{
  "id": "math/0410384",
  "version": "v1",
  "published": "2004-10-18T07:10:03.000Z",
  "updated": "2004-10-18T07:10:03.000Z",
  "title": "Hitting and return times in ergodic dynamical systems",
  "authors": [
    "N. Haydn",
    "Y. Lacroix",
    "S. Vaienti"
  ],
  "comment": "8 pages",
  "categories": [
    "math.DS",
    "math.PR"
  ],
  "abstract": "Given an ergodic dynamical system $(X,T,\\mu)$, and $U\\subset X$ measurable with $\\mu (U)>0$, let $\\mu (U)\\tau_U(x)$ denote the normalized hitting time of $x\\in X$ to $U$. We prove that given a sequence $(U_n)$ with $\\mu (U_n)\\to 0$, the distribution function of the normalized hitting times to $U_n$ converges weakly to some sub-probability distribution $F$ if and only if the distribution function of the normalized return time converges weakly to some distribution function $\\tilde F$, and that in the converging case, $$ F(t)=\\int_0^t(1-\\tilde F(s))ds, t\\ge 0.\\tag$\\diamondsuit$ $$ This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is too.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-18T07:10:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "37A50",
      "28D05"
    ],
    "keywords": [
      "ergodic dynamical system",
      "distribution function",
      "normalized hitting time",
      "sub-probability distribution",
      "normalized return time converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10384H"
    }
  }
}