论类多项式映射的大熵量支持

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-03 DOI:arxiv-2409.02039

Sardor Bazarbaev, Fabrizio Bianchi, Karim Rakhimov

{"title":"论类多项式映射的大熵量支持","authors":"Sardor Bazarbaev, Fabrizio Bianchi, Karim Rakhimov","doi":"arxiv-2409.02039","DOIUrl":null,"url":null,"abstract":"Let $f$ be a polynomial-like map with dominant topological degree $d_t\\geq 2$\nand let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the\nsupport of every ergodic measure whose measure-theoretic entropy is strictly\nlarger than $\\log \\sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., the\nsupport of the unique measure of maximal entropy $\\mu$. The proof is based on\nthe exponential speed of convergence of the measures $d_t^{-n}(f^n)^*\\delta_a$\ntowards $\\mu$, which is valid for a generic point $a$ and with a controlled\nerror bound depending on $a$. Our proof also gives a new proof of the same\nstatement in the setting of endomorphisms of $\\mathbb P^k(\\mathbb C)$ - a\nresult due to de Th\\'elin and Dinh - which does not rely on the existence of a\nGreen current.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"86 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the support of measures of large entropy for polynomial-like maps\",\"authors\":\"Sardor Bazarbaev, Fabrizio Bianchi, Karim Rakhimov\",\"doi\":\"arxiv-2409.02039\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $f$ be a polynomial-like map with dominant topological degree $d_t\\\\geq 2$\\nand let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the\\nsupport of every ergodic measure whose measure-theoretic entropy is strictly\\nlarger than $\\\\log \\\\sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., the\\nsupport of the unique measure of maximal entropy $\\\\mu$. The proof is based on\\nthe exponential speed of convergence of the measures $d_t^{-n}(f^n)^*\\\\delta_a$\\ntowards $\\\\mu$, which is valid for a generic point $a$ and with a controlled\\nerror bound depending on $a$. Our proof also gives a new proof of the same\\nstatement in the setting of endomorphisms of $\\\\mathbb P^k(\\\\mathbb C)$ - a\\nresult due to de Th\\\\'elin and Dinh - which does not rely on the existence of a\\nGreen current.\",\"PeriodicalId\":501035,\"journal\":{\"name\":\"arXiv - MATH - Dynamical Systems\",\"volume\":\"86 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Dynamical Systems\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.02039\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Dynamical Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.02039","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}