{
  "id": "1804.10235",
  "version": "v1",
  "published": "2018-04-26T18:33:14.000Z",
  "updated": "2018-04-26T18:33:14.000Z",
  "title": "On substitution tilings and Delone sets without finite local complexity",
  "authors": [
    "Jeong-Yup Lee",
    "Boris Solomyak"
  ],
  "comment": "33 pages, 3 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [Lee-Solomyak (2012)] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-26T18:33:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B50",
      "52C23"
    ],
    "keywords": [
      "finite local complexity",
      "substitution tilings",
      "delone sets",
      "tiling dynamical system",
      "meyer set property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}