A strong Borel--Cantelli lemma for recurrence

Abstract

Consider a mixing dynamical systems $([0,1], T, μ)$, for instance a piecewise expanding interval map with a Gibbs measure $μ$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $μ(B(x, r_k(x)) = m_k$. It is proved that for almost all $x$, the number of $k \leq n$ such that $T^k (x) \in B_k (x)$ is approximately equal to $m_1 + \ldots + m_n$. 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 $τ$ is the return time.