{
  "id": "1803.07338",
  "version": "v1",
  "published": "2018-03-20T09:52:19.000Z",
  "updated": "2018-03-20T09:52:19.000Z",
  "title": "The $β$-transformation with a hole at 0",
  "authors": [
    "Charlene Kalle",
    "Derong Kong",
    "Niels Langeveld",
    "Wenxia Li"
  ],
  "comment": "32 pages, 4 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "For $\\beta\\in(1,2]$ the $\\beta$-transformation $T_\\beta: [0,1) \\to [0,1)$ is defined by $T_\\beta ( x) = \\beta x \\pmod 1$. For $t\\in[0, 1)$ let $K_\\beta(t)$ be the survivor set of $T_\\beta$ with hole $(0,t)$ given by \\[K_\\beta(t):=\\{x\\in[0, 1): T_\\beta^n(x)\\not \\in (0, t) \\textrm{ for all }n\\ge 0\\}.\\] In this paper we characterise the bifurcation set $E_\\beta$ of all parameters $t\\in[0,1)$ for which the set valued function $t\\mapsto K_\\beta(t)$ is not locally constant. We show that $E_\\beta$ is a Lebesgue null set of full Hausdorff dimension for all $\\beta\\in(1,2)$. We prove that for Lebesgue almost every $\\beta\\in(1,2)$ the bifurcation set $E_\\beta$ contains both infinitely many isolated and accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\\beta\\in(1,2)$ for which $E_\\beta$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_2$, the bifurcation set of the doubling map. Finally, we give for each $\\beta \\in (1,2)$ a lower and upper bound for the value $\\tau_\\beta$, such that the Hausdorff dimension of $K_\\beta(t)$ is positive if and only if $t< \\tau_\\beta$. We show that $\\tau_\\beta \\le 1-\\frac1{\\beta}$ for all $\\beta \\in (1,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-20T09:52:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K55",
      "11A63",
      "68R15",
      "26A30",
      "28D05",
      "37B10",
      "37E05",
      "37E15"
    ],
    "keywords": [
      "bifurcation set",
      "transformation",
      "zero hausdorff dimension",
      "accumulation points arbitrarily close",
      "full hausdorff dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}