Dynamical Borel–Cantelli lemmas

Viktoria Xing

doi:10.3934/dcds.2020339

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2021, Volume 41, Issue 4: 1737-1754. Doi: 10.3934/dcds.2020339

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Dynamical Borel–Cantelli lemmas

Viktoria Xing

Lund University, Sweden

Received: April 30, 2020

Revised: June 30, 2020

Early access: October 2020

Published: April 2021

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Abstract

This paper is a study of Borel–Cantelli lemmas in dynamical systems. D. Kleinbock and G. Margulis [7] have given a very useful sufficient condition for strongly Borel–Cantelli sequences, which is based on the work of W. M. Schmidt [10], [11]. We will obtain a weaker sufficient condition for the strongly Borel–Cantelli sequences. Two versions of the dynamical Borel–Cantelli lemmas will be deduced by extending a theorem by W. M. Schmidt [11], W. J. LeVeque [8], and W. Philipp [9]. Some applications of our theorems will also be discussed. Firstly, a characterization of the strongly Borel–Cantelli sequences in one-dimensional Gibbs–Markov systems will be established. This will improve the theorem of C. Gupta, M. Nicol, and W. Ott in [4]. Secondly, N. Haydn, M. Nicol, T. Persson, and S. Vaienti [5] proved the strong Borel–Cantelli property in sequences of balls in terms of a polynomial decay of correlations for Lipschitz observables. Our theorems will then be applied to relax their inequality assumption.

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Mathematics Subject Classification: Primary: 37A05; Secondary: 60F15.

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References

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