{
  "id": "1704.02604",
  "version": "v1",
  "published": "2017-04-09T14:00:55.000Z",
  "updated": "2017-04-09T14:00:55.000Z",
  "title": "$β$-transformation on an interval with hole",
  "authors": [
    "Nikita Agarwal"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $T_{\\beta}$ be the expanding map of $[0,1]$ defined by $T_{\\beta}(x) = \\beta x\\ \\text{mod 1}$, where $\\beta$ is an integer atleast 2. Given $0\\leq a<b\\leq 1$, let $\\mathcal{W}_{\\beta}(a,b)=\\{x\\in [0,1]\\ \\vert \\ T_{\\beta}^nx\\notin (a,b), \\ n\\geq 0\\}$ be the maximal $T$-invariant subset of $[0,1]\\setminus (a,b)$. We characterize the intervals $(a,b)$ for which the Hausdorff dimension of $\\mathcal{W}_{\\beta}(a,b)$ is positive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-09T14:00:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A05",
      "28D05"
    ],
    "keywords": [
      "transformation",
      "integer atleast",
      "invariant subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}