Optimal transportation and stationary measures for Iterated Function Systems

TL;DR

This paper applies optimal transportation techniques to analyze stationary measures of Iterated Function Systems, establishing existence, uniqueness, moment bounds, convergence properties, and stability under perturbations.

Contribution

It introduces a novel approach using optimal transportation to study stationary measures of IFS, including new results on stability and response to perturbations.

Findings

Existence and uniqueness of stationary measures under contraction-on-average.

Generalized moment bounds and tail estimates for stationary measures.

Lipschitz continuity of stationary measures under system perturbations.

Abstract

In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalized moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a "linear response formula" at almost every parameter of the perturbation.