{
  "id": "1412.6384",
  "version": "v1",
  "published": "2014-12-19T15:40:25.000Z",
  "updated": "2014-12-19T15:40:25.000Z",
  "title": "Asymmetrical holes for the $β$-transformation",
  "authors": [
    "Lyndsey Clark"
  ],
  "comment": "20 pages, 6 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "This paper extends those of Glendinning and Sidorov [3] and of Hare and Sidorov [6] from the case of the doubling map to the more general $\\beta$-transformation. Let $\\beta \\in (1,2)$ and consider the $\\beta$-transformation $T_{\\beta}(x)=\\beta x \\pmod 1$. Let $\\mathcal{J}_{\\beta} (a,b) := \\{ x \\in (0,1) : T_{\\beta}^n(x) \\notin (a,b) \\text{ for all } n \\geq 0 \\}$. An integer $n$ is bad for $(a,b)$ if every $n$-cycle for $T_{\\beta}$ intersects $(a,b)$. Denote the set of all bad $n$ for $(a,b)$ by $B_\\beta(a,b)$. In this paper we completely describe the following sets: \\[ D_0(\\beta) = \\{ (a,b) \\in [0,1)^2 : \\mathcal{J}_{\\beta}(a,b) \\neq \\emptyset \\}, \\] \\[ D_1(\\beta) = \\{ (a,b) \\in [0,1)^2 : \\mathcal{J}_{\\beta}(a,b) \\text{ is uncountable} \\}, \\] \\[ D_2(\\beta) = \\{ (a,b) \\in [0,1)^2 : B_\\beta(a,b) \\text{ is finite} \\}. \\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-19T15:40:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28D05",
      "37B10",
      "68R15"
    ],
    "keywords": [
      "asymmetrical holes",
      "transformation",
      "paper extends",
      "intersects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.6384C"
    }
  }
}