Hitting and return times in ergodic dynamical systems

Given an ergodic dynamical system (X, T, μ), and U⊂X measurable with μ (U) >0, let μ (U) τU (x) denote the normalized hitting time of x∈X to U. We prove that given a sequence (Un) with μ (Un) →0, the distribution function of the normalized hitting times to Un 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, (⋄) F (t) =₀^t (1-F (s) ) \, ds, t0. 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.