{
  "id": "0904.1295",
  "version": "v1",
  "published": "2009-04-08T09:46:21.000Z",
  "updated": "2009-04-08T09:46:21.000Z",
  "title": "Entire functions with Julia sets of positive measure",
  "authors": [
    "Magnus Aspenberg",
    "Walter Bergweiler"
  ],
  "comment": "17 pages",
  "journal": "Math. Ann. 352 (2012), 27-54",
  "doi": "10.1007/s00208-010-0625-0",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-08T09:46:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "30D05",
      "30D15"
    ],
    "keywords": [
      "julia set",
      "positive measure",
      "transcendental entire function",
      "denjoy-carleman-ahlfors theorem implies",
      "asymptotic values"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.1295A"
    }
  }
}