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

Abstract

We employ techniques from optimal transport in order to prove the decay of transfer operators associated with 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 given fixed Hölder function, and denote by the Ruelle operator associated with A. We show that if is normalized (i.e. if ), then the dual transfer operator 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.