A recurrence-type strong Borel–Cantelli lemma for Axiom A diffeomorphisms

Pub Date : 2024-09-18 DOI:10.1017/etds.2024.64
ALEJANDRO RODRIGUEZ SPONHEIMER
{"title":"A recurrence-type strong Borel–Cantelli lemma for Axiom A diffeomorphisms","authors":"ALEJANDRO RODRIGUEZ SPONHEIMER","doi":"10.1017/etds.2024.64","DOIUrl":null,"url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000646_inline1.png\"/> <jats:tex-math> $(X,\\mu ,T,d)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000646_inline2.png\"/> <jats:tex-math> $(M_k)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> that converges to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000646_inline3.png\"/> <jats:tex-math> $0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000646_inline4.png\"/> <jats:tex-math> $\\mu $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-almost every <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000646_inline5.png\"/> <jats:tex-math> $x\\in X$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000646_eqnu1.png\"/> <jats:tex-math> $$ \\begin{align*} \\lim_{n \\to \\infty}\\frac{\\sum_{k=1}^{n} \\mathbf{1}_{B_k(x)}(T^{k}x)} {\\sum_{k=1}^{n} \\mu(B_k(x))} = 1, \\end{align*} $$ </jats:tex-math> </jats:alternatives> </jats:disp-formula>where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0143385724000646_inline6.png\"/> <jats:tex-math> $\\mu (B_k(x)) = M_k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/etds.2024.64","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract

Let $(X,\mu ,T,d)$ be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for $\mu $ -almost every $x\in X$ , $$ \begin{align*} \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \end{align*} $$ where $\mu (B_k(x)) = M_k$ . In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.
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公理 A 差分的递推型强博雷尔-康特利定理
让$(X,\mu ,T,d)$是一个度量保全的动力系统,对于利普齐兹连续观测值,三折相关性呈指数衰减。给定一个足够慢地收敛到 $0$ 的序列 $(M_k)$,我们会得到一个强动力学的 Borel-Cantelli 递归结果,即对于 $\mu $ - 几乎每一个 $x\in X$ , $$ \begin{align*}\limit_{n \to \infty}\frac{sum_{k=1}^{n}\mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n}\mu(B_k(x))} = 1, \end{align*}$$ 其中 $\mu (B_k(x)) = M_k$ 。我们特别指出,在某些假设条件下,这一结果对于公理 A 差分和平衡态都是成立的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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