{
  "id": "0907.2676",
  "version": "v3",
  "published": "2009-07-15T18:17:04.000Z",
  "updated": "2010-01-29T13:57:06.000Z",
  "title": "Beta-expansions, natural extensions and multiple tilings associated with Pisot units",
  "authors": [
    "Charlene Kalle",
    "Wolfgang Steiner"
  ],
  "journal": "Transactions of the American Mathematical Society 364, 5 (2012) 2281-2318",
  "doi": "10.1090/S0002-9947-2012-05362-1",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "abstract": "From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\\beta$ and the greedy $\\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\\beta$-expansions. Remarkably, the symmetric $\\beta$-transformation does not satisfy this condition when $\\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-01-29T13:57:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A63",
      "11R06",
      "28A80",
      "28D05",
      "37B10",
      "52C22",
      "52C23"
    ],
    "keywords": [
      "pisot unit",
      "multiple tilings",
      "natural extensions",
      "beta-expansions",
      "smallest pisot number"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS",
      "journal": "Trans. Amer. Math. Soc."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.2676K"
    }
  }
}