A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems

Content Provider	Scilit
Author	Haydn, Nicolai Nicol, Matthew Persson, Tomas Vaienti, Sandro
Copyright Year	2012
Abstract	Let $(B_{i}$) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(B_{i}$)=∞. The classical Borel–Cantelli lemma states that if the sets $B_{i}$ are independent, then $μ({x∈X:x∈B_{i}$ infinitely often})=1. Suppose (T,X,μ) is a dynamical system and $(B_{i}$) is a sequence of sets in X. We consider whether T^{i}$x∈B_{i}$ infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If T^{i}$x∈B_{i}$ infinitely often for μ almost every x, we call the sequence $(B_{i}$) a Borel–Cantelli sequence. If the sets $B_{i}$ $:=B(p,r_{i}$) are nested balls of radius $r_{i}$ about a point p, then the question of whether T^{i}$x∈B_{i}$ infinitely often for μ almost every x is often called the shrinking target problem. We show, under certain assumptions on the measure μ, that for balls $B_{i}$ if μ(B_{i}$)≥i^{−γ}$, 0<γ<1, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observables implies that the sequence is Borel–Cantelli. If $μ(B_{i})≥C_{1}$ /i, then exponential decay of correlations implies that the sequence is Borel–Cantelli. We give conditions in terms of return time statistics which quantify Borel–Cantelli results for sequences of balls such that $μ(B_{i}$)≥C/i. Corollaries of our results are that for planar dispersing billiards and Lozi maps, sequences of nested balls B(p,1/i) are Borel–Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.
Related Links	http://arxiv.org/pdf/1103.2113 https://www.cambridge.org/core/services/aop-cambridge-core/content/view/785672238877F8ED68D338467C754C2B/S014338571100099Xa.pdf/div-class-title-a-note-on-borel-cantelli-lemmas-for-non-uniformly-hyperbolic-dynamical-systems-div.pdf
Ending Page	498
Page Count	24
Starting Page	475
ISSN	01433857
e-ISSN	14694417
DOI	10.1017/s014338571100099x
Journal	Ergodic Theory and Dynamical Systems
Issue Number	2
Volume Number	33
Language	English
Publisher	Cambridge University Press (CUP)
Publisher Date	2013-04-01
Access Restriction	Open
Subject Keyword	Ergodic Theory and Dynamical Systems Dynamical Systems
Content Type	Text
Subject	Applied Mathematics