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We construct new coordinates for the Teichmüller space Teich of a punctured torus into R × R + . The coordinates depend on the representation of Teich as a space of marked Kleinian groups Gµ that depend holomorphically on a parameter µ varying in a simply connected domain in C. They describe the geometry of the hyperbolic manifold H 3 /Gµ; they reflect exactly the visual patterns one sees in the limit sets of the groups Gµ; and they are directly computable from the generators of Gµ.

Abstract. We define a class 2 of entire functions whose covering properties are similar to those of rational maps. The set 2 is closed under composition of functions, and we show that when regarded as dynamical systems of the plane, the elements of 2 share many properties with rational maps. In particular, they have finite dimensional spaces of quasiconformal deformations, and they contain no wandering domains in their stable sets. 0: IntroductionMotivated by recent activity in the theory of dynamics of rational maps, we consider the problem of iterating a transcendental entire map of the plane.Early in the 1900s Fatou and Julia independently developed a theory of rational maps as dynamical systems. Under iteration of a rational map R, the Riemann sphere decomposes into two completely invariant sets: a stable region, on which the behaviour of R is either dissipative or elliptic; and an unstable region, on which R is chaotic.In 1926, Fatou [7] observed that this decomposition has an analogue for transcendental entire maps. He then showed that, while the unstable set of an entire map shares many of the qualitative features of the unstable set of a rational map, the stable properties can be quite different.In [13] Sullivan proved a 'no wandering domain' theorem for rational maps that, combined with the work of Fatou and Julia, gives a complete classification of the behaviour of R on its stable set. This classification depends heavily on the covering properties of R, namely:(1) There is a strong relationship between the stable phenomena and the asymptotic behaviour of the values at which R fails to be a covering map.(2) The only values at which R fails to be a covering map are the critical values. Since the covering properties of a transcendental entire map E are more complicated than those of R, the classification theorem does not carry over.In this article, we define a class 2 of entire maps of finite type [see § 1] whose covering properties are close to the covering properties of a rational map. For this class we prove an essential finiteness theorem: THEOREM 3.1. If E:C-*C has finite type, then E has a finite dimensional space of quasiconformal deformations.

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