Strong stochastic stability for non-uniformly expanding maps

Abstract

We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in [AA03], where it was proved the convergence of the stationary measures of the random process to the SRB measure of the initial system in the weak * topology. Here, under slightly weaker assumptions on the random perturbations, we obtain a stronger version of stochastic stability: convergence of the densities of the stationary measures to the density of the SRB measure of the unperturbed system in the L 1 -norm. As an application of our results we obtain strong stochastic stability for two classes of non-uniformly expanding maps. The first one is an open class of local diffeomorphisms introduced in [ABV00] and the second one is the class of Viana maps.

On ergodicity of Markovian mostly expanding semi-group actions

“…Proof. We apply the change of variable formula for f n ω and analogous to Corollaries 4.10, 4.11 and 4.12 of [5], we conclude the result.…”

Ergodicity of non-autonomous discrete systems with non-uniform expansion

“…Additionally, every point (k, x) where x ∈Ã has infinitely many (δ, λ)-hyperbolic preballs. The rest of the proof follows the argument of [4,Prop. 2.13] which is inspired by [2] .…”

“…In the sequel, we want to study the local ergodicity of non-autonomous systems. We will review the theory of hyperbolic preballs and hyperbolic times introduced by Alves [1] for autonomous systems and extended by Alves and Vilarinho [4] for random maps under assumptions of non-uniform expansions. This theory has been deeply studied and generalized in many works as [2,3,24].…”

On stochastic stability of non-uniformly expanding interval maps

“…The idea of inducing is well known and powerful in the research on deterministic non-uniformly hyperbolic dynamics. Implementation of the idea in the random setting appeared in [1,2,5]. However, it seems more difficult to verify the assumptions adopted in [1,2] than to construct an inducing scheme directly, at least in the set-up of this paper.…”

“…Implementation of the idea in the random setting appeared in [1,2,5]. However, it seems more difficult to verify the assumptions adopted in [1,2] than to construct an inducing scheme directly, at least in the set-up of this paper. (In [5], stochastic stability was not discussed.…”