{
  "id": "math/0505652",
  "version": "v2",
  "published": "2005-05-30T15:24:36.000Z",
  "updated": "2006-05-17T15:15:15.000Z",
  "title": "On Newton's Method for Entire Functions",
  "authors": [
    "Johannes Rueckert",
    "Dierk Schleicher"
  ],
  "comment": "19 pages, 4 figures. Changes in Version 2: Sharpened the result in Section 4",
  "journal": "J London Math Soc (2) 75 (2007) 659-676",
  "doi": "10.1112/jlms/jdm046",
  "categories": [
    "math.DS"
  ],
  "abstract": "The Newton map N_f of an entire function f turns the roots of f into attracting fixed points. Let U be the immediate attracting basin for such a fixed point of N_f. We study the behavior of N_f in a component V of C\\U. If V can be surrounded by an invariant curve within U and satisfies the condition that each point in the extended plane has at most finitely many preimages in V, we show that V contains another immediate basin of N_f or a virtual immediate basin. A virtual immediate basin is an unbounded invariant Fatou component in which the dynamics converges to infty through an absorbing set.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-05-17T15:15:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30D05",
      "37F10",
      "37F20",
      "49M15"
    ],
    "keywords": [
      "entire function",
      "newtons method",
      "virtual immediate basin",
      "fixed point",
      "unbounded invariant fatou component"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......5652R"
    }
  }
}