{
  "id": "1512.04479",
  "version": "v1",
  "published": "2015-12-14T19:31:53.000Z",
  "updated": "2015-12-14T19:31:53.000Z",
  "title": "Patterns of Negative Shifts and Beta-Shifts",
  "authors": [
    "Sergi Elizalde",
    "Katherine Moore"
  ],
  "comment": "Preliminary Draft. 20 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "The $\\beta$-shift is the transformation from the unit interval to itself that maps $x$ to the fractional part of $\\beta x$. Permutations realized by the relative order of the elements in the orbits of these maps have been studied for positive integer values of $\\beta$ and for real values $\\beta>1$. In both cases, a combinatorial description of the smallest positive value of $\\beta$ needed to realize a permutation is provided. In this paper we extend these results to the case of negative $\\beta$, both in the integer and in the real case. Negative $\\beta$-shifts are related to digital expansions with negative real bases, studied by Ito and Sadahiro, and Liao and Steiner.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-14T19:31:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A05",
      "37M10"
    ],
    "keywords": [
      "negative shifts",
      "beta-shifts",
      "unit interval",
      "combinatorial description",
      "smallest positive value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}