Dynamical Borel-Cantelli lemma for recurrence theory

Abstract

We study the dynamical Borel-Cantelli lemma for recurrence sets in a measure preserving dynamical system (X, µ, T ) with a compatible metric d. We prove that, under some regularity conditions, the µ-measure of the following set R(ψ) = {x ∈ X : d(T n x, x) < ψ(n) for infinitely many n ∈ N} obeys a zero-full law according to the convergence or divergence of a certain series, where ψ : N → R + . Some of the applications of our main theorem include the continued fractions dynamical systems, the beta dynamical systems, and the homogeneous self-similar sets.

A strong Borel--Cantelli lemma for recurrence

“…Proposition 1. Suppose that µ is a probability measure and that (E ) is a sequence of sets that satisfy the estimate (7) µ…”

“…Recent improvements of Boshernitzan's result for particular classes of dynamical systems can be found in papers by Pawelec [12]; Chang, Wu and Wu [5]; Baker and Farmer [1]; Hussain, Li, Simmons and Wang [7]; and by Kirsebom, Kunde and Persson [9].…”