1. INTRODUCTION 3

wandering domains and revolutionized the field. Since then there has been an explo-

sion of applications in this area, see for instance [99, 78, 84] where quasiconformal

maps play an essential role in describing families of Julia sets and exhibiting the

self similarity of the Mandelbrot set. We also mention the work of Thurston, Mc-

Mullen and many others [106, 83] giving applications of quasiconformal mappings

in 3-dimensional topology and geometry.

The more general classes of mappings we shall soon meet, the so called "map-

pings of finite distortion", are even more flexible than quasiconformal mappings.

Among them are the solutions of degenerate Beltrami equations. Many construc-

tions in analysis, geometry and topology rely on limiting processes. The existence,

uniqueness and compactness properties of families of mappings with finite distor-

tion make them ideal tools for solving various problems in these areas. For instance

in studying deformations of elastic bodies and the related extremals for variational

integrals in certain degenerate settings, mappings of finite distortion are often the

natural candidates to consider because they are closed under uniform convergence,

whereas the limit of diffeomorphisms need not be smooth nor even a homeomor-

phism. Further applications can be found in our recent joint work [9] where a

first attempt to study extremal mappings in a more general setting with potential

applications to Teichmiiller theory. In fact more generally when one studies the

boundary of spaces of deformations of conformal dynamical systems one is quickly

led to consider mappings which are reasonably regular but are not quasiconformal,

such as mappings of finite distortion. Indeed David's results have already found im-

portant application in the theory of iteration of rational functions, see for instance

Haissinsky and Tan [46]. This is illustrated in Figure 2.

Figure 2. A family of filled in Julia set quasicircles degenerating to a Julia set

(the "cauliflower") of finite distortion1.

The first important results in the theory of mappings of finite, but unbounded,

distortion are due to Lehto in 1969 [72, 73]. He gave geometric conditions (in terms

of moduli of separating annuli) and analytic conditions (involving sophisticated

means of distortion functions) to determine sufficient conditions for existence and

uniqueness for solutions to the Beltrami equation (defined below at (2.3)) away from

a compact singular set where the distortion function may become unbounded. These

results are quite difficult to use in practise, though Lehto established a number of

the uniform estimates needed to develop the theory. Some years later David (1986)

This picture produced using R.L. Devaney's "Mandelbrot Set Explorer", see

http://math.bu.edu/DYSYS/explorer/