{
  "id": "1401.7027",
  "version": "v3",
  "published": "2014-01-27T21:22:25.000Z",
  "updated": "2015-03-05T13:34:56.000Z",
  "title": "Intermediate β-shifts of finite type",
  "authors": [
    "Bing Li",
    "Tuomas Sahlsten",
    "Tony Samuel"
  ],
  "comment": "v3: 19 pages, 6 figures, fixed typos and minor errors, to appear in Discrete Contin. Dyn. Syst. A",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $\\beta$-shift (transformation) and an intermediate $\\beta$-shift (transformation), for a fixed $\\beta \\in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{\\beta,\\alpha} \\colon x \\mapsto \\beta x + \\alpha \\bmod 1$ for which the corresponding intermediate $\\beta$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(\\beta,\\alpha)$ such that the intermediate $\\beta$-shift associated with $T_{\\beta, \\alpha}$ is a subshift of finite type. It is also proved that these maps $T_{\\beta,\\alpha}$ are not transitive. This is in contrast to the situation for the corresponding greedy and lazy $\\beta$-shifts and $\\beta$-transformations, for which both of the two properties do not hold.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-25T08:31:24.000Z",
      "abstract": "An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $\\beta$-shift (transformation) and an intermediate $\\beta$-shift (transformation), for a fixed $\\beta \\in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{\\beta,\\alpha} \\colon x \\mapsto \\beta x + \\alpha \\bmod 1$ for which the corresponding intermediate $\\beta$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(\\beta,\\alpha)$ such that the intermediate $\\beta$-shift associated with $T_{\\beta, \\alpha}$ is a subshift of finite type and where the map $T_{\\beta,\\alpha}$ is not transitive. This is in contrast to the situation for the corresponding greedy and lazy $\\beta$-shifts and $\\beta$-transformations, in that these two properties do not hold.",
      "comment": "18 pages, 6 figures; v2 includes the verification of an extra condition required in the proof of the second part of Theorem 1.1",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-03-05T13:34:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10",
      "58F17",
      "11A67",
      "11R06"
    ],
    "keywords": [
      "finite type",
      "transformation",
      "highlight dynamical differences",
      "linear maps",
      "classification"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.7027L"
    }
  }
}