.2.2.2.1.1.1.1" xref="S6.p2.20.m10.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S6.p2.20.m10.2.2.2.1.1.1.3" xref="S6.p2.20.m10.2.2.2.1.1.1.3.cmml">w</mi><mo id="S6.p2.20.m10.2.2.2.1.1.1.1a" xref="S6.p2.20.m10.2.2.2.1.1.1.1.cmml">⁢</mo><mi id="S6.p2.20.m10.2.2.2.1.1.1.4" xref="S6.p2.20.m10.2.2.2.1.1.1.4.cmml">v</mi></mrow><mo id="S6.p2.20.m10.2.2.2.1.1.3" stretchy="false" xref="S6.p2.20.m10.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.20.m10.2b"><apply id="S6.p2.20.m10.2.2.cmml" xref="S6.p2.20.m10.2.2"><eq id="S6.p2.20.m10.2.2.3.cmml" xref="S6.p2.20.m10.2.2.3"></eq><apply id="S6.p2.20.m10.1.1.1.cmml" xref="S6.p2.20.m10.1.1.1"><times id="S6.p2.20.m10.1.1.1.2.cmml" xref="S6.p2.20.m10.1.1.1.2"></times><ci id="S6.p2.20.m10.1.1.1.3.cmml" xref="S6.p2.20.m10.1.1.1.3">𝜎</ci><apply id="S6.p2.20.m10.1.1.1.1.1.1.cmml" xref="S6.p2.20.m10.1.1.1.1.1"><times id="S6.p2.20.m10.1.1.1.1.1.1.1.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.1"></times><apply id="S6.p2.20.m10.1.1.1.1.1.1.2.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.p2.20.m10.1.1.1.1.1.1.2.1.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.2">superscript</csymbol><ci id="S6.p2.20.m10.1.1.1.1.1.1.2.2.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.2.2">𝑢</ci><ci id="S6.p2.20.m10.1.1.1.1.1.1.2.3.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.2.3">′</ci></apply><apply id="S6.p2.20.m10.1.1.1.1.1.1.3.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.p2.20.m10.1.1.1.1.1.1.3.1.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.3">superscript</csymbol><ci id="S6.p2.20.m10.1.1.1.1.1.1.3.2.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.3.2">𝑤</ci><ci id="S6.p2.20.m10.1.1.1.1.1.1.3.3.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.3.3">′′</ci></apply><apply id="S6.p2.20.m10.1.1.1.1.1.1.4.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S6.p2.20.m10.1.1.1.1.1.1.4.1.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.4">superscript</csymbol><ci id="S6.p2.20.m10.1.1.1.1.1.1.4.2.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.4.2">𝑣</ci><ci id="S6.p2.20.m10.1.1.1.1.1.1.4.3.cmml" xref="S6.p2.20.m10.1.1.1.1.1.1.4.3">′</ci></apply></apply></apply><apply id="S6.p2.20.m10.2.2.2.cmml" xref="S6.p2.20.m10.2.2.2"><times id="S6.p2.20.m10.2.2.2.2.cmml" xref="S6.p2.20.m10.2.2.2.2"></times><ci id="S6.p2.20.m10.2.2.2.3.cmml" xref="S6.p2.20.m10.2.2.2.3">𝜎</ci><apply id="S6.p2.20.m10.2.2.2.1.1.1.cmml" xref="S6.p2.20.m10.2.2.2.1.1"><times id="S6.p2.20.m10.2.2.2.1.1.1.1.cmml" xref="S6.p2.20.m10.2.2.2.1.1.1.1"></times><ci id="S6.p2.20.m10.2.2.2.1.1.1.2.cmml" xref="S6.p2.20.m10.2.2.2.1.1.1.2">𝑢</ci><ci id="S6.p2.20.m10.2.2.2.1.1.1.3.cmml" xref="S6.p2.20.m10.2.2.2.1.1.1.3">𝑤</ci><ci id="S6.p2.20.m10.2.2.2.1.1.1.4.cmml" xref="S6.p2.20.m10.2.2.2.1.1.1.4">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.20.m10.2c">\sigma(u^{\prime}w^{\prime\prime}v^{\prime})=\sigma(uwv)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.20.m10.2d">italic_σ ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_w start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_σ ( italic_u italic_w italic_v )</annotation></semantics></math> one has <math alttext="w^{\prime\prime}=w" class="ltx_Math" display="inline" id="S6.p2.21.m11.1"><semantics id="S6.p2.21.m11.1a"><mrow id="S6.p2.21.m11.1.1" xref="S6.p2.21.m11.1.1.cmml"><msup id="S6.p2.21.m11.1.1.2" xref="S6.p2.21.m11.1.1.2.cmml"><mi id="S6.p2.21.m11.1.1.2.2" xref="S6.p2.21.m11.1.1.2.2.cmml">w</mi><mo id="S6.p2.21.m11.1.1.2.3" xref="S6.p2.21.m11.1.1.2.3.cmml">′′</mo></msup><mo id="S6.p2.21.m11.1.1.1" xref="S6.p2.21.m11.1.1.1.cmml">=</mo><mi id="S6.p2.21.m11.1.1.3" xref="S6.p2.21.m11.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.21.m11.1b"><apply id="S6.p2.21.m11.1.1.cmml" xref="S6.p2.21.m11.1.1"><eq id="S6.p2.21.m11.1.1.1.cmml" xref="S6.p2.21.m11.1.1.1"></eq><apply id="S6.p2.21.m11.1.1.2.cmml" xref="S6.p2.21.m11.1.1.2"><csymbol cd="ambiguous" id="S6.p2.21.m11.1.1.2.1.cmml" xref="S6.p2.21.m11.1.1.2">superscript</csymbol><ci id="S6.p2.21.m11.1.1.2.2.cmml" xref="S6.p2.21.m11.1.1.2.2">𝑤</ci><ci id="S6.p2.21.m11.1.1.2.3.cmml" xref="S6.p2.21.m11.1.1.2.3">′′</ci></apply><ci id="S6.p2.21.m11.1.1.3.cmml" xref="S6.p2.21.m11.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p2.21.m11.1c">w^{\prime\prime}=w</annotation><annotation encoding="application/x-llamapun" id="S6.p2.21.m11.1d">italic_w start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = italic_w</annotation></semantics></math> (and thus <math alttext="u^{\prime}w^{\prime\prime}v^{\prime}\in W_{n}(w)" class="ltx_Math" display="inline" id="S6.p2.22.m12.1"><semantics id="S6.p2.22.m12.1a"><mrow id="S6.p2.22.m12.1.2" xref="S6.p2.22.m12.1.2.cmml"><mrow id="S6.p2.22.m12.1.2.2" xref="S6.p2.22.m12.1.2.2.cmml"><msup id="S6.p2.22.m12.1.2.2.2" xref="S6.p2.22.m12.1.2.2.2.cmml"><mi id="S6.p2.22.m12.1.2.2.2.2" xref="S6.p2.22.m12.1.2.2.2.2.cmml">u</mi><mo id="S6.p2.22.m12.1.2.2.2.3" xref="S6.p2.22.m12.1.2.2.2.3.cmml">′</mo></msup><mo id="S6.p2.22.m12.1.2.2.1" xref="S6.p2.22.m12.1.2.2.1.cmml">⁢</mo><msup id="S6.p2.22.m12.1.2.2.3" xref="S6.p2.22.m12.1.2.2.3.cmml"><mi id="S6.p2.22.m12.1.2.2.3.2" xref="S6.p2.22.m12.1.2.2.3.2.cmml">w</mi><mo id="S6.p2.22.m12.1.2.2.3.3" xref="S6.p2.22.m12.1.2.2.3.3.cmml">′′</mo></msup><mo id="S6.p2.22.m12.1.2.2.1a" xref="S6.p2.22.m12.1.2.2.1.cmml">⁢</mo><msup id="S6.p2.22.m12.1.2.2.4" xref="S6.p2.22.m12.1.2.2.4.cmml"><mi id="S6.p2.22.m12.1.2.2.4.2" xref="S6.p2.22.m12.1.2.2.4.2.cmml">v</mi><mo id="S6.p2.22.m12.1.2.2.4.3" xref="S6.p2.22.m12.1.2.2.4.3.cmml">′</mo></msup></mrow><mo id="S6.p2.22.m12.1.2.1" xref="S6.p2.22.m12.1.2.1.cmml">∈</mo><mrow id="S6.p2.22.m12.1.2.3" xref="S6.p2.22.m12.1.2.3.cmml"><msub id="S6.p2.22.m12.1.2.3.2" xref="S6.p2.22.m12.1.2.3.2.cmml"><mi id="S6.p2.22.m12.1.2.3.2.2" xref="S6.p2.22.m12.1.2.3.2.2.cmml">W</mi><mi id="S6.p2.22.m12.1.2.3.2.3" xref="S6.p2.22.m12.1.2.3.2.3.cmml">n</mi></msub><mo id="S6.p2.22.m12.1.2.3.1" xref="S6.p2.22.m12.1.2.3.1.cmml">⁢</mo><mrow id="S6.p2.22.m12.1.2.3.3.2" xref="S6.p2.22.m12.1.2.3.cmml"><mo id="S6.p2.22.m12.1.2.3.3.2.1" stretchy="false" xref="S6.p2.22.m12.1.2.3.cmml">(</mo><mi id="S6.p2.22.m12.1.1" xref="S6.p2.22.m12.1.1.cmml">w</mi><mo id="S6.p2.22.m12.1.2.3.3.2.2" stretchy="false" xref="S6.p2.22.m12.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p2.22.m12.1b"><apply id="S6.p2.22.m12.1.2.cmml" xref="S6.p2.22.m12.1.2"><in id="S6.p2.22.m12.1.2.1.cmml" xref="S6.p2.22.m12.1.2.1"></in><apply id="S6.p2.22.m12.1.2.2.cmml" xref="S6.p2.22.m12.1.2.2"><times id="S6.p2.22.m12.1.2.2.1.cmml" xref="S6.p2.22.m12.1.2.2.1"></times><apply id="S6.p2.22.m12.1.2.2.2.cmml" xref="S6.p2.22.m12.1.2.2.2"><csymbol cd="ambiguous" id="S6.p2.22.m12.1.2.2.2.1.cmml" xref="S6.p2.22.m12.1.2.2.2">superscript</csymbol><ci id="S6.p2.22.m12.1.2.2.2.2.cmml" xref="S6.p2.22.m12.1.2.2.2.2">𝑢</ci><ci id="S6.p2.22.m12.1.2.2.2.3.cmml" xref="S6.p2.22.m12.1.2.2.2.3">′</ci></apply><apply id="S6.p2.22.m12.1.2.2.3.cmml" xref="S6.p2.22.m12.1.2.2.3"><csymbol cd="ambiguous" id="S6.p2.22.m12.1.2.2.3.1.cmml" xref="S6.p2.22.m12.1.2.2.3">superscript</csymbol><ci id="S6.p2.22.m12.1.2.2.3.2.cmml" xref="S6.p2.22.m12.1.2.2.3.2">𝑤</ci><ci 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encoding="application/x-tex" id="S6.p2.22.m12.1c">u^{\prime}w^{\prime\prime}v^{\prime}\in W_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.p2.22.m12.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_w start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math>).</p> </div> <div class="ltx_para" id="S6.p3"> <p class="ltx_p" id="S6.p3.3">We now fix some <math alttext="w\in\cal L(X)" class="ltx_Math" display="inline" id="S6.p3.1.m1.1"><semantics id="S6.p3.1.m1.1a"><mrow id="S6.p3.1.m1.1.2" xref="S6.p3.1.m1.1.2.cmml"><mi id="S6.p3.1.m1.1.2.2" xref="S6.p3.1.m1.1.2.2.cmml">w</mi><mo id="S6.p3.1.m1.1.2.1" xref="S6.p3.1.m1.1.2.1.cmml">∈</mo><mrow id="S6.p3.1.m1.1.2.3" xref="S6.p3.1.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p3.1.m1.1.2.3.2" xref="S6.p3.1.m1.1.2.3.2.cmml">ℒ</mi><mo id="S6.p3.1.m1.1.2.3.1" xref="S6.p3.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S6.p3.1.m1.1.2.3.3.2" xref="S6.p3.1.m1.1.2.3.cmml"><mo id="S6.p3.1.m1.1.2.3.3.2.1" stretchy="false" xref="S6.p3.1.m1.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p3.1.m1.1.1" xref="S6.p3.1.m1.1.1.cmml">𝒳</mi><mo id="S6.p3.1.m1.1.2.3.3.2.2" stretchy="false" xref="S6.p3.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.1.m1.1b"><apply id="S6.p3.1.m1.1.2.cmml" xref="S6.p3.1.m1.1.2"><in id="S6.p3.1.m1.1.2.1.cmml" xref="S6.p3.1.m1.1.2.1"></in><ci id="S6.p3.1.m1.1.2.2.cmml" xref="S6.p3.1.m1.1.2.2">𝑤</ci><apply id="S6.p3.1.m1.1.2.3.cmml" xref="S6.p3.1.m1.1.2.3"><times id="S6.p3.1.m1.1.2.3.1.cmml" xref="S6.p3.1.m1.1.2.3.1"></times><ci id="S6.p3.1.m1.1.2.3.2.cmml" xref="S6.p3.1.m1.1.2.3.2">ℒ</ci><ci id="S6.p3.1.m1.1.1.cmml" xref="S6.p3.1.m1.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.1.m1.1c">w\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p3.1.m1.1d">italic_w ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> and observe directly from the definition of the sets <math alttext="A_{n}(w)" class="ltx_Math" display="inline" id="S6.p3.2.m2.1"><semantics id="S6.p3.2.m2.1a"><mrow id="S6.p3.2.m2.1.2" xref="S6.p3.2.m2.1.2.cmml"><msub id="S6.p3.2.m2.1.2.2" xref="S6.p3.2.m2.1.2.2.cmml"><mi id="S6.p3.2.m2.1.2.2.2" xref="S6.p3.2.m2.1.2.2.2.cmml">A</mi><mi id="S6.p3.2.m2.1.2.2.3" xref="S6.p3.2.m2.1.2.2.3.cmml">n</mi></msub><mo id="S6.p3.2.m2.1.2.1" xref="S6.p3.2.m2.1.2.1.cmml">⁢</mo><mrow id="S6.p3.2.m2.1.2.3.2" xref="S6.p3.2.m2.1.2.cmml"><mo id="S6.p3.2.m2.1.2.3.2.1" stretchy="false" xref="S6.p3.2.m2.1.2.cmml">(</mo><mi id="S6.p3.2.m2.1.1" xref="S6.p3.2.m2.1.1.cmml">w</mi><mo id="S6.p3.2.m2.1.2.3.2.2" stretchy="false" xref="S6.p3.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p3.2.m2.1b"><apply id="S6.p3.2.m2.1.2.cmml" xref="S6.p3.2.m2.1.2"><times id="S6.p3.2.m2.1.2.1.cmml" xref="S6.p3.2.m2.1.2.1"></times><apply id="S6.p3.2.m2.1.2.2.cmml" xref="S6.p3.2.m2.1.2.2"><csymbol cd="ambiguous" id="S6.p3.2.m2.1.2.2.1.cmml" xref="S6.p3.2.m2.1.2.2">subscript</csymbol><ci id="S6.p3.2.m2.1.2.2.2.cmml" xref="S6.p3.2.m2.1.2.2.2">𝐴</ci><ci id="S6.p3.2.m2.1.2.2.3.cmml" xref="S6.p3.2.m2.1.2.2.3">𝑛</ci></apply><ci id="S6.p3.2.m2.1.1.cmml" xref="S6.p3.2.m2.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p3.2.m2.1c">A_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.p3.2.m2.1d">italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> that their elements define cylinders for which their union <math alttext="[A_{n}(w)]:=\bigcup\{[uwv]\mid uwv\in A_{n}(w)\}\subseteq X" class="ltx_Math" display="inline" id="S6.p3.3.m3.5"><semantics id="S6.p3.3.m3.5a"><mrow id="S6.p3.3.m3.5.5" 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xref="S6.Ex2.m1.3.3.1.1.2.3.3.2">𝑛</ci></apply></apply><apply id="S6.Ex2.m1.3.3.1.1.2.1.1.1.2.cmml" xref="S6.Ex2.m1.3.3.1.1.2.1.1.1.1"><csymbol cd="latexml" id="S6.Ex2.m1.3.3.1.1.2.1.1.1.2.1.cmml" xref="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.2">delimited-[]</csymbol><apply id="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.cmml" xref="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1"><times id="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.1.cmml" xref="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.1"></times><apply id="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.2.cmml" xref="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.2.1.cmml" xref="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.2">subscript</csymbol><ci id="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.2.2.cmml" xref="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.2.2">𝐴</ci><ci id="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.2.3.cmml" xref="S6.Ex2.m1.3.3.1.1.2.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S6.Ex2.m1.2.2.cmml" xref="S6.Ex2.m1.2.2">𝑤</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex2.m1.3c">T^{-n-1}([A_{n+1}(w)])\,\,\subseteq\,\,T^{-n}([A_{n}(w)])\,.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex2.m1.3d">italic_T start_POSTSUPERSCRIPT - italic_n - 1 end_POSTSUPERSCRIPT ( [ italic_A start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_w ) ] ) ⊆ italic_T start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ( [ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ] ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_theorem ltx_theorem_rem" id="S6.Thmthm2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm2.1.1.1">Remark 6.2</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm2.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm2.p1"> <p class="ltx_p" id="S6.Thmthm2.p1.5">(1) We denote by <math alttext="A_{\infty}(w)" 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xref="S6.Thmthm2.p1.1.m1.1.2"><times id="S6.Thmthm2.p1.1.m1.1.2.1.cmml" xref="S6.Thmthm2.p1.1.m1.1.2.1"></times><apply id="S6.Thmthm2.p1.1.m1.1.2.2.cmml" xref="S6.Thmthm2.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S6.Thmthm2.p1.1.m1.1.2.2.1.cmml" xref="S6.Thmthm2.p1.1.m1.1.2.2">subscript</csymbol><ci id="S6.Thmthm2.p1.1.m1.1.2.2.2.cmml" xref="S6.Thmthm2.p1.1.m1.1.2.2.2">𝐴</ci><infinity id="S6.Thmthm2.p1.1.m1.1.2.2.3.cmml" xref="S6.Thmthm2.p1.1.m1.1.2.2.3"></infinity></apply><ci id="S6.Thmthm2.p1.1.m1.1.1.cmml" xref="S6.Thmthm2.p1.1.m1.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p1.1.m1.1c">A_{\infty}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p1.1.m1.1d">italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> the intersection of the nested family of the <math alttext="T^{-n}([A_{n}(w)])" class="ltx_Math" display="inline" id="S6.Thmthm2.p1.2.m2.2"><semantics 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id="S6.Thmthm2.p1.2.m2.2.2.1.1.1.2.1.cmml" xref="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.2">delimited-[]</csymbol><apply id="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1"><times id="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.1"></times><apply id="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.2.1.cmml" xref="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.2">subscript</csymbol><ci id="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.2.2.cmml" xref="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.2.2">𝐴</ci><ci id="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.2.3.cmml" xref="S6.Thmthm2.p1.2.m2.2.2.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S6.Thmthm2.p1.2.m2.1.1.cmml" xref="S6.Thmthm2.p1.2.m2.1.1">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p1.2.m2.2c">T^{-n}([A_{n}(w)])</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p1.2.m2.2d">italic_T start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ( [ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ] )</annotation></semantics></math>. Since by definition we have <math alttext="A_{n}(w)\subseteq\cal L(X)" class="ltx_Math" display="inline" id="S6.Thmthm2.p1.3.m3.2"><semantics id="S6.Thmthm2.p1.3.m3.2a"><mrow id="S6.Thmthm2.p1.3.m3.2.3" xref="S6.Thmthm2.p1.3.m3.2.3.cmml"><mrow id="S6.Thmthm2.p1.3.m3.2.3.2" xref="S6.Thmthm2.p1.3.m3.2.3.2.cmml"><msub id="S6.Thmthm2.p1.3.m3.2.3.2.2" xref="S6.Thmthm2.p1.3.m3.2.3.2.2.cmml"><mi id="S6.Thmthm2.p1.3.m3.2.3.2.2.2" xref="S6.Thmthm2.p1.3.m3.2.3.2.2.2.cmml">A</mi><mi id="S6.Thmthm2.p1.3.m3.2.3.2.2.3" xref="S6.Thmthm2.p1.3.m3.2.3.2.2.3.cmml">n</mi></msub><mo id="S6.Thmthm2.p1.3.m3.2.3.2.1" xref="S6.Thmthm2.p1.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S6.Thmthm2.p1.3.m3.2.3.2.3.2" xref="S6.Thmthm2.p1.3.m3.2.3.2.cmml"><mo id="S6.Thmthm2.p1.3.m3.2.3.2.3.2.1" stretchy="false" xref="S6.Thmthm2.p1.3.m3.2.3.2.cmml">(</mo><mi id="S6.Thmthm2.p1.3.m3.1.1" xref="S6.Thmthm2.p1.3.m3.1.1.cmml">w</mi><mo id="S6.Thmthm2.p1.3.m3.2.3.2.3.2.2" stretchy="false" xref="S6.Thmthm2.p1.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm2.p1.3.m3.2.3.1" xref="S6.Thmthm2.p1.3.m3.2.3.1.cmml">⊆</mo><mrow id="S6.Thmthm2.p1.3.m3.2.3.3" xref="S6.Thmthm2.p1.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm2.p1.3.m3.2.3.3.2" xref="S6.Thmthm2.p1.3.m3.2.3.3.2.cmml">ℒ</mi><mo id="S6.Thmthm2.p1.3.m3.2.3.3.1" xref="S6.Thmthm2.p1.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S6.Thmthm2.p1.3.m3.2.3.3.3.2" xref="S6.Thmthm2.p1.3.m3.2.3.3.cmml"><mo id="S6.Thmthm2.p1.3.m3.2.3.3.3.2.1" stretchy="false" xref="S6.Thmthm2.p1.3.m3.2.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm2.p1.3.m3.2.2" xref="S6.Thmthm2.p1.3.m3.2.2.cmml">𝒳</mi><mo id="S6.Thmthm2.p1.3.m3.2.3.3.3.2.2" stretchy="false" xref="S6.Thmthm2.p1.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm2.p1.3.m3.2b"><apply id="S6.Thmthm2.p1.3.m3.2.3.cmml" xref="S6.Thmthm2.p1.3.m3.2.3"><subset id="S6.Thmthm2.p1.3.m3.2.3.1.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.1"></subset><apply id="S6.Thmthm2.p1.3.m3.2.3.2.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.2"><times id="S6.Thmthm2.p1.3.m3.2.3.2.1.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.2.1"></times><apply id="S6.Thmthm2.p1.3.m3.2.3.2.2.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="S6.Thmthm2.p1.3.m3.2.3.2.2.1.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.2.2">subscript</csymbol><ci id="S6.Thmthm2.p1.3.m3.2.3.2.2.2.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.2.2.2">𝐴</ci><ci id="S6.Thmthm2.p1.3.m3.2.3.2.2.3.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.2.2.3">𝑛</ci></apply><ci id="S6.Thmthm2.p1.3.m3.1.1.cmml" xref="S6.Thmthm2.p1.3.m3.1.1">𝑤</ci></apply><apply id="S6.Thmthm2.p1.3.m3.2.3.3.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.3"><times id="S6.Thmthm2.p1.3.m3.2.3.3.1.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.3.1"></times><ci id="S6.Thmthm2.p1.3.m3.2.3.3.2.cmml" xref="S6.Thmthm2.p1.3.m3.2.3.3.2">ℒ</ci><ci id="S6.Thmthm2.p1.3.m3.2.2.cmml" xref="S6.Thmthm2.p1.3.m3.2.2">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p1.3.m3.2c">A_{n}(w)\subseteq\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p1.3.m3.2d">italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ⊆ caligraphic_L ( caligraphic_X )</annotation></semantics></math> for all <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S6.Thmthm2.p1.4.m4.1"><semantics id="S6.Thmthm2.p1.4.m4.1a"><mrow id="S6.Thmthm2.p1.4.m4.1.1" xref="S6.Thmthm2.p1.4.m4.1.1.cmml"><mi id="S6.Thmthm2.p1.4.m4.1.1.2" xref="S6.Thmthm2.p1.4.m4.1.1.2.cmml">n</mi><mo id="S6.Thmthm2.p1.4.m4.1.1.1" xref="S6.Thmthm2.p1.4.m4.1.1.1.cmml">≥</mo><mn id="S6.Thmthm2.p1.4.m4.1.1.3" xref="S6.Thmthm2.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm2.p1.4.m4.1b"><apply id="S6.Thmthm2.p1.4.m4.1.1.cmml" xref="S6.Thmthm2.p1.4.m4.1.1"><geq id="S6.Thmthm2.p1.4.m4.1.1.1.cmml" xref="S6.Thmthm2.p1.4.m4.1.1.1"></geq><ci id="S6.Thmthm2.p1.4.m4.1.1.2.cmml" xref="S6.Thmthm2.p1.4.m4.1.1.2">𝑛</ci><cn id="S6.Thmthm2.p1.4.m4.1.1.3.cmml" type="integer" xref="S6.Thmthm2.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p1.4.m4.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p1.4.m4.1d">italic_n ≥ 0</annotation></semantics></math> we know that <math alttext="A_{\infty}(w)\subseteq X" class="ltx_Math" display="inline" id="S6.Thmthm2.p1.5.m5.1"><semantics id="S6.Thmthm2.p1.5.m5.1a"><mrow id="S6.Thmthm2.p1.5.m5.1.2" xref="S6.Thmthm2.p1.5.m5.1.2.cmml"><mrow id="S6.Thmthm2.p1.5.m5.1.2.2" xref="S6.Thmthm2.p1.5.m5.1.2.2.cmml"><msub id="S6.Thmthm2.p1.5.m5.1.2.2.2" xref="S6.Thmthm2.p1.5.m5.1.2.2.2.cmml"><mi id="S6.Thmthm2.p1.5.m5.1.2.2.2.2" xref="S6.Thmthm2.p1.5.m5.1.2.2.2.2.cmml">A</mi><mi id="S6.Thmthm2.p1.5.m5.1.2.2.2.3" mathvariant="normal" xref="S6.Thmthm2.p1.5.m5.1.2.2.2.3.cmml">∞</mi></msub><mo id="S6.Thmthm2.p1.5.m5.1.2.2.1" xref="S6.Thmthm2.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S6.Thmthm2.p1.5.m5.1.2.2.3.2" xref="S6.Thmthm2.p1.5.m5.1.2.2.cmml"><mo id="S6.Thmthm2.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S6.Thmthm2.p1.5.m5.1.2.2.cmml">(</mo><mi id="S6.Thmthm2.p1.5.m5.1.1" xref="S6.Thmthm2.p1.5.m5.1.1.cmml">w</mi><mo id="S6.Thmthm2.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S6.Thmthm2.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm2.p1.5.m5.1.2.1" xref="S6.Thmthm2.p1.5.m5.1.2.1.cmml">⊆</mo><mi id="S6.Thmthm2.p1.5.m5.1.2.3" xref="S6.Thmthm2.p1.5.m5.1.2.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm2.p1.5.m5.1b"><apply id="S6.Thmthm2.p1.5.m5.1.2.cmml" xref="S6.Thmthm2.p1.5.m5.1.2"><subset id="S6.Thmthm2.p1.5.m5.1.2.1.cmml" xref="S6.Thmthm2.p1.5.m5.1.2.1"></subset><apply id="S6.Thmthm2.p1.5.m5.1.2.2.cmml" xref="S6.Thmthm2.p1.5.m5.1.2.2"><times id="S6.Thmthm2.p1.5.m5.1.2.2.1.cmml" xref="S6.Thmthm2.p1.5.m5.1.2.2.1"></times><apply id="S6.Thmthm2.p1.5.m5.1.2.2.2.cmml" xref="S6.Thmthm2.p1.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S6.Thmthm2.p1.5.m5.1.2.2.2.1.cmml" xref="S6.Thmthm2.p1.5.m5.1.2.2.2">subscript</csymbol><ci id="S6.Thmthm2.p1.5.m5.1.2.2.2.2.cmml" xref="S6.Thmthm2.p1.5.m5.1.2.2.2.2">𝐴</ci><infinity id="S6.Thmthm2.p1.5.m5.1.2.2.2.3.cmml" xref="S6.Thmthm2.p1.5.m5.1.2.2.2.3"></infinity></apply><ci id="S6.Thmthm2.p1.5.m5.1.1.cmml" xref="S6.Thmthm2.p1.5.m5.1.1">𝑤</ci></apply><ci id="S6.Thmthm2.p1.5.m5.1.2.3.cmml" xref="S6.Thmthm2.p1.5.m5.1.2.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p1.5.m5.1c">A_{\infty}(w)\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p1.5.m5.1d">italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( italic_w ) ⊆ italic_X</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S6.Thmthm2.p2"> <p class="ltx_p" id="S6.Thmthm2.p2.4">(2) For any invariant measure <math alttext="\mu" class="ltx_Math" display="inline" id="S6.Thmthm2.p2.1.m1.1"><semantics id="S6.Thmthm2.p2.1.m1.1a"><mi id="S6.Thmthm2.p2.1.m1.1.1" xref="S6.Thmthm2.p2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm2.p2.1.m1.1b"><ci id="S6.Thmthm2.p2.1.m1.1.1.cmml" xref="S6.Thmthm2.p2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p2.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p2.1.m1.1d">italic_μ</annotation></semantics></math> on <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm2.p2.2.m2.1"><semantics id="S6.Thmthm2.p2.2.m2.1a"><mi id="S6.Thmthm2.p2.2.m2.1.1" xref="S6.Thmthm2.p2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm2.p2.2.m2.1b"><ci id="S6.Thmthm2.p2.2.m2.1.1.cmml" xref="S6.Thmthm2.p2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p2.2.m2.1d">italic_X</annotation></semantics></math> and any <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S6.Thmthm2.p2.3.m3.1"><semantics id="S6.Thmthm2.p2.3.m3.1a"><mrow id="S6.Thmthm2.p2.3.m3.1.1" xref="S6.Thmthm2.p2.3.m3.1.1.cmml"><mi id="S6.Thmthm2.p2.3.m3.1.1.2" xref="S6.Thmthm2.p2.3.m3.1.1.2.cmml">n</mi><mo id="S6.Thmthm2.p2.3.m3.1.1.1" xref="S6.Thmthm2.p2.3.m3.1.1.1.cmml">≥</mo><mn id="S6.Thmthm2.p2.3.m3.1.1.3" xref="S6.Thmthm2.p2.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm2.p2.3.m3.1b"><apply id="S6.Thmthm2.p2.3.m3.1.1.cmml" xref="S6.Thmthm2.p2.3.m3.1.1"><geq id="S6.Thmthm2.p2.3.m3.1.1.1.cmml" xref="S6.Thmthm2.p2.3.m3.1.1.1"></geq><ci id="S6.Thmthm2.p2.3.m3.1.1.2.cmml" xref="S6.Thmthm2.p2.3.m3.1.1.2">𝑛</ci><cn id="S6.Thmthm2.p2.3.m3.1.1.3.cmml" type="integer" xref="S6.Thmthm2.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p2.3.m3.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p2.3.m3.1d">italic_n ≥ 0</annotation></semantics></math> one has thus <math alttext="\mu([A_{n+1}(w)])\leq\mu([A_{n}(w)])" class="ltx_Math" display="inline" id="S6.Thmthm2.p2.4.m4.4"><semantics id="S6.Thmthm2.p2.4.m4.4a"><mrow id="S6.Thmthm2.p2.4.m4.4.4" xref="S6.Thmthm2.p2.4.m4.4.4.cmml"><mrow id="S6.Thmthm2.p2.4.m4.3.3.1" xref="S6.Thmthm2.p2.4.m4.3.3.1.cmml"><mi id="S6.Thmthm2.p2.4.m4.3.3.1.3" xref="S6.Thmthm2.p2.4.m4.3.3.1.3.cmml">μ</mi><mo id="S6.Thmthm2.p2.4.m4.3.3.1.2" xref="S6.Thmthm2.p2.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S6.Thmthm2.p2.4.m4.3.3.1.1.1" xref="S6.Thmthm2.p2.4.m4.3.3.1.cmml"><mo id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.2" stretchy="false" xref="S6.Thmthm2.p2.4.m4.3.3.1.cmml">(</mo><mrow id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.2.cmml"><mo id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.2" stretchy="false" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.2.1.cmml">[</mo><mrow id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.cmml"><msub id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.cmml"><mi id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.2" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.2.cmml">A</mi><mrow id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.cmml"><mi id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.2" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.1" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.3" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.1" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.3.2" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.cmml"><mo id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.cmml">(</mo><mi id="S6.Thmthm2.p2.4.m4.1.1" xref="S6.Thmthm2.p2.4.m4.1.1.cmml">w</mi><mo id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.3" stretchy="false" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.3" stretchy="false" xref="S6.Thmthm2.p2.4.m4.3.3.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm2.p2.4.m4.4.4.3" xref="S6.Thmthm2.p2.4.m4.4.4.3.cmml">≤</mo><mrow id="S6.Thmthm2.p2.4.m4.4.4.2" xref="S6.Thmthm2.p2.4.m4.4.4.2.cmml"><mi id="S6.Thmthm2.p2.4.m4.4.4.2.3" xref="S6.Thmthm2.p2.4.m4.4.4.2.3.cmml">μ</mi><mo id="S6.Thmthm2.p2.4.m4.4.4.2.2" xref="S6.Thmthm2.p2.4.m4.4.4.2.2.cmml">⁢</mo><mrow id="S6.Thmthm2.p2.4.m4.4.4.2.1.1" xref="S6.Thmthm2.p2.4.m4.4.4.2.cmml"><mo id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.2" stretchy="false" xref="S6.Thmthm2.p2.4.m4.4.4.2.cmml">(</mo><mrow id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.2.cmml"><mo id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.2" stretchy="false" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.2.1.cmml">[</mo><mrow id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.cmml"><msub id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.cmml"><mi id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.2" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.2.cmml">A</mi><mi id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.3" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.1" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.3.2" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.cmml"><mo id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.3.2.1" stretchy="false" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.cmml">(</mo><mi id="S6.Thmthm2.p2.4.m4.2.2" xref="S6.Thmthm2.p2.4.m4.2.2.cmml">w</mi><mo id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.3.2.2" stretchy="false" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.3" stretchy="false" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.2.1.cmml">]</mo></mrow><mo id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.3" stretchy="false" xref="S6.Thmthm2.p2.4.m4.4.4.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm2.p2.4.m4.4b"><apply id="S6.Thmthm2.p2.4.m4.4.4.cmml" xref="S6.Thmthm2.p2.4.m4.4.4"><leq id="S6.Thmthm2.p2.4.m4.4.4.3.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.3"></leq><apply id="S6.Thmthm2.p2.4.m4.3.3.1.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1"><times id="S6.Thmthm2.p2.4.m4.3.3.1.2.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.2"></times><ci id="S6.Thmthm2.p2.4.m4.3.3.1.3.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.3">𝜇</ci><apply id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.2.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1"><csymbol cd="latexml" id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.2.1.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.2">delimited-[]</csymbol><apply id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1"><times id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.1.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.1"></times><apply id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.1.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.2.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.2">𝐴</ci><apply id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3"><plus id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.1.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.1"></plus><ci id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.2.cmml" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.2">𝑛</ci><cn id="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S6.Thmthm2.p2.4.m4.3.3.1.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="S6.Thmthm2.p2.4.m4.1.1.cmml" xref="S6.Thmthm2.p2.4.m4.1.1">𝑤</ci></apply></apply></apply><apply id="S6.Thmthm2.p2.4.m4.4.4.2.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2"><times id="S6.Thmthm2.p2.4.m4.4.4.2.2.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.2"></times><ci id="S6.Thmthm2.p2.4.m4.4.4.2.3.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.3">𝜇</ci><apply id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.2.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1"><csymbol cd="latexml" id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.2.1.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.2">delimited-[]</csymbol><apply id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1"><times id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.1.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.1"></times><apply id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.1.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2">subscript</csymbol><ci id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.2.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.2">𝐴</ci><ci id="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.3.cmml" xref="S6.Thmthm2.p2.4.m4.4.4.2.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S6.Thmthm2.p2.4.m4.2.2.cmml" xref="S6.Thmthm2.p2.4.m4.2.2">𝑤</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm2.p2.4.m4.4c">\mu([A_{n+1}(w)])\leq\mu([A_{n}(w)])</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm2.p2.4.m4.4d">italic_μ ( [ italic_A start_POSTSUBSCRIPT italic_n + 1 end_POSTSUBSCRIPT ( italic_w ) ] ) ≤ italic_μ ( [ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ] )</annotation></semantics></math> and</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lim\mu([A_{n}(w)])=\mu(A_{\infty}(w))\,." class="ltx_Math" display="block" id="S6.Ex3.m1.3"><semantics id="S6.Ex3.m1.3a"><mrow id="S6.Ex3.m1.3.3.1" xref="S6.Ex3.m1.3.3.1.1.cmml"><mrow id="S6.Ex3.m1.3.3.1.1" xref="S6.Ex3.m1.3.3.1.1.cmml"><mrow id="S6.Ex3.m1.3.3.1.1.1" xref="S6.Ex3.m1.3.3.1.1.1.cmml"><mo id="S6.Ex3.m1.3.3.1.1.1.2" movablelimits="false" rspace="0.167em" xref="S6.Ex3.m1.3.3.1.1.1.2.cmml">lim</mo><mrow id="S6.Ex3.m1.3.3.1.1.1.1" xref="S6.Ex3.m1.3.3.1.1.1.1.cmml"><mi id="S6.Ex3.m1.3.3.1.1.1.1.3" xref="S6.Ex3.m1.3.3.1.1.1.1.3.cmml">μ</mi><mo id="S6.Ex3.m1.3.3.1.1.1.1.2" xref="S6.Ex3.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex3.m1.3.3.1.1.1.1.1.1" xref="S6.Ex3.m1.3.3.1.1.1.1.cmml"><mo id="S6.Ex3.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex3.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.2.cmml"><mo id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.2.1.cmml">[</mo><mrow id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><msub id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.2" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.2.2" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.2.3" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.1" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.3.2" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S6.Ex3.m1.1.1" xref="S6.Ex3.m1.1.1.cmml">w</mi><mo id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex3.m1.3.3.1.1.1.1.1.1.1.2.1.cmml">]</mo></mrow><mo id="S6.Ex3.m1.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex3.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Ex3.m1.3.3.1.1.3" xref="S6.Ex3.m1.3.3.1.1.3.cmml">=</mo><mrow id="S6.Ex3.m1.3.3.1.1.2" xref="S6.Ex3.m1.3.3.1.1.2.cmml"><mi id="S6.Ex3.m1.3.3.1.1.2.3" xref="S6.Ex3.m1.3.3.1.1.2.3.cmml">μ</mi><mo id="S6.Ex3.m1.3.3.1.1.2.2" xref="S6.Ex3.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="S6.Ex3.m1.3.3.1.1.2.1.1" xref="S6.Ex3.m1.3.3.1.1.2.1.1.1.cmml"><mo id="S6.Ex3.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="S6.Ex3.m1.3.3.1.1.2.1.1.1.cmml">(</mo><mrow id="S6.Ex3.m1.3.3.1.1.2.1.1.1" xref="S6.Ex3.m1.3.3.1.1.2.1.1.1.cmml"><msub 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id="S6.Ex3.m1.3.3.1.1.2.1.1.1.2.2.cmml" xref="S6.Ex3.m1.3.3.1.1.2.1.1.1.2.2">𝐴</ci><infinity id="S6.Ex3.m1.3.3.1.1.2.1.1.1.2.3.cmml" xref="S6.Ex3.m1.3.3.1.1.2.1.1.1.2.3"></infinity></apply><ci id="S6.Ex3.m1.2.2.cmml" xref="S6.Ex3.m1.2.2">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex3.m1.3c">\lim\mu([A_{n}(w)])=\mu(A_{\infty}(w))\,.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex3.m1.3d">roman_lim italic_μ ( [ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) ] ) = italic_μ ( italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( italic_w ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_lem" id="S6.Thmthm3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm3.1.1.1">Lemma 6.3</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm3.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm3.p1"> <p class="ltx_p" id="S6.Thmthm3.p1.4"><span class="ltx_text ltx_font_italic" id="S6.Thmthm3.p1.4.4">If <math alttext="\sigma" class="ltx_Math" display="inline" id="S6.Thmthm3.p1.1.1.m1.1"><semantics id="S6.Thmthm3.p1.1.1.m1.1a"><mi id="S6.Thmthm3.p1.1.1.m1.1.1" xref="S6.Thmthm3.p1.1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm3.p1.1.1.m1.1b"><ci id="S6.Thmthm3.p1.1.1.m1.1.1.cmml" xref="S6.Thmthm3.p1.1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm3.p1.1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm3.p1.1.1.m1.1d">italic_σ</annotation></semantics></math> is shift-orbit injective in <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm3.p1.2.2.m2.1"><semantics id="S6.Thmthm3.p1.2.2.m2.1a"><mi id="S6.Thmthm3.p1.2.2.m2.1.1" xref="S6.Thmthm3.p1.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm3.p1.2.2.m2.1b"><ci id="S6.Thmthm3.p1.2.2.m2.1.1.cmml" xref="S6.Thmthm3.p1.2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm3.p1.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm3.p1.2.2.m2.1d">italic_X</annotation></semantics></math>, then for any <math alttext="w\in\cal L(X)" class="ltx_Math" display="inline" id="S6.Thmthm3.p1.3.3.m3.1"><semantics id="S6.Thmthm3.p1.3.3.m3.1a"><mrow id="S6.Thmthm3.p1.3.3.m3.1.2" xref="S6.Thmthm3.p1.3.3.m3.1.2.cmml"><mi id="S6.Thmthm3.p1.3.3.m3.1.2.2" xref="S6.Thmthm3.p1.3.3.m3.1.2.2.cmml">w</mi><mo id="S6.Thmthm3.p1.3.3.m3.1.2.1" xref="S6.Thmthm3.p1.3.3.m3.1.2.1.cmml">∈</mo><mrow id="S6.Thmthm3.p1.3.3.m3.1.2.3" xref="S6.Thmthm3.p1.3.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm3.p1.3.3.m3.1.2.3.2" xref="S6.Thmthm3.p1.3.3.m3.1.2.3.2.cmml">ℒ</mi><mo id="S6.Thmthm3.p1.3.3.m3.1.2.3.1" xref="S6.Thmthm3.p1.3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S6.Thmthm3.p1.3.3.m3.1.2.3.3.2" xref="S6.Thmthm3.p1.3.3.m3.1.2.3.cmml"><mo id="S6.Thmthm3.p1.3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S6.Thmthm3.p1.3.3.m3.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm3.p1.3.3.m3.1.1" xref="S6.Thmthm3.p1.3.3.m3.1.1.cmml">𝒳</mi><mo id="S6.Thmthm3.p1.3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S6.Thmthm3.p1.3.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm3.p1.3.3.m3.1b"><apply id="S6.Thmthm3.p1.3.3.m3.1.2.cmml" xref="S6.Thmthm3.p1.3.3.m3.1.2"><in id="S6.Thmthm3.p1.3.3.m3.1.2.1.cmml" xref="S6.Thmthm3.p1.3.3.m3.1.2.1"></in><ci id="S6.Thmthm3.p1.3.3.m3.1.2.2.cmml" xref="S6.Thmthm3.p1.3.3.m3.1.2.2">𝑤</ci><apply id="S6.Thmthm3.p1.3.3.m3.1.2.3.cmml" xref="S6.Thmthm3.p1.3.3.m3.1.2.3"><times id="S6.Thmthm3.p1.3.3.m3.1.2.3.1.cmml" xref="S6.Thmthm3.p1.3.3.m3.1.2.3.1"></times><ci id="S6.Thmthm3.p1.3.3.m3.1.2.3.2.cmml" xref="S6.Thmthm3.p1.3.3.m3.1.2.3.2">ℒ</ci><ci id="S6.Thmthm3.p1.3.3.m3.1.1.cmml" xref="S6.Thmthm3.p1.3.3.m3.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm3.p1.3.3.m3.1c">w\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm3.p1.3.3.m3.1d">italic_w ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> the intersection <math alttext="A_{\infty}(w)" class="ltx_Math" display="inline" id="S6.Thmthm3.p1.4.4.m4.1"><semantics id="S6.Thmthm3.p1.4.4.m4.1a"><mrow id="S6.Thmthm3.p1.4.4.m4.1.2" xref="S6.Thmthm3.p1.4.4.m4.1.2.cmml"><msub id="S6.Thmthm3.p1.4.4.m4.1.2.2" xref="S6.Thmthm3.p1.4.4.m4.1.2.2.cmml"><mi id="S6.Thmthm3.p1.4.4.m4.1.2.2.2" xref="S6.Thmthm3.p1.4.4.m4.1.2.2.2.cmml">A</mi><mi id="S6.Thmthm3.p1.4.4.m4.1.2.2.3" mathvariant="normal" xref="S6.Thmthm3.p1.4.4.m4.1.2.2.3.cmml">∞</mi></msub><mo id="S6.Thmthm3.p1.4.4.m4.1.2.1" xref="S6.Thmthm3.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S6.Thmthm3.p1.4.4.m4.1.2.3.2" xref="S6.Thmthm3.p1.4.4.m4.1.2.cmml"><mo id="S6.Thmthm3.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="S6.Thmthm3.p1.4.4.m4.1.2.cmml">(</mo><mi id="S6.Thmthm3.p1.4.4.m4.1.1" xref="S6.Thmthm3.p1.4.4.m4.1.1.cmml">w</mi><mo id="S6.Thmthm3.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="S6.Thmthm3.p1.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm3.p1.4.4.m4.1b"><apply id="S6.Thmthm3.p1.4.4.m4.1.2.cmml" xref="S6.Thmthm3.p1.4.4.m4.1.2"><times id="S6.Thmthm3.p1.4.4.m4.1.2.1.cmml" xref="S6.Thmthm3.p1.4.4.m4.1.2.1"></times><apply id="S6.Thmthm3.p1.4.4.m4.1.2.2.cmml" xref="S6.Thmthm3.p1.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S6.Thmthm3.p1.4.4.m4.1.2.2.1.cmml" xref="S6.Thmthm3.p1.4.4.m4.1.2.2">subscript</csymbol><ci id="S6.Thmthm3.p1.4.4.m4.1.2.2.2.cmml" xref="S6.Thmthm3.p1.4.4.m4.1.2.2.2">𝐴</ci><infinity id="S6.Thmthm3.p1.4.4.m4.1.2.2.3.cmml" xref="S6.Thmthm3.p1.4.4.m4.1.2.2.3"></infinity></apply><ci id="S6.Thmthm3.p1.4.4.m4.1.1.cmml" xref="S6.Thmthm3.p1.4.4.m4.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm3.p1.4.4.m4.1c">A_{\infty}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm3.p1.4.4.m4.1d">italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> consists only of periodic words.</span></p> </div> </div> <div class="ltx_proof" id="S6.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.2.p1"> <p class="ltx_p" id="S6.2.p1.3">For any biinfinite word <math alttext="{\bf x}\in A_{\infty}(w)" class="ltx_Math" display="inline" id="S6.2.p1.1.m1.1"><semantics id="S6.2.p1.1.m1.1a"><mrow id="S6.2.p1.1.m1.1.2" xref="S6.2.p1.1.m1.1.2.cmml"><mi id="S6.2.p1.1.m1.1.2.2" xref="S6.2.p1.1.m1.1.2.2.cmml">𝐱</mi><mo id="S6.2.p1.1.m1.1.2.1" xref="S6.2.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S6.2.p1.1.m1.1.2.3" xref="S6.2.p1.1.m1.1.2.3.cmml"><msub id="S6.2.p1.1.m1.1.2.3.2" xref="S6.2.p1.1.m1.1.2.3.2.cmml"><mi id="S6.2.p1.1.m1.1.2.3.2.2" xref="S6.2.p1.1.m1.1.2.3.2.2.cmml">A</mi><mi id="S6.2.p1.1.m1.1.2.3.2.3" mathvariant="normal" xref="S6.2.p1.1.m1.1.2.3.2.3.cmml">∞</mi></msub><mo id="S6.2.p1.1.m1.1.2.3.1" xref="S6.2.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S6.2.p1.1.m1.1.2.3.3.2" xref="S6.2.p1.1.m1.1.2.3.cmml"><mo id="S6.2.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S6.2.p1.1.m1.1.2.3.cmml">(</mo><mi id="S6.2.p1.1.m1.1.1" xref="S6.2.p1.1.m1.1.1.cmml">w</mi><mo id="S6.2.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S6.2.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.1.m1.1b"><apply id="S6.2.p1.1.m1.1.2.cmml" xref="S6.2.p1.1.m1.1.2"><in id="S6.2.p1.1.m1.1.2.1.cmml" xref="S6.2.p1.1.m1.1.2.1"></in><ci id="S6.2.p1.1.m1.1.2.2.cmml" xref="S6.2.p1.1.m1.1.2.2">𝐱</ci><apply id="S6.2.p1.1.m1.1.2.3.cmml" xref="S6.2.p1.1.m1.1.2.3"><times id="S6.2.p1.1.m1.1.2.3.1.cmml" xref="S6.2.p1.1.m1.1.2.3.1"></times><apply id="S6.2.p1.1.m1.1.2.3.2.cmml" xref="S6.2.p1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S6.2.p1.1.m1.1.2.3.2.1.cmml" xref="S6.2.p1.1.m1.1.2.3.2">subscript</csymbol><ci id="S6.2.p1.1.m1.1.2.3.2.2.cmml" xref="S6.2.p1.1.m1.1.2.3.2.2">𝐴</ci><infinity id="S6.2.p1.1.m1.1.2.3.2.3.cmml" xref="S6.2.p1.1.m1.1.2.3.2.3"></infinity></apply><ci id="S6.2.p1.1.m1.1.1.cmml" xref="S6.2.p1.1.m1.1.1">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.1.m1.1c">{\bf x}\in A_{\infty}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.1.m1.1d">bold_x ∈ italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> there exists, by definition of <math alttext="A_{\infty}(w)" class="ltx_Math" display="inline" id="S6.2.p1.2.m2.1"><semantics id="S6.2.p1.2.m2.1a"><mrow id="S6.2.p1.2.m2.1.2" xref="S6.2.p1.2.m2.1.2.cmml"><msub id="S6.2.p1.2.m2.1.2.2" xref="S6.2.p1.2.m2.1.2.2.cmml"><mi id="S6.2.p1.2.m2.1.2.2.2" xref="S6.2.p1.2.m2.1.2.2.2.cmml">A</mi><mi id="S6.2.p1.2.m2.1.2.2.3" mathvariant="normal" xref="S6.2.p1.2.m2.1.2.2.3.cmml">∞</mi></msub><mo id="S6.2.p1.2.m2.1.2.1" xref="S6.2.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S6.2.p1.2.m2.1.2.3.2" xref="S6.2.p1.2.m2.1.2.cmml"><mo id="S6.2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S6.2.p1.2.m2.1.2.cmml">(</mo><mi id="S6.2.p1.2.m2.1.1" xref="S6.2.p1.2.m2.1.1.cmml">w</mi><mo id="S6.2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S6.2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.2.m2.1b"><apply id="S6.2.p1.2.m2.1.2.cmml" xref="S6.2.p1.2.m2.1.2"><times id="S6.2.p1.2.m2.1.2.1.cmml" xref="S6.2.p1.2.m2.1.2.1"></times><apply id="S6.2.p1.2.m2.1.2.2.cmml" xref="S6.2.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S6.2.p1.2.m2.1.2.2.1.cmml" xref="S6.2.p1.2.m2.1.2.2">subscript</csymbol><ci id="S6.2.p1.2.m2.1.2.2.2.cmml" xref="S6.2.p1.2.m2.1.2.2.2">𝐴</ci><infinity id="S6.2.p1.2.m2.1.2.2.3.cmml" xref="S6.2.p1.2.m2.1.2.2.3"></infinity></apply><ci id="S6.2.p1.2.m2.1.1.cmml" xref="S6.2.p1.2.m2.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.2.m2.1c">A_{\infty}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.2.m2.1d">italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math>, a sequence of words <math alttext="u_{n},v_{n},u^{\prime}_{n},v^{\prime}_{n},w^{\prime}_{n}\in\cal L(X)" class="ltx_Math" display="inline" id="S6.2.p1.3.m3.6"><semantics id="S6.2.p1.3.m3.6a"><mrow id="S6.2.p1.3.m3.6.6" xref="S6.2.p1.3.m3.6.6.cmml"><mrow id="S6.2.p1.3.m3.6.6.5.5" xref="S6.2.p1.3.m3.6.6.5.6.cmml"><msub id="S6.2.p1.3.m3.2.2.1.1.1" xref="S6.2.p1.3.m3.2.2.1.1.1.cmml"><mi id="S6.2.p1.3.m3.2.2.1.1.1.2" xref="S6.2.p1.3.m3.2.2.1.1.1.2.cmml">u</mi><mi id="S6.2.p1.3.m3.2.2.1.1.1.3" xref="S6.2.p1.3.m3.2.2.1.1.1.3.cmml">n</mi></msub><mo id="S6.2.p1.3.m3.6.6.5.5.6" xref="S6.2.p1.3.m3.6.6.5.6.cmml">,</mo><msub id="S6.2.p1.3.m3.3.3.2.2.2" xref="S6.2.p1.3.m3.3.3.2.2.2.cmml"><mi id="S6.2.p1.3.m3.3.3.2.2.2.2" xref="S6.2.p1.3.m3.3.3.2.2.2.2.cmml">v</mi><mi id="S6.2.p1.3.m3.3.3.2.2.2.3" xref="S6.2.p1.3.m3.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S6.2.p1.3.m3.6.6.5.5.7" xref="S6.2.p1.3.m3.6.6.5.6.cmml">,</mo><msubsup id="S6.2.p1.3.m3.4.4.3.3.3" xref="S6.2.p1.3.m3.4.4.3.3.3.cmml"><mi id="S6.2.p1.3.m3.4.4.3.3.3.2.2" xref="S6.2.p1.3.m3.4.4.3.3.3.2.2.cmml">u</mi><mi id="S6.2.p1.3.m3.4.4.3.3.3.3" xref="S6.2.p1.3.m3.4.4.3.3.3.3.cmml">n</mi><mo id="S6.2.p1.3.m3.4.4.3.3.3.2.3" xref="S6.2.p1.3.m3.4.4.3.3.3.2.3.cmml">′</mo></msubsup><mo id="S6.2.p1.3.m3.6.6.5.5.8" xref="S6.2.p1.3.m3.6.6.5.6.cmml">,</mo><msubsup id="S6.2.p1.3.m3.5.5.4.4.4" xref="S6.2.p1.3.m3.5.5.4.4.4.cmml"><mi id="S6.2.p1.3.m3.5.5.4.4.4.2.2" xref="S6.2.p1.3.m3.5.5.4.4.4.2.2.cmml">v</mi><mi id="S6.2.p1.3.m3.5.5.4.4.4.3" xref="S6.2.p1.3.m3.5.5.4.4.4.3.cmml">n</mi><mo id="S6.2.p1.3.m3.5.5.4.4.4.2.3" xref="S6.2.p1.3.m3.5.5.4.4.4.2.3.cmml">′</mo></msubsup><mo id="S6.2.p1.3.m3.6.6.5.5.9" xref="S6.2.p1.3.m3.6.6.5.6.cmml">,</mo><msubsup id="S6.2.p1.3.m3.6.6.5.5.5" xref="S6.2.p1.3.m3.6.6.5.5.5.cmml"><mi id="S6.2.p1.3.m3.6.6.5.5.5.2.2" xref="S6.2.p1.3.m3.6.6.5.5.5.2.2.cmml">w</mi><mi id="S6.2.p1.3.m3.6.6.5.5.5.3" xref="S6.2.p1.3.m3.6.6.5.5.5.3.cmml">n</mi><mo id="S6.2.p1.3.m3.6.6.5.5.5.2.3" xref="S6.2.p1.3.m3.6.6.5.5.5.2.3.cmml">′</mo></msubsup></mrow><mo id="S6.2.p1.3.m3.6.6.6" xref="S6.2.p1.3.m3.6.6.6.cmml">∈</mo><mrow id="S6.2.p1.3.m3.6.6.7" xref="S6.2.p1.3.m3.6.6.7.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.2.p1.3.m3.6.6.7.2" xref="S6.2.p1.3.m3.6.6.7.2.cmml">ℒ</mi><mo id="S6.2.p1.3.m3.6.6.7.1" xref="S6.2.p1.3.m3.6.6.7.1.cmml">⁢</mo><mrow id="S6.2.p1.3.m3.6.6.7.3.2" xref="S6.2.p1.3.m3.6.6.7.cmml"><mo id="S6.2.p1.3.m3.6.6.7.3.2.1" stretchy="false" xref="S6.2.p1.3.m3.6.6.7.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.2.p1.3.m3.1.1" xref="S6.2.p1.3.m3.1.1.cmml">𝒳</mi><mo id="S6.2.p1.3.m3.6.6.7.3.2.2" stretchy="false" xref="S6.2.p1.3.m3.6.6.7.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.3.m3.6b"><apply id="S6.2.p1.3.m3.6.6.cmml" xref="S6.2.p1.3.m3.6.6"><in id="S6.2.p1.3.m3.6.6.6.cmml" xref="S6.2.p1.3.m3.6.6.6"></in><list id="S6.2.p1.3.m3.6.6.5.6.cmml" xref="S6.2.p1.3.m3.6.6.5.5"><apply id="S6.2.p1.3.m3.2.2.1.1.1.cmml" xref="S6.2.p1.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.2.p1.3.m3.2.2.1.1.1.1.cmml" xref="S6.2.p1.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S6.2.p1.3.m3.2.2.1.1.1.2.cmml" xref="S6.2.p1.3.m3.2.2.1.1.1.2">𝑢</ci><ci id="S6.2.p1.3.m3.2.2.1.1.1.3.cmml" xref="S6.2.p1.3.m3.2.2.1.1.1.3">𝑛</ci></apply><apply id="S6.2.p1.3.m3.3.3.2.2.2.cmml" xref="S6.2.p1.3.m3.3.3.2.2.2"><csymbol cd="ambiguous" id="S6.2.p1.3.m3.3.3.2.2.2.1.cmml" xref="S6.2.p1.3.m3.3.3.2.2.2">subscript</csymbol><ci id="S6.2.p1.3.m3.3.3.2.2.2.2.cmml" xref="S6.2.p1.3.m3.3.3.2.2.2.2">𝑣</ci><ci id="S6.2.p1.3.m3.3.3.2.2.2.3.cmml" xref="S6.2.p1.3.m3.3.3.2.2.2.3">𝑛</ci></apply><apply id="S6.2.p1.3.m3.4.4.3.3.3.cmml" xref="S6.2.p1.3.m3.4.4.3.3.3"><csymbol cd="ambiguous" id="S6.2.p1.3.m3.4.4.3.3.3.1.cmml" xref="S6.2.p1.3.m3.4.4.3.3.3">subscript</csymbol><apply id="S6.2.p1.3.m3.4.4.3.3.3.2.cmml" xref="S6.2.p1.3.m3.4.4.3.3.3"><csymbol cd="ambiguous" id="S6.2.p1.3.m3.4.4.3.3.3.2.1.cmml" xref="S6.2.p1.3.m3.4.4.3.3.3">superscript</csymbol><ci id="S6.2.p1.3.m3.4.4.3.3.3.2.2.cmml" xref="S6.2.p1.3.m3.4.4.3.3.3.2.2">𝑢</ci><ci id="S6.2.p1.3.m3.4.4.3.3.3.2.3.cmml" xref="S6.2.p1.3.m3.4.4.3.3.3.2.3">′</ci></apply><ci id="S6.2.p1.3.m3.4.4.3.3.3.3.cmml" xref="S6.2.p1.3.m3.4.4.3.3.3.3">𝑛</ci></apply><apply id="S6.2.p1.3.m3.5.5.4.4.4.cmml" xref="S6.2.p1.3.m3.5.5.4.4.4"><csymbol cd="ambiguous" id="S6.2.p1.3.m3.5.5.4.4.4.1.cmml" xref="S6.2.p1.3.m3.5.5.4.4.4">subscript</csymbol><apply id="S6.2.p1.3.m3.5.5.4.4.4.2.cmml" xref="S6.2.p1.3.m3.5.5.4.4.4"><csymbol cd="ambiguous" id="S6.2.p1.3.m3.5.5.4.4.4.2.1.cmml" xref="S6.2.p1.3.m3.5.5.4.4.4">superscript</csymbol><ci id="S6.2.p1.3.m3.5.5.4.4.4.2.2.cmml" xref="S6.2.p1.3.m3.5.5.4.4.4.2.2">𝑣</ci><ci id="S6.2.p1.3.m3.5.5.4.4.4.2.3.cmml" xref="S6.2.p1.3.m3.5.5.4.4.4.2.3">′</ci></apply><ci id="S6.2.p1.3.m3.5.5.4.4.4.3.cmml" xref="S6.2.p1.3.m3.5.5.4.4.4.3">𝑛</ci></apply><apply id="S6.2.p1.3.m3.6.6.5.5.5.cmml" xref="S6.2.p1.3.m3.6.6.5.5.5"><csymbol cd="ambiguous" id="S6.2.p1.3.m3.6.6.5.5.5.1.cmml" xref="S6.2.p1.3.m3.6.6.5.5.5">subscript</csymbol><apply id="S6.2.p1.3.m3.6.6.5.5.5.2.cmml" xref="S6.2.p1.3.m3.6.6.5.5.5"><csymbol cd="ambiguous" id="S6.2.p1.3.m3.6.6.5.5.5.2.1.cmml" xref="S6.2.p1.3.m3.6.6.5.5.5">superscript</csymbol><ci id="S6.2.p1.3.m3.6.6.5.5.5.2.2.cmml" xref="S6.2.p1.3.m3.6.6.5.5.5.2.2">𝑤</ci><ci id="S6.2.p1.3.m3.6.6.5.5.5.2.3.cmml" xref="S6.2.p1.3.m3.6.6.5.5.5.2.3">′</ci></apply><ci id="S6.2.p1.3.m3.6.6.5.5.5.3.cmml" xref="S6.2.p1.3.m3.6.6.5.5.5.3">𝑛</ci></apply></list><apply id="S6.2.p1.3.m3.6.6.7.cmml" xref="S6.2.p1.3.m3.6.6.7"><times id="S6.2.p1.3.m3.6.6.7.1.cmml" xref="S6.2.p1.3.m3.6.6.7.1"></times><ci id="S6.2.p1.3.m3.6.6.7.2.cmml" xref="S6.2.p1.3.m3.6.6.7.2">ℒ</ci><ci id="S6.2.p1.3.m3.1.1.cmml" xref="S6.2.p1.3.m3.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.3.m3.6c">u_{n},v_{n},u^{\prime}_{n},v^{\prime}_{n},w^{\prime}_{n}\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.3.m3.6d">italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> which have the following properties:</p> <ol class="ltx_enumerate" id="S6.I1"> <li class="ltx_item" id="S6.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S6.I1.i1.p1"> <p class="ltx_p" id="S6.I1.i1.p1.1"><math alttext="|u_{n}|=|v_{n}|=|u^{\prime}_{n}|=|v^{\prime}_{n}|=n\quad\text{and }\quad|w^{% \prime}_{n}|=|w|" class="ltx_Math" display="inline" id="S6.I1.i1.p1.1.m1.4"><semantics id="S6.I1.i1.p1.1.m1.4a"><mrow id="S6.I1.i1.p1.1.m1.4.4.2" xref="S6.I1.i1.p1.1.m1.4.4.3.cmml"><mrow id="S6.I1.i1.p1.1.m1.3.3.1.1" xref="S6.I1.i1.p1.1.m1.3.3.1.1.cmml"><mrow id="S6.I1.i1.p1.1.m1.3.3.1.1.1.1" xref="S6.I1.i1.p1.1.m1.3.3.1.1.1.2.cmml"><mo id="S6.I1.i1.p1.1.m1.3.3.1.1.1.1.2" stretchy="false" xref="S6.I1.i1.p1.1.m1.3.3.1.1.1.2.1.cmml">|</mo><msub id="S6.I1.i1.p1.1.m1.3.3.1.1.1.1.1" 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id="S6.I1.i1.p1.1.m1.4c">|u_{n}|=|v_{n}|=|u^{\prime}_{n}|=|v^{\prime}_{n}|=n\quad\text{and }\quad|w^{% \prime}_{n}|=|w|</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.1.m1.4d">| italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = | italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = | italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = | italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = italic_n and | italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | = | italic_w |</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S6.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S6.I1.i2.p1"> <p class="ltx_p" id="S6.I1.i2.p1.2"><math alttext="u^{\prime}_{n}w^{\prime}_{n}v^{\prime}_{n}\in\cal L(X)" class="ltx_Math" 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id="S6.I1.i2.p1.1.m1.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> and <math alttext="u_{n}wv_{n}\in\cal L(X)" class="ltx_Math" display="inline" id="S6.I1.i2.p1.2.m2.1"><semantics id="S6.I1.i2.p1.2.m2.1a"><mrow id="S6.I1.i2.p1.2.m2.1.2" xref="S6.I1.i2.p1.2.m2.1.2.cmml"><mrow id="S6.I1.i2.p1.2.m2.1.2.2" xref="S6.I1.i2.p1.2.m2.1.2.2.cmml"><msub id="S6.I1.i2.p1.2.m2.1.2.2.2" xref="S6.I1.i2.p1.2.m2.1.2.2.2.cmml"><mi id="S6.I1.i2.p1.2.m2.1.2.2.2.2" xref="S6.I1.i2.p1.2.m2.1.2.2.2.2.cmml">u</mi><mi id="S6.I1.i2.p1.2.m2.1.2.2.2.3" xref="S6.I1.i2.p1.2.m2.1.2.2.2.3.cmml">n</mi></msub><mo id="S6.I1.i2.p1.2.m2.1.2.2.1" xref="S6.I1.i2.p1.2.m2.1.2.2.1.cmml">⁢</mo><mi id="S6.I1.i2.p1.2.m2.1.2.2.3" xref="S6.I1.i2.p1.2.m2.1.2.2.3.cmml">w</mi><mo id="S6.I1.i2.p1.2.m2.1.2.2.1a" xref="S6.I1.i2.p1.2.m2.1.2.2.1.cmml">⁢</mo><msub id="S6.I1.i2.p1.2.m2.1.2.2.4" xref="S6.I1.i2.p1.2.m2.1.2.2.4.cmml"><mi id="S6.I1.i2.p1.2.m2.1.2.2.4.2" xref="S6.I1.i2.p1.2.m2.1.2.2.4.2.cmml">v</mi><mi id="S6.I1.i2.p1.2.m2.1.2.2.4.3" xref="S6.I1.i2.p1.2.m2.1.2.2.4.3.cmml">n</mi></msub></mrow><mo id="S6.I1.i2.p1.2.m2.1.2.1" xref="S6.I1.i2.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="S6.I1.i2.p1.2.m2.1.2.3" xref="S6.I1.i2.p1.2.m2.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.I1.i2.p1.2.m2.1.2.3.2" xref="S6.I1.i2.p1.2.m2.1.2.3.2.cmml">ℒ</mi><mo id="S6.I1.i2.p1.2.m2.1.2.3.1" xref="S6.I1.i2.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S6.I1.i2.p1.2.m2.1.2.3.3.2" xref="S6.I1.i2.p1.2.m2.1.2.3.cmml"><mo id="S6.I1.i2.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S6.I1.i2.p1.2.m2.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.I1.i2.p1.2.m2.1.1" 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xref="S6.I1.i2.p1.2.m2.1.2.2.4"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.1.2.2.4.1.cmml" xref="S6.I1.i2.p1.2.m2.1.2.2.4">subscript</csymbol><ci id="S6.I1.i2.p1.2.m2.1.2.2.4.2.cmml" xref="S6.I1.i2.p1.2.m2.1.2.2.4.2">𝑣</ci><ci id="S6.I1.i2.p1.2.m2.1.2.2.4.3.cmml" xref="S6.I1.i2.p1.2.m2.1.2.2.4.3">𝑛</ci></apply></apply><apply id="S6.I1.i2.p1.2.m2.1.2.3.cmml" xref="S6.I1.i2.p1.2.m2.1.2.3"><times id="S6.I1.i2.p1.2.m2.1.2.3.1.cmml" xref="S6.I1.i2.p1.2.m2.1.2.3.1"></times><ci id="S6.I1.i2.p1.2.m2.1.2.3.2.cmml" xref="S6.I1.i2.p1.2.m2.1.2.3.2">ℒ</ci><ci id="S6.I1.i2.p1.2.m2.1.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.2.m2.1c">u_{n}wv_{n}\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.2.m2.1d">italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_w italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S6.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S6.I1.i3.p1"> <p class="ltx_p" id="S6.I1.i3.p1.1"><math alttext="\sigma(u_{n}wv_{n})=\sigma(u^{\prime}_{n}w^{\prime}_{n}v^{\prime}_{n})" class="ltx_Math" display="inline" id="S6.I1.i3.p1.1.m1.2"><semantics id="S6.I1.i3.p1.1.m1.2a"><mrow id="S6.I1.i3.p1.1.m1.2.2" xref="S6.I1.i3.p1.1.m1.2.2.cmml"><mrow id="S6.I1.i3.p1.1.m1.1.1.1" xref="S6.I1.i3.p1.1.m1.1.1.1.cmml"><mi id="S6.I1.i3.p1.1.m1.1.1.1.3" xref="S6.I1.i3.p1.1.m1.1.1.1.3.cmml">σ</mi><mo id="S6.I1.i3.p1.1.m1.1.1.1.2" xref="S6.I1.i3.p1.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.I1.i3.p1.1.m1.1.1.1.1.1" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S6.I1.i3.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.I1.i3.p1.1.m1.1.1.1.1.1.1" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.cmml"><msub 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xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.1.cmml">⁢</mo><msubsup id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.cmml"><mi id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.2.2" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.2.2.cmml">w</mi><mi id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.3" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.3.cmml">n</mi><mo id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.2.3" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.2.3.cmml">′</mo></msubsup><mo id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.1a" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.1.cmml">⁢</mo><msubsup id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.cmml"><mi id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.2.2" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.2.2.cmml">v</mi><mi id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.3" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.3.cmml">n</mi><mo id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.2.3" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.2.3.cmml">′</mo></msubsup></mrow><mo id="S6.I1.i3.p1.1.m1.2.2.2.1.1.3" stretchy="false" 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id="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.2.3.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.3">𝑤</ci><apply id="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.4.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.4.1.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.4">subscript</csymbol><ci id="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.4.2.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.4.2">𝑣</ci><ci id="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.4.3.cmml" xref="S6.I1.i3.p1.1.m1.1.1.1.1.1.1.4.3">𝑛</ci></apply></apply></apply><apply id="S6.I1.i3.p1.1.m1.2.2.2.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2"><times id="S6.I1.i3.p1.1.m1.2.2.2.2.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.2"></times><ci id="S6.I1.i3.p1.1.m1.2.2.2.3.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.3">𝜎</ci><apply id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1"><times id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.1.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.1"></times><apply id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.1.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2">subscript</csymbol><apply id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.2.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.2.1.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2">superscript</csymbol><ci id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.2.2.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.2.2">𝑢</ci><ci id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.2.3.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.2.3">′</ci></apply><ci id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.3.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.2.3">𝑛</ci></apply><apply id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3.1.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.3">subscript</csymbol><apply 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xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.2.2">𝑣</ci><ci id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.2.3.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.2.3">′</ci></apply><ci id="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.3.cmml" xref="S6.I1.i3.p1.1.m1.2.2.2.1.1.1.4.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i3.p1.1.m1.2c">\sigma(u_{n}wv_{n})=\sigma(u^{\prime}_{n}w^{\prime}_{n}v^{\prime}_{n})</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i3.p1.1.m1.2d">italic_σ ( italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_w italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_σ ( italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S6.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(4)</span> <div class="ltx_para" id="S6.I1.i4.p1"> <p class="ltx_p" id="S6.I1.i4.p1.1"><math alttext="w^{\prime}_{n}\neq w" class="ltx_Math" display="inline" id="S6.I1.i4.p1.1.m1.1"><semantics id="S6.I1.i4.p1.1.m1.1a"><mrow id="S6.I1.i4.p1.1.m1.1.1" xref="S6.I1.i4.p1.1.m1.1.1.cmml"><msubsup id="S6.I1.i4.p1.1.m1.1.1.2" xref="S6.I1.i4.p1.1.m1.1.1.2.cmml"><mi id="S6.I1.i4.p1.1.m1.1.1.2.2.2" xref="S6.I1.i4.p1.1.m1.1.1.2.2.2.cmml">w</mi><mi id="S6.I1.i4.p1.1.m1.1.1.2.3" xref="S6.I1.i4.p1.1.m1.1.1.2.3.cmml">n</mi><mo id="S6.I1.i4.p1.1.m1.1.1.2.2.3" xref="S6.I1.i4.p1.1.m1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S6.I1.i4.p1.1.m1.1.1.1" xref="S6.I1.i4.p1.1.m1.1.1.1.cmml">≠</mo><mi id="S6.I1.i4.p1.1.m1.1.1.3" xref="S6.I1.i4.p1.1.m1.1.1.3.cmml">w</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i4.p1.1.m1.1b"><apply id="S6.I1.i4.p1.1.m1.1.1.cmml" xref="S6.I1.i4.p1.1.m1.1.1"><neq id="S6.I1.i4.p1.1.m1.1.1.1.cmml" xref="S6.I1.i4.p1.1.m1.1.1.1"></neq><apply id="S6.I1.i4.p1.1.m1.1.1.2.cmml" xref="S6.I1.i4.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i4.p1.1.m1.1.1.2.1.cmml" xref="S6.I1.i4.p1.1.m1.1.1.2">subscript</csymbol><apply id="S6.I1.i4.p1.1.m1.1.1.2.2.cmml" xref="S6.I1.i4.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i4.p1.1.m1.1.1.2.2.1.cmml" xref="S6.I1.i4.p1.1.m1.1.1.2">superscript</csymbol><ci id="S6.I1.i4.p1.1.m1.1.1.2.2.2.cmml" xref="S6.I1.i4.p1.1.m1.1.1.2.2.2">𝑤</ci><ci id="S6.I1.i4.p1.1.m1.1.1.2.2.3.cmml" xref="S6.I1.i4.p1.1.m1.1.1.2.2.3">′</ci></apply><ci id="S6.I1.i4.p1.1.m1.1.1.2.3.cmml" xref="S6.I1.i4.p1.1.m1.1.1.2.3">𝑛</ci></apply><ci id="S6.I1.i4.p1.1.m1.1.1.3.cmml" xref="S6.I1.i4.p1.1.m1.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i4.p1.1.m1.1c">w^{\prime}_{n}\neq w</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i4.p1.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ≠ italic_w</annotation></semantics></math></p> </div> </li> <li class="ltx_item" id="S6.I1.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(5)</span> <div class="ltx_para" id="S6.I1.i5.p1"> <p class="ltx_p" id="S6.I1.i5.p1.2"><math alttext="\lim wv_{n}={\bf x}_{[1,+\infty)}" class="ltx_Math" display="inline" id="S6.I1.i5.p1.1.m1.2"><semantics id="S6.I1.i5.p1.1.m1.2a"><mrow id="S6.I1.i5.p1.1.m1.2.3" xref="S6.I1.i5.p1.1.m1.2.3.cmml"><mrow id="S6.I1.i5.p1.1.m1.2.3.2" xref="S6.I1.i5.p1.1.m1.2.3.2.cmml"><mo id="S6.I1.i5.p1.1.m1.2.3.2.1" rspace="0.167em" xref="S6.I1.i5.p1.1.m1.2.3.2.1.cmml">lim</mo><mrow id="S6.I1.i5.p1.1.m1.2.3.2.2" xref="S6.I1.i5.p1.1.m1.2.3.2.2.cmml"><mi id="S6.I1.i5.p1.1.m1.2.3.2.2.2" xref="S6.I1.i5.p1.1.m1.2.3.2.2.2.cmml">w</mi><mo id="S6.I1.i5.p1.1.m1.2.3.2.2.1" xref="S6.I1.i5.p1.1.m1.2.3.2.2.1.cmml">⁢</mo><msub id="S6.I1.i5.p1.1.m1.2.3.2.2.3" 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id="S6.I1.i5.p1.1.m1.2.3.2.2.3.2.cmml" xref="S6.I1.i5.p1.1.m1.2.3.2.2.3.2">𝑣</ci><ci id="S6.I1.i5.p1.1.m1.2.3.2.2.3.3.cmml" xref="S6.I1.i5.p1.1.m1.2.3.2.2.3.3">𝑛</ci></apply></apply></apply><apply id="S6.I1.i5.p1.1.m1.2.3.3.cmml" xref="S6.I1.i5.p1.1.m1.2.3.3"><csymbol cd="ambiguous" id="S6.I1.i5.p1.1.m1.2.3.3.1.cmml" xref="S6.I1.i5.p1.1.m1.2.3.3">subscript</csymbol><ci id="S6.I1.i5.p1.1.m1.2.3.3.2.cmml" xref="S6.I1.i5.p1.1.m1.2.3.3.2">𝐱</ci><interval closure="closed-open" id="S6.I1.i5.p1.1.m1.2.2.2.3.cmml" xref="S6.I1.i5.p1.1.m1.2.2.2.2"><cn id="S6.I1.i5.p1.1.m1.1.1.1.1.cmml" type="integer" xref="S6.I1.i5.p1.1.m1.1.1.1.1">1</cn><apply id="S6.I1.i5.p1.1.m1.2.2.2.2.1.cmml" xref="S6.I1.i5.p1.1.m1.2.2.2.2.1"><plus id="S6.I1.i5.p1.1.m1.2.2.2.2.1.1.cmml" xref="S6.I1.i5.p1.1.m1.2.2.2.2.1"></plus><infinity id="S6.I1.i5.p1.1.m1.2.2.2.2.1.2.cmml" xref="S6.I1.i5.p1.1.m1.2.2.2.2.1.2"></infinity></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i5.p1.1.m1.2c">\lim wv_{n}={\bf x}_{[1,+\infty)}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i5.p1.1.m1.2d">roman_lim italic_w italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_x start_POSTSUBSCRIPT [ 1 , + ∞ ) end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\lim u_{n}={\bf x}_{(-\infty,0]}" class="ltx_Math" display="inline" id="S6.I1.i5.p1.2.m2.2"><semantics id="S6.I1.i5.p1.2.m2.2a"><mrow id="S6.I1.i5.p1.2.m2.2.3" xref="S6.I1.i5.p1.2.m2.2.3.cmml"><mrow id="S6.I1.i5.p1.2.m2.2.3.2" xref="S6.I1.i5.p1.2.m2.2.3.2.cmml"><mo id="S6.I1.i5.p1.2.m2.2.3.2.1" rspace="0.167em" xref="S6.I1.i5.p1.2.m2.2.3.2.1.cmml">lim</mo><msub id="S6.I1.i5.p1.2.m2.2.3.2.2" xref="S6.I1.i5.p1.2.m2.2.3.2.2.cmml"><mi id="S6.I1.i5.p1.2.m2.2.3.2.2.2" xref="S6.I1.i5.p1.2.m2.2.3.2.2.2.cmml">u</mi><mi id="S6.I1.i5.p1.2.m2.2.3.2.2.3" xref="S6.I1.i5.p1.2.m2.2.3.2.2.3.cmml">n</mi></msub></mrow><mo id="S6.I1.i5.p1.2.m2.2.3.1" xref="S6.I1.i5.p1.2.m2.2.3.1.cmml">=</mo><msub id="S6.I1.i5.p1.2.m2.2.3.3" xref="S6.I1.i5.p1.2.m2.2.3.3.cmml"><mi id="S6.I1.i5.p1.2.m2.2.3.3.2" xref="S6.I1.i5.p1.2.m2.2.3.3.2.cmml">𝐱</mi><mrow id="S6.I1.i5.p1.2.m2.2.2.2.2" xref="S6.I1.i5.p1.2.m2.2.2.2.3.cmml"><mo id="S6.I1.i5.p1.2.m2.2.2.2.2.2" stretchy="false" xref="S6.I1.i5.p1.2.m2.2.2.2.3.cmml">(</mo><mrow id="S6.I1.i5.p1.2.m2.2.2.2.2.1" xref="S6.I1.i5.p1.2.m2.2.2.2.2.1.cmml"><mo id="S6.I1.i5.p1.2.m2.2.2.2.2.1a" xref="S6.I1.i5.p1.2.m2.2.2.2.2.1.cmml">−</mo><mi id="S6.I1.i5.p1.2.m2.2.2.2.2.1.2" mathvariant="normal" xref="S6.I1.i5.p1.2.m2.2.2.2.2.1.2.cmml">∞</mi></mrow><mo id="S6.I1.i5.p1.2.m2.2.2.2.2.3" xref="S6.I1.i5.p1.2.m2.2.2.2.3.cmml">,</mo><mn id="S6.I1.i5.p1.2.m2.1.1.1.1" xref="S6.I1.i5.p1.2.m2.1.1.1.1.cmml">0</mn><mo id="S6.I1.i5.p1.2.m2.2.2.2.2.4" stretchy="false" xref="S6.I1.i5.p1.2.m2.2.2.2.3.cmml">]</mo></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i5.p1.2.m2.2b"><apply id="S6.I1.i5.p1.2.m2.2.3.cmml" xref="S6.I1.i5.p1.2.m2.2.3"><eq id="S6.I1.i5.p1.2.m2.2.3.1.cmml" xref="S6.I1.i5.p1.2.m2.2.3.1"></eq><apply id="S6.I1.i5.p1.2.m2.2.3.2.cmml" xref="S6.I1.i5.p1.2.m2.2.3.2"><limit id="S6.I1.i5.p1.2.m2.2.3.2.1.cmml" xref="S6.I1.i5.p1.2.m2.2.3.2.1"></limit><apply id="S6.I1.i5.p1.2.m2.2.3.2.2.cmml" xref="S6.I1.i5.p1.2.m2.2.3.2.2"><csymbol cd="ambiguous" id="S6.I1.i5.p1.2.m2.2.3.2.2.1.cmml" xref="S6.I1.i5.p1.2.m2.2.3.2.2">subscript</csymbol><ci id="S6.I1.i5.p1.2.m2.2.3.2.2.2.cmml" xref="S6.I1.i5.p1.2.m2.2.3.2.2.2">𝑢</ci><ci id="S6.I1.i5.p1.2.m2.2.3.2.2.3.cmml" xref="S6.I1.i5.p1.2.m2.2.3.2.2.3">𝑛</ci></apply></apply><apply id="S6.I1.i5.p1.2.m2.2.3.3.cmml" xref="S6.I1.i5.p1.2.m2.2.3.3"><csymbol cd="ambiguous" id="S6.I1.i5.p1.2.m2.2.3.3.1.cmml" xref="S6.I1.i5.p1.2.m2.2.3.3">subscript</csymbol><ci id="S6.I1.i5.p1.2.m2.2.3.3.2.cmml" xref="S6.I1.i5.p1.2.m2.2.3.3.2">𝐱</ci><interval closure="open-closed" id="S6.I1.i5.p1.2.m2.2.2.2.3.cmml" xref="S6.I1.i5.p1.2.m2.2.2.2.2"><apply id="S6.I1.i5.p1.2.m2.2.2.2.2.1.cmml" xref="S6.I1.i5.p1.2.m2.2.2.2.2.1"><minus id="S6.I1.i5.p1.2.m2.2.2.2.2.1.1.cmml" xref="S6.I1.i5.p1.2.m2.2.2.2.2.1"></minus><infinity id="S6.I1.i5.p1.2.m2.2.2.2.2.1.2.cmml" xref="S6.I1.i5.p1.2.m2.2.2.2.2.1.2"></infinity></apply><cn id="S6.I1.i5.p1.2.m2.1.1.1.1.cmml" type="integer" xref="S6.I1.i5.p1.2.m2.1.1.1.1">0</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i5.p1.2.m2.2c">\lim u_{n}={\bf x}_{(-\infty,0]}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i5.p1.2.m2.2d">roman_lim italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_x start_POSTSUBSCRIPT ( - ∞ , 0 ] end_POSTSUBSCRIPT</annotation></semantics></math></p> </div> </li> </ol> <p class="ltx_p" id="S6.2.p1.9">The statement (5) needs a bit of interpretation: We think of the words <math alttext="wv_{n}" class="ltx_Math" display="inline" id="S6.2.p1.4.m1.1"><semantics id="S6.2.p1.4.m1.1a"><mrow id="S6.2.p1.4.m1.1.1" xref="S6.2.p1.4.m1.1.1.cmml"><mi id="S6.2.p1.4.m1.1.1.2" xref="S6.2.p1.4.m1.1.1.2.cmml">w</mi><mo id="S6.2.p1.4.m1.1.1.1" xref="S6.2.p1.4.m1.1.1.1.cmml">⁢</mo><msub id="S6.2.p1.4.m1.1.1.3" xref="S6.2.p1.4.m1.1.1.3.cmml"><mi id="S6.2.p1.4.m1.1.1.3.2" xref="S6.2.p1.4.m1.1.1.3.2.cmml">v</mi><mi id="S6.2.p1.4.m1.1.1.3.3" xref="S6.2.p1.4.m1.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.4.m1.1b"><apply id="S6.2.p1.4.m1.1.1.cmml" xref="S6.2.p1.4.m1.1.1"><times id="S6.2.p1.4.m1.1.1.1.cmml" xref="S6.2.p1.4.m1.1.1.1"></times><ci id="S6.2.p1.4.m1.1.1.2.cmml" xref="S6.2.p1.4.m1.1.1.2">𝑤</ci><apply id="S6.2.p1.4.m1.1.1.3.cmml" xref="S6.2.p1.4.m1.1.1.3"><csymbol cd="ambiguous" id="S6.2.p1.4.m1.1.1.3.1.cmml" xref="S6.2.p1.4.m1.1.1.3">subscript</csymbol><ci id="S6.2.p1.4.m1.1.1.3.2.cmml" xref="S6.2.p1.4.m1.1.1.3.2">𝑣</ci><ci id="S6.2.p1.4.m1.1.1.3.3.cmml" xref="S6.2.p1.4.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.4.m1.1c">wv_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.4.m1.1d">italic_w italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as being indexed from <math alttext="1" class="ltx_Math" display="inline" id="S6.2.p1.5.m2.1"><semantics id="S6.2.p1.5.m2.1a"><mn id="S6.2.p1.5.m2.1.1" xref="S6.2.p1.5.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.2.p1.5.m2.1b"><cn id="S6.2.p1.5.m2.1.1.cmml" type="integer" xref="S6.2.p1.5.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.5.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.5.m2.1d">1</annotation></semantics></math> to <math alttext="|wv_{n}|" class="ltx_Math" display="inline" id="S6.2.p1.6.m3.1"><semantics id="S6.2.p1.6.m3.1a"><mrow id="S6.2.p1.6.m3.1.1.1" xref="S6.2.p1.6.m3.1.1.2.cmml"><mo id="S6.2.p1.6.m3.1.1.1.2" stretchy="false" xref="S6.2.p1.6.m3.1.1.2.1.cmml">|</mo><mrow id="S6.2.p1.6.m3.1.1.1.1" xref="S6.2.p1.6.m3.1.1.1.1.cmml"><mi id="S6.2.p1.6.m3.1.1.1.1.2" xref="S6.2.p1.6.m3.1.1.1.1.2.cmml">w</mi><mo id="S6.2.p1.6.m3.1.1.1.1.1" xref="S6.2.p1.6.m3.1.1.1.1.1.cmml">⁢</mo><msub id="S6.2.p1.6.m3.1.1.1.1.3" xref="S6.2.p1.6.m3.1.1.1.1.3.cmml"><mi id="S6.2.p1.6.m3.1.1.1.1.3.2" xref="S6.2.p1.6.m3.1.1.1.1.3.2.cmml">v</mi><mi id="S6.2.p1.6.m3.1.1.1.1.3.3" xref="S6.2.p1.6.m3.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S6.2.p1.6.m3.1.1.1.3" stretchy="false" xref="S6.2.p1.6.m3.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.6.m3.1b"><apply id="S6.2.p1.6.m3.1.1.2.cmml" xref="S6.2.p1.6.m3.1.1.1"><abs id="S6.2.p1.6.m3.1.1.2.1.cmml" xref="S6.2.p1.6.m3.1.1.1.2"></abs><apply id="S6.2.p1.6.m3.1.1.1.1.cmml" xref="S6.2.p1.6.m3.1.1.1.1"><times id="S6.2.p1.6.m3.1.1.1.1.1.cmml" xref="S6.2.p1.6.m3.1.1.1.1.1"></times><ci id="S6.2.p1.6.m3.1.1.1.1.2.cmml" xref="S6.2.p1.6.m3.1.1.1.1.2">𝑤</ci><apply id="S6.2.p1.6.m3.1.1.1.1.3.cmml" xref="S6.2.p1.6.m3.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.2.p1.6.m3.1.1.1.1.3.1.cmml" xref="S6.2.p1.6.m3.1.1.1.1.3">subscript</csymbol><ci id="S6.2.p1.6.m3.1.1.1.1.3.2.cmml" xref="S6.2.p1.6.m3.1.1.1.1.3.2">𝑣</ci><ci id="S6.2.p1.6.m3.1.1.1.1.3.3.cmml" xref="S6.2.p1.6.m3.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.6.m3.1c">|wv_{n}|</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.6.m3.1d">| italic_w italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT |</annotation></semantics></math> and of <math alttext="u_{n}" class="ltx_Math" display="inline" id="S6.2.p1.7.m4.1"><semantics id="S6.2.p1.7.m4.1a"><msub id="S6.2.p1.7.m4.1.1" xref="S6.2.p1.7.m4.1.1.cmml"><mi id="S6.2.p1.7.m4.1.1.2" xref="S6.2.p1.7.m4.1.1.2.cmml">u</mi><mi id="S6.2.p1.7.m4.1.1.3" xref="S6.2.p1.7.m4.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.2.p1.7.m4.1b"><apply id="S6.2.p1.7.m4.1.1.cmml" xref="S6.2.p1.7.m4.1.1"><csymbol cd="ambiguous" id="S6.2.p1.7.m4.1.1.1.cmml" xref="S6.2.p1.7.m4.1.1">subscript</csymbol><ci id="S6.2.p1.7.m4.1.1.2.cmml" xref="S6.2.p1.7.m4.1.1.2">𝑢</ci><ci id="S6.2.p1.7.m4.1.1.3.cmml" xref="S6.2.p1.7.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.7.m4.1c">u_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.7.m4.1d">italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as being indexed from <math alttext="-|u_{n}|+1" class="ltx_Math" display="inline" id="S6.2.p1.8.m5.1"><semantics id="S6.2.p1.8.m5.1a"><mrow id="S6.2.p1.8.m5.1.1" xref="S6.2.p1.8.m5.1.1.cmml"><mrow id="S6.2.p1.8.m5.1.1.1" xref="S6.2.p1.8.m5.1.1.1.cmml"><mo id="S6.2.p1.8.m5.1.1.1a" xref="S6.2.p1.8.m5.1.1.1.cmml">−</mo><mrow id="S6.2.p1.8.m5.1.1.1.1.1" xref="S6.2.p1.8.m5.1.1.1.1.2.cmml"><mo id="S6.2.p1.8.m5.1.1.1.1.1.2" stretchy="false" xref="S6.2.p1.8.m5.1.1.1.1.2.1.cmml">|</mo><msub id="S6.2.p1.8.m5.1.1.1.1.1.1" xref="S6.2.p1.8.m5.1.1.1.1.1.1.cmml"><mi id="S6.2.p1.8.m5.1.1.1.1.1.1.2" xref="S6.2.p1.8.m5.1.1.1.1.1.1.2.cmml">u</mi><mi id="S6.2.p1.8.m5.1.1.1.1.1.1.3" xref="S6.2.p1.8.m5.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.2.p1.8.m5.1.1.1.1.1.3" stretchy="false" xref="S6.2.p1.8.m5.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S6.2.p1.8.m5.1.1.2" xref="S6.2.p1.8.m5.1.1.2.cmml">+</mo><mn id="S6.2.p1.8.m5.1.1.3" xref="S6.2.p1.8.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.2.p1.8.m5.1b"><apply id="S6.2.p1.8.m5.1.1.cmml" xref="S6.2.p1.8.m5.1.1"><plus id="S6.2.p1.8.m5.1.1.2.cmml" xref="S6.2.p1.8.m5.1.1.2"></plus><apply id="S6.2.p1.8.m5.1.1.1.cmml" xref="S6.2.p1.8.m5.1.1.1"><minus id="S6.2.p1.8.m5.1.1.1.2.cmml" xref="S6.2.p1.8.m5.1.1.1"></minus><apply id="S6.2.p1.8.m5.1.1.1.1.2.cmml" xref="S6.2.p1.8.m5.1.1.1.1.1"><abs id="S6.2.p1.8.m5.1.1.1.1.2.1.cmml" xref="S6.2.p1.8.m5.1.1.1.1.1.2"></abs><apply id="S6.2.p1.8.m5.1.1.1.1.1.1.cmml" xref="S6.2.p1.8.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.2.p1.8.m5.1.1.1.1.1.1.1.cmml" xref="S6.2.p1.8.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S6.2.p1.8.m5.1.1.1.1.1.1.2.cmml" xref="S6.2.p1.8.m5.1.1.1.1.1.1.2">𝑢</ci><ci id="S6.2.p1.8.m5.1.1.1.1.1.1.3.cmml" xref="S6.2.p1.8.m5.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply><cn id="S6.2.p1.8.m5.1.1.3.cmml" type="integer" xref="S6.2.p1.8.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.2.p1.8.m5.1c">-|u_{n}|+1</annotation><annotation encoding="application/x-llamapun" id="S6.2.p1.8.m5.1d">- | italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | + 1</annotation></semantics></math> to <math alttext="0" class="ltx_Math" display="inline" id="S6.2.p1.9.m6.1"><semantics id="S6.2.p1.9.m6.1a"><mn id="S6.2.p1.9.m6.1.1" xref="S6.2.p1.9.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.2.p1.9.m6.1b"><cn id="S6.2.p1.9.m6.1.1.cmml" type="integer" xref="S6.2.p1.9.m6.1.1">0</cn></annotation-xml></semantics></math>, and we pass to the limit while keeping the indices fixed. In other words, statement (5) is equivalent to</p> <ol class="ltx_enumerate" id="S6.I2"> <li class="ltx_item" id="S6.I2.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(5’)</span> <div class="ltx_para" id="S6.I2.ix1.p1"> <p class="ltx_p" id="S6.I2.ix1.p1.6"><math alttext="\lim{\bf y}(n)={\bf x}" class="ltx_Math" display="inline" id="S6.I2.ix1.p1.1.m1.1"><semantics id="S6.I2.ix1.p1.1.m1.1a"><mrow id="S6.I2.ix1.p1.1.m1.1.2" xref="S6.I2.ix1.p1.1.m1.1.2.cmml"><mrow id="S6.I2.ix1.p1.1.m1.1.2.2" xref="S6.I2.ix1.p1.1.m1.1.2.2.cmml"><mo id="S6.I2.ix1.p1.1.m1.1.2.2.1" rspace="0.167em" xref="S6.I2.ix1.p1.1.m1.1.2.2.1.cmml">lim</mo><mrow id="S6.I2.ix1.p1.1.m1.1.2.2.2" xref="S6.I2.ix1.p1.1.m1.1.2.2.2.cmml"><mi id="S6.I2.ix1.p1.1.m1.1.2.2.2.2" xref="S6.I2.ix1.p1.1.m1.1.2.2.2.2.cmml">𝐲</mi><mo id="S6.I2.ix1.p1.1.m1.1.2.2.2.1" xref="S6.I2.ix1.p1.1.m1.1.2.2.2.1.cmml">⁢</mo><mrow id="S6.I2.ix1.p1.1.m1.1.2.2.2.3.2" xref="S6.I2.ix1.p1.1.m1.1.2.2.2.cmml"><mo id="S6.I2.ix1.p1.1.m1.1.2.2.2.3.2.1" stretchy="false" xref="S6.I2.ix1.p1.1.m1.1.2.2.2.cmml">(</mo><mi id="S6.I2.ix1.p1.1.m1.1.1" xref="S6.I2.ix1.p1.1.m1.1.1.cmml">n</mi><mo id="S6.I2.ix1.p1.1.m1.1.2.2.2.3.2.2" stretchy="false" xref="S6.I2.ix1.p1.1.m1.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S6.I2.ix1.p1.1.m1.1.2.1" xref="S6.I2.ix1.p1.1.m1.1.2.1.cmml">=</mo><mi id="S6.I2.ix1.p1.1.m1.1.2.3" xref="S6.I2.ix1.p1.1.m1.1.2.3.cmml">𝐱</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.ix1.p1.1.m1.1b"><apply id="S6.I2.ix1.p1.1.m1.1.2.cmml" xref="S6.I2.ix1.p1.1.m1.1.2"><eq id="S6.I2.ix1.p1.1.m1.1.2.1.cmml" xref="S6.I2.ix1.p1.1.m1.1.2.1"></eq><apply id="S6.I2.ix1.p1.1.m1.1.2.2.cmml" xref="S6.I2.ix1.p1.1.m1.1.2.2"><limit id="S6.I2.ix1.p1.1.m1.1.2.2.1.cmml" xref="S6.I2.ix1.p1.1.m1.1.2.2.1"></limit><apply id="S6.I2.ix1.p1.1.m1.1.2.2.2.cmml" xref="S6.I2.ix1.p1.1.m1.1.2.2.2"><times id="S6.I2.ix1.p1.1.m1.1.2.2.2.1.cmml" xref="S6.I2.ix1.p1.1.m1.1.2.2.2.1"></times><ci id="S6.I2.ix1.p1.1.m1.1.2.2.2.2.cmml" xref="S6.I2.ix1.p1.1.m1.1.2.2.2.2">𝐲</ci><ci id="S6.I2.ix1.p1.1.m1.1.1.cmml" xref="S6.I2.ix1.p1.1.m1.1.1">𝑛</ci></apply></apply><ci id="S6.I2.ix1.p1.1.m1.1.2.3.cmml" xref="S6.I2.ix1.p1.1.m1.1.2.3">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.ix1.p1.1.m1.1c">\lim{\bf y}(n)={\bf x}</annotation><annotation encoding="application/x-llamapun" id="S6.I2.ix1.p1.1.m1.1d">roman_lim bold_y ( italic_n ) = bold_x</annotation></semantics></math> in <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.I2.ix1.p1.2.m2.1"><semantics id="S6.I2.ix1.p1.2.m2.1a"><msup id="S6.I2.ix1.p1.2.m2.1.1" xref="S6.I2.ix1.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.I2.ix1.p1.2.m2.1.1.2" xref="S6.I2.ix1.p1.2.m2.1.1.2.cmml">𝒜</mi><mi id="S6.I2.ix1.p1.2.m2.1.1.3" xref="S6.I2.ix1.p1.2.m2.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S6.I2.ix1.p1.2.m2.1b"><apply id="S6.I2.ix1.p1.2.m2.1.1.cmml" xref="S6.I2.ix1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.I2.ix1.p1.2.m2.1.1.1.cmml" xref="S6.I2.ix1.p1.2.m2.1.1">superscript</csymbol><ci id="S6.I2.ix1.p1.2.m2.1.1.2.cmml" xref="S6.I2.ix1.p1.2.m2.1.1.2">𝒜</ci><ci id="S6.I2.ix1.p1.2.m2.1.1.3.cmml" xref="S6.I2.ix1.p1.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.ix1.p1.2.m2.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.I2.ix1.p1.2.m2.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, where for any <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S6.I2.ix1.p1.3.m3.1"><semantics id="S6.I2.ix1.p1.3.m3.1a"><mrow id="S6.I2.ix1.p1.3.m3.1.1" xref="S6.I2.ix1.p1.3.m3.1.1.cmml"><mi id="S6.I2.ix1.p1.3.m3.1.1.2" xref="S6.I2.ix1.p1.3.m3.1.1.2.cmml">n</mi><mo id="S6.I2.ix1.p1.3.m3.1.1.1" xref="S6.I2.ix1.p1.3.m3.1.1.1.cmml">≥</mo><mn id="S6.I2.ix1.p1.3.m3.1.1.3" xref="S6.I2.ix1.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.ix1.p1.3.m3.1b"><apply id="S6.I2.ix1.p1.3.m3.1.1.cmml" xref="S6.I2.ix1.p1.3.m3.1.1"><geq id="S6.I2.ix1.p1.3.m3.1.1.1.cmml" xref="S6.I2.ix1.p1.3.m3.1.1.1"></geq><ci id="S6.I2.ix1.p1.3.m3.1.1.2.cmml" xref="S6.I2.ix1.p1.3.m3.1.1.2">𝑛</ci><cn id="S6.I2.ix1.p1.3.m3.1.1.3.cmml" type="integer" xref="S6.I2.ix1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.ix1.p1.3.m3.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.I2.ix1.p1.3.m3.1d">italic_n ≥ 0</annotation></semantics></math> the biinfinite word <math alttext="{\bf y}(n)" class="ltx_Math" display="inline" id="S6.I2.ix1.p1.4.m4.1"><semantics id="S6.I2.ix1.p1.4.m4.1a"><mrow id="S6.I2.ix1.p1.4.m4.1.2" xref="S6.I2.ix1.p1.4.m4.1.2.cmml"><mi id="S6.I2.ix1.p1.4.m4.1.2.2" xref="S6.I2.ix1.p1.4.m4.1.2.2.cmml">𝐲</mi><mo id="S6.I2.ix1.p1.4.m4.1.2.1" xref="S6.I2.ix1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S6.I2.ix1.p1.4.m4.1.2.3.2" xref="S6.I2.ix1.p1.4.m4.1.2.cmml"><mo id="S6.I2.ix1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S6.I2.ix1.p1.4.m4.1.2.cmml">(</mo><mi id="S6.I2.ix1.p1.4.m4.1.1" xref="S6.I2.ix1.p1.4.m4.1.1.cmml">n</mi><mo id="S6.I2.ix1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S6.I2.ix1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.ix1.p1.4.m4.1b"><apply id="S6.I2.ix1.p1.4.m4.1.2.cmml" xref="S6.I2.ix1.p1.4.m4.1.2"><times id="S6.I2.ix1.p1.4.m4.1.2.1.cmml" xref="S6.I2.ix1.p1.4.m4.1.2.1"></times><ci id="S6.I2.ix1.p1.4.m4.1.2.2.cmml" xref="S6.I2.ix1.p1.4.m4.1.2.2">𝐲</ci><ci id="S6.I2.ix1.p1.4.m4.1.1.cmml" xref="S6.I2.ix1.p1.4.m4.1.1">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.ix1.p1.4.m4.1c">{\bf y}(n)</annotation><annotation encoding="application/x-llamapun" id="S6.I2.ix1.p1.4.m4.1d">bold_y ( italic_n )</annotation></semantics></math> is defined through <math alttext="{\bf y}_{[1,\infty)}(n)=wv_{n}^{+\infty}" class="ltx_Math" display="inline" id="S6.I2.ix1.p1.5.m5.3"><semantics id="S6.I2.ix1.p1.5.m5.3a"><mrow id="S6.I2.ix1.p1.5.m5.3.4" xref="S6.I2.ix1.p1.5.m5.3.4.cmml"><mrow id="S6.I2.ix1.p1.5.m5.3.4.2" xref="S6.I2.ix1.p1.5.m5.3.4.2.cmml"><msub id="S6.I2.ix1.p1.5.m5.3.4.2.2" xref="S6.I2.ix1.p1.5.m5.3.4.2.2.cmml"><mi id="S6.I2.ix1.p1.5.m5.3.4.2.2.2" xref="S6.I2.ix1.p1.5.m5.3.4.2.2.2.cmml">𝐲</mi><mrow id="S6.I2.ix1.p1.5.m5.2.2.2.4" xref="S6.I2.ix1.p1.5.m5.2.2.2.3.cmml"><mo id="S6.I2.ix1.p1.5.m5.2.2.2.4.1" stretchy="false" xref="S6.I2.ix1.p1.5.m5.2.2.2.3.cmml">[</mo><mn id="S6.I2.ix1.p1.5.m5.1.1.1.1" xref="S6.I2.ix1.p1.5.m5.1.1.1.1.cmml">1</mn><mo id="S6.I2.ix1.p1.5.m5.2.2.2.4.2" xref="S6.I2.ix1.p1.5.m5.2.2.2.3.cmml">,</mo><mi id="S6.I2.ix1.p1.5.m5.2.2.2.2" mathvariant="normal" xref="S6.I2.ix1.p1.5.m5.2.2.2.2.cmml">∞</mi><mo id="S6.I2.ix1.p1.5.m5.2.2.2.4.3" stretchy="false" xref="S6.I2.ix1.p1.5.m5.2.2.2.3.cmml">)</mo></mrow></msub><mo id="S6.I2.ix1.p1.5.m5.3.4.2.1" xref="S6.I2.ix1.p1.5.m5.3.4.2.1.cmml">⁢</mo><mrow id="S6.I2.ix1.p1.5.m5.3.4.2.3.2" xref="S6.I2.ix1.p1.5.m5.3.4.2.cmml"><mo id="S6.I2.ix1.p1.5.m5.3.4.2.3.2.1" stretchy="false" xref="S6.I2.ix1.p1.5.m5.3.4.2.cmml">(</mo><mi id="S6.I2.ix1.p1.5.m5.3.3" xref="S6.I2.ix1.p1.5.m5.3.3.cmml">n</mi><mo id="S6.I2.ix1.p1.5.m5.3.4.2.3.2.2" stretchy="false" xref="S6.I2.ix1.p1.5.m5.3.4.2.cmml">)</mo></mrow></mrow><mo id="S6.I2.ix1.p1.5.m5.3.4.1" xref="S6.I2.ix1.p1.5.m5.3.4.1.cmml">=</mo><mrow id="S6.I2.ix1.p1.5.m5.3.4.3" xref="S6.I2.ix1.p1.5.m5.3.4.3.cmml"><mi id="S6.I2.ix1.p1.5.m5.3.4.3.2" xref="S6.I2.ix1.p1.5.m5.3.4.3.2.cmml">w</mi><mo id="S6.I2.ix1.p1.5.m5.3.4.3.1" xref="S6.I2.ix1.p1.5.m5.3.4.3.1.cmml">⁢</mo><msubsup id="S6.I2.ix1.p1.5.m5.3.4.3.3" xref="S6.I2.ix1.p1.5.m5.3.4.3.3.cmml"><mi id="S6.I2.ix1.p1.5.m5.3.4.3.3.2.2" xref="S6.I2.ix1.p1.5.m5.3.4.3.3.2.2.cmml">v</mi><mi id="S6.I2.ix1.p1.5.m5.3.4.3.3.2.3" 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id="S6.I2.ix1.p1.5.m5.3.4.3.3.2.2.cmml" xref="S6.I2.ix1.p1.5.m5.3.4.3.3.2.2">𝑣</ci><ci id="S6.I2.ix1.p1.5.m5.3.4.3.3.2.3.cmml" xref="S6.I2.ix1.p1.5.m5.3.4.3.3.2.3">𝑛</ci></apply><apply id="S6.I2.ix1.p1.5.m5.3.4.3.3.3.cmml" xref="S6.I2.ix1.p1.5.m5.3.4.3.3.3"><plus id="S6.I2.ix1.p1.5.m5.3.4.3.3.3.1.cmml" xref="S6.I2.ix1.p1.5.m5.3.4.3.3.3"></plus><infinity id="S6.I2.ix1.p1.5.m5.3.4.3.3.3.2.cmml" xref="S6.I2.ix1.p1.5.m5.3.4.3.3.3.2"></infinity></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.ix1.p1.5.m5.3c">{\bf y}_{[1,\infty)}(n)=wv_{n}^{+\infty}</annotation><annotation encoding="application/x-llamapun" id="S6.I2.ix1.p1.5.m5.3d">bold_y start_POSTSUBSCRIPT [ 1 , ∞ ) end_POSTSUBSCRIPT ( italic_n ) = italic_w italic_v start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="{\bf y}_{(\infty,0]}(n)=u_{n}^{-\infty}" class="ltx_Math" display="inline" 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id="S6.I2.ix1.p1.6.m6.3b"><apply id="S6.I2.ix1.p1.6.m6.3.4.cmml" xref="S6.I2.ix1.p1.6.m6.3.4"><eq id="S6.I2.ix1.p1.6.m6.3.4.1.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.1"></eq><apply id="S6.I2.ix1.p1.6.m6.3.4.2.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.2"><times id="S6.I2.ix1.p1.6.m6.3.4.2.1.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.2.1"></times><apply id="S6.I2.ix1.p1.6.m6.3.4.2.2.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.2.2"><csymbol cd="ambiguous" id="S6.I2.ix1.p1.6.m6.3.4.2.2.1.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.2.2">subscript</csymbol><ci id="S6.I2.ix1.p1.6.m6.3.4.2.2.2.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.2.2.2">𝐲</ci><interval closure="open-closed" id="S6.I2.ix1.p1.6.m6.2.2.2.3.cmml" xref="S6.I2.ix1.p1.6.m6.2.2.2.4"><infinity id="S6.I2.ix1.p1.6.m6.1.1.1.1.cmml" xref="S6.I2.ix1.p1.6.m6.1.1.1.1"></infinity><cn id="S6.I2.ix1.p1.6.m6.2.2.2.2.cmml" type="integer" xref="S6.I2.ix1.p1.6.m6.2.2.2.2">0</cn></interval></apply><ci id="S6.I2.ix1.p1.6.m6.3.3.cmml" xref="S6.I2.ix1.p1.6.m6.3.3">𝑛</ci></apply><apply id="S6.I2.ix1.p1.6.m6.3.4.3.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3"><csymbol cd="ambiguous" id="S6.I2.ix1.p1.6.m6.3.4.3.1.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3">superscript</csymbol><apply id="S6.I2.ix1.p1.6.m6.3.4.3.2.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3"><csymbol cd="ambiguous" id="S6.I2.ix1.p1.6.m6.3.4.3.2.1.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3">subscript</csymbol><ci id="S6.I2.ix1.p1.6.m6.3.4.3.2.2.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3.2.2">𝑢</ci><ci id="S6.I2.ix1.p1.6.m6.3.4.3.2.3.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3.2.3">𝑛</ci></apply><apply id="S6.I2.ix1.p1.6.m6.3.4.3.3.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3.3"><minus id="S6.I2.ix1.p1.6.m6.3.4.3.3.1.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3.3"></minus><infinity id="S6.I2.ix1.p1.6.m6.3.4.3.3.2.cmml" xref="S6.I2.ix1.p1.6.m6.3.4.3.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.ix1.p1.6.m6.3c">{\bf y}_{(\infty,0]}(n)=u_{n}^{-\infty}</annotation><annotation encoding="application/x-llamapun" id="S6.I2.ix1.p1.6.m6.3d">bold_y start_POSTSUBSCRIPT ( ∞ , 0 ] end_POSTSUBSCRIPT ( italic_n ) = italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> </ol> </div> <div class="ltx_para" id="S6.3.p2"> <p class="ltx_p" id="S6.3.p2.8">We now proceed analogously with the sequence of words <math alttext="w^{\prime}_{n}v^{\prime}_{n}" class="ltx_Math" display="inline" id="S6.3.p2.1.m1.1"><semantics id="S6.3.p2.1.m1.1a"><mrow id="S6.3.p2.1.m1.1.1" xref="S6.3.p2.1.m1.1.1.cmml"><msubsup id="S6.3.p2.1.m1.1.1.2" xref="S6.3.p2.1.m1.1.1.2.cmml"><mi id="S6.3.p2.1.m1.1.1.2.2.2" xref="S6.3.p2.1.m1.1.1.2.2.2.cmml">w</mi><mi id="S6.3.p2.1.m1.1.1.2.3" xref="S6.3.p2.1.m1.1.1.2.3.cmml">n</mi><mo id="S6.3.p2.1.m1.1.1.2.2.3" xref="S6.3.p2.1.m1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S6.3.p2.1.m1.1.1.1" xref="S6.3.p2.1.m1.1.1.1.cmml">⁢</mo><msubsup id="S6.3.p2.1.m1.1.1.3" xref="S6.3.p2.1.m1.1.1.3.cmml"><mi id="S6.3.p2.1.m1.1.1.3.2.2" xref="S6.3.p2.1.m1.1.1.3.2.2.cmml">v</mi><mi id="S6.3.p2.1.m1.1.1.3.3" xref="S6.3.p2.1.m1.1.1.3.3.cmml">n</mi><mo id="S6.3.p2.1.m1.1.1.3.2.3" xref="S6.3.p2.1.m1.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.1.m1.1b"><apply id="S6.3.p2.1.m1.1.1.cmml" xref="S6.3.p2.1.m1.1.1"><times id="S6.3.p2.1.m1.1.1.1.cmml" xref="S6.3.p2.1.m1.1.1.1"></times><apply id="S6.3.p2.1.m1.1.1.2.cmml" xref="S6.3.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.3.p2.1.m1.1.1.2.1.cmml" xref="S6.3.p2.1.m1.1.1.2">subscript</csymbol><apply id="S6.3.p2.1.m1.1.1.2.2.cmml" xref="S6.3.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.3.p2.1.m1.1.1.2.2.1.cmml" xref="S6.3.p2.1.m1.1.1.2">superscript</csymbol><ci id="S6.3.p2.1.m1.1.1.2.2.2.cmml" xref="S6.3.p2.1.m1.1.1.2.2.2">𝑤</ci><ci id="S6.3.p2.1.m1.1.1.2.2.3.cmml" xref="S6.3.p2.1.m1.1.1.2.2.3">′</ci></apply><ci id="S6.3.p2.1.m1.1.1.2.3.cmml" xref="S6.3.p2.1.m1.1.1.2.3">𝑛</ci></apply><apply id="S6.3.p2.1.m1.1.1.3.cmml" xref="S6.3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.3.p2.1.m1.1.1.3.1.cmml" xref="S6.3.p2.1.m1.1.1.3">subscript</csymbol><apply id="S6.3.p2.1.m1.1.1.3.2.cmml" xref="S6.3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.3.p2.1.m1.1.1.3.2.1.cmml" xref="S6.3.p2.1.m1.1.1.3">superscript</csymbol><ci id="S6.3.p2.1.m1.1.1.3.2.2.cmml" xref="S6.3.p2.1.m1.1.1.3.2.2">𝑣</ci><ci id="S6.3.p2.1.m1.1.1.3.2.3.cmml" xref="S6.3.p2.1.m1.1.1.3.2.3">′</ci></apply><ci id="S6.3.p2.1.m1.1.1.3.3.cmml" xref="S6.3.p2.1.m1.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.1.m1.1c">w^{\prime}_{n}v^{\prime}_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.1.m1.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> (indexed from <math alttext="1" class="ltx_Math" display="inline" id="S6.3.p2.2.m2.1"><semantics id="S6.3.p2.2.m2.1a"><mn id="S6.3.p2.2.m2.1.1" xref="S6.3.p2.2.m2.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S6.3.p2.2.m2.1b"><cn id="S6.3.p2.2.m2.1.1.cmml" type="integer" xref="S6.3.p2.2.m2.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.2.m2.1c">1</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.2.m2.1d">1</annotation></semantics></math> to <math alttext="|w^{\prime}_{n}v^{\prime}_{n}|" class="ltx_Math" display="inline" id="S6.3.p2.3.m3.1"><semantics id="S6.3.p2.3.m3.1a"><mrow id="S6.3.p2.3.m3.1.1.1" xref="S6.3.p2.3.m3.1.1.2.cmml"><mo id="S6.3.p2.3.m3.1.1.1.2" stretchy="false" xref="S6.3.p2.3.m3.1.1.2.1.cmml">|</mo><mrow id="S6.3.p2.3.m3.1.1.1.1" xref="S6.3.p2.3.m3.1.1.1.1.cmml"><msubsup id="S6.3.p2.3.m3.1.1.1.1.2" xref="S6.3.p2.3.m3.1.1.1.1.2.cmml"><mi id="S6.3.p2.3.m3.1.1.1.1.2.2.2" xref="S6.3.p2.3.m3.1.1.1.1.2.2.2.cmml">w</mi><mi id="S6.3.p2.3.m3.1.1.1.1.2.3" xref="S6.3.p2.3.m3.1.1.1.1.2.3.cmml">n</mi><mo id="S6.3.p2.3.m3.1.1.1.1.2.2.3" xref="S6.3.p2.3.m3.1.1.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S6.3.p2.3.m3.1.1.1.1.1" xref="S6.3.p2.3.m3.1.1.1.1.1.cmml">⁢</mo><msubsup id="S6.3.p2.3.m3.1.1.1.1.3" xref="S6.3.p2.3.m3.1.1.1.1.3.cmml"><mi id="S6.3.p2.3.m3.1.1.1.1.3.2.2" xref="S6.3.p2.3.m3.1.1.1.1.3.2.2.cmml">v</mi><mi id="S6.3.p2.3.m3.1.1.1.1.3.3" xref="S6.3.p2.3.m3.1.1.1.1.3.3.cmml">n</mi><mo id="S6.3.p2.3.m3.1.1.1.1.3.2.3" xref="S6.3.p2.3.m3.1.1.1.1.3.2.3.cmml">′</mo></msubsup></mrow><mo id="S6.3.p2.3.m3.1.1.1.3" stretchy="false" xref="S6.3.p2.3.m3.1.1.2.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.3.m3.1b"><apply id="S6.3.p2.3.m3.1.1.2.cmml" xref="S6.3.p2.3.m3.1.1.1"><abs id="S6.3.p2.3.m3.1.1.2.1.cmml" xref="S6.3.p2.3.m3.1.1.1.2"></abs><apply id="S6.3.p2.3.m3.1.1.1.1.cmml" xref="S6.3.p2.3.m3.1.1.1.1"><times id="S6.3.p2.3.m3.1.1.1.1.1.cmml" xref="S6.3.p2.3.m3.1.1.1.1.1"></times><apply id="S6.3.p2.3.m3.1.1.1.1.2.cmml" xref="S6.3.p2.3.m3.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.3.p2.3.m3.1.1.1.1.2.1.cmml" xref="S6.3.p2.3.m3.1.1.1.1.2">subscript</csymbol><apply id="S6.3.p2.3.m3.1.1.1.1.2.2.cmml" xref="S6.3.p2.3.m3.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.3.p2.3.m3.1.1.1.1.2.2.1.cmml" xref="S6.3.p2.3.m3.1.1.1.1.2">superscript</csymbol><ci id="S6.3.p2.3.m3.1.1.1.1.2.2.2.cmml" xref="S6.3.p2.3.m3.1.1.1.1.2.2.2">𝑤</ci><ci id="S6.3.p2.3.m3.1.1.1.1.2.2.3.cmml" xref="S6.3.p2.3.m3.1.1.1.1.2.2.3">′</ci></apply><ci id="S6.3.p2.3.m3.1.1.1.1.2.3.cmml" xref="S6.3.p2.3.m3.1.1.1.1.2.3">𝑛</ci></apply><apply id="S6.3.p2.3.m3.1.1.1.1.3.cmml" xref="S6.3.p2.3.m3.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.3.p2.3.m3.1.1.1.1.3.1.cmml" xref="S6.3.p2.3.m3.1.1.1.1.3">subscript</csymbol><apply id="S6.3.p2.3.m3.1.1.1.1.3.2.cmml" xref="S6.3.p2.3.m3.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.3.p2.3.m3.1.1.1.1.3.2.1.cmml" xref="S6.3.p2.3.m3.1.1.1.1.3">superscript</csymbol><ci id="S6.3.p2.3.m3.1.1.1.1.3.2.2.cmml" xref="S6.3.p2.3.m3.1.1.1.1.3.2.2">𝑣</ci><ci id="S6.3.p2.3.m3.1.1.1.1.3.2.3.cmml" xref="S6.3.p2.3.m3.1.1.1.1.3.2.3">′</ci></apply><ci id="S6.3.p2.3.m3.1.1.1.1.3.3.cmml" xref="S6.3.p2.3.m3.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.3.m3.1c">|w^{\prime}_{n}v^{\prime}_{n}|</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.3.m3.1d">| italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT |</annotation></semantics></math>) and the words <math alttext="u^{\prime}_{n}" class="ltx_Math" display="inline" id="S6.3.p2.4.m4.1"><semantics id="S6.3.p2.4.m4.1a"><msubsup id="S6.3.p2.4.m4.1.1" xref="S6.3.p2.4.m4.1.1.cmml"><mi id="S6.3.p2.4.m4.1.1.2.2" xref="S6.3.p2.4.m4.1.1.2.2.cmml">u</mi><mi id="S6.3.p2.4.m4.1.1.3" xref="S6.3.p2.4.m4.1.1.3.cmml">n</mi><mo id="S6.3.p2.4.m4.1.1.2.3" xref="S6.3.p2.4.m4.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S6.3.p2.4.m4.1b"><apply id="S6.3.p2.4.m4.1.1.cmml" xref="S6.3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S6.3.p2.4.m4.1.1.1.cmml" xref="S6.3.p2.4.m4.1.1">subscript</csymbol><apply id="S6.3.p2.4.m4.1.1.2.cmml" xref="S6.3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S6.3.p2.4.m4.1.1.2.1.cmml" xref="S6.3.p2.4.m4.1.1">superscript</csymbol><ci id="S6.3.p2.4.m4.1.1.2.2.cmml" xref="S6.3.p2.4.m4.1.1.2.2">𝑢</ci><ci id="S6.3.p2.4.m4.1.1.2.3.cmml" xref="S6.3.p2.4.m4.1.1.2.3">′</ci></apply><ci id="S6.3.p2.4.m4.1.1.3.cmml" xref="S6.3.p2.4.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.4.m4.1c">u^{\prime}_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.4.m4.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> (indexed from <math alttext="-|u^{\prime}_{n}|+1" class="ltx_Math" display="inline" id="S6.3.p2.5.m5.1"><semantics id="S6.3.p2.5.m5.1a"><mrow id="S6.3.p2.5.m5.1.1" xref="S6.3.p2.5.m5.1.1.cmml"><mrow id="S6.3.p2.5.m5.1.1.1" xref="S6.3.p2.5.m5.1.1.1.cmml"><mo id="S6.3.p2.5.m5.1.1.1a" xref="S6.3.p2.5.m5.1.1.1.cmml">−</mo><mrow id="S6.3.p2.5.m5.1.1.1.1.1" xref="S6.3.p2.5.m5.1.1.1.1.2.cmml"><mo id="S6.3.p2.5.m5.1.1.1.1.1.2" stretchy="false" xref="S6.3.p2.5.m5.1.1.1.1.2.1.cmml">|</mo><msubsup id="S6.3.p2.5.m5.1.1.1.1.1.1" xref="S6.3.p2.5.m5.1.1.1.1.1.1.cmml"><mi id="S6.3.p2.5.m5.1.1.1.1.1.1.2.2" xref="S6.3.p2.5.m5.1.1.1.1.1.1.2.2.cmml">u</mi><mi id="S6.3.p2.5.m5.1.1.1.1.1.1.3" xref="S6.3.p2.5.m5.1.1.1.1.1.1.3.cmml">n</mi><mo id="S6.3.p2.5.m5.1.1.1.1.1.1.2.3" xref="S6.3.p2.5.m5.1.1.1.1.1.1.2.3.cmml">′</mo></msubsup><mo id="S6.3.p2.5.m5.1.1.1.1.1.3" stretchy="false" xref="S6.3.p2.5.m5.1.1.1.1.2.1.cmml">|</mo></mrow></mrow><mo id="S6.3.p2.5.m5.1.1.2" xref="S6.3.p2.5.m5.1.1.2.cmml">+</mo><mn id="S6.3.p2.5.m5.1.1.3" xref="S6.3.p2.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.5.m5.1b"><apply id="S6.3.p2.5.m5.1.1.cmml" xref="S6.3.p2.5.m5.1.1"><plus id="S6.3.p2.5.m5.1.1.2.cmml" xref="S6.3.p2.5.m5.1.1.2"></plus><apply id="S6.3.p2.5.m5.1.1.1.cmml" xref="S6.3.p2.5.m5.1.1.1"><minus id="S6.3.p2.5.m5.1.1.1.2.cmml" xref="S6.3.p2.5.m5.1.1.1"></minus><apply id="S6.3.p2.5.m5.1.1.1.1.2.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1"><abs id="S6.3.p2.5.m5.1.1.1.1.2.1.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1.2"></abs><apply id="S6.3.p2.5.m5.1.1.1.1.1.1.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.3.p2.5.m5.1.1.1.1.1.1.1.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1.1">subscript</csymbol><apply id="S6.3.p2.5.m5.1.1.1.1.1.1.2.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.3.p2.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1.1">superscript</csymbol><ci id="S6.3.p2.5.m5.1.1.1.1.1.1.2.2.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1.1.2.2">𝑢</ci><ci id="S6.3.p2.5.m5.1.1.1.1.1.1.2.3.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1.1.2.3">′</ci></apply><ci id="S6.3.p2.5.m5.1.1.1.1.1.1.3.cmml" xref="S6.3.p2.5.m5.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply><cn id="S6.3.p2.5.m5.1.1.3.cmml" type="integer" xref="S6.3.p2.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.5.m5.1c">-|u^{\prime}_{n}|+1</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.5.m5.1d">- | italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | + 1</annotation></semantics></math> to <math alttext="0" class="ltx_Math" display="inline" id="S6.3.p2.6.m6.1"><semantics id="S6.3.p2.6.m6.1a"><mn id="S6.3.p2.6.m6.1.1" xref="S6.3.p2.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S6.3.p2.6.m6.1b"><cn id="S6.3.p2.6.m6.1.1.cmml" type="integer" xref="S6.3.p2.6.m6.1.1">0</cn></annotation-xml></semantics></math>), and pass to a subsequence of the integers <math alttext="n" class="ltx_Math" display="inline" id="S6.3.p2.7.m7.1"><semantics id="S6.3.p2.7.m7.1a"><mi id="S6.3.p2.7.m7.1.1" xref="S6.3.p2.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.7.m7.1b"><ci id="S6.3.p2.7.m7.1.1.cmml" xref="S6.3.p2.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.7.m7.1d">italic_n</annotation></semantics></math> such that there exists a biinfinite “limit word” <math alttext="{\bf z}\in\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.3.p2.8.m8.1"><semantics id="S6.3.p2.8.m8.1a"><mrow id="S6.3.p2.8.m8.1.1" xref="S6.3.p2.8.m8.1.1.cmml"><mi id="S6.3.p2.8.m8.1.1.2" xref="S6.3.p2.8.m8.1.1.2.cmml">𝐳</mi><mo id="S6.3.p2.8.m8.1.1.1" xref="S6.3.p2.8.m8.1.1.1.cmml">∈</mo><msup id="S6.3.p2.8.m8.1.1.3" xref="S6.3.p2.8.m8.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.3.p2.8.m8.1.1.3.2" xref="S6.3.p2.8.m8.1.1.3.2.cmml">𝒜</mi><mi id="S6.3.p2.8.m8.1.1.3.3" xref="S6.3.p2.8.m8.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.8.m8.1b"><apply id="S6.3.p2.8.m8.1.1.cmml" xref="S6.3.p2.8.m8.1.1"><in id="S6.3.p2.8.m8.1.1.1.cmml" xref="S6.3.p2.8.m8.1.1.1"></in><ci id="S6.3.p2.8.m8.1.1.2.cmml" xref="S6.3.p2.8.m8.1.1.2">𝐳</ci><apply id="S6.3.p2.8.m8.1.1.3.cmml" xref="S6.3.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S6.3.p2.8.m8.1.1.3.1.cmml" xref="S6.3.p2.8.m8.1.1.3">superscript</csymbol><ci id="S6.3.p2.8.m8.1.1.3.2.cmml" xref="S6.3.p2.8.m8.1.1.3.2">𝒜</ci><ci id="S6.3.p2.8.m8.1.1.3.3.cmml" xref="S6.3.p2.8.m8.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.8.m8.1c">{\bf z}\in\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.8.m8.1d">bold_z ∈ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> which satisfies</p> <ol class="ltx_enumerate" id="S6.I3"> <li class="ltx_item" id="S6.I3.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(6)</span> <div class="ltx_para" id="S6.I3.ix1.p1"> <p class="ltx_p" id="S6.I3.ix1.p1.2"><math alttext="\lim w^{\prime}_{n}v^{\prime}_{n}={\bf z}_{[1,+\infty)}" class="ltx_Math" display="inline" id="S6.I3.ix1.p1.1.m1.2"><semantics id="S6.I3.ix1.p1.1.m1.2a"><mrow id="S6.I3.ix1.p1.1.m1.2.3" xref="S6.I3.ix1.p1.1.m1.2.3.cmml"><mrow id="S6.I3.ix1.p1.1.m1.2.3.2" xref="S6.I3.ix1.p1.1.m1.2.3.2.cmml"><mo id="S6.I3.ix1.p1.1.m1.2.3.2.1" rspace="0.167em" xref="S6.I3.ix1.p1.1.m1.2.3.2.1.cmml">lim</mo><mrow id="S6.I3.ix1.p1.1.m1.2.3.2.2" xref="S6.I3.ix1.p1.1.m1.2.3.2.2.cmml"><msubsup id="S6.I3.ix1.p1.1.m1.2.3.2.2.2" xref="S6.I3.ix1.p1.1.m1.2.3.2.2.2.cmml"><mi 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xref="S6.I3.ix1.p1.1.m1.2.2.2.2"><cn id="S6.I3.ix1.p1.1.m1.1.1.1.1.cmml" type="integer" xref="S6.I3.ix1.p1.1.m1.1.1.1.1">1</cn><apply id="S6.I3.ix1.p1.1.m1.2.2.2.2.1.cmml" xref="S6.I3.ix1.p1.1.m1.2.2.2.2.1"><plus id="S6.I3.ix1.p1.1.m1.2.2.2.2.1.1.cmml" xref="S6.I3.ix1.p1.1.m1.2.2.2.2.1"></plus><infinity id="S6.I3.ix1.p1.1.m1.2.2.2.2.1.2.cmml" xref="S6.I3.ix1.p1.1.m1.2.2.2.2.1.2"></infinity></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.ix1.p1.1.m1.2c">\lim w^{\prime}_{n}v^{\prime}_{n}={\bf z}_{[1,+\infty)}</annotation><annotation encoding="application/x-llamapun" id="S6.I3.ix1.p1.1.m1.2d">roman_lim italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_z start_POSTSUBSCRIPT [ 1 , + ∞ ) end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\lim u^{\prime}_{n}={\bf 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id="S6.I3.ix1.p1.2.m2.3.3.1.1.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1"><eq id="S6.I3.ix1.p1.2.m2.3.3.1.1.1.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.1"></eq><apply id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2"><limit id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.1.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2.1"></limit><apply id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.1.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2">subscript</csymbol><apply id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.2.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.2.1.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2">superscript</csymbol><ci id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.2.2.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.2.2">𝑢</ci><ci id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.2.3.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.2.3">′</ci></apply><ci id="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.3.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.2.2.3">𝑛</ci></apply></apply><apply id="S6.I3.ix1.p1.2.m2.3.3.1.1.3.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.3"><csymbol cd="ambiguous" id="S6.I3.ix1.p1.2.m2.3.3.1.1.3.1.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.3">subscript</csymbol><ci id="S6.I3.ix1.p1.2.m2.3.3.1.1.3.2.cmml" xref="S6.I3.ix1.p1.2.m2.3.3.1.1.3.2">𝐳</ci><interval closure="open-closed" id="S6.I3.ix1.p1.2.m2.2.2.2.3.cmml" xref="S6.I3.ix1.p1.2.m2.2.2.2.2"><apply id="S6.I3.ix1.p1.2.m2.2.2.2.2.1.cmml" xref="S6.I3.ix1.p1.2.m2.2.2.2.2.1"><minus id="S6.I3.ix1.p1.2.m2.2.2.2.2.1.1.cmml" xref="S6.I3.ix1.p1.2.m2.2.2.2.2.1"></minus><infinity id="S6.I3.ix1.p1.2.m2.2.2.2.2.1.2.cmml" xref="S6.I3.ix1.p1.2.m2.2.2.2.2.1.2"></infinity></apply><cn id="S6.I3.ix1.p1.2.m2.1.1.1.1.cmml" type="integer" xref="S6.I3.ix1.p1.2.m2.1.1.1.1">0</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I3.ix1.p1.2.m2.3c">\lim u^{\prime}_{n}={\bf z}_{(-\infty,0]}\,.</annotation><annotation encoding="application/x-llamapun" id="S6.I3.ix1.p1.2.m2.3d">roman_lim italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = bold_z start_POSTSUBSCRIPT ( - ∞ , 0 ] end_POSTSUBSCRIPT .</annotation></semantics></math></p> </div> </li> </ol> <p class="ltx_p" id="S6.3.p2.27">Again, these limits are meant to be equivalent to the statement</p> <ol class="ltx_enumerate" id="S6.I4"> <li class="ltx_item" id="S6.I4.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(6’)</span> <div class="ltx_para" id="S6.I4.ix1.p1"> <p class="ltx_p" id="S6.I4.ix1.p1.5"><math alttext="\lim{\bf z}(n)={\bf z}" class="ltx_Math" display="inline" id="S6.I4.ix1.p1.1.m1.1"><semantics id="S6.I4.ix1.p1.1.m1.1a"><mrow id="S6.I4.ix1.p1.1.m1.1.2" xref="S6.I4.ix1.p1.1.m1.1.2.cmml"><mrow id="S6.I4.ix1.p1.1.m1.1.2.2" xref="S6.I4.ix1.p1.1.m1.1.2.2.cmml"><mo id="S6.I4.ix1.p1.1.m1.1.2.2.1" rspace="0.167em" xref="S6.I4.ix1.p1.1.m1.1.2.2.1.cmml">lim</mo><mrow id="S6.I4.ix1.p1.1.m1.1.2.2.2" xref="S6.I4.ix1.p1.1.m1.1.2.2.2.cmml"><mi id="S6.I4.ix1.p1.1.m1.1.2.2.2.2" xref="S6.I4.ix1.p1.1.m1.1.2.2.2.2.cmml">𝐳</mi><mo id="S6.I4.ix1.p1.1.m1.1.2.2.2.1" xref="S6.I4.ix1.p1.1.m1.1.2.2.2.1.cmml">⁢</mo><mrow id="S6.I4.ix1.p1.1.m1.1.2.2.2.3.2" xref="S6.I4.ix1.p1.1.m1.1.2.2.2.cmml"><mo id="S6.I4.ix1.p1.1.m1.1.2.2.2.3.2.1" stretchy="false" xref="S6.I4.ix1.p1.1.m1.1.2.2.2.cmml">(</mo><mi id="S6.I4.ix1.p1.1.m1.1.1" xref="S6.I4.ix1.p1.1.m1.1.1.cmml">n</mi><mo id="S6.I4.ix1.p1.1.m1.1.2.2.2.3.2.2" stretchy="false" xref="S6.I4.ix1.p1.1.m1.1.2.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="S6.I4.ix1.p1.1.m1.1.2.1" xref="S6.I4.ix1.p1.1.m1.1.2.1.cmml">=</mo><mi id="S6.I4.ix1.p1.1.m1.1.2.3" xref="S6.I4.ix1.p1.1.m1.1.2.3.cmml">𝐳</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.ix1.p1.1.m1.1b"><apply id="S6.I4.ix1.p1.1.m1.1.2.cmml" xref="S6.I4.ix1.p1.1.m1.1.2"><eq id="S6.I4.ix1.p1.1.m1.1.2.1.cmml" xref="S6.I4.ix1.p1.1.m1.1.2.1"></eq><apply id="S6.I4.ix1.p1.1.m1.1.2.2.cmml" xref="S6.I4.ix1.p1.1.m1.1.2.2"><limit id="S6.I4.ix1.p1.1.m1.1.2.2.1.cmml" xref="S6.I4.ix1.p1.1.m1.1.2.2.1"></limit><apply id="S6.I4.ix1.p1.1.m1.1.2.2.2.cmml" xref="S6.I4.ix1.p1.1.m1.1.2.2.2"><times id="S6.I4.ix1.p1.1.m1.1.2.2.2.1.cmml" xref="S6.I4.ix1.p1.1.m1.1.2.2.2.1"></times><ci id="S6.I4.ix1.p1.1.m1.1.2.2.2.2.cmml" xref="S6.I4.ix1.p1.1.m1.1.2.2.2.2">𝐳</ci><ci id="S6.I4.ix1.p1.1.m1.1.1.cmml" xref="S6.I4.ix1.p1.1.m1.1.1">𝑛</ci></apply></apply><ci id="S6.I4.ix1.p1.1.m1.1.2.3.cmml" xref="S6.I4.ix1.p1.1.m1.1.2.3">𝐳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.ix1.p1.1.m1.1c">\lim{\bf z}(n)={\bf z}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.ix1.p1.1.m1.1d">roman_lim bold_z ( italic_n ) = bold_z</annotation></semantics></math> in <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.I4.ix1.p1.2.m2.1"><semantics id="S6.I4.ix1.p1.2.m2.1a"><msup id="S6.I4.ix1.p1.2.m2.1.1" xref="S6.I4.ix1.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.I4.ix1.p1.2.m2.1.1.2" xref="S6.I4.ix1.p1.2.m2.1.1.2.cmml">𝒜</mi><mi id="S6.I4.ix1.p1.2.m2.1.1.3" xref="S6.I4.ix1.p1.2.m2.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S6.I4.ix1.p1.2.m2.1b"><apply id="S6.I4.ix1.p1.2.m2.1.1.cmml" xref="S6.I4.ix1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S6.I4.ix1.p1.2.m2.1.1.1.cmml" xref="S6.I4.ix1.p1.2.m2.1.1">superscript</csymbol><ci id="S6.I4.ix1.p1.2.m2.1.1.2.cmml" xref="S6.I4.ix1.p1.2.m2.1.1.2">𝒜</ci><ci id="S6.I4.ix1.p1.2.m2.1.1.3.cmml" xref="S6.I4.ix1.p1.2.m2.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.ix1.p1.2.m2.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.ix1.p1.2.m2.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>, where the <math alttext="{\bf z}(n)\in\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.I4.ix1.p1.3.m3.1"><semantics id="S6.I4.ix1.p1.3.m3.1a"><mrow id="S6.I4.ix1.p1.3.m3.1.2" xref="S6.I4.ix1.p1.3.m3.1.2.cmml"><mrow id="S6.I4.ix1.p1.3.m3.1.2.2" xref="S6.I4.ix1.p1.3.m3.1.2.2.cmml"><mi id="S6.I4.ix1.p1.3.m3.1.2.2.2" xref="S6.I4.ix1.p1.3.m3.1.2.2.2.cmml">𝐳</mi><mo id="S6.I4.ix1.p1.3.m3.1.2.2.1" xref="S6.I4.ix1.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S6.I4.ix1.p1.3.m3.1.2.2.3.2" xref="S6.I4.ix1.p1.3.m3.1.2.2.cmml"><mo id="S6.I4.ix1.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="S6.I4.ix1.p1.3.m3.1.2.2.cmml">(</mo><mi id="S6.I4.ix1.p1.3.m3.1.1" xref="S6.I4.ix1.p1.3.m3.1.1.cmml">n</mi><mo id="S6.I4.ix1.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="S6.I4.ix1.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.I4.ix1.p1.3.m3.1.2.1" xref="S6.I4.ix1.p1.3.m3.1.2.1.cmml">∈</mo><msup id="S6.I4.ix1.p1.3.m3.1.2.3" xref="S6.I4.ix1.p1.3.m3.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.I4.ix1.p1.3.m3.1.2.3.2" xref="S6.I4.ix1.p1.3.m3.1.2.3.2.cmml">𝒜</mi><mi id="S6.I4.ix1.p1.3.m3.1.2.3.3" xref="S6.I4.ix1.p1.3.m3.1.2.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.I4.ix1.p1.3.m3.1b"><apply id="S6.I4.ix1.p1.3.m3.1.2.cmml" xref="S6.I4.ix1.p1.3.m3.1.2"><in id="S6.I4.ix1.p1.3.m3.1.2.1.cmml" xref="S6.I4.ix1.p1.3.m3.1.2.1"></in><apply id="S6.I4.ix1.p1.3.m3.1.2.2.cmml" xref="S6.I4.ix1.p1.3.m3.1.2.2"><times id="S6.I4.ix1.p1.3.m3.1.2.2.1.cmml" xref="S6.I4.ix1.p1.3.m3.1.2.2.1"></times><ci id="S6.I4.ix1.p1.3.m3.1.2.2.2.cmml" xref="S6.I4.ix1.p1.3.m3.1.2.2.2">𝐳</ci><ci id="S6.I4.ix1.p1.3.m3.1.1.cmml" xref="S6.I4.ix1.p1.3.m3.1.1">𝑛</ci></apply><apply id="S6.I4.ix1.p1.3.m3.1.2.3.cmml" xref="S6.I4.ix1.p1.3.m3.1.2.3"><csymbol cd="ambiguous" id="S6.I4.ix1.p1.3.m3.1.2.3.1.cmml" xref="S6.I4.ix1.p1.3.m3.1.2.3">superscript</csymbol><ci id="S6.I4.ix1.p1.3.m3.1.2.3.2.cmml" xref="S6.I4.ix1.p1.3.m3.1.2.3.2">𝒜</ci><ci id="S6.I4.ix1.p1.3.m3.1.2.3.3.cmml" xref="S6.I4.ix1.p1.3.m3.1.2.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.ix1.p1.3.m3.1c">{\bf z}(n)\in\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.ix1.p1.3.m3.1d">bold_z ( italic_n ) ∈ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> are defined via <math alttext="{\bf z}_{[1,\infty)}(n)=w^{\prime}_{n}{v^{\prime}_{n}}^{+\infty}" class="ltx_Math" display="inline" id="S6.I4.ix1.p1.4.m4.3"><semantics id="S6.I4.ix1.p1.4.m4.3a"><mrow id="S6.I4.ix1.p1.4.m4.3.4" xref="S6.I4.ix1.p1.4.m4.3.4.cmml"><mrow id="S6.I4.ix1.p1.4.m4.3.4.2" xref="S6.I4.ix1.p1.4.m4.3.4.2.cmml"><msub id="S6.I4.ix1.p1.4.m4.3.4.2.2" xref="S6.I4.ix1.p1.4.m4.3.4.2.2.cmml"><mi id="S6.I4.ix1.p1.4.m4.3.4.2.2.2" xref="S6.I4.ix1.p1.4.m4.3.4.2.2.2.cmml">𝐳</mi><mrow id="S6.I4.ix1.p1.4.m4.2.2.2.4" xref="S6.I4.ix1.p1.4.m4.2.2.2.3.cmml"><mo id="S6.I4.ix1.p1.4.m4.2.2.2.4.1" stretchy="false" xref="S6.I4.ix1.p1.4.m4.2.2.2.3.cmml">[</mo><mn id="S6.I4.ix1.p1.4.m4.1.1.1.1" xref="S6.I4.ix1.p1.4.m4.1.1.1.1.cmml">1</mn><mo id="S6.I4.ix1.p1.4.m4.2.2.2.4.2" xref="S6.I4.ix1.p1.4.m4.2.2.2.3.cmml">,</mo><mi id="S6.I4.ix1.p1.4.m4.2.2.2.2" mathvariant="normal" xref="S6.I4.ix1.p1.4.m4.2.2.2.2.cmml">∞</mi><mo id="S6.I4.ix1.p1.4.m4.2.2.2.4.3" stretchy="false" xref="S6.I4.ix1.p1.4.m4.2.2.2.3.cmml">)</mo></mrow></msub><mo id="S6.I4.ix1.p1.4.m4.3.4.2.1" xref="S6.I4.ix1.p1.4.m4.3.4.2.1.cmml">⁢</mo><mrow id="S6.I4.ix1.p1.4.m4.3.4.2.3.2" xref="S6.I4.ix1.p1.4.m4.3.4.2.cmml"><mo id="S6.I4.ix1.p1.4.m4.3.4.2.3.2.1" stretchy="false" xref="S6.I4.ix1.p1.4.m4.3.4.2.cmml">(</mo><mi id="S6.I4.ix1.p1.4.m4.3.3" xref="S6.I4.ix1.p1.4.m4.3.3.cmml">n</mi><mo id="S6.I4.ix1.p1.4.m4.3.4.2.3.2.2" stretchy="false" xref="S6.I4.ix1.p1.4.m4.3.4.2.cmml">)</mo></mrow></mrow><mo id="S6.I4.ix1.p1.4.m4.3.4.1" xref="S6.I4.ix1.p1.4.m4.3.4.1.cmml">=</mo><mrow id="S6.I4.ix1.p1.4.m4.3.4.3" xref="S6.I4.ix1.p1.4.m4.3.4.3.cmml"><msubsup id="S6.I4.ix1.p1.4.m4.3.4.3.2" xref="S6.I4.ix1.p1.4.m4.3.4.3.2.cmml"><mi 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id="S6.I4.ix1.p1.5.m5.3.4.3.3.cmml" xref="S6.I4.ix1.p1.5.m5.3.4.3.3"><minus id="S6.I4.ix1.p1.5.m5.3.4.3.3.1.cmml" xref="S6.I4.ix1.p1.5.m5.3.4.3.3"></minus><infinity id="S6.I4.ix1.p1.5.m5.3.4.3.3.2.cmml" xref="S6.I4.ix1.p1.5.m5.3.4.3.3.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I4.ix1.p1.5.m5.3c">{\bf z}_{(\infty,0]}(n)={u^{\prime}_{n}}^{-\infty}</annotation><annotation encoding="application/x-llamapun" id="S6.I4.ix1.p1.5.m5.3d">bold_z start_POSTSUBSCRIPT ( ∞ , 0 ] end_POSTSUBSCRIPT ( italic_n ) = italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S6.3.p2.15">Since by property (2) we have <math alttext="u^{\prime}_{n}w^{\prime}_{n}v^{\prime}_{n}\in\cal L(X)" class="ltx_Math" display="inline" id="S6.3.p2.9.m1.1"><semantics id="S6.3.p2.9.m1.1a"><mrow 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xref="S6.3.p2.9.m1.1.2.2.4.2.2.cmml">v</mi><mi id="S6.3.p2.9.m1.1.2.2.4.3" xref="S6.3.p2.9.m1.1.2.2.4.3.cmml">n</mi><mo id="S6.3.p2.9.m1.1.2.2.4.2.3" xref="S6.3.p2.9.m1.1.2.2.4.2.3.cmml">′</mo></msubsup></mrow><mo id="S6.3.p2.9.m1.1.2.1" xref="S6.3.p2.9.m1.1.2.1.cmml">∈</mo><mrow id="S6.3.p2.9.m1.1.2.3" xref="S6.3.p2.9.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.3.p2.9.m1.1.2.3.2" xref="S6.3.p2.9.m1.1.2.3.2.cmml">ℒ</mi><mo id="S6.3.p2.9.m1.1.2.3.1" xref="S6.3.p2.9.m1.1.2.3.1.cmml">⁢</mo><mrow id="S6.3.p2.9.m1.1.2.3.3.2" xref="S6.3.p2.9.m1.1.2.3.cmml"><mo id="S6.3.p2.9.m1.1.2.3.3.2.1" stretchy="false" xref="S6.3.p2.9.m1.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.3.p2.9.m1.1.1" xref="S6.3.p2.9.m1.1.1.cmml">𝒳</mi><mo id="S6.3.p2.9.m1.1.2.3.3.2.2" stretchy="false" xref="S6.3.p2.9.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.9.m1.1b"><apply id="S6.3.p2.9.m1.1.2.cmml" xref="S6.3.p2.9.m1.1.2"><in id="S6.3.p2.9.m1.1.2.1.cmml" xref="S6.3.p2.9.m1.1.2.1"></in><apply id="S6.3.p2.9.m1.1.2.2.cmml" xref="S6.3.p2.9.m1.1.2.2"><times id="S6.3.p2.9.m1.1.2.2.1.cmml" xref="S6.3.p2.9.m1.1.2.2.1"></times><apply id="S6.3.p2.9.m1.1.2.2.2.cmml" xref="S6.3.p2.9.m1.1.2.2.2"><csymbol cd="ambiguous" id="S6.3.p2.9.m1.1.2.2.2.1.cmml" xref="S6.3.p2.9.m1.1.2.2.2">subscript</csymbol><apply id="S6.3.p2.9.m1.1.2.2.2.2.cmml" xref="S6.3.p2.9.m1.1.2.2.2"><csymbol cd="ambiguous" id="S6.3.p2.9.m1.1.2.2.2.2.1.cmml" xref="S6.3.p2.9.m1.1.2.2.2">superscript</csymbol><ci id="S6.3.p2.9.m1.1.2.2.2.2.2.cmml" xref="S6.3.p2.9.m1.1.2.2.2.2.2">𝑢</ci><ci id="S6.3.p2.9.m1.1.2.2.2.2.3.cmml" xref="S6.3.p2.9.m1.1.2.2.2.2.3">′</ci></apply><ci id="S6.3.p2.9.m1.1.2.2.2.3.cmml" xref="S6.3.p2.9.m1.1.2.2.2.3">𝑛</ci></apply><apply id="S6.3.p2.9.m1.1.2.2.3.cmml" xref="S6.3.p2.9.m1.1.2.2.3"><csymbol cd="ambiguous" id="S6.3.p2.9.m1.1.2.2.3.1.cmml" xref="S6.3.p2.9.m1.1.2.2.3">subscript</csymbol><apply id="S6.3.p2.9.m1.1.2.2.3.2.cmml" xref="S6.3.p2.9.m1.1.2.2.3"><csymbol cd="ambiguous" id="S6.3.p2.9.m1.1.2.2.3.2.1.cmml" xref="S6.3.p2.9.m1.1.2.2.3">superscript</csymbol><ci id="S6.3.p2.9.m1.1.2.2.3.2.2.cmml" xref="S6.3.p2.9.m1.1.2.2.3.2.2">𝑤</ci><ci id="S6.3.p2.9.m1.1.2.2.3.2.3.cmml" xref="S6.3.p2.9.m1.1.2.2.3.2.3">′</ci></apply><ci id="S6.3.p2.9.m1.1.2.2.3.3.cmml" xref="S6.3.p2.9.m1.1.2.2.3.3">𝑛</ci></apply><apply id="S6.3.p2.9.m1.1.2.2.4.cmml" xref="S6.3.p2.9.m1.1.2.2.4"><csymbol cd="ambiguous" id="S6.3.p2.9.m1.1.2.2.4.1.cmml" xref="S6.3.p2.9.m1.1.2.2.4">subscript</csymbol><apply id="S6.3.p2.9.m1.1.2.2.4.2.cmml" xref="S6.3.p2.9.m1.1.2.2.4"><csymbol cd="ambiguous" id="S6.3.p2.9.m1.1.2.2.4.2.1.cmml" xref="S6.3.p2.9.m1.1.2.2.4">superscript</csymbol><ci id="S6.3.p2.9.m1.1.2.2.4.2.2.cmml" xref="S6.3.p2.9.m1.1.2.2.4.2.2">𝑣</ci><ci id="S6.3.p2.9.m1.1.2.2.4.2.3.cmml" xref="S6.3.p2.9.m1.1.2.2.4.2.3">′</ci></apply><ci id="S6.3.p2.9.m1.1.2.2.4.3.cmml" xref="S6.3.p2.9.m1.1.2.2.4.3">𝑛</ci></apply></apply><apply id="S6.3.p2.9.m1.1.2.3.cmml" xref="S6.3.p2.9.m1.1.2.3"><times id="S6.3.p2.9.m1.1.2.3.1.cmml" xref="S6.3.p2.9.m1.1.2.3.1"></times><ci id="S6.3.p2.9.m1.1.2.3.2.cmml" xref="S6.3.p2.9.m1.1.2.3.2">ℒ</ci><ci id="S6.3.p2.9.m1.1.1.cmml" xref="S6.3.p2.9.m1.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.9.m1.1c">u^{\prime}_{n}w^{\prime}_{n}v^{\prime}_{n}\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.9.m1.1d">italic_u start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math>, for all indices <math alttext="n" class="ltx_Math" display="inline" id="S6.3.p2.10.m2.1"><semantics id="S6.3.p2.10.m2.1a"><mi id="S6.3.p2.10.m2.1.1" xref="S6.3.p2.10.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.10.m2.1b"><ci id="S6.3.p2.10.m2.1.1.cmml" xref="S6.3.p2.10.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.10.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.10.m2.1d">italic_n</annotation></semantics></math> in the limits (6) or (6’), we obtain <math alttext="{\bf z}\in X" class="ltx_Math" display="inline" id="S6.3.p2.11.m3.1"><semantics id="S6.3.p2.11.m3.1a"><mrow id="S6.3.p2.11.m3.1.1" xref="S6.3.p2.11.m3.1.1.cmml"><mi id="S6.3.p2.11.m3.1.1.2" xref="S6.3.p2.11.m3.1.1.2.cmml">𝐳</mi><mo id="S6.3.p2.11.m3.1.1.1" xref="S6.3.p2.11.m3.1.1.1.cmml">∈</mo><mi id="S6.3.p2.11.m3.1.1.3" xref="S6.3.p2.11.m3.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.11.m3.1b"><apply id="S6.3.p2.11.m3.1.1.cmml" xref="S6.3.p2.11.m3.1.1"><in id="S6.3.p2.11.m3.1.1.1.cmml" xref="S6.3.p2.11.m3.1.1.1"></in><ci id="S6.3.p2.11.m3.1.1.2.cmml" xref="S6.3.p2.11.m3.1.1.2">𝐳</ci><ci id="S6.3.p2.11.m3.1.1.3.cmml" xref="S6.3.p2.11.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.11.m3.1c">{\bf z}\in X</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.11.m3.1d">bold_z ∈ italic_X</annotation></semantics></math>. Furthermore, from the above limit set-up we know that the factor <math alttext="{\bf z}_{[1,|w|]}" class="ltx_Math" display="inline" id="S6.3.p2.12.m4.3"><semantics id="S6.3.p2.12.m4.3a"><msub id="S6.3.p2.12.m4.3.4" xref="S6.3.p2.12.m4.3.4.cmml"><mi id="S6.3.p2.12.m4.3.4.2" xref="S6.3.p2.12.m4.3.4.2.cmml">𝐳</mi><mrow id="S6.3.p2.12.m4.3.3.3.3" xref="S6.3.p2.12.m4.3.3.3.4.cmml"><mo id="S6.3.p2.12.m4.3.3.3.3.2" stretchy="false" xref="S6.3.p2.12.m4.3.3.3.4.cmml">[</mo><mn id="S6.3.p2.12.m4.2.2.2.2" xref="S6.3.p2.12.m4.2.2.2.2.cmml">1</mn><mo id="S6.3.p2.12.m4.3.3.3.3.3" xref="S6.3.p2.12.m4.3.3.3.4.cmml">,</mo><mrow id="S6.3.p2.12.m4.3.3.3.3.1.2" xref="S6.3.p2.12.m4.3.3.3.3.1.1.cmml"><mo id="S6.3.p2.12.m4.3.3.3.3.1.2.1" stretchy="false" xref="S6.3.p2.12.m4.3.3.3.3.1.1.1.cmml">|</mo><mi id="S6.3.p2.12.m4.1.1.1.1" xref="S6.3.p2.12.m4.1.1.1.1.cmml">w</mi><mo id="S6.3.p2.12.m4.3.3.3.3.1.2.2" stretchy="false" xref="S6.3.p2.12.m4.3.3.3.3.1.1.1.cmml">|</mo></mrow><mo id="S6.3.p2.12.m4.3.3.3.3.4" stretchy="false" xref="S6.3.p2.12.m4.3.3.3.4.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.3.p2.12.m4.3b"><apply id="S6.3.p2.12.m4.3.4.cmml" xref="S6.3.p2.12.m4.3.4"><csymbol cd="ambiguous" id="S6.3.p2.12.m4.3.4.1.cmml" xref="S6.3.p2.12.m4.3.4">subscript</csymbol><ci id="S6.3.p2.12.m4.3.4.2.cmml" xref="S6.3.p2.12.m4.3.4.2">𝐳</ci><interval closure="closed" id="S6.3.p2.12.m4.3.3.3.4.cmml" xref="S6.3.p2.12.m4.3.3.3.3"><cn id="S6.3.p2.12.m4.2.2.2.2.cmml" type="integer" xref="S6.3.p2.12.m4.2.2.2.2">1</cn><apply id="S6.3.p2.12.m4.3.3.3.3.1.1.cmml" xref="S6.3.p2.12.m4.3.3.3.3.1.2"><abs id="S6.3.p2.12.m4.3.3.3.3.1.1.1.cmml" xref="S6.3.p2.12.m4.3.3.3.3.1.2.1"></abs><ci id="S6.3.p2.12.m4.1.1.1.1.cmml" xref="S6.3.p2.12.m4.1.1.1.1">𝑤</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.12.m4.3c">{\bf z}_{[1,|w|]}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.12.m4.3d">bold_z start_POSTSUBSCRIPT [ 1 , | italic_w | ] end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\bf z" class="ltx_Math" display="inline" id="S6.3.p2.13.m5.1"><semantics id="S6.3.p2.13.m5.1a"><mi id="S6.3.p2.13.m5.1.1" xref="S6.3.p2.13.m5.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.13.m5.1b"><ci id="S6.3.p2.13.m5.1.1.cmml" xref="S6.3.p2.13.m5.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.13.m5.1c">\bf z</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.13.m5.1d">bold_z</annotation></semantics></math> is equal to some <math alttext="w^{\prime}_{n}" class="ltx_Math" display="inline" id="S6.3.p2.14.m6.1"><semantics id="S6.3.p2.14.m6.1a"><msubsup id="S6.3.p2.14.m6.1.1" xref="S6.3.p2.14.m6.1.1.cmml"><mi id="S6.3.p2.14.m6.1.1.2.2" xref="S6.3.p2.14.m6.1.1.2.2.cmml">w</mi><mi id="S6.3.p2.14.m6.1.1.3" xref="S6.3.p2.14.m6.1.1.3.cmml">n</mi><mo id="S6.3.p2.14.m6.1.1.2.3" xref="S6.3.p2.14.m6.1.1.2.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S6.3.p2.14.m6.1b"><apply id="S6.3.p2.14.m6.1.1.cmml" xref="S6.3.p2.14.m6.1.1"><csymbol cd="ambiguous" id="S6.3.p2.14.m6.1.1.1.cmml" xref="S6.3.p2.14.m6.1.1">subscript</csymbol><apply id="S6.3.p2.14.m6.1.1.2.cmml" xref="S6.3.p2.14.m6.1.1"><csymbol cd="ambiguous" id="S6.3.p2.14.m6.1.1.2.1.cmml" xref="S6.3.p2.14.m6.1.1">superscript</csymbol><ci id="S6.3.p2.14.m6.1.1.2.2.cmml" xref="S6.3.p2.14.m6.1.1.2.2">𝑤</ci><ci id="S6.3.p2.14.m6.1.1.2.3.cmml" xref="S6.3.p2.14.m6.1.1.2.3">′</ci></apply><ci id="S6.3.p2.14.m6.1.1.3.cmml" xref="S6.3.p2.14.m6.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.14.m6.1c">w^{\prime}_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.14.m6.1d">italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, and thus (by property (4)) distinct from <math alttext="w" class="ltx_Math" display="inline" id="S6.3.p2.15.m7.1"><semantics id="S6.3.p2.15.m7.1a"><mi id="S6.3.p2.15.m7.1.1" xref="S6.3.p2.15.m7.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.15.m7.1b"><ci id="S6.3.p2.15.m7.1.1.cmml" xref="S6.3.p2.15.m7.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.15.m7.1c">w</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.15.m7.1d">italic_w</annotation></semantics></math>. We thus deduce</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\bf z}\neq{\bf x}\,." class="ltx_Math" display="block" id="S6.Ex4.m1.1"><semantics id="S6.Ex4.m1.1a"><mrow id="S6.Ex4.m1.1.1.1" xref="S6.Ex4.m1.1.1.1.1.cmml"><mrow id="S6.Ex4.m1.1.1.1.1" xref="S6.Ex4.m1.1.1.1.1.cmml"><mi id="S6.Ex4.m1.1.1.1.1.2" xref="S6.Ex4.m1.1.1.1.1.2.cmml">𝐳</mi><mo id="S6.Ex4.m1.1.1.1.1.1" xref="S6.Ex4.m1.1.1.1.1.1.cmml">≠</mo><mi id="S6.Ex4.m1.1.1.1.1.3" xref="S6.Ex4.m1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo id="S6.Ex4.m1.1.1.1.2" lspace="0.170em" xref="S6.Ex4.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex4.m1.1b"><apply id="S6.Ex4.m1.1.1.1.1.cmml" xref="S6.Ex4.m1.1.1.1"><neq id="S6.Ex4.m1.1.1.1.1.1.cmml" xref="S6.Ex4.m1.1.1.1.1.1"></neq><ci id="S6.Ex4.m1.1.1.1.1.2.cmml" xref="S6.Ex4.m1.1.1.1.1.2">𝐳</ci><ci id="S6.Ex4.m1.1.1.1.1.3.cmml" xref="S6.Ex4.m1.1.1.1.1.3">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex4.m1.1c">{\bf z}\neq{\bf x}\,.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex4.m1.1d">bold_z ≠ bold_x .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.3.p2.28">On the other hand, we obtain from property (3) directly</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma({\bf z})=\lim\sigma({\bf z}_{n})=\lim\sigma({\bf y}_{n})=\sigma({\bf x}% )\,." class="ltx_Math" display="block" id="S6.Ex5.m1.3"><semantics id="S6.Ex5.m1.3a"><mrow id="S6.Ex5.m1.3.3.1" xref="S6.Ex5.m1.3.3.1.1.cmml"><mrow id="S6.Ex5.m1.3.3.1.1" xref="S6.Ex5.m1.3.3.1.1.cmml"><mrow id="S6.Ex5.m1.3.3.1.1.4" xref="S6.Ex5.m1.3.3.1.1.4.cmml"><mi id="S6.Ex5.m1.3.3.1.1.4.2" xref="S6.Ex5.m1.3.3.1.1.4.2.cmml">σ</mi><mo id="S6.Ex5.m1.3.3.1.1.4.1" xref="S6.Ex5.m1.3.3.1.1.4.1.cmml">⁢</mo><mrow id="S6.Ex5.m1.3.3.1.1.4.3.2" xref="S6.Ex5.m1.3.3.1.1.4.cmml"><mo id="S6.Ex5.m1.3.3.1.1.4.3.2.1" stretchy="false" xref="S6.Ex5.m1.3.3.1.1.4.cmml">(</mo><mi id="S6.Ex5.m1.1.1" xref="S6.Ex5.m1.1.1.cmml">𝐳</mi><mo id="S6.Ex5.m1.3.3.1.1.4.3.2.2" stretchy="false" xref="S6.Ex5.m1.3.3.1.1.4.cmml">)</mo></mrow></mrow><mo id="S6.Ex5.m1.3.3.1.1.5" rspace="0.1389em" xref="S6.Ex5.m1.3.3.1.1.5.cmml">=</mo><mrow id="S6.Ex5.m1.3.3.1.1.1" xref="S6.Ex5.m1.3.3.1.1.1.cmml"><mo id="S6.Ex5.m1.3.3.1.1.1.2" lspace="0.1389em" movablelimits="false" rspace="0.167em" xref="S6.Ex5.m1.3.3.1.1.1.2.cmml">lim</mo><mrow id="S6.Ex5.m1.3.3.1.1.1.1" xref="S6.Ex5.m1.3.3.1.1.1.1.cmml"><mi id="S6.Ex5.m1.3.3.1.1.1.1.3" xref="S6.Ex5.m1.3.3.1.1.1.1.3.cmml">σ</mi><mo 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id="S6.Ex5.m1.3.3.1.1.1.cmml" xref="S6.Ex5.m1.3.3.1.1.1"><limit id="S6.Ex5.m1.3.3.1.1.1.2.cmml" xref="S6.Ex5.m1.3.3.1.1.1.2"></limit><apply id="S6.Ex5.m1.3.3.1.1.1.1.cmml" xref="S6.Ex5.m1.3.3.1.1.1.1"><times id="S6.Ex5.m1.3.3.1.1.1.1.2.cmml" xref="S6.Ex5.m1.3.3.1.1.1.1.2"></times><ci id="S6.Ex5.m1.3.3.1.1.1.1.3.cmml" xref="S6.Ex5.m1.3.3.1.1.1.1.3">𝜎</ci><apply id="S6.Ex5.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S6.Ex5.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex5.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex5.m1.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S6.Ex5.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.Ex5.m1.3.3.1.1.1.1.1.1.1.2">𝐳</ci><ci id="S6.Ex5.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S6.Ex5.m1.3.3.1.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></apply><apply id="S6.Ex5.m1.3.3.1.1c.cmml" xref="S6.Ex5.m1.3.3.1"><eq id="S6.Ex5.m1.3.3.1.1.6.cmml" xref="S6.Ex5.m1.3.3.1.1.6"></eq><share href="https://arxiv.org/html/2211.11234v4#S6.Ex5.m1.3.3.1.1.1.cmml" id="S6.Ex5.m1.3.3.1.1d.cmml" 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id="S6.Ex5.m1.3.3.1.1f.cmml" xref="S6.Ex5.m1.3.3.1"></share><apply id="S6.Ex5.m1.3.3.1.1.8.cmml" xref="S6.Ex5.m1.3.3.1.1.8"><times id="S6.Ex5.m1.3.3.1.1.8.1.cmml" xref="S6.Ex5.m1.3.3.1.1.8.1"></times><ci id="S6.Ex5.m1.3.3.1.1.8.2.cmml" xref="S6.Ex5.m1.3.3.1.1.8.2">𝜎</ci><ci id="S6.Ex5.m1.2.2.cmml" xref="S6.Ex5.m1.2.2">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex5.m1.3c">\sigma({\bf z})=\lim\sigma({\bf z}_{n})=\lim\sigma({\bf y}_{n})=\sigma({\bf x}% )\,.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex5.m1.3d">italic_σ ( bold_z ) = roman_lim italic_σ ( bold_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = roman_lim italic_σ ( bold_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_σ ( bold_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.3.p2.26">We now use the assumption that <math alttext="\sigma" class="ltx_Math" display="inline" id="S6.3.p2.16.m1.1"><semantics id="S6.3.p2.16.m1.1a"><mi id="S6.3.p2.16.m1.1.1" xref="S6.3.p2.16.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.16.m1.1b"><ci id="S6.3.p2.16.m1.1.1.cmml" xref="S6.3.p2.16.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.16.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.16.m1.1d">italic_σ</annotation></semantics></math> is shift-orbit injective in <math alttext="X" class="ltx_Math" display="inline" id="S6.3.p2.17.m2.1"><semantics id="S6.3.p2.17.m2.1a"><mi id="S6.3.p2.17.m2.1.1" xref="S6.3.p2.17.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.17.m2.1b"><ci id="S6.3.p2.17.m2.1.1.cmml" xref="S6.3.p2.17.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.17.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.17.m2.1d">italic_X</annotation></semantics></math> to deduce that <math alttext="\bf x" class="ltx_Math" display="inline" id="S6.3.p2.18.m3.1"><semantics id="S6.3.p2.18.m3.1a"><mi id="S6.3.p2.18.m3.1.1" xref="S6.3.p2.18.m3.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.18.m3.1b"><ci id="S6.3.p2.18.m3.1.1.cmml" xref="S6.3.p2.18.m3.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.18.m3.1c">\bf x</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.18.m3.1d">bold_x</annotation></semantics></math> and <math alttext="\bf z" class="ltx_Math" display="inline" id="S6.3.p2.19.m4.1"><semantics id="S6.3.p2.19.m4.1a"><mi id="S6.3.p2.19.m4.1.1" xref="S6.3.p2.19.m4.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.19.m4.1b"><ci id="S6.3.p2.19.m4.1.1.cmml" xref="S6.3.p2.19.m4.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.19.m4.1c">\bf z</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.19.m4.1d">bold_z</annotation></semantics></math> are shift-translates of each other: There exists some integer <math alttext="k\neq 0" class="ltx_Math" display="inline" id="S6.3.p2.20.m5.1"><semantics id="S6.3.p2.20.m5.1a"><mrow id="S6.3.p2.20.m5.1.1" xref="S6.3.p2.20.m5.1.1.cmml"><mi id="S6.3.p2.20.m5.1.1.2" xref="S6.3.p2.20.m5.1.1.2.cmml">k</mi><mo id="S6.3.p2.20.m5.1.1.1" xref="S6.3.p2.20.m5.1.1.1.cmml">≠</mo><mn id="S6.3.p2.20.m5.1.1.3" xref="S6.3.p2.20.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.20.m5.1b"><apply id="S6.3.p2.20.m5.1.1.cmml" xref="S6.3.p2.20.m5.1.1"><neq id="S6.3.p2.20.m5.1.1.1.cmml" xref="S6.3.p2.20.m5.1.1.1"></neq><ci id="S6.3.p2.20.m5.1.1.2.cmml" xref="S6.3.p2.20.m5.1.1.2">𝑘</ci><cn id="S6.3.p2.20.m5.1.1.3.cmml" type="integer" xref="S6.3.p2.20.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.20.m5.1c">k\neq 0</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.20.m5.1d">italic_k ≠ 0</annotation></semantics></math> such that <math alttext="T^{k}({\bf z})={\bf x}" class="ltx_Math" display="inline" id="S6.3.p2.21.m6.1"><semantics id="S6.3.p2.21.m6.1a"><mrow id="S6.3.p2.21.m6.1.2" xref="S6.3.p2.21.m6.1.2.cmml"><mrow id="S6.3.p2.21.m6.1.2.2" xref="S6.3.p2.21.m6.1.2.2.cmml"><msup id="S6.3.p2.21.m6.1.2.2.2" xref="S6.3.p2.21.m6.1.2.2.2.cmml"><mi id="S6.3.p2.21.m6.1.2.2.2.2" xref="S6.3.p2.21.m6.1.2.2.2.2.cmml">T</mi><mi id="S6.3.p2.21.m6.1.2.2.2.3" xref="S6.3.p2.21.m6.1.2.2.2.3.cmml">k</mi></msup><mo id="S6.3.p2.21.m6.1.2.2.1" xref="S6.3.p2.21.m6.1.2.2.1.cmml">⁢</mo><mrow id="S6.3.p2.21.m6.1.2.2.3.2" xref="S6.3.p2.21.m6.1.2.2.cmml"><mo id="S6.3.p2.21.m6.1.2.2.3.2.1" stretchy="false" xref="S6.3.p2.21.m6.1.2.2.cmml">(</mo><mi id="S6.3.p2.21.m6.1.1" xref="S6.3.p2.21.m6.1.1.cmml">𝐳</mi><mo id="S6.3.p2.21.m6.1.2.2.3.2.2" stretchy="false" xref="S6.3.p2.21.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.3.p2.21.m6.1.2.1" xref="S6.3.p2.21.m6.1.2.1.cmml">=</mo><mi id="S6.3.p2.21.m6.1.2.3" xref="S6.3.p2.21.m6.1.2.3.cmml">𝐱</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.21.m6.1b"><apply id="S6.3.p2.21.m6.1.2.cmml" xref="S6.3.p2.21.m6.1.2"><eq id="S6.3.p2.21.m6.1.2.1.cmml" xref="S6.3.p2.21.m6.1.2.1"></eq><apply id="S6.3.p2.21.m6.1.2.2.cmml" xref="S6.3.p2.21.m6.1.2.2"><times id="S6.3.p2.21.m6.1.2.2.1.cmml" xref="S6.3.p2.21.m6.1.2.2.1"></times><apply id="S6.3.p2.21.m6.1.2.2.2.cmml" xref="S6.3.p2.21.m6.1.2.2.2"><csymbol cd="ambiguous" id="S6.3.p2.21.m6.1.2.2.2.1.cmml" xref="S6.3.p2.21.m6.1.2.2.2">superscript</csymbol><ci id="S6.3.p2.21.m6.1.2.2.2.2.cmml" xref="S6.3.p2.21.m6.1.2.2.2.2">𝑇</ci><ci id="S6.3.p2.21.m6.1.2.2.2.3.cmml" xref="S6.3.p2.21.m6.1.2.2.2.3">𝑘</ci></apply><ci id="S6.3.p2.21.m6.1.1.cmml" xref="S6.3.p2.21.m6.1.1">𝐳</ci></apply><ci id="S6.3.p2.21.m6.1.2.3.cmml" xref="S6.3.p2.21.m6.1.2.3">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.21.m6.1c">T^{k}({\bf z})={\bf x}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.21.m6.1d">italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( bold_z ) = bold_x</annotation></semantics></math>. We thus have <math alttext="T^{k}(\sigma({\bf x}))=T^{k}(\sigma({\bf z}))=\sigma(T^{k}({\bf z}))=\sigma({% \bf x})" class="ltx_Math" display="inline" id="S6.3.p2.22.m7.7"><semantics id="S6.3.p2.22.m7.7a"><mrow id="S6.3.p2.22.m7.7.7" xref="S6.3.p2.22.m7.7.7.cmml"><mrow id="S6.3.p2.22.m7.5.5.1" xref="S6.3.p2.22.m7.5.5.1.cmml"><msup id="S6.3.p2.22.m7.5.5.1.3" xref="S6.3.p2.22.m7.5.5.1.3.cmml"><mi id="S6.3.p2.22.m7.5.5.1.3.2" xref="S6.3.p2.22.m7.5.5.1.3.2.cmml">T</mi><mi id="S6.3.p2.22.m7.5.5.1.3.3" xref="S6.3.p2.22.m7.5.5.1.3.3.cmml">k</mi></msup><mo id="S6.3.p2.22.m7.5.5.1.2" xref="S6.3.p2.22.m7.5.5.1.2.cmml">⁢</mo><mrow id="S6.3.p2.22.m7.5.5.1.1.1" xref="S6.3.p2.22.m7.5.5.1.1.1.1.cmml"><mo id="S6.3.p2.22.m7.5.5.1.1.1.2" stretchy="false" xref="S6.3.p2.22.m7.5.5.1.1.1.1.cmml">(</mo><mrow id="S6.3.p2.22.m7.5.5.1.1.1.1" xref="S6.3.p2.22.m7.5.5.1.1.1.1.cmml"><mi id="S6.3.p2.22.m7.5.5.1.1.1.1.2" xref="S6.3.p2.22.m7.5.5.1.1.1.1.2.cmml">σ</mi><mo id="S6.3.p2.22.m7.5.5.1.1.1.1.1" xref="S6.3.p2.22.m7.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.3.p2.22.m7.5.5.1.1.1.1.3.2" xref="S6.3.p2.22.m7.5.5.1.1.1.1.cmml"><mo id="S6.3.p2.22.m7.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S6.3.p2.22.m7.5.5.1.1.1.1.cmml">(</mo><mi id="S6.3.p2.22.m7.1.1" xref="S6.3.p2.22.m7.1.1.cmml">𝐱</mi><mo id="S6.3.p2.22.m7.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S6.3.p2.22.m7.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.3.p2.22.m7.5.5.1.1.1.3" stretchy="false" xref="S6.3.p2.22.m7.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.3.p2.22.m7.7.7.5" xref="S6.3.p2.22.m7.7.7.5.cmml">=</mo><mrow id="S6.3.p2.22.m7.6.6.2" xref="S6.3.p2.22.m7.6.6.2.cmml"><msup id="S6.3.p2.22.m7.6.6.2.3" xref="S6.3.p2.22.m7.6.6.2.3.cmml"><mi id="S6.3.p2.22.m7.6.6.2.3.2" xref="S6.3.p2.22.m7.6.6.2.3.2.cmml">T</mi><mi id="S6.3.p2.22.m7.6.6.2.3.3" xref="S6.3.p2.22.m7.6.6.2.3.3.cmml">k</mi></msup><mo id="S6.3.p2.22.m7.6.6.2.2" xref="S6.3.p2.22.m7.6.6.2.2.cmml">⁢</mo><mrow id="S6.3.p2.22.m7.6.6.2.1.1" xref="S6.3.p2.22.m7.6.6.2.1.1.1.cmml"><mo id="S6.3.p2.22.m7.6.6.2.1.1.2" stretchy="false" xref="S6.3.p2.22.m7.6.6.2.1.1.1.cmml">(</mo><mrow id="S6.3.p2.22.m7.6.6.2.1.1.1" xref="S6.3.p2.22.m7.6.6.2.1.1.1.cmml"><mi id="S6.3.p2.22.m7.6.6.2.1.1.1.2" xref="S6.3.p2.22.m7.6.6.2.1.1.1.2.cmml">σ</mi><mo id="S6.3.p2.22.m7.6.6.2.1.1.1.1" xref="S6.3.p2.22.m7.6.6.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.3.p2.22.m7.6.6.2.1.1.1.3.2" xref="S6.3.p2.22.m7.6.6.2.1.1.1.cmml"><mo id="S6.3.p2.22.m7.6.6.2.1.1.1.3.2.1" stretchy="false" xref="S6.3.p2.22.m7.6.6.2.1.1.1.cmml">(</mo><mi id="S6.3.p2.22.m7.2.2" xref="S6.3.p2.22.m7.2.2.cmml">𝐳</mi><mo id="S6.3.p2.22.m7.6.6.2.1.1.1.3.2.2" stretchy="false" xref="S6.3.p2.22.m7.6.6.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.3.p2.22.m7.6.6.2.1.1.3" stretchy="false" xref="S6.3.p2.22.m7.6.6.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.3.p2.22.m7.7.7.6" xref="S6.3.p2.22.m7.7.7.6.cmml">=</mo><mrow id="S6.3.p2.22.m7.7.7.3" xref="S6.3.p2.22.m7.7.7.3.cmml"><mi id="S6.3.p2.22.m7.7.7.3.3" xref="S6.3.p2.22.m7.7.7.3.3.cmml">σ</mi><mo id="S6.3.p2.22.m7.7.7.3.2" xref="S6.3.p2.22.m7.7.7.3.2.cmml">⁢</mo><mrow id="S6.3.p2.22.m7.7.7.3.1.1" xref="S6.3.p2.22.m7.7.7.3.1.1.1.cmml"><mo id="S6.3.p2.22.m7.7.7.3.1.1.2" stretchy="false" xref="S6.3.p2.22.m7.7.7.3.1.1.1.cmml">(</mo><mrow id="S6.3.p2.22.m7.7.7.3.1.1.1" xref="S6.3.p2.22.m7.7.7.3.1.1.1.cmml"><msup id="S6.3.p2.22.m7.7.7.3.1.1.1.2" xref="S6.3.p2.22.m7.7.7.3.1.1.1.2.cmml"><mi id="S6.3.p2.22.m7.7.7.3.1.1.1.2.2" xref="S6.3.p2.22.m7.7.7.3.1.1.1.2.2.cmml">T</mi><mi id="S6.3.p2.22.m7.7.7.3.1.1.1.2.3" xref="S6.3.p2.22.m7.7.7.3.1.1.1.2.3.cmml">k</mi></msup><mo id="S6.3.p2.22.m7.7.7.3.1.1.1.1" xref="S6.3.p2.22.m7.7.7.3.1.1.1.1.cmml">⁢</mo><mrow id="S6.3.p2.22.m7.7.7.3.1.1.1.3.2" xref="S6.3.p2.22.m7.7.7.3.1.1.1.cmml"><mo id="S6.3.p2.22.m7.7.7.3.1.1.1.3.2.1" stretchy="false" xref="S6.3.p2.22.m7.7.7.3.1.1.1.cmml">(</mo><mi id="S6.3.p2.22.m7.3.3" xref="S6.3.p2.22.m7.3.3.cmml">𝐳</mi><mo 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xref="S6.3.p2.22.m7.7.7.7"></eq><share href="https://arxiv.org/html/2211.11234v4#S6.3.p2.22.m7.7.7.3.cmml" id="S6.3.p2.22.m7.7.7f.cmml" xref="S6.3.p2.22.m7.7.7"></share><apply id="S6.3.p2.22.m7.7.7.8.cmml" xref="S6.3.p2.22.m7.7.7.8"><times id="S6.3.p2.22.m7.7.7.8.1.cmml" xref="S6.3.p2.22.m7.7.7.8.1"></times><ci id="S6.3.p2.22.m7.7.7.8.2.cmml" xref="S6.3.p2.22.m7.7.7.8.2">𝜎</ci><ci id="S6.3.p2.22.m7.4.4.cmml" xref="S6.3.p2.22.m7.4.4">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.22.m7.7c">T^{k}(\sigma({\bf x}))=T^{k}(\sigma({\bf z}))=\sigma(T^{k}({\bf z}))=\sigma({% \bf x})</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.22.m7.7d">italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_σ ( bold_x ) ) = italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( italic_σ ( bold_z ) ) = italic_σ ( italic_T start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( bold_z ) ) = italic_σ ( bold_x )</annotation></semantics></math>. But any biinfinite word which is equal to a non-trivial shift-translate of itself must be periodic. Hence <math alttext="\sigma({\bf x})" class="ltx_Math" display="inline" id="S6.3.p2.23.m8.1"><semantics id="S6.3.p2.23.m8.1a"><mrow id="S6.3.p2.23.m8.1.2" xref="S6.3.p2.23.m8.1.2.cmml"><mi id="S6.3.p2.23.m8.1.2.2" xref="S6.3.p2.23.m8.1.2.2.cmml">σ</mi><mo id="S6.3.p2.23.m8.1.2.1" xref="S6.3.p2.23.m8.1.2.1.cmml">⁢</mo><mrow id="S6.3.p2.23.m8.1.2.3.2" xref="S6.3.p2.23.m8.1.2.cmml"><mo id="S6.3.p2.23.m8.1.2.3.2.1" stretchy="false" xref="S6.3.p2.23.m8.1.2.cmml">(</mo><mi id="S6.3.p2.23.m8.1.1" xref="S6.3.p2.23.m8.1.1.cmml">𝐱</mi><mo id="S6.3.p2.23.m8.1.2.3.2.2" stretchy="false" xref="S6.3.p2.23.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.3.p2.23.m8.1b"><apply id="S6.3.p2.23.m8.1.2.cmml" xref="S6.3.p2.23.m8.1.2"><times id="S6.3.p2.23.m8.1.2.1.cmml" xref="S6.3.p2.23.m8.1.2.1"></times><ci id="S6.3.p2.23.m8.1.2.2.cmml" xref="S6.3.p2.23.m8.1.2.2">𝜎</ci><ci id="S6.3.p2.23.m8.1.1.cmml" xref="S6.3.p2.23.m8.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.23.m8.1c">\sigma({\bf x})</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.23.m8.1d">italic_σ ( bold_x )</annotation></semantics></math> is periodic, and now we can employ Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm1" title="Lemma 6.1. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.1</span></a> in order to deduce that the biinfinite word <math alttext="{\bf x}" class="ltx_Math" display="inline" id="S6.3.p2.24.m9.1"><semantics id="S6.3.p2.24.m9.1a"><mi id="S6.3.p2.24.m9.1.1" xref="S6.3.p2.24.m9.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S6.3.p2.24.m9.1b"><ci id="S6.3.p2.24.m9.1.1.cmml" xref="S6.3.p2.24.m9.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.24.m9.1c">{\bf x}</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.24.m9.1d">bold_x</annotation></semantics></math> is periodic. <span class="ltx_text ltx_inline-block" id="S6.3.p2.25.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S6.3.p2.25.1.m1.1"><semantics id="S6.3.p2.25.1.m1.1a"><mo id="S6.3.p2.25.1.m1.1.1" xref="S6.3.p2.25.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S6.3.p2.25.1.m1.1b"><csymbol cd="latexml" id="S6.3.p2.25.1.m1.1.1.cmml" xref="S6.3.p2.25.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.25.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.25.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S6.3.p2.26.m10.1"><semantics id="S6.3.p2.26.m10.1a"><mo id="S6.3.p2.26.m10.1.1" xref="S6.3.p2.26.m10.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S6.3.p2.26.m10.1b"><csymbol cd="latexml" id="S6.3.p2.26.m10.1.1.cmml" xref="S6.3.p2.26.m10.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.3.p2.26.m10.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S6.3.p2.26.m10.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S6.p4"> <p class="ltx_p" id="S6.p4.8">We denote by <math alttext="\text{\rm Per}(X)" class="ltx_Math" display="inline" id="S6.p4.1.m1.1"><semantics id="S6.p4.1.m1.1a"><mrow id="S6.p4.1.m1.1.2" xref="S6.p4.1.m1.1.2.cmml"><mtext id="S6.p4.1.m1.1.2.2" xref="S6.p4.1.m1.1.2.2a.cmml">Per</mtext><mo id="S6.p4.1.m1.1.2.1" xref="S6.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S6.p4.1.m1.1.2.3.2" xref="S6.p4.1.m1.1.2.cmml"><mo id="S6.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S6.p4.1.m1.1.2.cmml">(</mo><mi id="S6.p4.1.m1.1.1" xref="S6.p4.1.m1.1.1.cmml">X</mi><mo id="S6.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S6.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.1.m1.1b"><apply id="S6.p4.1.m1.1.2.cmml" xref="S6.p4.1.m1.1.2"><times id="S6.p4.1.m1.1.2.1.cmml" xref="S6.p4.1.m1.1.2.1"></times><ci id="S6.p4.1.m1.1.2.2a.cmml" xref="S6.p4.1.m1.1.2.2"><mtext id="S6.p4.1.m1.1.2.2.cmml" xref="S6.p4.1.m1.1.2.2">Per</mtext></ci><ci id="S6.p4.1.m1.1.1.cmml" xref="S6.p4.1.m1.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.1.m1.1c">\text{\rm Per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p4.1.m1.1d">Per ( italic_X )</annotation></semantics></math> the subset of all biinfinite periodic words in <math alttext="X" class="ltx_Math" display="inline" id="S6.p4.2.m2.1"><semantics id="S6.p4.2.m2.1a"><mi id="S6.p4.2.m2.1.1" xref="S6.p4.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.p4.2.m2.1b"><ci id="S6.p4.2.m2.1.1.cmml" xref="S6.p4.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.p4.2.m2.1d">italic_X</annotation></semantics></math>. We observe that both, the countable set <math alttext="\text{\rm Per}(X)" class="ltx_Math" display="inline" id="S6.p4.3.m3.1"><semantics id="S6.p4.3.m3.1a"><mrow id="S6.p4.3.m3.1.2" xref="S6.p4.3.m3.1.2.cmml"><mtext id="S6.p4.3.m3.1.2.2" xref="S6.p4.3.m3.1.2.2a.cmml">Per</mtext><mo id="S6.p4.3.m3.1.2.1" xref="S6.p4.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.p4.3.m3.1.2.3.2" xref="S6.p4.3.m3.1.2.cmml"><mo id="S6.p4.3.m3.1.2.3.2.1" stretchy="false" xref="S6.p4.3.m3.1.2.cmml">(</mo><mi id="S6.p4.3.m3.1.1" xref="S6.p4.3.m3.1.1.cmml">X</mi><mo id="S6.p4.3.m3.1.2.3.2.2" stretchy="false" xref="S6.p4.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.3.m3.1b"><apply id="S6.p4.3.m3.1.2.cmml" xref="S6.p4.3.m3.1.2"><times id="S6.p4.3.m3.1.2.1.cmml" xref="S6.p4.3.m3.1.2.1"></times><ci id="S6.p4.3.m3.1.2.2a.cmml" xref="S6.p4.3.m3.1.2.2"><mtext id="S6.p4.3.m3.1.2.2.cmml" xref="S6.p4.3.m3.1.2.2">Per</mtext></ci><ci id="S6.p4.3.m3.1.1.cmml" xref="S6.p4.3.m3.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.3.m3.1c">\text{\rm Per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p4.3.m3.1d">Per ( italic_X )</annotation></semantics></math> and its complement <math alttext="X\smallsetminus\text{\rm Per}(X)" class="ltx_Math" display="inline" id="S6.p4.4.m4.1"><semantics id="S6.p4.4.m4.1a"><mrow id="S6.p4.4.m4.1.2" xref="S6.p4.4.m4.1.2.cmml"><mi id="S6.p4.4.m4.1.2.2" xref="S6.p4.4.m4.1.2.2.cmml">X</mi><mo id="S6.p4.4.m4.1.2.1" xref="S6.p4.4.m4.1.2.1.cmml">∖</mo><mrow id="S6.p4.4.m4.1.2.3" xref="S6.p4.4.m4.1.2.3.cmml"><mtext id="S6.p4.4.m4.1.2.3.2" xref="S6.p4.4.m4.1.2.3.2a.cmml">Per</mtext><mo id="S6.p4.4.m4.1.2.3.1" xref="S6.p4.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S6.p4.4.m4.1.2.3.3.2" xref="S6.p4.4.m4.1.2.3.cmml"><mo id="S6.p4.4.m4.1.2.3.3.2.1" stretchy="false" xref="S6.p4.4.m4.1.2.3.cmml">(</mo><mi id="S6.p4.4.m4.1.1" xref="S6.p4.4.m4.1.1.cmml">X</mi><mo id="S6.p4.4.m4.1.2.3.3.2.2" stretchy="false" xref="S6.p4.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.4.m4.1b"><apply id="S6.p4.4.m4.1.2.cmml" xref="S6.p4.4.m4.1.2"><setdiff id="S6.p4.4.m4.1.2.1.cmml" xref="S6.p4.4.m4.1.2.1"></setdiff><ci id="S6.p4.4.m4.1.2.2.cmml" xref="S6.p4.4.m4.1.2.2">𝑋</ci><apply id="S6.p4.4.m4.1.2.3.cmml" xref="S6.p4.4.m4.1.2.3"><times id="S6.p4.4.m4.1.2.3.1.cmml" xref="S6.p4.4.m4.1.2.3.1"></times><ci id="S6.p4.4.m4.1.2.3.2a.cmml" xref="S6.p4.4.m4.1.2.3.2"><mtext id="S6.p4.4.m4.1.2.3.2.cmml" xref="S6.p4.4.m4.1.2.3.2">Per</mtext></ci><ci id="S6.p4.4.m4.1.1.cmml" xref="S6.p4.4.m4.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.4.m4.1c">X\smallsetminus\text{\rm Per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p4.4.m4.1d">italic_X ∖ Per ( italic_X )</annotation></semantics></math> are measurable subsets of <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.p4.5.m5.1"><semantics id="S6.p4.5.m5.1a"><msup id="S6.p4.5.m5.1.1" xref="S6.p4.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p4.5.m5.1.1.2" xref="S6.p4.5.m5.1.1.2.cmml">𝒜</mi><mi id="S6.p4.5.m5.1.1.3" xref="S6.p4.5.m5.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S6.p4.5.m5.1b"><apply id="S6.p4.5.m5.1.1.cmml" xref="S6.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S6.p4.5.m5.1.1.1.cmml" xref="S6.p4.5.m5.1.1">superscript</csymbol><ci id="S6.p4.5.m5.1.1.2.cmml" xref="S6.p4.5.m5.1.1.2">𝒜</ci><ci id="S6.p4.5.m5.1.1.3.cmml" xref="S6.p4.5.m5.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.5.m5.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.5.m5.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>. Any measure <math alttext="\mu\in\cal M(X)" class="ltx_Math" display="inline" id="S6.p4.6.m6.1"><semantics id="S6.p4.6.m6.1a"><mrow id="S6.p4.6.m6.1.2" xref="S6.p4.6.m6.1.2.cmml"><mi id="S6.p4.6.m6.1.2.2" xref="S6.p4.6.m6.1.2.2.cmml">μ</mi><mo id="S6.p4.6.m6.1.2.1" xref="S6.p4.6.m6.1.2.1.cmml">∈</mo><mrow id="S6.p4.6.m6.1.2.3" xref="S6.p4.6.m6.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.p4.6.m6.1.2.3.2" xref="S6.p4.6.m6.1.2.3.2.cmml">ℳ</mi><mo id="S6.p4.6.m6.1.2.3.1" xref="S6.p4.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S6.p4.6.m6.1.2.3.3.2" xref="S6.p4.6.m6.1.2.3.cmml"><mo id="S6.p4.6.m6.1.2.3.3.2.1" stretchy="false" xref="S6.p4.6.m6.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.p4.6.m6.1.1" xref="S6.p4.6.m6.1.1.cmml">𝒳</mi><mo id="S6.p4.6.m6.1.2.3.3.2.2" stretchy="false" xref="S6.p4.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.6.m6.1b"><apply id="S6.p4.6.m6.1.2.cmml" xref="S6.p4.6.m6.1.2"><in id="S6.p4.6.m6.1.2.1.cmml" xref="S6.p4.6.m6.1.2.1"></in><ci id="S6.p4.6.m6.1.2.2.cmml" xref="S6.p4.6.m6.1.2.2">𝜇</ci><apply id="S6.p4.6.m6.1.2.3.cmml" xref="S6.p4.6.m6.1.2.3"><times id="S6.p4.6.m6.1.2.3.1.cmml" xref="S6.p4.6.m6.1.2.3.1"></times><ci id="S6.p4.6.m6.1.2.3.2.cmml" xref="S6.p4.6.m6.1.2.3.2">ℳ</ci><ci id="S6.p4.6.m6.1.1.cmml" xref="S6.p4.6.m6.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.6.m6.1c">\mu\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p4.6.m6.1d">italic_μ ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math> satisfies <math alttext="\mu(\text{\rm Per}(X))=0" class="ltx_Math" display="inline" id="S6.p4.7.m7.2"><semantics id="S6.p4.7.m7.2a"><mrow id="S6.p4.7.m7.2.2" xref="S6.p4.7.m7.2.2.cmml"><mrow id="S6.p4.7.m7.2.2.1" xref="S6.p4.7.m7.2.2.1.cmml"><mi id="S6.p4.7.m7.2.2.1.3" xref="S6.p4.7.m7.2.2.1.3.cmml">μ</mi><mo id="S6.p4.7.m7.2.2.1.2" xref="S6.p4.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S6.p4.7.m7.2.2.1.1.1" xref="S6.p4.7.m7.2.2.1.1.1.1.cmml"><mo id="S6.p4.7.m7.2.2.1.1.1.2" stretchy="false" xref="S6.p4.7.m7.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.p4.7.m7.2.2.1.1.1.1" xref="S6.p4.7.m7.2.2.1.1.1.1.cmml"><mtext id="S6.p4.7.m7.2.2.1.1.1.1.2" xref="S6.p4.7.m7.2.2.1.1.1.1.2a.cmml">Per</mtext><mo id="S6.p4.7.m7.2.2.1.1.1.1.1" xref="S6.p4.7.m7.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.p4.7.m7.2.2.1.1.1.1.3.2" xref="S6.p4.7.m7.2.2.1.1.1.1.cmml"><mo id="S6.p4.7.m7.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S6.p4.7.m7.2.2.1.1.1.1.cmml">(</mo><mi id="S6.p4.7.m7.1.1" xref="S6.p4.7.m7.1.1.cmml">X</mi><mo id="S6.p4.7.m7.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S6.p4.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.p4.7.m7.2.2.1.1.1.3" stretchy="false" xref="S6.p4.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.p4.7.m7.2.2.2" xref="S6.p4.7.m7.2.2.2.cmml">=</mo><mn id="S6.p4.7.m7.2.2.3" xref="S6.p4.7.m7.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.7.m7.2b"><apply id="S6.p4.7.m7.2.2.cmml" xref="S6.p4.7.m7.2.2"><eq id="S6.p4.7.m7.2.2.2.cmml" xref="S6.p4.7.m7.2.2.2"></eq><apply id="S6.p4.7.m7.2.2.1.cmml" xref="S6.p4.7.m7.2.2.1"><times id="S6.p4.7.m7.2.2.1.2.cmml" xref="S6.p4.7.m7.2.2.1.2"></times><ci id="S6.p4.7.m7.2.2.1.3.cmml" xref="S6.p4.7.m7.2.2.1.3">𝜇</ci><apply id="S6.p4.7.m7.2.2.1.1.1.1.cmml" xref="S6.p4.7.m7.2.2.1.1.1"><times id="S6.p4.7.m7.2.2.1.1.1.1.1.cmml" xref="S6.p4.7.m7.2.2.1.1.1.1.1"></times><ci id="S6.p4.7.m7.2.2.1.1.1.1.2a.cmml" xref="S6.p4.7.m7.2.2.1.1.1.1.2"><mtext id="S6.p4.7.m7.2.2.1.1.1.1.2.cmml" xref="S6.p4.7.m7.2.2.1.1.1.1.2">Per</mtext></ci><ci id="S6.p4.7.m7.1.1.cmml" xref="S6.p4.7.m7.1.1">𝑋</ci></apply></apply><cn id="S6.p4.7.m7.2.2.3.cmml" type="integer" xref="S6.p4.7.m7.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.7.m7.2c">\mu(\text{\rm Per}(X))=0</annotation><annotation encoding="application/x-llamapun" id="S6.p4.7.m7.2d">italic_μ ( Per ( italic_X ) ) = 0</annotation></semantics></math> if and only if <math alttext="\mu" class="ltx_Math" display="inline" id="S6.p4.8.m8.1"><semantics id="S6.p4.8.m8.1a"><mi id="S6.p4.8.m8.1.1" xref="S6.p4.8.m8.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.p4.8.m8.1b"><ci id="S6.p4.8.m8.1.1.cmml" xref="S6.p4.8.m8.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.8.m8.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.p4.8.m8.1d">italic_μ</annotation></semantics></math> is non-atomic. This gives rise to a decomposition</p> <table class="ltx_equation ltx_eqn_table" id="S6.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.5)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu\,\,=\,\,\mu^{per}+\mu^{na}" class="ltx_Math" display="block" id="S6.E5.m1.1"><semantics id="S6.E5.m1.1a"><mrow id="S6.E5.m1.1.1" xref="S6.E5.m1.1.1.cmml"><mi id="S6.E5.m1.1.1.2" xref="S6.E5.m1.1.1.2.cmml">μ</mi><mo id="S6.E5.m1.1.1.1" lspace="0.608em" rspace="0.608em" xref="S6.E5.m1.1.1.1.cmml">=</mo><mrow id="S6.E5.m1.1.1.3" xref="S6.E5.m1.1.1.3.cmml"><msup id="S6.E5.m1.1.1.3.2" xref="S6.E5.m1.1.1.3.2.cmml"><mi id="S6.E5.m1.1.1.3.2.2" xref="S6.E5.m1.1.1.3.2.2.cmml">μ</mi><mrow id="S6.E5.m1.1.1.3.2.3" xref="S6.E5.m1.1.1.3.2.3.cmml"><mi id="S6.E5.m1.1.1.3.2.3.2" xref="S6.E5.m1.1.1.3.2.3.2.cmml">p</mi><mo id="S6.E5.m1.1.1.3.2.3.1" xref="S6.E5.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S6.E5.m1.1.1.3.2.3.3" xref="S6.E5.m1.1.1.3.2.3.3.cmml">e</mi><mo id="S6.E5.m1.1.1.3.2.3.1a" xref="S6.E5.m1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S6.E5.m1.1.1.3.2.3.4" xref="S6.E5.m1.1.1.3.2.3.4.cmml">r</mi></mrow></msup><mo id="S6.E5.m1.1.1.3.1" xref="S6.E5.m1.1.1.3.1.cmml">+</mo><msup id="S6.E5.m1.1.1.3.3" xref="S6.E5.m1.1.1.3.3.cmml"><mi id="S6.E5.m1.1.1.3.3.2" xref="S6.E5.m1.1.1.3.3.2.cmml">μ</mi><mrow id="S6.E5.m1.1.1.3.3.3" xref="S6.E5.m1.1.1.3.3.3.cmml"><mi id="S6.E5.m1.1.1.3.3.3.2" xref="S6.E5.m1.1.1.3.3.3.2.cmml">n</mi><mo id="S6.E5.m1.1.1.3.3.3.1" xref="S6.E5.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S6.E5.m1.1.1.3.3.3.3" xref="S6.E5.m1.1.1.3.3.3.3.cmml">a</mi></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.E5.m1.1b"><apply id="S6.E5.m1.1.1.cmml" xref="S6.E5.m1.1.1"><eq id="S6.E5.m1.1.1.1.cmml" xref="S6.E5.m1.1.1.1"></eq><ci id="S6.E5.m1.1.1.2.cmml" xref="S6.E5.m1.1.1.2">𝜇</ci><apply id="S6.E5.m1.1.1.3.cmml" xref="S6.E5.m1.1.1.3"><plus id="S6.E5.m1.1.1.3.1.cmml" xref="S6.E5.m1.1.1.3.1"></plus><apply id="S6.E5.m1.1.1.3.2.cmml" xref="S6.E5.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.E5.m1.1.1.3.2.1.cmml" xref="S6.E5.m1.1.1.3.2">superscript</csymbol><ci id="S6.E5.m1.1.1.3.2.2.cmml" xref="S6.E5.m1.1.1.3.2.2">𝜇</ci><apply id="S6.E5.m1.1.1.3.2.3.cmml" xref="S6.E5.m1.1.1.3.2.3"><times id="S6.E5.m1.1.1.3.2.3.1.cmml" xref="S6.E5.m1.1.1.3.2.3.1"></times><ci id="S6.E5.m1.1.1.3.2.3.2.cmml" xref="S6.E5.m1.1.1.3.2.3.2">𝑝</ci><ci id="S6.E5.m1.1.1.3.2.3.3.cmml" xref="S6.E5.m1.1.1.3.2.3.3">𝑒</ci><ci id="S6.E5.m1.1.1.3.2.3.4.cmml" xref="S6.E5.m1.1.1.3.2.3.4">𝑟</ci></apply></apply><apply id="S6.E5.m1.1.1.3.3.cmml" xref="S6.E5.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.E5.m1.1.1.3.3.1.cmml" xref="S6.E5.m1.1.1.3.3">superscript</csymbol><ci id="S6.E5.m1.1.1.3.3.2.cmml" xref="S6.E5.m1.1.1.3.3.2">𝜇</ci><apply id="S6.E5.m1.1.1.3.3.3.cmml" xref="S6.E5.m1.1.1.3.3.3"><times id="S6.E5.m1.1.1.3.3.3.1.cmml" xref="S6.E5.m1.1.1.3.3.3.1"></times><ci id="S6.E5.m1.1.1.3.3.3.2.cmml" xref="S6.E5.m1.1.1.3.3.3.2">𝑛</ci><ci id="S6.E5.m1.1.1.3.3.3.3.cmml" xref="S6.E5.m1.1.1.3.3.3.3">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E5.m1.1c">\mu\,\,=\,\,\mu^{per}+\mu^{na}</annotation><annotation encoding="application/x-llamapun" id="S6.E5.m1.1d">italic_μ = italic_μ start_POSTSUPERSCRIPT italic_p italic_e italic_r end_POSTSUPERSCRIPT + italic_μ start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.p4.14">of <math alttext="\mu" class="ltx_Math" display="inline" id="S6.p4.9.m1.1"><semantics id="S6.p4.9.m1.1a"><mi id="S6.p4.9.m1.1.1" xref="S6.p4.9.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.p4.9.m1.1b"><ci id="S6.p4.9.m1.1.1.cmml" xref="S6.p4.9.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.9.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.p4.9.m1.1d">italic_μ</annotation></semantics></math> as sum of a non-atomic measure <math alttext="\mu^{na}" class="ltx_Math" display="inline" id="S6.p4.10.m2.1"><semantics id="S6.p4.10.m2.1a"><msup id="S6.p4.10.m2.1.1" xref="S6.p4.10.m2.1.1.cmml"><mi id="S6.p4.10.m2.1.1.2" xref="S6.p4.10.m2.1.1.2.cmml">μ</mi><mrow id="S6.p4.10.m2.1.1.3" xref="S6.p4.10.m2.1.1.3.cmml"><mi id="S6.p4.10.m2.1.1.3.2" xref="S6.p4.10.m2.1.1.3.2.cmml">n</mi><mo id="S6.p4.10.m2.1.1.3.1" xref="S6.p4.10.m2.1.1.3.1.cmml">⁢</mo><mi id="S6.p4.10.m2.1.1.3.3" xref="S6.p4.10.m2.1.1.3.3.cmml">a</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.p4.10.m2.1b"><apply id="S6.p4.10.m2.1.1.cmml" xref="S6.p4.10.m2.1.1"><csymbol cd="ambiguous" id="S6.p4.10.m2.1.1.1.cmml" xref="S6.p4.10.m2.1.1">superscript</csymbol><ci id="S6.p4.10.m2.1.1.2.cmml" xref="S6.p4.10.m2.1.1.2">𝜇</ci><apply id="S6.p4.10.m2.1.1.3.cmml" xref="S6.p4.10.m2.1.1.3"><times id="S6.p4.10.m2.1.1.3.1.cmml" xref="S6.p4.10.m2.1.1.3.1"></times><ci id="S6.p4.10.m2.1.1.3.2.cmml" xref="S6.p4.10.m2.1.1.3.2">𝑛</ci><ci id="S6.p4.10.m2.1.1.3.3.cmml" xref="S6.p4.10.m2.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.10.m2.1c">\mu^{na}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.10.m2.1d">italic_μ start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT</annotation></semantics></math> and a <span class="ltx_text ltx_font_italic" id="S6.p4.14.1">periodic</span> measure <math alttext="\mu^{per}" class="ltx_Math" display="inline" id="S6.p4.11.m3.1"><semantics id="S6.p4.11.m3.1a"><msup id="S6.p4.11.m3.1.1" xref="S6.p4.11.m3.1.1.cmml"><mi id="S6.p4.11.m3.1.1.2" xref="S6.p4.11.m3.1.1.2.cmml">μ</mi><mrow id="S6.p4.11.m3.1.1.3" xref="S6.p4.11.m3.1.1.3.cmml"><mi id="S6.p4.11.m3.1.1.3.2" xref="S6.p4.11.m3.1.1.3.2.cmml">p</mi><mo id="S6.p4.11.m3.1.1.3.1" xref="S6.p4.11.m3.1.1.3.1.cmml">⁢</mo><mi id="S6.p4.11.m3.1.1.3.3" xref="S6.p4.11.m3.1.1.3.3.cmml">e</mi><mo id="S6.p4.11.m3.1.1.3.1a" xref="S6.p4.11.m3.1.1.3.1.cmml">⁢</mo><mi id="S6.p4.11.m3.1.1.3.4" xref="S6.p4.11.m3.1.1.3.4.cmml">r</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.p4.11.m3.1b"><apply id="S6.p4.11.m3.1.1.cmml" xref="S6.p4.11.m3.1.1"><csymbol cd="ambiguous" id="S6.p4.11.m3.1.1.1.cmml" xref="S6.p4.11.m3.1.1">superscript</csymbol><ci id="S6.p4.11.m3.1.1.2.cmml" xref="S6.p4.11.m3.1.1.2">𝜇</ci><apply id="S6.p4.11.m3.1.1.3.cmml" xref="S6.p4.11.m3.1.1.3"><times id="S6.p4.11.m3.1.1.3.1.cmml" xref="S6.p4.11.m3.1.1.3.1"></times><ci id="S6.p4.11.m3.1.1.3.2.cmml" xref="S6.p4.11.m3.1.1.3.2">𝑝</ci><ci id="S6.p4.11.m3.1.1.3.3.cmml" xref="S6.p4.11.m3.1.1.3.3">𝑒</ci><ci id="S6.p4.11.m3.1.1.3.4.cmml" xref="S6.p4.11.m3.1.1.3.4">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.11.m3.1c">\mu^{per}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.11.m3.1d">italic_μ start_POSTSUPERSCRIPT italic_p italic_e italic_r end_POSTSUPERSCRIPT</annotation></semantics></math>, by which we mean that <math alttext="\mu^{per}" class="ltx_Math" display="inline" id="S6.p4.12.m4.1"><semantics id="S6.p4.12.m4.1a"><msup id="S6.p4.12.m4.1.1" xref="S6.p4.12.m4.1.1.cmml"><mi id="S6.p4.12.m4.1.1.2" xref="S6.p4.12.m4.1.1.2.cmml">μ</mi><mrow id="S6.p4.12.m4.1.1.3" xref="S6.p4.12.m4.1.1.3.cmml"><mi id="S6.p4.12.m4.1.1.3.2" xref="S6.p4.12.m4.1.1.3.2.cmml">p</mi><mo id="S6.p4.12.m4.1.1.3.1" xref="S6.p4.12.m4.1.1.3.1.cmml">⁢</mo><mi id="S6.p4.12.m4.1.1.3.3" xref="S6.p4.12.m4.1.1.3.3.cmml">e</mi><mo id="S6.p4.12.m4.1.1.3.1a" xref="S6.p4.12.m4.1.1.3.1.cmml">⁢</mo><mi id="S6.p4.12.m4.1.1.3.4" xref="S6.p4.12.m4.1.1.3.4.cmml">r</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.p4.12.m4.1b"><apply id="S6.p4.12.m4.1.1.cmml" xref="S6.p4.12.m4.1.1"><csymbol cd="ambiguous" id="S6.p4.12.m4.1.1.1.cmml" xref="S6.p4.12.m4.1.1">superscript</csymbol><ci id="S6.p4.12.m4.1.1.2.cmml" xref="S6.p4.12.m4.1.1.2">𝜇</ci><apply id="S6.p4.12.m4.1.1.3.cmml" xref="S6.p4.12.m4.1.1.3"><times id="S6.p4.12.m4.1.1.3.1.cmml" xref="S6.p4.12.m4.1.1.3.1"></times><ci id="S6.p4.12.m4.1.1.3.2.cmml" xref="S6.p4.12.m4.1.1.3.2">𝑝</ci><ci id="S6.p4.12.m4.1.1.3.3.cmml" xref="S6.p4.12.m4.1.1.3.3">𝑒</ci><ci id="S6.p4.12.m4.1.1.3.4.cmml" xref="S6.p4.12.m4.1.1.3.4">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.12.m4.1c">\mu^{per}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.12.m4.1d">italic_μ start_POSTSUPERSCRIPT italic_p italic_e italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> is a countable or finite sum of scalar multiples of the characteristic measures <math alttext="\mu_{w}" class="ltx_Math" display="inline" id="S6.p4.13.m5.1"><semantics id="S6.p4.13.m5.1a"><msub id="S6.p4.13.m5.1.1" xref="S6.p4.13.m5.1.1.cmml"><mi id="S6.p4.13.m5.1.1.2" xref="S6.p4.13.m5.1.1.2.cmml">μ</mi><mi id="S6.p4.13.m5.1.1.3" xref="S6.p4.13.m5.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S6.p4.13.m5.1b"><apply id="S6.p4.13.m5.1.1.cmml" xref="S6.p4.13.m5.1.1"><csymbol cd="ambiguous" id="S6.p4.13.m5.1.1.1.cmml" xref="S6.p4.13.m5.1.1">subscript</csymbol><ci id="S6.p4.13.m5.1.1.2.cmml" xref="S6.p4.13.m5.1.1.2">𝜇</ci><ci id="S6.p4.13.m5.1.1.3.cmml" xref="S6.p4.13.m5.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.13.m5.1c">\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S6.p4.13.m5.1d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> from (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E5" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.5</span></a>). Note that this decomposition is canonical, in that for any measurable set <math alttext="B\subseteq X" class="ltx_Math" display="inline" id="S6.p4.14.m6.1"><semantics id="S6.p4.14.m6.1a"><mrow id="S6.p4.14.m6.1.1" xref="S6.p4.14.m6.1.1.cmml"><mi id="S6.p4.14.m6.1.1.2" xref="S6.p4.14.m6.1.1.2.cmml">B</mi><mo id="S6.p4.14.m6.1.1.1" xref="S6.p4.14.m6.1.1.1.cmml">⊆</mo><mi id="S6.p4.14.m6.1.1.3" xref="S6.p4.14.m6.1.1.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p4.14.m6.1b"><apply id="S6.p4.14.m6.1.1.cmml" xref="S6.p4.14.m6.1.1"><subset id="S6.p4.14.m6.1.1.1.cmml" xref="S6.p4.14.m6.1.1.1"></subset><ci id="S6.p4.14.m6.1.1.2.cmml" xref="S6.p4.14.m6.1.1.2">𝐵</ci><ci id="S6.p4.14.m6.1.1.3.cmml" xref="S6.p4.14.m6.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p4.14.m6.1c">B\subseteq X</annotation><annotation encoding="application/x-llamapun" id="S6.p4.14.m6.1d">italic_B ⊆ italic_X</annotation></semantics></math> one has</p> <table class="ltx_equation ltx_eqn_table" id="S6.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.6)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu^{per}(B)\,\,=\,\,\mu(B\cap\text{\rm Per}(X))\quad\text{and}\quad\mu^{na}(B% )\,\,=\,\,\mu(B\cap(X\smallsetminus\text{\rm Per}(X)))\,." class="ltx_Math" display="block" id="S6.E6.m1.6"><semantics id="S6.E6.m1.6a"><mrow id="S6.E6.m1.6.6.1"><mrow id="S6.E6.m1.6.6.1.1.2" xref="S6.E6.m1.6.6.1.1.3.cmml"><mrow id="S6.E6.m1.6.6.1.1.1.1" xref="S6.E6.m1.6.6.1.1.1.1.cmml"><mrow id="S6.E6.m1.6.6.1.1.1.1.3" xref="S6.E6.m1.6.6.1.1.1.1.3.cmml"><msup id="S6.E6.m1.6.6.1.1.1.1.3.2" xref="S6.E6.m1.6.6.1.1.1.1.3.2.cmml"><mi id="S6.E6.m1.6.6.1.1.1.1.3.2.2" xref="S6.E6.m1.6.6.1.1.1.1.3.2.2.cmml">μ</mi><mrow 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id="S6.E6.m1.6.6.1.1.2.2.1.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1"><times id="S6.E6.m1.6.6.1.1.2.2.1.2.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.2"></times><ci id="S6.E6.m1.6.6.1.1.2.2.1.3.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.3">𝜇</ci><apply id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1"><intersect id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.2.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.2"></intersect><ci id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.3.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.3">𝐵</ci><apply id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1"><setdiff id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.1"></setdiff><ci id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.2">𝑋</ci><apply id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.3"><times id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.3.1.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.3.1"></times><ci id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.3.2a.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.3.2"><mtext id="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.3.2.cmml" xref="S6.E6.m1.6.6.1.1.2.2.1.1.1.1.1.1.1.3.2">Per</mtext></ci><ci id="S6.E6.m1.4.4.cmml" xref="S6.E6.m1.4.4">𝑋</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E6.m1.6c">\mu^{per}(B)\,\,=\,\,\mu(B\cap\text{\rm Per}(X))\quad\text{and}\quad\mu^{na}(B% )\,\,=\,\,\mu(B\cap(X\smallsetminus\text{\rm Per}(X)))\,.</annotation><annotation encoding="application/x-llamapun" id="S6.E6.m1.6d">italic_μ start_POSTSUPERSCRIPT italic_p italic_e italic_r end_POSTSUPERSCRIPT ( italic_B ) = italic_μ ( italic_B ∩ Per ( italic_X ) ) and italic_μ start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT ( italic_B ) = italic_μ ( italic_B ∩ ( italic_X ∖ Per ( italic_X ) ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_theorem ltx_theorem_prop" id="S6.Thmthm4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm4.1.1.1">Proposition 6.4</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm4.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm4.p1"> <p class="ltx_p" id="S6.Thmthm4.p1.4"><span class="ltx_text ltx_font_italic" id="S6.Thmthm4.p1.4.4">Let <math alttext="\mu" class="ltx_Math" display="inline" id="S6.Thmthm4.p1.1.1.m1.1"><semantics id="S6.Thmthm4.p1.1.1.m1.1a"><mi id="S6.Thmthm4.p1.1.1.m1.1.1" xref="S6.Thmthm4.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm4.p1.1.1.m1.1b"><ci id="S6.Thmthm4.p1.1.1.m1.1.1.cmml" xref="S6.Thmthm4.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm4.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm4.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> and <math alttext="\mu^{\prime}" class="ltx_Math" display="inline" id="S6.Thmthm4.p1.2.2.m2.1"><semantics id="S6.Thmthm4.p1.2.2.m2.1a"><msup id="S6.Thmthm4.p1.2.2.m2.1.1" xref="S6.Thmthm4.p1.2.2.m2.1.1.cmml"><mi id="S6.Thmthm4.p1.2.2.m2.1.1.2" xref="S6.Thmthm4.p1.2.2.m2.1.1.2.cmml">μ</mi><mo id="S6.Thmthm4.p1.2.2.m2.1.1.3" xref="S6.Thmthm4.p1.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S6.Thmthm4.p1.2.2.m2.1b"><apply id="S6.Thmthm4.p1.2.2.m2.1.1.cmml" xref="S6.Thmthm4.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.Thmthm4.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmthm4.p1.2.2.m2.1.1">superscript</csymbol><ci id="S6.Thmthm4.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmthm4.p1.2.2.m2.1.1.2">𝜇</ci><ci id="S6.Thmthm4.p1.2.2.m2.1.1.3.cmml" xref="S6.Thmthm4.p1.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm4.p1.2.2.m2.1c">\mu^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm4.p1.2.2.m2.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be two invariant measures on <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm4.p1.3.3.m3.1"><semantics id="S6.Thmthm4.p1.3.3.m3.1a"><mi id="S6.Thmthm4.p1.3.3.m3.1.1" xref="S6.Thmthm4.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm4.p1.3.3.m3.1b"><ci id="S6.Thmthm4.p1.3.3.m3.1.1.cmml" xref="S6.Thmthm4.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm4.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm4.p1.3.3.m3.1d">italic_X</annotation></semantics></math> which are both non-atomic. Assume furthermore that <math alttext="\sigma" class="ltx_Math" display="inline" id="S6.Thmthm4.p1.4.4.m4.1"><semantics id="S6.Thmthm4.p1.4.4.m4.1a"><mi id="S6.Thmthm4.p1.4.4.m4.1.1" xref="S6.Thmthm4.p1.4.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm4.p1.4.4.m4.1b"><ci id="S6.Thmthm4.p1.4.4.m4.1.1.cmml" xref="S6.Thmthm4.p1.4.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm4.p1.4.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm4.p1.4.4.m4.1d">italic_σ</annotation></semantics></math> is shift-orbit injective. Then one has</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma_{*}(\mu)=\sigma_{*}(\mu^{\prime})\quad\Longrightarrow\quad\mu=\mu^{% \prime}\,." class="ltx_Math" display="block" id="S6.Ex6.m1.3"><semantics id="S6.Ex6.m1.3a"><mrow id="S6.Ex6.m1.3.3.1"><mrow id="S6.Ex6.m1.3.3.1.1.2" xref="S6.Ex6.m1.3.3.1.1.3.cmml"><mrow id="S6.Ex6.m1.3.3.1.1.1.1" xref="S6.Ex6.m1.3.3.1.1.1.1.cmml"><mrow id="S6.Ex6.m1.3.3.1.1.1.1.3" xref="S6.Ex6.m1.3.3.1.1.1.1.3.cmml"><msub id="S6.Ex6.m1.3.3.1.1.1.1.3.2" xref="S6.Ex6.m1.3.3.1.1.1.1.3.2.cmml"><mi id="S6.Ex6.m1.3.3.1.1.1.1.3.2.2" xref="S6.Ex6.m1.3.3.1.1.1.1.3.2.2.cmml">σ</mi><mo id="S6.Ex6.m1.3.3.1.1.1.1.3.2.3" xref="S6.Ex6.m1.3.3.1.1.1.1.3.2.3.cmml">∗</mo></msub><mo id="S6.Ex6.m1.3.3.1.1.1.1.3.1" xref="S6.Ex6.m1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S6.Ex6.m1.3.3.1.1.1.1.3.3.2" xref="S6.Ex6.m1.3.3.1.1.1.1.3.cmml"><mo id="S6.Ex6.m1.3.3.1.1.1.1.3.3.2.1" stretchy="false" xref="S6.Ex6.m1.3.3.1.1.1.1.3.cmml">(</mo><mi id="S6.Ex6.m1.1.1" xref="S6.Ex6.m1.1.1.cmml">μ</mi><mo id="S6.Ex6.m1.3.3.1.1.1.1.3.3.2.2" stretchy="false" xref="S6.Ex6.m1.3.3.1.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex6.m1.3.3.1.1.1.1.2" xref="S6.Ex6.m1.3.3.1.1.1.1.2.cmml">=</mo><mrow id="S6.Ex6.m1.3.3.1.1.1.1.1.1" xref="S6.Ex6.m1.3.3.1.1.1.1.1.2.cmml"><mrow id="S6.Ex6.m1.3.3.1.1.1.1.1.1.1" xref="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.cmml"><msub id="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.3" xref="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.3.2" xref="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.3.2.cmml">σ</mi><mo id="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.3.3" xref="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.3.3.cmml">∗</mo></msub><mo id="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.2" xref="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex6.m1.3.3.1.1.1.1.1.1.1.1.1" 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xref="S6.Ex6.m1.3.3.1.1.2.2.3.2">𝜇</ci><ci id="S6.Ex6.m1.3.3.1.1.2.2.3.3.cmml" xref="S6.Ex6.m1.3.3.1.1.2.2.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex6.m1.3c">\sigma_{*}(\mu)=\sigma_{*}(\mu^{\prime})\quad\Longrightarrow\quad\mu=\mu^{% \prime}\,.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex6.m1.3d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) = italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ⟹ italic_μ = italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S6.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.4.p1"> <p class="ltx_p" id="S6.4.p1.6">We fix an arbitrary word <math alttext="w\in\cal L(X)" class="ltx_Math" display="inline" id="S6.4.p1.1.m1.1"><semantics id="S6.4.p1.1.m1.1a"><mrow id="S6.4.p1.1.m1.1.2" xref="S6.4.p1.1.m1.1.2.cmml"><mi id="S6.4.p1.1.m1.1.2.2" xref="S6.4.p1.1.m1.1.2.2.cmml">w</mi><mo id="S6.4.p1.1.m1.1.2.1" xref="S6.4.p1.1.m1.1.2.1.cmml">∈</mo><mrow id="S6.4.p1.1.m1.1.2.3" xref="S6.4.p1.1.m1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.4.p1.1.m1.1.2.3.2" xref="S6.4.p1.1.m1.1.2.3.2.cmml">ℒ</mi><mo id="S6.4.p1.1.m1.1.2.3.1" xref="S6.4.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S6.4.p1.1.m1.1.2.3.3.2" xref="S6.4.p1.1.m1.1.2.3.cmml"><mo id="S6.4.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S6.4.p1.1.m1.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.4.p1.1.m1.1.1" xref="S6.4.p1.1.m1.1.1.cmml">𝒳</mi><mo id="S6.4.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S6.4.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.1.m1.1b"><apply id="S6.4.p1.1.m1.1.2.cmml" xref="S6.4.p1.1.m1.1.2"><in id="S6.4.p1.1.m1.1.2.1.cmml" xref="S6.4.p1.1.m1.1.2.1"></in><ci id="S6.4.p1.1.m1.1.2.2.cmml" xref="S6.4.p1.1.m1.1.2.2">𝑤</ci><apply id="S6.4.p1.1.m1.1.2.3.cmml" xref="S6.4.p1.1.m1.1.2.3"><times id="S6.4.p1.1.m1.1.2.3.1.cmml" xref="S6.4.p1.1.m1.1.2.3.1"></times><ci id="S6.4.p1.1.m1.1.2.3.2.cmml" xref="S6.4.p1.1.m1.1.2.3.2">ℒ</ci><ci id="S6.4.p1.1.m1.1.1.cmml" xref="S6.4.p1.1.m1.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.1.m1.1c">w\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.1.m1.1d">italic_w ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> and consider for any <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S6.4.p1.2.m2.1"><semantics id="S6.4.p1.2.m2.1a"><mrow id="S6.4.p1.2.m2.1.1" xref="S6.4.p1.2.m2.1.1.cmml"><mi id="S6.4.p1.2.m2.1.1.2" xref="S6.4.p1.2.m2.1.1.2.cmml">n</mi><mo id="S6.4.p1.2.m2.1.1.1" xref="S6.4.p1.2.m2.1.1.1.cmml">≥</mo><mn id="S6.4.p1.2.m2.1.1.3" xref="S6.4.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.2.m2.1b"><apply id="S6.4.p1.2.m2.1.1.cmml" xref="S6.4.p1.2.m2.1.1"><geq id="S6.4.p1.2.m2.1.1.1.cmml" xref="S6.4.p1.2.m2.1.1.1"></geq><ci id="S6.4.p1.2.m2.1.1.2.cmml" xref="S6.4.p1.2.m2.1.1.2">𝑛</ci><cn id="S6.4.p1.2.m2.1.1.3.cmml" type="integer" xref="S6.4.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.2.m2.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.2.m2.1d">italic_n ≥ 0</annotation></semantics></math> the subsets <math alttext="W_{n}(w),U_{n}(w)" class="ltx_Math" display="inline" id="S6.4.p1.3.m3.4"><semantics id="S6.4.p1.3.m3.4a"><mrow id="S6.4.p1.3.m3.4.4.2" xref="S6.4.p1.3.m3.4.4.3.cmml"><mrow id="S6.4.p1.3.m3.3.3.1.1" xref="S6.4.p1.3.m3.3.3.1.1.cmml"><msub id="S6.4.p1.3.m3.3.3.1.1.2" xref="S6.4.p1.3.m3.3.3.1.1.2.cmml"><mi id="S6.4.p1.3.m3.3.3.1.1.2.2" xref="S6.4.p1.3.m3.3.3.1.1.2.2.cmml">W</mi><mi id="S6.4.p1.3.m3.3.3.1.1.2.3" xref="S6.4.p1.3.m3.3.3.1.1.2.3.cmml">n</mi></msub><mo id="S6.4.p1.3.m3.3.3.1.1.1" xref="S6.4.p1.3.m3.3.3.1.1.1.cmml">⁢</mo><mrow id="S6.4.p1.3.m3.3.3.1.1.3.2" xref="S6.4.p1.3.m3.3.3.1.1.cmml"><mo id="S6.4.p1.3.m3.3.3.1.1.3.2.1" stretchy="false" xref="S6.4.p1.3.m3.3.3.1.1.cmml">(</mo><mi id="S6.4.p1.3.m3.1.1" xref="S6.4.p1.3.m3.1.1.cmml">w</mi><mo id="S6.4.p1.3.m3.3.3.1.1.3.2.2" stretchy="false" xref="S6.4.p1.3.m3.3.3.1.1.cmml">)</mo></mrow></mrow><mo id="S6.4.p1.3.m3.4.4.2.3" xref="S6.4.p1.3.m3.4.4.3.cmml">,</mo><mrow id="S6.4.p1.3.m3.4.4.2.2" xref="S6.4.p1.3.m3.4.4.2.2.cmml"><msub id="S6.4.p1.3.m3.4.4.2.2.2" xref="S6.4.p1.3.m3.4.4.2.2.2.cmml"><mi id="S6.4.p1.3.m3.4.4.2.2.2.2" xref="S6.4.p1.3.m3.4.4.2.2.2.2.cmml">U</mi><mi id="S6.4.p1.3.m3.4.4.2.2.2.3" xref="S6.4.p1.3.m3.4.4.2.2.2.3.cmml">n</mi></msub><mo id="S6.4.p1.3.m3.4.4.2.2.1" xref="S6.4.p1.3.m3.4.4.2.2.1.cmml">⁢</mo><mrow id="S6.4.p1.3.m3.4.4.2.2.3.2" 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id="S6.4.p1.3.m3.4.4.2.2.cmml" xref="S6.4.p1.3.m3.4.4.2.2"><times id="S6.4.p1.3.m3.4.4.2.2.1.cmml" xref="S6.4.p1.3.m3.4.4.2.2.1"></times><apply id="S6.4.p1.3.m3.4.4.2.2.2.cmml" xref="S6.4.p1.3.m3.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.4.p1.3.m3.4.4.2.2.2.1.cmml" xref="S6.4.p1.3.m3.4.4.2.2.2">subscript</csymbol><ci id="S6.4.p1.3.m3.4.4.2.2.2.2.cmml" xref="S6.4.p1.3.m3.4.4.2.2.2.2">𝑈</ci><ci id="S6.4.p1.3.m3.4.4.2.2.2.3.cmml" xref="S6.4.p1.3.m3.4.4.2.2.2.3">𝑛</ci></apply><ci id="S6.4.p1.3.m3.2.2.cmml" xref="S6.4.p1.3.m3.2.2">𝑤</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.3.m3.4c">W_{n}(w),U_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.3.m3.4d">italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) , italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> and <math alttext="A_{n}(w)" class="ltx_Math" display="inline" id="S6.4.p1.4.m4.1"><semantics id="S6.4.p1.4.m4.1a"><mrow id="S6.4.p1.4.m4.1.2" xref="S6.4.p1.4.m4.1.2.cmml"><msub id="S6.4.p1.4.m4.1.2.2" xref="S6.4.p1.4.m4.1.2.2.cmml"><mi id="S6.4.p1.4.m4.1.2.2.2" xref="S6.4.p1.4.m4.1.2.2.2.cmml">A</mi><mi id="S6.4.p1.4.m4.1.2.2.3" xref="S6.4.p1.4.m4.1.2.2.3.cmml">n</mi></msub><mo id="S6.4.p1.4.m4.1.2.1" xref="S6.4.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S6.4.p1.4.m4.1.2.3.2" xref="S6.4.p1.4.m4.1.2.cmml"><mo id="S6.4.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S6.4.p1.4.m4.1.2.cmml">(</mo><mi id="S6.4.p1.4.m4.1.1" xref="S6.4.p1.4.m4.1.1.cmml">w</mi><mo id="S6.4.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S6.4.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.4.m4.1b"><apply id="S6.4.p1.4.m4.1.2.cmml" xref="S6.4.p1.4.m4.1.2"><times id="S6.4.p1.4.m4.1.2.1.cmml" xref="S6.4.p1.4.m4.1.2.1"></times><apply id="S6.4.p1.4.m4.1.2.2.cmml" xref="S6.4.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S6.4.p1.4.m4.1.2.2.1.cmml" xref="S6.4.p1.4.m4.1.2.2">subscript</csymbol><ci id="S6.4.p1.4.m4.1.2.2.2.cmml" xref="S6.4.p1.4.m4.1.2.2.2">𝐴</ci><ci id="S6.4.p1.4.m4.1.2.2.3.cmml" xref="S6.4.p1.4.m4.1.2.2.3">𝑛</ci></apply><ci id="S6.4.p1.4.m4.1.1.cmml" xref="S6.4.p1.4.m4.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.4.m4.1c">A_{n}(w)</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.4.m4.1d">italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w )</annotation></semantics></math> as defined in the equalities (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E1" title="In 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.1</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E3" title="In 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.3</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E4" title="In 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.4</span></a>), abbreviated here to <math alttext="W_{n},U_{n}" class="ltx_Math" display="inline" id="S6.4.p1.5.m5.2"><semantics id="S6.4.p1.5.m5.2a"><mrow id="S6.4.p1.5.m5.2.2.2" xref="S6.4.p1.5.m5.2.2.3.cmml"><msub id="S6.4.p1.5.m5.1.1.1.1" xref="S6.4.p1.5.m5.1.1.1.1.cmml"><mi id="S6.4.p1.5.m5.1.1.1.1.2" xref="S6.4.p1.5.m5.1.1.1.1.2.cmml">W</mi><mi id="S6.4.p1.5.m5.1.1.1.1.3" xref="S6.4.p1.5.m5.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.4.p1.5.m5.2.2.2.3" xref="S6.4.p1.5.m5.2.2.3.cmml">,</mo><msub id="S6.4.p1.5.m5.2.2.2.2" xref="S6.4.p1.5.m5.2.2.2.2.cmml"><mi id="S6.4.p1.5.m5.2.2.2.2.2" xref="S6.4.p1.5.m5.2.2.2.2.2.cmml">U</mi><mi id="S6.4.p1.5.m5.2.2.2.2.3" xref="S6.4.p1.5.m5.2.2.2.2.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.4.p1.5.m5.2b"><list id="S6.4.p1.5.m5.2.2.3.cmml" xref="S6.4.p1.5.m5.2.2.2"><apply id="S6.4.p1.5.m5.1.1.1.1.cmml" xref="S6.4.p1.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S6.4.p1.5.m5.1.1.1.1.1.cmml" xref="S6.4.p1.5.m5.1.1.1.1">subscript</csymbol><ci id="S6.4.p1.5.m5.1.1.1.1.2.cmml" xref="S6.4.p1.5.m5.1.1.1.1.2">𝑊</ci><ci id="S6.4.p1.5.m5.1.1.1.1.3.cmml" xref="S6.4.p1.5.m5.1.1.1.1.3">𝑛</ci></apply><apply id="S6.4.p1.5.m5.2.2.2.2.cmml" xref="S6.4.p1.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S6.4.p1.5.m5.2.2.2.2.1.cmml" xref="S6.4.p1.5.m5.2.2.2.2">subscript</csymbol><ci id="S6.4.p1.5.m5.2.2.2.2.2.cmml" xref="S6.4.p1.5.m5.2.2.2.2.2">𝑈</ci><ci id="S6.4.p1.5.m5.2.2.2.2.3.cmml" xref="S6.4.p1.5.m5.2.2.2.2.3">𝑛</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.5.m5.2c">W_{n},U_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.5.m5.2d">italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="A_{n}" class="ltx_Math" display="inline" id="S6.4.p1.6.m6.1"><semantics id="S6.4.p1.6.m6.1a"><msub id="S6.4.p1.6.m6.1.1" xref="S6.4.p1.6.m6.1.1.cmml"><mi id="S6.4.p1.6.m6.1.1.2" xref="S6.4.p1.6.m6.1.1.2.cmml">A</mi><mi id="S6.4.p1.6.m6.1.1.3" xref="S6.4.p1.6.m6.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.4.p1.6.m6.1b"><apply id="S6.4.p1.6.m6.1.1.cmml" xref="S6.4.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S6.4.p1.6.m6.1.1.1.cmml" xref="S6.4.p1.6.m6.1.1">subscript</csymbol><ci id="S6.4.p1.6.m6.1.1.2.cmml" xref="S6.4.p1.6.m6.1.1.2">𝐴</ci><ci id="S6.4.p1.6.m6.1.1.3.cmml" xref="S6.4.p1.6.m6.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.4.p1.6.m6.1c">A_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.4.p1.6.m6.1d">italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> respectively.</p> </div> <div class="ltx_para" id="S6.5.p2"> <p class="ltx_p" id="S6.5.p2.3">Let us also fix some integer <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S6.5.p2.1.m1.1"><semantics id="S6.5.p2.1.m1.1a"><mrow id="S6.5.p2.1.m1.1.1" xref="S6.5.p2.1.m1.1.1.cmml"><mi id="S6.5.p2.1.m1.1.1.2" xref="S6.5.p2.1.m1.1.1.2.cmml">n</mi><mo id="S6.5.p2.1.m1.1.1.1" xref="S6.5.p2.1.m1.1.1.1.cmml">≥</mo><mn id="S6.5.p2.1.m1.1.1.3" xref="S6.5.p2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.5.p2.1.m1.1b"><apply id="S6.5.p2.1.m1.1.1.cmml" xref="S6.5.p2.1.m1.1.1"><geq id="S6.5.p2.1.m1.1.1.1.cmml" xref="S6.5.p2.1.m1.1.1.1"></geq><ci id="S6.5.p2.1.m1.1.1.2.cmml" xref="S6.5.p2.1.m1.1.1.2">𝑛</ci><cn id="S6.5.p2.1.m1.1.1.3.cmml" type="integer" xref="S6.5.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p2.1.m1.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.5.p2.1.m1.1d">italic_n ≥ 0</annotation></semantics></math>. From the definition of a weight function (see Section <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.SS1" title="2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.1</span></a>) we see directly that for any measure <math alttext="\mu" class="ltx_Math" display="inline" id="S6.5.p2.2.m2.1"><semantics id="S6.5.p2.2.m2.1a"><mi id="S6.5.p2.2.m2.1.1" xref="S6.5.p2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.5.p2.2.m2.1b"><ci id="S6.5.p2.2.m2.1.1.cmml" xref="S6.5.p2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.5.p2.2.m2.1d">italic_μ</annotation></semantics></math> on <math alttext="X" class="ltx_Math" display="inline" id="S6.5.p2.3.m3.1"><semantics id="S6.5.p2.3.m3.1a"><mi id="S6.5.p2.3.m3.1.1" xref="S6.5.p2.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.5.p2.3.m3.1b"><ci id="S6.5.p2.3.m3.1.1.cmml" xref="S6.5.p2.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p2.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.5.p2.3.m3.1d">italic_X</annotation></semantics></math> the corresponding weight function satisfies</p> <table class="ltx_equation ltx_eqn_table" id="S6.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.7)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu(w)=\sum_{w^{\prime}\in W_{n}(w)}\mu(w^{\prime})\,." class="ltx_Math" display="block" id="S6.E7.m1.3"><semantics id="S6.E7.m1.3a"><mrow id="S6.E7.m1.3.3.1" xref="S6.E7.m1.3.3.1.1.cmml"><mrow id="S6.E7.m1.3.3.1.1" xref="S6.E7.m1.3.3.1.1.cmml"><mrow id="S6.E7.m1.3.3.1.1.3" xref="S6.E7.m1.3.3.1.1.3.cmml"><mi id="S6.E7.m1.3.3.1.1.3.2" xref="S6.E7.m1.3.3.1.1.3.2.cmml">μ</mi><mo id="S6.E7.m1.3.3.1.1.3.1" xref="S6.E7.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S6.E7.m1.3.3.1.1.3.3.2" xref="S6.E7.m1.3.3.1.1.3.cmml"><mo id="S6.E7.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S6.E7.m1.3.3.1.1.3.cmml">(</mo><mi id="S6.E7.m1.2.2" xref="S6.E7.m1.2.2.cmml">w</mi><mo id="S6.E7.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="S6.E7.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow><mo id="S6.E7.m1.3.3.1.1.2" rspace="0.111em" xref="S6.E7.m1.3.3.1.1.2.cmml">=</mo><mrow id="S6.E7.m1.3.3.1.1.1" xref="S6.E7.m1.3.3.1.1.1.cmml"><munder id="S6.E7.m1.3.3.1.1.1.2" xref="S6.E7.m1.3.3.1.1.1.2.cmml"><mo id="S6.E7.m1.3.3.1.1.1.2.2" movablelimits="false" xref="S6.E7.m1.3.3.1.1.1.2.2.cmml">∑</mo><mrow id="S6.E7.m1.1.1.1" xref="S6.E7.m1.1.1.1.cmml"><msup id="S6.E7.m1.1.1.1.3" xref="S6.E7.m1.1.1.1.3.cmml"><mi id="S6.E7.m1.1.1.1.3.2" xref="S6.E7.m1.1.1.1.3.2.cmml">w</mi><mo id="S6.E7.m1.1.1.1.3.3" xref="S6.E7.m1.1.1.1.3.3.cmml">′</mo></msup><mo id="S6.E7.m1.1.1.1.2" 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xref="S6.E7.m1.1.1.1.4.2">subscript</csymbol><ci id="S6.E7.m1.1.1.1.4.2.2.cmml" xref="S6.E7.m1.1.1.1.4.2.2">𝑊</ci><ci id="S6.E7.m1.1.1.1.4.2.3.cmml" xref="S6.E7.m1.1.1.1.4.2.3">𝑛</ci></apply><ci id="S6.E7.m1.1.1.1.1.cmml" xref="S6.E7.m1.1.1.1.1">𝑤</ci></apply></apply></apply><apply id="S6.E7.m1.3.3.1.1.1.1.cmml" xref="S6.E7.m1.3.3.1.1.1.1"><times id="S6.E7.m1.3.3.1.1.1.1.2.cmml" xref="S6.E7.m1.3.3.1.1.1.1.2"></times><ci id="S6.E7.m1.3.3.1.1.1.1.3.cmml" xref="S6.E7.m1.3.3.1.1.1.1.3">𝜇</ci><apply id="S6.E7.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S6.E7.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.E7.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S6.E7.m1.3.3.1.1.1.1.1.1">superscript</csymbol><ci id="S6.E7.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.E7.m1.3.3.1.1.1.1.1.1.1.2">𝑤</ci><ci id="S6.E7.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S6.E7.m1.3.3.1.1.1.1.1.1.1.3">′</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E7.m1.3c">\mu(w)=\sum_{w^{\prime}\in W_{n}(w)}\mu(w^{\prime})\,.</annotation><annotation encoding="application/x-llamapun" id="S6.E7.m1.3d">italic_μ ( italic_w ) = ∑ start_POSTSUBSCRIPT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_w ) end_POSTSUBSCRIPT italic_μ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.5.p2.6">From (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E2" title="In 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.2</span></a>) we know that the set <math alttext="W_{n}" class="ltx_Math" display="inline" id="S6.5.p2.4.m1.1"><semantics id="S6.5.p2.4.m1.1a"><msub id="S6.5.p2.4.m1.1.1" xref="S6.5.p2.4.m1.1.1.cmml"><mi id="S6.5.p2.4.m1.1.1.2" xref="S6.5.p2.4.m1.1.1.2.cmml">W</mi><mi id="S6.5.p2.4.m1.1.1.3" xref="S6.5.p2.4.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.5.p2.4.m1.1b"><apply id="S6.5.p2.4.m1.1.1.cmml" xref="S6.5.p2.4.m1.1.1"><csymbol cd="ambiguous" id="S6.5.p2.4.m1.1.1.1.cmml" xref="S6.5.p2.4.m1.1.1">subscript</csymbol><ci id="S6.5.p2.4.m1.1.1.2.cmml" xref="S6.5.p2.4.m1.1.1.2">𝑊</ci><ci id="S6.5.p2.4.m1.1.1.3.cmml" xref="S6.5.p2.4.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p2.4.m1.1c">W_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p2.4.m1.1d">italic_W start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> decomposes as disjoint union of <math alttext="U_{n}" class="ltx_Math" display="inline" id="S6.5.p2.5.m2.1"><semantics id="S6.5.p2.5.m2.1a"><msub id="S6.5.p2.5.m2.1.1" xref="S6.5.p2.5.m2.1.1.cmml"><mi id="S6.5.p2.5.m2.1.1.2" xref="S6.5.p2.5.m2.1.1.2.cmml">U</mi><mi id="S6.5.p2.5.m2.1.1.3" xref="S6.5.p2.5.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.5.p2.5.m2.1b"><apply id="S6.5.p2.5.m2.1.1.cmml" xref="S6.5.p2.5.m2.1.1"><csymbol cd="ambiguous" id="S6.5.p2.5.m2.1.1.1.cmml" xref="S6.5.p2.5.m2.1.1">subscript</csymbol><ci id="S6.5.p2.5.m2.1.1.2.cmml" xref="S6.5.p2.5.m2.1.1.2">𝑈</ci><ci id="S6.5.p2.5.m2.1.1.3.cmml" xref="S6.5.p2.5.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p2.5.m2.1c">U_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p2.5.m2.1d">italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="A_{n}" class="ltx_Math" display="inline" id="S6.5.p2.6.m3.1"><semantics id="S6.5.p2.6.m3.1a"><msub id="S6.5.p2.6.m3.1.1" xref="S6.5.p2.6.m3.1.1.cmml"><mi id="S6.5.p2.6.m3.1.1.2" xref="S6.5.p2.6.m3.1.1.2.cmml">A</mi><mi id="S6.5.p2.6.m3.1.1.3" xref="S6.5.p2.6.m3.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.5.p2.6.m3.1b"><apply id="S6.5.p2.6.m3.1.1.cmml" xref="S6.5.p2.6.m3.1.1"><csymbol cd="ambiguous" id="S6.5.p2.6.m3.1.1.1.cmml" xref="S6.5.p2.6.m3.1.1">subscript</csymbol><ci id="S6.5.p2.6.m3.1.1.2.cmml" xref="S6.5.p2.6.m3.1.1.2">𝐴</ci><ci id="S6.5.p2.6.m3.1.1.3.cmml" xref="S6.5.p2.6.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p2.6.m3.1c">A_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.5.p2.6.m3.1d">italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, so that (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E7" title="In Proof. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.7</span></a>) gives us</p> <table class="ltx_equation ltx_eqn_table" id="S6.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.8)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu(w)=\mu(U_{n})+\mu(A_{n})\,," class="ltx_Math" display="block" id="S6.E8.m1.2"><semantics id="S6.E8.m1.2a"><mrow id="S6.E8.m1.2.2.1" xref="S6.E8.m1.2.2.1.1.cmml"><mrow id="S6.E8.m1.2.2.1.1" xref="S6.E8.m1.2.2.1.1.cmml"><mrow id="S6.E8.m1.2.2.1.1.4" xref="S6.E8.m1.2.2.1.1.4.cmml"><mi id="S6.E8.m1.2.2.1.1.4.2" xref="S6.E8.m1.2.2.1.1.4.2.cmml">μ</mi><mo id="S6.E8.m1.2.2.1.1.4.1" xref="S6.E8.m1.2.2.1.1.4.1.cmml">⁢</mo><mrow id="S6.E8.m1.2.2.1.1.4.3.2" xref="S6.E8.m1.2.2.1.1.4.cmml"><mo id="S6.E8.m1.2.2.1.1.4.3.2.1" stretchy="false" xref="S6.E8.m1.2.2.1.1.4.cmml">(</mo><mi id="S6.E8.m1.1.1" xref="S6.E8.m1.1.1.cmml">w</mi><mo id="S6.E8.m1.2.2.1.1.4.3.2.2" stretchy="false" xref="S6.E8.m1.2.2.1.1.4.cmml">)</mo></mrow></mrow><mo id="S6.E8.m1.2.2.1.1.3" xref="S6.E8.m1.2.2.1.1.3.cmml">=</mo><mrow id="S6.E8.m1.2.2.1.1.2" xref="S6.E8.m1.2.2.1.1.2.cmml"><mrow id="S6.E8.m1.2.2.1.1.1.1" xref="S6.E8.m1.2.2.1.1.1.1.cmml"><mi id="S6.E8.m1.2.2.1.1.1.1.3" xref="S6.E8.m1.2.2.1.1.1.1.3.cmml">μ</mi><mo id="S6.E8.m1.2.2.1.1.1.1.2" xref="S6.E8.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.E8.m1.2.2.1.1.1.1.1.1" xref="S6.E8.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S6.E8.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S6.E8.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.E8.m1.2.2.1.1.1.1.1.1.1" xref="S6.E8.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="S6.E8.m1.2.2.1.1.1.1.1.1.1.2" xref="S6.E8.m1.2.2.1.1.1.1.1.1.1.2.cmml">U</mi><mi id="S6.E8.m1.2.2.1.1.1.1.1.1.1.3" xref="S6.E8.m1.2.2.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.E8.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="S6.E8.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.E8.m1.2.2.1.1.2.3" xref="S6.E8.m1.2.2.1.1.2.3.cmml">+</mo><mrow id="S6.E8.m1.2.2.1.1.2.2" xref="S6.E8.m1.2.2.1.1.2.2.cmml"><mi id="S6.E8.m1.2.2.1.1.2.2.3" xref="S6.E8.m1.2.2.1.1.2.2.3.cmml">μ</mi><mo id="S6.E8.m1.2.2.1.1.2.2.2" xref="S6.E8.m1.2.2.1.1.2.2.2.cmml">⁢</mo><mrow id="S6.E8.m1.2.2.1.1.2.2.1.1" xref="S6.E8.m1.2.2.1.1.2.2.1.1.1.cmml"><mo id="S6.E8.m1.2.2.1.1.2.2.1.1.2" stretchy="false" xref="S6.E8.m1.2.2.1.1.2.2.1.1.1.cmml">(</mo><msub id="S6.E8.m1.2.2.1.1.2.2.1.1.1" xref="S6.E8.m1.2.2.1.1.2.2.1.1.1.cmml"><mi id="S6.E8.m1.2.2.1.1.2.2.1.1.1.2" xref="S6.E8.m1.2.2.1.1.2.2.1.1.1.2.cmml">A</mi><mi id="S6.E8.m1.2.2.1.1.2.2.1.1.1.3" xref="S6.E8.m1.2.2.1.1.2.2.1.1.1.3.cmml">n</mi></msub><mo id="S6.E8.m1.2.2.1.1.2.2.1.1.3" rspace="0.170em" stretchy="false" xref="S6.E8.m1.2.2.1.1.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S6.E8.m1.2.2.1.2" xref="S6.E8.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.E8.m1.2b"><apply id="S6.E8.m1.2.2.1.1.cmml" xref="S6.E8.m1.2.2.1"><eq id="S6.E8.m1.2.2.1.1.3.cmml" xref="S6.E8.m1.2.2.1.1.3"></eq><apply id="S6.E8.m1.2.2.1.1.4.cmml" xref="S6.E8.m1.2.2.1.1.4"><times id="S6.E8.m1.2.2.1.1.4.1.cmml" xref="S6.E8.m1.2.2.1.1.4.1"></times><ci id="S6.E8.m1.2.2.1.1.4.2.cmml" xref="S6.E8.m1.2.2.1.1.4.2">𝜇</ci><ci id="S6.E8.m1.1.1.cmml" xref="S6.E8.m1.1.1">𝑤</ci></apply><apply id="S6.E8.m1.2.2.1.1.2.cmml" xref="S6.E8.m1.2.2.1.1.2"><plus id="S6.E8.m1.2.2.1.1.2.3.cmml" xref="S6.E8.m1.2.2.1.1.2.3"></plus><apply id="S6.E8.m1.2.2.1.1.1.1.cmml" xref="S6.E8.m1.2.2.1.1.1.1"><times id="S6.E8.m1.2.2.1.1.1.1.2.cmml" xref="S6.E8.m1.2.2.1.1.1.1.2"></times><ci id="S6.E8.m1.2.2.1.1.1.1.3.cmml" xref="S6.E8.m1.2.2.1.1.1.1.3">𝜇</ci><apply id="S6.E8.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S6.E8.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.E8.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S6.E8.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.E8.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S6.E8.m1.2.2.1.1.1.1.1.1.1.2">𝑈</ci><ci id="S6.E8.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S6.E8.m1.2.2.1.1.1.1.1.1.1.3">𝑛</ci></apply></apply><apply id="S6.E8.m1.2.2.1.1.2.2.cmml" xref="S6.E8.m1.2.2.1.1.2.2"><times id="S6.E8.m1.2.2.1.1.2.2.2.cmml" xref="S6.E8.m1.2.2.1.1.2.2.2"></times><ci id="S6.E8.m1.2.2.1.1.2.2.3.cmml" xref="S6.E8.m1.2.2.1.1.2.2.3">𝜇</ci><apply id="S6.E8.m1.2.2.1.1.2.2.1.1.1.cmml" xref="S6.E8.m1.2.2.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S6.E8.m1.2.2.1.1.2.2.1.1.1.1.cmml" xref="S6.E8.m1.2.2.1.1.2.2.1.1">subscript</csymbol><ci id="S6.E8.m1.2.2.1.1.2.2.1.1.1.2.cmml" xref="S6.E8.m1.2.2.1.1.2.2.1.1.1.2">𝐴</ci><ci id="S6.E8.m1.2.2.1.1.2.2.1.1.1.3.cmml" xref="S6.E8.m1.2.2.1.1.2.2.1.1.1.3">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E8.m1.2c">\mu(w)=\mu(U_{n})+\mu(A_{n})\,,</annotation><annotation encoding="application/x-llamapun" id="S6.E8.m1.2d">italic_μ ( italic_w ) = italic_μ ( italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) + italic_μ ( italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.5.p2.8">where we set <math alttext="\mu(U_{n}):=\underset{w^{\prime}\in U_{n}}{\sum}\mu(w^{\prime})" class="ltx_Math" display="inline" id="S6.5.p2.7.m1.2"><semantics id="S6.5.p2.7.m1.2a"><mrow id="S6.5.p2.7.m1.2.2" xref="S6.5.p2.7.m1.2.2.cmml"><mrow id="S6.5.p2.7.m1.1.1.1" xref="S6.5.p2.7.m1.1.1.1.cmml"><mi id="S6.5.p2.7.m1.1.1.1.3" xref="S6.5.p2.7.m1.1.1.1.3.cmml">μ</mi><mo id="S6.5.p2.7.m1.1.1.1.2" xref="S6.5.p2.7.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.5.p2.7.m1.1.1.1.1.1" xref="S6.5.p2.7.m1.1.1.1.1.1.1.cmml"><mo id="S6.5.p2.7.m1.1.1.1.1.1.2" stretchy="false" xref="S6.5.p2.7.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.5.p2.7.m1.1.1.1.1.1.1" xref="S6.5.p2.7.m1.1.1.1.1.1.1.cmml"><mi id="S6.5.p2.7.m1.1.1.1.1.1.1.2" xref="S6.5.p2.7.m1.1.1.1.1.1.1.2.cmml">U</mi><mi id="S6.5.p2.7.m1.1.1.1.1.1.1.3" xref="S6.5.p2.7.m1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.5.p2.7.m1.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S6.5.p2.7.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.5.p2.7.m1.2.2.3" rspace="0.111em" xref="S6.5.p2.7.m1.2.2.3.cmml">:=</mo><mrow id="S6.5.p2.7.m1.2.2.2" xref="S6.5.p2.7.m1.2.2.2.cmml"><munder accentunder="true" id="S6.5.p2.7.m1.2.2.2.3" xref="S6.5.p2.7.m1.2.2.2.3.cmml"><mo id="S6.5.p2.7.m1.2.2.2.3.2" xref="S6.5.p2.7.m1.2.2.2.3.2.cmml">∑</mo><mrow id="S6.5.p2.7.m1.2.2.2.3.1" xref="S6.5.p2.7.m1.2.2.2.3.1.cmml"><msup id="S6.5.p2.7.m1.2.2.2.3.1.2" xref="S6.5.p2.7.m1.2.2.2.3.1.2.cmml"><mi id="S6.5.p2.7.m1.2.2.2.3.1.2.2" xref="S6.5.p2.7.m1.2.2.2.3.1.2.2.cmml">w</mi><mo id="S6.5.p2.7.m1.2.2.2.3.1.2.3" xref="S6.5.p2.7.m1.2.2.2.3.1.2.3.cmml">′</mo></msup><mo id="S6.5.p2.7.m1.2.2.2.3.1.1" xref="S6.5.p2.7.m1.2.2.2.3.1.1.cmml">∈</mo><msub id="S6.5.p2.7.m1.2.2.2.3.1.3" xref="S6.5.p2.7.m1.2.2.2.3.1.3.cmml"><mi id="S6.5.p2.7.m1.2.2.2.3.1.3.2" xref="S6.5.p2.7.m1.2.2.2.3.1.3.2.cmml">U</mi><mi id="S6.5.p2.7.m1.2.2.2.3.1.3.3" xref="S6.5.p2.7.m1.2.2.2.3.1.3.3.cmml">n</mi></msub></mrow></munder><mo id="S6.5.p2.7.m1.2.2.2.2" xref="S6.5.p2.7.m1.2.2.2.2.cmml">⁢</mo><mi id="S6.5.p2.7.m1.2.2.2.4" xref="S6.5.p2.7.m1.2.2.2.4.cmml">μ</mi><mo id="S6.5.p2.7.m1.2.2.2.2a" xref="S6.5.p2.7.m1.2.2.2.2.cmml">⁢</mo><mrow id="S6.5.p2.7.m1.2.2.2.1.1" xref="S6.5.p2.7.m1.2.2.2.1.1.1.cmml"><mo id="S6.5.p2.7.m1.2.2.2.1.1.2" stretchy="false" xref="S6.5.p2.7.m1.2.2.2.1.1.1.cmml">(</mo><msup id="S6.5.p2.7.m1.2.2.2.1.1.1" xref="S6.5.p2.7.m1.2.2.2.1.1.1.cmml"><mi id="S6.5.p2.7.m1.2.2.2.1.1.1.2" xref="S6.5.p2.7.m1.2.2.2.1.1.1.2.cmml">w</mi><mo id="S6.5.p2.7.m1.2.2.2.1.1.1.3" xref="S6.5.p2.7.m1.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S6.5.p2.7.m1.2.2.2.1.1.3" stretchy="false" xref="S6.5.p2.7.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.5.p2.7.m1.2b"><apply id="S6.5.p2.7.m1.2.2.cmml" xref="S6.5.p2.7.m1.2.2"><csymbol cd="latexml" id="S6.5.p2.7.m1.2.2.3.cmml" xref="S6.5.p2.7.m1.2.2.3">assign</csymbol><apply id="S6.5.p2.7.m1.1.1.1.cmml" xref="S6.5.p2.7.m1.1.1.1"><times id="S6.5.p2.7.m1.1.1.1.2.cmml" xref="S6.5.p2.7.m1.1.1.1.2"></times><ci id="S6.5.p2.7.m1.1.1.1.3.cmml" xref="S6.5.p2.7.m1.1.1.1.3">𝜇</ci><apply id="S6.5.p2.7.m1.1.1.1.1.1.1.cmml" xref="S6.5.p2.7.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.5.p2.7.m1.1.1.1.1.1.1.1.cmml" xref="S6.5.p2.7.m1.1.1.1.1.1">subscript</csymbol><ci id="S6.5.p2.7.m1.1.1.1.1.1.1.2.cmml" xref="S6.5.p2.7.m1.1.1.1.1.1.1.2">𝑈</ci><ci id="S6.5.p2.7.m1.1.1.1.1.1.1.3.cmml" xref="S6.5.p2.7.m1.1.1.1.1.1.1.3">𝑛</ci></apply></apply><apply id="S6.5.p2.7.m1.2.2.2.cmml" xref="S6.5.p2.7.m1.2.2.2"><times id="S6.5.p2.7.m1.2.2.2.2.cmml" xref="S6.5.p2.7.m1.2.2.2.2"></times><apply id="S6.5.p2.7.m1.2.2.2.3.cmml" xref="S6.5.p2.7.m1.2.2.2.3"><apply id="S6.5.p2.7.m1.2.2.2.3.1.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1"><in id="S6.5.p2.7.m1.2.2.2.3.1.1.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.1"></in><apply id="S6.5.p2.7.m1.2.2.2.3.1.2.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.2"><csymbol cd="ambiguous" id="S6.5.p2.7.m1.2.2.2.3.1.2.1.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.2">superscript</csymbol><ci id="S6.5.p2.7.m1.2.2.2.3.1.2.2.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.2.2">𝑤</ci><ci id="S6.5.p2.7.m1.2.2.2.3.1.2.3.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.2.3">′</ci></apply><apply id="S6.5.p2.7.m1.2.2.2.3.1.3.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.3"><csymbol cd="ambiguous" id="S6.5.p2.7.m1.2.2.2.3.1.3.1.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.3">subscript</csymbol><ci id="S6.5.p2.7.m1.2.2.2.3.1.3.2.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.3.2">𝑈</ci><ci id="S6.5.p2.7.m1.2.2.2.3.1.3.3.cmml" xref="S6.5.p2.7.m1.2.2.2.3.1.3.3">𝑛</ci></apply></apply><sum id="S6.5.p2.7.m1.2.2.2.3.2.cmml" xref="S6.5.p2.7.m1.2.2.2.3.2"></sum></apply><ci id="S6.5.p2.7.m1.2.2.2.4.cmml" xref="S6.5.p2.7.m1.2.2.2.4">𝜇</ci><apply id="S6.5.p2.7.m1.2.2.2.1.1.1.cmml" xref="S6.5.p2.7.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.5.p2.7.m1.2.2.2.1.1.1.1.cmml" xref="S6.5.p2.7.m1.2.2.2.1.1">superscript</csymbol><ci id="S6.5.p2.7.m1.2.2.2.1.1.1.2.cmml" xref="S6.5.p2.7.m1.2.2.2.1.1.1.2">𝑤</ci><ci id="S6.5.p2.7.m1.2.2.2.1.1.1.3.cmml" xref="S6.5.p2.7.m1.2.2.2.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p2.7.m1.2c">\mu(U_{n}):=\underset{w^{\prime}\in U_{n}}{\sum}\mu(w^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.5.p2.7.m1.2d">italic_μ ( italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) := start_UNDERACCENT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_UNDERACCENT start_ARG ∑ end_ARG italic_μ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\mu(A_{n}):=\underset{w^{\prime}\in A_{n}}{\sum}\mu(w^{\prime})" class="ltx_Math" display="inline" id="S6.5.p2.8.m2.2"><semantics id="S6.5.p2.8.m2.2a"><mrow id="S6.5.p2.8.m2.2.2" xref="S6.5.p2.8.m2.2.2.cmml"><mrow id="S6.5.p2.8.m2.1.1.1" xref="S6.5.p2.8.m2.1.1.1.cmml"><mi id="S6.5.p2.8.m2.1.1.1.3" xref="S6.5.p2.8.m2.1.1.1.3.cmml">μ</mi><mo id="S6.5.p2.8.m2.1.1.1.2" xref="S6.5.p2.8.m2.1.1.1.2.cmml">⁢</mo><mrow id="S6.5.p2.8.m2.1.1.1.1.1" xref="S6.5.p2.8.m2.1.1.1.1.1.1.cmml"><mo id="S6.5.p2.8.m2.1.1.1.1.1.2" stretchy="false" xref="S6.5.p2.8.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S6.5.p2.8.m2.1.1.1.1.1.1" xref="S6.5.p2.8.m2.1.1.1.1.1.1.cmml"><mi id="S6.5.p2.8.m2.1.1.1.1.1.1.2" xref="S6.5.p2.8.m2.1.1.1.1.1.1.2.cmml">A</mi><mi id="S6.5.p2.8.m2.1.1.1.1.1.1.3" xref="S6.5.p2.8.m2.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.5.p2.8.m2.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S6.5.p2.8.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.5.p2.8.m2.2.2.3" rspace="0.111em" xref="S6.5.p2.8.m2.2.2.3.cmml">:=</mo><mrow id="S6.5.p2.8.m2.2.2.2" xref="S6.5.p2.8.m2.2.2.2.cmml"><munder accentunder="true" id="S6.5.p2.8.m2.2.2.2.3" xref="S6.5.p2.8.m2.2.2.2.3.cmml"><mo id="S6.5.p2.8.m2.2.2.2.3.2" xref="S6.5.p2.8.m2.2.2.2.3.2.cmml">∑</mo><mrow id="S6.5.p2.8.m2.2.2.2.3.1" xref="S6.5.p2.8.m2.2.2.2.3.1.cmml"><msup id="S6.5.p2.8.m2.2.2.2.3.1.2" xref="S6.5.p2.8.m2.2.2.2.3.1.2.cmml"><mi id="S6.5.p2.8.m2.2.2.2.3.1.2.2" xref="S6.5.p2.8.m2.2.2.2.3.1.2.2.cmml">w</mi><mo id="S6.5.p2.8.m2.2.2.2.3.1.2.3" xref="S6.5.p2.8.m2.2.2.2.3.1.2.3.cmml">′</mo></msup><mo id="S6.5.p2.8.m2.2.2.2.3.1.1" xref="S6.5.p2.8.m2.2.2.2.3.1.1.cmml">∈</mo><msub id="S6.5.p2.8.m2.2.2.2.3.1.3" xref="S6.5.p2.8.m2.2.2.2.3.1.3.cmml"><mi id="S6.5.p2.8.m2.2.2.2.3.1.3.2" xref="S6.5.p2.8.m2.2.2.2.3.1.3.2.cmml">A</mi><mi id="S6.5.p2.8.m2.2.2.2.3.1.3.3" xref="S6.5.p2.8.m2.2.2.2.3.1.3.3.cmml">n</mi></msub></mrow></munder><mo id="S6.5.p2.8.m2.2.2.2.2" xref="S6.5.p2.8.m2.2.2.2.2.cmml">⁢</mo><mi id="S6.5.p2.8.m2.2.2.2.4" xref="S6.5.p2.8.m2.2.2.2.4.cmml">μ</mi><mo id="S6.5.p2.8.m2.2.2.2.2a" xref="S6.5.p2.8.m2.2.2.2.2.cmml">⁢</mo><mrow id="S6.5.p2.8.m2.2.2.2.1.1" xref="S6.5.p2.8.m2.2.2.2.1.1.1.cmml"><mo id="S6.5.p2.8.m2.2.2.2.1.1.2" stretchy="false" xref="S6.5.p2.8.m2.2.2.2.1.1.1.cmml">(</mo><msup id="S6.5.p2.8.m2.2.2.2.1.1.1" xref="S6.5.p2.8.m2.2.2.2.1.1.1.cmml"><mi id="S6.5.p2.8.m2.2.2.2.1.1.1.2" xref="S6.5.p2.8.m2.2.2.2.1.1.1.2.cmml">w</mi><mo id="S6.5.p2.8.m2.2.2.2.1.1.1.3" xref="S6.5.p2.8.m2.2.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S6.5.p2.8.m2.2.2.2.1.1.3" stretchy="false" xref="S6.5.p2.8.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.5.p2.8.m2.2b"><apply id="S6.5.p2.8.m2.2.2.cmml" xref="S6.5.p2.8.m2.2.2"><csymbol cd="latexml" id="S6.5.p2.8.m2.2.2.3.cmml" xref="S6.5.p2.8.m2.2.2.3">assign</csymbol><apply id="S6.5.p2.8.m2.1.1.1.cmml" xref="S6.5.p2.8.m2.1.1.1"><times id="S6.5.p2.8.m2.1.1.1.2.cmml" xref="S6.5.p2.8.m2.1.1.1.2"></times><ci id="S6.5.p2.8.m2.1.1.1.3.cmml" xref="S6.5.p2.8.m2.1.1.1.3">𝜇</ci><apply id="S6.5.p2.8.m2.1.1.1.1.1.1.cmml" xref="S6.5.p2.8.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.5.p2.8.m2.1.1.1.1.1.1.1.cmml" xref="S6.5.p2.8.m2.1.1.1.1.1">subscript</csymbol><ci id="S6.5.p2.8.m2.1.1.1.1.1.1.2.cmml" xref="S6.5.p2.8.m2.1.1.1.1.1.1.2">𝐴</ci><ci id="S6.5.p2.8.m2.1.1.1.1.1.1.3.cmml" xref="S6.5.p2.8.m2.1.1.1.1.1.1.3">𝑛</ci></apply></apply><apply id="S6.5.p2.8.m2.2.2.2.cmml" xref="S6.5.p2.8.m2.2.2.2"><times id="S6.5.p2.8.m2.2.2.2.2.cmml" xref="S6.5.p2.8.m2.2.2.2.2"></times><apply id="S6.5.p2.8.m2.2.2.2.3.cmml" xref="S6.5.p2.8.m2.2.2.2.3"><apply id="S6.5.p2.8.m2.2.2.2.3.1.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1"><in id="S6.5.p2.8.m2.2.2.2.3.1.1.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.1"></in><apply id="S6.5.p2.8.m2.2.2.2.3.1.2.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.2"><csymbol cd="ambiguous" id="S6.5.p2.8.m2.2.2.2.3.1.2.1.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.2">superscript</csymbol><ci id="S6.5.p2.8.m2.2.2.2.3.1.2.2.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.2.2">𝑤</ci><ci id="S6.5.p2.8.m2.2.2.2.3.1.2.3.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.2.3">′</ci></apply><apply id="S6.5.p2.8.m2.2.2.2.3.1.3.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.3"><csymbol cd="ambiguous" id="S6.5.p2.8.m2.2.2.2.3.1.3.1.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.3">subscript</csymbol><ci id="S6.5.p2.8.m2.2.2.2.3.1.3.2.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.3.2">𝐴</ci><ci id="S6.5.p2.8.m2.2.2.2.3.1.3.3.cmml" xref="S6.5.p2.8.m2.2.2.2.3.1.3.3">𝑛</ci></apply></apply><sum id="S6.5.p2.8.m2.2.2.2.3.2.cmml" xref="S6.5.p2.8.m2.2.2.2.3.2"></sum></apply><ci id="S6.5.p2.8.m2.2.2.2.4.cmml" xref="S6.5.p2.8.m2.2.2.2.4">𝜇</ci><apply id="S6.5.p2.8.m2.2.2.2.1.1.1.cmml" xref="S6.5.p2.8.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.5.p2.8.m2.2.2.2.1.1.1.1.cmml" xref="S6.5.p2.8.m2.2.2.2.1.1">superscript</csymbol><ci id="S6.5.p2.8.m2.2.2.2.1.1.1.2.cmml" xref="S6.5.p2.8.m2.2.2.2.1.1.1.2">𝑤</ci><ci id="S6.5.p2.8.m2.2.2.2.1.1.1.3.cmml" xref="S6.5.p2.8.m2.2.2.2.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.5.p2.8.m2.2c">\mu(A_{n}):=\underset{w^{\prime}\in A_{n}}{\sum}\mu(w^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S6.5.p2.8.m2.2d">italic_μ ( italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) := start_UNDERACCENT italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_UNDERACCENT start_ARG ∑ end_ARG italic_μ ( italic_w start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.6.p3"> <p class="ltx_p" id="S6.6.p3.7">We now define <math alttext="V_{n}:=\sigma(U_{n})" class="ltx_Math" display="inline" id="S6.6.p3.1.m1.1"><semantics id="S6.6.p3.1.m1.1a"><mrow id="S6.6.p3.1.m1.1.1" xref="S6.6.p3.1.m1.1.1.cmml"><msub id="S6.6.p3.1.m1.1.1.3" xref="S6.6.p3.1.m1.1.1.3.cmml"><mi id="S6.6.p3.1.m1.1.1.3.2" xref="S6.6.p3.1.m1.1.1.3.2.cmml">V</mi><mi id="S6.6.p3.1.m1.1.1.3.3" xref="S6.6.p3.1.m1.1.1.3.3.cmml">n</mi></msub><mo id="S6.6.p3.1.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S6.6.p3.1.m1.1.1.2.cmml">:=</mo><mrow id="S6.6.p3.1.m1.1.1.1" xref="S6.6.p3.1.m1.1.1.1.cmml"><mi id="S6.6.p3.1.m1.1.1.1.3" xref="S6.6.p3.1.m1.1.1.1.3.cmml">σ</mi><mo id="S6.6.p3.1.m1.1.1.1.2" xref="S6.6.p3.1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.6.p3.1.m1.1.1.1.1.1" xref="S6.6.p3.1.m1.1.1.1.1.1.1.cmml"><mo id="S6.6.p3.1.m1.1.1.1.1.1.2" stretchy="false" xref="S6.6.p3.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.6.p3.1.m1.1.1.1.1.1.1" xref="S6.6.p3.1.m1.1.1.1.1.1.1.cmml"><mi id="S6.6.p3.1.m1.1.1.1.1.1.1.2" xref="S6.6.p3.1.m1.1.1.1.1.1.1.2.cmml">U</mi><mi id="S6.6.p3.1.m1.1.1.1.1.1.1.3" xref="S6.6.p3.1.m1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.6.p3.1.m1.1.1.1.1.1.3" stretchy="false" xref="S6.6.p3.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p3.1.m1.1b"><apply id="S6.6.p3.1.m1.1.1.cmml" xref="S6.6.p3.1.m1.1.1"><csymbol cd="latexml" id="S6.6.p3.1.m1.1.1.2.cmml" xref="S6.6.p3.1.m1.1.1.2">assign</csymbol><apply id="S6.6.p3.1.m1.1.1.3.cmml" xref="S6.6.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.6.p3.1.m1.1.1.3.1.cmml" xref="S6.6.p3.1.m1.1.1.3">subscript</csymbol><ci id="S6.6.p3.1.m1.1.1.3.2.cmml" xref="S6.6.p3.1.m1.1.1.3.2">𝑉</ci><ci id="S6.6.p3.1.m1.1.1.3.3.cmml" xref="S6.6.p3.1.m1.1.1.3.3">𝑛</ci></apply><apply id="S6.6.p3.1.m1.1.1.1.cmml" xref="S6.6.p3.1.m1.1.1.1"><times id="S6.6.p3.1.m1.1.1.1.2.cmml" xref="S6.6.p3.1.m1.1.1.1.2"></times><ci id="S6.6.p3.1.m1.1.1.1.3.cmml" xref="S6.6.p3.1.m1.1.1.1.3">𝜎</ci><apply id="S6.6.p3.1.m1.1.1.1.1.1.1.cmml" xref="S6.6.p3.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.6.p3.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.6.p3.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S6.6.p3.1.m1.1.1.1.1.1.1.2.cmml" xref="S6.6.p3.1.m1.1.1.1.1.1.1.2">𝑈</ci><ci id="S6.6.p3.1.m1.1.1.1.1.1.1.3.cmml" xref="S6.6.p3.1.m1.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p3.1.m1.1c">V_{n}:=\sigma(U_{n})</annotation><annotation encoding="application/x-llamapun" id="S6.6.p3.1.m1.1d">italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT := italic_σ ( italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> and observe from the definition of <math alttext="U_{n}" class="ltx_Math" display="inline" id="S6.6.p3.2.m2.1"><semantics id="S6.6.p3.2.m2.1a"><msub id="S6.6.p3.2.m2.1.1" xref="S6.6.p3.2.m2.1.1.cmml"><mi id="S6.6.p3.2.m2.1.1.2" xref="S6.6.p3.2.m2.1.1.2.cmml">U</mi><mi id="S6.6.p3.2.m2.1.1.3" xref="S6.6.p3.2.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.6.p3.2.m2.1b"><apply id="S6.6.p3.2.m2.1.1.cmml" xref="S6.6.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S6.6.p3.2.m2.1.1.1.cmml" xref="S6.6.p3.2.m2.1.1">subscript</csymbol><ci id="S6.6.p3.2.m2.1.1.2.cmml" xref="S6.6.p3.2.m2.1.1.2">𝑈</ci><ci id="S6.6.p3.2.m2.1.1.3.cmml" xref="S6.6.p3.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p3.2.m2.1c">U_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.6.p3.2.m2.1d">italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> that one has <math alttext="\sigma^{-1}(V_{n})\cap\cal L(X)=U_{n}" class="ltx_Math" display="inline" id="S6.6.p3.3.m3.2"><semantics id="S6.6.p3.3.m3.2a"><mrow id="S6.6.p3.3.m3.2.2" xref="S6.6.p3.3.m3.2.2.cmml"><mrow id="S6.6.p3.3.m3.2.2.1" xref="S6.6.p3.3.m3.2.2.1.cmml"><mrow id="S6.6.p3.3.m3.2.2.1.1" xref="S6.6.p3.3.m3.2.2.1.1.cmml"><msup id="S6.6.p3.3.m3.2.2.1.1.3" xref="S6.6.p3.3.m3.2.2.1.1.3.cmml"><mi id="S6.6.p3.3.m3.2.2.1.1.3.2" xref="S6.6.p3.3.m3.2.2.1.1.3.2.cmml">σ</mi><mrow id="S6.6.p3.3.m3.2.2.1.1.3.3" xref="S6.6.p3.3.m3.2.2.1.1.3.3.cmml"><mo id="S6.6.p3.3.m3.2.2.1.1.3.3a" xref="S6.6.p3.3.m3.2.2.1.1.3.3.cmml">−</mo><mn id="S6.6.p3.3.m3.2.2.1.1.3.3.2" xref="S6.6.p3.3.m3.2.2.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S6.6.p3.3.m3.2.2.1.1.2" xref="S6.6.p3.3.m3.2.2.1.1.2.cmml">⁢</mo><mrow id="S6.6.p3.3.m3.2.2.1.1.1.1" xref="S6.6.p3.3.m3.2.2.1.1.1.1.1.cmml"><mo id="S6.6.p3.3.m3.2.2.1.1.1.1.2" stretchy="false" xref="S6.6.p3.3.m3.2.2.1.1.1.1.1.cmml">(</mo><msub id="S6.6.p3.3.m3.2.2.1.1.1.1.1" xref="S6.6.p3.3.m3.2.2.1.1.1.1.1.cmml"><mi id="S6.6.p3.3.m3.2.2.1.1.1.1.1.2" xref="S6.6.p3.3.m3.2.2.1.1.1.1.1.2.cmml">V</mi><mi id="S6.6.p3.3.m3.2.2.1.1.1.1.1.3" xref="S6.6.p3.3.m3.2.2.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.6.p3.3.m3.2.2.1.1.1.1.3" stretchy="false" xref="S6.6.p3.3.m3.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.6.p3.3.m3.2.2.1.2" xref="S6.6.p3.3.m3.2.2.1.2.cmml">∩</mo><mrow id="S6.6.p3.3.m3.2.2.1.3" xref="S6.6.p3.3.m3.2.2.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.6.p3.3.m3.2.2.1.3.2" xref="S6.6.p3.3.m3.2.2.1.3.2.cmml">ℒ</mi><mo id="S6.6.p3.3.m3.2.2.1.3.1" xref="S6.6.p3.3.m3.2.2.1.3.1.cmml">⁢</mo><mrow id="S6.6.p3.3.m3.2.2.1.3.3.2" xref="S6.6.p3.3.m3.2.2.1.3.cmml"><mo id="S6.6.p3.3.m3.2.2.1.3.3.2.1" stretchy="false" xref="S6.6.p3.3.m3.2.2.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.6.p3.3.m3.1.1" xref="S6.6.p3.3.m3.1.1.cmml">𝒳</mi><mo id="S6.6.p3.3.m3.2.2.1.3.3.2.2" stretchy="false" xref="S6.6.p3.3.m3.2.2.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S6.6.p3.3.m3.2.2.2" xref="S6.6.p3.3.m3.2.2.2.cmml">=</mo><msub id="S6.6.p3.3.m3.2.2.3" xref="S6.6.p3.3.m3.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.6.p3.3.m3.2.2.3.2" xref="S6.6.p3.3.m3.2.2.3.2.cmml">𝒰</mi><mi class="ltx_font_mathcaligraphic" id="S6.6.p3.3.m3.2.2.3.3" xref="S6.6.p3.3.m3.2.2.3.3.cmml">𝓃</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p3.3.m3.2b"><apply id="S6.6.p3.3.m3.2.2.cmml" xref="S6.6.p3.3.m3.2.2"><eq id="S6.6.p3.3.m3.2.2.2.cmml" xref="S6.6.p3.3.m3.2.2.2"></eq><apply id="S6.6.p3.3.m3.2.2.1.cmml" xref="S6.6.p3.3.m3.2.2.1"><intersect id="S6.6.p3.3.m3.2.2.1.2.cmml" xref="S6.6.p3.3.m3.2.2.1.2"></intersect><apply id="S6.6.p3.3.m3.2.2.1.1.cmml" xref="S6.6.p3.3.m3.2.2.1.1"><times id="S6.6.p3.3.m3.2.2.1.1.2.cmml" xref="S6.6.p3.3.m3.2.2.1.1.2"></times><apply id="S6.6.p3.3.m3.2.2.1.1.3.cmml" xref="S6.6.p3.3.m3.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.6.p3.3.m3.2.2.1.1.3.1.cmml" xref="S6.6.p3.3.m3.2.2.1.1.3">superscript</csymbol><ci id="S6.6.p3.3.m3.2.2.1.1.3.2.cmml" xref="S6.6.p3.3.m3.2.2.1.1.3.2">𝜎</ci><apply id="S6.6.p3.3.m3.2.2.1.1.3.3.cmml" xref="S6.6.p3.3.m3.2.2.1.1.3.3"><minus id="S6.6.p3.3.m3.2.2.1.1.3.3.1.cmml" xref="S6.6.p3.3.m3.2.2.1.1.3.3"></minus><cn id="S6.6.p3.3.m3.2.2.1.1.3.3.2.cmml" type="integer" xref="S6.6.p3.3.m3.2.2.1.1.3.3.2">1</cn></apply></apply><apply id="S6.6.p3.3.m3.2.2.1.1.1.1.1.cmml" xref="S6.6.p3.3.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.6.p3.3.m3.2.2.1.1.1.1.1.1.cmml" xref="S6.6.p3.3.m3.2.2.1.1.1.1">subscript</csymbol><ci id="S6.6.p3.3.m3.2.2.1.1.1.1.1.2.cmml" xref="S6.6.p3.3.m3.2.2.1.1.1.1.1.2">𝑉</ci><ci id="S6.6.p3.3.m3.2.2.1.1.1.1.1.3.cmml" xref="S6.6.p3.3.m3.2.2.1.1.1.1.1.3">𝑛</ci></apply></apply><apply id="S6.6.p3.3.m3.2.2.1.3.cmml" xref="S6.6.p3.3.m3.2.2.1.3"><times id="S6.6.p3.3.m3.2.2.1.3.1.cmml" xref="S6.6.p3.3.m3.2.2.1.3.1"></times><ci id="S6.6.p3.3.m3.2.2.1.3.2.cmml" xref="S6.6.p3.3.m3.2.2.1.3.2">ℒ</ci><ci id="S6.6.p3.3.m3.1.1.cmml" xref="S6.6.p3.3.m3.1.1">𝒳</ci></apply></apply><apply id="S6.6.p3.3.m3.2.2.3.cmml" xref="S6.6.p3.3.m3.2.2.3"><csymbol cd="ambiguous" id="S6.6.p3.3.m3.2.2.3.1.cmml" xref="S6.6.p3.3.m3.2.2.3">subscript</csymbol><ci id="S6.6.p3.3.m3.2.2.3.2.cmml" xref="S6.6.p3.3.m3.2.2.3.2">𝒰</ci><ci id="S6.6.p3.3.m3.2.2.3.3.cmml" xref="S6.6.p3.3.m3.2.2.3.3">𝓃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p3.3.m3.2c">\sigma^{-1}(V_{n})\cap\cal L(X)=U_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.6.p3.3.m3.2d">italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) ∩ caligraphic_L ( caligraphic_X ) = caligraphic_U start_POSTSUBSCRIPT caligraphic_n end_POSTSUBSCRIPT</annotation></semantics></math>. Hence we deduce from the definition of the push-forward measure <math alttext="\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.6.p3.4.m4.1"><semantics id="S6.6.p3.4.m4.1a"><mrow id="S6.6.p3.4.m4.1.2" xref="S6.6.p3.4.m4.1.2.cmml"><msub id="S6.6.p3.4.m4.1.2.2" xref="S6.6.p3.4.m4.1.2.2.cmml"><mi id="S6.6.p3.4.m4.1.2.2.2" xref="S6.6.p3.4.m4.1.2.2.2.cmml">σ</mi><mo id="S6.6.p3.4.m4.1.2.2.3" xref="S6.6.p3.4.m4.1.2.2.3.cmml">∗</mo></msub><mo id="S6.6.p3.4.m4.1.2.1" xref="S6.6.p3.4.m4.1.2.1.cmml">⁢</mo><mrow id="S6.6.p3.4.m4.1.2.3.2" xref="S6.6.p3.4.m4.1.2.cmml"><mo id="S6.6.p3.4.m4.1.2.3.2.1" stretchy="false" xref="S6.6.p3.4.m4.1.2.cmml">(</mo><mi id="S6.6.p3.4.m4.1.1" xref="S6.6.p3.4.m4.1.1.cmml">μ</mi><mo id="S6.6.p3.4.m4.1.2.3.2.2" stretchy="false" xref="S6.6.p3.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p3.4.m4.1b"><apply id="S6.6.p3.4.m4.1.2.cmml" xref="S6.6.p3.4.m4.1.2"><times id="S6.6.p3.4.m4.1.2.1.cmml" xref="S6.6.p3.4.m4.1.2.1"></times><apply id="S6.6.p3.4.m4.1.2.2.cmml" xref="S6.6.p3.4.m4.1.2.2"><csymbol cd="ambiguous" id="S6.6.p3.4.m4.1.2.2.1.cmml" xref="S6.6.p3.4.m4.1.2.2">subscript</csymbol><ci id="S6.6.p3.4.m4.1.2.2.2.cmml" xref="S6.6.p3.4.m4.1.2.2.2">𝜎</ci><times id="S6.6.p3.4.m4.1.2.2.3.cmml" xref="S6.6.p3.4.m4.1.2.2.3"></times></apply><ci id="S6.6.p3.4.m4.1.1.cmml" xref="S6.6.p3.4.m4.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p3.4.m4.1c">\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.6.p3.4.m4.1d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math>, together with the fact that by definition of <math alttext="\mu\in\cal M(X)" class="ltx_Math" display="inline" id="S6.6.p3.5.m5.1"><semantics id="S6.6.p3.5.m5.1a"><mrow id="S6.6.p3.5.m5.1.2" xref="S6.6.p3.5.m5.1.2.cmml"><mi id="S6.6.p3.5.m5.1.2.2" xref="S6.6.p3.5.m5.1.2.2.cmml">μ</mi><mo id="S6.6.p3.5.m5.1.2.1" xref="S6.6.p3.5.m5.1.2.1.cmml">∈</mo><mrow id="S6.6.p3.5.m5.1.2.3" xref="S6.6.p3.5.m5.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.6.p3.5.m5.1.2.3.2" xref="S6.6.p3.5.m5.1.2.3.2.cmml">ℳ</mi><mo id="S6.6.p3.5.m5.1.2.3.1" xref="S6.6.p3.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S6.6.p3.5.m5.1.2.3.3.2" xref="S6.6.p3.5.m5.1.2.3.cmml"><mo id="S6.6.p3.5.m5.1.2.3.3.2.1" stretchy="false" xref="S6.6.p3.5.m5.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.6.p3.5.m5.1.1" xref="S6.6.p3.5.m5.1.1.cmml">𝒳</mi><mo id="S6.6.p3.5.m5.1.2.3.3.2.2" stretchy="false" xref="S6.6.p3.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p3.5.m5.1b"><apply id="S6.6.p3.5.m5.1.2.cmml" xref="S6.6.p3.5.m5.1.2"><in id="S6.6.p3.5.m5.1.2.1.cmml" xref="S6.6.p3.5.m5.1.2.1"></in><ci id="S6.6.p3.5.m5.1.2.2.cmml" xref="S6.6.p3.5.m5.1.2.2">𝜇</ci><apply id="S6.6.p3.5.m5.1.2.3.cmml" xref="S6.6.p3.5.m5.1.2.3"><times id="S6.6.p3.5.m5.1.2.3.1.cmml" xref="S6.6.p3.5.m5.1.2.3.1"></times><ci id="S6.6.p3.5.m5.1.2.3.2.cmml" xref="S6.6.p3.5.m5.1.2.3.2">ℳ</ci><ci id="S6.6.p3.5.m5.1.1.cmml" xref="S6.6.p3.5.m5.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p3.5.m5.1c">\mu\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S6.6.p3.5.m5.1d">italic_μ ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math> one has <math alttext="\mu(w_{0})=0" class="ltx_Math" display="inline" id="S6.6.p3.6.m6.1"><semantics id="S6.6.p3.6.m6.1a"><mrow id="S6.6.p3.6.m6.1.1" xref="S6.6.p3.6.m6.1.1.cmml"><mrow id="S6.6.p3.6.m6.1.1.1" xref="S6.6.p3.6.m6.1.1.1.cmml"><mi id="S6.6.p3.6.m6.1.1.1.3" xref="S6.6.p3.6.m6.1.1.1.3.cmml">μ</mi><mo id="S6.6.p3.6.m6.1.1.1.2" xref="S6.6.p3.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S6.6.p3.6.m6.1.1.1.1.1" xref="S6.6.p3.6.m6.1.1.1.1.1.1.cmml"><mo id="S6.6.p3.6.m6.1.1.1.1.1.2" stretchy="false" xref="S6.6.p3.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="S6.6.p3.6.m6.1.1.1.1.1.1" xref="S6.6.p3.6.m6.1.1.1.1.1.1.cmml"><mi id="S6.6.p3.6.m6.1.1.1.1.1.1.2" xref="S6.6.p3.6.m6.1.1.1.1.1.1.2.cmml">w</mi><mn id="S6.6.p3.6.m6.1.1.1.1.1.1.3" xref="S6.6.p3.6.m6.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.6.p3.6.m6.1.1.1.1.1.3" stretchy="false" xref="S6.6.p3.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.6.p3.6.m6.1.1.2" xref="S6.6.p3.6.m6.1.1.2.cmml">=</mo><mn id="S6.6.p3.6.m6.1.1.3" xref="S6.6.p3.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p3.6.m6.1b"><apply id="S6.6.p3.6.m6.1.1.cmml" xref="S6.6.p3.6.m6.1.1"><eq id="S6.6.p3.6.m6.1.1.2.cmml" xref="S6.6.p3.6.m6.1.1.2"></eq><apply id="S6.6.p3.6.m6.1.1.1.cmml" xref="S6.6.p3.6.m6.1.1.1"><times id="S6.6.p3.6.m6.1.1.1.2.cmml" xref="S6.6.p3.6.m6.1.1.1.2"></times><ci id="S6.6.p3.6.m6.1.1.1.3.cmml" xref="S6.6.p3.6.m6.1.1.1.3">𝜇</ci><apply id="S6.6.p3.6.m6.1.1.1.1.1.1.cmml" xref="S6.6.p3.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.6.p3.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.6.p3.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S6.6.p3.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.6.p3.6.m6.1.1.1.1.1.1.2">𝑤</ci><cn id="S6.6.p3.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.6.p3.6.m6.1.1.1.1.1.1.3">0</cn></apply></apply><cn id="S6.6.p3.6.m6.1.1.3.cmml" type="integer" xref="S6.6.p3.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p3.6.m6.1c">\mu(w_{0})=0</annotation><annotation encoding="application/x-llamapun" id="S6.6.p3.6.m6.1d">italic_μ ( italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> for any <math alttext="w_{0}\notin\cal L(X)" class="ltx_Math" display="inline" id="S6.6.p3.7.m7.1"><semantics id="S6.6.p3.7.m7.1a"><mrow id="S6.6.p3.7.m7.1.2" xref="S6.6.p3.7.m7.1.2.cmml"><msub id="S6.6.p3.7.m7.1.2.2" xref="S6.6.p3.7.m7.1.2.2.cmml"><mi id="S6.6.p3.7.m7.1.2.2.2" xref="S6.6.p3.7.m7.1.2.2.2.cmml">w</mi><mn id="S6.6.p3.7.m7.1.2.2.3" xref="S6.6.p3.7.m7.1.2.2.3.cmml">0</mn></msub><mo id="S6.6.p3.7.m7.1.2.1" xref="S6.6.p3.7.m7.1.2.1.cmml">∉</mo><mrow id="S6.6.p3.7.m7.1.2.3" xref="S6.6.p3.7.m7.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.6.p3.7.m7.1.2.3.2" xref="S6.6.p3.7.m7.1.2.3.2.cmml">ℒ</mi><mo id="S6.6.p3.7.m7.1.2.3.1" xref="S6.6.p3.7.m7.1.2.3.1.cmml">⁢</mo><mrow id="S6.6.p3.7.m7.1.2.3.3.2" xref="S6.6.p3.7.m7.1.2.3.cmml"><mo id="S6.6.p3.7.m7.1.2.3.3.2.1" stretchy="false" xref="S6.6.p3.7.m7.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.6.p3.7.m7.1.1" xref="S6.6.p3.7.m7.1.1.cmml">𝒳</mi><mo id="S6.6.p3.7.m7.1.2.3.3.2.2" stretchy="false" xref="S6.6.p3.7.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.6.p3.7.m7.1b"><apply id="S6.6.p3.7.m7.1.2.cmml" xref="S6.6.p3.7.m7.1.2"><notin id="S6.6.p3.7.m7.1.2.1.cmml" xref="S6.6.p3.7.m7.1.2.1"></notin><apply id="S6.6.p3.7.m7.1.2.2.cmml" xref="S6.6.p3.7.m7.1.2.2"><csymbol cd="ambiguous" id="S6.6.p3.7.m7.1.2.2.1.cmml" xref="S6.6.p3.7.m7.1.2.2">subscript</csymbol><ci id="S6.6.p3.7.m7.1.2.2.2.cmml" xref="S6.6.p3.7.m7.1.2.2.2">𝑤</ci><cn id="S6.6.p3.7.m7.1.2.2.3.cmml" type="integer" xref="S6.6.p3.7.m7.1.2.2.3">0</cn></apply><apply id="S6.6.p3.7.m7.1.2.3.cmml" xref="S6.6.p3.7.m7.1.2.3"><times id="S6.6.p3.7.m7.1.2.3.1.cmml" xref="S6.6.p3.7.m7.1.2.3.1"></times><ci id="S6.6.p3.7.m7.1.2.3.2.cmml" xref="S6.6.p3.7.m7.1.2.3.2">ℒ</ci><ci id="S6.6.p3.7.m7.1.1.cmml" xref="S6.6.p3.7.m7.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.6.p3.7.m7.1c">w_{0}\notin\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.6.p3.7.m7.1d">italic_w start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∉ caligraphic_L ( caligraphic_X )</annotation></semantics></math>, that</p> <table class="ltx_equation ltx_eqn_table" id="S6.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.9)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu(U_{n})=\sum_{v\in V_{n}}\sigma_{*}(\mu)(v)=:\sigma_{*}(\mu)(V_{n})\,." class="ltx_math_unparsed" display="block" id="S6.E9.m1.2"><semantics id="S6.E9.m1.2a"><mrow id="S6.E9.m1.2b"><mi id="S6.E9.m1.2.3">μ</mi><mrow id="S6.E9.m1.2.4"><mo id="S6.E9.m1.2.4.1" stretchy="false">(</mo><msub id="S6.E9.m1.2.4.2"><mi id="S6.E9.m1.2.4.2.2">U</mi><mi id="S6.E9.m1.2.4.2.3">n</mi></msub><mo id="S6.E9.m1.2.4.3" stretchy="false">)</mo></mrow><mo id="S6.E9.m1.2.5" rspace="0.111em">=</mo><munder id="S6.E9.m1.2.6"><mo id="S6.E9.m1.2.6.2" movablelimits="false">∑</mo><mrow id="S6.E9.m1.2.6.3"><mi id="S6.E9.m1.2.6.3.2">v</mi><mo id="S6.E9.m1.2.6.3.1">∈</mo><msub id="S6.E9.m1.2.6.3.3"><mi id="S6.E9.m1.2.6.3.3.2">V</mi><mi id="S6.E9.m1.2.6.3.3.3">n</mi></msub></mrow></munder><msub id="S6.E9.m1.2.7"><mi id="S6.E9.m1.2.7.2">σ</mi><mo id="S6.E9.m1.2.7.3">∗</mo></msub><mrow id="S6.E9.m1.2.8"><mo id="S6.E9.m1.2.8.1" stretchy="false">(</mo><mi id="S6.E9.m1.1.1">μ</mi><mo id="S6.E9.m1.2.8.2" stretchy="false">)</mo></mrow><mrow id="S6.E9.m1.2.9"><mo id="S6.E9.m1.2.9.1" stretchy="false">(</mo><mi id="S6.E9.m1.2.2">v</mi><mo id="S6.E9.m1.2.9.2" stretchy="false">)</mo></mrow><mo id="S6.E9.m1.2.10" rspace="0em">=</mo><mo id="S6.E9.m1.2.11" rspace="0.278em">:</mo><msub id="S6.E9.m1.2.12"><mi id="S6.E9.m1.2.12.2">σ</mi><mo id="S6.E9.m1.2.12.3">∗</mo></msub><mrow id="S6.E9.m1.2.13"><mo id="S6.E9.m1.2.13.1" stretchy="false">(</mo><mi id="S6.E9.m1.2.13.2">μ</mi><mo id="S6.E9.m1.2.13.3" stretchy="false">)</mo></mrow><mrow id="S6.E9.m1.2.14"><mo id="S6.E9.m1.2.14.1" stretchy="false">(</mo><msub id="S6.E9.m1.2.14.2"><mi id="S6.E9.m1.2.14.2.2">V</mi><mi id="S6.E9.m1.2.14.2.3">n</mi></msub><mo id="S6.E9.m1.2.14.3" stretchy="false">)</mo></mrow><mo id="S6.E9.m1.2.15" lspace="0.170em">.</mo></mrow><annotation encoding="application/x-tex" id="S6.E9.m1.2c">\mu(U_{n})=\sum_{v\in V_{n}}\sigma_{*}(\mu)(v)=:\sigma_{*}(\mu)(V_{n})\,.</annotation><annotation encoding="application/x-llamapun" id="S6.E9.m1.2d">italic_μ ( italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_v ∈ italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( italic_v ) = : italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S6.7.p4"> <p class="ltx_p" id="S6.7.p4.1">We thus deduce from the equalities (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E8" title="In Proof. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E9" title="In Proof. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.9</span></a>) that for any integer <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S6.7.p4.1.m1.1"><semantics id="S6.7.p4.1.m1.1a"><mrow id="S6.7.p4.1.m1.1.1" xref="S6.7.p4.1.m1.1.1.cmml"><mi id="S6.7.p4.1.m1.1.1.2" xref="S6.7.p4.1.m1.1.1.2.cmml">n</mi><mo id="S6.7.p4.1.m1.1.1.1" xref="S6.7.p4.1.m1.1.1.1.cmml">≥</mo><mn id="S6.7.p4.1.m1.1.1.3" xref="S6.7.p4.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.1.m1.1b"><apply id="S6.7.p4.1.m1.1.1.cmml" xref="S6.7.p4.1.m1.1.1"><geq id="S6.7.p4.1.m1.1.1.1.cmml" xref="S6.7.p4.1.m1.1.1.1"></geq><ci id="S6.7.p4.1.m1.1.1.2.cmml" xref="S6.7.p4.1.m1.1.1.2">𝑛</ci><cn id="S6.7.p4.1.m1.1.1.3.cmml" type="integer" xref="S6.7.p4.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.1.m1.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.1.m1.1d">italic_n ≥ 0</annotation></semantics></math> one has</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu(w)=\mu(U_{n})+\mu(A_{n})=\sigma_{*}(\mu)(V_{n})+\mu(A_{n})\,." class="ltx_Math" display="block" id="S6.Ex7.m1.3"><semantics id="S6.Ex7.m1.3a"><mrow id="S6.Ex7.m1.3.3.1" xref="S6.Ex7.m1.3.3.1.1.cmml"><mrow id="S6.Ex7.m1.3.3.1.1" xref="S6.Ex7.m1.3.3.1.1.cmml"><mrow id="S6.Ex7.m1.3.3.1.1.6" xref="S6.Ex7.m1.3.3.1.1.6.cmml"><mi id="S6.Ex7.m1.3.3.1.1.6.2" xref="S6.Ex7.m1.3.3.1.1.6.2.cmml">μ</mi><mo id="S6.Ex7.m1.3.3.1.1.6.1" xref="S6.Ex7.m1.3.3.1.1.6.1.cmml">⁢</mo><mrow id="S6.Ex7.m1.3.3.1.1.6.3.2" xref="S6.Ex7.m1.3.3.1.1.6.cmml"><mo id="S6.Ex7.m1.3.3.1.1.6.3.2.1" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.6.cmml">(</mo><mi id="S6.Ex7.m1.1.1" xref="S6.Ex7.m1.1.1.cmml">w</mi><mo id="S6.Ex7.m1.3.3.1.1.6.3.2.2" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.6.cmml">)</mo></mrow></mrow><mo id="S6.Ex7.m1.3.3.1.1.7" xref="S6.Ex7.m1.3.3.1.1.7.cmml">=</mo><mrow id="S6.Ex7.m1.3.3.1.1.2" xref="S6.Ex7.m1.3.3.1.1.2.cmml"><mrow id="S6.Ex7.m1.3.3.1.1.1.1" xref="S6.Ex7.m1.3.3.1.1.1.1.cmml"><mi id="S6.Ex7.m1.3.3.1.1.1.1.3" xref="S6.Ex7.m1.3.3.1.1.1.1.3.cmml">μ</mi><mo id="S6.Ex7.m1.3.3.1.1.1.1.2" xref="S6.Ex7.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex7.m1.3.3.1.1.1.1.1.1" xref="S6.Ex7.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex7.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex7.m1.3.3.1.1.1.1.1.1.1" xref="S6.Ex7.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex7.m1.3.3.1.1.1.1.1.1.1.2" xref="S6.Ex7.m1.3.3.1.1.1.1.1.1.1.2.cmml">U</mi><mi id="S6.Ex7.m1.3.3.1.1.1.1.1.1.1.3" xref="S6.Ex7.m1.3.3.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.Ex7.m1.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex7.m1.3.3.1.1.2.3" xref="S6.Ex7.m1.3.3.1.1.2.3.cmml">+</mo><mrow id="S6.Ex7.m1.3.3.1.1.2.2" xref="S6.Ex7.m1.3.3.1.1.2.2.cmml"><mi id="S6.Ex7.m1.3.3.1.1.2.2.3" xref="S6.Ex7.m1.3.3.1.1.2.2.3.cmml">μ</mi><mo id="S6.Ex7.m1.3.3.1.1.2.2.2" xref="S6.Ex7.m1.3.3.1.1.2.2.2.cmml">⁢</mo><mrow id="S6.Ex7.m1.3.3.1.1.2.2.1.1" xref="S6.Ex7.m1.3.3.1.1.2.2.1.1.1.cmml"><mo id="S6.Ex7.m1.3.3.1.1.2.2.1.1.2" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.2.2.1.1.1.cmml">(</mo><msub id="S6.Ex7.m1.3.3.1.1.2.2.1.1.1" xref="S6.Ex7.m1.3.3.1.1.2.2.1.1.1.cmml"><mi id="S6.Ex7.m1.3.3.1.1.2.2.1.1.1.2" xref="S6.Ex7.m1.3.3.1.1.2.2.1.1.1.2.cmml">A</mi><mi id="S6.Ex7.m1.3.3.1.1.2.2.1.1.1.3" xref="S6.Ex7.m1.3.3.1.1.2.2.1.1.1.3.cmml">n</mi></msub><mo id="S6.Ex7.m1.3.3.1.1.2.2.1.1.3" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Ex7.m1.3.3.1.1.8" xref="S6.Ex7.m1.3.3.1.1.8.cmml">=</mo><mrow id="S6.Ex7.m1.3.3.1.1.4" xref="S6.Ex7.m1.3.3.1.1.4.cmml"><mrow id="S6.Ex7.m1.3.3.1.1.3.1" xref="S6.Ex7.m1.3.3.1.1.3.1.cmml"><msub id="S6.Ex7.m1.3.3.1.1.3.1.3" xref="S6.Ex7.m1.3.3.1.1.3.1.3.cmml"><mi id="S6.Ex7.m1.3.3.1.1.3.1.3.2" xref="S6.Ex7.m1.3.3.1.1.3.1.3.2.cmml">σ</mi><mo id="S6.Ex7.m1.3.3.1.1.3.1.3.3" xref="S6.Ex7.m1.3.3.1.1.3.1.3.3.cmml">∗</mo></msub><mo id="S6.Ex7.m1.3.3.1.1.3.1.2" xref="S6.Ex7.m1.3.3.1.1.3.1.2.cmml">⁢</mo><mrow id="S6.Ex7.m1.3.3.1.1.3.1.4.2" xref="S6.Ex7.m1.3.3.1.1.3.1.cmml"><mo id="S6.Ex7.m1.3.3.1.1.3.1.4.2.1" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.3.1.cmml">(</mo><mi id="S6.Ex7.m1.2.2" xref="S6.Ex7.m1.2.2.cmml">μ</mi><mo id="S6.Ex7.m1.3.3.1.1.3.1.4.2.2" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.3.1.cmml">)</mo></mrow><mo id="S6.Ex7.m1.3.3.1.1.3.1.2a" xref="S6.Ex7.m1.3.3.1.1.3.1.2.cmml">⁢</mo><mrow id="S6.Ex7.m1.3.3.1.1.3.1.1.1" xref="S6.Ex7.m1.3.3.1.1.3.1.1.1.1.cmml"><mo id="S6.Ex7.m1.3.3.1.1.3.1.1.1.2" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.3.1.1.1.1.cmml">(</mo><msub id="S6.Ex7.m1.3.3.1.1.3.1.1.1.1" xref="S6.Ex7.m1.3.3.1.1.3.1.1.1.1.cmml"><mi id="S6.Ex7.m1.3.3.1.1.3.1.1.1.1.2" xref="S6.Ex7.m1.3.3.1.1.3.1.1.1.1.2.cmml">V</mi><mi id="S6.Ex7.m1.3.3.1.1.3.1.1.1.1.3" xref="S6.Ex7.m1.3.3.1.1.3.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.Ex7.m1.3.3.1.1.3.1.1.1.3" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex7.m1.3.3.1.1.4.3" xref="S6.Ex7.m1.3.3.1.1.4.3.cmml">+</mo><mrow id="S6.Ex7.m1.3.3.1.1.4.2" xref="S6.Ex7.m1.3.3.1.1.4.2.cmml"><mi id="S6.Ex7.m1.3.3.1.1.4.2.3" xref="S6.Ex7.m1.3.3.1.1.4.2.3.cmml">μ</mi><mo id="S6.Ex7.m1.3.3.1.1.4.2.2" xref="S6.Ex7.m1.3.3.1.1.4.2.2.cmml">⁢</mo><mrow id="S6.Ex7.m1.3.3.1.1.4.2.1.1" xref="S6.Ex7.m1.3.3.1.1.4.2.1.1.1.cmml"><mo id="S6.Ex7.m1.3.3.1.1.4.2.1.1.2" stretchy="false" xref="S6.Ex7.m1.3.3.1.1.4.2.1.1.1.cmml">(</mo><msub id="S6.Ex7.m1.3.3.1.1.4.2.1.1.1" xref="S6.Ex7.m1.3.3.1.1.4.2.1.1.1.cmml"><mi 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.7.p4.8">We now pass to the limit for <math alttext="n\to\infty" class="ltx_Math" display="inline" id="S6.7.p4.2.m1.1"><semantics id="S6.7.p4.2.m1.1a"><mrow id="S6.7.p4.2.m1.1.1" xref="S6.7.p4.2.m1.1.1.cmml"><mi id="S6.7.p4.2.m1.1.1.2" xref="S6.7.p4.2.m1.1.1.2.cmml">n</mi><mo id="S6.7.p4.2.m1.1.1.1" stretchy="false" xref="S6.7.p4.2.m1.1.1.1.cmml">→</mo><mi id="S6.7.p4.2.m1.1.1.3" mathvariant="normal" xref="S6.7.p4.2.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.2.m1.1b"><apply id="S6.7.p4.2.m1.1.1.cmml" xref="S6.7.p4.2.m1.1.1"><ci id="S6.7.p4.2.m1.1.1.1.cmml" xref="S6.7.p4.2.m1.1.1.1">→</ci><ci id="S6.7.p4.2.m1.1.1.2.cmml" xref="S6.7.p4.2.m1.1.1.2">𝑛</ci><infinity id="S6.7.p4.2.m1.1.1.3.cmml" xref="S6.7.p4.2.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.2.m1.1c">n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.2.m1.1d">italic_n → ∞</annotation></semantics></math> and recall from Remark <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm2" title="Remark 6.2. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.2</span></a> (2) that <math alttext="\lim\mu(A_{n})=\mu(A_{\infty}(w))" class="ltx_Math" display="inline" id="S6.7.p4.3.m2.3"><semantics id="S6.7.p4.3.m2.3a"><mrow id="S6.7.p4.3.m2.3.3" xref="S6.7.p4.3.m2.3.3.cmml"><mrow id="S6.7.p4.3.m2.2.2.1" xref="S6.7.p4.3.m2.2.2.1.cmml"><mo id="S6.7.p4.3.m2.2.2.1.2" rspace="0.167em" xref="S6.7.p4.3.m2.2.2.1.2.cmml">lim</mo><mrow id="S6.7.p4.3.m2.2.2.1.1" xref="S6.7.p4.3.m2.2.2.1.1.cmml"><mi id="S6.7.p4.3.m2.2.2.1.1.3" xref="S6.7.p4.3.m2.2.2.1.1.3.cmml">μ</mi><mo id="S6.7.p4.3.m2.2.2.1.1.2" xref="S6.7.p4.3.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S6.7.p4.3.m2.2.2.1.1.1.1" xref="S6.7.p4.3.m2.2.2.1.1.1.1.1.cmml"><mo id="S6.7.p4.3.m2.2.2.1.1.1.1.2" stretchy="false" xref="S6.7.p4.3.m2.2.2.1.1.1.1.1.cmml">(</mo><msub id="S6.7.p4.3.m2.2.2.1.1.1.1.1" xref="S6.7.p4.3.m2.2.2.1.1.1.1.1.cmml"><mi id="S6.7.p4.3.m2.2.2.1.1.1.1.1.2" xref="S6.7.p4.3.m2.2.2.1.1.1.1.1.2.cmml">A</mi><mi id="S6.7.p4.3.m2.2.2.1.1.1.1.1.3" xref="S6.7.p4.3.m2.2.2.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.7.p4.3.m2.2.2.1.1.1.1.3" stretchy="false" xref="S6.7.p4.3.m2.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.7.p4.3.m2.3.3.3" xref="S6.7.p4.3.m2.3.3.3.cmml">=</mo><mrow id="S6.7.p4.3.m2.3.3.2" xref="S6.7.p4.3.m2.3.3.2.cmml"><mi id="S6.7.p4.3.m2.3.3.2.3" xref="S6.7.p4.3.m2.3.3.2.3.cmml">μ</mi><mo id="S6.7.p4.3.m2.3.3.2.2" xref="S6.7.p4.3.m2.3.3.2.2.cmml">⁢</mo><mrow id="S6.7.p4.3.m2.3.3.2.1.1" xref="S6.7.p4.3.m2.3.3.2.1.1.1.cmml"><mo id="S6.7.p4.3.m2.3.3.2.1.1.2" stretchy="false" xref="S6.7.p4.3.m2.3.3.2.1.1.1.cmml">(</mo><mrow id="S6.7.p4.3.m2.3.3.2.1.1.1" xref="S6.7.p4.3.m2.3.3.2.1.1.1.cmml"><msub id="S6.7.p4.3.m2.3.3.2.1.1.1.2" xref="S6.7.p4.3.m2.3.3.2.1.1.1.2.cmml"><mi id="S6.7.p4.3.m2.3.3.2.1.1.1.2.2" xref="S6.7.p4.3.m2.3.3.2.1.1.1.2.2.cmml">A</mi><mi id="S6.7.p4.3.m2.3.3.2.1.1.1.2.3" mathvariant="normal" xref="S6.7.p4.3.m2.3.3.2.1.1.1.2.3.cmml">∞</mi></msub><mo id="S6.7.p4.3.m2.3.3.2.1.1.1.1" xref="S6.7.p4.3.m2.3.3.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.7.p4.3.m2.3.3.2.1.1.1.3.2" xref="S6.7.p4.3.m2.3.3.2.1.1.1.cmml"><mo id="S6.7.p4.3.m2.3.3.2.1.1.1.3.2.1" stretchy="false" xref="S6.7.p4.3.m2.3.3.2.1.1.1.cmml">(</mo><mi id="S6.7.p4.3.m2.1.1" xref="S6.7.p4.3.m2.1.1.cmml">w</mi><mo id="S6.7.p4.3.m2.3.3.2.1.1.1.3.2.2" stretchy="false" xref="S6.7.p4.3.m2.3.3.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.7.p4.3.m2.3.3.2.1.1.3" stretchy="false" xref="S6.7.p4.3.m2.3.3.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.3.m2.3b"><apply id="S6.7.p4.3.m2.3.3.cmml" xref="S6.7.p4.3.m2.3.3"><eq id="S6.7.p4.3.m2.3.3.3.cmml" xref="S6.7.p4.3.m2.3.3.3"></eq><apply id="S6.7.p4.3.m2.2.2.1.cmml" xref="S6.7.p4.3.m2.2.2.1"><limit id="S6.7.p4.3.m2.2.2.1.2.cmml" xref="S6.7.p4.3.m2.2.2.1.2"></limit><apply id="S6.7.p4.3.m2.2.2.1.1.cmml" xref="S6.7.p4.3.m2.2.2.1.1"><times id="S6.7.p4.3.m2.2.2.1.1.2.cmml" xref="S6.7.p4.3.m2.2.2.1.1.2"></times><ci id="S6.7.p4.3.m2.2.2.1.1.3.cmml" xref="S6.7.p4.3.m2.2.2.1.1.3">𝜇</ci><apply id="S6.7.p4.3.m2.2.2.1.1.1.1.1.cmml" xref="S6.7.p4.3.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.7.p4.3.m2.2.2.1.1.1.1.1.1.cmml" xref="S6.7.p4.3.m2.2.2.1.1.1.1">subscript</csymbol><ci id="S6.7.p4.3.m2.2.2.1.1.1.1.1.2.cmml" xref="S6.7.p4.3.m2.2.2.1.1.1.1.1.2">𝐴</ci><ci id="S6.7.p4.3.m2.2.2.1.1.1.1.1.3.cmml" xref="S6.7.p4.3.m2.2.2.1.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="S6.7.p4.3.m2.3.3.2.cmml" xref="S6.7.p4.3.m2.3.3.2"><times id="S6.7.p4.3.m2.3.3.2.2.cmml" xref="S6.7.p4.3.m2.3.3.2.2"></times><ci id="S6.7.p4.3.m2.3.3.2.3.cmml" xref="S6.7.p4.3.m2.3.3.2.3">𝜇</ci><apply id="S6.7.p4.3.m2.3.3.2.1.1.1.cmml" xref="S6.7.p4.3.m2.3.3.2.1.1"><times id="S6.7.p4.3.m2.3.3.2.1.1.1.1.cmml" xref="S6.7.p4.3.m2.3.3.2.1.1.1.1"></times><apply id="S6.7.p4.3.m2.3.3.2.1.1.1.2.cmml" xref="S6.7.p4.3.m2.3.3.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.7.p4.3.m2.3.3.2.1.1.1.2.1.cmml" xref="S6.7.p4.3.m2.3.3.2.1.1.1.2">subscript</csymbol><ci id="S6.7.p4.3.m2.3.3.2.1.1.1.2.2.cmml" xref="S6.7.p4.3.m2.3.3.2.1.1.1.2.2">𝐴</ci><infinity id="S6.7.p4.3.m2.3.3.2.1.1.1.2.3.cmml" xref="S6.7.p4.3.m2.3.3.2.1.1.1.2.3"></infinity></apply><ci id="S6.7.p4.3.m2.1.1.cmml" xref="S6.7.p4.3.m2.1.1">𝑤</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.3.m2.3c">\lim\mu(A_{n})=\mu(A_{\infty}(w))</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.3.m2.3d">roman_lim italic_μ ( italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_μ ( italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( italic_w ) )</annotation></semantics></math>. From Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm3" title="Lemma 6.3. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.3</span></a> we know <math alttext="A_{\infty}(w)\subseteq\text{\rm Per}(X)" class="ltx_Math" display="inline" id="S6.7.p4.4.m3.2"><semantics id="S6.7.p4.4.m3.2a"><mrow id="S6.7.p4.4.m3.2.3" xref="S6.7.p4.4.m3.2.3.cmml"><mrow id="S6.7.p4.4.m3.2.3.2" xref="S6.7.p4.4.m3.2.3.2.cmml"><msub id="S6.7.p4.4.m3.2.3.2.2" xref="S6.7.p4.4.m3.2.3.2.2.cmml"><mi id="S6.7.p4.4.m3.2.3.2.2.2" xref="S6.7.p4.4.m3.2.3.2.2.2.cmml">A</mi><mi id="S6.7.p4.4.m3.2.3.2.2.3" mathvariant="normal" xref="S6.7.p4.4.m3.2.3.2.2.3.cmml">∞</mi></msub><mo id="S6.7.p4.4.m3.2.3.2.1" xref="S6.7.p4.4.m3.2.3.2.1.cmml">⁢</mo><mrow id="S6.7.p4.4.m3.2.3.2.3.2" xref="S6.7.p4.4.m3.2.3.2.cmml"><mo id="S6.7.p4.4.m3.2.3.2.3.2.1" stretchy="false" xref="S6.7.p4.4.m3.2.3.2.cmml">(</mo><mi id="S6.7.p4.4.m3.1.1" xref="S6.7.p4.4.m3.1.1.cmml">w</mi><mo id="S6.7.p4.4.m3.2.3.2.3.2.2" stretchy="false" xref="S6.7.p4.4.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.7.p4.4.m3.2.3.1" xref="S6.7.p4.4.m3.2.3.1.cmml">⊆</mo><mrow id="S6.7.p4.4.m3.2.3.3" xref="S6.7.p4.4.m3.2.3.3.cmml"><mtext id="S6.7.p4.4.m3.2.3.3.2" xref="S6.7.p4.4.m3.2.3.3.2a.cmml">Per</mtext><mo id="S6.7.p4.4.m3.2.3.3.1" xref="S6.7.p4.4.m3.2.3.3.1.cmml">⁢</mo><mrow id="S6.7.p4.4.m3.2.3.3.3.2" xref="S6.7.p4.4.m3.2.3.3.cmml"><mo id="S6.7.p4.4.m3.2.3.3.3.2.1" stretchy="false" xref="S6.7.p4.4.m3.2.3.3.cmml">(</mo><mi id="S6.7.p4.4.m3.2.2" xref="S6.7.p4.4.m3.2.2.cmml">X</mi><mo id="S6.7.p4.4.m3.2.3.3.3.2.2" stretchy="false" xref="S6.7.p4.4.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.4.m3.2b"><apply id="S6.7.p4.4.m3.2.3.cmml" xref="S6.7.p4.4.m3.2.3"><subset id="S6.7.p4.4.m3.2.3.1.cmml" xref="S6.7.p4.4.m3.2.3.1"></subset><apply id="S6.7.p4.4.m3.2.3.2.cmml" xref="S6.7.p4.4.m3.2.3.2"><times id="S6.7.p4.4.m3.2.3.2.1.cmml" xref="S6.7.p4.4.m3.2.3.2.1"></times><apply id="S6.7.p4.4.m3.2.3.2.2.cmml" xref="S6.7.p4.4.m3.2.3.2.2"><csymbol cd="ambiguous" id="S6.7.p4.4.m3.2.3.2.2.1.cmml" xref="S6.7.p4.4.m3.2.3.2.2">subscript</csymbol><ci id="S6.7.p4.4.m3.2.3.2.2.2.cmml" xref="S6.7.p4.4.m3.2.3.2.2.2">𝐴</ci><infinity id="S6.7.p4.4.m3.2.3.2.2.3.cmml" xref="S6.7.p4.4.m3.2.3.2.2.3"></infinity></apply><ci id="S6.7.p4.4.m3.1.1.cmml" xref="S6.7.p4.4.m3.1.1">𝑤</ci></apply><apply id="S6.7.p4.4.m3.2.3.3.cmml" xref="S6.7.p4.4.m3.2.3.3"><times id="S6.7.p4.4.m3.2.3.3.1.cmml" xref="S6.7.p4.4.m3.2.3.3.1"></times><ci id="S6.7.p4.4.m3.2.3.3.2a.cmml" xref="S6.7.p4.4.m3.2.3.3.2"><mtext id="S6.7.p4.4.m3.2.3.3.2.cmml" xref="S6.7.p4.4.m3.2.3.3.2">Per</mtext></ci><ci id="S6.7.p4.4.m3.2.2.cmml" xref="S6.7.p4.4.m3.2.2">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.4.m3.2c">A_{\infty}(w)\subseteq\text{\rm Per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.4.m3.2d">italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( italic_w ) ⊆ Per ( italic_X )</annotation></semantics></math>, so that our assumption that <math alttext="\mu" class="ltx_Math" display="inline" id="S6.7.p4.5.m4.1"><semantics id="S6.7.p4.5.m4.1a"><mi id="S6.7.p4.5.m4.1.1" xref="S6.7.p4.5.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.7.p4.5.m4.1b"><ci id="S6.7.p4.5.m4.1.1.cmml" xref="S6.7.p4.5.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.5.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.5.m4.1d">italic_μ</annotation></semantics></math> is non-atomic and hence zero on <math alttext="\text{\rm Per}(X)" class="ltx_Math" display="inline" id="S6.7.p4.6.m5.1"><semantics id="S6.7.p4.6.m5.1a"><mrow id="S6.7.p4.6.m5.1.2" xref="S6.7.p4.6.m5.1.2.cmml"><mtext id="S6.7.p4.6.m5.1.2.2" xref="S6.7.p4.6.m5.1.2.2a.cmml">Per</mtext><mo id="S6.7.p4.6.m5.1.2.1" xref="S6.7.p4.6.m5.1.2.1.cmml">⁢</mo><mrow id="S6.7.p4.6.m5.1.2.3.2" xref="S6.7.p4.6.m5.1.2.cmml"><mo id="S6.7.p4.6.m5.1.2.3.2.1" stretchy="false" xref="S6.7.p4.6.m5.1.2.cmml">(</mo><mi id="S6.7.p4.6.m5.1.1" xref="S6.7.p4.6.m5.1.1.cmml">X</mi><mo id="S6.7.p4.6.m5.1.2.3.2.2" stretchy="false" xref="S6.7.p4.6.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.6.m5.1b"><apply id="S6.7.p4.6.m5.1.2.cmml" xref="S6.7.p4.6.m5.1.2"><times id="S6.7.p4.6.m5.1.2.1.cmml" xref="S6.7.p4.6.m5.1.2.1"></times><ci id="S6.7.p4.6.m5.1.2.2a.cmml" xref="S6.7.p4.6.m5.1.2.2"><mtext id="S6.7.p4.6.m5.1.2.2.cmml" xref="S6.7.p4.6.m5.1.2.2">Per</mtext></ci><ci id="S6.7.p4.6.m5.1.1.cmml" xref="S6.7.p4.6.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.6.m5.1c">\text{\rm Per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.6.m5.1d">Per ( italic_X )</annotation></semantics></math> implies <math alttext="\lim\mu(A_{n})=0" class="ltx_Math" display="inline" id="S6.7.p4.7.m6.1"><semantics id="S6.7.p4.7.m6.1a"><mrow id="S6.7.p4.7.m6.1.1" xref="S6.7.p4.7.m6.1.1.cmml"><mrow id="S6.7.p4.7.m6.1.1.1" xref="S6.7.p4.7.m6.1.1.1.cmml"><mo id="S6.7.p4.7.m6.1.1.1.2" rspace="0.167em" xref="S6.7.p4.7.m6.1.1.1.2.cmml">lim</mo><mrow id="S6.7.p4.7.m6.1.1.1.1" xref="S6.7.p4.7.m6.1.1.1.1.cmml"><mi id="S6.7.p4.7.m6.1.1.1.1.3" xref="S6.7.p4.7.m6.1.1.1.1.3.cmml">μ</mi><mo id="S6.7.p4.7.m6.1.1.1.1.2" xref="S6.7.p4.7.m6.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.7.p4.7.m6.1.1.1.1.1.1" xref="S6.7.p4.7.m6.1.1.1.1.1.1.1.cmml"><mo id="S6.7.p4.7.m6.1.1.1.1.1.1.2" stretchy="false" xref="S6.7.p4.7.m6.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.7.p4.7.m6.1.1.1.1.1.1.1" xref="S6.7.p4.7.m6.1.1.1.1.1.1.1.cmml"><mi id="S6.7.p4.7.m6.1.1.1.1.1.1.1.2" xref="S6.7.p4.7.m6.1.1.1.1.1.1.1.2.cmml">A</mi><mi id="S6.7.p4.7.m6.1.1.1.1.1.1.1.3" xref="S6.7.p4.7.m6.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.7.p4.7.m6.1.1.1.1.1.1.3" stretchy="false" xref="S6.7.p4.7.m6.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.7.p4.7.m6.1.1.2" xref="S6.7.p4.7.m6.1.1.2.cmml">=</mo><mn id="S6.7.p4.7.m6.1.1.3" xref="S6.7.p4.7.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.7.m6.1b"><apply id="S6.7.p4.7.m6.1.1.cmml" xref="S6.7.p4.7.m6.1.1"><eq id="S6.7.p4.7.m6.1.1.2.cmml" xref="S6.7.p4.7.m6.1.1.2"></eq><apply id="S6.7.p4.7.m6.1.1.1.cmml" xref="S6.7.p4.7.m6.1.1.1"><limit id="S6.7.p4.7.m6.1.1.1.2.cmml" xref="S6.7.p4.7.m6.1.1.1.2"></limit><apply id="S6.7.p4.7.m6.1.1.1.1.cmml" xref="S6.7.p4.7.m6.1.1.1.1"><times id="S6.7.p4.7.m6.1.1.1.1.2.cmml" xref="S6.7.p4.7.m6.1.1.1.1.2"></times><ci id="S6.7.p4.7.m6.1.1.1.1.3.cmml" xref="S6.7.p4.7.m6.1.1.1.1.3">𝜇</ci><apply id="S6.7.p4.7.m6.1.1.1.1.1.1.1.cmml" xref="S6.7.p4.7.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.7.p4.7.m6.1.1.1.1.1.1.1.1.cmml" xref="S6.7.p4.7.m6.1.1.1.1.1.1">subscript</csymbol><ci id="S6.7.p4.7.m6.1.1.1.1.1.1.1.2.cmml" xref="S6.7.p4.7.m6.1.1.1.1.1.1.1.2">𝐴</ci><ci id="S6.7.p4.7.m6.1.1.1.1.1.1.1.3.cmml" xref="S6.7.p4.7.m6.1.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply><cn id="S6.7.p4.7.m6.1.1.3.cmml" type="integer" xref="S6.7.p4.7.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.7.m6.1c">\lim\mu(A_{n})=0</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.7.m6.1d">roman_lim italic_μ ( italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>. Hence <math alttext="\lim\sigma_{*}(\mu)(V_{n})" class="ltx_Math" display="inline" id="S6.7.p4.8.m7.2"><semantics id="S6.7.p4.8.m7.2a"><mrow id="S6.7.p4.8.m7.2.2" xref="S6.7.p4.8.m7.2.2.cmml"><mo id="S6.7.p4.8.m7.2.2.2" rspace="0.167em" xref="S6.7.p4.8.m7.2.2.2.cmml">lim</mo><mrow id="S6.7.p4.8.m7.2.2.1" xref="S6.7.p4.8.m7.2.2.1.cmml"><msub id="S6.7.p4.8.m7.2.2.1.3" xref="S6.7.p4.8.m7.2.2.1.3.cmml"><mi id="S6.7.p4.8.m7.2.2.1.3.2" xref="S6.7.p4.8.m7.2.2.1.3.2.cmml">σ</mi><mo id="S6.7.p4.8.m7.2.2.1.3.3" xref="S6.7.p4.8.m7.2.2.1.3.3.cmml">∗</mo></msub><mo id="S6.7.p4.8.m7.2.2.1.2" xref="S6.7.p4.8.m7.2.2.1.2.cmml">⁢</mo><mrow id="S6.7.p4.8.m7.2.2.1.4.2" xref="S6.7.p4.8.m7.2.2.1.cmml"><mo id="S6.7.p4.8.m7.2.2.1.4.2.1" stretchy="false" xref="S6.7.p4.8.m7.2.2.1.cmml">(</mo><mi id="S6.7.p4.8.m7.1.1" xref="S6.7.p4.8.m7.1.1.cmml">μ</mi><mo id="S6.7.p4.8.m7.2.2.1.4.2.2" stretchy="false" xref="S6.7.p4.8.m7.2.2.1.cmml">)</mo></mrow><mo id="S6.7.p4.8.m7.2.2.1.2a" xref="S6.7.p4.8.m7.2.2.1.2.cmml">⁢</mo><mrow id="S6.7.p4.8.m7.2.2.1.1.1" xref="S6.7.p4.8.m7.2.2.1.1.1.1.cmml"><mo id="S6.7.p4.8.m7.2.2.1.1.1.2" stretchy="false" xref="S6.7.p4.8.m7.2.2.1.1.1.1.cmml">(</mo><msub id="S6.7.p4.8.m7.2.2.1.1.1.1" xref="S6.7.p4.8.m7.2.2.1.1.1.1.cmml"><mi id="S6.7.p4.8.m7.2.2.1.1.1.1.2" xref="S6.7.p4.8.m7.2.2.1.1.1.1.2.cmml">V</mi><mi id="S6.7.p4.8.m7.2.2.1.1.1.1.3" xref="S6.7.p4.8.m7.2.2.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.7.p4.8.m7.2.2.1.1.1.3" stretchy="false" xref="S6.7.p4.8.m7.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.8.m7.2b"><apply id="S6.7.p4.8.m7.2.2.cmml" xref="S6.7.p4.8.m7.2.2"><limit id="S6.7.p4.8.m7.2.2.2.cmml" xref="S6.7.p4.8.m7.2.2.2"></limit><apply id="S6.7.p4.8.m7.2.2.1.cmml" xref="S6.7.p4.8.m7.2.2.1"><times id="S6.7.p4.8.m7.2.2.1.2.cmml" xref="S6.7.p4.8.m7.2.2.1.2"></times><apply id="S6.7.p4.8.m7.2.2.1.3.cmml" xref="S6.7.p4.8.m7.2.2.1.3"><csymbol cd="ambiguous" id="S6.7.p4.8.m7.2.2.1.3.1.cmml" xref="S6.7.p4.8.m7.2.2.1.3">subscript</csymbol><ci id="S6.7.p4.8.m7.2.2.1.3.2.cmml" xref="S6.7.p4.8.m7.2.2.1.3.2">𝜎</ci><times id="S6.7.p4.8.m7.2.2.1.3.3.cmml" xref="S6.7.p4.8.m7.2.2.1.3.3"></times></apply><ci id="S6.7.p4.8.m7.1.1.cmml" xref="S6.7.p4.8.m7.1.1">𝜇</ci><apply id="S6.7.p4.8.m7.2.2.1.1.1.1.cmml" xref="S6.7.p4.8.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.7.p4.8.m7.2.2.1.1.1.1.1.cmml" xref="S6.7.p4.8.m7.2.2.1.1.1">subscript</csymbol><ci id="S6.7.p4.8.m7.2.2.1.1.1.1.2.cmml" xref="S6.7.p4.8.m7.2.2.1.1.1.1.2">𝑉</ci><ci id="S6.7.p4.8.m7.2.2.1.1.1.1.3.cmml" xref="S6.7.p4.8.m7.2.2.1.1.1.1.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.8.m7.2c">\lim\sigma_{*}(\mu)(V_{n})</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.8.m7.2d">roman_lim italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> does exist and satisfies</p> <table class="ltx_equation ltx_eqn_table" id="S6.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_left">(6.10)</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lim_{n\to\infty}\sigma_{*}(\mu)(V_{n})=\mu(w)\,." class="ltx_Math" display="block" id="S6.E10.m1.3"><semantics id="S6.E10.m1.3a"><mrow id="S6.E10.m1.3.3.1" xref="S6.E10.m1.3.3.1.1.cmml"><mrow id="S6.E10.m1.3.3.1.1" xref="S6.E10.m1.3.3.1.1.cmml"><mrow id="S6.E10.m1.3.3.1.1.1" xref="S6.E10.m1.3.3.1.1.1.cmml"><munder id="S6.E10.m1.3.3.1.1.1.2" xref="S6.E10.m1.3.3.1.1.1.2.cmml"><mo id="S6.E10.m1.3.3.1.1.1.2.2" movablelimits="false" xref="S6.E10.m1.3.3.1.1.1.2.2.cmml">lim</mo><mrow id="S6.E10.m1.3.3.1.1.1.2.3" xref="S6.E10.m1.3.3.1.1.1.2.3.cmml"><mi id="S6.E10.m1.3.3.1.1.1.2.3.2" xref="S6.E10.m1.3.3.1.1.1.2.3.2.cmml">n</mi><mo id="S6.E10.m1.3.3.1.1.1.2.3.1" stretchy="false" xref="S6.E10.m1.3.3.1.1.1.2.3.1.cmml">→</mo><mi id="S6.E10.m1.3.3.1.1.1.2.3.3" mathvariant="normal" xref="S6.E10.m1.3.3.1.1.1.2.3.3.cmml">∞</mi></mrow></munder><mrow id="S6.E10.m1.3.3.1.1.1.1" xref="S6.E10.m1.3.3.1.1.1.1.cmml"><msub id="S6.E10.m1.3.3.1.1.1.1.3" xref="S6.E10.m1.3.3.1.1.1.1.3.cmml"><mi id="S6.E10.m1.3.3.1.1.1.1.3.2" xref="S6.E10.m1.3.3.1.1.1.1.3.2.cmml">σ</mi><mo id="S6.E10.m1.3.3.1.1.1.1.3.3" xref="S6.E10.m1.3.3.1.1.1.1.3.3.cmml">∗</mo></msub><mo id="S6.E10.m1.3.3.1.1.1.1.2" xref="S6.E10.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.E10.m1.3.3.1.1.1.1.4.2" xref="S6.E10.m1.3.3.1.1.1.1.cmml"><mo id="S6.E10.m1.3.3.1.1.1.1.4.2.1" stretchy="false" xref="S6.E10.m1.3.3.1.1.1.1.cmml">(</mo><mi id="S6.E10.m1.1.1" xref="S6.E10.m1.1.1.cmml">μ</mi><mo id="S6.E10.m1.3.3.1.1.1.1.4.2.2" stretchy="false" xref="S6.E10.m1.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S6.E10.m1.3.3.1.1.1.1.2a" xref="S6.E10.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.E10.m1.3.3.1.1.1.1.1.1" xref="S6.E10.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S6.E10.m1.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.E10.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.E10.m1.3.3.1.1.1.1.1.1.1" xref="S6.E10.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S6.E10.m1.3.3.1.1.1.1.1.1.1.2" xref="S6.E10.m1.3.3.1.1.1.1.1.1.1.2.cmml">V</mi><mi id="S6.E10.m1.3.3.1.1.1.1.1.1.1.3" xref="S6.E10.m1.3.3.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S6.E10.m1.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S6.E10.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.E10.m1.3.3.1.1.2" xref="S6.E10.m1.3.3.1.1.2.cmml">=</mo><mrow id="S6.E10.m1.3.3.1.1.3" xref="S6.E10.m1.3.3.1.1.3.cmml"><mi id="S6.E10.m1.3.3.1.1.3.2" xref="S6.E10.m1.3.3.1.1.3.2.cmml">μ</mi><mo id="S6.E10.m1.3.3.1.1.3.1" xref="S6.E10.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S6.E10.m1.3.3.1.1.3.3.2" xref="S6.E10.m1.3.3.1.1.3.cmml"><mo id="S6.E10.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S6.E10.m1.3.3.1.1.3.cmml">(</mo><mi id="S6.E10.m1.2.2" xref="S6.E10.m1.2.2.cmml">w</mi><mo id="S6.E10.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="S6.E10.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S6.E10.m1.3.3.1.2" lspace="0.170em" xref="S6.E10.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.E10.m1.3b"><apply id="S6.E10.m1.3.3.1.1.cmml" xref="S6.E10.m1.3.3.1"><eq id="S6.E10.m1.3.3.1.1.2.cmml" xref="S6.E10.m1.3.3.1.1.2"></eq><apply id="S6.E10.m1.3.3.1.1.1.cmml" xref="S6.E10.m1.3.3.1.1.1"><apply id="S6.E10.m1.3.3.1.1.1.2.cmml" xref="S6.E10.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S6.E10.m1.3.3.1.1.1.2.1.cmml" xref="S6.E10.m1.3.3.1.1.1.2">subscript</csymbol><limit id="S6.E10.m1.3.3.1.1.1.2.2.cmml" xref="S6.E10.m1.3.3.1.1.1.2.2"></limit><apply id="S6.E10.m1.3.3.1.1.1.2.3.cmml" xref="S6.E10.m1.3.3.1.1.1.2.3"><ci id="S6.E10.m1.3.3.1.1.1.2.3.1.cmml" xref="S6.E10.m1.3.3.1.1.1.2.3.1">→</ci><ci id="S6.E10.m1.3.3.1.1.1.2.3.2.cmml" xref="S6.E10.m1.3.3.1.1.1.2.3.2">𝑛</ci><infinity id="S6.E10.m1.3.3.1.1.1.2.3.3.cmml" xref="S6.E10.m1.3.3.1.1.1.2.3.3"></infinity></apply></apply><apply id="S6.E10.m1.3.3.1.1.1.1.cmml" xref="S6.E10.m1.3.3.1.1.1.1"><times id="S6.E10.m1.3.3.1.1.1.1.2.cmml" xref="S6.E10.m1.3.3.1.1.1.1.2"></times><apply id="S6.E10.m1.3.3.1.1.1.1.3.cmml" xref="S6.E10.m1.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.E10.m1.3.3.1.1.1.1.3.1.cmml" xref="S6.E10.m1.3.3.1.1.1.1.3">subscript</csymbol><ci id="S6.E10.m1.3.3.1.1.1.1.3.2.cmml" xref="S6.E10.m1.3.3.1.1.1.1.3.2">𝜎</ci><times id="S6.E10.m1.3.3.1.1.1.1.3.3.cmml" xref="S6.E10.m1.3.3.1.1.1.1.3.3"></times></apply><ci id="S6.E10.m1.1.1.cmml" xref="S6.E10.m1.1.1">𝜇</ci><apply id="S6.E10.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S6.E10.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.E10.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S6.E10.m1.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S6.E10.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.E10.m1.3.3.1.1.1.1.1.1.1.2">𝑉</ci><ci id="S6.E10.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S6.E10.m1.3.3.1.1.1.1.1.1.1.3">𝑛</ci></apply></apply></apply><apply id="S6.E10.m1.3.3.1.1.3.cmml" xref="S6.E10.m1.3.3.1.1.3"><times id="S6.E10.m1.3.3.1.1.3.1.cmml" xref="S6.E10.m1.3.3.1.1.3.1"></times><ci id="S6.E10.m1.3.3.1.1.3.2.cmml" xref="S6.E10.m1.3.3.1.1.3.2">𝜇</ci><ci id="S6.E10.m1.2.2.cmml" xref="S6.E10.m1.2.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E10.m1.3c">\lim_{n\to\infty}\sigma_{*}(\mu)(V_{n})=\mu(w)\,.</annotation><annotation encoding="application/x-llamapun" id="S6.E10.m1.3d">roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) = italic_μ ( italic_w ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.7.p4.15">Since the sets <math alttext="V_{n}" class="ltx_Math" display="inline" id="S6.7.p4.9.m1.1"><semantics id="S6.7.p4.9.m1.1a"><msub id="S6.7.p4.9.m1.1.1" xref="S6.7.p4.9.m1.1.1.cmml"><mi id="S6.7.p4.9.m1.1.1.2" xref="S6.7.p4.9.m1.1.1.2.cmml">V</mi><mi id="S6.7.p4.9.m1.1.1.3" xref="S6.7.p4.9.m1.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S6.7.p4.9.m1.1b"><apply id="S6.7.p4.9.m1.1.1.cmml" xref="S6.7.p4.9.m1.1.1"><csymbol cd="ambiguous" id="S6.7.p4.9.m1.1.1.1.cmml" xref="S6.7.p4.9.m1.1.1">subscript</csymbol><ci id="S6.7.p4.9.m1.1.1.2.cmml" xref="S6.7.p4.9.m1.1.1.2">𝑉</ci><ci id="S6.7.p4.9.m1.1.1.3.cmml" xref="S6.7.p4.9.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.9.m1.1c">V_{n}</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.9.m1.1d">italic_V start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> depend only on <math alttext="w" class="ltx_Math" display="inline" id="S6.7.p4.10.m2.1"><semantics id="S6.7.p4.10.m2.1a"><mi id="S6.7.p4.10.m2.1.1" xref="S6.7.p4.10.m2.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S6.7.p4.10.m2.1b"><ci id="S6.7.p4.10.m2.1.1.cmml" xref="S6.7.p4.10.m2.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.10.m2.1c">w</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.10.m2.1d">italic_w</annotation></semantics></math> and not on <math alttext="\mu" class="ltx_Math" display="inline" id="S6.7.p4.11.m3.1"><semantics id="S6.7.p4.11.m3.1a"><mi id="S6.7.p4.11.m3.1.1" xref="S6.7.p4.11.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.7.p4.11.m3.1b"><ci id="S6.7.p4.11.m3.1.1.cmml" xref="S6.7.p4.11.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.11.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.11.m3.1d">italic_μ</annotation></semantics></math>, equality (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E10" title="In Proof. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.10</span></a>) shows that <math alttext="\mu(w)" class="ltx_Math" display="inline" id="S6.7.p4.12.m4.1"><semantics id="S6.7.p4.12.m4.1a"><mrow id="S6.7.p4.12.m4.1.2" xref="S6.7.p4.12.m4.1.2.cmml"><mi id="S6.7.p4.12.m4.1.2.2" xref="S6.7.p4.12.m4.1.2.2.cmml">μ</mi><mo id="S6.7.p4.12.m4.1.2.1" xref="S6.7.p4.12.m4.1.2.1.cmml">⁢</mo><mrow id="S6.7.p4.12.m4.1.2.3.2" xref="S6.7.p4.12.m4.1.2.cmml"><mo id="S6.7.p4.12.m4.1.2.3.2.1" stretchy="false" xref="S6.7.p4.12.m4.1.2.cmml">(</mo><mi id="S6.7.p4.12.m4.1.1" xref="S6.7.p4.12.m4.1.1.cmml">w</mi><mo id="S6.7.p4.12.m4.1.2.3.2.2" stretchy="false" xref="S6.7.p4.12.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.12.m4.1b"><apply id="S6.7.p4.12.m4.1.2.cmml" xref="S6.7.p4.12.m4.1.2"><times id="S6.7.p4.12.m4.1.2.1.cmml" xref="S6.7.p4.12.m4.1.2.1"></times><ci id="S6.7.p4.12.m4.1.2.2.cmml" xref="S6.7.p4.12.m4.1.2.2">𝜇</ci><ci id="S6.7.p4.12.m4.1.1.cmml" xref="S6.7.p4.12.m4.1.1">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.12.m4.1c">\mu(w)</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.12.m4.1d">italic_μ ( italic_w )</annotation></semantics></math> is entirely determined by the values of the induced measure <math alttext="\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.7.p4.13.m5.1"><semantics id="S6.7.p4.13.m5.1a"><mrow id="S6.7.p4.13.m5.1.2" xref="S6.7.p4.13.m5.1.2.cmml"><msub id="S6.7.p4.13.m5.1.2.2" xref="S6.7.p4.13.m5.1.2.2.cmml"><mi id="S6.7.p4.13.m5.1.2.2.2" xref="S6.7.p4.13.m5.1.2.2.2.cmml">σ</mi><mo id="S6.7.p4.13.m5.1.2.2.3" xref="S6.7.p4.13.m5.1.2.2.3.cmml">∗</mo></msub><mo id="S6.7.p4.13.m5.1.2.1" xref="S6.7.p4.13.m5.1.2.1.cmml">⁢</mo><mrow id="S6.7.p4.13.m5.1.2.3.2" xref="S6.7.p4.13.m5.1.2.cmml"><mo id="S6.7.p4.13.m5.1.2.3.2.1" stretchy="false" xref="S6.7.p4.13.m5.1.2.cmml">(</mo><mi id="S6.7.p4.13.m5.1.1" xref="S6.7.p4.13.m5.1.1.cmml">μ</mi><mo id="S6.7.p4.13.m5.1.2.3.2.2" stretchy="false" xref="S6.7.p4.13.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.7.p4.13.m5.1b"><apply id="S6.7.p4.13.m5.1.2.cmml" xref="S6.7.p4.13.m5.1.2"><times id="S6.7.p4.13.m5.1.2.1.cmml" xref="S6.7.p4.13.m5.1.2.1"></times><apply id="S6.7.p4.13.m5.1.2.2.cmml" xref="S6.7.p4.13.m5.1.2.2"><csymbol cd="ambiguous" id="S6.7.p4.13.m5.1.2.2.1.cmml" xref="S6.7.p4.13.m5.1.2.2">subscript</csymbol><ci id="S6.7.p4.13.m5.1.2.2.2.cmml" xref="S6.7.p4.13.m5.1.2.2.2">𝜎</ci><times id="S6.7.p4.13.m5.1.2.2.3.cmml" xref="S6.7.p4.13.m5.1.2.2.3"></times></apply><ci id="S6.7.p4.13.m5.1.1.cmml" xref="S6.7.p4.13.m5.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.13.m5.1c">\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.13.m5.1d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math>, which proves our claim. <span class="ltx_text ltx_inline-block" id="S6.7.p4.14.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S6.7.p4.14.1.m1.1"><semantics id="S6.7.p4.14.1.m1.1a"><mo id="S6.7.p4.14.1.m1.1.1" xref="S6.7.p4.14.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S6.7.p4.14.1.m1.1b"><csymbol cd="latexml" id="S6.7.p4.14.1.m1.1.1.cmml" xref="S6.7.p4.14.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.14.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.14.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S6.7.p4.15.m6.1"><semantics id="S6.7.p4.15.m6.1a"><mo id="S6.7.p4.15.m6.1.1" xref="S6.7.p4.15.m6.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S6.7.p4.15.m6.1b"><csymbol cd="latexml" id="S6.7.p4.15.m6.1.1.cmml" xref="S6.7.p4.15.m6.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.7.p4.15.m6.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S6.7.p4.15.m6.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_lem" id="S6.Thmthm5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm5.1.1.1">Lemma 6.5</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm5.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm5.p1"> <p class="ltx_p" id="S6.Thmthm5.p1.2"><span class="ltx_text ltx_font_italic" id="S6.Thmthm5.p1.2.2">Let <math alttext="\mu" class="ltx_Math" display="inline" id="S6.Thmthm5.p1.1.1.m1.1"><semantics id="S6.Thmthm5.p1.1.1.m1.1a"><mi id="S6.Thmthm5.p1.1.1.m1.1.1" xref="S6.Thmthm5.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm5.p1.1.1.m1.1b"><ci id="S6.Thmthm5.p1.1.1.m1.1.1.cmml" xref="S6.Thmthm5.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm5.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm5.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> be any shift-invariant measure on the subshift <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm5.p1.2.2.m2.1"><semantics id="S6.Thmthm5.p1.2.2.m2.1a"><mi id="S6.Thmthm5.p1.2.2.m2.1.1" xref="S6.Thmthm5.p1.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm5.p1.2.2.m2.1b"><ci id="S6.Thmthm5.p1.2.2.m2.1.1.cmml" xref="S6.Thmthm5.p1.2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm5.p1.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm5.p1.2.2.m2.1d">italic_X</annotation></semantics></math>. Then we have:</span></p> <ol class="ltx_enumerate" id="S6.I5"> <li class="ltx_item" id="S6.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S6.I5.i1.p1"> <p class="ltx_p" id="S6.I5.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S6.I5.i1.p1.4.1">If </span><math alttext="\mu" class="ltx_Math" display="inline" id="S6.I5.i1.p1.1.m1.1"><semantics id="S6.I5.i1.p1.1.m1.1a"><mi id="S6.I5.i1.p1.1.m1.1.1" xref="S6.I5.i1.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.I5.i1.p1.1.m1.1b"><ci id="S6.I5.i1.p1.1.m1.1.1.cmml" xref="S6.I5.i1.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i1.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i1.p1.1.m1.1d">italic_μ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i1.p1.4.2"> is periodic (i.e. </span><math alttext="\mu(X\smallsetminus\text{\rm Per}(X)=0" class="ltx_math_unparsed" display="inline" id="S6.I5.i1.p1.2.m2.1"><semantics id="S6.I5.i1.p1.2.m2.1a"><mrow id="S6.I5.i1.p1.2.m2.1b"><mi id="S6.I5.i1.p1.2.m2.1.2">μ</mi><mrow id="S6.I5.i1.p1.2.m2.1.3"><mo id="S6.I5.i1.p1.2.m2.1.3.1" stretchy="false">(</mo><mi id="S6.I5.i1.p1.2.m2.1.3.2">X</mi><mo id="S6.I5.i1.p1.2.m2.1.3.3">∖</mo><mtext id="S6.I5.i1.p1.2.m2.1.3.4">Per</mtext><mrow id="S6.I5.i1.p1.2.m2.1.3.5"><mo id="S6.I5.i1.p1.2.m2.1.3.5.1" stretchy="false">(</mo><mi id="S6.I5.i1.p1.2.m2.1.1">X</mi><mo id="S6.I5.i1.p1.2.m2.1.3.5.2" stretchy="false">)</mo></mrow><mo id="S6.I5.i1.p1.2.m2.1.3.6">=</mo><mn id="S6.I5.i1.p1.2.m2.1.3.7">0</mn></mrow></mrow><annotation encoding="application/x-tex" id="S6.I5.i1.p1.2.m2.1c">\mu(X\smallsetminus\text{\rm Per}(X)=0</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i1.p1.2.m2.1d">italic_μ ( italic_X ∖ Per ( italic_X ) = 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i1.p1.4.3">), then </span><math alttext="\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.I5.i1.p1.3.m3.1"><semantics id="S6.I5.i1.p1.3.m3.1a"><mrow id="S6.I5.i1.p1.3.m3.1.2" xref="S6.I5.i1.p1.3.m3.1.2.cmml"><msub id="S6.I5.i1.p1.3.m3.1.2.2" xref="S6.I5.i1.p1.3.m3.1.2.2.cmml"><mi id="S6.I5.i1.p1.3.m3.1.2.2.2" xref="S6.I5.i1.p1.3.m3.1.2.2.2.cmml">σ</mi><mo id="S6.I5.i1.p1.3.m3.1.2.2.3" xref="S6.I5.i1.p1.3.m3.1.2.2.3.cmml">∗</mo></msub><mo id="S6.I5.i1.p1.3.m3.1.2.1" xref="S6.I5.i1.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.I5.i1.p1.3.m3.1.2.3.2" xref="S6.I5.i1.p1.3.m3.1.2.cmml"><mo id="S6.I5.i1.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S6.I5.i1.p1.3.m3.1.2.cmml">(</mo><mi id="S6.I5.i1.p1.3.m3.1.1" xref="S6.I5.i1.p1.3.m3.1.1.cmml">μ</mi><mo id="S6.I5.i1.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S6.I5.i1.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I5.i1.p1.3.m3.1b"><apply id="S6.I5.i1.p1.3.m3.1.2.cmml" xref="S6.I5.i1.p1.3.m3.1.2"><times id="S6.I5.i1.p1.3.m3.1.2.1.cmml" xref="S6.I5.i1.p1.3.m3.1.2.1"></times><apply id="S6.I5.i1.p1.3.m3.1.2.2.cmml" xref="S6.I5.i1.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S6.I5.i1.p1.3.m3.1.2.2.1.cmml" xref="S6.I5.i1.p1.3.m3.1.2.2">subscript</csymbol><ci id="S6.I5.i1.p1.3.m3.1.2.2.2.cmml" xref="S6.I5.i1.p1.3.m3.1.2.2.2">𝜎</ci><times id="S6.I5.i1.p1.3.m3.1.2.2.3.cmml" xref="S6.I5.i1.p1.3.m3.1.2.2.3"></times></apply><ci id="S6.I5.i1.p1.3.m3.1.1.cmml" xref="S6.I5.i1.p1.3.m3.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i1.p1.3.m3.1c">\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i1.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i1.p1.4.4"> is also periodic (i.e. </span><math alttext="\sigma_{*}(\mu)(\sigma(X)\smallsetminus\text{\rm Per}(\sigma(X))=0" class="ltx_math_unparsed" display="inline" id="S6.I5.i1.p1.4.m4.3"><semantics id="S6.I5.i1.p1.4.m4.3a"><mrow id="S6.I5.i1.p1.4.m4.3b"><msub id="S6.I5.i1.p1.4.m4.3.4"><mi id="S6.I5.i1.p1.4.m4.3.4.2">σ</mi><mo id="S6.I5.i1.p1.4.m4.3.4.3">∗</mo></msub><mrow id="S6.I5.i1.p1.4.m4.3.5"><mo id="S6.I5.i1.p1.4.m4.3.5.1" stretchy="false">(</mo><mi id="S6.I5.i1.p1.4.m4.1.1">μ</mi><mo id="S6.I5.i1.p1.4.m4.3.5.2" stretchy="false">)</mo></mrow><mrow id="S6.I5.i1.p1.4.m4.3.6"><mo id="S6.I5.i1.p1.4.m4.3.6.1" stretchy="false">(</mo><mi id="S6.I5.i1.p1.4.m4.3.6.2">σ</mi><mrow id="S6.I5.i1.p1.4.m4.3.6.3"><mo id="S6.I5.i1.p1.4.m4.3.6.3.1" stretchy="false">(</mo><mi id="S6.I5.i1.p1.4.m4.2.2">X</mi><mo id="S6.I5.i1.p1.4.m4.3.6.3.2" stretchy="false">)</mo></mrow><mo id="S6.I5.i1.p1.4.m4.3.6.4">∖</mo><mtext id="S6.I5.i1.p1.4.m4.3.6.5">Per</mtext><mrow id="S6.I5.i1.p1.4.m4.3.6.6"><mo id="S6.I5.i1.p1.4.m4.3.6.6.1" stretchy="false">(</mo><mi id="S6.I5.i1.p1.4.m4.3.6.6.2">σ</mi><mrow id="S6.I5.i1.p1.4.m4.3.6.6.3"><mo id="S6.I5.i1.p1.4.m4.3.6.6.3.1" stretchy="false">(</mo><mi id="S6.I5.i1.p1.4.m4.3.3">X</mi><mo id="S6.I5.i1.p1.4.m4.3.6.6.3.2" stretchy="false">)</mo></mrow><mo id="S6.I5.i1.p1.4.m4.3.6.6.4" stretchy="false">)</mo></mrow><mo id="S6.I5.i1.p1.4.m4.3.6.7">=</mo><mn id="S6.I5.i1.p1.4.m4.3.6.8">0</mn></mrow></mrow><annotation encoding="application/x-tex" id="S6.I5.i1.p1.4.m4.3c">\sigma_{*}(\mu)(\sigma(X)\smallsetminus\text{\rm Per}(\sigma(X))=0</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i1.p1.4.m4.3d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( italic_σ ( italic_X ) ∖ Per ( italic_σ ( italic_X ) ) = 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i1.p1.4.5">).</span></p> </div> </li> <li class="ltx_item" id="S6.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S6.I5.i2.p1"> <p class="ltx_p" id="S6.I5.i2.p1.5"><span class="ltx_text ltx_font_italic" id="S6.I5.i2.p1.5.1">Under the additional hypothesis that </span><math alttext="\sigma" class="ltx_Math" display="inline" id="S6.I5.i2.p1.1.m1.1"><semantics id="S6.I5.i2.p1.1.m1.1a"><mi id="S6.I5.i2.p1.1.m1.1.1" xref="S6.I5.i2.p1.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.1.m1.1b"><ci id="S6.I5.i2.p1.1.m1.1.1.cmml" xref="S6.I5.i2.p1.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.1.m1.1d">italic_σ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i2.p1.5.2"> is shift-orbit injective, we also have: If </span><math alttext="\mu" class="ltx_Math" display="inline" id="S6.I5.i2.p1.2.m2.1"><semantics id="S6.I5.i2.p1.2.m2.1a"><mi id="S6.I5.i2.p1.2.m2.1.1" xref="S6.I5.i2.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.2.m2.1b"><ci id="S6.I5.i2.p1.2.m2.1.1.cmml" xref="S6.I5.i2.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.2.m2.1d">italic_μ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i2.p1.5.3"> is non-atomic (i.e. </span><math alttext="\mu(\text{\rm Per}(X))=0" class="ltx_Math" display="inline" id="S6.I5.i2.p1.3.m3.2"><semantics id="S6.I5.i2.p1.3.m3.2a"><mrow id="S6.I5.i2.p1.3.m3.2.2" xref="S6.I5.i2.p1.3.m3.2.2.cmml"><mrow id="S6.I5.i2.p1.3.m3.2.2.1" xref="S6.I5.i2.p1.3.m3.2.2.1.cmml"><mi id="S6.I5.i2.p1.3.m3.2.2.1.3" xref="S6.I5.i2.p1.3.m3.2.2.1.3.cmml">μ</mi><mo id="S6.I5.i2.p1.3.m3.2.2.1.2" xref="S6.I5.i2.p1.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S6.I5.i2.p1.3.m3.2.2.1.1.1" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="S6.I5.i2.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.cmml"><mtext id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.2" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.2a.cmml">Per</mtext><mo id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.1" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.3.2" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.cmml">(</mo><mi id="S6.I5.i2.p1.3.m3.1.1" xref="S6.I5.i2.p1.3.m3.1.1.cmml">X</mi><mo id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I5.i2.p1.3.m3.2.2.1.1.1.3" stretchy="false" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I5.i2.p1.3.m3.2.2.2" xref="S6.I5.i2.p1.3.m3.2.2.2.cmml">=</mo><mn id="S6.I5.i2.p1.3.m3.2.2.3" xref="S6.I5.i2.p1.3.m3.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.3.m3.2b"><apply id="S6.I5.i2.p1.3.m3.2.2.cmml" xref="S6.I5.i2.p1.3.m3.2.2"><eq id="S6.I5.i2.p1.3.m3.2.2.2.cmml" xref="S6.I5.i2.p1.3.m3.2.2.2"></eq><apply id="S6.I5.i2.p1.3.m3.2.2.1.cmml" xref="S6.I5.i2.p1.3.m3.2.2.1"><times id="S6.I5.i2.p1.3.m3.2.2.1.2.cmml" xref="S6.I5.i2.p1.3.m3.2.2.1.2"></times><ci id="S6.I5.i2.p1.3.m3.2.2.1.3.cmml" xref="S6.I5.i2.p1.3.m3.2.2.1.3">𝜇</ci><apply id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.cmml" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1"><times id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.1.cmml" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.1"></times><ci id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.2a.cmml" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.2"><mtext id="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.2.cmml" xref="S6.I5.i2.p1.3.m3.2.2.1.1.1.1.2">Per</mtext></ci><ci id="S6.I5.i2.p1.3.m3.1.1.cmml" xref="S6.I5.i2.p1.3.m3.1.1">𝑋</ci></apply></apply><cn id="S6.I5.i2.p1.3.m3.2.2.3.cmml" type="integer" xref="S6.I5.i2.p1.3.m3.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.3.m3.2c">\mu(\text{\rm Per}(X))=0</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.3.m3.2d">italic_μ ( Per ( italic_X ) ) = 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i2.p1.5.4">), then </span><math alttext="\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.I5.i2.p1.4.m4.1"><semantics id="S6.I5.i2.p1.4.m4.1a"><mrow id="S6.I5.i2.p1.4.m4.1.2" xref="S6.I5.i2.p1.4.m4.1.2.cmml"><msub id="S6.I5.i2.p1.4.m4.1.2.2" xref="S6.I5.i2.p1.4.m4.1.2.2.cmml"><mi id="S6.I5.i2.p1.4.m4.1.2.2.2" xref="S6.I5.i2.p1.4.m4.1.2.2.2.cmml">σ</mi><mo id="S6.I5.i2.p1.4.m4.1.2.2.3" xref="S6.I5.i2.p1.4.m4.1.2.2.3.cmml">∗</mo></msub><mo id="S6.I5.i2.p1.4.m4.1.2.1" xref="S6.I5.i2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S6.I5.i2.p1.4.m4.1.2.3.2" xref="S6.I5.i2.p1.4.m4.1.2.cmml"><mo id="S6.I5.i2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S6.I5.i2.p1.4.m4.1.2.cmml">(</mo><mi id="S6.I5.i2.p1.4.m4.1.1" xref="S6.I5.i2.p1.4.m4.1.1.cmml">μ</mi><mo id="S6.I5.i2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S6.I5.i2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.4.m4.1b"><apply id="S6.I5.i2.p1.4.m4.1.2.cmml" xref="S6.I5.i2.p1.4.m4.1.2"><times id="S6.I5.i2.p1.4.m4.1.2.1.cmml" xref="S6.I5.i2.p1.4.m4.1.2.1"></times><apply id="S6.I5.i2.p1.4.m4.1.2.2.cmml" xref="S6.I5.i2.p1.4.m4.1.2.2"><csymbol cd="ambiguous" id="S6.I5.i2.p1.4.m4.1.2.2.1.cmml" xref="S6.I5.i2.p1.4.m4.1.2.2">subscript</csymbol><ci id="S6.I5.i2.p1.4.m4.1.2.2.2.cmml" xref="S6.I5.i2.p1.4.m4.1.2.2.2">𝜎</ci><times id="S6.I5.i2.p1.4.m4.1.2.2.3.cmml" xref="S6.I5.i2.p1.4.m4.1.2.2.3"></times></apply><ci id="S6.I5.i2.p1.4.m4.1.1.cmml" xref="S6.I5.i2.p1.4.m4.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.4.m4.1c">\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.4.m4.1d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i2.p1.5.5"> is also non-atomic (i.e. </span><math alttext="\sigma_{*}(\mu)(\text{\rm Per}(\sigma(X)))=0" class="ltx_Math" display="inline" id="S6.I5.i2.p1.5.m5.3"><semantics id="S6.I5.i2.p1.5.m5.3a"><mrow id="S6.I5.i2.p1.5.m5.3.3" xref="S6.I5.i2.p1.5.m5.3.3.cmml"><mrow id="S6.I5.i2.p1.5.m5.3.3.1" xref="S6.I5.i2.p1.5.m5.3.3.1.cmml"><msub id="S6.I5.i2.p1.5.m5.3.3.1.3" xref="S6.I5.i2.p1.5.m5.3.3.1.3.cmml"><mi id="S6.I5.i2.p1.5.m5.3.3.1.3.2" xref="S6.I5.i2.p1.5.m5.3.3.1.3.2.cmml">σ</mi><mo id="S6.I5.i2.p1.5.m5.3.3.1.3.3" xref="S6.I5.i2.p1.5.m5.3.3.1.3.3.cmml">∗</mo></msub><mo id="S6.I5.i2.p1.5.m5.3.3.1.2" xref="S6.I5.i2.p1.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="S6.I5.i2.p1.5.m5.3.3.1.4.2" xref="S6.I5.i2.p1.5.m5.3.3.1.cmml"><mo id="S6.I5.i2.p1.5.m5.3.3.1.4.2.1" stretchy="false" xref="S6.I5.i2.p1.5.m5.3.3.1.cmml">(</mo><mi id="S6.I5.i2.p1.5.m5.1.1" xref="S6.I5.i2.p1.5.m5.1.1.cmml">μ</mi><mo id="S6.I5.i2.p1.5.m5.3.3.1.4.2.2" stretchy="false" xref="S6.I5.i2.p1.5.m5.3.3.1.cmml">)</mo></mrow><mo id="S6.I5.i2.p1.5.m5.3.3.1.2a" xref="S6.I5.i2.p1.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="S6.I5.i2.p1.5.m5.3.3.1.1.1" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.cmml"><mo id="S6.I5.i2.p1.5.m5.3.3.1.1.1.2" stretchy="false" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.cmml"><mtext id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.3" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.3a.cmml">Per</mtext><mo id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.2" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml"><mo id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml"><mi id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.2" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.1" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.3.2" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml"><mo id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml">(</mo><mi id="S6.I5.i2.p1.5.m5.2.2" xref="S6.I5.i2.p1.5.m5.2.2.cmml">X</mi><mo id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I5.i2.p1.5.m5.3.3.1.1.1.3" stretchy="false" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.I5.i2.p1.5.m5.3.3.2" xref="S6.I5.i2.p1.5.m5.3.3.2.cmml">=</mo><mn id="S6.I5.i2.p1.5.m5.3.3.3" xref="S6.I5.i2.p1.5.m5.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I5.i2.p1.5.m5.3b"><apply id="S6.I5.i2.p1.5.m5.3.3.cmml" xref="S6.I5.i2.p1.5.m5.3.3"><eq id="S6.I5.i2.p1.5.m5.3.3.2.cmml" xref="S6.I5.i2.p1.5.m5.3.3.2"></eq><apply id="S6.I5.i2.p1.5.m5.3.3.1.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1"><times id="S6.I5.i2.p1.5.m5.3.3.1.2.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.2"></times><apply id="S6.I5.i2.p1.5.m5.3.3.1.3.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.3"><csymbol cd="ambiguous" id="S6.I5.i2.p1.5.m5.3.3.1.3.1.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.3">subscript</csymbol><ci id="S6.I5.i2.p1.5.m5.3.3.1.3.2.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.3.2">𝜎</ci><times id="S6.I5.i2.p1.5.m5.3.3.1.3.3.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.3.3"></times></apply><ci id="S6.I5.i2.p1.5.m5.1.1.cmml" xref="S6.I5.i2.p1.5.m5.1.1">𝜇</ci><apply id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1"><times id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.2.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.2"></times><ci id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.3a.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.3"><mtext id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.3.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.3">Per</mtext></ci><apply id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1"><times id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.1.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.1"></times><ci id="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.I5.i2.p1.5.m5.3.3.1.1.1.1.1.1.1.2">𝜎</ci><ci id="S6.I5.i2.p1.5.m5.2.2.cmml" xref="S6.I5.i2.p1.5.m5.2.2">𝑋</ci></apply></apply></apply><cn id="S6.I5.i2.p1.5.m5.3.3.3.cmml" type="integer" xref="S6.I5.i2.p1.5.m5.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I5.i2.p1.5.m5.3c">\sigma_{*}(\mu)(\text{\rm Per}(\sigma(X)))=0</annotation><annotation encoding="application/x-llamapun" id="S6.I5.i2.p1.5.m5.3d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( Per ( italic_σ ( italic_X ) ) ) = 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I5.i2.p1.5.6">).</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S6.9"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.8.p1"> <p class="ltx_p" id="S6.8.p1.7">(1) If <math alttext="\mu(X\smallsetminus\text{\rm Per}(X))=0" class="ltx_Math" display="inline" id="S6.8.p1.1.m1.2"><semantics id="S6.8.p1.1.m1.2a"><mrow id="S6.8.p1.1.m1.2.2" xref="S6.8.p1.1.m1.2.2.cmml"><mrow id="S6.8.p1.1.m1.2.2.1" xref="S6.8.p1.1.m1.2.2.1.cmml"><mi id="S6.8.p1.1.m1.2.2.1.3" xref="S6.8.p1.1.m1.2.2.1.3.cmml">μ</mi><mo id="S6.8.p1.1.m1.2.2.1.2" xref="S6.8.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S6.8.p1.1.m1.2.2.1.1.1" xref="S6.8.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S6.8.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S6.8.p1.1.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.8.p1.1.m1.2.2.1.1.1.1" xref="S6.8.p1.1.m1.2.2.1.1.1.1.cmml"><mi id="S6.8.p1.1.m1.2.2.1.1.1.1.2" xref="S6.8.p1.1.m1.2.2.1.1.1.1.2.cmml">X</mi><mo id="S6.8.p1.1.m1.2.2.1.1.1.1.1" xref="S6.8.p1.1.m1.2.2.1.1.1.1.1.cmml">∖</mo><mrow id="S6.8.p1.1.m1.2.2.1.1.1.1.3" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.cmml"><mtext id="S6.8.p1.1.m1.2.2.1.1.1.1.3.2" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.2a.cmml">Per</mtext><mo id="S6.8.p1.1.m1.2.2.1.1.1.1.3.1" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S6.8.p1.1.m1.2.2.1.1.1.1.3.3.2" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.cmml"><mo id="S6.8.p1.1.m1.2.2.1.1.1.1.3.3.2.1" stretchy="false" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.cmml">(</mo><mi id="S6.8.p1.1.m1.1.1" xref="S6.8.p1.1.m1.1.1.cmml">X</mi><mo id="S6.8.p1.1.m1.2.2.1.1.1.1.3.3.2.2" stretchy="false" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S6.8.p1.1.m1.2.2.1.1.1.3" stretchy="false" xref="S6.8.p1.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.8.p1.1.m1.2.2.2" xref="S6.8.p1.1.m1.2.2.2.cmml">=</mo><mn id="S6.8.p1.1.m1.2.2.3" xref="S6.8.p1.1.m1.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.8.p1.1.m1.2b"><apply id="S6.8.p1.1.m1.2.2.cmml" xref="S6.8.p1.1.m1.2.2"><eq id="S6.8.p1.1.m1.2.2.2.cmml" xref="S6.8.p1.1.m1.2.2.2"></eq><apply id="S6.8.p1.1.m1.2.2.1.cmml" xref="S6.8.p1.1.m1.2.2.1"><times id="S6.8.p1.1.m1.2.2.1.2.cmml" xref="S6.8.p1.1.m1.2.2.1.2"></times><ci id="S6.8.p1.1.m1.2.2.1.3.cmml" xref="S6.8.p1.1.m1.2.2.1.3">𝜇</ci><apply id="S6.8.p1.1.m1.2.2.1.1.1.1.cmml" xref="S6.8.p1.1.m1.2.2.1.1.1"><setdiff id="S6.8.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S6.8.p1.1.m1.2.2.1.1.1.1.1"></setdiff><ci id="S6.8.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="S6.8.p1.1.m1.2.2.1.1.1.1.2">𝑋</ci><apply id="S6.8.p1.1.m1.2.2.1.1.1.1.3.cmml" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3"><times id="S6.8.p1.1.m1.2.2.1.1.1.1.3.1.cmml" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.1"></times><ci id="S6.8.p1.1.m1.2.2.1.1.1.1.3.2a.cmml" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.2"><mtext id="S6.8.p1.1.m1.2.2.1.1.1.1.3.2.cmml" xref="S6.8.p1.1.m1.2.2.1.1.1.1.3.2">Per</mtext></ci><ci id="S6.8.p1.1.m1.1.1.cmml" xref="S6.8.p1.1.m1.1.1">𝑋</ci></apply></apply></apply><cn id="S6.8.p1.1.m1.2.2.3.cmml" type="integer" xref="S6.8.p1.1.m1.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.1.m1.2c">\mu(X\smallsetminus\text{\rm Per}(X))=0</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.1.m1.2d">italic_μ ( italic_X ∖ Per ( italic_X ) ) = 0</annotation></semantics></math>, then <math alttext="\mu" class="ltx_Math" display="inline" id="S6.8.p1.2.m2.1"><semantics id="S6.8.p1.2.m2.1a"><mi id="S6.8.p1.2.m2.1.1" xref="S6.8.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.8.p1.2.m2.1b"><ci id="S6.8.p1.2.m2.1.1.cmml" xref="S6.8.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.2.m2.1d">italic_μ</annotation></semantics></math> is carried entirely by the (countable) set of periodic words in <math alttext="X" class="ltx_Math" display="inline" id="S6.8.p1.3.m3.1"><semantics id="S6.8.p1.3.m3.1a"><mi id="S6.8.p1.3.m3.1.1" xref="S6.8.p1.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.8.p1.3.m3.1b"><ci id="S6.8.p1.3.m3.1.1.cmml" xref="S6.8.p1.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.3.m3.1d">italic_X</annotation></semantics></math>: There exist a coefficient <math alttext="\lambda_{w}\geq 0" class="ltx_Math" display="inline" id="S6.8.p1.4.m4.1"><semantics id="S6.8.p1.4.m4.1a"><mrow id="S6.8.p1.4.m4.1.1" xref="S6.8.p1.4.m4.1.1.cmml"><msub id="S6.8.p1.4.m4.1.1.2" xref="S6.8.p1.4.m4.1.1.2.cmml"><mi id="S6.8.p1.4.m4.1.1.2.2" xref="S6.8.p1.4.m4.1.1.2.2.cmml">λ</mi><mi id="S6.8.p1.4.m4.1.1.2.3" xref="S6.8.p1.4.m4.1.1.2.3.cmml">w</mi></msub><mo id="S6.8.p1.4.m4.1.1.1" xref="S6.8.p1.4.m4.1.1.1.cmml">≥</mo><mn id="S6.8.p1.4.m4.1.1.3" xref="S6.8.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.8.p1.4.m4.1b"><apply id="S6.8.p1.4.m4.1.1.cmml" xref="S6.8.p1.4.m4.1.1"><geq id="S6.8.p1.4.m4.1.1.1.cmml" xref="S6.8.p1.4.m4.1.1.1"></geq><apply id="S6.8.p1.4.m4.1.1.2.cmml" xref="S6.8.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S6.8.p1.4.m4.1.1.2.1.cmml" xref="S6.8.p1.4.m4.1.1.2">subscript</csymbol><ci id="S6.8.p1.4.m4.1.1.2.2.cmml" xref="S6.8.p1.4.m4.1.1.2.2">𝜆</ci><ci id="S6.8.p1.4.m4.1.1.2.3.cmml" xref="S6.8.p1.4.m4.1.1.2.3">𝑤</ci></apply><cn id="S6.8.p1.4.m4.1.1.3.cmml" type="integer" xref="S6.8.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.4.m4.1c">\lambda_{w}\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.4.m4.1d">italic_λ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> for any <math alttext="w\in\cal L(X)" class="ltx_Math" display="inline" id="S6.8.p1.5.m5.1"><semantics id="S6.8.p1.5.m5.1a"><mrow id="S6.8.p1.5.m5.1.2" xref="S6.8.p1.5.m5.1.2.cmml"><mi id="S6.8.p1.5.m5.1.2.2" xref="S6.8.p1.5.m5.1.2.2.cmml">w</mi><mo id="S6.8.p1.5.m5.1.2.1" xref="S6.8.p1.5.m5.1.2.1.cmml">∈</mo><mrow id="S6.8.p1.5.m5.1.2.3" xref="S6.8.p1.5.m5.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.8.p1.5.m5.1.2.3.2" xref="S6.8.p1.5.m5.1.2.3.2.cmml">ℒ</mi><mo id="S6.8.p1.5.m5.1.2.3.1" xref="S6.8.p1.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S6.8.p1.5.m5.1.2.3.3.2" xref="S6.8.p1.5.m5.1.2.3.cmml"><mo id="S6.8.p1.5.m5.1.2.3.3.2.1" stretchy="false" xref="S6.8.p1.5.m5.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.8.p1.5.m5.1.1" xref="S6.8.p1.5.m5.1.1.cmml">𝒳</mi><mo id="S6.8.p1.5.m5.1.2.3.3.2.2" stretchy="false" xref="S6.8.p1.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.8.p1.5.m5.1b"><apply id="S6.8.p1.5.m5.1.2.cmml" xref="S6.8.p1.5.m5.1.2"><in id="S6.8.p1.5.m5.1.2.1.cmml" xref="S6.8.p1.5.m5.1.2.1"></in><ci id="S6.8.p1.5.m5.1.2.2.cmml" xref="S6.8.p1.5.m5.1.2.2">𝑤</ci><apply id="S6.8.p1.5.m5.1.2.3.cmml" xref="S6.8.p1.5.m5.1.2.3"><times id="S6.8.p1.5.m5.1.2.3.1.cmml" xref="S6.8.p1.5.m5.1.2.3.1"></times><ci id="S6.8.p1.5.m5.1.2.3.2.cmml" xref="S6.8.p1.5.m5.1.2.3.2">ℒ</ci><ci id="S6.8.p1.5.m5.1.1.cmml" xref="S6.8.p1.5.m5.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.5.m5.1c">w\in\cal L(X)</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.5.m5.1d">italic_w ∈ caligraphic_L ( caligraphic_X )</annotation></semantics></math> such that, using the characteristic measures <math alttext="\mu_{w}" class="ltx_Math" display="inline" id="S6.8.p1.6.m6.1"><semantics id="S6.8.p1.6.m6.1a"><msub id="S6.8.p1.6.m6.1.1" xref="S6.8.p1.6.m6.1.1.cmml"><mi id="S6.8.p1.6.m6.1.1.2" xref="S6.8.p1.6.m6.1.1.2.cmml">μ</mi><mi id="S6.8.p1.6.m6.1.1.3" xref="S6.8.p1.6.m6.1.1.3.cmml">w</mi></msub><annotation-xml encoding="MathML-Content" id="S6.8.p1.6.m6.1b"><apply id="S6.8.p1.6.m6.1.1.cmml" xref="S6.8.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S6.8.p1.6.m6.1.1.1.cmml" xref="S6.8.p1.6.m6.1.1">subscript</csymbol><ci id="S6.8.p1.6.m6.1.1.2.cmml" xref="S6.8.p1.6.m6.1.1.2">𝜇</ci><ci id="S6.8.p1.6.m6.1.1.3.cmml" xref="S6.8.p1.6.m6.1.1.3">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.6.m6.1c">\mu_{w}</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.6.m6.1d">italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT</annotation></semantics></math> from (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S2.E5" title="In 2.1. Standard terminology and well known facts ‣ 2. Notation and conventions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">2.5</span></a>), the measure <math alttext="\mu" class="ltx_Math" display="inline" id="S6.8.p1.7.m7.1"><semantics id="S6.8.p1.7.m7.1a"><mi id="S6.8.p1.7.m7.1.1" xref="S6.8.p1.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.8.p1.7.m7.1b"><ci id="S6.8.p1.7.m7.1.1.cmml" xref="S6.8.p1.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.7.m7.1d">italic_μ</annotation></semantics></math> can be expressed as countable sum</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mu=\sum_{w\in\cal L(X)}\lambda_{w}\,\mu_{w}\,." class="ltx_Math" display="block" id="S6.Ex8.m1.2"><semantics id="S6.Ex8.m1.2a"><mrow id="S6.Ex8.m1.2.2.1" xref="S6.Ex8.m1.2.2.1.1.cmml"><mrow id="S6.Ex8.m1.2.2.1.1" xref="S6.Ex8.m1.2.2.1.1.cmml"><mi id="S6.Ex8.m1.2.2.1.1.2" xref="S6.Ex8.m1.2.2.1.1.2.cmml">μ</mi><mo id="S6.Ex8.m1.2.2.1.1.1" rspace="0.111em" xref="S6.Ex8.m1.2.2.1.1.1.cmml">=</mo><mrow id="S6.Ex8.m1.2.2.1.1.3" xref="S6.Ex8.m1.2.2.1.1.3.cmml"><munder id="S6.Ex8.m1.2.2.1.1.3.1" xref="S6.Ex8.m1.2.2.1.1.3.1.cmml"><mo id="S6.Ex8.m1.2.2.1.1.3.1.2" movablelimits="false" xref="S6.Ex8.m1.2.2.1.1.3.1.2.cmml">∑</mo><mrow id="S6.Ex8.m1.1.1.1" xref="S6.Ex8.m1.1.1.1.cmml"><mi id="S6.Ex8.m1.1.1.1.3" xref="S6.Ex8.m1.1.1.1.3.cmml">w</mi><mo id="S6.Ex8.m1.1.1.1.2" xref="S6.Ex8.m1.1.1.1.2.cmml">∈</mo><mrow id="S6.Ex8.m1.1.1.1.4" xref="S6.Ex8.m1.1.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex8.m1.1.1.1.4.2" xref="S6.Ex8.m1.1.1.1.4.2.cmml">ℒ</mi><mo id="S6.Ex8.m1.1.1.1.4.1" xref="S6.Ex8.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S6.Ex8.m1.1.1.1.4.3.2" xref="S6.Ex8.m1.1.1.1.4.cmml"><mo id="S6.Ex8.m1.1.1.1.4.3.2.1" stretchy="false" xref="S6.Ex8.m1.1.1.1.4.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex8.m1.1.1.1.1" xref="S6.Ex8.m1.1.1.1.1.cmml">𝒳</mi><mo id="S6.Ex8.m1.1.1.1.4.3.2.2" stretchy="false" xref="S6.Ex8.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mrow id="S6.Ex8.m1.2.2.1.1.3.2" xref="S6.Ex8.m1.2.2.1.1.3.2.cmml"><msub id="S6.Ex8.m1.2.2.1.1.3.2.2" xref="S6.Ex8.m1.2.2.1.1.3.2.2.cmml"><mi id="S6.Ex8.m1.2.2.1.1.3.2.2.2" xref="S6.Ex8.m1.2.2.1.1.3.2.2.2.cmml">λ</mi><mi id="S6.Ex8.m1.2.2.1.1.3.2.2.3" xref="S6.Ex8.m1.2.2.1.1.3.2.2.3.cmml">w</mi></msub><mo id="S6.Ex8.m1.2.2.1.1.3.2.1" xref="S6.Ex8.m1.2.2.1.1.3.2.1.cmml">⁢</mo><msub id="S6.Ex8.m1.2.2.1.1.3.2.3" xref="S6.Ex8.m1.2.2.1.1.3.2.3.cmml"><mi id="S6.Ex8.m1.2.2.1.1.3.2.3.2" xref="S6.Ex8.m1.2.2.1.1.3.2.3.2.cmml">μ</mi><mi id="S6.Ex8.m1.2.2.1.1.3.2.3.3" xref="S6.Ex8.m1.2.2.1.1.3.2.3.3.cmml">w</mi></msub></mrow></mrow></mrow><mo id="S6.Ex8.m1.2.2.1.2" lspace="0em" xref="S6.Ex8.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex8.m1.2b"><apply id="S6.Ex8.m1.2.2.1.1.cmml" xref="S6.Ex8.m1.2.2.1"><eq id="S6.Ex8.m1.2.2.1.1.1.cmml" xref="S6.Ex8.m1.2.2.1.1.1"></eq><ci id="S6.Ex8.m1.2.2.1.1.2.cmml" xref="S6.Ex8.m1.2.2.1.1.2">𝜇</ci><apply id="S6.Ex8.m1.2.2.1.1.3.cmml" xref="S6.Ex8.m1.2.2.1.1.3"><apply id="S6.Ex8.m1.2.2.1.1.3.1.cmml" xref="S6.Ex8.m1.2.2.1.1.3.1"><csymbol cd="ambiguous" id="S6.Ex8.m1.2.2.1.1.3.1.1.cmml" xref="S6.Ex8.m1.2.2.1.1.3.1">subscript</csymbol><sum id="S6.Ex8.m1.2.2.1.1.3.1.2.cmml" xref="S6.Ex8.m1.2.2.1.1.3.1.2"></sum><apply id="S6.Ex8.m1.1.1.1.cmml" xref="S6.Ex8.m1.1.1.1"><in id="S6.Ex8.m1.1.1.1.2.cmml" xref="S6.Ex8.m1.1.1.1.2"></in><ci id="S6.Ex8.m1.1.1.1.3.cmml" xref="S6.Ex8.m1.1.1.1.3">𝑤</ci><apply id="S6.Ex8.m1.1.1.1.4.cmml" xref="S6.Ex8.m1.1.1.1.4"><times id="S6.Ex8.m1.1.1.1.4.1.cmml" xref="S6.Ex8.m1.1.1.1.4.1"></times><ci id="S6.Ex8.m1.1.1.1.4.2.cmml" xref="S6.Ex8.m1.1.1.1.4.2">ℒ</ci><ci id="S6.Ex8.m1.1.1.1.1.cmml" xref="S6.Ex8.m1.1.1.1.1">𝒳</ci></apply></apply></apply><apply id="S6.Ex8.m1.2.2.1.1.3.2.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2"><times id="S6.Ex8.m1.2.2.1.1.3.2.1.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.1"></times><apply id="S6.Ex8.m1.2.2.1.1.3.2.2.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.2"><csymbol cd="ambiguous" id="S6.Ex8.m1.2.2.1.1.3.2.2.1.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.2">subscript</csymbol><ci id="S6.Ex8.m1.2.2.1.1.3.2.2.2.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.2.2">𝜆</ci><ci id="S6.Ex8.m1.2.2.1.1.3.2.2.3.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.2.3">𝑤</ci></apply><apply id="S6.Ex8.m1.2.2.1.1.3.2.3.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.3"><csymbol cd="ambiguous" id="S6.Ex8.m1.2.2.1.1.3.2.3.1.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.3">subscript</csymbol><ci id="S6.Ex8.m1.2.2.1.1.3.2.3.2.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.3.2">𝜇</ci><ci id="S6.Ex8.m1.2.2.1.1.3.2.3.3.cmml" xref="S6.Ex8.m1.2.2.1.1.3.2.3.3">𝑤</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex8.m1.2c">\mu=\sum_{w\in\cal L(X)}\lambda_{w}\,\mu_{w}\,.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex8.m1.2d">italic_μ = ∑ start_POSTSUBSCRIPT italic_w ∈ caligraphic_L ( caligraphic_X ) end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.8.p1.9">But then Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S3.Thmthm7" title="Lemma 3.7. ‣ 3.4. Basic properties of the measure transfer map ‣ 3. The measure transfer ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">3.7</span></a> (d) gives directly <math alttext="\sigma_{*}(\mu)=\underset{w\in\cal L(X)}{\sum}\lambda_{w}\,\mu_{\sigma(w)}\," class="ltx_Math" display="inline" id="S6.8.p1.8.m1.3"><semantics id="S6.8.p1.8.m1.3a"><mrow id="S6.8.p1.8.m1.3.4" xref="S6.8.p1.8.m1.3.4.cmml"><mrow id="S6.8.p1.8.m1.3.4.2" xref="S6.8.p1.8.m1.3.4.2.cmml"><msub id="S6.8.p1.8.m1.3.4.2.2" xref="S6.8.p1.8.m1.3.4.2.2.cmml"><mi id="S6.8.p1.8.m1.3.4.2.2.2" xref="S6.8.p1.8.m1.3.4.2.2.2.cmml">σ</mi><mo id="S6.8.p1.8.m1.3.4.2.2.3" xref="S6.8.p1.8.m1.3.4.2.2.3.cmml">∗</mo></msub><mo id="S6.8.p1.8.m1.3.4.2.1" xref="S6.8.p1.8.m1.3.4.2.1.cmml">⁢</mo><mrow id="S6.8.p1.8.m1.3.4.2.3.2" xref="S6.8.p1.8.m1.3.4.2.cmml"><mo id="S6.8.p1.8.m1.3.4.2.3.2.1" stretchy="false" xref="S6.8.p1.8.m1.3.4.2.cmml">(</mo><mi id="S6.8.p1.8.m1.3.3" xref="S6.8.p1.8.m1.3.3.cmml">μ</mi><mo id="S6.8.p1.8.m1.3.4.2.3.2.2" stretchy="false" xref="S6.8.p1.8.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S6.8.p1.8.m1.3.4.1" rspace="0.111em" xref="S6.8.p1.8.m1.3.4.1.cmml">=</mo><mrow id="S6.8.p1.8.m1.3.4.3" xref="S6.8.p1.8.m1.3.4.3.cmml"><munder accentunder="true" id="S6.8.p1.8.m1.1.1" xref="S6.8.p1.8.m1.1.1.cmml"><mo id="S6.8.p1.8.m1.1.1.2" xref="S6.8.p1.8.m1.1.1.2.cmml">∑</mo><mrow id="S6.8.p1.8.m1.1.1.1" xref="S6.8.p1.8.m1.1.1.1.cmml"><mi id="S6.8.p1.8.m1.1.1.1.3" xref="S6.8.p1.8.m1.1.1.1.3.cmml">w</mi><mo id="S6.8.p1.8.m1.1.1.1.2" xref="S6.8.p1.8.m1.1.1.1.2.cmml">∈</mo><mrow id="S6.8.p1.8.m1.1.1.1.4" xref="S6.8.p1.8.m1.1.1.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.8.p1.8.m1.1.1.1.4.2" xref="S6.8.p1.8.m1.1.1.1.4.2.cmml">ℒ</mi><mo id="S6.8.p1.8.m1.1.1.1.4.1" xref="S6.8.p1.8.m1.1.1.1.4.1.cmml">⁢</mo><mrow id="S6.8.p1.8.m1.1.1.1.4.3.2" xref="S6.8.p1.8.m1.1.1.1.4.cmml"><mo id="S6.8.p1.8.m1.1.1.1.4.3.2.1" stretchy="false" xref="S6.8.p1.8.m1.1.1.1.4.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.8.p1.8.m1.1.1.1.1" xref="S6.8.p1.8.m1.1.1.1.1.cmml">𝒳</mi><mo id="S6.8.p1.8.m1.1.1.1.4.3.2.2" stretchy="false" xref="S6.8.p1.8.m1.1.1.1.4.cmml">)</mo></mrow></mrow></mrow></munder><mo id="S6.8.p1.8.m1.3.4.3.1" xref="S6.8.p1.8.m1.3.4.3.1.cmml">⁢</mo><msub id="S6.8.p1.8.m1.3.4.3.2" xref="S6.8.p1.8.m1.3.4.3.2.cmml"><mi id="S6.8.p1.8.m1.3.4.3.2.2" xref="S6.8.p1.8.m1.3.4.3.2.2.cmml">λ</mi><mi class="ltx_font_mathcaligraphic" id="S6.8.p1.8.m1.3.4.3.2.3" xref="S6.8.p1.8.m1.3.4.3.2.3.cmml">𝓌</mi></msub><mo id="S6.8.p1.8.m1.3.4.3.1a" xref="S6.8.p1.8.m1.3.4.3.1.cmml">⁢</mo><msub id="S6.8.p1.8.m1.3.4.3.3" xref="S6.8.p1.8.m1.3.4.3.3.cmml"><mi id="S6.8.p1.8.m1.3.4.3.3.2" xref="S6.8.p1.8.m1.3.4.3.3.2.cmml">μ</mi><mrow id="S6.8.p1.8.m1.2.2.1" xref="S6.8.p1.8.m1.2.2.1.cmml"><mi id="S6.8.p1.8.m1.2.2.1.3" xref="S6.8.p1.8.m1.2.2.1.3.cmml">σ</mi><mo id="S6.8.p1.8.m1.2.2.1.2" xref="S6.8.p1.8.m1.2.2.1.2.cmml">⁢</mo><mrow id="S6.8.p1.8.m1.2.2.1.4.2" xref="S6.8.p1.8.m1.2.2.1.cmml"><mo id="S6.8.p1.8.m1.2.2.1.4.2.1" stretchy="false" xref="S6.8.p1.8.m1.2.2.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.8.p1.8.m1.2.2.1.1" xref="S6.8.p1.8.m1.2.2.1.1.cmml">𝓌</mi><mo id="S6.8.p1.8.m1.2.2.1.4.2.2" stretchy="false" xref="S6.8.p1.8.m1.2.2.1.cmml">)</mo></mrow></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.8.p1.8.m1.3b"><apply id="S6.8.p1.8.m1.3.4.cmml" xref="S6.8.p1.8.m1.3.4"><eq id="S6.8.p1.8.m1.3.4.1.cmml" xref="S6.8.p1.8.m1.3.4.1"></eq><apply id="S6.8.p1.8.m1.3.4.2.cmml" xref="S6.8.p1.8.m1.3.4.2"><times id="S6.8.p1.8.m1.3.4.2.1.cmml" xref="S6.8.p1.8.m1.3.4.2.1"></times><apply id="S6.8.p1.8.m1.3.4.2.2.cmml" xref="S6.8.p1.8.m1.3.4.2.2"><csymbol cd="ambiguous" id="S6.8.p1.8.m1.3.4.2.2.1.cmml" xref="S6.8.p1.8.m1.3.4.2.2">subscript</csymbol><ci id="S6.8.p1.8.m1.3.4.2.2.2.cmml" xref="S6.8.p1.8.m1.3.4.2.2.2">𝜎</ci><times id="S6.8.p1.8.m1.3.4.2.2.3.cmml" xref="S6.8.p1.8.m1.3.4.2.2.3"></times></apply><ci id="S6.8.p1.8.m1.3.3.cmml" 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id="S6.8.p1.8.m1.3.4.3.2.2.cmml" xref="S6.8.p1.8.m1.3.4.3.2.2">𝜆</ci><ci id="S6.8.p1.8.m1.3.4.3.2.3.cmml" xref="S6.8.p1.8.m1.3.4.3.2.3">𝓌</ci></apply><apply id="S6.8.p1.8.m1.3.4.3.3.cmml" xref="S6.8.p1.8.m1.3.4.3.3"><csymbol cd="ambiguous" id="S6.8.p1.8.m1.3.4.3.3.1.cmml" xref="S6.8.p1.8.m1.3.4.3.3">subscript</csymbol><ci id="S6.8.p1.8.m1.3.4.3.3.2.cmml" xref="S6.8.p1.8.m1.3.4.3.3.2">𝜇</ci><apply id="S6.8.p1.8.m1.2.2.1.cmml" xref="S6.8.p1.8.m1.2.2.1"><times id="S6.8.p1.8.m1.2.2.1.2.cmml" xref="S6.8.p1.8.m1.2.2.1.2"></times><ci id="S6.8.p1.8.m1.2.2.1.3.cmml" xref="S6.8.p1.8.m1.2.2.1.3">𝜎</ci><ci id="S6.8.p1.8.m1.2.2.1.1.cmml" xref="S6.8.p1.8.m1.2.2.1.1">𝓌</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.8.m1.3c">\sigma_{*}(\mu)=\underset{w\in\cal L(X)}{\sum}\lambda_{w}\,\mu_{\sigma(w)}\,</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.8.m1.3d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) = start_UNDERACCENT italic_w ∈ caligraphic_L ( caligraphic_X ) end_UNDERACCENT start_ARG ∑ end_ARG italic_λ start_POSTSUBSCRIPT caligraphic_w end_POSTSUBSCRIPT italic_μ start_POSTSUBSCRIPT italic_σ ( caligraphic_w ) end_POSTSUBSCRIPT</annotation></semantics></math>, which is clearly the zero-measure outside of <math alttext="\text{\rm Per}(\sigma(X))" class="ltx_Math" display="inline" id="S6.8.p1.9.m2.2"><semantics id="S6.8.p1.9.m2.2a"><mrow id="S6.8.p1.9.m2.2.2" xref="S6.8.p1.9.m2.2.2.cmml"><mtext id="S6.8.p1.9.m2.2.2.3" xref="S6.8.p1.9.m2.2.2.3a.cmml">Per</mtext><mo id="S6.8.p1.9.m2.2.2.2" xref="S6.8.p1.9.m2.2.2.2.cmml">⁢</mo><mrow id="S6.8.p1.9.m2.2.2.1.1" xref="S6.8.p1.9.m2.2.2.1.1.1.cmml"><mo id="S6.8.p1.9.m2.2.2.1.1.2" stretchy="false" xref="S6.8.p1.9.m2.2.2.1.1.1.cmml">(</mo><mrow id="S6.8.p1.9.m2.2.2.1.1.1" xref="S6.8.p1.9.m2.2.2.1.1.1.cmml"><mi id="S6.8.p1.9.m2.2.2.1.1.1.2" xref="S6.8.p1.9.m2.2.2.1.1.1.2.cmml">σ</mi><mo id="S6.8.p1.9.m2.2.2.1.1.1.1" xref="S6.8.p1.9.m2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.8.p1.9.m2.2.2.1.1.1.3.2" xref="S6.8.p1.9.m2.2.2.1.1.1.cmml"><mo id="S6.8.p1.9.m2.2.2.1.1.1.3.2.1" stretchy="false" xref="S6.8.p1.9.m2.2.2.1.1.1.cmml">(</mo><mi id="S6.8.p1.9.m2.1.1" xref="S6.8.p1.9.m2.1.1.cmml">X</mi><mo id="S6.8.p1.9.m2.2.2.1.1.1.3.2.2" stretchy="false" xref="S6.8.p1.9.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.8.p1.9.m2.2.2.1.1.3" stretchy="false" xref="S6.8.p1.9.m2.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.8.p1.9.m2.2b"><apply id="S6.8.p1.9.m2.2.2.cmml" xref="S6.8.p1.9.m2.2.2"><times id="S6.8.p1.9.m2.2.2.2.cmml" xref="S6.8.p1.9.m2.2.2.2"></times><ci id="S6.8.p1.9.m2.2.2.3a.cmml" xref="S6.8.p1.9.m2.2.2.3"><mtext id="S6.8.p1.9.m2.2.2.3.cmml" xref="S6.8.p1.9.m2.2.2.3">Per</mtext></ci><apply id="S6.8.p1.9.m2.2.2.1.1.1.cmml" xref="S6.8.p1.9.m2.2.2.1.1"><times id="S6.8.p1.9.m2.2.2.1.1.1.1.cmml" xref="S6.8.p1.9.m2.2.2.1.1.1.1"></times><ci id="S6.8.p1.9.m2.2.2.1.1.1.2.cmml" xref="S6.8.p1.9.m2.2.2.1.1.1.2">𝜎</ci><ci id="S6.8.p1.9.m2.1.1.cmml" xref="S6.8.p1.9.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.8.p1.9.m2.2c">\text{\rm Per}(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S6.8.p1.9.m2.2d">Per ( italic_σ ( italic_X ) )</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S6.9.p2"> <p class="ltx_p" id="S6.9.p2.8">(2) For any <math alttext="{\bf x}\in\text{\rm Per}(\sigma(X))" class="ltx_Math" display="inline" id="S6.9.p2.1.m1.2"><semantics id="S6.9.p2.1.m1.2a"><mrow id="S6.9.p2.1.m1.2.2" xref="S6.9.p2.1.m1.2.2.cmml"><mi id="S6.9.p2.1.m1.2.2.3" xref="S6.9.p2.1.m1.2.2.3.cmml">𝐱</mi><mo id="S6.9.p2.1.m1.2.2.2" xref="S6.9.p2.1.m1.2.2.2.cmml">∈</mo><mrow id="S6.9.p2.1.m1.2.2.1" xref="S6.9.p2.1.m1.2.2.1.cmml"><mtext id="S6.9.p2.1.m1.2.2.1.3" xref="S6.9.p2.1.m1.2.2.1.3a.cmml">Per</mtext><mo id="S6.9.p2.1.m1.2.2.1.2" xref="S6.9.p2.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S6.9.p2.1.m1.2.2.1.1.1" xref="S6.9.p2.1.m1.2.2.1.1.1.1.cmml"><mo id="S6.9.p2.1.m1.2.2.1.1.1.2" stretchy="false" xref="S6.9.p2.1.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.9.p2.1.m1.2.2.1.1.1.1" xref="S6.9.p2.1.m1.2.2.1.1.1.1.cmml"><mi id="S6.9.p2.1.m1.2.2.1.1.1.1.2" xref="S6.9.p2.1.m1.2.2.1.1.1.1.2.cmml">σ</mi><mo id="S6.9.p2.1.m1.2.2.1.1.1.1.1" xref="S6.9.p2.1.m1.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.9.p2.1.m1.2.2.1.1.1.1.3.2" xref="S6.9.p2.1.m1.2.2.1.1.1.1.cmml"><mo id="S6.9.p2.1.m1.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S6.9.p2.1.m1.2.2.1.1.1.1.cmml">(</mo><mi id="S6.9.p2.1.m1.1.1" xref="S6.9.p2.1.m1.1.1.cmml">X</mi><mo id="S6.9.p2.1.m1.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S6.9.p2.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.9.p2.1.m1.2.2.1.1.1.3" stretchy="false" xref="S6.9.p2.1.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.9.p2.1.m1.2b"><apply id="S6.9.p2.1.m1.2.2.cmml" xref="S6.9.p2.1.m1.2.2"><in id="S6.9.p2.1.m1.2.2.2.cmml" xref="S6.9.p2.1.m1.2.2.2"></in><ci id="S6.9.p2.1.m1.2.2.3.cmml" xref="S6.9.p2.1.m1.2.2.3">𝐱</ci><apply id="S6.9.p2.1.m1.2.2.1.cmml" xref="S6.9.p2.1.m1.2.2.1"><times id="S6.9.p2.1.m1.2.2.1.2.cmml" xref="S6.9.p2.1.m1.2.2.1.2"></times><ci id="S6.9.p2.1.m1.2.2.1.3a.cmml" xref="S6.9.p2.1.m1.2.2.1.3"><mtext id="S6.9.p2.1.m1.2.2.1.3.cmml" xref="S6.9.p2.1.m1.2.2.1.3">Per</mtext></ci><apply id="S6.9.p2.1.m1.2.2.1.1.1.1.cmml" xref="S6.9.p2.1.m1.2.2.1.1.1"><times id="S6.9.p2.1.m1.2.2.1.1.1.1.1.cmml" xref="S6.9.p2.1.m1.2.2.1.1.1.1.1"></times><ci id="S6.9.p2.1.m1.2.2.1.1.1.1.2.cmml" xref="S6.9.p2.1.m1.2.2.1.1.1.1.2">𝜎</ci><ci id="S6.9.p2.1.m1.1.1.cmml" xref="S6.9.p2.1.m1.1.1">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.9.p2.1.m1.2c">{\bf x}\in\text{\rm Per}(\sigma(X))</annotation><annotation encoding="application/x-llamapun" id="S6.9.p2.1.m1.2d">bold_x ∈ Per ( italic_σ ( italic_X ) )</annotation></semantics></math> we know from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm1" title="Lemma 6.1. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.1</span></a> that <math alttext="\sigma^{-1}({\bf x})" class="ltx_Math" display="inline" id="S6.9.p2.2.m2.1"><semantics id="S6.9.p2.2.m2.1a"><mrow id="S6.9.p2.2.m2.1.2" xref="S6.9.p2.2.m2.1.2.cmml"><msup id="S6.9.p2.2.m2.1.2.2" xref="S6.9.p2.2.m2.1.2.2.cmml"><mi id="S6.9.p2.2.m2.1.2.2.2" xref="S6.9.p2.2.m2.1.2.2.2.cmml">σ</mi><mrow id="S6.9.p2.2.m2.1.2.2.3" xref="S6.9.p2.2.m2.1.2.2.3.cmml"><mo id="S6.9.p2.2.m2.1.2.2.3a" xref="S6.9.p2.2.m2.1.2.2.3.cmml">−</mo><mn id="S6.9.p2.2.m2.1.2.2.3.2" xref="S6.9.p2.2.m2.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S6.9.p2.2.m2.1.2.1" xref="S6.9.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S6.9.p2.2.m2.1.2.3.2" xref="S6.9.p2.2.m2.1.2.cmml"><mo id="S6.9.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S6.9.p2.2.m2.1.2.cmml">(</mo><mi id="S6.9.p2.2.m2.1.1" xref="S6.9.p2.2.m2.1.1.cmml">𝐱</mi><mo id="S6.9.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S6.9.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.9.p2.2.m2.1b"><apply id="S6.9.p2.2.m2.1.2.cmml" xref="S6.9.p2.2.m2.1.2"><times id="S6.9.p2.2.m2.1.2.1.cmml" xref="S6.9.p2.2.m2.1.2.1"></times><apply id="S6.9.p2.2.m2.1.2.2.cmml" xref="S6.9.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S6.9.p2.2.m2.1.2.2.1.cmml" xref="S6.9.p2.2.m2.1.2.2">superscript</csymbol><ci id="S6.9.p2.2.m2.1.2.2.2.cmml" xref="S6.9.p2.2.m2.1.2.2.2">𝜎</ci><apply id="S6.9.p2.2.m2.1.2.2.3.cmml" xref="S6.9.p2.2.m2.1.2.2.3"><minus id="S6.9.p2.2.m2.1.2.2.3.1.cmml" xref="S6.9.p2.2.m2.1.2.2.3"></minus><cn id="S6.9.p2.2.m2.1.2.2.3.2.cmml" type="integer" xref="S6.9.p2.2.m2.1.2.2.3.2">1</cn></apply></apply><ci id="S6.9.p2.2.m2.1.1.cmml" xref="S6.9.p2.2.m2.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.9.p2.2.m2.1c">\sigma^{-1}({\bf x})</annotation><annotation encoding="application/x-llamapun" id="S6.9.p2.2.m2.1d">italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x )</annotation></semantics></math> consist of periodic words only. Hence the assumption <math alttext="\mu(\text{\rm Per}(X))=0" class="ltx_Math" display="inline" id="S6.9.p2.3.m3.2"><semantics id="S6.9.p2.3.m3.2a"><mrow id="S6.9.p2.3.m3.2.2" xref="S6.9.p2.3.m3.2.2.cmml"><mrow id="S6.9.p2.3.m3.2.2.1" xref="S6.9.p2.3.m3.2.2.1.cmml"><mi id="S6.9.p2.3.m3.2.2.1.3" xref="S6.9.p2.3.m3.2.2.1.3.cmml">μ</mi><mo id="S6.9.p2.3.m3.2.2.1.2" xref="S6.9.p2.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S6.9.p2.3.m3.2.2.1.1.1" xref="S6.9.p2.3.m3.2.2.1.1.1.1.cmml"><mo id="S6.9.p2.3.m3.2.2.1.1.1.2" stretchy="false" xref="S6.9.p2.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.9.p2.3.m3.2.2.1.1.1.1" xref="S6.9.p2.3.m3.2.2.1.1.1.1.cmml"><mtext id="S6.9.p2.3.m3.2.2.1.1.1.1.2" xref="S6.9.p2.3.m3.2.2.1.1.1.1.2a.cmml">Per</mtext><mo id="S6.9.p2.3.m3.2.2.1.1.1.1.1" xref="S6.9.p2.3.m3.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.9.p2.3.m3.2.2.1.1.1.1.3.2" xref="S6.9.p2.3.m3.2.2.1.1.1.1.cmml"><mo id="S6.9.p2.3.m3.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S6.9.p2.3.m3.2.2.1.1.1.1.cmml">(</mo><mi id="S6.9.p2.3.m3.1.1" xref="S6.9.p2.3.m3.1.1.cmml">X</mi><mo id="S6.9.p2.3.m3.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S6.9.p2.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.9.p2.3.m3.2.2.1.1.1.3" stretchy="false" xref="S6.9.p2.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.9.p2.3.m3.2.2.2" xref="S6.9.p2.3.m3.2.2.2.cmml">=</mo><mn id="S6.9.p2.3.m3.2.2.3" xref="S6.9.p2.3.m3.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.9.p2.3.m3.2b"><apply id="S6.9.p2.3.m3.2.2.cmml" xref="S6.9.p2.3.m3.2.2"><eq id="S6.9.p2.3.m3.2.2.2.cmml" xref="S6.9.p2.3.m3.2.2.2"></eq><apply id="S6.9.p2.3.m3.2.2.1.cmml" xref="S6.9.p2.3.m3.2.2.1"><times id="S6.9.p2.3.m3.2.2.1.2.cmml" xref="S6.9.p2.3.m3.2.2.1.2"></times><ci id="S6.9.p2.3.m3.2.2.1.3.cmml" xref="S6.9.p2.3.m3.2.2.1.3">𝜇</ci><apply id="S6.9.p2.3.m3.2.2.1.1.1.1.cmml" xref="S6.9.p2.3.m3.2.2.1.1.1"><times id="S6.9.p2.3.m3.2.2.1.1.1.1.1.cmml" xref="S6.9.p2.3.m3.2.2.1.1.1.1.1"></times><ci id="S6.9.p2.3.m3.2.2.1.1.1.1.2a.cmml" xref="S6.9.p2.3.m3.2.2.1.1.1.1.2"><mtext id="S6.9.p2.3.m3.2.2.1.1.1.1.2.cmml" xref="S6.9.p2.3.m3.2.2.1.1.1.1.2">Per</mtext></ci><ci id="S6.9.p2.3.m3.1.1.cmml" xref="S6.9.p2.3.m3.1.1">𝑋</ci></apply></apply><cn id="S6.9.p2.3.m3.2.2.3.cmml" type="integer" xref="S6.9.p2.3.m3.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.9.p2.3.m3.2c">\mu(\text{\rm Per}(X))=0</annotation><annotation encoding="application/x-llamapun" id="S6.9.p2.3.m3.2d">italic_μ ( Per ( italic_X ) ) = 0</annotation></semantics></math> implies that <math alttext="\sigma_{*}(\mu)({\bf x})=\mu(\sigma^{-1}({\bf x}))=0" class="ltx_Math" display="inline" id="S6.9.p2.4.m4.4"><semantics id="S6.9.p2.4.m4.4a"><mrow id="S6.9.p2.4.m4.4.4" xref="S6.9.p2.4.m4.4.4.cmml"><mrow id="S6.9.p2.4.m4.4.4.3" xref="S6.9.p2.4.m4.4.4.3.cmml"><msub id="S6.9.p2.4.m4.4.4.3.2" xref="S6.9.p2.4.m4.4.4.3.2.cmml"><mi id="S6.9.p2.4.m4.4.4.3.2.2" xref="S6.9.p2.4.m4.4.4.3.2.2.cmml">σ</mi><mo id="S6.9.p2.4.m4.4.4.3.2.3" xref="S6.9.p2.4.m4.4.4.3.2.3.cmml">∗</mo></msub><mo id="S6.9.p2.4.m4.4.4.3.1" xref="S6.9.p2.4.m4.4.4.3.1.cmml">⁢</mo><mrow id="S6.9.p2.4.m4.4.4.3.3.2" xref="S6.9.p2.4.m4.4.4.3.cmml"><mo id="S6.9.p2.4.m4.4.4.3.3.2.1" stretchy="false" xref="S6.9.p2.4.m4.4.4.3.cmml">(</mo><mi id="S6.9.p2.4.m4.1.1" xref="S6.9.p2.4.m4.1.1.cmml">μ</mi><mo id="S6.9.p2.4.m4.4.4.3.3.2.2" stretchy="false" xref="S6.9.p2.4.m4.4.4.3.cmml">)</mo></mrow><mo id="S6.9.p2.4.m4.4.4.3.1a" xref="S6.9.p2.4.m4.4.4.3.1.cmml">⁢</mo><mrow id="S6.9.p2.4.m4.4.4.3.4.2" xref="S6.9.p2.4.m4.4.4.3.cmml"><mo id="S6.9.p2.4.m4.4.4.3.4.2.1" stretchy="false" xref="S6.9.p2.4.m4.4.4.3.cmml">(</mo><mi id="S6.9.p2.4.m4.2.2" xref="S6.9.p2.4.m4.2.2.cmml">𝐱</mi><mo id="S6.9.p2.4.m4.4.4.3.4.2.2" stretchy="false" xref="S6.9.p2.4.m4.4.4.3.cmml">)</mo></mrow></mrow><mo id="S6.9.p2.4.m4.4.4.4" xref="S6.9.p2.4.m4.4.4.4.cmml">=</mo><mrow id="S6.9.p2.4.m4.4.4.1" xref="S6.9.p2.4.m4.4.4.1.cmml"><mi id="S6.9.p2.4.m4.4.4.1.3" xref="S6.9.p2.4.m4.4.4.1.3.cmml">μ</mi><mo id="S6.9.p2.4.m4.4.4.1.2" xref="S6.9.p2.4.m4.4.4.1.2.cmml">⁢</mo><mrow id="S6.9.p2.4.m4.4.4.1.1.1" xref="S6.9.p2.4.m4.4.4.1.1.1.1.cmml"><mo id="S6.9.p2.4.m4.4.4.1.1.1.2" stretchy="false" xref="S6.9.p2.4.m4.4.4.1.1.1.1.cmml">(</mo><mrow id="S6.9.p2.4.m4.4.4.1.1.1.1" xref="S6.9.p2.4.m4.4.4.1.1.1.1.cmml"><msup id="S6.9.p2.4.m4.4.4.1.1.1.1.2" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.cmml"><mi id="S6.9.p2.4.m4.4.4.1.1.1.1.2.2" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.2.cmml">σ</mi><mrow id="S6.9.p2.4.m4.4.4.1.1.1.1.2.3" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.3.cmml"><mo id="S6.9.p2.4.m4.4.4.1.1.1.1.2.3a" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.3.cmml">−</mo><mn id="S6.9.p2.4.m4.4.4.1.1.1.1.2.3.2" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S6.9.p2.4.m4.4.4.1.1.1.1.1" xref="S6.9.p2.4.m4.4.4.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.9.p2.4.m4.4.4.1.1.1.1.3.2" xref="S6.9.p2.4.m4.4.4.1.1.1.1.cmml"><mo id="S6.9.p2.4.m4.4.4.1.1.1.1.3.2.1" stretchy="false" xref="S6.9.p2.4.m4.4.4.1.1.1.1.cmml">(</mo><mi id="S6.9.p2.4.m4.3.3" xref="S6.9.p2.4.m4.3.3.cmml">𝐱</mi><mo id="S6.9.p2.4.m4.4.4.1.1.1.1.3.2.2" stretchy="false" xref="S6.9.p2.4.m4.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.9.p2.4.m4.4.4.1.1.1.3" stretchy="false" xref="S6.9.p2.4.m4.4.4.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.9.p2.4.m4.4.4.5" xref="S6.9.p2.4.m4.4.4.5.cmml">=</mo><mn id="S6.9.p2.4.m4.4.4.6" xref="S6.9.p2.4.m4.4.4.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.9.p2.4.m4.4b"><apply id="S6.9.p2.4.m4.4.4.cmml" xref="S6.9.p2.4.m4.4.4"><and id="S6.9.p2.4.m4.4.4a.cmml" xref="S6.9.p2.4.m4.4.4"></and><apply id="S6.9.p2.4.m4.4.4b.cmml" xref="S6.9.p2.4.m4.4.4"><eq id="S6.9.p2.4.m4.4.4.4.cmml" xref="S6.9.p2.4.m4.4.4.4"></eq><apply id="S6.9.p2.4.m4.4.4.3.cmml" xref="S6.9.p2.4.m4.4.4.3"><times id="S6.9.p2.4.m4.4.4.3.1.cmml" xref="S6.9.p2.4.m4.4.4.3.1"></times><apply id="S6.9.p2.4.m4.4.4.3.2.cmml" xref="S6.9.p2.4.m4.4.4.3.2"><csymbol cd="ambiguous" id="S6.9.p2.4.m4.4.4.3.2.1.cmml" xref="S6.9.p2.4.m4.4.4.3.2">subscript</csymbol><ci id="S6.9.p2.4.m4.4.4.3.2.2.cmml" xref="S6.9.p2.4.m4.4.4.3.2.2">𝜎</ci><times id="S6.9.p2.4.m4.4.4.3.2.3.cmml" xref="S6.9.p2.4.m4.4.4.3.2.3"></times></apply><ci id="S6.9.p2.4.m4.1.1.cmml" xref="S6.9.p2.4.m4.1.1">𝜇</ci><ci id="S6.9.p2.4.m4.2.2.cmml" xref="S6.9.p2.4.m4.2.2">𝐱</ci></apply><apply id="S6.9.p2.4.m4.4.4.1.cmml" xref="S6.9.p2.4.m4.4.4.1"><times id="S6.9.p2.4.m4.4.4.1.2.cmml" xref="S6.9.p2.4.m4.4.4.1.2"></times><ci id="S6.9.p2.4.m4.4.4.1.3.cmml" xref="S6.9.p2.4.m4.4.4.1.3">𝜇</ci><apply id="S6.9.p2.4.m4.4.4.1.1.1.1.cmml" xref="S6.9.p2.4.m4.4.4.1.1.1"><times id="S6.9.p2.4.m4.4.4.1.1.1.1.1.cmml" xref="S6.9.p2.4.m4.4.4.1.1.1.1.1"></times><apply id="S6.9.p2.4.m4.4.4.1.1.1.1.2.cmml" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.9.p2.4.m4.4.4.1.1.1.1.2.1.cmml" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2">superscript</csymbol><ci id="S6.9.p2.4.m4.4.4.1.1.1.1.2.2.cmml" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.2">𝜎</ci><apply id="S6.9.p2.4.m4.4.4.1.1.1.1.2.3.cmml" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.3"><minus id="S6.9.p2.4.m4.4.4.1.1.1.1.2.3.1.cmml" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.3"></minus><cn id="S6.9.p2.4.m4.4.4.1.1.1.1.2.3.2.cmml" type="integer" xref="S6.9.p2.4.m4.4.4.1.1.1.1.2.3.2">1</cn></apply></apply><ci id="S6.9.p2.4.m4.3.3.cmml" xref="S6.9.p2.4.m4.3.3">𝐱</ci></apply></apply></apply><apply id="S6.9.p2.4.m4.4.4c.cmml" xref="S6.9.p2.4.m4.4.4"><eq id="S6.9.p2.4.m4.4.4.5.cmml" xref="S6.9.p2.4.m4.4.4.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S6.9.p2.4.m4.4.4.1.cmml" id="S6.9.p2.4.m4.4.4d.cmml" xref="S6.9.p2.4.m4.4.4"></share><cn id="S6.9.p2.4.m4.4.4.6.cmml" type="integer" xref="S6.9.p2.4.m4.4.4.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.9.p2.4.m4.4c">\sigma_{*}(\mu)({\bf x})=\mu(\sigma^{-1}({\bf x}))=0</annotation><annotation encoding="application/x-llamapun" id="S6.9.p2.4.m4.4d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( bold_x ) = italic_μ ( italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x ) ) = 0</annotation></semantics></math>. Since <math alttext="\text{\rm Per}(X)" class="ltx_Math" display="inline" id="S6.9.p2.5.m5.1"><semantics id="S6.9.p2.5.m5.1a"><mrow id="S6.9.p2.5.m5.1.2" xref="S6.9.p2.5.m5.1.2.cmml"><mtext id="S6.9.p2.5.m5.1.2.2" xref="S6.9.p2.5.m5.1.2.2a.cmml">Per</mtext><mo id="S6.9.p2.5.m5.1.2.1" xref="S6.9.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S6.9.p2.5.m5.1.2.3.2" xref="S6.9.p2.5.m5.1.2.cmml"><mo id="S6.9.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S6.9.p2.5.m5.1.2.cmml">(</mo><mi id="S6.9.p2.5.m5.1.1" xref="S6.9.p2.5.m5.1.1.cmml">X</mi><mo id="S6.9.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S6.9.p2.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.9.p2.5.m5.1b"><apply id="S6.9.p2.5.m5.1.2.cmml" xref="S6.9.p2.5.m5.1.2"><times id="S6.9.p2.5.m5.1.2.1.cmml" xref="S6.9.p2.5.m5.1.2.1"></times><ci id="S6.9.p2.5.m5.1.2.2a.cmml" xref="S6.9.p2.5.m5.1.2.2"><mtext id="S6.9.p2.5.m5.1.2.2.cmml" xref="S6.9.p2.5.m5.1.2.2">Per</mtext></ci><ci id="S6.9.p2.5.m5.1.1.cmml" xref="S6.9.p2.5.m5.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.9.p2.5.m5.1c">\text{\rm Per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.9.p2.5.m5.1d">Per ( italic_X )</annotation></semantics></math> is countable, this implies <math alttext="\sigma_{*}(\mu)(\text{\rm Per}(X))=0" class="ltx_Math" display="inline" id="S6.9.p2.6.m6.3"><semantics id="S6.9.p2.6.m6.3a"><mrow id="S6.9.p2.6.m6.3.3" xref="S6.9.p2.6.m6.3.3.cmml"><mrow id="S6.9.p2.6.m6.3.3.1" xref="S6.9.p2.6.m6.3.3.1.cmml"><msub id="S6.9.p2.6.m6.3.3.1.3" xref="S6.9.p2.6.m6.3.3.1.3.cmml"><mi id="S6.9.p2.6.m6.3.3.1.3.2" xref="S6.9.p2.6.m6.3.3.1.3.2.cmml">σ</mi><mo id="S6.9.p2.6.m6.3.3.1.3.3" xref="S6.9.p2.6.m6.3.3.1.3.3.cmml">∗</mo></msub><mo id="S6.9.p2.6.m6.3.3.1.2" xref="S6.9.p2.6.m6.3.3.1.2.cmml">⁢</mo><mrow id="S6.9.p2.6.m6.3.3.1.4.2" xref="S6.9.p2.6.m6.3.3.1.cmml"><mo id="S6.9.p2.6.m6.3.3.1.4.2.1" stretchy="false" xref="S6.9.p2.6.m6.3.3.1.cmml">(</mo><mi id="S6.9.p2.6.m6.1.1" xref="S6.9.p2.6.m6.1.1.cmml">μ</mi><mo id="S6.9.p2.6.m6.3.3.1.4.2.2" stretchy="false" xref="S6.9.p2.6.m6.3.3.1.cmml">)</mo></mrow><mo id="S6.9.p2.6.m6.3.3.1.2a" xref="S6.9.p2.6.m6.3.3.1.2.cmml">⁢</mo><mrow id="S6.9.p2.6.m6.3.3.1.1.1" xref="S6.9.p2.6.m6.3.3.1.1.1.1.cmml"><mo id="S6.9.p2.6.m6.3.3.1.1.1.2" stretchy="false" xref="S6.9.p2.6.m6.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.9.p2.6.m6.3.3.1.1.1.1" xref="S6.9.p2.6.m6.3.3.1.1.1.1.cmml"><mtext id="S6.9.p2.6.m6.3.3.1.1.1.1.2" xref="S6.9.p2.6.m6.3.3.1.1.1.1.2a.cmml">Per</mtext><mo id="S6.9.p2.6.m6.3.3.1.1.1.1.1" xref="S6.9.p2.6.m6.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.9.p2.6.m6.3.3.1.1.1.1.3.2" xref="S6.9.p2.6.m6.3.3.1.1.1.1.cmml"><mo id="S6.9.p2.6.m6.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S6.9.p2.6.m6.3.3.1.1.1.1.cmml">(</mo><mi id="S6.9.p2.6.m6.2.2" xref="S6.9.p2.6.m6.2.2.cmml">X</mi><mo id="S6.9.p2.6.m6.3.3.1.1.1.1.3.2.2" stretchy="false" xref="S6.9.p2.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.9.p2.6.m6.3.3.1.1.1.3" stretchy="false" xref="S6.9.p2.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.9.p2.6.m6.3.3.2" xref="S6.9.p2.6.m6.3.3.2.cmml">=</mo><mn id="S6.9.p2.6.m6.3.3.3" xref="S6.9.p2.6.m6.3.3.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.9.p2.6.m6.3b"><apply id="S6.9.p2.6.m6.3.3.cmml" xref="S6.9.p2.6.m6.3.3"><eq id="S6.9.p2.6.m6.3.3.2.cmml" xref="S6.9.p2.6.m6.3.3.2"></eq><apply id="S6.9.p2.6.m6.3.3.1.cmml" xref="S6.9.p2.6.m6.3.3.1"><times id="S6.9.p2.6.m6.3.3.1.2.cmml" xref="S6.9.p2.6.m6.3.3.1.2"></times><apply id="S6.9.p2.6.m6.3.3.1.3.cmml" xref="S6.9.p2.6.m6.3.3.1.3"><csymbol cd="ambiguous" id="S6.9.p2.6.m6.3.3.1.3.1.cmml" xref="S6.9.p2.6.m6.3.3.1.3">subscript</csymbol><ci id="S6.9.p2.6.m6.3.3.1.3.2.cmml" xref="S6.9.p2.6.m6.3.3.1.3.2">𝜎</ci><times id="S6.9.p2.6.m6.3.3.1.3.3.cmml" xref="S6.9.p2.6.m6.3.3.1.3.3"></times></apply><ci id="S6.9.p2.6.m6.1.1.cmml" xref="S6.9.p2.6.m6.1.1">𝜇</ci><apply id="S6.9.p2.6.m6.3.3.1.1.1.1.cmml" xref="S6.9.p2.6.m6.3.3.1.1.1"><times id="S6.9.p2.6.m6.3.3.1.1.1.1.1.cmml" xref="S6.9.p2.6.m6.3.3.1.1.1.1.1"></times><ci id="S6.9.p2.6.m6.3.3.1.1.1.1.2a.cmml" xref="S6.9.p2.6.m6.3.3.1.1.1.1.2"><mtext id="S6.9.p2.6.m6.3.3.1.1.1.1.2.cmml" xref="S6.9.p2.6.m6.3.3.1.1.1.1.2">Per</mtext></ci><ci id="S6.9.p2.6.m6.2.2.cmml" xref="S6.9.p2.6.m6.2.2">𝑋</ci></apply></apply><cn id="S6.9.p2.6.m6.3.3.3.cmml" type="integer" xref="S6.9.p2.6.m6.3.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.9.p2.6.m6.3c">\sigma_{*}(\mu)(\text{\rm Per}(X))=0</annotation><annotation encoding="application/x-llamapun" id="S6.9.p2.6.m6.3d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( Per ( italic_X ) ) = 0</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S6.9.p2.7.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S6.9.p2.7.1.m1.1"><semantics id="S6.9.p2.7.1.m1.1a"><mo id="S6.9.p2.7.1.m1.1.1" xref="S6.9.p2.7.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S6.9.p2.7.1.m1.1b"><csymbol cd="latexml" id="S6.9.p2.7.1.m1.1.1.cmml" xref="S6.9.p2.7.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.9.p2.7.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S6.9.p2.7.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S6.9.p2.8.m7.1"><semantics id="S6.9.p2.8.m7.1a"><mo id="S6.9.p2.8.m7.1.1" xref="S6.9.p2.8.m7.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S6.9.p2.8.m7.1b"><csymbol cd="latexml" id="S6.9.p2.8.m7.1.1.cmml" xref="S6.9.p2.8.m7.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.9.p2.8.m7.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S6.9.p2.8.m7.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_prop" id="S6.Thmthm6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm6.1.1.1">Proposition 6.6</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm6.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm6.p1"> <p class="ltx_p" id="S6.Thmthm6.p1.4"><span class="ltx_text ltx_font_italic" id="S6.Thmthm6.p1.4.4">For the subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.Thmthm6.p1.1.1.m1.1"><semantics id="S6.Thmthm6.p1.1.1.m1.1a"><mrow id="S6.Thmthm6.p1.1.1.m1.1.1" xref="S6.Thmthm6.p1.1.1.m1.1.1.cmml"><mi id="S6.Thmthm6.p1.1.1.m1.1.1.2" xref="S6.Thmthm6.p1.1.1.m1.1.1.2.cmml">X</mi><mo id="S6.Thmthm6.p1.1.1.m1.1.1.1" xref="S6.Thmthm6.p1.1.1.m1.1.1.1.cmml">⊆</mo><msup id="S6.Thmthm6.p1.1.1.m1.1.1.3" xref="S6.Thmthm6.p1.1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.1.1.m1.1.1.3.2" xref="S6.Thmthm6.p1.1.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S6.Thmthm6.p1.1.1.m1.1.1.3.3" xref="S6.Thmthm6.p1.1.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p1.1.1.m1.1b"><apply id="S6.Thmthm6.p1.1.1.m1.1.1.cmml" xref="S6.Thmthm6.p1.1.1.m1.1.1"><subset id="S6.Thmthm6.p1.1.1.m1.1.1.1.cmml" xref="S6.Thmthm6.p1.1.1.m1.1.1.1"></subset><ci id="S6.Thmthm6.p1.1.1.m1.1.1.2.cmml" xref="S6.Thmthm6.p1.1.1.m1.1.1.2">𝑋</ci><apply id="S6.Thmthm6.p1.1.1.m1.1.1.3.cmml" xref="S6.Thmthm6.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmthm6.p1.1.1.m1.1.1.3.1.cmml" xref="S6.Thmthm6.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S6.Thmthm6.p1.1.1.m1.1.1.3.2.cmml" xref="S6.Thmthm6.p1.1.1.m1.1.1.3.2">𝒜</ci><ci id="S6.Thmthm6.p1.1.1.m1.1.1.3.3.cmml" xref="S6.Thmthm6.p1.1.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p1.1.1.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p1.1.1.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> let <math alttext="\cal M_{per}(X)\subseteq\cal M(X)" class="ltx_Math" display="inline" id="S6.Thmthm6.p1.2.2.m2.2"><semantics id="S6.Thmthm6.p1.2.2.m2.2a"><mrow id="S6.Thmthm6.p1.2.2.m2.2.3" xref="S6.Thmthm6.p1.2.2.m2.2.3.cmml"><mrow id="S6.Thmthm6.p1.2.2.m2.2.3.2" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.cmml"><msub id="S6.Thmthm6.p1.2.2.m2.2.3.2.2" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.2" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.2.cmml">ℳ</mi><mrow id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.2" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.2.cmml">𝓅</mi><mo id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.1" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.3" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.3.cmml">ℯ</mi><mo id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.1a" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.4" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.4.cmml">𝓇</mi></mrow></msub><mo id="S6.Thmthm6.p1.2.2.m2.2.3.2.1" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p1.2.2.m2.2.3.2.3.2" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.cmml"><mo id="S6.Thmthm6.p1.2.2.m2.2.3.2.3.2.1" stretchy="false" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.2.2.m2.1.1" xref="S6.Thmthm6.p1.2.2.m2.1.1.cmml">𝒳</mi><mo id="S6.Thmthm6.p1.2.2.m2.2.3.2.3.2.2" stretchy="false" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm6.p1.2.2.m2.2.3.1" xref="S6.Thmthm6.p1.2.2.m2.2.3.1.cmml">⊆</mo><mrow id="S6.Thmthm6.p1.2.2.m2.2.3.3" xref="S6.Thmthm6.p1.2.2.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.2.2.m2.2.3.3.2" xref="S6.Thmthm6.p1.2.2.m2.2.3.3.2.cmml">ℳ</mi><mo id="S6.Thmthm6.p1.2.2.m2.2.3.3.1" xref="S6.Thmthm6.p1.2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p1.2.2.m2.2.3.3.3.2" xref="S6.Thmthm6.p1.2.2.m2.2.3.3.cmml"><mo id="S6.Thmthm6.p1.2.2.m2.2.3.3.3.2.1" stretchy="false" xref="S6.Thmthm6.p1.2.2.m2.2.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.2.2.m2.2.2" xref="S6.Thmthm6.p1.2.2.m2.2.2.cmml">𝒳</mi><mo id="S6.Thmthm6.p1.2.2.m2.2.3.3.3.2.2" stretchy="false" xref="S6.Thmthm6.p1.2.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p1.2.2.m2.2b"><apply id="S6.Thmthm6.p1.2.2.m2.2.3.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3"><subset id="S6.Thmthm6.p1.2.2.m2.2.3.1.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.1"></subset><apply id="S6.Thmthm6.p1.2.2.m2.2.3.2.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2"><times id="S6.Thmthm6.p1.2.2.m2.2.3.2.1.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.1"></times><apply id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2"><csymbol cd="ambiguous" id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.1.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2">subscript</csymbol><ci id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.2.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.2">ℳ</ci><apply id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3"><times id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.1.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.1"></times><ci id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.2.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.2">𝓅</ci><ci id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.3.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.3">ℯ</ci><ci id="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.4.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.2.2.3.4">𝓇</ci></apply></apply><ci id="S6.Thmthm6.p1.2.2.m2.1.1.cmml" xref="S6.Thmthm6.p1.2.2.m2.1.1">𝒳</ci></apply><apply id="S6.Thmthm6.p1.2.2.m2.2.3.3.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.3"><times id="S6.Thmthm6.p1.2.2.m2.2.3.3.1.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.3.1"></times><ci id="S6.Thmthm6.p1.2.2.m2.2.3.3.2.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.3.3.2">ℳ</ci><ci id="S6.Thmthm6.p1.2.2.m2.2.2.cmml" xref="S6.Thmthm6.p1.2.2.m2.2.2">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p1.2.2.m2.2c">\cal M_{per}(X)\subseteq\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p1.2.2.m2.2d">caligraphic_M start_POSTSUBSCRIPT caligraphic_p caligraphic_e caligraphic_r end_POSTSUBSCRIPT ( caligraphic_X ) ⊆ caligraphic_M ( caligraphic_X )</annotation></semantics></math> and <math alttext="\cal M_{na}(X)\subseteq\cal M(X)" class="ltx_Math" display="inline" id="S6.Thmthm6.p1.3.3.m3.2"><semantics id="S6.Thmthm6.p1.3.3.m3.2a"><mrow id="S6.Thmthm6.p1.3.3.m3.2.3" xref="S6.Thmthm6.p1.3.3.m3.2.3.cmml"><mrow id="S6.Thmthm6.p1.3.3.m3.2.3.2" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.cmml"><msub id="S6.Thmthm6.p1.3.3.m3.2.3.2.2" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.2" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.2.cmml">ℳ</mi><mrow id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.2" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.2.cmml">𝓃</mi><mo id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.1" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.3" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.3.cmml">𝒶</mi></mrow></msub><mo id="S6.Thmthm6.p1.3.3.m3.2.3.2.1" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p1.3.3.m3.2.3.2.3.2" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.cmml"><mo id="S6.Thmthm6.p1.3.3.m3.2.3.2.3.2.1" stretchy="false" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.3.3.m3.1.1" xref="S6.Thmthm6.p1.3.3.m3.1.1.cmml">𝒳</mi><mo id="S6.Thmthm6.p1.3.3.m3.2.3.2.3.2.2" stretchy="false" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm6.p1.3.3.m3.2.3.1" xref="S6.Thmthm6.p1.3.3.m3.2.3.1.cmml">⊆</mo><mrow id="S6.Thmthm6.p1.3.3.m3.2.3.3" xref="S6.Thmthm6.p1.3.3.m3.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.3.3.m3.2.3.3.2" xref="S6.Thmthm6.p1.3.3.m3.2.3.3.2.cmml">ℳ</mi><mo id="S6.Thmthm6.p1.3.3.m3.2.3.3.1" xref="S6.Thmthm6.p1.3.3.m3.2.3.3.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p1.3.3.m3.2.3.3.3.2" xref="S6.Thmthm6.p1.3.3.m3.2.3.3.cmml"><mo id="S6.Thmthm6.p1.3.3.m3.2.3.3.3.2.1" stretchy="false" xref="S6.Thmthm6.p1.3.3.m3.2.3.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p1.3.3.m3.2.2" xref="S6.Thmthm6.p1.3.3.m3.2.2.cmml">𝒳</mi><mo id="S6.Thmthm6.p1.3.3.m3.2.3.3.3.2.2" stretchy="false" xref="S6.Thmthm6.p1.3.3.m3.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p1.3.3.m3.2b"><apply id="S6.Thmthm6.p1.3.3.m3.2.3.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3"><subset id="S6.Thmthm6.p1.3.3.m3.2.3.1.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.1"></subset><apply id="S6.Thmthm6.p1.3.3.m3.2.3.2.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2"><times id="S6.Thmthm6.p1.3.3.m3.2.3.2.1.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.1"></times><apply id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2"><csymbol cd="ambiguous" id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.1.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2">subscript</csymbol><ci id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.2.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.2">ℳ</ci><apply id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3"><times id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.1.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.1"></times><ci id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.2.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.2">𝓃</ci><ci id="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.3.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.2.2.3.3">𝒶</ci></apply></apply><ci id="S6.Thmthm6.p1.3.3.m3.1.1.cmml" xref="S6.Thmthm6.p1.3.3.m3.1.1">𝒳</ci></apply><apply id="S6.Thmthm6.p1.3.3.m3.2.3.3.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.3"><times id="S6.Thmthm6.p1.3.3.m3.2.3.3.1.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.3.1"></times><ci id="S6.Thmthm6.p1.3.3.m3.2.3.3.2.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.3.3.2">ℳ</ci><ci id="S6.Thmthm6.p1.3.3.m3.2.2.cmml" xref="S6.Thmthm6.p1.3.3.m3.2.2">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p1.3.3.m3.2c">\cal M_{na}(X)\subseteq\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p1.3.3.m3.2d">caligraphic_M start_POSTSUBSCRIPT caligraphic_n caligraphic_a end_POSTSUBSCRIPT ( caligraphic_X ) ⊆ caligraphic_M ( caligraphic_X )</annotation></semantics></math> be the subsets of periodic and of non-atomic invariant measures on <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm6.p1.4.4.m4.1"><semantics id="S6.Thmthm6.p1.4.4.m4.1a"><mi id="S6.Thmthm6.p1.4.4.m4.1.1" xref="S6.Thmthm6.p1.4.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p1.4.4.m4.1b"><ci id="S6.Thmthm6.p1.4.4.m4.1.1.cmml" xref="S6.Thmthm6.p1.4.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p1.4.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p1.4.4.m4.1d">italic_X</annotation></semantics></math> respectively.</span></p> </div> <div class="ltx_para" id="S6.Thmthm6.p2"> <p class="ltx_p" id="S6.Thmthm6.p2.5"><span class="ltx_text ltx_font_italic" id="S6.Thmthm6.p2.5.5">For any shift-orbit injective morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S6.Thmthm6.p2.1.1.m1.1"><semantics id="S6.Thmthm6.p2.1.1.m1.1a"><mrow id="S6.Thmthm6.p2.1.1.m1.1.1" xref="S6.Thmthm6.p2.1.1.m1.1.1.cmml"><mi id="S6.Thmthm6.p2.1.1.m1.1.1.2" xref="S6.Thmthm6.p2.1.1.m1.1.1.2.cmml">σ</mi><mo id="S6.Thmthm6.p2.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmthm6.p2.1.1.m1.1.1.1.cmml">:</mo><mrow id="S6.Thmthm6.p2.1.1.m1.1.1.3" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.cmml"><msup id="S6.Thmthm6.p2.1.1.m1.1.1.3.2" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.1.1.m1.1.1.3.2.2" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S6.Thmthm6.p2.1.1.m1.1.1.3.2.3" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S6.Thmthm6.p2.1.1.m1.1.1.3.1" stretchy="false" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S6.Thmthm6.p2.1.1.m1.1.1.3.3" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.1.1.m1.1.1.3.3.2" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S6.Thmthm6.p2.1.1.m1.1.1.3.3.3" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p2.1.1.m1.1b"><apply id="S6.Thmthm6.p2.1.1.m1.1.1.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1"><ci id="S6.Thmthm6.p2.1.1.m1.1.1.1.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.1">:</ci><ci id="S6.Thmthm6.p2.1.1.m1.1.1.2.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.2">𝜎</ci><apply id="S6.Thmthm6.p2.1.1.m1.1.1.3.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3"><ci id="S6.Thmthm6.p2.1.1.m1.1.1.3.1.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.1">→</ci><apply id="S6.Thmthm6.p2.1.1.m1.1.1.3.2.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.1.1.m1.1.1.3.2.1.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S6.Thmthm6.p2.1.1.m1.1.1.3.2.2.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S6.Thmthm6.p2.1.1.m1.1.1.3.2.3.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.2.3"></times></apply><apply id="S6.Thmthm6.p2.1.1.m1.1.1.3.3.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.1.1.m1.1.1.3.3.1.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S6.Thmthm6.p2.1.1.m1.1.1.3.3.2.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S6.Thmthm6.p2.1.1.m1.1.1.3.3.3.cmml" xref="S6.Thmthm6.p2.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p2.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p2.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> the injectivity of the push-forward map <math alttext="\sigma_{*}^{X}:\cal M(X)\to\cal M(\sigma(X)),\,\mu\mapsto\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.Thmthm6.p2.2.2.m2.5"><semantics id="S6.Thmthm6.p2.2.2.m2.5a"><mrow id="S6.Thmthm6.p2.2.2.m2.5.5" xref="S6.Thmthm6.p2.2.2.m2.5.5.cmml"><msubsup id="S6.Thmthm6.p2.2.2.m2.5.5.4" xref="S6.Thmthm6.p2.2.2.m2.5.5.4.cmml"><mi id="S6.Thmthm6.p2.2.2.m2.5.5.4.2.2" xref="S6.Thmthm6.p2.2.2.m2.5.5.4.2.2.cmml">σ</mi><mo id="S6.Thmthm6.p2.2.2.m2.5.5.4.2.3" xref="S6.Thmthm6.p2.2.2.m2.5.5.4.2.3.cmml">∗</mo><mi id="S6.Thmthm6.p2.2.2.m2.5.5.4.3" xref="S6.Thmthm6.p2.2.2.m2.5.5.4.3.cmml">X</mi></msubsup><mo id="S6.Thmthm6.p2.2.2.m2.5.5.3" lspace="0.278em" rspace="0.278em" xref="S6.Thmthm6.p2.2.2.m2.5.5.3.cmml">:</mo><mrow id="S6.Thmthm6.p2.2.2.m2.5.5.2.2" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.3.cmml"><mrow id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.cmml"><mrow id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.2" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.2.cmml">ℳ</mi><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.1" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.3.2" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.cmml"><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.3.2.1" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.2.2.m2.1.1" xref="S6.Thmthm6.p2.2.2.m2.1.1.cmml">𝒳</mi><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.3.2.2" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.2" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.2.cmml">→</mo><mrow id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.3" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.3.cmml">ℳ</mi><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.2" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.cmml"><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.2" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.cmml"><mi id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.2" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.1" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.3.2" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.cmml"><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.2.2.m2.2.2" xref="S6.Thmthm6.p2.2.2.m2.2.2.cmml">𝒳</mi><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.3" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.3" rspace="0.337em" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.3a.cmml">,</mo><mrow id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.cmml"><mi id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.2" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.2.cmml">μ</mi><mo id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.1" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.1.cmml">↦</mo><mrow id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.cmml"><msub id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.cmml"><mi id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.2" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.2.cmml">σ</mi><mo id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.3" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.3.cmml">∗</mo></msub><mo id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.1" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.3.2" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.cmml"><mo id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.3.2.1" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.cmml">(</mo><mi id="S6.Thmthm6.p2.2.2.m2.3.3" xref="S6.Thmthm6.p2.2.2.m2.3.3.cmml">μ</mi><mo id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.3.2.2" stretchy="false" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p2.2.2.m2.5b"><apply id="S6.Thmthm6.p2.2.2.m2.5.5.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5"><ci id="S6.Thmthm6.p2.2.2.m2.5.5.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.3">:</ci><apply id="S6.Thmthm6.p2.2.2.m2.5.5.4.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.4"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.2.2.m2.5.5.4.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.4">superscript</csymbol><apply id="S6.Thmthm6.p2.2.2.m2.5.5.4.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.4"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.2.2.m2.5.5.4.2.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.4">subscript</csymbol><ci id="S6.Thmthm6.p2.2.2.m2.5.5.4.2.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.4.2.2">𝜎</ci><times id="S6.Thmthm6.p2.2.2.m2.5.5.4.2.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.4.2.3"></times></apply><ci id="S6.Thmthm6.p2.2.2.m2.5.5.4.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.4.3">𝑋</ci></apply><apply id="S6.Thmthm6.p2.2.2.m2.5.5.2.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.2.2.m2.5.5.2.3a.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.3">formulae-sequence</csymbol><apply id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1"><ci id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.2">→</ci><apply id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3"><times id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.1"></times><ci id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.3.2">ℳ</ci><ci id="S6.Thmthm6.p2.2.2.m2.1.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.1.1">𝒳</ci></apply><apply id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1"><times id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.2"></times><ci id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.3">ℳ</ci><apply id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1"><times id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.1"></times><ci id="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.4.4.1.1.1.1.1.1.1.2">𝜎</ci><ci id="S6.Thmthm6.p2.2.2.m2.2.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.2.2">𝒳</ci></apply></apply></apply><apply id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2"><csymbol cd="latexml" id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.1">maps-to</csymbol><ci id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.2">𝜇</ci><apply id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3"><times id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.1"></times><apply id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.1.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2">subscript</csymbol><ci id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.2.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.2">𝜎</ci><times id="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.5.5.2.2.2.3.2.3"></times></apply><ci id="S6.Thmthm6.p2.2.2.m2.3.3.cmml" xref="S6.Thmthm6.p2.2.2.m2.3.3">𝜇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p2.2.2.m2.5c">\sigma_{*}^{X}:\cal M(X)\to\cal M(\sigma(X)),\,\mu\mapsto\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p2.2.2.m2.5d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT : caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) ) , italic_μ ↦ italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math> follows if one proves the injectivity for the restrictions of <math alttext="\sigma_{*}^{X}" class="ltx_Math" display="inline" id="S6.Thmthm6.p2.3.3.m3.1"><semantics id="S6.Thmthm6.p2.3.3.m3.1a"><msubsup id="S6.Thmthm6.p2.3.3.m3.1.1" xref="S6.Thmthm6.p2.3.3.m3.1.1.cmml"><mi id="S6.Thmthm6.p2.3.3.m3.1.1.2.2" xref="S6.Thmthm6.p2.3.3.m3.1.1.2.2.cmml">σ</mi><mo id="S6.Thmthm6.p2.3.3.m3.1.1.2.3" xref="S6.Thmthm6.p2.3.3.m3.1.1.2.3.cmml">∗</mo><mi id="S6.Thmthm6.p2.3.3.m3.1.1.3" xref="S6.Thmthm6.p2.3.3.m3.1.1.3.cmml">X</mi></msubsup><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p2.3.3.m3.1b"><apply id="S6.Thmthm6.p2.3.3.m3.1.1.cmml" xref="S6.Thmthm6.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.3.3.m3.1.1.1.cmml" xref="S6.Thmthm6.p2.3.3.m3.1.1">superscript</csymbol><apply id="S6.Thmthm6.p2.3.3.m3.1.1.2.cmml" xref="S6.Thmthm6.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.3.3.m3.1.1.2.1.cmml" xref="S6.Thmthm6.p2.3.3.m3.1.1">subscript</csymbol><ci id="S6.Thmthm6.p2.3.3.m3.1.1.2.2.cmml" xref="S6.Thmthm6.p2.3.3.m3.1.1.2.2">𝜎</ci><times id="S6.Thmthm6.p2.3.3.m3.1.1.2.3.cmml" xref="S6.Thmthm6.p2.3.3.m3.1.1.2.3"></times></apply><ci id="S6.Thmthm6.p2.3.3.m3.1.1.3.cmml" xref="S6.Thmthm6.p2.3.3.m3.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p2.3.3.m3.1c">\sigma_{*}^{X}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p2.3.3.m3.1d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT</annotation></semantics></math> to both, <math alttext="\cal M_{per}(X)" class="ltx_Math" display="inline" id="S6.Thmthm6.p2.4.4.m4.1"><semantics id="S6.Thmthm6.p2.4.4.m4.1a"><mrow id="S6.Thmthm6.p2.4.4.m4.1.2" xref="S6.Thmthm6.p2.4.4.m4.1.2.cmml"><msub id="S6.Thmthm6.p2.4.4.m4.1.2.2" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.4.4.m4.1.2.2.2" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.2.cmml">ℳ</mi><mrow id="S6.Thmthm6.p2.4.4.m4.1.2.2.3" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.2" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.2.cmml">𝓅</mi><mo id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.1" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.3" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.3.cmml">ℯ</mi><mo id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.1a" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.4" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.4.cmml">𝓇</mi></mrow></msub><mo id="S6.Thmthm6.p2.4.4.m4.1.2.1" xref="S6.Thmthm6.p2.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p2.4.4.m4.1.2.3.2" xref="S6.Thmthm6.p2.4.4.m4.1.2.cmml"><mo id="S6.Thmthm6.p2.4.4.m4.1.2.3.2.1" stretchy="false" xref="S6.Thmthm6.p2.4.4.m4.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.4.4.m4.1.1" xref="S6.Thmthm6.p2.4.4.m4.1.1.cmml">𝒳</mi><mo id="S6.Thmthm6.p2.4.4.m4.1.2.3.2.2" stretchy="false" xref="S6.Thmthm6.p2.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p2.4.4.m4.1b"><apply id="S6.Thmthm6.p2.4.4.m4.1.2.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2"><times id="S6.Thmthm6.p2.4.4.m4.1.2.1.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.1"></times><apply id="S6.Thmthm6.p2.4.4.m4.1.2.2.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.2"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.4.4.m4.1.2.2.1.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.2">subscript</csymbol><ci id="S6.Thmthm6.p2.4.4.m4.1.2.2.2.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.2">ℳ</ci><apply id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3"><times id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.1.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.1"></times><ci id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.2.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.2">𝓅</ci><ci id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.3.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.3">ℯ</ci><ci id="S6.Thmthm6.p2.4.4.m4.1.2.2.3.4.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.2.2.3.4">𝓇</ci></apply></apply><ci id="S6.Thmthm6.p2.4.4.m4.1.1.cmml" xref="S6.Thmthm6.p2.4.4.m4.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p2.4.4.m4.1c">\cal M_{per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p2.4.4.m4.1d">caligraphic_M start_POSTSUBSCRIPT caligraphic_p caligraphic_e caligraphic_r end_POSTSUBSCRIPT ( caligraphic_X )</annotation></semantics></math> and <math alttext="\cal M_{na}(X)" class="ltx_Math" display="inline" id="S6.Thmthm6.p2.5.5.m5.1"><semantics id="S6.Thmthm6.p2.5.5.m5.1a"><mrow id="S6.Thmthm6.p2.5.5.m5.1.2" xref="S6.Thmthm6.p2.5.5.m5.1.2.cmml"><msub id="S6.Thmthm6.p2.5.5.m5.1.2.2" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.5.5.m5.1.2.2.2" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.2.cmml">ℳ</mi><mrow id="S6.Thmthm6.p2.5.5.m5.1.2.2.3" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.5.5.m5.1.2.2.3.2" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.3.2.cmml">𝓃</mi><mo id="S6.Thmthm6.p2.5.5.m5.1.2.2.3.1" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.5.5.m5.1.2.2.3.3" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.3.3.cmml">𝒶</mi></mrow></msub><mo id="S6.Thmthm6.p2.5.5.m5.1.2.1" xref="S6.Thmthm6.p2.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S6.Thmthm6.p2.5.5.m5.1.2.3.2" xref="S6.Thmthm6.p2.5.5.m5.1.2.cmml"><mo id="S6.Thmthm6.p2.5.5.m5.1.2.3.2.1" stretchy="false" xref="S6.Thmthm6.p2.5.5.m5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm6.p2.5.5.m5.1.1" xref="S6.Thmthm6.p2.5.5.m5.1.1.cmml">𝒳</mi><mo id="S6.Thmthm6.p2.5.5.m5.1.2.3.2.2" stretchy="false" xref="S6.Thmthm6.p2.5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm6.p2.5.5.m5.1b"><apply id="S6.Thmthm6.p2.5.5.m5.1.2.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2"><times id="S6.Thmthm6.p2.5.5.m5.1.2.1.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2.1"></times><apply id="S6.Thmthm6.p2.5.5.m5.1.2.2.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2.2"><csymbol cd="ambiguous" id="S6.Thmthm6.p2.5.5.m5.1.2.2.1.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2.2">subscript</csymbol><ci id="S6.Thmthm6.p2.5.5.m5.1.2.2.2.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.2">ℳ</ci><apply id="S6.Thmthm6.p2.5.5.m5.1.2.2.3.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.3"><times id="S6.Thmthm6.p2.5.5.m5.1.2.2.3.1.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.3.1"></times><ci id="S6.Thmthm6.p2.5.5.m5.1.2.2.3.2.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.3.2">𝓃</ci><ci id="S6.Thmthm6.p2.5.5.m5.1.2.2.3.3.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.2.2.3.3">𝒶</ci></apply></apply><ci id="S6.Thmthm6.p2.5.5.m5.1.1.cmml" xref="S6.Thmthm6.p2.5.5.m5.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm6.p2.5.5.m5.1c">\cal M_{na}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm6.p2.5.5.m5.1d">caligraphic_M start_POSTSUBSCRIPT caligraphic_n caligraphic_a end_POSTSUBSCRIPT ( caligraphic_X )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S6.10"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.10.p1"> <p class="ltx_p" id="S6.10.p1.16">Let <math alttext="\mu_{1},\mu_{2}\in\cal M(X)" class="ltx_Math" display="inline" id="S6.10.p1.1.m1.3"><semantics id="S6.10.p1.1.m1.3a"><mrow id="S6.10.p1.1.m1.3.3" xref="S6.10.p1.1.m1.3.3.cmml"><mrow id="S6.10.p1.1.m1.3.3.2.2" xref="S6.10.p1.1.m1.3.3.2.3.cmml"><msub id="S6.10.p1.1.m1.2.2.1.1.1" xref="S6.10.p1.1.m1.2.2.1.1.1.cmml"><mi id="S6.10.p1.1.m1.2.2.1.1.1.2" xref="S6.10.p1.1.m1.2.2.1.1.1.2.cmml">μ</mi><mn id="S6.10.p1.1.m1.2.2.1.1.1.3" xref="S6.10.p1.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S6.10.p1.1.m1.3.3.2.2.3" xref="S6.10.p1.1.m1.3.3.2.3.cmml">,</mo><msub id="S6.10.p1.1.m1.3.3.2.2.2" xref="S6.10.p1.1.m1.3.3.2.2.2.cmml"><mi id="S6.10.p1.1.m1.3.3.2.2.2.2" xref="S6.10.p1.1.m1.3.3.2.2.2.2.cmml">μ</mi><mn id="S6.10.p1.1.m1.3.3.2.2.2.3" xref="S6.10.p1.1.m1.3.3.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S6.10.p1.1.m1.3.3.3" xref="S6.10.p1.1.m1.3.3.3.cmml">∈</mo><mrow id="S6.10.p1.1.m1.3.3.4" xref="S6.10.p1.1.m1.3.3.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.1.m1.3.3.4.2" xref="S6.10.p1.1.m1.3.3.4.2.cmml">ℳ</mi><mo id="S6.10.p1.1.m1.3.3.4.1" xref="S6.10.p1.1.m1.3.3.4.1.cmml">⁢</mo><mrow id="S6.10.p1.1.m1.3.3.4.3.2" xref="S6.10.p1.1.m1.3.3.4.cmml"><mo id="S6.10.p1.1.m1.3.3.4.3.2.1" stretchy="false" xref="S6.10.p1.1.m1.3.3.4.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.1.m1.1.1" xref="S6.10.p1.1.m1.1.1.cmml">𝒳</mi><mo id="S6.10.p1.1.m1.3.3.4.3.2.2" stretchy="false" xref="S6.10.p1.1.m1.3.3.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.10.p1.1.m1.3b"><apply id="S6.10.p1.1.m1.3.3.cmml" xref="S6.10.p1.1.m1.3.3"><in id="S6.10.p1.1.m1.3.3.3.cmml" xref="S6.10.p1.1.m1.3.3.3"></in><list id="S6.10.p1.1.m1.3.3.2.3.cmml" xref="S6.10.p1.1.m1.3.3.2.2"><apply id="S6.10.p1.1.m1.2.2.1.1.1.cmml" xref="S6.10.p1.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.10.p1.1.m1.2.2.1.1.1.1.cmml" xref="S6.10.p1.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S6.10.p1.1.m1.2.2.1.1.1.2.cmml" xref="S6.10.p1.1.m1.2.2.1.1.1.2">𝜇</ci><cn id="S6.10.p1.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S6.10.p1.1.m1.2.2.1.1.1.3">1</cn></apply><apply id="S6.10.p1.1.m1.3.3.2.2.2.cmml" xref="S6.10.p1.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S6.10.p1.1.m1.3.3.2.2.2.1.cmml" xref="S6.10.p1.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S6.10.p1.1.m1.3.3.2.2.2.2.cmml" xref="S6.10.p1.1.m1.3.3.2.2.2.2">𝜇</ci><cn id="S6.10.p1.1.m1.3.3.2.2.2.3.cmml" type="integer" xref="S6.10.p1.1.m1.3.3.2.2.2.3">2</cn></apply></list><apply id="S6.10.p1.1.m1.3.3.4.cmml" xref="S6.10.p1.1.m1.3.3.4"><times id="S6.10.p1.1.m1.3.3.4.1.cmml" xref="S6.10.p1.1.m1.3.3.4.1"></times><ci id="S6.10.p1.1.m1.3.3.4.2.cmml" xref="S6.10.p1.1.m1.3.3.4.2">ℳ</ci><ci id="S6.10.p1.1.m1.1.1.cmml" xref="S6.10.p1.1.m1.1.1">𝒳</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.1.m1.3c">\mu_{1},\mu_{2}\in\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.1.m1.3d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ caligraphic_M ( caligraphic_X )</annotation></semantics></math> be invariant measures which satisfy <math alttext="\sigma_{*}(\mu_{1})=\sigma_{*}(\mu_{1})=:\mu_{0}" class="ltx_math_unparsed" display="inline" id="S6.10.p1.2.m2.1"><semantics id="S6.10.p1.2.m2.1a"><mrow id="S6.10.p1.2.m2.1b"><msub id="S6.10.p1.2.m2.1.1"><mi id="S6.10.p1.2.m2.1.1.2">σ</mi><mo id="S6.10.p1.2.m2.1.1.3">∗</mo></msub><mrow id="S6.10.p1.2.m2.1.2"><mo id="S6.10.p1.2.m2.1.2.1" stretchy="false">(</mo><msub id="S6.10.p1.2.m2.1.2.2"><mi id="S6.10.p1.2.m2.1.2.2.2">μ</mi><mn id="S6.10.p1.2.m2.1.2.2.3">1</mn></msub><mo id="S6.10.p1.2.m2.1.2.3" stretchy="false">)</mo></mrow><mo id="S6.10.p1.2.m2.1.3">=</mo><msub id="S6.10.p1.2.m2.1.4"><mi id="S6.10.p1.2.m2.1.4.2">σ</mi><mo id="S6.10.p1.2.m2.1.4.3">∗</mo></msub><mrow id="S6.10.p1.2.m2.1.5"><mo id="S6.10.p1.2.m2.1.5.1" stretchy="false">(</mo><msub id="S6.10.p1.2.m2.1.5.2"><mi id="S6.10.p1.2.m2.1.5.2.2">μ</mi><mn id="S6.10.p1.2.m2.1.5.2.3">1</mn></msub><mo id="S6.10.p1.2.m2.1.5.3" stretchy="false">)</mo></mrow><mo id="S6.10.p1.2.m2.1.6" rspace="0em">=</mo><mo id="S6.10.p1.2.m2.1.7" rspace="0.278em">:</mo><msub id="S6.10.p1.2.m2.1.8"><mi id="S6.10.p1.2.m2.1.8.2">μ</mi><mn id="S6.10.p1.2.m2.1.8.3">0</mn></msub></mrow><annotation encoding="application/x-tex" id="S6.10.p1.2.m2.1c">\sigma_{*}(\mu_{1})=\sigma_{*}(\mu_{1})=:\mu_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.2.m2.1d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = : italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Consider the canonical decompositions into a periodic and an non-atomic measure from (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E5" title="In 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.5</span></a>) given by <math alttext="\mu_{1}=\mu_{1}^{per}+\mu_{1}^{na}" class="ltx_Math" display="inline" id="S6.10.p1.3.m3.1"><semantics id="S6.10.p1.3.m3.1a"><mrow id="S6.10.p1.3.m3.1.1" xref="S6.10.p1.3.m3.1.1.cmml"><msub id="S6.10.p1.3.m3.1.1.2" xref="S6.10.p1.3.m3.1.1.2.cmml"><mi id="S6.10.p1.3.m3.1.1.2.2" xref="S6.10.p1.3.m3.1.1.2.2.cmml">μ</mi><mn id="S6.10.p1.3.m3.1.1.2.3" xref="S6.10.p1.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S6.10.p1.3.m3.1.1.1" xref="S6.10.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S6.10.p1.3.m3.1.1.3" xref="S6.10.p1.3.m3.1.1.3.cmml"><msubsup id="S6.10.p1.3.m3.1.1.3.2" xref="S6.10.p1.3.m3.1.1.3.2.cmml"><mi id="S6.10.p1.3.m3.1.1.3.2.2.2" xref="S6.10.p1.3.m3.1.1.3.2.2.2.cmml">μ</mi><mn id="S6.10.p1.3.m3.1.1.3.2.2.3" xref="S6.10.p1.3.m3.1.1.3.2.2.3.cmml">1</mn><mrow 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xref="S6.10.p1.3.m3.1.1.3.3.3.1.cmml">⁢</mo><mi id="S6.10.p1.3.m3.1.1.3.3.3.3" xref="S6.10.p1.3.m3.1.1.3.3.3.3.cmml">a</mi></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.10.p1.3.m3.1b"><apply id="S6.10.p1.3.m3.1.1.cmml" xref="S6.10.p1.3.m3.1.1"><eq id="S6.10.p1.3.m3.1.1.1.cmml" xref="S6.10.p1.3.m3.1.1.1"></eq><apply id="S6.10.p1.3.m3.1.1.2.cmml" xref="S6.10.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S6.10.p1.3.m3.1.1.2.1.cmml" xref="S6.10.p1.3.m3.1.1.2">subscript</csymbol><ci id="S6.10.p1.3.m3.1.1.2.2.cmml" xref="S6.10.p1.3.m3.1.1.2.2">𝜇</ci><cn id="S6.10.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S6.10.p1.3.m3.1.1.2.3">1</cn></apply><apply id="S6.10.p1.3.m3.1.1.3.cmml" xref="S6.10.p1.3.m3.1.1.3"><plus id="S6.10.p1.3.m3.1.1.3.1.cmml" xref="S6.10.p1.3.m3.1.1.3.1"></plus><apply id="S6.10.p1.3.m3.1.1.3.2.cmml" xref="S6.10.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S6.10.p1.3.m3.1.1.3.2.1.cmml" xref="S6.10.p1.3.m3.1.1.3.2">superscript</csymbol><apply 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start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mu_{0}=\mu_{0}^{per}+\mu_{0}^{na}" class="ltx_Math" display="inline" id="S6.10.p1.5.m5.1"><semantics id="S6.10.p1.5.m5.1a"><mrow id="S6.10.p1.5.m5.1.1" xref="S6.10.p1.5.m5.1.1.cmml"><msub id="S6.10.p1.5.m5.1.1.2" xref="S6.10.p1.5.m5.1.1.2.cmml"><mi id="S6.10.p1.5.m5.1.1.2.2" xref="S6.10.p1.5.m5.1.1.2.2.cmml">μ</mi><mn id="S6.10.p1.5.m5.1.1.2.3" xref="S6.10.p1.5.m5.1.1.2.3.cmml">0</mn></msub><mo id="S6.10.p1.5.m5.1.1.1" xref="S6.10.p1.5.m5.1.1.1.cmml">=</mo><mrow id="S6.10.p1.5.m5.1.1.3" xref="S6.10.p1.5.m5.1.1.3.cmml"><msubsup id="S6.10.p1.5.m5.1.1.3.2" xref="S6.10.p1.5.m5.1.1.3.2.cmml"><mi id="S6.10.p1.5.m5.1.1.3.2.2.2" xref="S6.10.p1.5.m5.1.1.3.2.2.2.cmml">μ</mi><mn id="S6.10.p1.5.m5.1.1.3.2.2.3" xref="S6.10.p1.5.m5.1.1.3.2.2.3.cmml">0</mn><mrow id="S6.10.p1.5.m5.1.1.3.2.3" xref="S6.10.p1.5.m5.1.1.3.2.3.cmml"><mi id="S6.10.p1.5.m5.1.1.3.2.3.2" xref="S6.10.p1.5.m5.1.1.3.2.3.2.cmml">p</mi><mo id="S6.10.p1.5.m5.1.1.3.2.3.1" xref="S6.10.p1.5.m5.1.1.3.2.3.1.cmml">⁢</mo><mi id="S6.10.p1.5.m5.1.1.3.2.3.3" xref="S6.10.p1.5.m5.1.1.3.2.3.3.cmml">e</mi><mo id="S6.10.p1.5.m5.1.1.3.2.3.1a" xref="S6.10.p1.5.m5.1.1.3.2.3.1.cmml">⁢</mo><mi id="S6.10.p1.5.m5.1.1.3.2.3.4" xref="S6.10.p1.5.m5.1.1.3.2.3.4.cmml">r</mi></mrow></msubsup><mo id="S6.10.p1.5.m5.1.1.3.1" xref="S6.10.p1.5.m5.1.1.3.1.cmml">+</mo><msubsup id="S6.10.p1.5.m5.1.1.3.3" xref="S6.10.p1.5.m5.1.1.3.3.cmml"><mi id="S6.10.p1.5.m5.1.1.3.3.2.2" xref="S6.10.p1.5.m5.1.1.3.3.2.2.cmml">μ</mi><mn id="S6.10.p1.5.m5.1.1.3.3.2.3" xref="S6.10.p1.5.m5.1.1.3.3.2.3.cmml">0</mn><mrow id="S6.10.p1.5.m5.1.1.3.3.3" xref="S6.10.p1.5.m5.1.1.3.3.3.cmml"><mi id="S6.10.p1.5.m5.1.1.3.3.3.2" xref="S6.10.p1.5.m5.1.1.3.3.3.2.cmml">n</mi><mo id="S6.10.p1.5.m5.1.1.3.3.3.1" xref="S6.10.p1.5.m5.1.1.3.3.3.1.cmml">⁢</mo><mi id="S6.10.p1.5.m5.1.1.3.3.3.3" xref="S6.10.p1.5.m5.1.1.3.3.3.3.cmml">a</mi></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.10.p1.5.m5.1b"><apply id="S6.10.p1.5.m5.1.1.cmml" xref="S6.10.p1.5.m5.1.1"><eq id="S6.10.p1.5.m5.1.1.1.cmml" xref="S6.10.p1.5.m5.1.1.1"></eq><apply id="S6.10.p1.5.m5.1.1.2.cmml" xref="S6.10.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S6.10.p1.5.m5.1.1.2.1.cmml" xref="S6.10.p1.5.m5.1.1.2">subscript</csymbol><ci id="S6.10.p1.5.m5.1.1.2.2.cmml" xref="S6.10.p1.5.m5.1.1.2.2">𝜇</ci><cn id="S6.10.p1.5.m5.1.1.2.3.cmml" type="integer" xref="S6.10.p1.5.m5.1.1.2.3">0</cn></apply><apply id="S6.10.p1.5.m5.1.1.3.cmml" xref="S6.10.p1.5.m5.1.1.3"><plus id="S6.10.p1.5.m5.1.1.3.1.cmml" xref="S6.10.p1.5.m5.1.1.3.1"></plus><apply id="S6.10.p1.5.m5.1.1.3.2.cmml" xref="S6.10.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S6.10.p1.5.m5.1.1.3.2.1.cmml" xref="S6.10.p1.5.m5.1.1.3.2">superscript</csymbol><apply id="S6.10.p1.5.m5.1.1.3.2.2.cmml" xref="S6.10.p1.5.m5.1.1.3.2"><csymbol 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xref="S6.10.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S6.10.p1.5.m5.1.1.3.3.2.2.cmml" xref="S6.10.p1.5.m5.1.1.3.3.2.2">𝜇</ci><cn id="S6.10.p1.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="S6.10.p1.5.m5.1.1.3.3.2.3">0</cn></apply><apply id="S6.10.p1.5.m5.1.1.3.3.3.cmml" xref="S6.10.p1.5.m5.1.1.3.3.3"><times id="S6.10.p1.5.m5.1.1.3.3.3.1.cmml" xref="S6.10.p1.5.m5.1.1.3.3.3.1"></times><ci id="S6.10.p1.5.m5.1.1.3.3.3.2.cmml" xref="S6.10.p1.5.m5.1.1.3.3.3.2">𝑛</ci><ci id="S6.10.p1.5.m5.1.1.3.3.3.3.cmml" xref="S6.10.p1.5.m5.1.1.3.3.3.3">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.5.m5.1c">\mu_{0}=\mu_{0}^{per}+\mu_{0}^{na}</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.5.m5.1d">italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p italic_e italic_r end_POSTSUPERSCRIPT + italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT</annotation></semantics></math>. From Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm5" title="Lemma 6.5. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.5</span></a> and the uniqueness of the canonical decomposition for <math alttext="\mu_{0}" class="ltx_Math" display="inline" id="S6.10.p1.6.m6.1"><semantics id="S6.10.p1.6.m6.1a"><msub id="S6.10.p1.6.m6.1.1" xref="S6.10.p1.6.m6.1.1.cmml"><mi id="S6.10.p1.6.m6.1.1.2" xref="S6.10.p1.6.m6.1.1.2.cmml">μ</mi><mn id="S6.10.p1.6.m6.1.1.3" xref="S6.10.p1.6.m6.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S6.10.p1.6.m6.1b"><apply id="S6.10.p1.6.m6.1.1.cmml" xref="S6.10.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S6.10.p1.6.m6.1.1.1.cmml" xref="S6.10.p1.6.m6.1.1">subscript</csymbol><ci id="S6.10.p1.6.m6.1.1.2.cmml" xref="S6.10.p1.6.m6.1.1.2">𝜇</ci><cn id="S6.10.p1.6.m6.1.1.3.cmml" type="integer" xref="S6.10.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.6.m6.1c">\mu_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.6.m6.1d">italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> (see the equalities (<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.E6" title="In 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.6</span></a>)) we derive that <math alttext="\sigma_{*}(\mu_{1}^{per})=\sigma_{*}(\mu_{2}^{per})=\mu_{0}^{per}" class="ltx_Math" display="inline" id="S6.10.p1.7.m7.2"><semantics id="S6.10.p1.7.m7.2a"><mrow id="S6.10.p1.7.m7.2.2" xref="S6.10.p1.7.m7.2.2.cmml"><mrow id="S6.10.p1.7.m7.1.1.1" xref="S6.10.p1.7.m7.1.1.1.cmml"><msub id="S6.10.p1.7.m7.1.1.1.3" xref="S6.10.p1.7.m7.1.1.1.3.cmml"><mi id="S6.10.p1.7.m7.1.1.1.3.2" xref="S6.10.p1.7.m7.1.1.1.3.2.cmml">σ</mi><mo id="S6.10.p1.7.m7.1.1.1.3.3" xref="S6.10.p1.7.m7.1.1.1.3.3.cmml">∗</mo></msub><mo id="S6.10.p1.7.m7.1.1.1.2" xref="S6.10.p1.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S6.10.p1.7.m7.1.1.1.1.1" xref="S6.10.p1.7.m7.1.1.1.1.1.1.cmml"><mo id="S6.10.p1.7.m7.1.1.1.1.1.2" stretchy="false" xref="S6.10.p1.7.m7.1.1.1.1.1.1.cmml">(</mo><msubsup id="S6.10.p1.7.m7.1.1.1.1.1.1" xref="S6.10.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S6.10.p1.7.m7.1.1.1.1.1.1.2.2" xref="S6.10.p1.7.m7.1.1.1.1.1.1.2.2.cmml">μ</mi><mn id="S6.10.p1.7.m7.1.1.1.1.1.1.2.3" xref="S6.10.p1.7.m7.1.1.1.1.1.1.2.3.cmml">1</mn><mrow id="S6.10.p1.7.m7.1.1.1.1.1.1.3" xref="S6.10.p1.7.m7.1.1.1.1.1.1.3.cmml"><mi id="S6.10.p1.7.m7.1.1.1.1.1.1.3.2" xref="S6.10.p1.7.m7.1.1.1.1.1.1.3.2.cmml">p</mi><mo id="S6.10.p1.7.m7.1.1.1.1.1.1.3.1" xref="S6.10.p1.7.m7.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S6.10.p1.7.m7.1.1.1.1.1.1.3.3" xref="S6.10.p1.7.m7.1.1.1.1.1.1.3.3.cmml">e</mi><mo id="S6.10.p1.7.m7.1.1.1.1.1.1.3.1a" xref="S6.10.p1.7.m7.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S6.10.p1.7.m7.1.1.1.1.1.1.3.4" xref="S6.10.p1.7.m7.1.1.1.1.1.1.3.4.cmml">r</mi></mrow></msubsup><mo id="S6.10.p1.7.m7.1.1.1.1.1.3" stretchy="false" xref="S6.10.p1.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.10.p1.7.m7.2.2.4" xref="S6.10.p1.7.m7.2.2.4.cmml">=</mo><mrow id="S6.10.p1.7.m7.2.2.2" xref="S6.10.p1.7.m7.2.2.2.cmml"><msub id="S6.10.p1.7.m7.2.2.2.3" xref="S6.10.p1.7.m7.2.2.2.3.cmml"><mi id="S6.10.p1.7.m7.2.2.2.3.2" xref="S6.10.p1.7.m7.2.2.2.3.2.cmml">σ</mi><mo id="S6.10.p1.7.m7.2.2.2.3.3" xref="S6.10.p1.7.m7.2.2.2.3.3.cmml">∗</mo></msub><mo id="S6.10.p1.7.m7.2.2.2.2" xref="S6.10.p1.7.m7.2.2.2.2.cmml">⁢</mo><mrow id="S6.10.p1.7.m7.2.2.2.1.1" xref="S6.10.p1.7.m7.2.2.2.1.1.1.cmml"><mo id="S6.10.p1.7.m7.2.2.2.1.1.2" stretchy="false" xref="S6.10.p1.7.m7.2.2.2.1.1.1.cmml">(</mo><msubsup id="S6.10.p1.7.m7.2.2.2.1.1.1" xref="S6.10.p1.7.m7.2.2.2.1.1.1.cmml"><mi id="S6.10.p1.7.m7.2.2.2.1.1.1.2.2" xref="S6.10.p1.7.m7.2.2.2.1.1.1.2.2.cmml">μ</mi><mn id="S6.10.p1.7.m7.2.2.2.1.1.1.2.3" xref="S6.10.p1.7.m7.2.2.2.1.1.1.2.3.cmml">2</mn><mrow id="S6.10.p1.7.m7.2.2.2.1.1.1.3" xref="S6.10.p1.7.m7.2.2.2.1.1.1.3.cmml"><mi id="S6.10.p1.7.m7.2.2.2.1.1.1.3.2" xref="S6.10.p1.7.m7.2.2.2.1.1.1.3.2.cmml">p</mi><mo id="S6.10.p1.7.m7.2.2.2.1.1.1.3.1" 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xref="S6.10.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.10.p1.8.m8.1.1.1.1.1.1.1.cmml" xref="S6.10.p1.8.m8.1.1.1.1.1">superscript</csymbol><apply id="S6.10.p1.8.m8.1.1.1.1.1.1.2.cmml" xref="S6.10.p1.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.10.p1.8.m8.1.1.1.1.1.1.2.1.cmml" xref="S6.10.p1.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S6.10.p1.8.m8.1.1.1.1.1.1.2.2.cmml" xref="S6.10.p1.8.m8.1.1.1.1.1.1.2.2">𝜇</ci><cn id="S6.10.p1.8.m8.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.10.p1.8.m8.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S6.10.p1.8.m8.1.1.1.1.1.1.3.cmml" xref="S6.10.p1.8.m8.1.1.1.1.1.1.3"><times id="S6.10.p1.8.m8.1.1.1.1.1.1.3.1.cmml" xref="S6.10.p1.8.m8.1.1.1.1.1.1.3.1"></times><ci id="S6.10.p1.8.m8.1.1.1.1.1.1.3.2.cmml" xref="S6.10.p1.8.m8.1.1.1.1.1.1.3.2">𝑛</ci><ci id="S6.10.p1.8.m8.1.1.1.1.1.1.3.3.cmml" xref="S6.10.p1.8.m8.1.1.1.1.1.1.3.3">𝑎</ci></apply></apply></apply><apply id="S6.10.p1.8.m8.2.2.2.cmml" xref="S6.10.p1.8.m8.2.2.2"><times id="S6.10.p1.8.m8.2.2.2.2.cmml" xref="S6.10.p1.8.m8.2.2.2.2"></times><apply id="S6.10.p1.8.m8.2.2.2.3.cmml" xref="S6.10.p1.8.m8.2.2.2.3"><csymbol cd="ambiguous" id="S6.10.p1.8.m8.2.2.2.3.1.cmml" xref="S6.10.p1.8.m8.2.2.2.3">subscript</csymbol><ci id="S6.10.p1.8.m8.2.2.2.3.2.cmml" xref="S6.10.p1.8.m8.2.2.2.3.2">𝜎</ci><times id="S6.10.p1.8.m8.2.2.2.3.3.cmml" xref="S6.10.p1.8.m8.2.2.2.3.3"></times></apply><apply id="S6.10.p1.8.m8.2.2.2.1.1.1.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.10.p1.8.m8.2.2.2.1.1.1.1.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1">superscript</csymbol><apply id="S6.10.p1.8.m8.2.2.2.1.1.1.2.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.10.p1.8.m8.2.2.2.1.1.1.2.1.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1">subscript</csymbol><ci id="S6.10.p1.8.m8.2.2.2.1.1.1.2.2.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1.1.2.2">𝜇</ci><cn id="S6.10.p1.8.m8.2.2.2.1.1.1.2.3.cmml" type="integer" xref="S6.10.p1.8.m8.2.2.2.1.1.1.2.3">2</cn></apply><apply id="S6.10.p1.8.m8.2.2.2.1.1.1.3.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1.1.3"><times id="S6.10.p1.8.m8.2.2.2.1.1.1.3.1.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1.1.3.1"></times><ci id="S6.10.p1.8.m8.2.2.2.1.1.1.3.2.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1.1.3.2">𝑛</ci><ci id="S6.10.p1.8.m8.2.2.2.1.1.1.3.3.cmml" xref="S6.10.p1.8.m8.2.2.2.1.1.1.3.3">𝑎</ci></apply></apply></apply></apply><apply id="S6.10.p1.8.m8.2.2c.cmml" xref="S6.10.p1.8.m8.2.2"><eq id="S6.10.p1.8.m8.2.2.5.cmml" xref="S6.10.p1.8.m8.2.2.5"></eq><share href="https://arxiv.org/html/2211.11234v4#S6.10.p1.8.m8.2.2.2.cmml" id="S6.10.p1.8.m8.2.2d.cmml" xref="S6.10.p1.8.m8.2.2"></share><apply id="S6.10.p1.8.m8.2.2.6.cmml" xref="S6.10.p1.8.m8.2.2.6"><csymbol cd="ambiguous" id="S6.10.p1.8.m8.2.2.6.1.cmml" xref="S6.10.p1.8.m8.2.2.6">superscript</csymbol><apply id="S6.10.p1.8.m8.2.2.6.2.cmml" xref="S6.10.p1.8.m8.2.2.6"><csymbol cd="ambiguous" id="S6.10.p1.8.m8.2.2.6.2.1.cmml" xref="S6.10.p1.8.m8.2.2.6">subscript</csymbol><ci id="S6.10.p1.8.m8.2.2.6.2.2.cmml" xref="S6.10.p1.8.m8.2.2.6.2.2">𝜇</ci><cn id="S6.10.p1.8.m8.2.2.6.2.3.cmml" type="integer" xref="S6.10.p1.8.m8.2.2.6.2.3">0</cn></apply><apply id="S6.10.p1.8.m8.2.2.6.3.cmml" xref="S6.10.p1.8.m8.2.2.6.3"><times id="S6.10.p1.8.m8.2.2.6.3.1.cmml" xref="S6.10.p1.8.m8.2.2.6.3.1"></times><ci id="S6.10.p1.8.m8.2.2.6.3.2.cmml" xref="S6.10.p1.8.m8.2.2.6.3.2">𝑛</ci><ci id="S6.10.p1.8.m8.2.2.6.3.3.cmml" xref="S6.10.p1.8.m8.2.2.6.3.3">𝑎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.8.m8.2c">\sigma_{*}(\mu_{1}^{na})=\sigma_{*}(\mu_{2}^{na})=\mu_{0}^{na}</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.8.m8.2d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT ) = italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT ) = italic_μ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT</annotation></semantics></math>. Injectivity of <math alttext="\sigma_{*}^{X}" class="ltx_Math" display="inline" id="S6.10.p1.9.m9.1"><semantics id="S6.10.p1.9.m9.1a"><msubsup id="S6.10.p1.9.m9.1.1" xref="S6.10.p1.9.m9.1.1.cmml"><mi id="S6.10.p1.9.m9.1.1.2.2" xref="S6.10.p1.9.m9.1.1.2.2.cmml">σ</mi><mo id="S6.10.p1.9.m9.1.1.2.3" xref="S6.10.p1.9.m9.1.1.2.3.cmml">∗</mo><mi id="S6.10.p1.9.m9.1.1.3" xref="S6.10.p1.9.m9.1.1.3.cmml">X</mi></msubsup><annotation-xml encoding="MathML-Content" id="S6.10.p1.9.m9.1b"><apply id="S6.10.p1.9.m9.1.1.cmml" xref="S6.10.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S6.10.p1.9.m9.1.1.1.cmml" xref="S6.10.p1.9.m9.1.1">superscript</csymbol><apply id="S6.10.p1.9.m9.1.1.2.cmml" xref="S6.10.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S6.10.p1.9.m9.1.1.2.1.cmml" xref="S6.10.p1.9.m9.1.1">subscript</csymbol><ci id="S6.10.p1.9.m9.1.1.2.2.cmml" xref="S6.10.p1.9.m9.1.1.2.2">𝜎</ci><times id="S6.10.p1.9.m9.1.1.2.3.cmml" xref="S6.10.p1.9.m9.1.1.2.3"></times></apply><ci id="S6.10.p1.9.m9.1.1.3.cmml" xref="S6.10.p1.9.m9.1.1.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.9.m9.1c">\sigma_{*}^{X}</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.9.m9.1d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_X end_POSTSUPERSCRIPT</annotation></semantics></math> on <math alttext="\cal M_{per}(X)" class="ltx_Math" display="inline" id="S6.10.p1.10.m10.1"><semantics id="S6.10.p1.10.m10.1a"><mrow id="S6.10.p1.10.m10.1.2" xref="S6.10.p1.10.m10.1.2.cmml"><msub id="S6.10.p1.10.m10.1.2.2" xref="S6.10.p1.10.m10.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.10.m10.1.2.2.2" xref="S6.10.p1.10.m10.1.2.2.2.cmml">ℳ</mi><mrow id="S6.10.p1.10.m10.1.2.2.3" xref="S6.10.p1.10.m10.1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.10.m10.1.2.2.3.2" xref="S6.10.p1.10.m10.1.2.2.3.2.cmml">𝓅</mi><mo id="S6.10.p1.10.m10.1.2.2.3.1" xref="S6.10.p1.10.m10.1.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.10.m10.1.2.2.3.3" xref="S6.10.p1.10.m10.1.2.2.3.3.cmml">ℯ</mi><mo id="S6.10.p1.10.m10.1.2.2.3.1a" xref="S6.10.p1.10.m10.1.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.10.m10.1.2.2.3.4" xref="S6.10.p1.10.m10.1.2.2.3.4.cmml">𝓇</mi></mrow></msub><mo id="S6.10.p1.10.m10.1.2.1" xref="S6.10.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S6.10.p1.10.m10.1.2.3.2" xref="S6.10.p1.10.m10.1.2.cmml"><mo id="S6.10.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S6.10.p1.10.m10.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.10.m10.1.1" xref="S6.10.p1.10.m10.1.1.cmml">𝒳</mi><mo id="S6.10.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S6.10.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.10.p1.10.m10.1b"><apply id="S6.10.p1.10.m10.1.2.cmml" xref="S6.10.p1.10.m10.1.2"><times id="S6.10.p1.10.m10.1.2.1.cmml" xref="S6.10.p1.10.m10.1.2.1"></times><apply id="S6.10.p1.10.m10.1.2.2.cmml" xref="S6.10.p1.10.m10.1.2.2"><csymbol cd="ambiguous" id="S6.10.p1.10.m10.1.2.2.1.cmml" xref="S6.10.p1.10.m10.1.2.2">subscript</csymbol><ci id="S6.10.p1.10.m10.1.2.2.2.cmml" xref="S6.10.p1.10.m10.1.2.2.2">ℳ</ci><apply id="S6.10.p1.10.m10.1.2.2.3.cmml" xref="S6.10.p1.10.m10.1.2.2.3"><times id="S6.10.p1.10.m10.1.2.2.3.1.cmml" xref="S6.10.p1.10.m10.1.2.2.3.1"></times><ci id="S6.10.p1.10.m10.1.2.2.3.2.cmml" xref="S6.10.p1.10.m10.1.2.2.3.2">𝓅</ci><ci id="S6.10.p1.10.m10.1.2.2.3.3.cmml" xref="S6.10.p1.10.m10.1.2.2.3.3">ℯ</ci><ci id="S6.10.p1.10.m10.1.2.2.3.4.cmml" xref="S6.10.p1.10.m10.1.2.2.3.4">𝓇</ci></apply></apply><ci id="S6.10.p1.10.m10.1.1.cmml" xref="S6.10.p1.10.m10.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.10.m10.1c">\cal M_{per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.10.m10.1d">caligraphic_M start_POSTSUBSCRIPT caligraphic_p caligraphic_e caligraphic_r end_POSTSUBSCRIPT ( caligraphic_X )</annotation></semantics></math> and <math alttext="\cal M_{na}(X)" class="ltx_Math" display="inline" id="S6.10.p1.11.m11.1"><semantics id="S6.10.p1.11.m11.1a"><mrow id="S6.10.p1.11.m11.1.2" xref="S6.10.p1.11.m11.1.2.cmml"><msub id="S6.10.p1.11.m11.1.2.2" xref="S6.10.p1.11.m11.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.11.m11.1.2.2.2" xref="S6.10.p1.11.m11.1.2.2.2.cmml">ℳ</mi><mrow id="S6.10.p1.11.m11.1.2.2.3" xref="S6.10.p1.11.m11.1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.11.m11.1.2.2.3.2" xref="S6.10.p1.11.m11.1.2.2.3.2.cmml">𝓃</mi><mo id="S6.10.p1.11.m11.1.2.2.3.1" xref="S6.10.p1.11.m11.1.2.2.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.11.m11.1.2.2.3.3" xref="S6.10.p1.11.m11.1.2.2.3.3.cmml">𝒶</mi></mrow></msub><mo id="S6.10.p1.11.m11.1.2.1" xref="S6.10.p1.11.m11.1.2.1.cmml">⁢</mo><mrow id="S6.10.p1.11.m11.1.2.3.2" xref="S6.10.p1.11.m11.1.2.cmml"><mo id="S6.10.p1.11.m11.1.2.3.2.1" stretchy="false" xref="S6.10.p1.11.m11.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.10.p1.11.m11.1.1" xref="S6.10.p1.11.m11.1.1.cmml">𝒳</mi><mo id="S6.10.p1.11.m11.1.2.3.2.2" stretchy="false" xref="S6.10.p1.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.10.p1.11.m11.1b"><apply id="S6.10.p1.11.m11.1.2.cmml" xref="S6.10.p1.11.m11.1.2"><times id="S6.10.p1.11.m11.1.2.1.cmml" xref="S6.10.p1.11.m11.1.2.1"></times><apply id="S6.10.p1.11.m11.1.2.2.cmml" xref="S6.10.p1.11.m11.1.2.2"><csymbol cd="ambiguous" id="S6.10.p1.11.m11.1.2.2.1.cmml" xref="S6.10.p1.11.m11.1.2.2">subscript</csymbol><ci id="S6.10.p1.11.m11.1.2.2.2.cmml" xref="S6.10.p1.11.m11.1.2.2.2">ℳ</ci><apply id="S6.10.p1.11.m11.1.2.2.3.cmml" xref="S6.10.p1.11.m11.1.2.2.3"><times id="S6.10.p1.11.m11.1.2.2.3.1.cmml" xref="S6.10.p1.11.m11.1.2.2.3.1"></times><ci id="S6.10.p1.11.m11.1.2.2.3.2.cmml" xref="S6.10.p1.11.m11.1.2.2.3.2">𝓃</ci><ci id="S6.10.p1.11.m11.1.2.2.3.3.cmml" xref="S6.10.p1.11.m11.1.2.2.3.3">𝒶</ci></apply></apply><ci id="S6.10.p1.11.m11.1.1.cmml" xref="S6.10.p1.11.m11.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.11.m11.1c">\cal M_{na}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.11.m11.1d">caligraphic_M start_POSTSUBSCRIPT caligraphic_n caligraphic_a end_POSTSUBSCRIPT ( caligraphic_X )</annotation></semantics></math> implies <math alttext="\mu_{1}^{per}=\mu_{2}^{per}" class="ltx_Math" display="inline" id="S6.10.p1.12.m12.1"><semantics id="S6.10.p1.12.m12.1a"><mrow id="S6.10.p1.12.m12.1.1" xref="S6.10.p1.12.m12.1.1.cmml"><msubsup id="S6.10.p1.12.m12.1.1.2" xref="S6.10.p1.12.m12.1.1.2.cmml"><mi id="S6.10.p1.12.m12.1.1.2.2.2" xref="S6.10.p1.12.m12.1.1.2.2.2.cmml">μ</mi><mn id="S6.10.p1.12.m12.1.1.2.2.3" xref="S6.10.p1.12.m12.1.1.2.2.3.cmml">1</mn><mrow id="S6.10.p1.12.m12.1.1.2.3" xref="S6.10.p1.12.m12.1.1.2.3.cmml"><mi id="S6.10.p1.12.m12.1.1.2.3.2" xref="S6.10.p1.12.m12.1.1.2.3.2.cmml">p</mi><mo id="S6.10.p1.12.m12.1.1.2.3.1" xref="S6.10.p1.12.m12.1.1.2.3.1.cmml">⁢</mo><mi id="S6.10.p1.12.m12.1.1.2.3.3" xref="S6.10.p1.12.m12.1.1.2.3.3.cmml">e</mi><mo id="S6.10.p1.12.m12.1.1.2.3.1a" xref="S6.10.p1.12.m12.1.1.2.3.1.cmml">⁢</mo><mi id="S6.10.p1.12.m12.1.1.2.3.4" xref="S6.10.p1.12.m12.1.1.2.3.4.cmml">r</mi></mrow></msubsup><mo id="S6.10.p1.12.m12.1.1.1" xref="S6.10.p1.12.m12.1.1.1.cmml">=</mo><msubsup id="S6.10.p1.12.m12.1.1.3" xref="S6.10.p1.12.m12.1.1.3.cmml"><mi id="S6.10.p1.12.m12.1.1.3.2.2" xref="S6.10.p1.12.m12.1.1.3.2.2.cmml">μ</mi><mn id="S6.10.p1.12.m12.1.1.3.2.3" xref="S6.10.p1.12.m12.1.1.3.2.3.cmml">2</mn><mrow id="S6.10.p1.12.m12.1.1.3.3" xref="S6.10.p1.12.m12.1.1.3.3.cmml"><mi id="S6.10.p1.12.m12.1.1.3.3.2" xref="S6.10.p1.12.m12.1.1.3.3.2.cmml">p</mi><mo id="S6.10.p1.12.m12.1.1.3.3.1" xref="S6.10.p1.12.m12.1.1.3.3.1.cmml">⁢</mo><mi id="S6.10.p1.12.m12.1.1.3.3.3" xref="S6.10.p1.12.m12.1.1.3.3.3.cmml">e</mi><mo id="S6.10.p1.12.m12.1.1.3.3.1a" xref="S6.10.p1.12.m12.1.1.3.3.1.cmml">⁢</mo><mi id="S6.10.p1.12.m12.1.1.3.3.4" xref="S6.10.p1.12.m12.1.1.3.3.4.cmml">r</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.10.p1.12.m12.1b"><apply id="S6.10.p1.12.m12.1.1.cmml" xref="S6.10.p1.12.m12.1.1"><eq id="S6.10.p1.12.m12.1.1.1.cmml" xref="S6.10.p1.12.m12.1.1.1"></eq><apply id="S6.10.p1.12.m12.1.1.2.cmml" xref="S6.10.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="S6.10.p1.12.m12.1.1.2.1.cmml" xref="S6.10.p1.12.m12.1.1.2">superscript</csymbol><apply id="S6.10.p1.12.m12.1.1.2.2.cmml" xref="S6.10.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="S6.10.p1.12.m12.1.1.2.2.1.cmml" xref="S6.10.p1.12.m12.1.1.2">subscript</csymbol><ci id="S6.10.p1.12.m12.1.1.2.2.2.cmml" xref="S6.10.p1.12.m12.1.1.2.2.2">𝜇</ci><cn id="S6.10.p1.12.m12.1.1.2.2.3.cmml" type="integer" xref="S6.10.p1.12.m12.1.1.2.2.3">1</cn></apply><apply id="S6.10.p1.12.m12.1.1.2.3.cmml" xref="S6.10.p1.12.m12.1.1.2.3"><times id="S6.10.p1.12.m12.1.1.2.3.1.cmml" xref="S6.10.p1.12.m12.1.1.2.3.1"></times><ci id="S6.10.p1.12.m12.1.1.2.3.2.cmml" xref="S6.10.p1.12.m12.1.1.2.3.2">𝑝</ci><ci id="S6.10.p1.12.m12.1.1.2.3.3.cmml" xref="S6.10.p1.12.m12.1.1.2.3.3">𝑒</ci><ci id="S6.10.p1.12.m12.1.1.2.3.4.cmml" xref="S6.10.p1.12.m12.1.1.2.3.4">𝑟</ci></apply></apply><apply id="S6.10.p1.12.m12.1.1.3.cmml" xref="S6.10.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S6.10.p1.12.m12.1.1.3.1.cmml" xref="S6.10.p1.12.m12.1.1.3">superscript</csymbol><apply id="S6.10.p1.12.m12.1.1.3.2.cmml" xref="S6.10.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S6.10.p1.12.m12.1.1.3.2.1.cmml" xref="S6.10.p1.12.m12.1.1.3">subscript</csymbol><ci id="S6.10.p1.12.m12.1.1.3.2.2.cmml" xref="S6.10.p1.12.m12.1.1.3.2.2">𝜇</ci><cn id="S6.10.p1.12.m12.1.1.3.2.3.cmml" type="integer" xref="S6.10.p1.12.m12.1.1.3.2.3">2</cn></apply><apply id="S6.10.p1.12.m12.1.1.3.3.cmml" xref="S6.10.p1.12.m12.1.1.3.3"><times id="S6.10.p1.12.m12.1.1.3.3.1.cmml" xref="S6.10.p1.12.m12.1.1.3.3.1"></times><ci id="S6.10.p1.12.m12.1.1.3.3.2.cmml" xref="S6.10.p1.12.m12.1.1.3.3.2">𝑝</ci><ci id="S6.10.p1.12.m12.1.1.3.3.3.cmml" xref="S6.10.p1.12.m12.1.1.3.3.3">𝑒</ci><ci id="S6.10.p1.12.m12.1.1.3.3.4.cmml" xref="S6.10.p1.12.m12.1.1.3.3.4">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.12.m12.1c">\mu_{1}^{per}=\mu_{2}^{per}</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.12.m12.1d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p italic_e italic_r end_POSTSUPERSCRIPT = italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p italic_e italic_r end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mu_{1}^{na}=\mu_{2}^{na}" class="ltx_Math" display="inline" id="S6.10.p1.13.m13.1"><semantics id="S6.10.p1.13.m13.1a"><mrow id="S6.10.p1.13.m13.1.1" xref="S6.10.p1.13.m13.1.1.cmml"><msubsup id="S6.10.p1.13.m13.1.1.2" xref="S6.10.p1.13.m13.1.1.2.cmml"><mi id="S6.10.p1.13.m13.1.1.2.2.2" xref="S6.10.p1.13.m13.1.1.2.2.2.cmml">μ</mi><mn id="S6.10.p1.13.m13.1.1.2.2.3" xref="S6.10.p1.13.m13.1.1.2.2.3.cmml">1</mn><mrow id="S6.10.p1.13.m13.1.1.2.3" xref="S6.10.p1.13.m13.1.1.2.3.cmml"><mi id="S6.10.p1.13.m13.1.1.2.3.2" xref="S6.10.p1.13.m13.1.1.2.3.2.cmml">n</mi><mo id="S6.10.p1.13.m13.1.1.2.3.1" xref="S6.10.p1.13.m13.1.1.2.3.1.cmml">⁢</mo><mi id="S6.10.p1.13.m13.1.1.2.3.3" xref="S6.10.p1.13.m13.1.1.2.3.3.cmml">a</mi></mrow></msubsup><mo id="S6.10.p1.13.m13.1.1.1" xref="S6.10.p1.13.m13.1.1.1.cmml">=</mo><msubsup id="S6.10.p1.13.m13.1.1.3" xref="S6.10.p1.13.m13.1.1.3.cmml"><mi id="S6.10.p1.13.m13.1.1.3.2.2" xref="S6.10.p1.13.m13.1.1.3.2.2.cmml">μ</mi><mn id="S6.10.p1.13.m13.1.1.3.2.3" xref="S6.10.p1.13.m13.1.1.3.2.3.cmml">2</mn><mrow id="S6.10.p1.13.m13.1.1.3.3" xref="S6.10.p1.13.m13.1.1.3.3.cmml"><mi id="S6.10.p1.13.m13.1.1.3.3.2" xref="S6.10.p1.13.m13.1.1.3.3.2.cmml">n</mi><mo id="S6.10.p1.13.m13.1.1.3.3.1" xref="S6.10.p1.13.m13.1.1.3.3.1.cmml">⁢</mo><mi id="S6.10.p1.13.m13.1.1.3.3.3" xref="S6.10.p1.13.m13.1.1.3.3.3.cmml">a</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.10.p1.13.m13.1b"><apply id="S6.10.p1.13.m13.1.1.cmml" xref="S6.10.p1.13.m13.1.1"><eq id="S6.10.p1.13.m13.1.1.1.cmml" xref="S6.10.p1.13.m13.1.1.1"></eq><apply id="S6.10.p1.13.m13.1.1.2.cmml" xref="S6.10.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S6.10.p1.13.m13.1.1.2.1.cmml" xref="S6.10.p1.13.m13.1.1.2">superscript</csymbol><apply id="S6.10.p1.13.m13.1.1.2.2.cmml" xref="S6.10.p1.13.m13.1.1.2"><csymbol cd="ambiguous" id="S6.10.p1.13.m13.1.1.2.2.1.cmml" xref="S6.10.p1.13.m13.1.1.2">subscript</csymbol><ci id="S6.10.p1.13.m13.1.1.2.2.2.cmml" xref="S6.10.p1.13.m13.1.1.2.2.2">𝜇</ci><cn id="S6.10.p1.13.m13.1.1.2.2.3.cmml" type="integer" xref="S6.10.p1.13.m13.1.1.2.2.3">1</cn></apply><apply id="S6.10.p1.13.m13.1.1.2.3.cmml" xref="S6.10.p1.13.m13.1.1.2.3"><times id="S6.10.p1.13.m13.1.1.2.3.1.cmml" xref="S6.10.p1.13.m13.1.1.2.3.1"></times><ci id="S6.10.p1.13.m13.1.1.2.3.2.cmml" xref="S6.10.p1.13.m13.1.1.2.3.2">𝑛</ci><ci id="S6.10.p1.13.m13.1.1.2.3.3.cmml" xref="S6.10.p1.13.m13.1.1.2.3.3">𝑎</ci></apply></apply><apply id="S6.10.p1.13.m13.1.1.3.cmml" xref="S6.10.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S6.10.p1.13.m13.1.1.3.1.cmml" xref="S6.10.p1.13.m13.1.1.3">superscript</csymbol><apply id="S6.10.p1.13.m13.1.1.3.2.cmml" xref="S6.10.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S6.10.p1.13.m13.1.1.3.2.1.cmml" xref="S6.10.p1.13.m13.1.1.3">subscript</csymbol><ci id="S6.10.p1.13.m13.1.1.3.2.2.cmml" xref="S6.10.p1.13.m13.1.1.3.2.2">𝜇</ci><cn id="S6.10.p1.13.m13.1.1.3.2.3.cmml" type="integer" xref="S6.10.p1.13.m13.1.1.3.2.3">2</cn></apply><apply id="S6.10.p1.13.m13.1.1.3.3.cmml" xref="S6.10.p1.13.m13.1.1.3.3"><times id="S6.10.p1.13.m13.1.1.3.3.1.cmml" xref="S6.10.p1.13.m13.1.1.3.3.1"></times><ci id="S6.10.p1.13.m13.1.1.3.3.2.cmml" xref="S6.10.p1.13.m13.1.1.3.3.2">𝑛</ci><ci id="S6.10.p1.13.m13.1.1.3.3.3.cmml" xref="S6.10.p1.13.m13.1.1.3.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.13.m13.1c">\mu_{1}^{na}=\mu_{2}^{na}</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.13.m13.1d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT = italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_a end_POSTSUPERSCRIPT</annotation></semantics></math>, which shows <math alttext="\mu_{1}=\mu_{2}" class="ltx_Math" display="inline" id="S6.10.p1.14.m14.1"><semantics id="S6.10.p1.14.m14.1a"><mrow id="S6.10.p1.14.m14.1.1" xref="S6.10.p1.14.m14.1.1.cmml"><msub id="S6.10.p1.14.m14.1.1.2" xref="S6.10.p1.14.m14.1.1.2.cmml"><mi id="S6.10.p1.14.m14.1.1.2.2" xref="S6.10.p1.14.m14.1.1.2.2.cmml">μ</mi><mn id="S6.10.p1.14.m14.1.1.2.3" xref="S6.10.p1.14.m14.1.1.2.3.cmml">1</mn></msub><mo id="S6.10.p1.14.m14.1.1.1" xref="S6.10.p1.14.m14.1.1.1.cmml">=</mo><msub id="S6.10.p1.14.m14.1.1.3" xref="S6.10.p1.14.m14.1.1.3.cmml"><mi id="S6.10.p1.14.m14.1.1.3.2" xref="S6.10.p1.14.m14.1.1.3.2.cmml">μ</mi><mn id="S6.10.p1.14.m14.1.1.3.3" xref="S6.10.p1.14.m14.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.10.p1.14.m14.1b"><apply id="S6.10.p1.14.m14.1.1.cmml" xref="S6.10.p1.14.m14.1.1"><eq id="S6.10.p1.14.m14.1.1.1.cmml" xref="S6.10.p1.14.m14.1.1.1"></eq><apply id="S6.10.p1.14.m14.1.1.2.cmml" xref="S6.10.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S6.10.p1.14.m14.1.1.2.1.cmml" xref="S6.10.p1.14.m14.1.1.2">subscript</csymbol><ci id="S6.10.p1.14.m14.1.1.2.2.cmml" xref="S6.10.p1.14.m14.1.1.2.2">𝜇</ci><cn id="S6.10.p1.14.m14.1.1.2.3.cmml" type="integer" xref="S6.10.p1.14.m14.1.1.2.3">1</cn></apply><apply id="S6.10.p1.14.m14.1.1.3.cmml" xref="S6.10.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S6.10.p1.14.m14.1.1.3.1.cmml" xref="S6.10.p1.14.m14.1.1.3">subscript</csymbol><ci id="S6.10.p1.14.m14.1.1.3.2.cmml" xref="S6.10.p1.14.m14.1.1.3.2">𝜇</ci><cn id="S6.10.p1.14.m14.1.1.3.3.cmml" type="integer" xref="S6.10.p1.14.m14.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.14.m14.1c">\mu_{1}=\mu_{2}</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.14.m14.1d">italic_μ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_μ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S6.10.p1.15.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S6.10.p1.15.1.m1.1"><semantics id="S6.10.p1.15.1.m1.1a"><mo id="S6.10.p1.15.1.m1.1.1" xref="S6.10.p1.15.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S6.10.p1.15.1.m1.1b"><csymbol cd="latexml" id="S6.10.p1.15.1.m1.1.1.cmml" xref="S6.10.p1.15.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.15.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.15.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S6.10.p1.16.m15.1"><semantics id="S6.10.p1.16.m15.1a"><mo id="S6.10.p1.16.m15.1.1" xref="S6.10.p1.16.m15.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S6.10.p1.16.m15.1b"><csymbol cd="latexml" id="S6.10.p1.16.m15.1.1.cmml" xref="S6.10.p1.16.m15.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.10.p1.16.m15.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S6.10.p1.16.m15.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_theorem ltx_theorem_thm" id="S6.Thmthm7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm7.1.1.1">Theorem 6.7</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm7.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm7.p1"> <p class="ltx_p" id="S6.Thmthm7.p1.5"><span class="ltx_text ltx_font_italic" id="S6.Thmthm7.p1.5.5">For any letter-to-letter morphism <math alttext="\sigma:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S6.Thmthm7.p1.1.1.m1.1"><semantics id="S6.Thmthm7.p1.1.1.m1.1a"><mrow id="S6.Thmthm7.p1.1.1.m1.1.1" xref="S6.Thmthm7.p1.1.1.m1.1.1.cmml"><mi id="S6.Thmthm7.p1.1.1.m1.1.1.2" xref="S6.Thmthm7.p1.1.1.m1.1.1.2.cmml">σ</mi><mo id="S6.Thmthm7.p1.1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmthm7.p1.1.1.m1.1.1.1.cmml">:</mo><mrow id="S6.Thmthm7.p1.1.1.m1.1.1.3" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.cmml"><msup id="S6.Thmthm7.p1.1.1.m1.1.1.3.2" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm7.p1.1.1.m1.1.1.3.2.2" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.2.2.cmml">𝒜</mi><mo id="S6.Thmthm7.p1.1.1.m1.1.1.3.2.3" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.2.3.cmml">∗</mo></msup><mo id="S6.Thmthm7.p1.1.1.m1.1.1.3.1" stretchy="false" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.1.cmml">→</mo><msup id="S6.Thmthm7.p1.1.1.m1.1.1.3.3" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm7.p1.1.1.m1.1.1.3.3.2" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.3.2.cmml">ℬ</mi><mo id="S6.Thmthm7.p1.1.1.m1.1.1.3.3.3" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm7.p1.1.1.m1.1b"><apply id="S6.Thmthm7.p1.1.1.m1.1.1.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1"><ci id="S6.Thmthm7.p1.1.1.m1.1.1.1.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.1">:</ci><ci id="S6.Thmthm7.p1.1.1.m1.1.1.2.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.2">𝜎</ci><apply id="S6.Thmthm7.p1.1.1.m1.1.1.3.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3"><ci id="S6.Thmthm7.p1.1.1.m1.1.1.3.1.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.1">→</ci><apply id="S6.Thmthm7.p1.1.1.m1.1.1.3.2.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.Thmthm7.p1.1.1.m1.1.1.3.2.1.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.2">superscript</csymbol><ci id="S6.Thmthm7.p1.1.1.m1.1.1.3.2.2.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.2.2">𝒜</ci><times id="S6.Thmthm7.p1.1.1.m1.1.1.3.2.3.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.2.3"></times></apply><apply id="S6.Thmthm7.p1.1.1.m1.1.1.3.3.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.Thmthm7.p1.1.1.m1.1.1.3.3.1.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.3">superscript</csymbol><ci id="S6.Thmthm7.p1.1.1.m1.1.1.3.3.2.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.3.2">ℬ</ci><times id="S6.Thmthm7.p1.1.1.m1.1.1.3.3.3.cmml" xref="S6.Thmthm7.p1.1.1.m1.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm7.p1.1.1.m1.1c">\sigma:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm7.p1.1.1.m1.1d">italic_σ : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> and any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.Thmthm7.p1.2.2.m2.1"><semantics id="S6.Thmthm7.p1.2.2.m2.1a"><mrow id="S6.Thmthm7.p1.2.2.m2.1.1" xref="S6.Thmthm7.p1.2.2.m2.1.1.cmml"><mi id="S6.Thmthm7.p1.2.2.m2.1.1.2" xref="S6.Thmthm7.p1.2.2.m2.1.1.2.cmml">X</mi><mo id="S6.Thmthm7.p1.2.2.m2.1.1.1" xref="S6.Thmthm7.p1.2.2.m2.1.1.1.cmml">⊆</mo><msup id="S6.Thmthm7.p1.2.2.m2.1.1.3" xref="S6.Thmthm7.p1.2.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm7.p1.2.2.m2.1.1.3.2" xref="S6.Thmthm7.p1.2.2.m2.1.1.3.2.cmml">𝒜</mi><mi id="S6.Thmthm7.p1.2.2.m2.1.1.3.3" xref="S6.Thmthm7.p1.2.2.m2.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm7.p1.2.2.m2.1b"><apply id="S6.Thmthm7.p1.2.2.m2.1.1.cmml" xref="S6.Thmthm7.p1.2.2.m2.1.1"><subset id="S6.Thmthm7.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmthm7.p1.2.2.m2.1.1.1"></subset><ci id="S6.Thmthm7.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmthm7.p1.2.2.m2.1.1.2">𝑋</ci><apply id="S6.Thmthm7.p1.2.2.m2.1.1.3.cmml" xref="S6.Thmthm7.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.Thmthm7.p1.2.2.m2.1.1.3.1.cmml" xref="S6.Thmthm7.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S6.Thmthm7.p1.2.2.m2.1.1.3.2.cmml" xref="S6.Thmthm7.p1.2.2.m2.1.1.3.2">𝒜</ci><ci id="S6.Thmthm7.p1.2.2.m2.1.1.3.3.cmml" xref="S6.Thmthm7.p1.2.2.m2.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm7.p1.2.2.m2.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm7.p1.2.2.m2.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> one has: If <math alttext="\sigma" class="ltx_Math" display="inline" id="S6.Thmthm7.p1.3.3.m3.1"><semantics id="S6.Thmthm7.p1.3.3.m3.1a"><mi id="S6.Thmthm7.p1.3.3.m3.1.1" xref="S6.Thmthm7.p1.3.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm7.p1.3.3.m3.1b"><ci id="S6.Thmthm7.p1.3.3.m3.1.1.cmml" xref="S6.Thmthm7.p1.3.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm7.p1.3.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm7.p1.3.3.m3.1d">italic_σ</annotation></semantics></math> is shift-orbit injective in <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm7.p1.4.4.m4.1"><semantics id="S6.Thmthm7.p1.4.4.m4.1a"><mi id="S6.Thmthm7.p1.4.4.m4.1.1" xref="S6.Thmthm7.p1.4.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm7.p1.4.4.m4.1b"><ci id="S6.Thmthm7.p1.4.4.m4.1.1.cmml" xref="S6.Thmthm7.p1.4.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm7.p1.4.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm7.p1.4.4.m4.1d">italic_X</annotation></semantics></math>, then the push-forward map <math alttext="\cal M(X)\to\cal M(\sigma(X)),\,\mu\mapsto\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.Thmthm7.p1.5.5.m5.5"><semantics id="S6.Thmthm7.p1.5.5.m5.5a"><mrow id="S6.Thmthm7.p1.5.5.m5.5.5.2" xref="S6.Thmthm7.p1.5.5.m5.5.5.3.cmml"><mrow id="S6.Thmthm7.p1.5.5.m5.4.4.1.1" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.cmml"><mrow id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.2" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.2.cmml">ℳ</mi><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.1" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.3.2" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.cmml"><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.3.2.1" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm7.p1.5.5.m5.1.1" xref="S6.Thmthm7.p1.5.5.m5.1.1.cmml">𝒳</mi><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.3.2.2" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.2" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.2.cmml">→</mo><mrow id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.3" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.3.cmml">ℳ</mi><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.2" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml"><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.2" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml"><mi id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.2" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.2.cmml">σ</mi><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.1" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.3.2" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml"><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm7.p1.5.5.m5.2.2" xref="S6.Thmthm7.p1.5.5.m5.2.2.cmml">𝒳</mi><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.3" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Thmthm7.p1.5.5.m5.5.5.2.3" rspace="0.337em" xref="S6.Thmthm7.p1.5.5.m5.5.5.3a.cmml">,</mo><mrow id="S6.Thmthm7.p1.5.5.m5.5.5.2.2" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.cmml"><mi id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.2" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.2.cmml">μ</mi><mo id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.1" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.1.cmml">↦</mo><mrow id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.cmml"><msub id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.cmml"><mi id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.2" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.2.cmml">σ</mi><mo id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.3" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.3.cmml">∗</mo></msub><mo id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.1" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.1.cmml">⁢</mo><mrow id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.3.2" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.cmml"><mo id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.3.2.1" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.cmml">(</mo><mi id="S6.Thmthm7.p1.5.5.m5.3.3" xref="S6.Thmthm7.p1.5.5.m5.3.3.cmml">μ</mi><mo id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.3.2.2" stretchy="false" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm7.p1.5.5.m5.5b"><apply id="S6.Thmthm7.p1.5.5.m5.5.5.3.cmml" xref="S6.Thmthm7.p1.5.5.m5.5.5.2"><csymbol cd="ambiguous" id="S6.Thmthm7.p1.5.5.m5.5.5.3a.cmml" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.3">formulae-sequence</csymbol><apply id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.cmml" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1"><ci id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.2.cmml" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.2">→</ci><apply id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.cmml" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3"><times id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.1.cmml" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.1"></times><ci id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.2.cmml" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.3.2">ℳ</ci><ci id="S6.Thmthm7.p1.5.5.m5.1.1.cmml" xref="S6.Thmthm7.p1.5.5.m5.1.1">𝒳</ci></apply><apply id="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1.cmml" xref="S6.Thmthm7.p1.5.5.m5.4.4.1.1.1"><times 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id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.1.cmml" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.1"></times><apply id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.cmml" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2"><csymbol cd="ambiguous" id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.1.cmml" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2">subscript</csymbol><ci id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.2.cmml" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.2">𝜎</ci><times id="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.3.cmml" xref="S6.Thmthm7.p1.5.5.m5.5.5.2.2.3.2.3"></times></apply><ci id="S6.Thmthm7.p1.5.5.m5.3.3.cmml" xref="S6.Thmthm7.p1.5.5.m5.3.3">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm7.p1.5.5.m5.5c">\cal M(X)\to\cal M(\sigma(X)),\,\mu\mapsto\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm7.p1.5.5.m5.5d">caligraphic_M ( caligraphic_X ) → caligraphic_M ( italic_σ ( caligraphic_X ) ) , italic_μ ↦ italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math> is injective.</span></p> </div> </div> <div class="ltx_proof" id="S6.12"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.11.p1"> <p class="ltx_p" id="S6.11.p1.3">From Proposition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm6" title="Proposition 6.6. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.6</span></a> we know that it suffices to prove the injectivity of the measure map <math alttext="\mu\mapsto\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.11.p1.1.m1.1"><semantics id="S6.11.p1.1.m1.1a"><mrow id="S6.11.p1.1.m1.1.2" xref="S6.11.p1.1.m1.1.2.cmml"><mi id="S6.11.p1.1.m1.1.2.2" xref="S6.11.p1.1.m1.1.2.2.cmml">μ</mi><mo id="S6.11.p1.1.m1.1.2.1" stretchy="false" xref="S6.11.p1.1.m1.1.2.1.cmml">↦</mo><mrow id="S6.11.p1.1.m1.1.2.3" xref="S6.11.p1.1.m1.1.2.3.cmml"><msub id="S6.11.p1.1.m1.1.2.3.2" xref="S6.11.p1.1.m1.1.2.3.2.cmml"><mi id="S6.11.p1.1.m1.1.2.3.2.2" xref="S6.11.p1.1.m1.1.2.3.2.2.cmml">σ</mi><mo id="S6.11.p1.1.m1.1.2.3.2.3" xref="S6.11.p1.1.m1.1.2.3.2.3.cmml">∗</mo></msub><mo id="S6.11.p1.1.m1.1.2.3.1" xref="S6.11.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S6.11.p1.1.m1.1.2.3.3.2" xref="S6.11.p1.1.m1.1.2.3.cmml"><mo id="S6.11.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="S6.11.p1.1.m1.1.2.3.cmml">(</mo><mi id="S6.11.p1.1.m1.1.1" xref="S6.11.p1.1.m1.1.1.cmml">μ</mi><mo id="S6.11.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="S6.11.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.11.p1.1.m1.1b"><apply id="S6.11.p1.1.m1.1.2.cmml" xref="S6.11.p1.1.m1.1.2"><csymbol cd="latexml" id="S6.11.p1.1.m1.1.2.1.cmml" xref="S6.11.p1.1.m1.1.2.1">maps-to</csymbol><ci id="S6.11.p1.1.m1.1.2.2.cmml" xref="S6.11.p1.1.m1.1.2.2">𝜇</ci><apply id="S6.11.p1.1.m1.1.2.3.cmml" xref="S6.11.p1.1.m1.1.2.3"><times id="S6.11.p1.1.m1.1.2.3.1.cmml" xref="S6.11.p1.1.m1.1.2.3.1"></times><apply id="S6.11.p1.1.m1.1.2.3.2.cmml" xref="S6.11.p1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="S6.11.p1.1.m1.1.2.3.2.1.cmml" xref="S6.11.p1.1.m1.1.2.3.2">subscript</csymbol><ci id="S6.11.p1.1.m1.1.2.3.2.2.cmml" xref="S6.11.p1.1.m1.1.2.3.2.2">𝜎</ci><times id="S6.11.p1.1.m1.1.2.3.2.3.cmml" xref="S6.11.p1.1.m1.1.2.3.2.3"></times></apply><ci id="S6.11.p1.1.m1.1.1.cmml" xref="S6.11.p1.1.m1.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.11.p1.1.m1.1c">\mu\mapsto\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.11.p1.1.m1.1d">italic_μ ↦ italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math> for the two cases, where <math alttext="\mu" class="ltx_Math" display="inline" id="S6.11.p1.2.m2.1"><semantics id="S6.11.p1.2.m2.1a"><mi id="S6.11.p1.2.m2.1.1" xref="S6.11.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.11.p1.2.m2.1b"><ci id="S6.11.p1.2.m2.1.1.cmml" xref="S6.11.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.11.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.11.p1.2.m2.1d">italic_μ</annotation></semantics></math> is periodic or where <math alttext="\mu" class="ltx_Math" display="inline" id="S6.11.p1.3.m3.1"><semantics id="S6.11.p1.3.m3.1a"><mi id="S6.11.p1.3.m3.1.1" xref="S6.11.p1.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.11.p1.3.m3.1b"><ci id="S6.11.p1.3.m3.1.1.cmml" xref="S6.11.p1.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.11.p1.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.11.p1.3.m3.1d">italic_μ</annotation></semantics></math> is non-atomic.</p> </div> <div class="ltx_para" id="S6.12.p2"> <p class="ltx_p" id="S6.12.p2.14">The latter case is already dealt with in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm4" title="Proposition 6.4. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.4</span></a>, so that we can from now on assume that <math alttext="\mu" class="ltx_Math" display="inline" id="S6.12.p2.1.m1.1"><semantics id="S6.12.p2.1.m1.1a"><mi id="S6.12.p2.1.m1.1.1" xref="S6.12.p2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.12.p2.1.m1.1b"><ci id="S6.12.p2.1.m1.1.1.cmml" xref="S6.12.p2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.1.m1.1d">italic_μ</annotation></semantics></math> is carried by <math alttext="\text{\rm Per}(X)" class="ltx_Math" display="inline" id="S6.12.p2.2.m2.1"><semantics id="S6.12.p2.2.m2.1a"><mrow id="S6.12.p2.2.m2.1.2" xref="S6.12.p2.2.m2.1.2.cmml"><mtext id="S6.12.p2.2.m2.1.2.2" xref="S6.12.p2.2.m2.1.2.2a.cmml">Per</mtext><mo id="S6.12.p2.2.m2.1.2.1" xref="S6.12.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S6.12.p2.2.m2.1.2.3.2" xref="S6.12.p2.2.m2.1.2.cmml"><mo id="S6.12.p2.2.m2.1.2.3.2.1" stretchy="false" xref="S6.12.p2.2.m2.1.2.cmml">(</mo><mi id="S6.12.p2.2.m2.1.1" xref="S6.12.p2.2.m2.1.1.cmml">X</mi><mo id="S6.12.p2.2.m2.1.2.3.2.2" stretchy="false" xref="S6.12.p2.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.12.p2.2.m2.1b"><apply id="S6.12.p2.2.m2.1.2.cmml" xref="S6.12.p2.2.m2.1.2"><times id="S6.12.p2.2.m2.1.2.1.cmml" xref="S6.12.p2.2.m2.1.2.1"></times><ci id="S6.12.p2.2.m2.1.2.2a.cmml" xref="S6.12.p2.2.m2.1.2.2"><mtext id="S6.12.p2.2.m2.1.2.2.cmml" xref="S6.12.p2.2.m2.1.2.2">Per</mtext></ci><ci id="S6.12.p2.2.m2.1.1.cmml" xref="S6.12.p2.2.m2.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.2.m2.1c">\text{\rm Per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.2.m2.1d">Per ( italic_X )</annotation></semantics></math>. But in this case the injectivity of the map <math alttext="\mu\mapsto\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.12.p2.3.m3.1"><semantics id="S6.12.p2.3.m3.1a"><mrow id="S6.12.p2.3.m3.1.2" xref="S6.12.p2.3.m3.1.2.cmml"><mi id="S6.12.p2.3.m3.1.2.2" xref="S6.12.p2.3.m3.1.2.2.cmml">μ</mi><mo id="S6.12.p2.3.m3.1.2.1" stretchy="false" xref="S6.12.p2.3.m3.1.2.1.cmml">↦</mo><mrow id="S6.12.p2.3.m3.1.2.3" xref="S6.12.p2.3.m3.1.2.3.cmml"><msub id="S6.12.p2.3.m3.1.2.3.2" xref="S6.12.p2.3.m3.1.2.3.2.cmml"><mi id="S6.12.p2.3.m3.1.2.3.2.2" xref="S6.12.p2.3.m3.1.2.3.2.2.cmml">σ</mi><mo id="S6.12.p2.3.m3.1.2.3.2.3" xref="S6.12.p2.3.m3.1.2.3.2.3.cmml">∗</mo></msub><mo id="S6.12.p2.3.m3.1.2.3.1" xref="S6.12.p2.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S6.12.p2.3.m3.1.2.3.3.2" xref="S6.12.p2.3.m3.1.2.3.cmml"><mo id="S6.12.p2.3.m3.1.2.3.3.2.1" stretchy="false" xref="S6.12.p2.3.m3.1.2.3.cmml">(</mo><mi id="S6.12.p2.3.m3.1.1" xref="S6.12.p2.3.m3.1.1.cmml">μ</mi><mo id="S6.12.p2.3.m3.1.2.3.3.2.2" stretchy="false" xref="S6.12.p2.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.12.p2.3.m3.1b"><apply id="S6.12.p2.3.m3.1.2.cmml" xref="S6.12.p2.3.m3.1.2"><csymbol cd="latexml" id="S6.12.p2.3.m3.1.2.1.cmml" xref="S6.12.p2.3.m3.1.2.1">maps-to</csymbol><ci id="S6.12.p2.3.m3.1.2.2.cmml" xref="S6.12.p2.3.m3.1.2.2">𝜇</ci><apply id="S6.12.p2.3.m3.1.2.3.cmml" xref="S6.12.p2.3.m3.1.2.3"><times id="S6.12.p2.3.m3.1.2.3.1.cmml" xref="S6.12.p2.3.m3.1.2.3.1"></times><apply id="S6.12.p2.3.m3.1.2.3.2.cmml" xref="S6.12.p2.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S6.12.p2.3.m3.1.2.3.2.1.cmml" xref="S6.12.p2.3.m3.1.2.3.2">subscript</csymbol><ci id="S6.12.p2.3.m3.1.2.3.2.2.cmml" xref="S6.12.p2.3.m3.1.2.3.2.2">𝜎</ci><times id="S6.12.p2.3.m3.1.2.3.2.3.cmml" xref="S6.12.p2.3.m3.1.2.3.2.3"></times></apply><ci id="S6.12.p2.3.m3.1.1.cmml" xref="S6.12.p2.3.m3.1.1">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.3.m3.1c">\mu\mapsto\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.3.m3.1d">italic_μ ↦ italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math> is trivial, since by the injectivity assumption on the shift-orbits together with Lemma <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm1" title="Lemma 6.1. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.1</span></a> we know that for every periodic word <math alttext="{\bf x}\in\sigma(X)" class="ltx_Math" display="inline" id="S6.12.p2.4.m4.1"><semantics id="S6.12.p2.4.m4.1a"><mrow id="S6.12.p2.4.m4.1.2" xref="S6.12.p2.4.m4.1.2.cmml"><mi id="S6.12.p2.4.m4.1.2.2" xref="S6.12.p2.4.m4.1.2.2.cmml">𝐱</mi><mo id="S6.12.p2.4.m4.1.2.1" xref="S6.12.p2.4.m4.1.2.1.cmml">∈</mo><mrow id="S6.12.p2.4.m4.1.2.3" xref="S6.12.p2.4.m4.1.2.3.cmml"><mi id="S6.12.p2.4.m4.1.2.3.2" xref="S6.12.p2.4.m4.1.2.3.2.cmml">σ</mi><mo id="S6.12.p2.4.m4.1.2.3.1" xref="S6.12.p2.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S6.12.p2.4.m4.1.2.3.3.2" xref="S6.12.p2.4.m4.1.2.3.cmml"><mo id="S6.12.p2.4.m4.1.2.3.3.2.1" stretchy="false" xref="S6.12.p2.4.m4.1.2.3.cmml">(</mo><mi id="S6.12.p2.4.m4.1.1" xref="S6.12.p2.4.m4.1.1.cmml">X</mi><mo id="S6.12.p2.4.m4.1.2.3.3.2.2" stretchy="false" xref="S6.12.p2.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.12.p2.4.m4.1b"><apply id="S6.12.p2.4.m4.1.2.cmml" xref="S6.12.p2.4.m4.1.2"><in id="S6.12.p2.4.m4.1.2.1.cmml" xref="S6.12.p2.4.m4.1.2.1"></in><ci id="S6.12.p2.4.m4.1.2.2.cmml" xref="S6.12.p2.4.m4.1.2.2">𝐱</ci><apply id="S6.12.p2.4.m4.1.2.3.cmml" xref="S6.12.p2.4.m4.1.2.3"><times id="S6.12.p2.4.m4.1.2.3.1.cmml" xref="S6.12.p2.4.m4.1.2.3.1"></times><ci id="S6.12.p2.4.m4.1.2.3.2.cmml" xref="S6.12.p2.4.m4.1.2.3.2">𝜎</ci><ci id="S6.12.p2.4.m4.1.1.cmml" xref="S6.12.p2.4.m4.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.4.m4.1c">{\bf x}\in\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.4.m4.1d">bold_x ∈ italic_σ ( italic_X )</annotation></semantics></math> the set <math alttext="\sigma^{-1}({\bf x})\cap X" class="ltx_Math" display="inline" id="S6.12.p2.5.m5.1"><semantics id="S6.12.p2.5.m5.1a"><mrow id="S6.12.p2.5.m5.1.2" xref="S6.12.p2.5.m5.1.2.cmml"><mrow id="S6.12.p2.5.m5.1.2.2" xref="S6.12.p2.5.m5.1.2.2.cmml"><msup id="S6.12.p2.5.m5.1.2.2.2" xref="S6.12.p2.5.m5.1.2.2.2.cmml"><mi id="S6.12.p2.5.m5.1.2.2.2.2" xref="S6.12.p2.5.m5.1.2.2.2.2.cmml">σ</mi><mrow id="S6.12.p2.5.m5.1.2.2.2.3" xref="S6.12.p2.5.m5.1.2.2.2.3.cmml"><mo id="S6.12.p2.5.m5.1.2.2.2.3a" xref="S6.12.p2.5.m5.1.2.2.2.3.cmml">−</mo><mn id="S6.12.p2.5.m5.1.2.2.2.3.2" xref="S6.12.p2.5.m5.1.2.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S6.12.p2.5.m5.1.2.2.1" xref="S6.12.p2.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S6.12.p2.5.m5.1.2.2.3.2" xref="S6.12.p2.5.m5.1.2.2.cmml"><mo id="S6.12.p2.5.m5.1.2.2.3.2.1" stretchy="false" xref="S6.12.p2.5.m5.1.2.2.cmml">(</mo><mi id="S6.12.p2.5.m5.1.1" xref="S6.12.p2.5.m5.1.1.cmml">𝐱</mi><mo id="S6.12.p2.5.m5.1.2.2.3.2.2" stretchy="false" xref="S6.12.p2.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.12.p2.5.m5.1.2.1" xref="S6.12.p2.5.m5.1.2.1.cmml">∩</mo><mi id="S6.12.p2.5.m5.1.2.3" xref="S6.12.p2.5.m5.1.2.3.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.12.p2.5.m5.1b"><apply id="S6.12.p2.5.m5.1.2.cmml" xref="S6.12.p2.5.m5.1.2"><intersect id="S6.12.p2.5.m5.1.2.1.cmml" xref="S6.12.p2.5.m5.1.2.1"></intersect><apply id="S6.12.p2.5.m5.1.2.2.cmml" xref="S6.12.p2.5.m5.1.2.2"><times id="S6.12.p2.5.m5.1.2.2.1.cmml" xref="S6.12.p2.5.m5.1.2.2.1"></times><apply id="S6.12.p2.5.m5.1.2.2.2.cmml" xref="S6.12.p2.5.m5.1.2.2.2"><csymbol cd="ambiguous" id="S6.12.p2.5.m5.1.2.2.2.1.cmml" xref="S6.12.p2.5.m5.1.2.2.2">superscript</csymbol><ci id="S6.12.p2.5.m5.1.2.2.2.2.cmml" xref="S6.12.p2.5.m5.1.2.2.2.2">𝜎</ci><apply id="S6.12.p2.5.m5.1.2.2.2.3.cmml" xref="S6.12.p2.5.m5.1.2.2.2.3"><minus id="S6.12.p2.5.m5.1.2.2.2.3.1.cmml" xref="S6.12.p2.5.m5.1.2.2.2.3"></minus><cn id="S6.12.p2.5.m5.1.2.2.2.3.2.cmml" type="integer" xref="S6.12.p2.5.m5.1.2.2.2.3.2">1</cn></apply></apply><ci id="S6.12.p2.5.m5.1.1.cmml" xref="S6.12.p2.5.m5.1.1">𝐱</ci></apply><ci id="S6.12.p2.5.m5.1.2.3.cmml" xref="S6.12.p2.5.m5.1.2.3">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.5.m5.1c">\sigma^{-1}({\bf x})\cap X</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.5.m5.1d">italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x ) ∩ italic_X</annotation></semantics></math> consists of words <math alttext="{\bf y}_{i}" class="ltx_Math" display="inline" id="S6.12.p2.6.m6.1"><semantics id="S6.12.p2.6.m6.1a"><msub id="S6.12.p2.6.m6.1.1" xref="S6.12.p2.6.m6.1.1.cmml"><mi id="S6.12.p2.6.m6.1.1.2" xref="S6.12.p2.6.m6.1.1.2.cmml">𝐲</mi><mi id="S6.12.p2.6.m6.1.1.3" xref="S6.12.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.12.p2.6.m6.1b"><apply id="S6.12.p2.6.m6.1.1.cmml" xref="S6.12.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S6.12.p2.6.m6.1.1.1.cmml" xref="S6.12.p2.6.m6.1.1">subscript</csymbol><ci id="S6.12.p2.6.m6.1.1.2.cmml" xref="S6.12.p2.6.m6.1.1.2">𝐲</ci><ci id="S6.12.p2.6.m6.1.1.3.cmml" xref="S6.12.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.6.m6.1c">{\bf y}_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.6.m6.1d">bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> which are periodic as well and belong all to a single shift-orbit <math alttext="\cal O({\bf y}_{i})" class="ltx_Math" display="inline" id="S6.12.p2.7.m7.1"><semantics id="S6.12.p2.7.m7.1a"><mrow id="S6.12.p2.7.m7.1.1" xref="S6.12.p2.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.12.p2.7.m7.1.1.3" xref="S6.12.p2.7.m7.1.1.3.cmml">𝒪</mi><mo id="S6.12.p2.7.m7.1.1.2" xref="S6.12.p2.7.m7.1.1.2.cmml">⁢</mo><mrow id="S6.12.p2.7.m7.1.1.1.1" xref="S6.12.p2.7.m7.1.1.1.1.1.cmml"><mo id="S6.12.p2.7.m7.1.1.1.1.2" stretchy="false" xref="S6.12.p2.7.m7.1.1.1.1.1.cmml">(</mo><msub id="S6.12.p2.7.m7.1.1.1.1.1" xref="S6.12.p2.7.m7.1.1.1.1.1.cmml"><mi id="S6.12.p2.7.m7.1.1.1.1.1.2" xref="S6.12.p2.7.m7.1.1.1.1.1.2.cmml">𝐲</mi><mi class="ltx_font_mathcaligraphic" id="S6.12.p2.7.m7.1.1.1.1.1.3" xref="S6.12.p2.7.m7.1.1.1.1.1.3.cmml">𝒾</mi></msub><mo id="S6.12.p2.7.m7.1.1.1.1.3" stretchy="false" xref="S6.12.p2.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.12.p2.7.m7.1b"><apply id="S6.12.p2.7.m7.1.1.cmml" xref="S6.12.p2.7.m7.1.1"><times id="S6.12.p2.7.m7.1.1.2.cmml" xref="S6.12.p2.7.m7.1.1.2"></times><ci id="S6.12.p2.7.m7.1.1.3.cmml" xref="S6.12.p2.7.m7.1.1.3">𝒪</ci><apply id="S6.12.p2.7.m7.1.1.1.1.1.cmml" xref="S6.12.p2.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S6.12.p2.7.m7.1.1.1.1.1.1.cmml" xref="S6.12.p2.7.m7.1.1.1.1">subscript</csymbol><ci id="S6.12.p2.7.m7.1.1.1.1.1.2.cmml" xref="S6.12.p2.7.m7.1.1.1.1.1.2">𝐲</ci><ci id="S6.12.p2.7.m7.1.1.1.1.1.3.cmml" xref="S6.12.p2.7.m7.1.1.1.1.1.3">𝒾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.7.m7.1c">\cal O({\bf y}_{i})</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.7.m7.1d">caligraphic_O ( bold_y start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT )</annotation></semantics></math>. Hence we have <math alttext="\mu(\cal O({\bf y}_{i}))=\sigma_{*}(\mu)(\cal O({\bf x}))" class="ltx_Math" display="inline" id="S6.12.p2.8.m8.4"><semantics id="S6.12.p2.8.m8.4a"><mrow id="S6.12.p2.8.m8.4.4" xref="S6.12.p2.8.m8.4.4.cmml"><mrow id="S6.12.p2.8.m8.3.3.1" xref="S6.12.p2.8.m8.3.3.1.cmml"><mi id="S6.12.p2.8.m8.3.3.1.3" xref="S6.12.p2.8.m8.3.3.1.3.cmml">μ</mi><mo id="S6.12.p2.8.m8.3.3.1.2" xref="S6.12.p2.8.m8.3.3.1.2.cmml">⁢</mo><mrow id="S6.12.p2.8.m8.3.3.1.1.1" xref="S6.12.p2.8.m8.3.3.1.1.1.1.cmml"><mo id="S6.12.p2.8.m8.3.3.1.1.1.2" stretchy="false" xref="S6.12.p2.8.m8.3.3.1.1.1.1.cmml">(</mo><mrow id="S6.12.p2.8.m8.3.3.1.1.1.1" xref="S6.12.p2.8.m8.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.12.p2.8.m8.3.3.1.1.1.1.3" xref="S6.12.p2.8.m8.3.3.1.1.1.1.3.cmml">𝒪</mi><mo id="S6.12.p2.8.m8.3.3.1.1.1.1.2" xref="S6.12.p2.8.m8.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.cmml"><mo id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.2" stretchy="false" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.cmml"><mi id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.2" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.2.cmml">𝐲</mi><mi class="ltx_font_mathcaligraphic" id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.3" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.3.cmml">𝒾</mi></msub><mo id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.3" stretchy="false" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.12.p2.8.m8.3.3.1.1.1.3" stretchy="false" xref="S6.12.p2.8.m8.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.12.p2.8.m8.4.4.3" xref="S6.12.p2.8.m8.4.4.3.cmml">=</mo><mrow id="S6.12.p2.8.m8.4.4.2" xref="S6.12.p2.8.m8.4.4.2.cmml"><msub id="S6.12.p2.8.m8.4.4.2.3" xref="S6.12.p2.8.m8.4.4.2.3.cmml"><mi id="S6.12.p2.8.m8.4.4.2.3.2" xref="S6.12.p2.8.m8.4.4.2.3.2.cmml">σ</mi><mo id="S6.12.p2.8.m8.4.4.2.3.3" xref="S6.12.p2.8.m8.4.4.2.3.3.cmml">∗</mo></msub><mo id="S6.12.p2.8.m8.4.4.2.2" xref="S6.12.p2.8.m8.4.4.2.2.cmml">⁢</mo><mrow id="S6.12.p2.8.m8.4.4.2.4.2" xref="S6.12.p2.8.m8.4.4.2.cmml"><mo id="S6.12.p2.8.m8.4.4.2.4.2.1" stretchy="false" xref="S6.12.p2.8.m8.4.4.2.cmml">(</mo><mi id="S6.12.p2.8.m8.1.1" xref="S6.12.p2.8.m8.1.1.cmml">μ</mi><mo id="S6.12.p2.8.m8.4.4.2.4.2.2" stretchy="false" xref="S6.12.p2.8.m8.4.4.2.cmml">)</mo></mrow><mo id="S6.12.p2.8.m8.4.4.2.2a" xref="S6.12.p2.8.m8.4.4.2.2.cmml">⁢</mo><mrow id="S6.12.p2.8.m8.4.4.2.1.1" xref="S6.12.p2.8.m8.4.4.2.1.1.1.cmml"><mo id="S6.12.p2.8.m8.4.4.2.1.1.2" stretchy="false" xref="S6.12.p2.8.m8.4.4.2.1.1.1.cmml">(</mo><mrow id="S6.12.p2.8.m8.4.4.2.1.1.1" xref="S6.12.p2.8.m8.4.4.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.12.p2.8.m8.4.4.2.1.1.1.2" xref="S6.12.p2.8.m8.4.4.2.1.1.1.2.cmml">𝒪</mi><mo id="S6.12.p2.8.m8.4.4.2.1.1.1.1" xref="S6.12.p2.8.m8.4.4.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.12.p2.8.m8.4.4.2.1.1.1.3.2" xref="S6.12.p2.8.m8.4.4.2.1.1.1.cmml"><mo id="S6.12.p2.8.m8.4.4.2.1.1.1.3.2.1" stretchy="false" xref="S6.12.p2.8.m8.4.4.2.1.1.1.cmml">(</mo><mi id="S6.12.p2.8.m8.2.2" xref="S6.12.p2.8.m8.2.2.cmml">𝐱</mi><mo id="S6.12.p2.8.m8.4.4.2.1.1.1.3.2.2" stretchy="false" xref="S6.12.p2.8.m8.4.4.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.12.p2.8.m8.4.4.2.1.1.3" stretchy="false" xref="S6.12.p2.8.m8.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.12.p2.8.m8.4b"><apply id="S6.12.p2.8.m8.4.4.cmml" xref="S6.12.p2.8.m8.4.4"><eq id="S6.12.p2.8.m8.4.4.3.cmml" xref="S6.12.p2.8.m8.4.4.3"></eq><apply id="S6.12.p2.8.m8.3.3.1.cmml" xref="S6.12.p2.8.m8.3.3.1"><times id="S6.12.p2.8.m8.3.3.1.2.cmml" xref="S6.12.p2.8.m8.3.3.1.2"></times><ci id="S6.12.p2.8.m8.3.3.1.3.cmml" xref="S6.12.p2.8.m8.3.3.1.3">𝜇</ci><apply id="S6.12.p2.8.m8.3.3.1.1.1.1.cmml" xref="S6.12.p2.8.m8.3.3.1.1.1"><times id="S6.12.p2.8.m8.3.3.1.1.1.1.2.cmml" xref="S6.12.p2.8.m8.3.3.1.1.1.1.2"></times><ci id="S6.12.p2.8.m8.3.3.1.1.1.1.3.cmml" xref="S6.12.p2.8.m8.3.3.1.1.1.1.3">𝒪</ci><apply id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.cmml" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.1.cmml" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1">subscript</csymbol><ci id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.2.cmml" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.2">𝐲</ci><ci id="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.3.cmml" xref="S6.12.p2.8.m8.3.3.1.1.1.1.1.1.1.3">𝒾</ci></apply></apply></apply><apply id="S6.12.p2.8.m8.4.4.2.cmml" xref="S6.12.p2.8.m8.4.4.2"><times id="S6.12.p2.8.m8.4.4.2.2.cmml" xref="S6.12.p2.8.m8.4.4.2.2"></times><apply id="S6.12.p2.8.m8.4.4.2.3.cmml" xref="S6.12.p2.8.m8.4.4.2.3"><csymbol cd="ambiguous" id="S6.12.p2.8.m8.4.4.2.3.1.cmml" xref="S6.12.p2.8.m8.4.4.2.3">subscript</csymbol><ci id="S6.12.p2.8.m8.4.4.2.3.2.cmml" xref="S6.12.p2.8.m8.4.4.2.3.2">𝜎</ci><times id="S6.12.p2.8.m8.4.4.2.3.3.cmml" xref="S6.12.p2.8.m8.4.4.2.3.3"></times></apply><ci id="S6.12.p2.8.m8.1.1.cmml" xref="S6.12.p2.8.m8.1.1">𝜇</ci><apply id="S6.12.p2.8.m8.4.4.2.1.1.1.cmml" xref="S6.12.p2.8.m8.4.4.2.1.1"><times id="S6.12.p2.8.m8.4.4.2.1.1.1.1.cmml" xref="S6.12.p2.8.m8.4.4.2.1.1.1.1"></times><ci id="S6.12.p2.8.m8.4.4.2.1.1.1.2.cmml" xref="S6.12.p2.8.m8.4.4.2.1.1.1.2">𝒪</ci><ci id="S6.12.p2.8.m8.2.2.cmml" xref="S6.12.p2.8.m8.2.2">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.8.m8.4c">\mu(\cal O({\bf y}_{i}))=\sigma_{*}(\mu)(\cal O({\bf x}))</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.8.m8.4d">italic_μ ( caligraphic_O ( bold_y start_POSTSUBSCRIPT caligraphic_i end_POSTSUBSCRIPT ) ) = italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ ) ( caligraphic_O ( bold_x ) )</annotation></semantics></math>. Since any invariant measure on <math alttext="X" class="ltx_Math" display="inline" id="S6.12.p2.9.m9.1"><semantics id="S6.12.p2.9.m9.1a"><mi id="S6.12.p2.9.m9.1.1" xref="S6.12.p2.9.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.12.p2.9.m9.1b"><ci id="S6.12.p2.9.m9.1.1.cmml" xref="S6.12.p2.9.m9.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.9.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.9.m9.1d">italic_X</annotation></semantics></math> which is carried by <math alttext="\text{\rm Per}(X)" class="ltx_Math" display="inline" id="S6.12.p2.10.m10.1"><semantics id="S6.12.p2.10.m10.1a"><mrow id="S6.12.p2.10.m10.1.2" xref="S6.12.p2.10.m10.1.2.cmml"><mtext id="S6.12.p2.10.m10.1.2.2" xref="S6.12.p2.10.m10.1.2.2a.cmml">Per</mtext><mo id="S6.12.p2.10.m10.1.2.1" xref="S6.12.p2.10.m10.1.2.1.cmml">⁢</mo><mrow id="S6.12.p2.10.m10.1.2.3.2" xref="S6.12.p2.10.m10.1.2.cmml"><mo id="S6.12.p2.10.m10.1.2.3.2.1" stretchy="false" xref="S6.12.p2.10.m10.1.2.cmml">(</mo><mi id="S6.12.p2.10.m10.1.1" xref="S6.12.p2.10.m10.1.1.cmml">X</mi><mo id="S6.12.p2.10.m10.1.2.3.2.2" stretchy="false" xref="S6.12.p2.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.12.p2.10.m10.1b"><apply id="S6.12.p2.10.m10.1.2.cmml" xref="S6.12.p2.10.m10.1.2"><times id="S6.12.p2.10.m10.1.2.1.cmml" xref="S6.12.p2.10.m10.1.2.1"></times><ci id="S6.12.p2.10.m10.1.2.2a.cmml" xref="S6.12.p2.10.m10.1.2.2"><mtext id="S6.12.p2.10.m10.1.2.2.cmml" xref="S6.12.p2.10.m10.1.2.2">Per</mtext></ci><ci id="S6.12.p2.10.m10.1.1.cmml" xref="S6.12.p2.10.m10.1.1">𝑋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.10.m10.1c">\text{\rm Per}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.10.m10.1d">Per ( italic_X )</annotation></semantics></math> is determined by knowing its evaluation on every periodic orbit, it follows that <math alttext="\mu" class="ltx_Math" display="inline" id="S6.12.p2.11.m11.1"><semantics id="S6.12.p2.11.m11.1a"><mi id="S6.12.p2.11.m11.1.1" xref="S6.12.p2.11.m11.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S6.12.p2.11.m11.1b"><ci id="S6.12.p2.11.m11.1.1.cmml" xref="S6.12.p2.11.m11.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.11.m11.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.11.m11.1d">italic_μ</annotation></semantics></math> is entirely determined by <math alttext="\sigma_{*}(\mu)" class="ltx_Math" display="inline" id="S6.12.p2.12.m12.1"><semantics id="S6.12.p2.12.m12.1a"><mrow id="S6.12.p2.12.m12.1.2" xref="S6.12.p2.12.m12.1.2.cmml"><msub id="S6.12.p2.12.m12.1.2.2" xref="S6.12.p2.12.m12.1.2.2.cmml"><mi id="S6.12.p2.12.m12.1.2.2.2" xref="S6.12.p2.12.m12.1.2.2.2.cmml">σ</mi><mo id="S6.12.p2.12.m12.1.2.2.3" xref="S6.12.p2.12.m12.1.2.2.3.cmml">∗</mo></msub><mo id="S6.12.p2.12.m12.1.2.1" xref="S6.12.p2.12.m12.1.2.1.cmml">⁢</mo><mrow id="S6.12.p2.12.m12.1.2.3.2" xref="S6.12.p2.12.m12.1.2.cmml"><mo id="S6.12.p2.12.m12.1.2.3.2.1" stretchy="false" xref="S6.12.p2.12.m12.1.2.cmml">(</mo><mi id="S6.12.p2.12.m12.1.1" xref="S6.12.p2.12.m12.1.1.cmml">μ</mi><mo id="S6.12.p2.12.m12.1.2.3.2.2" stretchy="false" xref="S6.12.p2.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.12.p2.12.m12.1b"><apply id="S6.12.p2.12.m12.1.2.cmml" xref="S6.12.p2.12.m12.1.2"><times id="S6.12.p2.12.m12.1.2.1.cmml" xref="S6.12.p2.12.m12.1.2.1"></times><apply id="S6.12.p2.12.m12.1.2.2.cmml" xref="S6.12.p2.12.m12.1.2.2"><csymbol cd="ambiguous" id="S6.12.p2.12.m12.1.2.2.1.cmml" xref="S6.12.p2.12.m12.1.2.2">subscript</csymbol><ci id="S6.12.p2.12.m12.1.2.2.2.cmml" xref="S6.12.p2.12.m12.1.2.2.2">𝜎</ci><times id="S6.12.p2.12.m12.1.2.2.3.cmml" xref="S6.12.p2.12.m12.1.2.2.3"></times></apply><ci id="S6.12.p2.12.m12.1.1.cmml" xref="S6.12.p2.12.m12.1.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.12.m12.1c">\sigma_{*}(\mu)</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.12.m12.1d">italic_σ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_μ )</annotation></semantics></math>. <span class="ltx_text ltx_inline-block" id="S6.12.p2.13.1" style="width:0.0pt;"><math alttext="\sqcup" class="ltx_Math" display="inline" id="S6.12.p2.13.1.m1.1"><semantics id="S6.12.p2.13.1.m1.1a"><mo id="S6.12.p2.13.1.m1.1.1" xref="S6.12.p2.13.1.m1.1.1.cmml">⊔</mo><annotation-xml encoding="MathML-Content" id="S6.12.p2.13.1.m1.1b"><csymbol cd="latexml" id="S6.12.p2.13.1.m1.1.1.cmml" xref="S6.12.p2.13.1.m1.1.1">square-union</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.13.1.m1.1c">\sqcup</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.13.1.m1.1d">⊔</annotation></semantics></math></span><math alttext="\sqcap" class="ltx_Math" display="inline" id="S6.12.p2.14.m13.1"><semantics id="S6.12.p2.14.m13.1a"><mo id="S6.12.p2.14.m13.1.1" xref="S6.12.p2.14.m13.1.1.cmml">⊓</mo><annotation-xml encoding="MathML-Content" id="S6.12.p2.14.m13.1b"><csymbol cd="latexml" id="S6.12.p2.14.m13.1.1.cmml" xref="S6.12.p2.14.m13.1.1">square-intersection</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.12.p2.14.m13.1c">\sqcap</annotation><annotation encoding="application/x-llamapun" id="S6.12.p2.14.m13.1d">⊓</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S6.p5"> <p class="ltx_p" id="S6.p5.1">The following remark serves as continuation of the discussion started at the end of last section:</p> </div> <div class="ltx_theorem ltx_theorem_rem" id="S6.Thmthm8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm8.1.1.1">Remark 6.8</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm8.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm8.p1"> <p class="ltx_p" id="S6.Thmthm8.p1.7">(1) We would like to point out a subtlety in the last proof: Although the preimage set <math alttext="\sigma^{-1}(\cal O({\bf x}))" class="ltx_Math" display="inline" id="S6.Thmthm8.p1.1.m1.2"><semantics id="S6.Thmthm8.p1.1.m1.2a"><mrow id="S6.Thmthm8.p1.1.m1.2.2" xref="S6.Thmthm8.p1.1.m1.2.2.cmml"><msup id="S6.Thmthm8.p1.1.m1.2.2.3" xref="S6.Thmthm8.p1.1.m1.2.2.3.cmml"><mi id="S6.Thmthm8.p1.1.m1.2.2.3.2" xref="S6.Thmthm8.p1.1.m1.2.2.3.2.cmml">σ</mi><mrow id="S6.Thmthm8.p1.1.m1.2.2.3.3" xref="S6.Thmthm8.p1.1.m1.2.2.3.3.cmml"><mo id="S6.Thmthm8.p1.1.m1.2.2.3.3a" xref="S6.Thmthm8.p1.1.m1.2.2.3.3.cmml">−</mo><mn id="S6.Thmthm8.p1.1.m1.2.2.3.3.2" xref="S6.Thmthm8.p1.1.m1.2.2.3.3.2.cmml">1</mn></mrow></msup><mo id="S6.Thmthm8.p1.1.m1.2.2.2" xref="S6.Thmthm8.p1.1.m1.2.2.2.cmml">⁢</mo><mrow id="S6.Thmthm8.p1.1.m1.2.2.1.1" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.cmml"><mo id="S6.Thmthm8.p1.1.m1.2.2.1.1.2" stretchy="false" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.cmml">(</mo><mrow id="S6.Thmthm8.p1.1.m1.2.2.1.1.1" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm8.p1.1.m1.2.2.1.1.1.2" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.2.cmml">𝒪</mi><mo id="S6.Thmthm8.p1.1.m1.2.2.1.1.1.1" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmthm8.p1.1.m1.2.2.1.1.1.3.2" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.cmml"><mo id="S6.Thmthm8.p1.1.m1.2.2.1.1.1.3.2.1" stretchy="false" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.cmml">(</mo><mi id="S6.Thmthm8.p1.1.m1.1.1" xref="S6.Thmthm8.p1.1.m1.1.1.cmml">𝐱</mi><mo id="S6.Thmthm8.p1.1.m1.2.2.1.1.1.3.2.2" stretchy="false" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm8.p1.1.m1.2.2.1.1.3" stretchy="false" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p1.1.m1.2b"><apply id="S6.Thmthm8.p1.1.m1.2.2.cmml" xref="S6.Thmthm8.p1.1.m1.2.2"><times id="S6.Thmthm8.p1.1.m1.2.2.2.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.2"></times><apply id="S6.Thmthm8.p1.1.m1.2.2.3.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.3"><csymbol cd="ambiguous" id="S6.Thmthm8.p1.1.m1.2.2.3.1.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.3">superscript</csymbol><ci id="S6.Thmthm8.p1.1.m1.2.2.3.2.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.3.2">𝜎</ci><apply id="S6.Thmthm8.p1.1.m1.2.2.3.3.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.3.3"><minus id="S6.Thmthm8.p1.1.m1.2.2.3.3.1.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.3.3"></minus><cn id="S6.Thmthm8.p1.1.m1.2.2.3.3.2.cmml" type="integer" xref="S6.Thmthm8.p1.1.m1.2.2.3.3.2">1</cn></apply></apply><apply id="S6.Thmthm8.p1.1.m1.2.2.1.1.1.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.1.1"><times id="S6.Thmthm8.p1.1.m1.2.2.1.1.1.1.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.1"></times><ci id="S6.Thmthm8.p1.1.m1.2.2.1.1.1.2.cmml" xref="S6.Thmthm8.p1.1.m1.2.2.1.1.1.2">𝒪</ci><ci id="S6.Thmthm8.p1.1.m1.1.1.cmml" xref="S6.Thmthm8.p1.1.m1.1.1">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p1.1.m1.2c">\sigma^{-1}(\cal O({\bf x}))</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p1.1.m1.2d">italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( caligraphic_O ( bold_x ) )</annotation></semantics></math> of any periodic orbit <math alttext="\cal O({\bf x})" class="ltx_Math" display="inline" id="S6.Thmthm8.p1.2.m2.1"><semantics id="S6.Thmthm8.p1.2.m2.1a"><mrow id="S6.Thmthm8.p1.2.m2.1.2" xref="S6.Thmthm8.p1.2.m2.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm8.p1.2.m2.1.2.2" xref="S6.Thmthm8.p1.2.m2.1.2.2.cmml">𝒪</mi><mo id="S6.Thmthm8.p1.2.m2.1.2.1" xref="S6.Thmthm8.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S6.Thmthm8.p1.2.m2.1.2.3.2" xref="S6.Thmthm8.p1.2.m2.1.2.cmml"><mo id="S6.Thmthm8.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S6.Thmthm8.p1.2.m2.1.2.cmml">(</mo><mi id="S6.Thmthm8.p1.2.m2.1.1" xref="S6.Thmthm8.p1.2.m2.1.1.cmml">𝐱</mi><mo id="S6.Thmthm8.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S6.Thmthm8.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p1.2.m2.1b"><apply id="S6.Thmthm8.p1.2.m2.1.2.cmml" xref="S6.Thmthm8.p1.2.m2.1.2"><times id="S6.Thmthm8.p1.2.m2.1.2.1.cmml" xref="S6.Thmthm8.p1.2.m2.1.2.1"></times><ci id="S6.Thmthm8.p1.2.m2.1.2.2.cmml" xref="S6.Thmthm8.p1.2.m2.1.2.2">𝒪</ci><ci id="S6.Thmthm8.p1.2.m2.1.1.cmml" xref="S6.Thmthm8.p1.2.m2.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p1.2.m2.1c">\cal O({\bf x})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p1.2.m2.1d">caligraphic_O ( bold_x )</annotation></semantics></math> is a single periodic orbit, it is in general not true that the preimage set <math alttext="\sigma^{-1}({\bf x})" class="ltx_Math" display="inline" id="S6.Thmthm8.p1.3.m3.1"><semantics id="S6.Thmthm8.p1.3.m3.1a"><mrow id="S6.Thmthm8.p1.3.m3.1.2" xref="S6.Thmthm8.p1.3.m3.1.2.cmml"><msup id="S6.Thmthm8.p1.3.m3.1.2.2" xref="S6.Thmthm8.p1.3.m3.1.2.2.cmml"><mi id="S6.Thmthm8.p1.3.m3.1.2.2.2" xref="S6.Thmthm8.p1.3.m3.1.2.2.2.cmml">σ</mi><mrow id="S6.Thmthm8.p1.3.m3.1.2.2.3" xref="S6.Thmthm8.p1.3.m3.1.2.2.3.cmml"><mo id="S6.Thmthm8.p1.3.m3.1.2.2.3a" xref="S6.Thmthm8.p1.3.m3.1.2.2.3.cmml">−</mo><mn id="S6.Thmthm8.p1.3.m3.1.2.2.3.2" xref="S6.Thmthm8.p1.3.m3.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S6.Thmthm8.p1.3.m3.1.2.1" xref="S6.Thmthm8.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S6.Thmthm8.p1.3.m3.1.2.3.2" xref="S6.Thmthm8.p1.3.m3.1.2.cmml"><mo id="S6.Thmthm8.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S6.Thmthm8.p1.3.m3.1.2.cmml">(</mo><mi id="S6.Thmthm8.p1.3.m3.1.1" xref="S6.Thmthm8.p1.3.m3.1.1.cmml">𝐱</mi><mo id="S6.Thmthm8.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S6.Thmthm8.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p1.3.m3.1b"><apply id="S6.Thmthm8.p1.3.m3.1.2.cmml" xref="S6.Thmthm8.p1.3.m3.1.2"><times id="S6.Thmthm8.p1.3.m3.1.2.1.cmml" xref="S6.Thmthm8.p1.3.m3.1.2.1"></times><apply id="S6.Thmthm8.p1.3.m3.1.2.2.cmml" xref="S6.Thmthm8.p1.3.m3.1.2.2"><csymbol cd="ambiguous" id="S6.Thmthm8.p1.3.m3.1.2.2.1.cmml" xref="S6.Thmthm8.p1.3.m3.1.2.2">superscript</csymbol><ci id="S6.Thmthm8.p1.3.m3.1.2.2.2.cmml" xref="S6.Thmthm8.p1.3.m3.1.2.2.2">𝜎</ci><apply id="S6.Thmthm8.p1.3.m3.1.2.2.3.cmml" xref="S6.Thmthm8.p1.3.m3.1.2.2.3"><minus id="S6.Thmthm8.p1.3.m3.1.2.2.3.1.cmml" xref="S6.Thmthm8.p1.3.m3.1.2.2.3"></minus><cn id="S6.Thmthm8.p1.3.m3.1.2.2.3.2.cmml" type="integer" xref="S6.Thmthm8.p1.3.m3.1.2.2.3.2">1</cn></apply></apply><ci id="S6.Thmthm8.p1.3.m3.1.1.cmml" xref="S6.Thmthm8.p1.3.m3.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p1.3.m3.1c">\sigma^{-1}({\bf x})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p1.3.m3.1d">italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x )</annotation></semantics></math> consists of a single biinfinite word <math alttext="{\bf y}_{i}\in\sigma^{-1}(\cal O({\bf x}))" class="ltx_Math" display="inline" id="S6.Thmthm8.p1.4.m4.2"><semantics id="S6.Thmthm8.p1.4.m4.2a"><mrow id="S6.Thmthm8.p1.4.m4.2.2" xref="S6.Thmthm8.p1.4.m4.2.2.cmml"><msub id="S6.Thmthm8.p1.4.m4.2.2.3" xref="S6.Thmthm8.p1.4.m4.2.2.3.cmml"><mi id="S6.Thmthm8.p1.4.m4.2.2.3.2" xref="S6.Thmthm8.p1.4.m4.2.2.3.2.cmml">𝐲</mi><mi id="S6.Thmthm8.p1.4.m4.2.2.3.3" xref="S6.Thmthm8.p1.4.m4.2.2.3.3.cmml">i</mi></msub><mo id="S6.Thmthm8.p1.4.m4.2.2.2" xref="S6.Thmthm8.p1.4.m4.2.2.2.cmml">∈</mo><mrow id="S6.Thmthm8.p1.4.m4.2.2.1" xref="S6.Thmthm8.p1.4.m4.2.2.1.cmml"><msup id="S6.Thmthm8.p1.4.m4.2.2.1.3" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.cmml"><mi id="S6.Thmthm8.p1.4.m4.2.2.1.3.2" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.2.cmml">σ</mi><mrow id="S6.Thmthm8.p1.4.m4.2.2.1.3.3" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.3.cmml"><mo id="S6.Thmthm8.p1.4.m4.2.2.1.3.3a" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.3.cmml">−</mo><mn id="S6.Thmthm8.p1.4.m4.2.2.1.3.3.2" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S6.Thmthm8.p1.4.m4.2.2.1.2" xref="S6.Thmthm8.p1.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S6.Thmthm8.p1.4.m4.2.2.1.1.1" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.cmml"><mo id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.2" stretchy="false" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.2" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.2.cmml">𝒪</mi><mo id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.1" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.3.2" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.cmml"><mo id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.cmml">(</mo><mi id="S6.Thmthm8.p1.4.m4.1.1" xref="S6.Thmthm8.p1.4.m4.1.1.cmml">𝐱</mi><mo id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.3" stretchy="false" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p1.4.m4.2b"><apply id="S6.Thmthm8.p1.4.m4.2.2.cmml" xref="S6.Thmthm8.p1.4.m4.2.2"><in id="S6.Thmthm8.p1.4.m4.2.2.2.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.2"></in><apply id="S6.Thmthm8.p1.4.m4.2.2.3.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.3"><csymbol cd="ambiguous" id="S6.Thmthm8.p1.4.m4.2.2.3.1.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.3">subscript</csymbol><ci id="S6.Thmthm8.p1.4.m4.2.2.3.2.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.3.2">𝐲</ci><ci id="S6.Thmthm8.p1.4.m4.2.2.3.3.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.3.3">𝑖</ci></apply><apply id="S6.Thmthm8.p1.4.m4.2.2.1.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1"><times id="S6.Thmthm8.p1.4.m4.2.2.1.2.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.2"></times><apply id="S6.Thmthm8.p1.4.m4.2.2.1.3.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.3"><csymbol cd="ambiguous" id="S6.Thmthm8.p1.4.m4.2.2.1.3.1.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.3">superscript</csymbol><ci id="S6.Thmthm8.p1.4.m4.2.2.1.3.2.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.2">𝜎</ci><apply id="S6.Thmthm8.p1.4.m4.2.2.1.3.3.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.3"><minus id="S6.Thmthm8.p1.4.m4.2.2.1.3.3.1.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.3"></minus><cn id="S6.Thmthm8.p1.4.m4.2.2.1.3.3.2.cmml" type="integer" xref="S6.Thmthm8.p1.4.m4.2.2.1.3.3.2">1</cn></apply></apply><apply id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1"><times id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.1"></times><ci id="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="S6.Thmthm8.p1.4.m4.2.2.1.1.1.1.2">𝒪</ci><ci id="S6.Thmthm8.p1.4.m4.1.1.cmml" xref="S6.Thmthm8.p1.4.m4.1.1">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p1.4.m4.2c">{\bf y}_{i}\in\sigma^{-1}(\cal O({\bf x}))</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p1.4.m4.2d">bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( caligraphic_O ( bold_x ) )</annotation></semantics></math>. It is here that the missing assumption “shift-period preserving” materializes, compared to the case where we assume “<math alttext="\sigma" class="ltx_Math" display="inline" id="S6.Thmthm8.p1.5.m5.1"><semantics id="S6.Thmthm8.p1.5.m5.1a"><mi id="S6.Thmthm8.p1.5.m5.1.1" xref="S6.Thmthm8.p1.5.m5.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p1.5.m5.1b"><ci id="S6.Thmthm8.p1.5.m5.1.1.cmml" xref="S6.Thmthm8.p1.5.m5.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p1.5.m5.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p1.5.m5.1d">italic_σ</annotation></semantics></math> recognizable in <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm8.p1.6.m6.1"><semantics id="S6.Thmthm8.p1.6.m6.1a"><mi id="S6.Thmthm8.p1.6.m6.1.1" xref="S6.Thmthm8.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p1.6.m6.1b"><ci id="S6.Thmthm8.p1.6.m6.1.1.cmml" xref="S6.Thmthm8.p1.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p1.6.m6.1d">italic_X</annotation></semantics></math>” instead of our weaker assumption “<math alttext="\sigma" class="ltx_Math" display="inline" id="S6.Thmthm8.p1.7.m7.1"><semantics id="S6.Thmthm8.p1.7.m7.1a"><mi id="S6.Thmthm8.p1.7.m7.1.1" xref="S6.Thmthm8.p1.7.m7.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p1.7.m7.1b"><ci id="S6.Thmthm8.p1.7.m7.1.1.cmml" xref="S6.Thmthm8.p1.7.m7.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p1.7.m7.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p1.7.m7.1d">italic_σ</annotation></semantics></math> is shift-orbit injective”.</p> </div> <div class="ltx_para ltx_noindent" id="S6.Thmthm8.p2"> <p class="ltx_p" id="S6.Thmthm8.p2.3">(2) On the other hand, as pointed out at the end of last section, “shift-orbit injective” is a slightly stronger property than “recognizable for aperiodic points”, and as we have observed in the last section, the latter doesn’t suffice to deduce the injectivity of the push-forward map on <math alttext="\cal M(X)" class="ltx_Math" display="inline" id="S6.Thmthm8.p2.1.m1.1"><semantics id="S6.Thmthm8.p2.1.m1.1a"><mrow id="S6.Thmthm8.p2.1.m1.1.2" xref="S6.Thmthm8.p2.1.m1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm8.p2.1.m1.1.2.2" xref="S6.Thmthm8.p2.1.m1.1.2.2.cmml">ℳ</mi><mo id="S6.Thmthm8.p2.1.m1.1.2.1" xref="S6.Thmthm8.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S6.Thmthm8.p2.1.m1.1.2.3.2" xref="S6.Thmthm8.p2.1.m1.1.2.cmml"><mo id="S6.Thmthm8.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S6.Thmthm8.p2.1.m1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm8.p2.1.m1.1.1" xref="S6.Thmthm8.p2.1.m1.1.1.cmml">𝒳</mi><mo id="S6.Thmthm8.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S6.Thmthm8.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p2.1.m1.1b"><apply id="S6.Thmthm8.p2.1.m1.1.2.cmml" xref="S6.Thmthm8.p2.1.m1.1.2"><times id="S6.Thmthm8.p2.1.m1.1.2.1.cmml" xref="S6.Thmthm8.p2.1.m1.1.2.1"></times><ci id="S6.Thmthm8.p2.1.m1.1.2.2.cmml" xref="S6.Thmthm8.p2.1.m1.1.2.2">ℳ</ci><ci id="S6.Thmthm8.p2.1.m1.1.1.cmml" xref="S6.Thmthm8.p2.1.m1.1.1">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p2.1.m1.1c">\cal M(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p2.1.m1.1d">caligraphic_M ( caligraphic_X )</annotation></semantics></math>. In fact, this injectivity fails in general already for non-atomic measures on the given subshift <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm8.p2.2.m2.1"><semantics id="S6.Thmthm8.p2.2.m2.1a"><mi id="S6.Thmthm8.p2.2.m2.1.1" xref="S6.Thmthm8.p2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p2.2.m2.1b"><ci id="S6.Thmthm8.p2.2.m2.1.1.cmml" xref="S6.Thmthm8.p2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p2.2.m2.1d">italic_X</annotation></semantics></math>, as can be seen easily from the morphism which sends all letters of an alphabet of size <math alttext="\geq 2" class="ltx_Math" display="inline" id="S6.Thmthm8.p2.3.m3.1"><semantics id="S6.Thmthm8.p2.3.m3.1a"><mrow id="S6.Thmthm8.p2.3.m3.1.1" xref="S6.Thmthm8.p2.3.m3.1.1.cmml"><mi id="S6.Thmthm8.p2.3.m3.1.1.2" xref="S6.Thmthm8.p2.3.m3.1.1.2.cmml"></mi><mo id="S6.Thmthm8.p2.3.m3.1.1.1" xref="S6.Thmthm8.p2.3.m3.1.1.1.cmml">≥</mo><mn id="S6.Thmthm8.p2.3.m3.1.1.3" xref="S6.Thmthm8.p2.3.m3.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm8.p2.3.m3.1b"><apply id="S6.Thmthm8.p2.3.m3.1.1.cmml" xref="S6.Thmthm8.p2.3.m3.1.1"><geq id="S6.Thmthm8.p2.3.m3.1.1.1.cmml" xref="S6.Thmthm8.p2.3.m3.1.1.1"></geq><csymbol cd="latexml" id="S6.Thmthm8.p2.3.m3.1.1.2.cmml" xref="S6.Thmthm8.p2.3.m3.1.1.2">absent</csymbol><cn id="S6.Thmthm8.p2.3.m3.1.1.3.cmml" type="integer" xref="S6.Thmthm8.p2.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm8.p2.3.m3.1c">\geq 2</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm8.p2.3.m3.1d">≥ 2</annotation></semantics></math> to a single letter of the image alphabet; any such morphism is a forteriori recognizable for aperiodic points in the full shift (see Warning <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm8" title="Warning 5.8. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.8</span></a>).</p> </div> </div> <div class="ltx_para" id="S6.p6"> <p class="ltx_p" id="S6.p6.4">We terminate this section by considering some concrete examples how Theorem <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S6.Thmthm7" title="Theorem 6.7. ‣ 6. The injectivity of the measure transfer for letter-to-letter morphisms ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">6.7</span></a> or rather Theorem <a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#S5.Thmthm6" title="Theorem 5.6. ‣ 5. Shift-orbit injectivity and related notions ‣ The measure transfer for subshifts induced by a morphism of free monoids"><span class="ltx_text ltx_ref_tag">5.6</span></a> can be put to use. An immediate application, for <math alttext="\sigma,X" class="ltx_Math" display="inline" id="S6.p6.1.m1.2"><semantics id="S6.p6.1.m1.2a"><mrow id="S6.p6.1.m1.2.3.2" xref="S6.p6.1.m1.2.3.1.cmml"><mi id="S6.p6.1.m1.1.1" xref="S6.p6.1.m1.1.1.cmml">σ</mi><mo id="S6.p6.1.m1.2.3.2.1" xref="S6.p6.1.m1.2.3.1.cmml">,</mo><mi id="S6.p6.1.m1.2.2" xref="S6.p6.1.m1.2.2.cmml">X</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.1.m1.2b"><list id="S6.p6.1.m1.2.3.1.cmml" xref="S6.p6.1.m1.2.3.2"><ci id="S6.p6.1.m1.1.1.cmml" xref="S6.p6.1.m1.1.1">𝜎</ci><ci id="S6.p6.1.m1.2.2.cmml" xref="S6.p6.1.m1.2.2">𝑋</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.1.m1.2c">\sigma,X</annotation><annotation encoding="application/x-llamapun" id="S6.p6.1.m1.2d">italic_σ , italic_X</annotation></semantics></math> and <math alttext="Y=\sigma(X)" class="ltx_Math" display="inline" id="S6.p6.2.m2.1"><semantics id="S6.p6.2.m2.1a"><mrow id="S6.p6.2.m2.1.2" xref="S6.p6.2.m2.1.2.cmml"><mi id="S6.p6.2.m2.1.2.2" xref="S6.p6.2.m2.1.2.2.cmml">Y</mi><mo id="S6.p6.2.m2.1.2.1" xref="S6.p6.2.m2.1.2.1.cmml">=</mo><mrow id="S6.p6.2.m2.1.2.3" xref="S6.p6.2.m2.1.2.3.cmml"><mi id="S6.p6.2.m2.1.2.3.2" xref="S6.p6.2.m2.1.2.3.2.cmml">σ</mi><mo id="S6.p6.2.m2.1.2.3.1" xref="S6.p6.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S6.p6.2.m2.1.2.3.3.2" xref="S6.p6.2.m2.1.2.3.cmml"><mo id="S6.p6.2.m2.1.2.3.3.2.1" stretchy="false" xref="S6.p6.2.m2.1.2.3.cmml">(</mo><mi id="S6.p6.2.m2.1.1" xref="S6.p6.2.m2.1.1.cmml">X</mi><mo id="S6.p6.2.m2.1.2.3.3.2.2" stretchy="false" xref="S6.p6.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.p6.2.m2.1b"><apply id="S6.p6.2.m2.1.2.cmml" xref="S6.p6.2.m2.1.2"><eq id="S6.p6.2.m2.1.2.1.cmml" xref="S6.p6.2.m2.1.2.1"></eq><ci id="S6.p6.2.m2.1.2.2.cmml" xref="S6.p6.2.m2.1.2.2">𝑌</ci><apply id="S6.p6.2.m2.1.2.3.cmml" xref="S6.p6.2.m2.1.2.3"><times id="S6.p6.2.m2.1.2.3.1.cmml" xref="S6.p6.2.m2.1.2.3.1"></times><ci id="S6.p6.2.m2.1.2.3.2.cmml" xref="S6.p6.2.m2.1.2.3.2">𝜎</ci><ci id="S6.p6.2.m2.1.1.cmml" xref="S6.p6.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.2.m2.1c">Y=\sigma(X)</annotation><annotation encoding="application/x-llamapun" id="S6.p6.2.m2.1d">italic_Y = italic_σ ( italic_X )</annotation></semantics></math> as above, would be to deduce from the assumption that <math alttext="Y" class="ltx_Math" display="inline" id="S6.p6.3.m3.1"><semantics id="S6.p6.3.m3.1a"><mi id="S6.p6.3.m3.1.1" xref="S6.p6.3.m3.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S6.p6.3.m3.1b"><ci id="S6.p6.3.m3.1.1.cmml" xref="S6.p6.3.m3.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.3.m3.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S6.p6.3.m3.1d">italic_Y</annotation></semantics></math> is uniquely ergodic the conclusion that <math alttext="X" class="ltx_Math" display="inline" id="S6.p6.4.m4.1"><semantics id="S6.p6.4.m4.1a"><mi id="S6.p6.4.m4.1.1" xref="S6.p6.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.p6.4.m4.1b"><ci id="S6.p6.4.m4.1.1.cmml" xref="S6.p6.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.p6.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.p6.4.m4.1d">italic_X</annotation></semantics></math> is also uniquely ergodic. Below we exhibit something slightly more elaborate:</p> </div> <div class="ltx_theorem ltx_theorem_example" id="S6.Thmthm9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmthm9.1.1.1">Example 6.9</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmthm9.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmthm9.p1"> <p class="ltx_p" id="S6.Thmthm9.p1.5">(1) Let us first consider for <math alttext="\cal A=\{a,b\}" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.1.m1.2"><semantics id="S6.Thmthm9.p1.1.m1.2a"><mrow id="S6.Thmthm9.p1.1.m1.2.3" xref="S6.Thmthm9.p1.1.m1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.1.m1.2.3.2" xref="S6.Thmthm9.p1.1.m1.2.3.2.cmml">𝒜</mi><mo id="S6.Thmthm9.p1.1.m1.2.3.1" xref="S6.Thmthm9.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S6.Thmthm9.p1.1.m1.2.3.3.2" xref="S6.Thmthm9.p1.1.m1.2.3.3.1.cmml"><mo id="S6.Thmthm9.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S6.Thmthm9.p1.1.m1.2.3.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.1.m1.1.1" xref="S6.Thmthm9.p1.1.m1.1.1.cmml">𝒶</mi><mo id="S6.Thmthm9.p1.1.m1.2.3.3.2.2" xref="S6.Thmthm9.p1.1.m1.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.1.m1.2.2" xref="S6.Thmthm9.p1.1.m1.2.2.cmml">𝒷</mi><mo id="S6.Thmthm9.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S6.Thmthm9.p1.1.m1.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.1.m1.2b"><apply id="S6.Thmthm9.p1.1.m1.2.3.cmml" xref="S6.Thmthm9.p1.1.m1.2.3"><eq id="S6.Thmthm9.p1.1.m1.2.3.1.cmml" xref="S6.Thmthm9.p1.1.m1.2.3.1"></eq><ci id="S6.Thmthm9.p1.1.m1.2.3.2.cmml" xref="S6.Thmthm9.p1.1.m1.2.3.2">𝒜</ci><set id="S6.Thmthm9.p1.1.m1.2.3.3.1.cmml" xref="S6.Thmthm9.p1.1.m1.2.3.3.2"><ci id="S6.Thmthm9.p1.1.m1.1.1.cmml" xref="S6.Thmthm9.p1.1.m1.1.1">𝒶</ci><ci id="S6.Thmthm9.p1.1.m1.2.2.cmml" xref="S6.Thmthm9.p1.1.m1.2.2">𝒷</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.1.m1.2c">\cal A=\{a,b\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.1.m1.2d">caligraphic_A = { caligraphic_a , caligraphic_b }</annotation></semantics></math> and <math alttext="\cal B=\{c,d\}" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.2.m2.2"><semantics id="S6.Thmthm9.p1.2.m2.2a"><mrow id="S6.Thmthm9.p1.2.m2.2.3" xref="S6.Thmthm9.p1.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.2.m2.2.3.2" xref="S6.Thmthm9.p1.2.m2.2.3.2.cmml">ℬ</mi><mo id="S6.Thmthm9.p1.2.m2.2.3.1" xref="S6.Thmthm9.p1.2.m2.2.3.1.cmml">=</mo><mrow id="S6.Thmthm9.p1.2.m2.2.3.3.2" xref="S6.Thmthm9.p1.2.m2.2.3.3.1.cmml"><mo id="S6.Thmthm9.p1.2.m2.2.3.3.2.1" stretchy="false" xref="S6.Thmthm9.p1.2.m2.2.3.3.1.cmml">{</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.2.m2.1.1" xref="S6.Thmthm9.p1.2.m2.1.1.cmml">𝒸</mi><mo id="S6.Thmthm9.p1.2.m2.2.3.3.2.2" xref="S6.Thmthm9.p1.2.m2.2.3.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.2.m2.2.2" xref="S6.Thmthm9.p1.2.m2.2.2.cmml">𝒹</mi><mo id="S6.Thmthm9.p1.2.m2.2.3.3.2.3" stretchy="false" xref="S6.Thmthm9.p1.2.m2.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.2.m2.2b"><apply id="S6.Thmthm9.p1.2.m2.2.3.cmml" xref="S6.Thmthm9.p1.2.m2.2.3"><eq id="S6.Thmthm9.p1.2.m2.2.3.1.cmml" xref="S6.Thmthm9.p1.2.m2.2.3.1"></eq><ci id="S6.Thmthm9.p1.2.m2.2.3.2.cmml" xref="S6.Thmthm9.p1.2.m2.2.3.2">ℬ</ci><set id="S6.Thmthm9.p1.2.m2.2.3.3.1.cmml" xref="S6.Thmthm9.p1.2.m2.2.3.3.2"><ci id="S6.Thmthm9.p1.2.m2.1.1.cmml" xref="S6.Thmthm9.p1.2.m2.1.1">𝒸</ci><ci id="S6.Thmthm9.p1.2.m2.2.2.cmml" xref="S6.Thmthm9.p1.2.m2.2.2">𝒹</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.2.m2.2c">\cal B=\{c,d\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.2.m2.2d">caligraphic_B = { caligraphic_c , caligraphic_d }</annotation></semantics></math> the morphisms <math alttext="\sigma_{i}:\cal A^{*}\to\cal B^{*}" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.3.m3.1"><semantics id="S6.Thmthm9.p1.3.m3.1a"><mrow id="S6.Thmthm9.p1.3.m3.1.1" xref="S6.Thmthm9.p1.3.m3.1.1.cmml"><msub id="S6.Thmthm9.p1.3.m3.1.1.2" xref="S6.Thmthm9.p1.3.m3.1.1.2.cmml"><mi id="S6.Thmthm9.p1.3.m3.1.1.2.2" xref="S6.Thmthm9.p1.3.m3.1.1.2.2.cmml">σ</mi><mi id="S6.Thmthm9.p1.3.m3.1.1.2.3" xref="S6.Thmthm9.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S6.Thmthm9.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.Thmthm9.p1.3.m3.1.1.1.cmml">:</mo><mrow id="S6.Thmthm9.p1.3.m3.1.1.3" xref="S6.Thmthm9.p1.3.m3.1.1.3.cmml"><msup id="S6.Thmthm9.p1.3.m3.1.1.3.2" xref="S6.Thmthm9.p1.3.m3.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.3.m3.1.1.3.2.2" xref="S6.Thmthm9.p1.3.m3.1.1.3.2.2.cmml">𝒜</mi><mo id="S6.Thmthm9.p1.3.m3.1.1.3.2.3" xref="S6.Thmthm9.p1.3.m3.1.1.3.2.3.cmml">∗</mo></msup><mo id="S6.Thmthm9.p1.3.m3.1.1.3.1" stretchy="false" xref="S6.Thmthm9.p1.3.m3.1.1.3.1.cmml">→</mo><msup id="S6.Thmthm9.p1.3.m3.1.1.3.3" xref="S6.Thmthm9.p1.3.m3.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.3.m3.1.1.3.3.2" xref="S6.Thmthm9.p1.3.m3.1.1.3.3.2.cmml">ℬ</mi><mo id="S6.Thmthm9.p1.3.m3.1.1.3.3.3" xref="S6.Thmthm9.p1.3.m3.1.1.3.3.3.cmml">∗</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.3.m3.1b"><apply id="S6.Thmthm9.p1.3.m3.1.1.cmml" xref="S6.Thmthm9.p1.3.m3.1.1"><ci id="S6.Thmthm9.p1.3.m3.1.1.1.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.1">:</ci><apply id="S6.Thmthm9.p1.3.m3.1.1.2.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.3.m3.1.1.2.1.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.2">subscript</csymbol><ci id="S6.Thmthm9.p1.3.m3.1.1.2.2.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.2.2">𝜎</ci><ci id="S6.Thmthm9.p1.3.m3.1.1.2.3.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S6.Thmthm9.p1.3.m3.1.1.3.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3"><ci id="S6.Thmthm9.p1.3.m3.1.1.3.1.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.1">→</ci><apply id="S6.Thmthm9.p1.3.m3.1.1.3.2.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.3.m3.1.1.3.2.1.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.2">superscript</csymbol><ci id="S6.Thmthm9.p1.3.m3.1.1.3.2.2.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.2.2">𝒜</ci><times id="S6.Thmthm9.p1.3.m3.1.1.3.2.3.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.2.3"></times></apply><apply id="S6.Thmthm9.p1.3.m3.1.1.3.3.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.3.m3.1.1.3.3.1.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.3">superscript</csymbol><ci id="S6.Thmthm9.p1.3.m3.1.1.3.3.2.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.3.2">ℬ</ci><times id="S6.Thmthm9.p1.3.m3.1.1.3.3.3.cmml" xref="S6.Thmthm9.p1.3.m3.1.1.3.3.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.3.m3.1c">\sigma_{i}:\cal A^{*}\to\cal B^{*}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT : caligraphic_A start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT → caligraphic_B start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> (for <math alttext="i=1" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.4.m4.1"><semantics id="S6.Thmthm9.p1.4.m4.1a"><mrow id="S6.Thmthm9.p1.4.m4.1.1" xref="S6.Thmthm9.p1.4.m4.1.1.cmml"><mi id="S6.Thmthm9.p1.4.m4.1.1.2" xref="S6.Thmthm9.p1.4.m4.1.1.2.cmml">i</mi><mo id="S6.Thmthm9.p1.4.m4.1.1.1" xref="S6.Thmthm9.p1.4.m4.1.1.1.cmml">=</mo><mn id="S6.Thmthm9.p1.4.m4.1.1.3" xref="S6.Thmthm9.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.4.m4.1b"><apply id="S6.Thmthm9.p1.4.m4.1.1.cmml" xref="S6.Thmthm9.p1.4.m4.1.1"><eq id="S6.Thmthm9.p1.4.m4.1.1.1.cmml" xref="S6.Thmthm9.p1.4.m4.1.1.1"></eq><ci id="S6.Thmthm9.p1.4.m4.1.1.2.cmml" xref="S6.Thmthm9.p1.4.m4.1.1.2">𝑖</ci><cn id="S6.Thmthm9.p1.4.m4.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.4.m4.1c">i=1</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.4.m4.1d">italic_i = 1</annotation></semantics></math> and <math alttext="i=2" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.5.m5.1"><semantics id="S6.Thmthm9.p1.5.m5.1a"><mrow id="S6.Thmthm9.p1.5.m5.1.1" xref="S6.Thmthm9.p1.5.m5.1.1.cmml"><mi id="S6.Thmthm9.p1.5.m5.1.1.2" xref="S6.Thmthm9.p1.5.m5.1.1.2.cmml">i</mi><mo id="S6.Thmthm9.p1.5.m5.1.1.1" xref="S6.Thmthm9.p1.5.m5.1.1.1.cmml">=</mo><mn id="S6.Thmthm9.p1.5.m5.1.1.3" xref="S6.Thmthm9.p1.5.m5.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.5.m5.1b"><apply id="S6.Thmthm9.p1.5.m5.1.1.cmml" xref="S6.Thmthm9.p1.5.m5.1.1"><eq id="S6.Thmthm9.p1.5.m5.1.1.1.cmml" xref="S6.Thmthm9.p1.5.m5.1.1.1"></eq><ci id="S6.Thmthm9.p1.5.m5.1.1.2.cmml" xref="S6.Thmthm9.p1.5.m5.1.1.2">𝑖</ci><cn id="S6.Thmthm9.p1.5.m5.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p1.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.5.m5.1c">i=2</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.5.m5.1d">italic_i = 2</annotation></semantics></math>) given by</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma_{1}(a)=c^{2}\,,\,\,\sigma_{1}(b)=d\quad\text{and}\quad\sigma_{2}(a)=cdc% \,,\,\,\sigma_{2}(b)=dcd\,." class="ltx_Math" display="block" id="S6.Ex9.m1.7"><semantics id="S6.Ex9.m1.7a"><mrow id="S6.Ex9.m1.7.7.1"><mrow id="S6.Ex9.m1.7.7.1.1.2" xref="S6.Ex9.m1.7.7.1.1.3.cmml"><mrow id="S6.Ex9.m1.7.7.1.1.1.1" xref="S6.Ex9.m1.7.7.1.1.1.1.cmml"><mrow id="S6.Ex9.m1.7.7.1.1.1.1.2" xref="S6.Ex9.m1.7.7.1.1.1.1.2.cmml"><msub id="S6.Ex9.m1.7.7.1.1.1.1.2.2" xref="S6.Ex9.m1.7.7.1.1.1.1.2.2.cmml"><mi id="S6.Ex9.m1.7.7.1.1.1.1.2.2.2" xref="S6.Ex9.m1.7.7.1.1.1.1.2.2.2.cmml">σ</mi><mn id="S6.Ex9.m1.7.7.1.1.1.1.2.2.3" xref="S6.Ex9.m1.7.7.1.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S6.Ex9.m1.7.7.1.1.1.1.2.1" xref="S6.Ex9.m1.7.7.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S6.Ex9.m1.7.7.1.1.1.1.2.3.2" xref="S6.Ex9.m1.7.7.1.1.1.1.2.cmml"><mo id="S6.Ex9.m1.7.7.1.1.1.1.2.3.2.1" stretchy="false" xref="S6.Ex9.m1.7.7.1.1.1.1.2.cmml">(</mo><mi id="S6.Ex9.m1.1.1" xref="S6.Ex9.m1.1.1.cmml">a</mi><mo id="S6.Ex9.m1.7.7.1.1.1.1.2.3.2.2" stretchy="false" xref="S6.Ex9.m1.7.7.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S6.Ex9.m1.7.7.1.1.1.1.1" 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id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.1.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.1"></times><apply id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.2.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.2.1.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.2">subscript</csymbol><ci id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.2.2.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.2.2">𝜎</ci><cn id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.2.3.cmml" type="integer" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.2.2.3">2</cn></apply><ci id="S6.Ex9.m1.4.4.cmml" xref="S6.Ex9.m1.4.4">𝑏</ci></apply><apply id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3"><times id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.1.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.1"></times><ci id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.2.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.2">𝑑</ci><ci id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.3.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.3">𝑐</ci><ci id="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.4.cmml" xref="S6.Ex9.m1.7.7.1.1.2.2.2.2.2.2.3.4">𝑑</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex9.m1.7c">\sigma_{1}(a)=c^{2}\,,\,\,\sigma_{1}(b)=d\quad\text{and}\quad\sigma_{2}(a)=cdc% \,,\,\,\sigma_{2}(b)=dcd\,.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex9.m1.7d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a ) = italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_b ) = italic_d and italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_a ) = italic_c italic_d italic_c , italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_b ) = italic_d italic_c italic_d .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmthm9.p1.12">Neither of these morphisms is recognizable in the full shift, since <math alttext="c^{\pm\infty}" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.6.m1.1"><semantics id="S6.Thmthm9.p1.6.m1.1a"><msup id="S6.Thmthm9.p1.6.m1.1.1" xref="S6.Thmthm9.p1.6.m1.1.1.cmml"><mi id="S6.Thmthm9.p1.6.m1.1.1.2" xref="S6.Thmthm9.p1.6.m1.1.1.2.cmml">c</mi><mrow id="S6.Thmthm9.p1.6.m1.1.1.3" xref="S6.Thmthm9.p1.6.m1.1.1.3.cmml"><mo id="S6.Thmthm9.p1.6.m1.1.1.3a" xref="S6.Thmthm9.p1.6.m1.1.1.3.cmml">±</mo><mi id="S6.Thmthm9.p1.6.m1.1.1.3.2" mathvariant="normal" xref="S6.Thmthm9.p1.6.m1.1.1.3.2.cmml">∞</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.6.m1.1b"><apply id="S6.Thmthm9.p1.6.m1.1.1.cmml" xref="S6.Thmthm9.p1.6.m1.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.6.m1.1.1.1.cmml" xref="S6.Thmthm9.p1.6.m1.1.1">superscript</csymbol><ci id="S6.Thmthm9.p1.6.m1.1.1.2.cmml" xref="S6.Thmthm9.p1.6.m1.1.1.2">𝑐</ci><apply id="S6.Thmthm9.p1.6.m1.1.1.3.cmml" xref="S6.Thmthm9.p1.6.m1.1.1.3"><csymbol cd="latexml" id="S6.Thmthm9.p1.6.m1.1.1.3.1.cmml" xref="S6.Thmthm9.p1.6.m1.1.1.3">plus-or-minus</csymbol><infinity id="S6.Thmthm9.p1.6.m1.1.1.3.2.cmml" xref="S6.Thmthm9.p1.6.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.6.m1.1c">c^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.6.m1.1d">italic_c start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> admits two different desubstitutions with respect to <math alttext="\sigma_{1}\," class="ltx_Math" display="inline" id="S6.Thmthm9.p1.7.m2.1"><semantics id="S6.Thmthm9.p1.7.m2.1a"><msub id="S6.Thmthm9.p1.7.m2.1.1" xref="S6.Thmthm9.p1.7.m2.1.1.cmml"><mi id="S6.Thmthm9.p1.7.m2.1.1.2" xref="S6.Thmthm9.p1.7.m2.1.1.2.cmml">σ</mi><mn id="S6.Thmthm9.p1.7.m2.1.1.3" xref="S6.Thmthm9.p1.7.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.7.m2.1b"><apply id="S6.Thmthm9.p1.7.m2.1.1.cmml" xref="S6.Thmthm9.p1.7.m2.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.7.m2.1.1.1.cmml" xref="S6.Thmthm9.p1.7.m2.1.1">subscript</csymbol><ci id="S6.Thmthm9.p1.7.m2.1.1.2.cmml" xref="S6.Thmthm9.p1.7.m2.1.1.2">𝜎</ci><cn id="S6.Thmthm9.p1.7.m2.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p1.7.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.7.m2.1c">\sigma_{1}\,</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.7.m2.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="(cd)^{\pm\infty}" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.8.m3.1"><semantics id="S6.Thmthm9.p1.8.m3.1a"><msup id="S6.Thmthm9.p1.8.m3.1.1" xref="S6.Thmthm9.p1.8.m3.1.1.cmml"><mrow id="S6.Thmthm9.p1.8.m3.1.1.1.1" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.cmml"><mo id="S6.Thmthm9.p1.8.m3.1.1.1.1.2" stretchy="false" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmthm9.p1.8.m3.1.1.1.1.1" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.cmml"><mi id="S6.Thmthm9.p1.8.m3.1.1.1.1.1.2" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.2.cmml">c</mi><mo id="S6.Thmthm9.p1.8.m3.1.1.1.1.1.1" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S6.Thmthm9.p1.8.m3.1.1.1.1.1.3" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.3.cmml">d</mi></mrow><mo id="S6.Thmthm9.p1.8.m3.1.1.1.1.3" stretchy="false" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S6.Thmthm9.p1.8.m3.1.1.3" xref="S6.Thmthm9.p1.8.m3.1.1.3.cmml"><mo id="S6.Thmthm9.p1.8.m3.1.1.3a" xref="S6.Thmthm9.p1.8.m3.1.1.3.cmml">±</mo><mi id="S6.Thmthm9.p1.8.m3.1.1.3.2" mathvariant="normal" xref="S6.Thmthm9.p1.8.m3.1.1.3.2.cmml">∞</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.8.m3.1b"><apply id="S6.Thmthm9.p1.8.m3.1.1.cmml" xref="S6.Thmthm9.p1.8.m3.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.8.m3.1.1.2.cmml" xref="S6.Thmthm9.p1.8.m3.1.1">superscript</csymbol><apply id="S6.Thmthm9.p1.8.m3.1.1.1.1.1.cmml" xref="S6.Thmthm9.p1.8.m3.1.1.1.1"><times id="S6.Thmthm9.p1.8.m3.1.1.1.1.1.1.cmml" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.1"></times><ci id="S6.Thmthm9.p1.8.m3.1.1.1.1.1.2.cmml" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.2">𝑐</ci><ci id="S6.Thmthm9.p1.8.m3.1.1.1.1.1.3.cmml" xref="S6.Thmthm9.p1.8.m3.1.1.1.1.1.3">𝑑</ci></apply><apply id="S6.Thmthm9.p1.8.m3.1.1.3.cmml" xref="S6.Thmthm9.p1.8.m3.1.1.3"><csymbol cd="latexml" id="S6.Thmthm9.p1.8.m3.1.1.3.1.cmml" xref="S6.Thmthm9.p1.8.m3.1.1.3">plus-or-minus</csymbol><infinity id="S6.Thmthm9.p1.8.m3.1.1.3.2.cmml" xref="S6.Thmthm9.p1.8.m3.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.8.m3.1c">(cd)^{\pm\infty}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.8.m3.1d">( italic_c italic_d ) start_POSTSUPERSCRIPT ± ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> admits three different desubstitutions with respect to <math alttext="\sigma_{2}\," class="ltx_Math" display="inline" id="S6.Thmthm9.p1.9.m4.1"><semantics id="S6.Thmthm9.p1.9.m4.1a"><msub id="S6.Thmthm9.p1.9.m4.1.1" xref="S6.Thmthm9.p1.9.m4.1.1.cmml"><mi id="S6.Thmthm9.p1.9.m4.1.1.2" xref="S6.Thmthm9.p1.9.m4.1.1.2.cmml">σ</mi><mn id="S6.Thmthm9.p1.9.m4.1.1.3" xref="S6.Thmthm9.p1.9.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.9.m4.1b"><apply id="S6.Thmthm9.p1.9.m4.1.1.cmml" xref="S6.Thmthm9.p1.9.m4.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.9.m4.1.1.1.cmml" xref="S6.Thmthm9.p1.9.m4.1.1">subscript</csymbol><ci id="S6.Thmthm9.p1.9.m4.1.1.2.cmml" xref="S6.Thmthm9.p1.9.m4.1.1.2">𝜎</ci><cn id="S6.Thmthm9.p1.9.m4.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p1.9.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.9.m4.1c">\sigma_{2}\,</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.9.m4.1d">italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Nevertheless, it is easy to verify by elementary arguments that both, <math alttext="\sigma_{1}" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.10.m5.1"><semantics id="S6.Thmthm9.p1.10.m5.1a"><msub id="S6.Thmthm9.p1.10.m5.1.1" xref="S6.Thmthm9.p1.10.m5.1.1.cmml"><mi id="S6.Thmthm9.p1.10.m5.1.1.2" xref="S6.Thmthm9.p1.10.m5.1.1.2.cmml">σ</mi><mn id="S6.Thmthm9.p1.10.m5.1.1.3" xref="S6.Thmthm9.p1.10.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.10.m5.1b"><apply id="S6.Thmthm9.p1.10.m5.1.1.cmml" xref="S6.Thmthm9.p1.10.m5.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.10.m5.1.1.1.cmml" xref="S6.Thmthm9.p1.10.m5.1.1">subscript</csymbol><ci id="S6.Thmthm9.p1.10.m5.1.1.2.cmml" xref="S6.Thmthm9.p1.10.m5.1.1.2">𝜎</ci><cn id="S6.Thmthm9.p1.10.m5.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p1.10.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.10.m5.1c">\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.10.m5.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{2}\," class="ltx_Math" display="inline" id="S6.Thmthm9.p1.11.m6.1"><semantics id="S6.Thmthm9.p1.11.m6.1a"><msub id="S6.Thmthm9.p1.11.m6.1.1" xref="S6.Thmthm9.p1.11.m6.1.1.cmml"><mi id="S6.Thmthm9.p1.11.m6.1.1.2" xref="S6.Thmthm9.p1.11.m6.1.1.2.cmml">σ</mi><mn id="S6.Thmthm9.p1.11.m6.1.1.3" xref="S6.Thmthm9.p1.11.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.11.m6.1b"><apply id="S6.Thmthm9.p1.11.m6.1.1.cmml" xref="S6.Thmthm9.p1.11.m6.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.11.m6.1.1.1.cmml" xref="S6.Thmthm9.p1.11.m6.1.1">subscript</csymbol><ci id="S6.Thmthm9.p1.11.m6.1.1.2.cmml" xref="S6.Thmthm9.p1.11.m6.1.1.2">𝜎</ci><cn id="S6.Thmthm9.p1.11.m6.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p1.11.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.11.m6.1c">\sigma_{2}\,</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.11.m6.1d">italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, are 1-1 on the set of shift-orbits of <math alttext="\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.Thmthm9.p1.12.m7.1"><semantics id="S6.Thmthm9.p1.12.m7.1a"><msup id="S6.Thmthm9.p1.12.m7.1.1" xref="S6.Thmthm9.p1.12.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p1.12.m7.1.1.2" xref="S6.Thmthm9.p1.12.m7.1.1.2.cmml">𝒜</mi><mi id="S6.Thmthm9.p1.12.m7.1.1.3" xref="S6.Thmthm9.p1.12.m7.1.1.3.cmml">ℤ</mi></msup><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p1.12.m7.1b"><apply id="S6.Thmthm9.p1.12.m7.1.1.cmml" xref="S6.Thmthm9.p1.12.m7.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p1.12.m7.1.1.1.cmml" xref="S6.Thmthm9.p1.12.m7.1.1">superscript</csymbol><ci id="S6.Thmthm9.p1.12.m7.1.1.2.cmml" xref="S6.Thmthm9.p1.12.m7.1.1.2">𝒜</ci><ci id="S6.Thmthm9.p1.12.m7.1.1.3.cmml" xref="S6.Thmthm9.p1.12.m7.1.1.3">ℤ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p1.12.m7.1c">\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p1.12.m7.1d">caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para ltx_noindent" id="S6.Thmthm9.p2"> <p class="ltx_p" id="S6.Thmthm9.p2.12">(2) Consider now any subshift <math alttext="X\subseteq\cal A^{\mathbb{Z}}" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.1.m1.1"><semantics id="S6.Thmthm9.p2.1.m1.1a"><mrow id="S6.Thmthm9.p2.1.m1.1.1" xref="S6.Thmthm9.p2.1.m1.1.1.cmml"><mi id="S6.Thmthm9.p2.1.m1.1.1.2" xref="S6.Thmthm9.p2.1.m1.1.1.2.cmml">X</mi><mo id="S6.Thmthm9.p2.1.m1.1.1.1" xref="S6.Thmthm9.p2.1.m1.1.1.1.cmml">⊆</mo><msup id="S6.Thmthm9.p2.1.m1.1.1.3" xref="S6.Thmthm9.p2.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Thmthm9.p2.1.m1.1.1.3.2" xref="S6.Thmthm9.p2.1.m1.1.1.3.2.cmml">𝒜</mi><mi id="S6.Thmthm9.p2.1.m1.1.1.3.3" xref="S6.Thmthm9.p2.1.m1.1.1.3.3.cmml">ℤ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.1.m1.1b"><apply id="S6.Thmthm9.p2.1.m1.1.1.cmml" xref="S6.Thmthm9.p2.1.m1.1.1"><subset id="S6.Thmthm9.p2.1.m1.1.1.1.cmml" xref="S6.Thmthm9.p2.1.m1.1.1.1"></subset><ci id="S6.Thmthm9.p2.1.m1.1.1.2.cmml" xref="S6.Thmthm9.p2.1.m1.1.1.2">𝑋</ci><apply id="S6.Thmthm9.p2.1.m1.1.1.3.cmml" xref="S6.Thmthm9.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.1.m1.1.1.3.1.cmml" xref="S6.Thmthm9.p2.1.m1.1.1.3">superscript</csymbol><ci id="S6.Thmthm9.p2.1.m1.1.1.3.2.cmml" xref="S6.Thmthm9.p2.1.m1.1.1.3.2">𝒜</ci><ci id="S6.Thmthm9.p2.1.m1.1.1.3.3.cmml" xref="S6.Thmthm9.p2.1.m1.1.1.3.3">ℤ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.1.m1.1c">X\subseteq\cal A^{\mathbb{Z}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.1.m1.1d">italic_X ⊆ caligraphic_A start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT</annotation></semantics></math> with image <math alttext="Y_{i}=\sigma_{i}(X)" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.2.m2.1"><semantics id="S6.Thmthm9.p2.2.m2.1a"><mrow id="S6.Thmthm9.p2.2.m2.1.2" xref="S6.Thmthm9.p2.2.m2.1.2.cmml"><msub id="S6.Thmthm9.p2.2.m2.1.2.2" xref="S6.Thmthm9.p2.2.m2.1.2.2.cmml"><mi id="S6.Thmthm9.p2.2.m2.1.2.2.2" xref="S6.Thmthm9.p2.2.m2.1.2.2.2.cmml">Y</mi><mi id="S6.Thmthm9.p2.2.m2.1.2.2.3" xref="S6.Thmthm9.p2.2.m2.1.2.2.3.cmml">i</mi></msub><mo id="S6.Thmthm9.p2.2.m2.1.2.1" xref="S6.Thmthm9.p2.2.m2.1.2.1.cmml">=</mo><mrow id="S6.Thmthm9.p2.2.m2.1.2.3" xref="S6.Thmthm9.p2.2.m2.1.2.3.cmml"><msub id="S6.Thmthm9.p2.2.m2.1.2.3.2" xref="S6.Thmthm9.p2.2.m2.1.2.3.2.cmml"><mi id="S6.Thmthm9.p2.2.m2.1.2.3.2.2" xref="S6.Thmthm9.p2.2.m2.1.2.3.2.2.cmml">σ</mi><mi id="S6.Thmthm9.p2.2.m2.1.2.3.2.3" xref="S6.Thmthm9.p2.2.m2.1.2.3.2.3.cmml">i</mi></msub><mo id="S6.Thmthm9.p2.2.m2.1.2.3.1" xref="S6.Thmthm9.p2.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S6.Thmthm9.p2.2.m2.1.2.3.3.2" xref="S6.Thmthm9.p2.2.m2.1.2.3.cmml"><mo id="S6.Thmthm9.p2.2.m2.1.2.3.3.2.1" stretchy="false" xref="S6.Thmthm9.p2.2.m2.1.2.3.cmml">(</mo><mi id="S6.Thmthm9.p2.2.m2.1.1" xref="S6.Thmthm9.p2.2.m2.1.1.cmml">X</mi><mo id="S6.Thmthm9.p2.2.m2.1.2.3.3.2.2" stretchy="false" xref="S6.Thmthm9.p2.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.2.m2.1b"><apply id="S6.Thmthm9.p2.2.m2.1.2.cmml" xref="S6.Thmthm9.p2.2.m2.1.2"><eq id="S6.Thmthm9.p2.2.m2.1.2.1.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.1"></eq><apply id="S6.Thmthm9.p2.2.m2.1.2.2.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.2"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.2.m2.1.2.2.1.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.2">subscript</csymbol><ci id="S6.Thmthm9.p2.2.m2.1.2.2.2.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.2.2">𝑌</ci><ci id="S6.Thmthm9.p2.2.m2.1.2.2.3.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.2.3">𝑖</ci></apply><apply id="S6.Thmthm9.p2.2.m2.1.2.3.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.3"><times id="S6.Thmthm9.p2.2.m2.1.2.3.1.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.3.1"></times><apply id="S6.Thmthm9.p2.2.m2.1.2.3.2.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.2.m2.1.2.3.2.1.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.3.2">subscript</csymbol><ci id="S6.Thmthm9.p2.2.m2.1.2.3.2.2.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.3.2.2">𝜎</ci><ci id="S6.Thmthm9.p2.2.m2.1.2.3.2.3.cmml" xref="S6.Thmthm9.p2.2.m2.1.2.3.2.3">𝑖</ci></apply><ci id="S6.Thmthm9.p2.2.m2.1.1.cmml" xref="S6.Thmthm9.p2.2.m2.1.1">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.2.m2.1c">Y_{i}=\sigma_{i}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.2.m2.1d">italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_X )</annotation></semantics></math>, and assume that <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.3.m3.1"><semantics id="S6.Thmthm9.p2.3.m3.1a"><mi id="S6.Thmthm9.p2.3.m3.1.1" xref="S6.Thmthm9.p2.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.3.m3.1b"><ci id="S6.Thmthm9.p2.3.m3.1.1.cmml" xref="S6.Thmthm9.p2.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.3.m3.1d">italic_X</annotation></semantics></math> admits <math alttext="k\geq 2" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.4.m4.1"><semantics id="S6.Thmthm9.p2.4.m4.1a"><mrow id="S6.Thmthm9.p2.4.m4.1.1" xref="S6.Thmthm9.p2.4.m4.1.1.cmml"><mi id="S6.Thmthm9.p2.4.m4.1.1.2" xref="S6.Thmthm9.p2.4.m4.1.1.2.cmml">k</mi><mo id="S6.Thmthm9.p2.4.m4.1.1.1" xref="S6.Thmthm9.p2.4.m4.1.1.1.cmml">≥</mo><mn id="S6.Thmthm9.p2.4.m4.1.1.3" xref="S6.Thmthm9.p2.4.m4.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.4.m4.1b"><apply id="S6.Thmthm9.p2.4.m4.1.1.cmml" xref="S6.Thmthm9.p2.4.m4.1.1"><geq id="S6.Thmthm9.p2.4.m4.1.1.1.cmml" xref="S6.Thmthm9.p2.4.m4.1.1.1"></geq><ci id="S6.Thmthm9.p2.4.m4.1.1.2.cmml" xref="S6.Thmthm9.p2.4.m4.1.1.2">𝑘</ci><cn id="S6.Thmthm9.p2.4.m4.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p2.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.4.m4.1c">k\geq 2</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.4.m4.1d">italic_k ≥ 2</annotation></semantics></math> distinct ergodic probability measures <math alttext="\mu_{j}" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.5.m5.1"><semantics id="S6.Thmthm9.p2.5.m5.1a"><msub id="S6.Thmthm9.p2.5.m5.1.1" xref="S6.Thmthm9.p2.5.m5.1.1.cmml"><mi id="S6.Thmthm9.p2.5.m5.1.1.2" xref="S6.Thmthm9.p2.5.m5.1.1.2.cmml">μ</mi><mi id="S6.Thmthm9.p2.5.m5.1.1.3" xref="S6.Thmthm9.p2.5.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.5.m5.1b"><apply id="S6.Thmthm9.p2.5.m5.1.1.cmml" xref="S6.Thmthm9.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.5.m5.1.1.1.cmml" xref="S6.Thmthm9.p2.5.m5.1.1">subscript</csymbol><ci id="S6.Thmthm9.p2.5.m5.1.1.2.cmml" xref="S6.Thmthm9.p2.5.m5.1.1.2">𝜇</ci><ci id="S6.Thmthm9.p2.5.m5.1.1.3.cmml" xref="S6.Thmthm9.p2.5.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.5.m5.1c">\mu_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.5.m5.1d">italic_μ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Then the <math alttext="\mu_{j}^{\sigma_{i}}" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.6.m6.1"><semantics id="S6.Thmthm9.p2.6.m6.1a"><msubsup id="S6.Thmthm9.p2.6.m6.1.1" xref="S6.Thmthm9.p2.6.m6.1.1.cmml"><mi id="S6.Thmthm9.p2.6.m6.1.1.2.2" xref="S6.Thmthm9.p2.6.m6.1.1.2.2.cmml">μ</mi><mi id="S6.Thmthm9.p2.6.m6.1.1.2.3" xref="S6.Thmthm9.p2.6.m6.1.1.2.3.cmml">j</mi><msub id="S6.Thmthm9.p2.6.m6.1.1.3" xref="S6.Thmthm9.p2.6.m6.1.1.3.cmml"><mi id="S6.Thmthm9.p2.6.m6.1.1.3.2" xref="S6.Thmthm9.p2.6.m6.1.1.3.2.cmml">σ</mi><mi id="S6.Thmthm9.p2.6.m6.1.1.3.3" xref="S6.Thmthm9.p2.6.m6.1.1.3.3.cmml">i</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.6.m6.1b"><apply id="S6.Thmthm9.p2.6.m6.1.1.cmml" xref="S6.Thmthm9.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.6.m6.1.1.1.cmml" xref="S6.Thmthm9.p2.6.m6.1.1">superscript</csymbol><apply id="S6.Thmthm9.p2.6.m6.1.1.2.cmml" xref="S6.Thmthm9.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.6.m6.1.1.2.1.cmml" xref="S6.Thmthm9.p2.6.m6.1.1">subscript</csymbol><ci id="S6.Thmthm9.p2.6.m6.1.1.2.2.cmml" xref="S6.Thmthm9.p2.6.m6.1.1.2.2">𝜇</ci><ci id="S6.Thmthm9.p2.6.m6.1.1.2.3.cmml" xref="S6.Thmthm9.p2.6.m6.1.1.2.3">𝑗</ci></apply><apply id="S6.Thmthm9.p2.6.m6.1.1.3.cmml" xref="S6.Thmthm9.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.6.m6.1.1.3.1.cmml" xref="S6.Thmthm9.p2.6.m6.1.1.3">subscript</csymbol><ci id="S6.Thmthm9.p2.6.m6.1.1.3.2.cmml" xref="S6.Thmthm9.p2.6.m6.1.1.3.2">𝜎</ci><ci id="S6.Thmthm9.p2.6.m6.1.1.3.3.cmml" xref="S6.Thmthm9.p2.6.m6.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.6.m6.1c">\mu_{j}^{\sigma_{i}}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.6.m6.1d">italic_μ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> give <math alttext="k" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.7.m7.1"><semantics id="S6.Thmthm9.p2.7.m7.1a"><mi id="S6.Thmthm9.p2.7.m7.1.1" xref="S6.Thmthm9.p2.7.m7.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.7.m7.1b"><ci id="S6.Thmthm9.p2.7.m7.1.1.cmml" xref="S6.Thmthm9.p2.7.m7.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.7.m7.1c">k</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.7.m7.1d">italic_k</annotation></semantics></math> distinct ergodic measures on <math alttext="Y_{i}\," class="ltx_Math" display="inline" id="S6.Thmthm9.p2.8.m8.1"><semantics id="S6.Thmthm9.p2.8.m8.1a"><msub id="S6.Thmthm9.p2.8.m8.1.1" xref="S6.Thmthm9.p2.8.m8.1.1.cmml"><mi id="S6.Thmthm9.p2.8.m8.1.1.2" xref="S6.Thmthm9.p2.8.m8.1.1.2.cmml">Y</mi><mi id="S6.Thmthm9.p2.8.m8.1.1.3" xref="S6.Thmthm9.p2.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.8.m8.1b"><apply id="S6.Thmthm9.p2.8.m8.1.1.cmml" xref="S6.Thmthm9.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.8.m8.1.1.1.cmml" xref="S6.Thmthm9.p2.8.m8.1.1">subscript</csymbol><ci id="S6.Thmthm9.p2.8.m8.1.1.2.cmml" xref="S6.Thmthm9.p2.8.m8.1.1.2">𝑌</ci><ci id="S6.Thmthm9.p2.8.m8.1.1.3.cmml" xref="S6.Thmthm9.p2.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.8.m8.1c">Y_{i}\,</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.8.m8.1d">italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and after normalization we obtain <math alttext="k" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.9.m9.1"><semantics id="S6.Thmthm9.p2.9.m9.1a"><mi id="S6.Thmthm9.p2.9.m9.1.1" xref="S6.Thmthm9.p2.9.m9.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.9.m9.1b"><ci id="S6.Thmthm9.p2.9.m9.1.1.cmml" xref="S6.Thmthm9.p2.9.m9.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.9.m9.1c">k</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.9.m9.1d">italic_k</annotation></semantics></math> distinct ergodic probability measures on <math alttext="Y_{i}\," class="ltx_Math" display="inline" id="S6.Thmthm9.p2.10.m10.1"><semantics id="S6.Thmthm9.p2.10.m10.1a"><msub id="S6.Thmthm9.p2.10.m10.1.1" xref="S6.Thmthm9.p2.10.m10.1.1.cmml"><mi id="S6.Thmthm9.p2.10.m10.1.1.2" xref="S6.Thmthm9.p2.10.m10.1.1.2.cmml">Y</mi><mi id="S6.Thmthm9.p2.10.m10.1.1.3" xref="S6.Thmthm9.p2.10.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.10.m10.1b"><apply id="S6.Thmthm9.p2.10.m10.1.1.cmml" xref="S6.Thmthm9.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.10.m10.1.1.1.cmml" xref="S6.Thmthm9.p2.10.m10.1.1">subscript</csymbol><ci id="S6.Thmthm9.p2.10.m10.1.1.2.cmml" xref="S6.Thmthm9.p2.10.m10.1.1.2">𝑌</ci><ci id="S6.Thmthm9.p2.10.m10.1.1.3.cmml" xref="S6.Thmthm9.p2.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.10.m10.1c">Y_{i}\,</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.10.m10.1d">italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which are the only ones on <math alttext="Y_{i}" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.11.m11.1"><semantics id="S6.Thmthm9.p2.11.m11.1a"><msub id="S6.Thmthm9.p2.11.m11.1.1" xref="S6.Thmthm9.p2.11.m11.1.1.cmml"><mi id="S6.Thmthm9.p2.11.m11.1.1.2" xref="S6.Thmthm9.p2.11.m11.1.1.2.cmml">Y</mi><mi id="S6.Thmthm9.p2.11.m11.1.1.3" xref="S6.Thmthm9.p2.11.m11.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.11.m11.1b"><apply id="S6.Thmthm9.p2.11.m11.1.1.cmml" xref="S6.Thmthm9.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p2.11.m11.1.1.1.cmml" xref="S6.Thmthm9.p2.11.m11.1.1">subscript</csymbol><ci id="S6.Thmthm9.p2.11.m11.1.1.2.cmml" xref="S6.Thmthm9.p2.11.m11.1.1.2">𝑌</ci><ci id="S6.Thmthm9.p2.11.m11.1.1.3.cmml" xref="S6.Thmthm9.p2.11.m11.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.11.m11.1c">Y_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.11.m11.1d">italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (by the surjectivity of the measure transfer map <math alttext="\sigma M" class="ltx_Math" display="inline" id="S6.Thmthm9.p2.12.m12.1"><semantics id="S6.Thmthm9.p2.12.m12.1a"><mrow id="S6.Thmthm9.p2.12.m12.1.1" xref="S6.Thmthm9.p2.12.m12.1.1.cmml"><mi id="S6.Thmthm9.p2.12.m12.1.1.2" xref="S6.Thmthm9.p2.12.m12.1.1.2.cmml">σ</mi><mo id="S6.Thmthm9.p2.12.m12.1.1.1" xref="S6.Thmthm9.p2.12.m12.1.1.1.cmml">⁢</mo><mi id="S6.Thmthm9.p2.12.m12.1.1.3" xref="S6.Thmthm9.p2.12.m12.1.1.3.cmml">M</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p2.12.m12.1b"><apply id="S6.Thmthm9.p2.12.m12.1.1.cmml" xref="S6.Thmthm9.p2.12.m12.1.1"><times id="S6.Thmthm9.p2.12.m12.1.1.1.cmml" xref="S6.Thmthm9.p2.12.m12.1.1.1"></times><ci id="S6.Thmthm9.p2.12.m12.1.1.2.cmml" xref="S6.Thmthm9.p2.12.m12.1.1.2">𝜎</ci><ci id="S6.Thmthm9.p2.12.m12.1.1.3.cmml" xref="S6.Thmthm9.p2.12.m12.1.1.3">𝑀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p2.12.m12.1c">\sigma M</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p2.12.m12.1d">italic_σ italic_M</annotation></semantics></math>).</p> </div> <div class="ltx_para" id="S6.Thmthm9.p3"> <p class="ltx_p" id="S6.Thmthm9.p3.3">A non-evident case occurs if <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmthm9.p3.1.m1.1"><semantics id="S6.Thmthm9.p3.1.m1.1a"><mi id="S6.Thmthm9.p3.1.m1.1.1" xref="S6.Thmthm9.p3.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p3.1.m1.1b"><ci id="S6.Thmthm9.p3.1.m1.1.1.cmml" xref="S6.Thmthm9.p3.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p3.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p3.1.m1.1d">italic_X</annotation></semantics></math> is one of the subshifts exhibited in Theorem 7.4 of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2211.11234v4#bib.bib3" title="">3</a>]</cite> which (i) are minimal, (ii) have topological entropy <math alttext="h_{X}=0" class="ltx_Math" display="inline" id="S6.Thmthm9.p3.2.m2.1"><semantics id="S6.Thmthm9.p3.2.m2.1a"><mrow id="S6.Thmthm9.p3.2.m2.1.1" xref="S6.Thmthm9.p3.2.m2.1.1.cmml"><msub id="S6.Thmthm9.p3.2.m2.1.1.2" xref="S6.Thmthm9.p3.2.m2.1.1.2.cmml"><mi id="S6.Thmthm9.p3.2.m2.1.1.2.2" xref="S6.Thmthm9.p3.2.m2.1.1.2.2.cmml">h</mi><mi id="S6.Thmthm9.p3.2.m2.1.1.2.3" xref="S6.Thmthm9.p3.2.m2.1.1.2.3.cmml">X</mi></msub><mo id="S6.Thmthm9.p3.2.m2.1.1.1" xref="S6.Thmthm9.p3.2.m2.1.1.1.cmml">=</mo><mn id="S6.Thmthm9.p3.2.m2.1.1.3" xref="S6.Thmthm9.p3.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p3.2.m2.1b"><apply id="S6.Thmthm9.p3.2.m2.1.1.cmml" xref="S6.Thmthm9.p3.2.m2.1.1"><eq id="S6.Thmthm9.p3.2.m2.1.1.1.cmml" xref="S6.Thmthm9.p3.2.m2.1.1.1"></eq><apply id="S6.Thmthm9.p3.2.m2.1.1.2.cmml" xref="S6.Thmthm9.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.Thmthm9.p3.2.m2.1.1.2.1.cmml" xref="S6.Thmthm9.p3.2.m2.1.1.2">subscript</csymbol><ci id="S6.Thmthm9.p3.2.m2.1.1.2.2.cmml" xref="S6.Thmthm9.p3.2.m2.1.1.2.2">ℎ</ci><ci id="S6.Thmthm9.p3.2.m2.1.1.2.3.cmml" xref="S6.Thmthm9.p3.2.m2.1.1.2.3">𝑋</ci></apply><cn id="S6.Thmthm9.p3.2.m2.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p3.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p3.2.m2.1c">h_{X}=0</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p3.2.m2.1d">italic_h start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = 0</annotation></semantics></math>, and (iii) possess infinitely many pairwise different ergodic probability measures. It follows that the <math alttext="Y_{i}" class="ltx_Math" display="inline" id="S6.Thmthm9.p3.3.m3.1"><semantics id="S6.Thmthm9.p3.3.m3.1a"><msub id="S6.Thmthm9.p3.3.m3.1.1" xref="S6.Thmthm9.p3.3.m3.1.1.cmml"><mi id="S6.Thmthm9.p3.3.m3.1.1.2" xref="S6.Thmthm9.p3.3.m3.1.1.2.cmml">Y</mi><mi id="S6.Thmthm9.p3.3.m3.1.1.3" xref="S6.Thmthm9.p3.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p3.3.m3.1b"><apply id="S6.Thmthm9.p3.3.m3.1.1.cmml" xref="S6.Thmthm9.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p3.3.m3.1.1.1.cmml" xref="S6.Thmthm9.p3.3.m3.1.1">subscript</csymbol><ci id="S6.Thmthm9.p3.3.m3.1.1.2.cmml" xref="S6.Thmthm9.p3.3.m3.1.1.2">𝑌</ci><ci id="S6.Thmthm9.p3.3.m3.1.1.3.cmml" xref="S6.Thmthm9.p3.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p3.3.m3.1c">Y_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p3.3.m3.1d">italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> also have all of the properties (i), (ii) and (iii).</p> </div> <div class="ltx_para ltx_noindent" id="S6.Thmthm9.p4"> <p class="ltx_p" id="S6.Thmthm9.p4.3">(3) There are many direct generalizations of the morphisms <math alttext="\sigma_{i}" class="ltx_Math" display="inline" id="S6.Thmthm9.p4.1.m1.1"><semantics id="S6.Thmthm9.p4.1.m1.1a"><msub id="S6.Thmthm9.p4.1.m1.1.1" xref="S6.Thmthm9.p4.1.m1.1.1.cmml"><mi id="S6.Thmthm9.p4.1.m1.1.1.2" xref="S6.Thmthm9.p4.1.m1.1.1.2.cmml">σ</mi><mi id="S6.Thmthm9.p4.1.m1.1.1.3" xref="S6.Thmthm9.p4.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p4.1.m1.1b"><apply id="S6.Thmthm9.p4.1.m1.1.1.cmml" xref="S6.Thmthm9.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p4.1.m1.1.1.1.cmml" xref="S6.Thmthm9.p4.1.m1.1.1">subscript</csymbol><ci id="S6.Thmthm9.p4.1.m1.1.1.2.cmml" xref="S6.Thmthm9.p4.1.m1.1.1.2">𝜎</ci><ci id="S6.Thmthm9.p4.1.m1.1.1.3.cmml" xref="S6.Thmthm9.p4.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p4.1.m1.1c">\sigma_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p4.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as given above, for instance those given for any <math alttext="n\geq 2" class="ltx_Math" display="inline" id="S6.Thmthm9.p4.2.m2.1"><semantics id="S6.Thmthm9.p4.2.m2.1a"><mrow id="S6.Thmthm9.p4.2.m2.1.1" xref="S6.Thmthm9.p4.2.m2.1.1.cmml"><mi id="S6.Thmthm9.p4.2.m2.1.1.2" xref="S6.Thmthm9.p4.2.m2.1.1.2.cmml">n</mi><mo id="S6.Thmthm9.p4.2.m2.1.1.1" xref="S6.Thmthm9.p4.2.m2.1.1.1.cmml">≥</mo><mn id="S6.Thmthm9.p4.2.m2.1.1.3" xref="S6.Thmthm9.p4.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p4.2.m2.1b"><apply id="S6.Thmthm9.p4.2.m2.1.1.cmml" xref="S6.Thmthm9.p4.2.m2.1.1"><geq id="S6.Thmthm9.p4.2.m2.1.1.1.cmml" xref="S6.Thmthm9.p4.2.m2.1.1.1"></geq><ci id="S6.Thmthm9.p4.2.m2.1.1.2.cmml" xref="S6.Thmthm9.p4.2.m2.1.1.2">𝑛</ci><cn id="S6.Thmthm9.p4.2.m2.1.1.3.cmml" type="integer" xref="S6.Thmthm9.p4.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p4.2.m2.1c">n\geq 2</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p4.2.m2.1d">italic_n ≥ 2</annotation></semantics></math> and <math alttext="1\leq k\leq n" class="ltx_Math" display="inline" id="S6.Thmthm9.p4.3.m3.1"><semantics id="S6.Thmthm9.p4.3.m3.1a"><mrow id="S6.Thmthm9.p4.3.m3.1.1" xref="S6.Thmthm9.p4.3.m3.1.1.cmml"><mn id="S6.Thmthm9.p4.3.m3.1.1.2" xref="S6.Thmthm9.p4.3.m3.1.1.2.cmml">1</mn><mo id="S6.Thmthm9.p4.3.m3.1.1.3" xref="S6.Thmthm9.p4.3.m3.1.1.3.cmml">≤</mo><mi id="S6.Thmthm9.p4.3.m3.1.1.4" xref="S6.Thmthm9.p4.3.m3.1.1.4.cmml">k</mi><mo id="S6.Thmthm9.p4.3.m3.1.1.5" xref="S6.Thmthm9.p4.3.m3.1.1.5.cmml">≤</mo><mi id="S6.Thmthm9.p4.3.m3.1.1.6" xref="S6.Thmthm9.p4.3.m3.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p4.3.m3.1b"><apply id="S6.Thmthm9.p4.3.m3.1.1.cmml" xref="S6.Thmthm9.p4.3.m3.1.1"><and id="S6.Thmthm9.p4.3.m3.1.1a.cmml" xref="S6.Thmthm9.p4.3.m3.1.1"></and><apply id="S6.Thmthm9.p4.3.m3.1.1b.cmml" xref="S6.Thmthm9.p4.3.m3.1.1"><leq id="S6.Thmthm9.p4.3.m3.1.1.3.cmml" xref="S6.Thmthm9.p4.3.m3.1.1.3"></leq><cn id="S6.Thmthm9.p4.3.m3.1.1.2.cmml" type="integer" xref="S6.Thmthm9.p4.3.m3.1.1.2">1</cn><ci id="S6.Thmthm9.p4.3.m3.1.1.4.cmml" xref="S6.Thmthm9.p4.3.m3.1.1.4">𝑘</ci></apply><apply id="S6.Thmthm9.p4.3.m3.1.1c.cmml" xref="S6.Thmthm9.p4.3.m3.1.1"><leq id="S6.Thmthm9.p4.3.m3.1.1.5.cmml" xref="S6.Thmthm9.p4.3.m3.1.1.5"></leq><share href="https://arxiv.org/html/2211.11234v4#S6.Thmthm9.p4.3.m3.1.1.4.cmml" id="S6.Thmthm9.p4.3.m3.1.1d.cmml" xref="S6.Thmthm9.p4.3.m3.1.1"></share><ci id="S6.Thmthm9.p4.3.m3.1.1.6.cmml" xref="S6.Thmthm9.p4.3.m3.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p4.3.m3.1c">1\leq k\leq n</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p4.3.m3.1d">1 ≤ italic_k ≤ italic_n</annotation></semantics></math> by</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\sigma^{\prime}_{1}(a_{k})=a_{k}^{j_{k}}\,\,(j_{k}\geq 1)\quad\text{and}\quad% \sigma^{\prime}_{2}(a_{k})=a_{k}a_{k+1}\ldots a_{n}a_{1}a_{2}\ldots a_{k-1}a_{% k}\,," class="ltx_Math" display="block" id="S6.Ex10.m1.2"><semantics id="S6.Ex10.m1.2a"><mrow id="S6.Ex10.m1.2.2.1"><mrow id="S6.Ex10.m1.2.2.1.1.2" xref="S6.Ex10.m1.2.2.1.1.3.cmml"><mrow id="S6.Ex10.m1.2.2.1.1.1.1" xref="S6.Ex10.m1.2.2.1.1.1.1.cmml"><mrow id="S6.Ex10.m1.2.2.1.1.1.1.1" 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\sigma^{\prime}_{2}(a_{k})=a_{k}a_{k+1}\ldots a_{n}a_{1}a_{2}\ldots a_{k-1}a_{% k}\,,</annotation><annotation encoding="application/x-llamapun" id="S6.Ex10.m1.2d">italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_j start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≥ 1 ) and italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT … italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_a start_POSTSUBSCRIPT italic_k - 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmthm9.p4.4">or by suitable compositions of those (and of morphisms which are recognizable in the full shift). For each of these composed morphisms the above observation yields subshifts <math alttext="Y_{i}" class="ltx_Math" display="inline" id="S6.Thmthm9.p4.4.m1.1"><semantics id="S6.Thmthm9.p4.4.m1.1a"><msub id="S6.Thmthm9.p4.4.m1.1.1" xref="S6.Thmthm9.p4.4.m1.1.1.cmml"><mi id="S6.Thmthm9.p4.4.m1.1.1.2" xref="S6.Thmthm9.p4.4.m1.1.1.2.cmml">Y</mi><mi id="S6.Thmthm9.p4.4.m1.1.1.3" xref="S6.Thmthm9.p4.4.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S6.Thmthm9.p4.4.m1.1b"><apply id="S6.Thmthm9.p4.4.m1.1.1.cmml" xref="S6.Thmthm9.p4.4.m1.1.1"><csymbol cd="ambiguous" id="S6.Thmthm9.p4.4.m1.1.1.1.cmml" xref="S6.Thmthm9.p4.4.m1.1.1">subscript</csymbol><ci id="S6.Thmthm9.p4.4.m1.1.1.2.cmml" xref="S6.Thmthm9.p4.4.m1.1.1.2">𝑌</ci><ci id="S6.Thmthm9.p4.4.m1.1.1.3.cmml" xref="S6.Thmthm9.p4.4.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmthm9.p4.4.m1.1c">Y_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmthm9.p4.4.m1.1d">italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as in (2) above. This has lead us to pose the following question:</p> </div> <div class="ltx_para" id="S6.Thmthm9.p5"> <p class="ltx_p" id="S6.Thmthm9.p5.1"><span class="ltx_text ltx_font_italic" id="S6.Thmthm9.p5.1.1">Does every infinite transitive sofic subshift contain a subshift with all the properties (i), (ii) and (iii) from part (2) above ?</span></p> </div> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> L. Barreira, <span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">Ergodic theory, hyperbolic dynamics and dimension theory.</span> Universitext, Springer-Verlag, Berlin - New York 2012 </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> N. Bédaride, A. Hilion and M. 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