{
  "id": "1103.2113",
  "version": "v1",
  "published": "2011-03-10T19:55:03.000Z",
  "updated": "2011-03-10T19:55:03.000Z",
  "title": "A note on Borel--Cantelli lemmas for non-uniformly hyperbolic dynamical systems",
  "authors": [
    "N. Haydn",
    "M. Nicol",
    "T. Persson",
    "S. Vaienti"
  ],
  "comment": "20 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(B_{i})$ be a sequence of measurable sets in a probability space $(X,\\mathcal{B}, \\mu)$ such that $\\sum_{n=1}^{\\infty} \\mu (B_{i}) = \\infty$. The classical Borel-Cantelli lemma states that if the sets $B_{i}$ are independent, then $\\mu (\\{x \\in X : x \\in B_{i} \\text{infinitely often (i.o.)}) = 1$. Suppose $(T,X,\\mu)$ is a dynamical system and $(B_i)$ is a sequence of sets in $X$. We consider whether $T^i x\\in B_i$ for $\\mu$ a.e.\\ $x\\in X$ and if so, is there an asymptotic estimate on the rate of entry. If $T^i x\\in B_i$ infinitely often for $\\mu$ a.e.\\ $x$ we call the sequence $B_i$ a Borel--Cantelli sequence. If the sets $B_i:= B(p,r_i)$ are nested balls about a point $p$ then the question of whether $T^i x\\in B_i$ infinitely often for $\\mu$ a.e.\\ $x$ is often called the shrinking target problem. We show, under certain assumptions on the measure $\\mu$, that for balls $B_i$ if $\\mu (B_i)\\ge i^{-\\gamma}$, $0<\\gamma <1$, then a sufficiently high polynomial rate of decay of correlations for Lipschitz observations implies that the sequence is Borel-Cantelli. If $\\mu (B_i)\\ge \\frac{C\\log i}{i}$ then exponential decay of correlations implies that the sequence is Borel-Cantelli. If it is only assumed that $\\mu (B_i) \\ge \\frac{1}{i}$ then we give conditions in terms of return time statistics which imply that for $\\mu$ a.e.\\ $p$ sequences of nested balls $B(p,1/i)$ are Borel-Cantelli. Corollaries of our results are that for planar dispersing billiards and Lozi maps $\\mu$ a.e.\\ $p$ sequences of nested balls $B(p,1/i)$ are Borel-Cantelli. We also give applications of these results to a variety of non-uniformly hyperbolic dynamical systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-10T19:55:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-uniformly hyperbolic dynamical systems",
      "nested balls",
      "return time statistics",
      "lipschitz observations implies",
      "classical borel-cantelli lemma states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.2113H"
    }
  }
}