Random perturbations of non-uniformly expanding maps

TL;DR

This paper establishes conditions for the stochastic stability of non-uniformly expanding maps, analyzing how small random perturbations affect their statistical behavior and measure structure.

Contribution

It provides both necessary and sufficient conditions for stochastic stability and bounds the number of invariant measures under small noise levels.

Findings

Stochastic stability characterized for non-uniformly expanding maps.

Number of invariant measures is bounded by SRB measures under small noise.

Proved stability for specific classes of non-uniformly expanding maps.

Abstract

We give both sufficient conditions and necessary conditions for the stochastic stability of non-uniformly expanding maps either with or without critical sets. We also show that the number of probability measures describing the statistical asymptotic behaviour of random orbits is bounded by the number of SRB measures if the noise level is small enough. As an application of these results we prove the stochastic stability of certain classes of non-uniformly expanding maps introduced in \cite{V} and \cite{ABV}.