{
  "id": "1504.03094",
  "version": "v1",
  "published": "2015-04-13T08:17:24.000Z",
  "updated": "2015-04-13T08:17:24.000Z",
  "title": "Dynamics of semigroups of entire maps in $\\mathbb{C}^k$",
  "authors": [
    "Sayani Bera",
    "Ratna Pal"
  ],
  "comment": "12 pages",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\\mathbb{C}^k,\\; k \\ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of $\\mathbb{C}^k,\\; k \\ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in $\\mathbb{C}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of $\\mathbb{C}^k.$ This generalizes a theorem of Fornaess-Sibony. Secondly, we define recurrent domains for semigroups and provide a description of such domains under some conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-13T08:17:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32H02",
      "32H50"
    ],
    "keywords": [
      "entire maps",
      "holomorphic endomorphisms",
      "julia set",
      "define recurrent domains",
      "maximal generic rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}