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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#topological_properties'>Topological properties</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>Mandelbrot set</em> is the <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a> on those points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{C}</annotation></semantics></math> on which the iteration of the operation “square and add <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>” does not diverge.</p> <p>This is a famous example of a <a class="existingWikiWord" href="/nlab/show/fractal">fractal</a>.</p> <h2 id="definition">Definition</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, consider the <a class="existingWikiWord" href="/nlab/show/function">function</a> on the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a> that squares its argument and adds <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> to the result:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℂ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>f</mi> <mi>c</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>ℂ</mi></mtd></mtr> <mtr><mtd><mi>z</mi></mtd> <mtd><mover><mo>↦</mo><mrow><mphantom><mi>AA</mi></mphantom><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><msup><mi>z</mi> <mn>2</mn></msup><mo>+</mo><mi>c</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{C} &\overset{\phantom{AA}f_c\phantom{AA}}{\longrightarrow}& \mathbb{C} \\ z &\overset{\phantom{AA}\phantom{AA}}{\mapsto}& z^2 + c } \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo>≔</mo><munder><munder><mrow><mi>f</mi><mo>∘</mo><mi>⋯</mi><mo>∘</mo><mi>f</mi><mo>∘</mo><mi>f</mi></mrow><mo>⏟</mo></munder><mrow><mi>n</mi><mspace width="thinmathspace"></mspace><mtext>factors</mtext></mrow></munder></mrow><annotation encoding="application/x-tex"> f_c^n \coloneqq \underset{n\,\text{factors}}{\underbrace{ f \circ \cdots \circ f \circ f }} </annotation></semantics></math></div> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">f_c</annotation></semantics></math> with itself (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mi>c</mi> <mn>0</mn></msubsup><mo>≔</mo><msub><mi>id</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">f_c^0 \coloneqq id_{\mathbb{C}}</annotation></semantics></math>).</p> <p>Starting with value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">0 \in \mathbb{C}</annotation></semantics></math>, this defines a <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> of points in the complex plane <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>ℂ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(f_c^n(0))_{i \in \mathbb{C}}</annotation></semantics></math> for each <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{C}</annotation></semantics></math>.</p> <p>The <em>Mandelbrot set</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mdlbrt</mi><mo>⊂</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">Mdlbrt \subset \mathcal{C}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a> on those values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> for which the corresponding <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>∈</mo><mi>𝔹</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(f_c^n)_{n \in \mathbb{B}}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/bounded+set">bounded</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Mdlbrt</mi><mo>≔</mo><mrow><mo>{</mo><mi>c</mi><mo>∈</mo><mi>ℂ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><mspace width="thinmathspace"></mspace><mtext>is bounded</mtext><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><mi>ℂ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Mdlbrt \coloneqq \left\{ c \in \mathbb{C} \;\colon\; (f_c^n(0))_{n \in \mathbb{N}}\, \text{is bounded} \right\} \;\subset\; \mathbb{C} \,. </annotation></semantics></math></div> <p>Globally, at low resolution, the Mandelbrot set looks like this:</p> <p><img src="https://ncatlab.org/nlab/files/MandelbrotSet.png" width="400" /></p> <h2 id="properties">Properties</h2> <h3 id="topological_properties">Topological properties</h3> <div class="num_defn" id="MandelbrotSpace"> <h6 id="definition_2">Definition</h6> <p><strong>(Mandelbrot space)</strong></p> <p>Regard the Mandelbrot set as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Mdlbrt</mi><mo>,</mo><msub><mi>τ</mi> <mi>sub</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (Mdlbrt, \tau_{sub}) </annotation></semantics></math></div> <p>via the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>sub</mi></msub></mrow><annotation encoding="application/x-tex">\tau_{sub}</annotation></semantics></math> inherited from the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo>≃</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C} \simeq \mathbb{R}^2</annotation></semantics></math>.</p> </div> <div class="num_prop" id="MndlbrtIsCompact"> <h6 id="proposition">Proposition</h6> <p>The Mandelbrot space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Mdlbrot</mi><mo>,</mo><msub><mi>τ</mi> <mi>sub</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Mdlbrot, \tau_{sub})</annotation></semantics></math> (def. <a class="maruku-ref" href="#MandelbrotSpace"></a>) is a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a>.</p> </div> <p>We <strong>prove</strong> this <a href="#ProofCompactMandelbrot">below</a>, after the following lemma:</p> <div class="num_lemma" id="EscapeRadius"> <h6 id="lemma">Lemma</h6> <p><strong>(escape radius)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\vert c\vert} \gt 2</annotation></semantics></math> then the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(f_c^n(0))_{n \in \mathbb{N}}</annotation></semantics></math> is not bounded, hence the sequence of <a class="existingWikiWord" href="/nlab/show/absolute+values">absolute values</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex">( {\vert f_c^n(0)\vert} )_{n \in \mathbb{N}}</annotation></semantics></math> diverges for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\vert c \vert} \gt 2</annotation></semantics></math>.</p> <p>In fact in this case the absolute values increase monotonically:</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\vert c\vert} \gt 2</annotation></semantics></math> then for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \gt 0</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>></mo><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\vert f_c^{n+1}(0)\vert } \gt {\vert f_c^n(0)\vert} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>So assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\vert c \vert} \gt 2</annotation></semantics></math>.</p> <p>We prove the last statement by <a class="existingWikiWord" href="/nlab/show/induction">induction</a>.</p> <p>Observe that it is true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math>, where we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mtd> <mtd><mo>=</mo><mrow><mo stretchy="false">|</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>+</mo><mi>c</mi><mo stretchy="false">|</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≥</mo><mrow><mo stretchy="false">|</mo><msup><mi>c</mi> <mn>2</mn></msup><mo stretchy="false">|</mo></mrow><mo>−</mo><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><munder><munder><mrow><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>></mo><mn>1</mn></mrow></munder></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>></mo><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo stretchy="false">|</mo><msub><mi>f</mi> <mi>c</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} {\vert f_c^2(0) \vert} & = {\vert c^2 + c \vert } \\ & \geq {\vert c^2\vert } - {\vert c \vert } \\ & = {\vert c\vert}\underset{\gt 1}{\underbrace{({\vert c\vert}-1)}} \\ & \gt {\vert c\vert } \\ & = {\vert f_c(0) \vert } \end{aligned} \,. </annotation></semantics></math></div> <p>Now assume that there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>></mo><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert f_c^n(0) \vert} \gt {\vert c\vert}</annotation></semantics></math>. Then it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mfrac><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mfrac></mtd> <mtd><mo>=</mo><mfrac><mrow><mo stretchy="false">|</mo><mo stretchy="false">(</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><mrow><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≥</mo><mfrac><mrow><msup><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>−</mo><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow></mrow><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>−</mo><mfrac><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mfrac></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>></mo><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo><mo>−</mo><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>></mo><mn>1</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \frac{ {\vert f_c^{n+1}(0)\vert} }{ \vert f_c^n(0) \vert } & = \frac{ {\vert (f_c^n(0))^2 + c \vert} }{ f_c^n(0) } \\ & \geq \frac{ {\vert f_c^n(0)\vert}^2 - {\vert c \vert} }{ {\vert f_c^n(0)\vert} } \\ & = {\vert f_c^n(0) \vert} - \frac{ {\vert c \vert} }{ {\vert f_c^n(0) \vert} } \\ & \gt \vert c \vert - 1 \\ & \gt 1 \end{aligned} \,. </annotation></semantics></math></div> <p>Here the first inequality is due to the <a class="existingWikiWord" href="/nlab/show/triangle+inequality">triangle inequality</a>, the second is due to the induction assumption, and the last one is due to the initial assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>c</mi><mo stretchy="false">|</mo></mrow><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\vert c \vert} \gt 2</annotation></semantics></math>.</p> </div> <div class="proof" id="ProofCompactMandelbrot"> <h6 id="proof_2">Proof</h6> <p>that the Mandelbrot space is compact (prop. <a class="maruku-ref" href="#MndlbrtIsCompact"></a>)</p> <p>By lemma <a class="maruku-ref" href="#EscapeRadius"></a> the Mandelbrot set is a <a class="existingWikiWord" href="/nlab/show/bounded+subset">bounded subset</a> of 2d <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>. Hence by the <a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a> is is now sufficient to show that it is a <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, this will imply that it is compact.</p> <p>The subset is closed if for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℂ</mi><mo>\</mo><mi>Mdlbrt</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{C} \backslash Mdlbrt</annotation></semantics></math> not contained in the Mandelbrot set there is an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> which still does not intersect the Mandelbrot set.</p> <p>Now that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∉</mo><mi>Mdlbrt</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">c \notin Mdlbrt \subset \mathbb{R}^2</annotation></semantics></math> means by definition that for every positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> there is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>></mo><mi>r</mi></mrow><annotation encoding="application/x-tex">\vert f_c^n(0)\vert \gt r</annotation></semantics></math>.</p> <p>Pick such an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">r = 2</annotation></semantics></math>. Let then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>≔</mo><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>−</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \epsilon \coloneqq {\vert f_c^n(0) \vert} - 2 \,. </annotation></semantics></math></div> <p>and consider the subset</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>c</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>≔</mo><mrow><mo>{</mo><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mo>(</mo><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>−</mo><mi>ϵ</mi><mo>)</mo></mrow><mo><</mo><mrow><mo>(</mo><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mrow><mo stretchy="false">|</mo><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>+</mo><mi>ϵ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U_{c,n} \coloneqq \left\{ z \in \mathbb{C} \;\vert\; \left( \left({\vert f_c^n(0) \vert} - \epsilon \right) \lt \left( {\vert z\vert} \lt {\vert f_c^n(0) \vert} + \epsilon \right) \right) \right\} \,. </annotation></semantics></math></div> <p>This is clearly an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_c^n(0)</annotation></semantics></math>. Hence by continuity of the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mi>c</mi> <mi>n</mi></msubsup><mo lspace="verythinmathspace">:</mo><mi>ℂ</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">f_c^n \colon \mathbb{C} \to \mathbb{C}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/pre-image">pre-image</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>f</mi> <mi>c</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><mi>c</mi><mo>,</mo><mi>n</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (f_c^{n-1})^{-1}(U_{c,n}) </annotation></semantics></math></div> <p>is an open neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{C}</annotation></semantics></math>, and by lemma <a class="maruku-ref" href="#EscapeRadius"></a> this does not intersect the Mandelbrot set.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Julia+set">Julia set</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 16, 2017 at 17:58:08. 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