A strong Borel–Cantelli lemma for recurrence
Volume 268 / 2023
Abstract
Consider a dynamical systems 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.