# On the liftability of expanding stationary measures

## TL;DR

This paper demonstrates that expanding stationary measures for certain dynamical systems can be lifted via a random Gibbs-Markov-Young structure, and that small stochastic perturbations preserve the expanding nature of these measures.

## Contribution

It introduces a method to lift expanding stationary measures using a random Gibbs-Markov-Young structure and shows stability of expansion under small stochastic perturbations.

## Key findings

- Existence of a random Gibbs-Markov-Young structure for expanding measures.
- Stability of expanding measures under small stochastic perturbations.
- Preservation of expansion for measures when the original system has finitely many expanding invariant measures.

## Abstract

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is a random Gibbs-Markov-Young structure which can be used to lift that measure. We also prove that if the original map admits a finite number of expanding invariant measures then the stationary measures of a sufficiently small stochastic perturbation are expanding.