Title

A Central Limit Theorem for Recurrence

Abstract

The famous Poincaré Recurrence Theorem states that, under some mild assumptions, almost every point in a probability-preserving dynamical system is recurrent, that is, the orbit of almost every point x returns arbitrarily close to x. Although a very general result, it gives us no quantitative information about the recurrence. In recent years, there have been several results establishing almost-sure limit laws for the number of close returns, which give information about the rate of recurrence as well as how close the close returns really are. (I have presented some of these results in previous PhD seminars.)

In this talk, I will briefly explain what a probability-preserving dynamical system is and present a distributional limit law for the number of close returns, which gives further quantitative information about the recurrence. I will briefly explain the role of decay of correlations and short returns estimates, which can be interpreted as dynamical 'long-term' and 'short-term' analogues of independence in probability theory.