Limit Laws for Poincaré Recurrence and the Shrinking Target Problem

Abstract

We establish distributional laws for Poincaré recurrence in measure-preserving systems $(X,T,μ)$ satisfying an exponential multiple decorrelation condition and a short returns condition. When the measure is absolutely continuous, the sum $\sum_{k=1}^{n} \mathbf{1}_{B(x,r_k)}(T^{k}x) - μ(B(x,r_k))$ 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 $μ$. 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.