Hitting and return times in ergodic dynamical systems

Abstract

Given an ergodic dynamical system (X,T,μ), and U⊂X measurable with μ(U)>0, let μ(U)τ_U(x) denote the normalized hitting time of x∈X to U. We prove that given a sequence (U_n) with μ(U_n)→0, the distribution function of the normalized hitting times to U_n converges weakly to some subprobability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution function F̃, and that in the converging case, $$(⋄)\qquad F(t)=\int_{0}^{t}\bigl(1-\tilde{F}(s)\bigr)\,ds,\qquad t\ge0.$$ This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also.