Return times distribution for Markov towers with decay of correlations

TL;DR

This paper proves that for certain dynamical systems with decay of correlations, return times are Poisson distributed for non-periodic points, extending previous results to infinite partitions and systems with infinite entropy.

Contribution

It establishes Poisson distribution of return times for Markov towers with decay of correlations, improving prior results by relaxing entropy and partition finiteness conditions.

Findings

Return times are Poisson distributed for non-periodic points.

The results apply to systems with infinite partitions and infinite entropy.

Rates of convergence to the Poisson distribution are obtained.

Abstract

In this paper we prove two results. First we show that dynamical systems with a $\phi$-mixing measure have in the limit Poisson distributed return times almost everywhere. We use the Chen-Stein method to also obtain rates of convergence. Our theorem improves on previous results by allowing for infinite partitions and dropping the requirement that the invariant measure have finite entropy with respect to the given partition. As has been shown elsewhere, the limiting distribution at periodic points is not Poissonian (but compound Poissonian). Here we show that for all non-periodic points the return times are in the limit Poisson distributed. In the second part we prove that Lai-Sang Young's Markov Towers have Poisson distributed return times if the correlations decay for observables that are H\"older continous and $\mathscr{L}^\infty$ bounded.