{
  "id": "1412.0848",
  "version": "v1",
  "published": "2014-12-02T10:35:32.000Z",
  "updated": "2014-12-02T10:35:32.000Z",
  "title": "Contraction in the Casserstein metric for some Markov chains, and applications to the dynamics of expanding maps",
  "authors": [
    "Artur Lopes",
    "Manuel Stadlbauer",
    "Benoit Kloeckner"
  ],
  "categories": [
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-02T10:35:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D20",
      "37D35",
      "49Q20"
    ],
    "keywords": [
      "expanding maps",
      "markov chains",
      "casserstein metric",
      "contraction",
      "applications"
    ],
    "publication": {
      "doi": "10.1088/0951-7715/28/11/4117",
      "journal": "Nonlinearity",
      "year": 2015,
      "month": "Oct",
      "volume": 28,
      "number": 11,
      "pages": 4117
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015Nonli..28.4117K"
    }
  }
}