Limit Laws for Poincaré Recurrence and the Shrinking Target Problem

Let $(X,T,\mu ,d)$ be a metric measure-preserving system. If $B(x,r_n(x))$ is a sequence of balls such that, for each $n$, the measure of $B(x,r_n(x))$ is constant, then we obtain a self-norming CLT for recurrence for systems satisfying a multiple decorrelation property. When $\mu$ is absolutely continuous, we obtain a distributional limit law for recurrence for the sequence of balls $B(x,r_n)$. In the latter case, the density of the limiting distribution is an average over Gaussian densities. An important assumption in the CLT for recurrence is that the CLT holds for the shrinking target problem. Because of this, we also prove an ASIP for expanding and Axiom A systems for non-autonomous Hölder observables and apply it to the shrinking target problem, thereby obtaining a CLT.