{
  "id": "1807.02330",
  "version": "v1",
  "published": "2018-07-06T09:47:56.000Z",
  "updated": "2018-07-06T09:47:56.000Z",
  "title": "On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard Maps",
  "authors": [
    "Viviane Baladi",
    "Mark Demers"
  ],
  "comment": "50 pages, 1 Figure",
  "categories": [
    "math.DS",
    "math-ph",
    "math.FA",
    "math.MP",
    "math.SP",
    "nlin.CD"
  ],
  "abstract": "The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the value given by the variational principle, and that it is compatible with the definitions of Bowen using spanning or separating sets. If $h_* \\ge \\log 2$ (our actual condition is weaker), then we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure $\\mu_*$ of maximal entropy for $T$ (i.e., $h_{\\mu_*}(T)=h_*$), we show that $\\mu_*$ has full support and is Bernoulli, and we prove that $\\mu_*$ is different from the smooth invariant measure except if all non grazing periodic orbits have multiplier equal to $h_*$. Second, $h_*$ is compatible with the Bowen-Pesin-Pitskel topological entropy of the restriction of $T$ to a non-compact domain of continuity. Last, applying results of Lima and Matheus, the map $T$ has at least $C e^{pnh_*}$ periodic points of period $pn$ for all $n$ and some $p\\ge 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-06T09:47:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D50",
      "37C30",
      "37B40",
      "37A25",
      "46E35",
      "47B38"
    ],
    "keywords": [
      "finite horizon sinai billiard maps",
      "maximal entropy",
      "periodic lorentz gas",
      "topological entropy",
      "non grazing periodic orbits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}