{
  "id": "1212.1320",
  "version": "v2",
  "published": "2012-12-06T13:21:48.000Z",
  "updated": "2014-01-08T16:02:16.000Z",
  "title": "Complexity as a homeomorphism invariant for tiling spaces",
  "authors": [
    "Antoine Julien"
  ],
  "comment": "Added a the result on the repetitivity function; rearranged some parts of the article; corrected a few minor mistakes. What was previously the last part was shrunk to an outlook. Links to deformations, groupoids and groupoid cohomology will be fully addressed in a future paper",
  "categories": [
    "math.DS"
  ],
  "abstract": "It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the complexity is polynomial, that the exponent of the leading term is preserved by homeomorphism). This theorem can be reworded in terms of $d$-dimensional infinite words: if two $\\mathbb{Z}^d$-subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent. An analogue theorem is proved for the repetitivity function, which is a quantitative measure of the recurrence of orbits in the tiling space. How this result relates to the theory of tilings deformations is outlined in the last part.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-08T16:02:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B50",
      "37B10"
    ],
    "keywords": [
      "homeomorphism invariant",
      "finite local complexity",
      "dimensional infinite words",
      "tilings deformations",
      "homeomorphic tiling spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.1320J"
    }
  }
}