Entry times distribution for mixing systems

TL;DR

This paper studies the distribution of return times to Bowen balls in mixing dynamical systems, demonstrating that higher order return times follow a Poisson distribution and providing asymptotic recurrence time results.

Contribution

It establishes the Poisson distribution of higher order return times to Bowen balls in mixing systems and offers a general asymptotic recurrence time result for ergodic and systems with specification.

Findings

Higher order return times are Poisson distributed in the limit

Recurrence times for Bowen balls have specific asymptotic behavior

Results apply to systems with Young tower structures

Abstract

We consider the return times dynamics to Bowen balls for continuous maps on metric spaces which have invariant probability measures with certain mixing properties. These mixing properties are satisfied for instance by systems that allow Young tower constructions. We show that the higher order return times to Bowen balls are in the limit Poisson distributed. We also provide a general result for the asymptotic behavior of the recurrence time for Bowen balls for ergodic systems and those with specification.