Quasiconformal homeomorphisms and dynamics. III: The Teichmüller space of a holomorphic dynamical system
type: Item
instance of: scholarly article
title: Quasiconformal homeomorphisms and dynamics. III: The Teichmüller space of a holomorphic dynamical system
zbMATH Open document ID: 0926.30028
author: Curtis T. McMullen
author: Dennis Sullivan
published in: Advances in Mathematics
publication date: 1999-11-24T00:00:00Z
full work available at URL: urn-3:HUL.InstRepos:3446030
review text: [For part II see \textit{D. Sullivan} in Acta Math. 155, 243-260 (1985; Zbl 0606.30044).] This paper is an expanded version of a preprint from 1983 by D. Sullivan with the same title. The goal of the paper is to explicate and complete the foundational material required to use the method of quasiconformal maps to study holomorphic dynamical systems in one variable. It achieves that goal while at the same time giving a very nice overview of the remarkable advances achieved in this subject in the intervening 15 years. While not completely self contained, it is certainly accesible to a reader with a basic knowledge of functions of a complex variable as well as the vocabulary of holomorphic dynamical systems. Let \(f:\widehat{\mathbb C}\to\widehat{\mathbb C}\) be a rational map. A rational map defines, by iteration a dynamical system on the Riemann sphere. Evidently maps which are conjugate, by a homeomorphism have qualitatively identical dynamics. In order to understand the dynamics of such maps the authors consider when two such maps are quasiconformally conjugate. Motivated by the deformation theory of Riemann surfaces, they go on to construct a Teichmüller space consisting of ``marked'' quasiconformal conjugacy classes of rational maps. The dynamics of \(f\) divide the sphere into a compact, usually fractal set \(J\), the Julia set, on which the dynamics are quite chaotic and its complement \(\Omega\), the Fatou set consisting of countably many stable regions each of which is eventually periodic under forward iteration. The first result proved in the paper is the classification of the stable regions into five types: parabolic, attractive and superattractive basins, Siegel disks and Herman rings. The authors then give a very general definition of a Teichmüller space, modular group and (pre)metric for a holomorphic dynamical system. They quickly review the more classical case of a Kleinian group before considering the Teichmüller space of a rational map. They prove tha…
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 37F30
Mathematics Subject Classification ID: 30C62
Mathematics Subject Classification ID: 30D05
zbMATH DE Number: 1211558
zbMATH Keywords: holomorphic dynamics
zbMATH Keywords: quasiconformal
zbMATH Keywords: Teichmüller space
zbMATH Keywords: rational maps
zbMATH Keywords: Fatou set
zbMATH Keywords: Julia set
Wikidata QID: Q101088249
reviewed by: C. P. Please
MaRDI profile type: Publication
OpenAlex ID: W1975292209
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DOI: 10.1006/AIMA.1998.1726
description: scientific article; zbMATH DE number 1211558