{
  "id": "1903.08331",
  "version": "v1",
  "published": "2019-03-20T03:48:51.000Z",
  "updated": "2019-03-20T03:48:51.000Z",
  "title": "On the specification property and synchronisation of unique $q$-expansions",
  "authors": [
    "Rafael Alcaraz Barrera"
  ],
  "comment": "44 pages",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "We consider unique expansions of a real numbers in base $q \\in (1,M+1]$. We study some dynamical properties of the naturally ocurring subshift $(\\mathbf{V}_q, \\sigma)$. In particular, we characterise the set of bases $q \\subset (1,M+1]$ such that $(\\mathbf{V}_q, \\sigma)$ has the specification property and the set of bases $q \\in (1, M+1]$ such that $(\\mathbf{V}_q, \\sigma)$ is a synchronised subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of $q$. We also calculate the size of such classes giving similar results to those shown by Schmeling in (Ergodic Theory and Dynamical Systems, 17:675--694, 6 1997) in the context of $\\beta$-transformations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-20T03:48:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "11A63",
      "37B40",
      "68R15"
    ],
    "keywords": [
      "specification property",
      "synchronisation",
      "dynamical properties",
      "classes giving similar results",
      "real numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}