On hyperbolic metric and asymptotically finite invariant differentials in holomorphic dynamics

2015/09/23 by Carlos Cabrera, Cabrera, Carlos, Peter Makienko +1
Mathematics · #32A70 #37F10 #37F30 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.DS #msc:32A70 #msc:37F10 #msc:37F30

paper · pdf · doi:10.48550/arxiv.1509.07105

10 pages

arxiv created 2015/09/23 · openalex publication_date 2015/09/23 · arxiv updated 2015/09/24 · openalex created_date 2022/09/03 · openalex updated_date 2026/07/28

Abstract

Given a rational map R, we consider the complement of the postcritical set S_R. In this paper we discuss the existence of invariant Beltrami differentials supported on a R invariant subset A of S_R. Under some geometrical restrictions, either on the hyperbolic geometry of A or on the asymptotic behavior of infinitesimal geodesics of the Teichmüller space of S_R, we show the absence of invariant Beltrami differentials supported on A. In particular, we show that if A has finite hyperbolic area, then A can not support invariant Beltrami differentials except in the case where R is a Lattès map.

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