Limit Laws for Poincaré Recurrence and the Shrinking Target Problem

Abstract

We establish distributional laws for Poincaré recurrence in measure-preserving systems $(X,T,\hat{I}¼)$ satisfying an exponential multiple decorrelation condition and a short returns condition. When the measure is absolutely continuous, the sum ${\sum }_{k=1}^n{1}_{B(x,r_k)}(T^{k}x)-\hat{I}¼(B(x,r_k))$ does not in general obey a CLT; instead, it converges to a non-standard distribution that is an average of Gaussian laws weighted by the density of $\hat{I}¼$. By considering a version of the sum where we appropriately rescale the radii of the balls, we recover the CLT. A key assumption in our recurrence theorems is that the corresponding hitting sums satisfy the CLT. We verify this assumption for Axiom A systems by establishing the stronger ASIP for the shrinking target problem, extending Haydn, Nicol, Török and Vaienti [Trans. Amer. Math. Soc. 2017] and related results. Systems for which our results apply include piecewise expanding systems on the interval, and Axiom A systems. The results highlight the difference between recurrence and hitting behaviour.

79 pages, 3 figures. Restructured and improved presentation of paper. Section and result labelling changed. Combined Theorem 2.3 and Corollary 2.2 of previous version, appearing as Theorem 4.4 in the current version. This fixes an oversight in the previous version