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

Abstract

Let $(B_{i})$ be a sequence of measurable sets in a probability space $(X,\mathcal{B}, \mu)$ such that $\sum_{n=1}^{\infty} \mu (B_{i}) = \infty$. The classical Borel-Cantelli lemma states that if the sets $B_{i}$ are independent, then $\mu (\{x \in X : x \in B_{i} \text{infinitely often (i.o.)}) = 1$. Suppose $(T,X,\mu)$ is a dynamical system and $(B_i)$ is a sequence of sets in $X$. We consider whether $T^i x\in B_i$ for $\mu$ a.e.\ $x\in X$ and if so, is there an asymptotic estimate on the rate of entry. If $T^i x\in B_i$ infinitely often for $\mu$ a.e.\ $x$ we call the sequence $B_i$ a Borel--Cantelli sequence. If the sets $B_i:= B(p,r_i)$ are nested balls about a point $p$ then the question of whether $T^i x\in B_i$ infinitely often for $\mu$ a.e.\ $x$ is often called the shrinking target problem. We show, under certain assumptions on the measure $\mu$, that for balls $B_i$ if $\mu (B_i)\ge i^{-\gamma}$, $0<\gamma <1$, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observations implies that the sequence is Borel-Cantelli. If $\mu (B_i)\ge \frac{C\log i}{i}$ then exponential decay of correlations implies that the sequence is Borel-Cantelli. If it is only assumed that $\mu (B_i) \ge \frac{1}{i}$ then we give conditions in terms of return time statistics which imply that for $\mu$ a.e.\ $p$ sequences of nested balls $B(p,1/i)$ are Borel-Cantelli. Corollaries of our results are that for planar dispersing billiards and Lozi maps $\mu$ a.e.\ $p$ 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.