{
  "id": "1102.4933",
  "version": "v1",
  "published": "2011-02-24T09:08:20.000Z",
  "updated": "2011-02-24T09:08:20.000Z",
  "title": "Hausdorff measure of escaping and Julia sets for bounded type functions of finite order",
  "authors": [
    "Jörn Peter"
  ],
  "comment": "21 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order $\\rho$ become 'smaller' as $\\rho\\to\\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function $h_\\gamma(t)=t^2g(1/t)^\\gamma$, where $g$ is the inverse of a linearizer of some exponential map and $\\gamma\\geq(\\log\\rho(f)+K_1)/c$, but for $\\rho$ large enough, there exists a function $f_\\rho$ of bounded type with order $\\rho$ such that the Hausdorff measures of the escaping set and the Julia set of $f_\\rho$ with respect to $h_{\\gamma'}$ are zero whenever $\\gamma'\\leq(\\log\\rho-K_2)/c$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-24T09:08:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30D05",
      "37F10"
    ],
    "keywords": [
      "hausdorff measure",
      "julia set",
      "bounded type functions",
      "finite order",
      "bounded type transcendental entire functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.4933P"
    }
  }
}