Document Zbl 1451.28005

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Math. Anal. Appl.</a> <a href="/?q=in%3A454433" title="Articles in this Issue">491, No. 2, Article ID 124362, 14 p. (2020)</a>. </div> <div class="abstract">The present paper is devoted to investigations of multifractal properties for a class of homogeneous Cantor sets with Bernoulli measures. <br class="zbmathjax-paragraph">One can note authors’ abstract:<br class="zbmathjax-paragraph">“We study multifractal properties of Bernoulli measure $\mu_p$ supported on homogeneous Cantor set determined by $([0, 1], (2)_{k\ge 1}, (c_k)_{k\ge 1})$, where $0 < p < 1/2$ and $c_k$ is either $d_1$ or $d_2$ with $0 < d_1 < d_2 < 1/2$. Let $\underline{\theta}$ and $\overline{\theta}$ be the lower and upper limits of the occurrence frequency of $d_1$ in $(c_k)_{k\ge 1}$, respectively. It is proved by <span class="zbmathjax-textit">M. Wu</span> [Sci. China Ser. A 48, No. 8, 1097–1112 (2005; <a href="/1126.28010">Zbl 1126.28010</a>)] that, if $\underline{\theta}=\overline{\theta}$ then the multifractal formalism holds. After given some conditions on $(c_k)_{k\ge 1}$, we get multifractal spectra for any $0 \le \underline{\theta} < \overline{\theta} \le 1$. We show that if $\underline{\theta}=0$ and $\overline{\theta}=1$, the refined formalism does not hold, while Olsen’s concave multifractal formalism holds in some cases. This answers a question asked by <span class="zbmathjax-textit">L. Olsen</span> [Adv. Math. 116, No. 1, 82–196 (1995; <a href="/0841.28012">Zbl 0841.28012</a>)]. We also show that if $0 < \underline{\theta} < \overline{\theta} < 1$, Olsen’s concave multifractal formalism does not hold.”<br class="zbmathjax-paragraph">The notion of level sets is considered. The authors note that computing dimensions of level sets is an important problem in multifractal analysis. Special attention is given to the multifractal, singularity spectra of the Borel probability measure on a metric space. For completeness of this research, the notions of a homogeneous Moran set and of a homogeneous Cantor set are recalled, some known results are noted, and several technical lemmas are proven. Conditions under which the Borel probability measure on a metric space satisfies the multifractal formalism are discussed. <br class="zbmathjax-paragraph">Connections between investigations presented in this paper and other researches are explained.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A17527">Symon Serbenyuk (Kyïv)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7265021">Cited in <strong>9</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A28A80" title="MSC2020">28A80</a> </td> <td class="space"> Fractals </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ahomogeneous+Cantor+set">homogeneous Cantor set</a>; <a href="/?q=ut%3Amultifractal+analysis">multifractal analysis</a>; <a href="/?q=ut%3Amultifractal+formalism">multifractal formalism</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1126.28010">Zbl 1126.28010</a>; <a href="/0841.28012">Zbl 0841.28012</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1451.28005" data-ciurl="/ci/07265021" data-biburl="/bibtex/07265021.bib" data-amsurl="/amsrefs/07265021.bib" data-xmlurl="/xml/07265021.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07265021.pdf" title="Zbl 1451.28005 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jmaa.2020.124362" aria-label="DOI for “Multifractal analysis of Bernoulli measures on a class of homogeneous Cantor sets”" title="10.1016/j.jmaa.2020.124362">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Barral, J., Inverse problems in multifractal analysis of measures, Ann. Sci. Éc. Norm. Supér. (4), 48, 6, 1457-1510 (2015) · <a href="/1334.28015" class="nowrap">Zbl 1334.28015</a></td> </tr><tr> <td>[2]</td> <td class="space">Ben Nash, F.; Bhouri, I.; Heurteaux, Y., The validity of the multifractal formalism: results and examples, Adv. Math., 165, 2, 264-284 (2002) · <a href="/1020.28005" class="nowrap">Zbl 1020.28005</a></td> </tr><tr> <td>[3]</td> <td class="space">Ben Nash, F.; Peyrière, J., Revisiting the multifractal analysis of measures, Rev. Mat. Iberoam., 29, 1, 315-328 (2013) · <a href="/1273.28008" class="nowrap">Zbl 1273.28008</a></td> </tr><tr> <td>[4]</td> <td class="space">Cawley, R.; Mauldin, R. D., Multifractal decompositions of Moran fractals, Adv. Math., 92, 2, 196-236 (1992) · <a href="/0763.58018" class="nowrap">Zbl 0763.58018</a></td> </tr><tr> <td>[5]</td> <td class="space">Edgar, G. A.; Mauldin, R. D., Multifractal decompositions of digraph recursive fractals, Proc. Lond. Math. Soc., 3-65, 3, 604-628 (1992) · <a href="/0764.28007" class="nowrap">Zbl 0764.28007</a></td> </tr><tr> <td>[6]</td> <td class="space">Falconer, K., Fractal Geometry: Mathematical Foundations and Applications (1990), John Wiley & Sons · <a href="/0689.28003" class="nowrap">Zbl 0689.28003</a></td> </tr><tr> <td>[7]</td> <td class="space">Falconer, K., Techniques in Fractal Geometry (1997), John Wiley & Sons · <a href="/0869.28003" class="nowrap">Zbl 0869.28003</a></td> </tr><tr> <td>[8]</td> <td class="space">Feng, D. J.; Wen, Z. Y.; Wu, J., Some dimensional results for homogeneous Moran sets, Sci. China Ser. A, 40, 5, 475-482 (1999) · <a href="/0881.28003" class="nowrap">Zbl 0881.28003</a></td> </tr><tr> <td>[9]</td> <td class="space">Hua, S.; Rao, H.; Wen, Z. Y.; Wu, J., On the structures and dimensions of Moran sets, Sci. China Ser. A, 43, 8, 836-852 (2000) · <a href="/0976.28002" class="nowrap">Zbl 0976.28002</a></td> </tr><tr> <td>[10]</td> <td class="space">Olsen, L., A multifractal formalism, Adv. Math., 116, 1, 82-196 (1995) · <a href="/0841.28012" class="nowrap">Zbl 0841.28012</a></td> </tr><tr> <td>[11]</td> <td class="space">Pesin, Y.; Weiss, H., The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples, Chaos, 7, 89-106 (1997) · <a href="/0933.37020" class="nowrap">Zbl 0933.37020</a></td> </tr><tr> <td>[12]</td> <td class="space">Rand, D., The singularity spectrum $f(\alpha)$ for cookie-cutters, Ergod. Theory Dyn. Syst., 9, 527-541 (1989) · <a href="/0664.58022" class="nowrap">Zbl 0664.58022</a></td> </tr><tr> <td>[13]</td> <td class="space">Shen, S., Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen’s b and B functions, J. Stat. Phys., 159, 5, 1216-1235 (2015) · <a href="/1325.28012" class="nowrap">Zbl 1325.28012</a></td> </tr><tr> <td>[14]</td> <td class="space">Wu, M., The multifractal spectrum of some Moran measures, Sci. China Ser. A, 48, 8, 1097-1112 (2005) · <a href="/1126.28010" class="nowrap">Zbl 1126.28010</a></td> </tr><tr> <td>[15]</td> <td class="space">Yuan, Z. H., Multifractal spectra of Moran measures without local dimension, Nonlinearity, 32, 12, 5060-5086 (2019) · <a href="/1425.28002" class="nowrap">Zbl 1425.28002</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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