Quasiconformal homeomorphisms and dynamics. III: The Teichmüller space of a holomorphic dynamical system
[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 that the Teichmüller space of a rational map is a finite dimensional complex orbifold, naturally describable as a product of three dynamically defined factors. Using this result they compute the number of moduli for the quasiconformal equivalence class of a rational map and give a new proof of the ''No wandering domains theorem:'' There does not exist a component \(\Omega_0\) of the Fatou set such that the forward iterates \(\{f^i(\Omega_0)\}\) are disjoint. This section closes with a proof of the discreteness of the modular group. The next topic considered is the density of ``stable parameters''. Here one considers a holomorphic family \(\{f_\lambda(z)\}\) of rational maps parametrized by a complex manifold \(X.\) A parameter \(\alpha\in X\) is stable if there is an open set \(U\subset X\) containing \(\alpha\) such that for every \(\beta\in U\) the map \(f_\beta\) is conjugate to \(f_\alpha.\) Different types of stability (topological, quasiconformal, etc.) are obtained by requiring the conjugacy to have different types of regularity (continuous, quasiconformal, etc.). They prove that in any holomorphic family of rational maps the topologically stable parameters are open and dense and that various notions of stability are equivalent. These results are proved using the ``extensions of holomorphic motions'' provided by the ``Harmonic \(\lambda\)-Lemma''. A rational map is called hyperbolic if \(| f'(z)| >1\) for all \(z\) in the Julia set. A central outstanding problem in the subject is the conjecture that hyperbolic maps are an open and dense subset among all rational maps of a given degree. The final result in the paper shows that this conjecture follows from the ``No invariant line fields'' conjecture: A rational map \(f\) carries no invariant line fields on its Julia set, except when \(f\) is double covered by an integral torus endomorphism. In the final section of the paper the results in the previous sections are applied to the case of quadratic polynomials.
scientific article; zbMATH DE number 1211558
scientific article