{
  "id": "2010.07714",
  "version": "v1",
  "published": "2020-10-15T12:44:11.000Z",
  "updated": "2020-10-15T12:44:11.000Z",
  "title": "On eventually always hitting points",
  "authors": [
    "Charis Ganotaki",
    "Tomas Persson"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider dynamical systems $(X,T,\\mu)$ which have exponential decay of correlations for either H\\\"older continuous functions or functions of bounded variation. Given a sequence of balls $(B_n)_{n=1}^\\infty$, we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points $x$ such that for all large enough $m$, there is a $k < m$ with $T^k (x) \\in B_m$. We also give an asymptotic estimate as $m \\to \\infty$ on the number of $k < m$ with $T^k (x) \\in B_m$. As an application, we prove for almost every point $x$ an asymptotic estimate on the number of $k \\leq m$ such that $a_k \\geq m^t$, where $t \\in (0,1)$ and $a_k$ are the continued fraction coefficients of $x$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-15T12:44:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A50",
      "37E05",
      "37D20",
      "11J70"
    ],
    "keywords": [
      "hitting points",
      "asymptotic estimate",
      "full measure",
      "exponential decay",
      "sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}