A Borel–Cantelli lemma for nonuniformly expanding dynamical systems

Author: Gupta Chinmaya   Nicol Matthew   Ott William  

Publisher: IOP Publishing

ISSN: 0951-7715

Source: Nonlinearity, Vol.23, Iss.8, 2010-08, pp. : 1991-2008

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Abstract

Let &(A_{n})_{n=1}^{infty} ; be a sequence of sets in a probability space &(X,mathscr{B},mu) ; such that &sum_{n=1}^{infty} mu (A_{n}) = infty ;. The classical Borel–Cantelli (BC) lemma states that if the sets An are independent, then μ({xX : xAn for infinitely many values of n}) = 1. We present analogous dynamical BC lemmas for certain sequences of sets (An) in X (including nested balls) for a class of deterministic dynamical systems T : XX with invariant probability measures. Our results apply to a class of Gibbs–Markov maps and one-dimensional nonuniformly expanding systems modelled by Young towers. We discuss some applications of our results to the extreme value theory of deterministic dynamical systems.