{
  "id": "1602.07809",
  "version": "v1",
  "published": "2016-02-25T05:58:52.000Z",
  "updated": "2016-02-25T05:58:52.000Z",
  "title": "Entropy of Isometries Semi-groups of Hyperbolic space",
  "authors": [
    "Paul Mercat"
  ],
  "comment": "in French",
  "categories": [
    "math.MG",
    "math.DS",
    "math.GR",
    "math.NT"
  ],
  "abstract": "We give a generalization to convex co-compact semigroups of a beautiful theorem of Patterson-Sullivan, telling that the critical exponent (that is the exponential growth rate) equals the Hausdorff dimension of the limit set (that is the smallest closed non-empty invariant subset), for a isometries discrete group of a proper hyperbolic space with compact boundary. To do that, we introduce a notion of entropy, which generalize the notion of critical exponent of discrete groups, and we show that it is equal to the upper bound of critical exponents of Schottky sub-semigroups (which are semigroups having the simplest dynamic). We obtains several others corollaries, such that the lower semi-continuity of the entropy, the fact that the critical exponent of a separate semigroup, that is defined as an upper limit, is in fact a true limit, and we obtain the existence of \"big\" Schottky sub-semigroups in discrete groups of isometries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-25T05:58:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic space",
      "isometries semi-groups",
      "critical exponent",
      "schottky sub-semigroups",
      "smallest closed non-empty invariant subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "fr",
      "license": "arXiv",
      "status": "editable"
    }
  }
}