1994/12/08 by Mikhail Lyubich, Yair N. Minsky, Lyubich, Mikhail +1 · 6 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.math/9412233
openalex publication_date 1994/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.