{
  "id": "1512.09205",
  "version": "v1",
  "published": "2015-12-31T03:10:32.000Z",
  "updated": "2015-12-31T03:10:32.000Z",
  "title": "Multifractal analysis of the divergence points of Birkhoff averages in $beta$-dynamical systems",
  "authors": [
    "Yuanhong Chen",
    "Zhenliang Zhang",
    "Xiaojun Zhao"
  ],
  "comment": "non",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $\\beta$-expansions. More precisely, let $([0,1),T_{\\beta})$ be the $\\beta$-dynamical system for a general $\\beta>1$ and $\\psi:[0,1]\\mapsto\\mathbb{R}$ be a continuous function. Denote by $\\textsf{A}(\\psi,x)$ all the accumulation points of $\\Big\\{\\frac{1}{n}\\sum_{j=0}^{n-1}\\psi(T^jx): n\\ge 1\\Big\\}$. The Hausdorff dimensions of the sets $$\\Big\\{x:\\textsf{A}(\\psi,x)\\supset[a,b]\\Big\\},\\ \\ \\Big\\{x:\\textsf{A}(\\psi,x)=[a,b]\\Big\\}, \\ \\Big\\{x:\\textsf{A}(\\psi,x)\\subset[a,b]\\Big\\}$$ i.e., the points for which the Birkhoff averages of $\\psi$ do not exist but behave in a certain prescribed way, are determined completely for any continuous function $\\psi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-31T03:10:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "28A80"
    ],
    "keywords": [
      "birkhoff averages",
      "divergence points",
      "multifractal analysis",
      "dynamical system",
      "continuous function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151209205C"
    }
  }
}