{
  "id": "1011.6011",
  "version": "v3",
  "published": "2010-11-28T02:54:30.000Z",
  "updated": "2015-11-20T04:56:53.000Z",
  "title": "Hyperbolic Periodic Points and Hyperbolic Measures with Dominated Splitting",
  "authors": [
    "Xueting Tian"
  ],
  "comment": "This paper is covered by an updated version \"Diffeomorphisms with Liao-Pesin set, arXiv:1004.0486v3\"",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper we consider a non-atomic invariant hyperbolic measure $\\mu$ of a $C^1$ diffeomorphsim on a compact manifold, in whose Oseledec splitting the stable bundle dominates the unstable bundle on $\\mu$ a.e. points. We show an \\textit{exponentially} shadowing and an \\textit{exponentially} closing lemma, and as applications we show two classical results. One is that there exists a hyperbolic periodic point such that the closure of its unstable manifold has \\textit{positive} measure and it has a homoclinic point from which one can deduce a horseshoe. Moreover, such hyperbolic periodic points are dense in the support $supp(\\mu)$ of the given hyperbolic measure. Another is to show Livshitz Theorem.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-12-17T02:15:59.000Z",
      "comment": "19pages, hyperbolic measure, dominated splitting, exponentially shadowing and closing lemma, stable manifold, hyperbolic periodic points,Livshitz Theorem",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-11-20T04:56:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D25",
      "37D30",
      "37D10",
      "37C50",
      "37C25"
    ],
    "keywords": [
      "hyperbolic periodic point",
      "dominated splitting",
      "non-atomic invariant hyperbolic measure",
      "homoclinic point",
      "livshitz theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.6011T"
    }
  }
}