Alejandro Rodriguez Sponheimer: How often and how close – limit laws for Poincaré recurrence

Abstract: The Poincaré recurrence theorem states that, under mild conditions, the orbit of almost every point of a measure-preserving dynamical system returns arbitrarily close to its initial point and does so infinitely often. Its main limitation is that it is a purely qualitative result; that is, it does not quantify how frequent or how close the returns are. In a pioneering paper in 1993, Boshernitzan quantified how close under slightly stronger conditions. If we assume stronger conditions, say a multiple decorrelation property, can we obtain more information?

In this talk, I will discuss recent results that quantify both how close and how frequently by counting returns to shrinking balls around the initial point. I will first briefly discuss strong Borel–Cantelli lemmas, which have received considerable attention in recent years. Given these strong laws, it is natural to ask whether a distributional law holds as well. I will explain how such a result can be obtained, and why the limit is not Gaussian, but instead a mixture of Gaussian distributions. Finally, I will comment briefly on how the question can be modified to recover a central limit theorem.