{
  "id": "1809.08021",
  "version": "v1",
  "published": "2018-09-21T10:22:19.000Z",
  "updated": "2018-09-21T10:22:19.000Z",
  "title": "Convergence to $α$-stable Lévy motion for chaotic billiards with several cusps at flat points",
  "authors": [
    "Paul Jungand Françoise Pène",
    "Hong-Kun Zhang"
  ],
  "comment": "18 pages, 1 figure",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider billiards with several cusps at flat points; the case of a single cusp was studied previously in \\cite{Z2016b} and \\cite{JZ17}. In particular, we show that properly normalized Birkorff sums of H\\\"older observables, with respect to the billiard map, converge in Skorokhod's $M_1$-topology to an $\\alpha$-stable L\\'evy motion, with a skewness parameter which depends on the values of the observable at the different flat points. This extends the main result of \\cite{JZ17} which proved convergence of the one-point marginals to totally skewed $\\alpha$-stable distributions, when there is a single cusp. We also show that convergence in the stronger $J_1$-topology is not possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-21T10:22:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "flat points",
      "stable lévy motion",
      "chaotic billiards",
      "convergence",
      "single cusp"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}