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1. INTRODUCTION

extensions of Morrey's classical result [87] on the existence of homeomorphic

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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