Kalendarium
10
June
PhD seminar: Alejandro Rodriguez Sponheimer
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.