{
  "id": "1001.2027",
  "version": "v2",
  "published": "2010-01-12T21:23:38.000Z",
  "updated": "2018-07-06T21:10:10.000Z",
  "title": "Homological Pisot Substitutions and Exact Regularity",
  "authors": [
    "Marcy Barge",
    "Henk Bruin",
    "Leslie Jones",
    "Lorenzo Sadun"
  ],
  "comment": "16 pages, LaTeX, no figures",
  "journal": "Israel Journal of Mathematics 188 (2012), 281-300",
  "categories": [
    "math.DS",
    "math.AT"
  ],
  "abstract": "We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such \"homological Pisot\" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-12T21:23:38.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-07-06T21:10:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B50",
      "54H20",
      "11R06",
      "37B10",
      "55N05",
      "55N35",
      "52C23"
    ],
    "keywords": [
      "homological pisot substitutions",
      "first rational cech cohomology",
      "pure discrete spectra",
      "exact regularity property",
      "one-dimensional substitution tiling spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1001.2027B"
    }
  }
}