{
  "id": "1004.2281",
  "version": "v2",
  "published": "2010-04-13T22:31:39.000Z",
  "updated": "2018-07-06T21:07:28.000Z",
  "title": "Exact Regularity and the Cohomology of Tiling Spaces",
  "authors": [
    "Lorenzo Sadun"
  ],
  "comment": "Updated to published version",
  "journal": "Ergodic Theory and Dynamical Systems 31 (2011) 1819-1834",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The Exact Regularity Property was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider exact regularity for arbitrary tiling spaces. Let ${T}$ be a $d$ dimensional repetitive tiling, and let $\\Omega_{{T}}$ be its hull. If $\\check H^d(\\Omega_{{T}}, Q) = Q^k$, then there exist $k$ patches whose appearance govern the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling ${T}$ comes from a substitution, then we can quantify that convergence rate. If ${T}$ is also one-dimensional, we put constraints on the measure of any cylinder set in $\\Omega_{{T}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-13T22:31:39.000Z",
      "abstract": "The Exact Regularity Property was introduced recently as a property of homological Pisot substitutions in one dimension. In this paper, we consider exact regularity for arbitrary tiling spaces. Let $\\bT$ be a $d$ dimensional repetitive tiling, and let $\\Omega_{\\bT}$ be its hull. If $\\check H^d(\\Omega_{\\bT}, \\Q) = \\Q^k$, then there exist $k$ patches whose appearance govern the number of appearances of every other patch. This gives uniform estimates on the convergence of all patch frequencies to the ergodic limit. If the tiling $\\bT$ comes from a substitution, then we can quantify that convergence rate. If $\\bT$ is also one-dimensional, we put constraints on the measure of any cylinder set in $\\Omega_{\\bT}$.",
      "comment": "15 pages, including 3 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-07-06T21:07:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B50",
      "54H20",
      "37B10",
      "55N05",
      "55N35",
      "52C23"
    ],
    "keywords": [
      "cohomology",
      "exact regularity property",
      "cylinder set",
      "homological pisot substitutions",
      "convergence rate"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 852084,
      "adsabs": "2010arXiv1004.2281S"
    }
  }
}