{
  "id": "1306.0821",
  "version": "v2",
  "published": "2013-06-04T14:53:18.000Z",
  "updated": "2014-12-06T01:06:50.000Z",
  "title": "Poincaré-Birkhoff theorems in random dynamics",
  "authors": [
    "Álvaro Pelayo",
    "Fraydoun Rezakhanlou"
  ],
  "comment": "35 pages, 5 figures. Presentation improved and added a new appendix explaining the relation between the classical theory and the random theory proposed in this paper",
  "categories": [
    "math.DS",
    "math.PR",
    "math.SG"
  ],
  "abstract": "We propose a generalization of the Poincar\\'e-Birkhoff Theorem on area-preserving twist maps to area-preserving twist maps that are random with respect to an ergodic probability measure. The classical theory is a particular instance of the random theory we propose.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-04T14:53:18.000Z",
      "abstract": "The Poincar\\'e-Birkhoff Theorem states that an area-preserving periodic twist map R x [-1,1] -> R x [-1,1] has two geometrically distinct fixed points. We generalize it to area-preserving twist maps F : (R x [-1,1]) x \\Omega -> R x [-1,1] that are random with respect to a \\tau-invariant ergodic probability measure P on a separable metric space \\Omega, where \\tau is a continuous R-action on \\Omega. We will prove, in particular, that the probability that the area-preserving twist F(-,-,;\\omega), \\omega \\in \\Omega, has fixed points, is one. The proofs are based on the notion of \"random generating function\", and we use a calculus suited for their study.",
      "comment": "28 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-06T01:06:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random dynamics",
      "poincaré-birkhoff theorems",
      "poincare-birkhoff theorem states",
      "ergodic probability measure",
      "area-preserving periodic twist map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.0821P"
    }
  }
}