{
  "id": "1202.4902",
  "version": "v2",
  "published": "2012-02-22T13:23:34.000Z",
  "updated": "2013-01-18T13:13:50.000Z",
  "title": "Metrics on tiling spaces, local isomorphism and an application of Brown's Lemma",
  "authors": [
    "Rui Pacheco",
    "Helder Vilarinho"
  ],
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "abstract": "We give an application of a topological dynamics version of multidimensional Brown's lemma to tiling theory: given a tiling of an Euclidean space and a finite geometric pattern of points $F$, one can find a patch such that, for each scale factor $\\lambda$, there is a vector $\\vec{t}_\\lambda$ so that copies of this patch appear in the tilling \"nearly\" centered on $\\lambda F+\\vec{t}_\\lambda$ once we allow \"bounded perturbations\" in the structure of the homothetic copies of $F$. Furthermore, we introduce a new unifying setting for the study of tiling spaces which allows rather general group \"actions\" on patches and we discuss the local isomorphism property of tilings within this setting.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-18T13:13:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tiling spaces",
      "application",
      "local isomorphism property",
      "multidimensional browns lemma",
      "euclidean space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.4902P"
    }
  }
}