{"title":"A strong Borel–Cantelli lemma for recurrence","authors":"T. Persson","doi":"10.4064/sm220216-2-7","DOIUrl":null,"url":null,"abstract":"ABSTRACT. Consider a mixing dynamical systems ([0, 1], T, μ), for instance a piecewise expanding interval map with a Gibbs measure μ. Given a non-summable sequence (mk) of non-negative numbers, one may define rk(x) such that μ(B(x, rk(x)) = mk. It is proved that for almost all x, the number of k ≤ n such that Tk(x) ∈ Bk(x) is approximately equal to m1 + . . . + mn. This is a sort of strong Borel–Cantelli lemma for recurrence. A consequence is that","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studia Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4064/sm220216-2-7","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
ABSTRACT. Consider a mixing dynamical systems ([0, 1], T, μ), for instance a piecewise expanding interval map with a Gibbs measure μ. Given a non-summable sequence (mk) of non-negative numbers, one may define rk(x) such that μ(B(x, rk(x)) = mk. It is proved that for almost all x, the number of k ≤ n such that Tk(x) ∈ Bk(x) is approximately equal to m1 + . . . + mn. This is a sort of strong Borel–Cantelli lemma for recurrence. A consequence is that
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