Alejandro Rodriguez Sponheimer

A Central Limit Theorem for Recurrence

A version of the famous Poincaré recurrence theorem states that, for a measure-preserving system with finite measure, almost every point has an orbit that returns arbitrarily close to the initial point. This result is qualitative in nature; it gives us no quantitative information about the recurrence. In a pioneering paper in 1993, Boshernitzan gives a quantitative bound for how close the returns are, under some weak assumptions on the measure-preserving system. Since Boshernitzan's result, there have been several improvements.

In the past few years, under speed of mixing assumptions, there have been various papers establishing strong Borel--Cantelli lemmas for recurrence, which are essentially strong laws of large numbers for recurrence. These results give quantitative estimates for rate of recurrence. In this talk, I will present a distributional law for recurrence and give a quick sketch of the proof, explaining where multiple decorrelation and short return time estimates come into play.