{
  "id": "1004.3441",
  "version": "v2",
  "published": "2010-04-20T13:17:33.000Z",
  "updated": "2010-08-24T12:28:04.000Z",
  "title": "Dominated Splitting and Pesin's Entropy Formula",
  "authors": [
    "Wenxiang Sun",
    "Xueting Tian"
  ],
  "journal": "Discrete & Continuous Dynamical Systems-A vol. 32-4 April 2012; 1421-1434",
  "doi": "10.3934/dcds.2012.32.1421",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP",
    "math.ST",
    "physics.data-an",
    "stat.TH"
  ],
  "abstract": "Let $M$ be a compact manifold and $f:\\,M\\to M$ be a $C^1$ diffeomorphism on $M$. If $\\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\\mu$ $a.\\,\\,e.\\,\\,x\\in M,$ there is a dominated splitting $T_{orb(x)}M=E\\oplus F$ on its orbit $orb(x)$, then we give an estimation through Lyapunov characteristic exponents from below in Pesin's entropy formula, i.e., the metric entropy $h_\\mu(f)$ satisfies $$h_{\\mu}(f)\\geq\\int \\chi(x)d\\mu,$$ where $\\chi(x)=\\sum_{i=1}^{dim\\,F(x)}\\lambda_i(x)$ and $\\lambda_1(x)\\geq\\lambda_2(x)\\geq...\\geq\\lambda_{dim\\,M}(x)$ are the Lyapunov exponents at $x$ with respect to $\\mu.$ Consequently, by using a dichotomy for generic volume-preserving diffeomorphism we show that Pesin's entropy formula holds for generic volume-preserving diffeomorphisms, which generalizes a result of Tahzibi in dimension 2.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-08-24T12:28:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37A05",
      "37A35",
      "37D25",
      "37D30"
    ],
    "keywords": [
      "dominated splitting",
      "generic volume-preserving diffeomorphism",
      "pesins entropy formula holds",
      "invariant probability measure",
      "lyapunov characteristic exponents"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.3441S"
    }
  }
}