{
  "id": "2307.12928",
  "version": "v1",
  "published": "2023-07-24T16:41:05.000Z",
  "updated": "2023-07-24T16:41:05.000Z",
  "title": "A Recurrence-type Strong Borel--Cantelli Lemma for Axiom A Diffeomorphisms",
  "authors": [
    "Alejandro Rodriguez Sponheimer"
  ],
  "comment": "18 pages, 0 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(X,\\mu,T,d)$ be a metric measure-preserving dynamical system such that $3$-fold correlations decay exponentially for H\\\"older continuous observables. Given a sequence $(M_k)$ that converges to $0$ slowly enough we obtain a strong dynamical Borel--Cantelli result for recurrence, i.e., for $\\mu$-a.e. $x\\in X$ \\[ \\lim_{n \\to \\infty}\\frac{\\sum_{k=1}^{n} \\mathbf{1}_{B_k(x)}(T^{k}x)} {\\sum_{k=1}^{n} \\mu(B_k(x))} = 1, \\] where $\\mu(B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and certain equilibrium states.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-24T16:41:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D20",
      "37A05",
      "37B20"
    ],
    "keywords": [
      "recurrence-type strong borel-cantelli lemma",
      "diffeomorphisms",
      "strong dynamical borel-cantelli result",
      "fold correlations decay",
      "metric measure-preserving dynamical system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}