Strong Borel--Cantelli lemmas for recurrence

The study of recurrence in dynamical systems can be dated back to the famous Poincaré Recurrence Theorem. One formulation of it is that, under some mild assumptions on the measure-preserving dynamical system (𝑋,𝑇,𝜇,𝑑), almost every point is recurrent, that is, 

liminf_𝑛→∞ 𝑑(𝑇^𝑛𝑥,𝑥) = 0 

for 𝜇-a.e. 𝑥. A shortcoming of this result is that we do not obtain any quantitative information about the recurrence. For example, what can be said about the rate of convergence? What about the rate of recurrence?

In this talk, I will give answers to these questions in the context of mixing dynamical systems by presenting a strong Borel-Cantelli lemma for recurrence. I will explain the main step of the proof, which is establishing correlation estimates for the sets 

𝐸_𝑘 = {𝑥 : 𝑑(𝑇^𝑘𝑥,𝑥) < 𝑟_𝑘(𝑥)}. 

In order to obtain the estimates, we use 'decay of correlations' -  an important tool used for establishing statistical properties of dynamical systems.