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 sub-probability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution functionF , and that in the converging case, F (t) = t 0 2000 Mathematics Subject Classification 37A05, 37A50, 60F05, 28D05.

Extreme Value Laws in Dynamical Systems for Non-smooth Observations

“…The existence of exponential HTS is equivalent to the existence of exponential RTS. In fact, according to [HLV05], a system has HTS G if and only if it has RTSG with G(t) = t 0 (1 −G(s)) ds. In [FFT10], the authors established a relation between the existence of HTS for balls and EVL for the stochastic processes defined in (1.1) arising from the stationary sequence of random variables given by (1.2).…”

The Rényi entropy function and the large deviation of short return times

“…let us put τ n (x) = min y∈An(x) τ An(x) (y), where A n (x) denotes the n-cylinder that contains x. This quantity arose in several circumstances: -Since it controls the short returns, it plays a crucial role to establish the asymptotic (exponential) distribution of the return times function τ A (x) when the measure of the set A goes to zero [21,3,1,2,26,25,24]. -It has been used to define the recurrence dimension since it served as the gauge set function to construct a suitable Carathéodory measure [5,30,7].…”

Decay of correlations and laws of rare events for transitive random maps

“…The relation between the existence of HTS and that of RTS is given by the Main Theorem in [19]. It states that a system has HTS G if and only if it has RTSG and…”