{
  "id": "2209.08882",
  "version": "v1",
  "published": "2022-09-19T09:36:22.000Z",
  "updated": "2022-09-19T09:36:22.000Z",
  "title": "Matching of orbits of certain $N$-expansions with a finite set of digits",
  "authors": [
    "Yufei Chen",
    "Cor Kraaikamp"
  ],
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "In this paper we consider a class of continued fraction expansions: the so-called $N$-expansions with a finite digit set, where $N\\geq 2$ is an integer. These \\emph{$N$-expansions with a finite digit set} were introduced in [KL,L], and further studied in [dJKN,S]. For $N$ fixed they are steered by a parameter $\\alpha\\in (0,\\sqrt{N}-1]$. In [KL], for $N=2$ an explicit interval $[A,B]$ was determined, such that for all $\\alpha\\in [A,B]$ the entropy $h(T_{\\alpha})$ of the underlying Gauss-map $T_{\\alpha}$ is equal. In this paper we show that for all $N\\in \\mathbb N$, $N\\geq 2$, such plateaux exist. In order to show that the entropy is constant on such plateaux, we obtain the underlying planar natural extension of the maps $T_{\\alpha}$, the $T_{\\alpha}$-invariant measure, ergodicity, and we show that for any two $\\alpha,\\alpha'$ from the same plateau, the natural extensions are metrically isomorphic, and the isomorphism is given explicitly. The plateaux are found by a property called matching.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-19T09:36:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite set",
      "finite digit set",
      "planar natural extension",
      "explicit interval",
      "continued fraction expansions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}