Degenerate random perturbations of Anosov diffeomorphisms

TL;DR

This paper investigates how rank k random perturbations of Anosov diffeomorphisms influence invariant measures, showing that under certain conditions, these measures are absolutely continuous with respect to the Riemannian measure.

Contribution

It establishes conditions under which invariant measures for degenerate random perturbations of Anosov diffeomorphisms are absolutely continuous, extending understanding of stochastic stability in hyperbolic systems.

Findings

Invariant measures are absolutely continuous under certain conditions.

Conditions relate the support of perturbations to foliations in Anosov systems.

Generic rank k perturbations typically yield absolutely continuous invariant measures.

Abstract

This paper deals with random perturbations of diffeomorphisms on n-dimensional Riemannian manifolds with distributions supported on k-dimensional disks, where k<n. First we demonstrate general but not very intuitive conditions which guarantee that all invariant measures for rank k random perturbations of $C^2$ diffeomorphisms are absolutely continuous with respect to the Riemannian measure on M. For two subclasses of Anosov diffeomorphisms: hyperbolic toral automorphisms and Anosov diffeomorphisms with codimension 1 stable manifolds, the above conditions are modified in order to relate k-dimensional disks that support the distributions to certain foliations that arise from Anosov diffeomorphisms. We conclude that generic rank k random perturbations have absolutely continuous invariant measures.