{
  "id": "2202.07344",
  "version": "v1",
  "published": "2022-02-15T12:07:12.000Z",
  "updated": "2022-02-15T12:07:12.000Z",
  "title": "A strong Borel--Cantelli lemma for recurrence",
  "authors": [
    "Tomas Persson"
  ],
  "comment": "14 pages, 0 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Consider a mixing dynamical systems $([0,1], T, \\mu)$, for instance a piecewise expanding interval map with a Gibbs measure $\\mu$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $\\mu (B(x, r_k(x)) = m_k$. It is proved that for almost all $x$, the number of $k \\leq n$ such that $T^k (x) \\in B_k (x)$ is approximately equal to $m_1 + \\ldots + m_n$. This is a sort of strong Borel--Cantelli lemma for recurrence. A consequence is that \\[ \\lim_{r \\to 0} \\frac{\\log \\tau_{B(x,r)} (x)}{- \\log \\mu (B (x,r))} = 1 \\] for almost every $x$, where $\\tau$ is the return time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-15T12:07:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "37A05",
      "37B20"
    ],
    "keywords": [
      "strong borel-cantelli lemma",
      "recurrence",
      "return time",
      "gibbs measure",
      "piecewise expanding interval map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}