Title

Strong Borel–Cantelli Lemmas for Recurrence

Abstract

The study of recurrence in dynamical systems dates back to the famous Poincare recurrence theorem, which states that almost every point of a measure preserving dynamical system is recurrent. That is, if you start at a point and let the system evolve, you will, with probability 1, eventually come back close to your starting point.

In this talk, I will present improvements that give more precise information about the rate of recurrence when the system is highly
chaotic. The improvements can be interpreted as 'dynamical strong Borel–Cantelli lemmas' — a sort of strong law of large numbers.

I will discuss two main differences when working with chaotic deterministic systems as opposed to random systems; for deterministic
systems, one uses the idea of 'long term independence' and one must tackle the issue of points with 'short return times.' If time permits, I will also present applications of dynamical Strong Borel--Cantelli lemmas to recurrence rates and the dimension of the probability measure.