Adapted random perturbations for non-uniformly expanding maps

TL;DR

This paper proves stochastic stability for certain non-uniformly expanding maps using adapted random perturbations, even with infinitely many critical points, extending stability results to higher dimensions.

Contribution

It introduces a method to construct perturbations that preserve hyperbolic times, enabling stochastic stability for a broad class of maps with minimal integrability conditions.

Findings

Stochastic stability holds for one-dimensional non-uniformly expanding maps with L^p hyperbolic time integrability.

The method applies to maps with infinitely many critical points.

Results extend to higher-dimensional endomorphisms under stronger assumptions.

Abstract

We obtain stochastic stability of C2 non-uniformly expanding one-dimensional endomorphisms, requiring only that the first hyperbolic time map be L^{p}-integrable for p>3. We show that, under this condition (which depends only on the unperturbed dynamics), we can construct a random perturbation that preserves the original hyperbolic times of the unperturbed map and, therefore, to obtain non-uniform expansion for random orbits. This ensures that the first hyperbolic time map is uniformly integrable for all small enough noise levels, which is known to imply stochastic stability. The method enables us to obtain stochastic stability for a class of maps with infinitely many critical points. For higher dimensional endomorphisms, a similar result is obtained, but under stronger assumptions.