{
  "id": "math/9412233",
  "version": "v1",
  "published": "1994-12-08T00:00:00.000Z",
  "updated": "1994-12-08T00:00:00.000Z",
  "title": "Laminations in holomorphic dynamics",
  "authors": [
    "Mikhail Lyubich",
    "Yair Minsky"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We suggest a way to associate to a rational map of the Riemann sphere a three dimensional object called a hyperbolic orbifold 3-lamination. The relation of this object to the map is analogous to the relation of a hyperbolic 3-manifold to a Kleinian group. In order to construct the 3-lamination we analyze the natural extension of a rational map and the complex affine structure on the canonical 2-dimensional leaf space contained in it. In this paper the construction is carried out in full for post-critically finite maps. We show that the corresponding laminations have a compact convex core. As a first application we give a three-dimensional proof of Thurston's rigidity for post-critically finite mappings, via the \"lamination extension\" of the proofs of the Mostow and Marden rigidity and isomorphism theorems for hyperbolic 3-manifolds. An Ahlfors-type argument for zero measure of the Julia set is applied along the way. This approach also provides a new point of view on the Lattes deformable examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1994-12-08T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "holomorphic dynamics",
      "rational map",
      "complex affine structure",
      "compact convex core",
      "zero measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1994math.....12233L"
    }
  }
}