{
  "id": "2004.13386",
  "version": "v1",
  "published": "2020-04-28T09:32:17.000Z",
  "updated": "2020-04-28T09:32:17.000Z",
  "title": "Periodic intermediate $β$-expansions of Pisot numbers",
  "authors": [
    "Blaine Quackenbush",
    "Tony Samuel",
    "Matthew A. West"
  ],
  "comment": "13 pages, 1 figure",
  "categories": [
    "math.DS"
  ],
  "abstract": "The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are $\\beta$-shifts, namely transformations of the form $T_{\\beta, \\alpha} \\colon x \\mapsto \\beta x + \\alpha \\bmod{1}$ acting on $[-\\alpha/(\\beta - 1), (1-\\alpha)/(\\beta - 1)]$, where $(\\beta, \\alpha) \\in \\Delta$ is fixed and where $\\Delta = \\{ (\\beta, \\alpha) \\in \\mathbb{R}^{2} \\colon \\beta \\in (1,2) \\; \\text{and} \\; 0 \\leq \\alpha \\leq 2-\\beta \\}$. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019), that the set of $(\\beta, \\alpha)$ such that $T_{\\beta, \\alpha}$ has the subshift of finite type property is dense in the parameter space $\\Delta$. Here, they proposed the following question. Given a fixed $\\beta \\in (1, 2)$ which is the $n$-th root of a Perron number, does there exists a dense set of $\\alpha$ in the fiber $\\{\\beta\\} \\times (0, 2- \\beta)$, so that $T_{\\beta, \\alpha}$ has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the property of beginning sofic (that is a factor of a subshift of finite). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269-278, 1980) from the case when $\\alpha = 0$ to the case when $\\alpha \\in (0, 2 - \\beta)$. That is, we examine the structure of the set of eventually periodic points of $T_{\\beta, \\alpha}$ when $\\beta$ is a Pisot number and when $\\beta$ is the $n$-th root of a Pisot number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-28T09:32:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "37B10",
      "11A67",
      "11R06"
    ],
    "keywords": [
      "pisot number",
      "finite type property",
      "periodic intermediate",
      "th root",
      "expansions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}