{"title":"On the support of measures of large entropy for polynomial-like maps","authors":"Sardor Bazarbaev, Fabrizio Bianchi, Karim Rakhimov","doi":"10.1007/s13324-025-01071-9","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>f</i> be a polynomial-like map with dominant topological degree <span>\\(d_t\\ge 2\\)</span> and let <span>\\(d_{k-1}<d_t\\)</span> be its dynamical degree of order <span>\\(k-1\\)</span>. We show that every ergodic measure whose measure-theoretic entropy is strictly larger than <span>\\(\\log \\sqrt{d_{k-1} d_t}\\)</span> is supported on the Julia set, i.e., the support of the unique measure of maximal entropy <span>\\(\\mu \\)</span>. The proof is based on the exponential speed of convergence of the measures<span>\\(d_t^{-n}(f^n)^*\\delta _a\\)</span> towards <span>\\(\\mu \\)</span>, which is valid for a generic point <i>a</i> and with a controlled error bound depending on <i>a</i>. Our proof also gives a new proof of the same statement in the setting of endomorphisms of <span>\\(\\mathbb P^k(\\mathbb C)\\)</span> – a result due to de Thélin and Dinh – which does not rely on the existence of a Green current.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"15 3","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s13324-025-01071-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let f be a polynomial-like map with dominant topological degree \(d_t\ge 2\) and let \(d_{k-1}<d_t\) be its dynamical degree of order \(k-1\). We show that every ergodic measure whose measure-theoretic entropy is strictly larger than \(\log \sqrt{d_{k-1} d_t}\) is supported on the Julia set, i.e., the support of the unique measure of maximal entropy \(\mu \). The proof is based on the exponential speed of convergence of the measures\(d_t^{-n}(f^n)^*\delta _a\) towards \(\mu \), which is valid for a generic point a and with a controlled error bound depending on a. Our proof also gives a new proof of the same statement in the setting of endomorphisms of \(\mathbb P^k(\mathbb C)\) – a result due to de Thélin and Dinh – which does not rely on the existence of a Green current.
期刊介绍:
Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.