{
  "id": "math/0608246",
  "version": "v1",
  "published": "2006-08-10T08:33:56.000Z",
  "updated": "2006-08-10T08:33:56.000Z",
  "title": "Numeration systems as dynamical systems -- introduction",
  "authors": [
    "Teturo Kamae"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/074921706000000220 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "IMS Lecture Notes--Monograph Series 2006, Vol. 48, 198-211",
  "doi": "10.1214/074921706000000220",
  "categories": [
    "math.DS"
  ],
  "abstract": "A numeration system originally implies a digitization of real numbers, but in this paper it rather implies a compactification of real numbers as a result of the digitization. By definition, a numeration system with $G$, where $G$ is a nontrivial closed multiplicative subgroup of ${\\mathbb{R}}_+$, is a nontrivial compact metrizable space $\\Omega$ admitting a continuous $(\\lambda\\omega+t)$-action of $(\\lambda,t)\\in G\\times{\\mathbb{R}}$ to $\\omega\\in\\Omega$, such that the $(\\omega+t)$-action is strictly ergodic with the unique invariant probability measure $\\mu_{\\Omega}$, which is the unique $G$-invariant probability measure attaining the topological entropy $|\\log\\lambda|$ of the transformation $\\omega\\mapsto\\lambda\\omega$ for any $\\lambda\\ne 1$. We construct a class of numeration systems coming from weighted substitutions, which contains those coming from substitutions or $\\beta$-expansions with algebraic $\\beta$. It also contains those with $G={\\mathbb{R}}_+$. We obtained an exact formula for the $\\zeta$-function of the numeration systems coming from weighted substitutions and studied the properties. We found a lot of applications of the numeration systems to the $\\beta$-expansions, Fractal geometry or the deterministic self-similar processes which are seen in \\cite{K4}. This paper is based on \\cite{K3} changing the way of presentation. The complete version of this paper is in \\cite{K4}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-10T08:33:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B10"
    ],
    "keywords": [
      "dynamical systems",
      "introduction",
      "numeration systems coming",
      "real numbers",
      "unique invariant probability measure"
    ],
    "tags": [
      "monograph",
      "journal article",
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8246K"
    }
  }
}