{
  "id": "1509.04900",
  "version": "v1",
  "published": "2015-09-16T12:18:15.000Z",
  "updated": "2015-09-16T12:18:15.000Z",
  "title": "Subshift of finite type and self-similar sets",
  "authors": [
    "Karma Dajani",
    "Kan Jiang"
  ],
  "comment": "28 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $K\\subset \\mathbb{R}$ be a self-similar set generated by some iterated function system. In this paper we prove, under some assumptions, that $K$ can be identified with a subshift of finite type. With this identification, we can calculate the Hausdorff dimension of $K$ as well as the set of elements in $K$ with unique codings using the machinery of Mauldin and Williams \\cite{MW}. We give three different applications of our main result. Firstly, we calculate the Hausdorff dimension of the set of points of $K$ with multiple codings. Secondly, in the setting of $\\beta$-expansions, when the set of all the unique codings is not a subshift of finite type, we can calculate in some cases the Hausdorff dimension of the univoque set. This application generalizes a result of de Vries and Komornik \\cite{MK}. Thirdly, for the doubling map with asymmetrical holes, we give a sufficient condition such that the attractor can be identified with a subshift of finite type. The third application partially answers a problem posed by Barrera \\cite{Barrera}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-16T12:18:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite type",
      "self-similar set",
      "hausdorff dimension",
      "unique codings",
      "third application partially answers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150904900D"
    }
  }
}