Limit laws for Poincaré recurrence – quantifying how often and how close

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 1993, Boshernitzan quantified how close under slightly stronger conditions. As for how frequently, if the system is ergodic, the Birkhoff ergodic theorem implies that, for any fixed region, almost every orbit spends a proportion of its time there equal to the measure of the region. Neither result, however, quantifies both how frequently and how close the orbits return.

In this talk, I will discuss recent results that do exactly this, by counting returns to shrinking balls around the initial point. I will first briefly discuss strong Borel–Cantelli lemmas, which are strong laws of large numbers and 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.

No prior knowledge of ergodic theory is required.