{
  "id": "1806.06166",
  "version": "v1",
  "published": "2018-06-16T01:38:27.000Z",
  "updated": "2018-06-16T01:38:27.000Z",
  "title": "$α$-expansions with odd partial quotients",
  "authors": [
    "Florin P. Boca",
    "Claire Merriman"
  ],
  "comment": "13 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider an analogue of Nakada's $\\alpha$-continued fraction transformations in the setting of continued fractions with odd partial quotients. More precisely, given $\\alpha \\in [\\frac{1}{2}(\\sqrt{5}-1),\\frac{1}{2}(\\sqrt{5}+1)]$, we show that every irrational number $x\\in [\\alpha-2,\\alpha)$ can be uniquely represented as $ x=\\frac{e_1}{d_1+\\frac{e_2}{d_2+\\frac{e_3}{d_3+\\cdots}}} $, with $e_i =e_i(x;\\alpha) \\in \\{ \\pm 1\\}$ and $d_i =d_i(x;\\alpha) \\in 2{\\mathbb N} -1$ determined by the iterates of the transformation $\\varphi_\\alpha (x) := \\frac{1}{| x|} - 2 \\left[ \\frac{1}{2| x|} +\\frac{1-\\alpha}{2} \\right]-1$ of $[\\alpha -2,\\alpha)$. We also describe the natural extension of $\\varphi_\\alpha$ and prove that the endomorphism $\\varphi_\\alpha$ is exact.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-16T01:38:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E05",
      "11J70",
      "11K50",
      "37A35"
    ],
    "keywords": [
      "odd partial quotients",
      "expansions",
      "continued fraction transformations",
      "natural extension",
      "irrational number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}