{
  "id": "0901.0574",
  "version": "v3",
  "published": "2009-01-05T23:23:31.000Z",
  "updated": "2009-07-14T14:01:45.000Z",
  "title": "Lorenz like flows: exponential decay of correlations for the Poincaré map, logarithm law, quantitative recurrence",
  "authors": [
    "S. Galatolo",
    "M. J. Pacifico"
  ],
  "comment": "Revision, after some advices",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "abstract": "In this paper we prove that the Poincar\\'e map associated to a Lorenz like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time $\\tau_r(x,x_0)$ is the time needed for the orbit of a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered at $x_0$, with small radius $r$. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the SRB measure at $x_0$: for each $x_0$ such that the local dimension $d_{\\mu}(x_0)$ exists, \\lim_{r\\to 0} \\frac{\\log \\tau_r(x,x_0)}{-\\log r} = d_{\\mu}(x_0)-1 holds for $\\mu$ almost each $x$. In a similar way it is possible to consider a quantitative recurrence indicator quantifying the speed of coming back of an orbit to its starting point. Similar results holds for this recurrence indicator.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-07-14T14:01:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C40",
      "37C10",
      "37A25"
    ],
    "keywords": [
      "logarithm law",
      "exponential decay",
      "correlations",
      "local dimension",
      "similar results holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0901.0574G"
    }
  }
}