{
  "id": "1101.4209",
  "version": "v2",
  "published": "2011-01-21T19:28:51.000Z",
  "updated": "2012-01-26T07:58:20.000Z",
  "title": "Brushing the hairs of transcendental entire functions",
  "authors": [
    "Krzysztof Barański",
    "Xavier Jarque",
    "Lasse Rempe"
  ],
  "comment": "19 pages. V2: Small number of minor corrections made from V1",
  "journal": "Topology Appl. 159 (2012), no. 8, 2102-2114",
  "doi": "10.1016/j.topol.2012.02.004",
  "categories": [
    "math.DS",
    "math.CV",
    "math.GN"
  ],
  "abstract": "Let f be a hyperbolic transcendental entire function of finite order in the Eremenko-Lyubich class (or a finite composition of such maps), and suppose that f has a unique Fatou component. We show that the Julia set of $f$ is a Cantor bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of Aarts and Oversteegen. In particular, we show that any two such Julia sets are ambiently homeomorphic. We also show that if $f\\in\\B$ has finite order (or is a finite composition of such maps), but is not necessarily hyperbolic, then the Julia set of f contains a Cantor bouquet. As part of our proof, we describe, for an arbitrary function $f\\in\\B$, a natural compactification of the dynamical plane by adding a \"circle of addresses\" at infinity.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-26T07:58:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "30D05"
    ],
    "keywords": [
      "julia set",
      "finite order",
      "cantor bouquet",
      "hyperbolic transcendental entire function",
      "finite composition"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.4209B"
    }
  }
}