CONJUGACIES BETWEEN RATIONAL MAPS AND EXTREMAL QUASICONFORMAL MAPS

Abstract

Abstract. We show that two rational maps which are K-quasiconformally combinatorially equivalent are K-quasiconformally conjugate. We also study the relationship between the boundary dilatation of a combinatorial equivalence and the dilatation of a conjugacy. The Teichmüller theory is a powerful tool in the study of complex analytic dynamics. In addition to the work of D. Sullivan [Su] and W. Thurston (refer [DH2]), where the classical theory of the Teichmüller spaces plays a crucial role, a beautiful theorem about extremal quasiconformal maps between open Riemann surfaces, due to R. Strebel ([S]), was employed in the work of C. McMullen [Mc2]. In this paper, we give new applications of this theorem and a theorem of H. Ohtake ([Oh]) about lifts of extremal quasiconformal maps. Suppose that f,g are rational maps of degree bigger than one which are quasiconformally combinatorially equivalent, i.e. there exist quasiconformal maps φ and φ1 from Ĉ to itself such that φ is isotopic to φ1 rel P (f) andφf = gφ1. Here P (f) = � f n (Cf) n>0 is the post-critical set of f, where Cf is the set of the critical points of f. It is known that there exists a quasiconformal conjugacy ψ from f to g, whichis isotopic to φ rel P (f) ([Mc3]). In the proof, ψ was constructed by the Douady-Earle extension and hence its maximal dilatation might be bigger than the maximal dilatation of φ. Applying extremal quasiconformal maps, we improve the theorem to show that ψ can be chosen such that its maximal dilatation is less than or equal to the maximal dilatation of φ (Theorem 1) and we give an application (Theorem 2) relating extremal and boundary dilatation. At first, let us recall Strebel’s Theorem. Let φ: R → R ′ be a quasiconformal map between open Riemann surfaces. The Beltrami differential of φ is µφ(z) d¯z ∂¯zφ(z) dz ∂zφ(z) Received by the editors August 21, 1999