{
  "id": "0905.0739",
  "version": "v2",
  "published": "2009-05-06T13:05:31.000Z",
  "updated": "2012-05-02T21:03:21.000Z",
  "title": "Irregular sets, the $β$-transformation and the almost specification property",
  "authors": [
    "Daniel J. Thompson"
  ],
  "comment": "To appear in Transactions of the American Mathematical Society. Final version was submitted to TAMS on November 24th 2010, and it appears I did not update the arXiv version at the time! Changes from version 1 are mostly minor - typos fixed, references improved. The most significant change is to section 5.1, where the results have been improved",
  "categories": [
    "math.DS"
  ],
  "abstract": "Let $(X,d)$ be a compact metric space, $f:X \\mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $\\phi$ be a continuous function on $X$. We show that the set of points for which the Birkhoff average of $\\phi$ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every $\\beta$-shift satisfies almost specification and we show that the irregular set for any $\\beta$-shift or $\\beta$-transformation is either empty or has full topological entropy and Hausdorff dimension.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-02T21:03:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37C45"
    ],
    "keywords": [
      "irregular set",
      "specification property",
      "transformation",
      "full topological entropy",
      "compact metric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.0739T"
    }
  }
}