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

TL;DR

This paper investigates Borel-Cantelli lemmas within non-uniformly hyperbolic dynamical systems, establishing conditions under which sequences of sets are visited infinitely often, with implications for shrinking target problems and specific systems like billiards and Lozi maps.

Contribution

It provides new criteria linking decay of correlations and return time statistics to Borel-Cantelli properties in non-uniformly hyperbolic systems.

Findings

Polynomial decay of correlations implies Borel-Cantelli for sets with measure ≥ i^{-eta}.

Exponential decay of correlations ensures Borel-Cantelli for sets with measure ≥ (log i)/i.

Return time statistics conditions guarantee Borel-Cantelli for nested balls in certain 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.