A strong Borel--Cantelli lemma for recurrence

TL;DR

This paper establishes a strong Borel--Cantelli type result for recurrence in mixing dynamical systems, showing that the number of visits to shrinking neighborhoods aligns with the sum of prescribed measures, and relates return times to measure decay.

Contribution

It introduces a strong recurrence lemma for mixing systems with non-summable sequences, extending classical Borel--Cantelli results to dynamical recurrence.

Findings

Almost sure recurrence count matches the sum of measures

Logarithm of return time scales with measure decay

Provides a limit relation for return times and measures

Abstract

Consider a mixing dynamical systems $([0,1], T, \mu)$, for instance a piecewise expanding interval map with a Gibbs measure $\mu$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $\mu (B(x, r_k(x)) = m_k$. 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 \[ \lim_{r \to 0} \frac{\log \tau_{B(x,r)} (x)}{- \log \mu (B (x,r))} = 1 \] for almost every $x$, where $\tau$ is the return time.