{
  "id": "2102.08627",
  "version": "v1",
  "published": "2021-02-17T08:36:48.000Z",
  "updated": "2021-02-17T08:36:48.000Z",
  "title": "Dynamical behavior of alternate base expansions",
  "authors": [
    "Émilie Charlier",
    "Célia Cisternino",
    "Karma Dajani"
  ],
  "comment": "28 pages, 15 figures",
  "categories": [
    "math.DS",
    "cs.DM",
    "math.RT"
  ],
  "abstract": "We generalize the greedy and lazy $\\beta$-transformations for a real base $\\beta$ to the setting of alternate bases $\\boldsymbol{\\beta}=(\\beta_0,\\ldots,\\beta_{p-1})$, which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_\\boldsymbol{\\beta}$ and $L_\\boldsymbol{\\beta}$ respectively, can be iterated in order to generate the digits of the greedy and lazy $\\boldsymbol{\\beta}$-expansions of real numbers. The aim of this paper is to describe the dynamical behaviors of $T_\\boldsymbol{\\beta}$ and $L_\\boldsymbol{\\beta}$. We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the $p$-Lebesgue measure) $T_\\boldsymbol{\\beta}$-invariant measure. We then show that this unique measure is in fact equivalent to the $p$-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $\\frac{1}{p}\\log(\\beta_{p-1}\\cdots \\beta_0)$. We then express the density of this measure and compute the frequencies of letters in the greedy $\\boldsymbol{\\beta}$-expansions. We obtain the dynamical properties of $L_\\boldsymbol{\\beta}$ by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $\\beta$-shift. Finally, we show that the $\\boldsymbol{\\beta}$-expansions can be seen as $(\\beta_{p-1}\\cdots \\beta_0)$-representations over general digit sets and we compare both frameworks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-17T08:36:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "37E05",
      "37A45",
      "28D05"
    ],
    "keywords": [
      "alternate base expansions",
      "dynamical behavior",
      "dynamical system",
      "general digit sets",
      "real base case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}