No abstract

In the early 1980's Thurston gave a topological characterization of rational maps whose critical points have finite iterated orbits ([Th, DH1]): given a topological branched covering F of the two sphere with finite critical orbits, if F has no Thurston obstructions then F possesses an invariant complex structure (up to isotopy), and is combinatorially equivalent to a rational map.We extend this theory to the setting of rational maps with infinite critical orbits, assuming a certain kind of hyperbolicity. Our study includes also holomorphic dynamical systems that arise as coverings over disconnected Riemann surfaces of finite type. The obstructions we encounter are similar to those of Thurston. We give concrete criteria for verifying whether or not such obstructions exist.Among many possible applications, these results can be used for example to construct holomorphic maps with prescribed dynamical properties; or to give a parameter description, both local and global, of bifurcations of complex dynamical systems.

We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.

One fundamental theorem in the theory of holomorphic dynamics is Thurston's topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston's theorem (the marked Thurston's theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein.