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

Published online by Cambridge University Press:  16 February 2012

NICOLAI HAYDN ,
MATTHEW NICOL ,
TOMAS PERSSON  and
SANDRO VAIENTI
NICOLAI HAYDN
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA (email: [email protected])
MATTHEW NICOL
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA (email: [email protected])
TOMAS PERSSON
Affiliation:
Centre for Mathematical Sciences, Lund University, Box 118, 22100 Lund, Sweden (email: [email protected])
SANDRO VAIENTI
Affiliation:
UMR-6207 Centre de Physique Théorique, CNRS, Universités d’Aix-Marseille I, II, Université du Sud, Toulon-Var and FRUMAM, Fédéderation de Recherche des Unités de Mathématiques de Marseille, CPT, Luminy Case 907, 13288 Marseille Cedex 9, France (email: [email protected])
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Abstract

Let (Bi) be a sequence of measurable sets in a probability space (X,ℬ,μ) such that ∑ ∞n=1μ(Bi)=∞. The classical Borel–Cantelli lemma states that if the sets Bi are independent, then μ({x∈X:x∈Bi infinitely often})=1. Suppose (T,X,μ) is a dynamical system and (Bi) is a sequence of sets in X. We consider whether Tix∈Bi infinitely often for μ almost every x∈X and, if so, is there an asymptotic estimate on the rate of entry? If Tix∈Bi infinitely often for μ almost every x, we call the sequence (Bi) a Borel–Cantelli sequence. If the sets Bi :=B(p,ri) are nested balls of radius ri about a point p, then the question of whether Tix∈Bi 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 Bi if μ(Bi)≥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(Bi)≥C1 /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(Bi)≥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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 2 , April 2013 , pp. 475 - 498
Copyright
©2012 Cambridge University Press

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