{
  "id": "1609.02122",
  "version": "v1",
  "published": "2016-09-07T19:20:56.000Z",
  "updated": "2016-09-07T19:20:56.000Z",
  "title": "Entropy, Topological transitivity, and Dimensional properties of unique $q$-expansions",
  "authors": [
    "Rafael Alcaraz Barrera",
    "Simon Baker",
    "Derong Kong"
  ],
  "comment": "56 pages, 7 figures",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "Let $M$ be a positive integer and $q \\in(1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\\{0,\\ldots, M\\}$. In particular, we study the set $\\mathcal{U}_{q}$ of real numbers with a unique $q$-expansion, and the set $\\mathcal{U}_q'$ of corresponding sequences. It was shown by Komornik, Kong and Li (2015) that the function $H$, which associates to each $q\\in(1, M+1]$ the topological entropy of $\\mathcal{U}_q'$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $H$, and characterize the set $\\mathcal{E}$ of $q$'s where the function $H$ is increasing. Moreover, we show that the set $\\mathcal{E}$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $(\\widetilde{\\mathcal{V}}_q', \\sigma),$ which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of $\\mathcal{U}_q$ coincide for all $q\\in(1,M+1]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-07T19:20:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "37B10",
      "37B40",
      "11K55",
      "68R15"
    ],
    "keywords": [
      "topological transitivity",
      "dimensional properties",
      "real numbers",
      "full hausdorff dimension",
      "devils staircase"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}