{
  "id": "0810.1673",
  "version": "v3",
  "published": "2008-10-09T15:47:19.000Z",
  "updated": "2010-05-02T14:20:18.000Z",
  "title": "Topology of Fatou components for endomorphisms of CP^k: Linking with the Green's Current",
  "authors": [
    "Suzanne Lynch Hruska",
    "Roland K. W. Roeder"
  ],
  "comment": "Revision 3: Added improvements to the discussion of skew products. 21 pages. Comments welcome",
  "categories": [
    "math.DS",
    "math.GT"
  ],
  "abstract": "Little is known about the global topology of the Fatou set $U(f)$ for holomorphic endomorphisms $f: \\mathbb{CP}^k \\to \\mathbb{CP}^k$, when $k >1$. Classical theory describes $U(f)$ as the complement in $ \\mathbb{CP}^k$ of the support of a dynamically-defined closed positive $(1,1)$ current. Given any closed positive $(1,1)$ current $S$ on $ \\mathbb{CP}^k$, we give a definition of linking number between closed loops in $\\mathbb{CP}^k \\setminus \\supp S$ and the current $S$. It has the property that if $lk(\\gamma,S) \\neq 0$, then $\\gamma$ represents a non-trivial homology element in $H_1(\\mathbb{CP}^k \\setminus \\supp S)$. As an application, we use these linking numbers to establish that many classes of endomorphisms of $\\mathbb{CP}^2$ have Fatou components with infinitely generated first homology. For example, we prove that the Fatou set has infinitely generated first homology for any polynomial endomorphism of $\\mathbb{CP}^2$ for which the restriction to the line at infinity is hyperbolic and has disconnected Julia set. In addition we show that a polynomial skew product of $\\mathbb{CP}^2$ has Fatou set with infinitely generated first homology if some vertical Julia set is disconnected. We then conclude with a section of concrete examples and questions for further study.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-05-02T14:20:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F20",
      "37F45",
      "32Q55",
      "32H50"
    ],
    "keywords": [
      "fatou components",
      "infinitely generated first homology",
      "greens current",
      "fatou set",
      "julia set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.1673L"
    }
  }
}