{
  "id": "1305.7164",
  "version": "v1",
  "published": "2013-05-30T17:10:59.000Z",
  "updated": "2013-05-30T17:10:59.000Z",
  "title": "On Poincaré extensions of rational maps",
  "authors": [
    "Carlos Cabrera",
    "Peter Makienko",
    "Guillermo Sienra"
  ],
  "comment": "25 pages, 1 figure",
  "categories": [
    "math.DS",
    "math.CV"
  ],
  "abstract": "There is a classical extension, of M\\\"obius automorphisms of the Riemann sphere into isometries of the hyperbolic space $\\mathbb{H}^3$, which is called the Poincar\\'e extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of $\\mathbb{H}^3$ exploiting the fact that any holomorphic covering between Riemann surfaces is M\\\"obius for a suitable choice of coordinates. We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a product in $\\mathbb{H}^3$ that allows to construct a visual extension of any given rational map.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-30T17:10:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational map",
      "riemann sphere",
      "extensions define conformally natural homomorphisms",
      "construct extensions",
      "visual extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.7164C"
    }
  }
}