{
  "id": "1302.2486",
  "version": "v2",
  "published": "2013-02-11T14:37:30.000Z",
  "updated": "2013-09-11T12:15:00.000Z",
  "title": "The doubling map with asymmetrical holes",
  "authors": [
    "Paul Glendinning",
    "Nikita Sidorov"
  ],
  "comment": "26 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $0<a<b<1$ and let $T$ be the doubling map. Set $\\mathcal J(a,b):=\\{x\\in[0,1] : T^nx\\notin (a,b), n\\ge0\\}$. In this paper we completely characterize the holes $(a,b)$ for which any of the following scenarios holds: {enumerate} $\\mathcal J(a,b)$ contains a point $x\\in(0,1)$; $\\mathcal J(a,b)\\cap [\\de,1-\\de]$ is infinite for any fixed $\\de>0$; $\\mathcal J(a,b)$ is uncountable of zero Hausdorff dimension; $\\mathcal J(a,b)$ is of positive Hausdorff dimension. {enumerate} In particular, we show that (iv) is always the case if \\[ b-a<\\frac14\\prod_{n=1}^\\infty \\bigl(1-2^{-2^n}\\bigr)\\approx 0.175092 \\] and that this bound is sharp. As a corollary, we give a full description of first and second order critical holes introduced in \\cite{SSC} for the doubling map. Furthermore, we show that our model yields a continuum of \"routes to chaos\" via arbitrary sequences of products of natural numbers, thus generalizing the standard route to chaos via period doubling.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-11T12:15:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28D05",
      "37B10"
    ],
    "keywords": [
      "doubling map",
      "asymmetrical holes",
      "zero hausdorff dimension",
      "second order critical holes",
      "scenarios holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.2486G"
    }
  }
}