Dynamical Borel-Cantelli Lemma for Recurrence under Lipschitz Twists

Abstract

In the study of some dynamical systems the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this paper, we introduce a generalized definition that can specialize into the shrinking targets and recurrence; our approach gives a unified proof of the zero-one laws for the two problems.

30 pages; the main theorem of the previous version is slightly weakened, several new theorems added

References in corpus (2)

• On shrinking targets and self-returning points
• Diophantine analysis of the expansions of a fixed point under continuum many bases

Cited by in corpus (2)

• A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms
• Quantitative recurrence problem on some Bedford-McMullen carpets