2 1. INTRODUCTION extensions of Morrey's classical result [87] on the existence of homeomorphic L2 solutions to the Beltrami equation, what has come to be known as the "measur- able Riemann mapping theorem" and extend more recent work of David [33] and Brakolova-Jenkins [26] which we discuss below. Quasiconformal mappings (the homeomorphic solutions of Beltrami equations) provide a class of mappings which lie "between" homeomorphisms and diffeomor- phisms and yet retain many of the features of analytic functions. They were first used in a significant way by Ahlfors in 1935 [1] when they proved to be an essen- tial ingredient of his geometric development of Nevanlinna theory via the theory of covering spaces and Riemann surfaces. Using the Beltrami equation, Bers [19] gave a careful exposition of the method of quasiconformal mappings applied to the classical problems of uniformization in complex analysis. Bers gave a proof that every homotopy class contains a locally quasiconformal homeomorphism of one Riemann surface onto another and he also gave complete and independent proofs of the Limit Circle Theorem (uniformization of a closed surface with signature), the Principal Circle Theorem (closed surfaces with symmetries), and the Retrosection Theorem (Schottky uniformization). Addi- tionally he showed how to construct a simultaneous uniformization of two surfaces by means of a quasi-Fuchsian group. Teichmuller theory shows how deformations, or parameterized families, of discrete groups are related to each other via quasi- conformal mappings. Implying for instance that the Cantor limit set of a finitely generated Schottky group is quasiconformally equivalent to the usual middle thirds Cantor set. As illustrated in Figure 1. these Cantor sets can be quite complicated. Figure 1. A Cantor limit set of a two-generator Schottky group. About twenty years ago Sullivan [104] introduced the Beltrami equation into the theory of holomorphic dynamics in his solution of the Fatou-Julia problem on
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