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Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 2013

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

N. Haydn
  • Fonction : Auteur
M. Nicol
  • Fonction : Auteur
T. Persson
  • Fonction : Auteur
S. Vaienti

Résumé

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.

Dates et versions

hal-00975710 , version 1 (08-04-2014)

Identifiants

Citer

N. Haydn, M. Nicol, T. Persson, S. Vaienti. A note on Borel--Cantelli lemmas for non-uniformly hyperbolic dynamical systems. Ergodic Theory and Dynamical Systems, 2013, 33, pp.475-498. ⟨10.1017/S014338571100099X⟩. ⟨hal-00975710⟩
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