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

Abstract

We show that a uniformly continuous random perturbation of a transitive map defines an aperiodic Harris chain which also satisfies Doeblin’s condition. As a result, we get exponential decay of correlations for suitable random perturbations of such systems. We also prove that, for transitive maps, the limiting distribution for extreme value laws and hitting/return time statistics is standard exponential. Moreover, we show that the rare event point process converges in distribution to a standard Poisson process.