Almost sure mixing rates for non-uniformly expanding maps

Abstract

We consider random perturbations of non-uniformly expanding maps, possibly having a non-degenerate critical set. We prove that, if the Lebesgue measure of the set of points failing the non-uniform expansion or the slow recurrence to the critical set at a certain time, for almost all random orbits, decays in a (stretched) exponential fashion, then the decay of correlations along random orbits is stretched exponential, up to some waiting time. As applications, we obtain almost sure stretched exponential decay of random correlations for Viana maps, as for a class of non-uniformly expanding local diffeomorphisms and a quadratic family of interval maps.

Almost sure rates of mixing for partially hyperbolic attractors

“…Recently, there has been a significant interest in obtaining probabilistic limit theorems for random dynamical systems [1,12,13,31,32,33,34,38,39,54], related time dependent systems [22,40,41,45] and for stochastic flows [30,21]. As in the case of deterministic dynamical systems, some knowledge on the correlation decay rate of the underlying system is required to obtain probabilistic limit laws for random dynamical systems.…”

Quenched Decay of Correlations for One-dimensional Random Lorenz Maps

“…This appendix is based on the work and results of [9,14]. See also the related work of [6,16] on random towers. Using the notation and definitions in Section 3, we can introduce a random tower for almost every ω as follows:…”

Quenched decay of correlations for one dimensional random Lorenz maps

“…This appendix is based on the work and results of [8,13]. See also the related work of [5,15] on random towers. Using the notation and definitions in section 3, we can introduce a random tower for almost every ω as follows:…”