{
  "id": "1707.09321",
  "version": "v1",
  "published": "2017-07-28T16:44:24.000Z",
  "updated": "2017-07-28T16:44:24.000Z",
  "title": "Natural Extensions for Nakada's alpha-expansions: descending from 1 to g^2",
  "authors": [
    "Jaap de Jonge",
    "Cor Kraaikamp"
  ],
  "comment": "29 pages, 20 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "By means of singularisations and insertions in Nakada's alpha-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map T_alpha is given for (\\sqrt{10}-2)/3\\leq\\alpha<1. From our construction it follows that \\Omega_\\alpha, the domain of the natural extension of T_\\alpha, is metrically isomorphic to \\Omega_g for \\alpha \\in [g^2,g), where g is the small golden mean. Finally, although \\Omega_\\alpha proves to be very intricate and unmanageable for \\alpha \\in [g^2, (\\sqrt{10}-2)/3), the \\alpha-Legendre constant L(\\alpha) on this interval is explicitly given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-28T16:44:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J70",
      "11K50"
    ],
    "keywords": [
      "natural extension",
      "nakadas alpha-expansions",
      "nakadas continued fraction map",
      "small golden mean",
      "introducing partial quotients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}