A strong Borel--Cantelli lemma for recurrence

Abstract

Consider a mixing dynamical systems $([0,1],T,\hat{I}¼)$, for instance a piecewise expanding interval map with a Gibbs measure $\hat{I}¼$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k(x)$ such that $\hat{I}¼(B(x,r_k(x))=m_k$. It is proved that for almost all $x$, the number of $k\le n$ such that $T^k(x)\in B_k(x)$ is approximately equal to $m_1+\dots +m_n$. This is a sort of strong Borel--Cantelli lemma for recurrence. A consequence is that \[ \lim_{r \to 0} \frac{\log τ_{B(x,r)} (x)}{- \log μ(B (x,r))} = 1 \] for almost every $x$, where $\ddot{I}„$ is the return time.

16 pages, 0 figures. Minor corrections, in particular to the proof of Proposition 1