The dynamical Borel-Cantelli lemma and the waiting time problems

TL;DR

This paper explores the relationship between the dynamical Borel-Cantelli lemma and waiting time problems, establishing conditions under which one implies the other in dynamical systems, with applications to systems like axiom A and interval exchanges.

Contribution

It proves a bi-conditional relationship between the dynamical Borel-Cantelli property and waiting time behavior, extending results to systems like axiom A and generic interval exchanges.

Findings

Dynamical Borel-Cantelli property implies inverse scaling of waiting times.

Waiting time behavior can imply Borel-Cantelli properties for certain sequences.

Results apply to systems such as axiom A and interval exchanges.

Abstract

We investigate the connection between the dynamical Borel-Cantelli and waiting time results. We prove that if a system has the dynamical Borel-Cantelli property, then the time needed to enter for the first time in a sequence of small balls scales as the inverse of the measure of the balls. Conversely if we know the waiting time behavior of a system we can prove that certain sequences of decreasing balls satisfies the Borel-Cantelli property. This allows to obtain Borel-Cantelli like results in systems like axiom A and generic interval exchanges.