Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps

TL;DR

This paper uses optimal transport techniques to prove exponential contraction of transfer operators for expanding maps, leading to new insights into the stability of Gibbs measures and avoiding traditional inequalities.

Contribution

It introduces a novel proof of transfer operator contraction using Wasserstein metrics, applicable to a broad class of systems called Iterated Contraction Systems.

Findings

Exponential contraction of the dual transfer operator in Wasserstein metric.

Lipschitz dependence of Gibbs measures on potential variations.

Extension of methods to general Iterated Contraction Systems.

Abstract

We employ techniques from optimal transport in order to prove decay of transfer operators associated to iterated functions systems and expanding maps, giving rise to a new proof without requiring a Doeblin-Fortet (or Lasota-Yorke) inequality. Our main result is the following. Suppose $T$ is an expanding transformation acting on a compact metric space $M$ and $A: M \to \mathbb{R}$ a given fixed H{\"o}lder function, and denote by $L$ the Ruelle operator associated to $A$. We show that if $L$ is normalized (i.e. if $L(1)=1$), then the dual transfer operator $L^*$ is an exponential contraction on the set of probability measures on $M$ with the $1$-Wasserstein metric.Our approach is flexible and extends to a relatively general setting, which we name Iterated Contraction Systems. We also derive from our main result several dynamical consequences; for example we show that Gibbs measures depends in a Lipschitz-continuous way on variations of the potential.