Decay of random correlation functions for unimodal maps

Abstract

Since the pioneering results of Jakobson and subsequent work by Benedicks-Carleson and others, it is known that quadratic maps tf a ( χ) = a − χ 2 admit a unique absolutely continuous invariant measure for a positive measure set of parameters a. For topologically mixing tf a , Young and Keller-Nowicki independently proved exponential decay of correlation functions for this a.c.i.m. and smooth observables. We consider random compositions of small perturbations tf + ω t , with tf = tf a or another unimodal map satisfying certain nonuniform hyperbolicity axioms, and ω t chosen independently and identically in [−ϵ, ϵ]. Baladi-Viana showed exponential mixing of the associated Markov chain, i.e., averaging over all random itineraries. We obtain stretched exponential bounds for the random correlation functions of Lipschitz observables for the sample measure μ ω of almost every itinerary.