A strong Borel–Cantelli lemma for recurrence
Volume 268 / 2023
Studia Mathematica 268 (2023), 75-89
MSC: Primary 37E05; Secondary 37A05, 37B20.
DOI: 10.4064/sm220216-2-7
Published online: 8 August 2022
Abstract
Consider a dynamical systems $([0,1], T, \mu )$ which is exponentially mixing for $L^1$ against bounded variation. 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 $m_1 + \cdots + m_n$. 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.
