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

الأصل · EN

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 → R a given fixed Hölder 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.