Dynamical Borel-Cantelli Lemma for Recurrence under Lipschitz Twists

TL;DR

This paper introduces a unified approach to prove zero-one laws for shrinking targets and recurrence in dynamical systems, using a generalized Borel-Cantelli lemma that simplifies and unifies previous separate proofs.

Contribution

The paper develops a generalized Borel-Cantelli lemma that unifies the zero-one laws for shrinking targets and recurrence in dynamical systems.

Findings

Unified proof for zero-one laws in dynamical systems

Generalized Borel-Cantelli lemma applicable to multiple problems

Simplification of existing proofs for recurrence and shrinking targets

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.