Entry and return times distribution

Abstract

This paper reviews recent progress that has been made in the study of return times distribution. We discuss recent results on the first return time for systems with various mixing properties and then distribution results for higher order returns. We survey different techniques that have been used to obtain limiting results for higher order returns such as the moment method, the Chen–Stein method and total variation method. We also cover results that look at the limiting distribution at periodic points. Finally, we also look at the connection to recurrence times and results of the kind proven by Ornstein and Weiss two decades ago. Only positive entropy systems are considered.

Keywords:

• entry times
• return times
• induced map
• mixing measures
• decay of correlations
• recurrence time
• Rényi entropy

Notes

1. In fact f can be computed recursively:

2. Denote by and let f be the pointwise limit of f _n as n → ∞. It is then required that .