• DOI:10.1088/0951-7715/28/11/4117
• Corpus ID: 28831122

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

@article{Kloeckner2014ContractionIT,
  title={Contraction in the Wasserstein metric for some Markov chains, and applications to the dynamics of expanding maps},
  author={Beno{\^i}t R. Kloeckner and Artur O. Lopes and Manuel Stadlbauer},
  journal={arXiv: Dynamical Systems},
  year={2014}
}

• Benoît R. Kloeckner, A. Lopes, M. Stadlbauer
• Published 2 December 2014
• Mathematics
• arXiv: Dynamical Systems

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.