{
  "id": "1610.01872",
  "version": "v1",
  "published": "2016-10-06T13:49:53.000Z",
  "updated": "2016-10-06T13:49:53.000Z",
  "title": "Matching for generalised $β$-transformations",
  "authors": [
    "Henk Bruin",
    "Carlo Carminati",
    "Charlene Kalle"
  ],
  "comment": "16 pages, 11 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate matching for the family $T_\\alpha(x) = \\beta x + \\alpha \\pmod 1$, $\\alpha \\in [0,1]$, for fixed $\\beta > 1$. Matching refers to the property that there is an $n \\in \\mathbb N$ such that $T_\\alpha^n(0) = T_\\alpha^n(1)$. We show that for various Pisot numbers $\\beta$, matching occurs on an open dense set of $\\alpha \\in [0,1]$ and we compute the Hausdorff dimension of its complement. Numerical evidence shows more cases where matching is prevalent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-06T13:49:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37E10",
      "11R06",
      "37E05",
      "37E45",
      "37A45"
    ],
    "keywords": [
      "transformations",
      "open dense set",
      "pisot numbers",
      "hausdorff dimension",
      "matching occurs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}