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A strong Borel–Cantelli lemma for recurrence

Volume 268 / 2023

Tomas Persson 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 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.

Authors

  • Tomas PerssonCentre for Mathematical Sciences
    Lund University
    Box 118
    221 00 Lund, Sweden
    e-mail

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