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.

Introduction 1. Preliminaries 2. Statement of the Results and Examples 3. General Analysis on Branched Coverings and Matings 4. Proof of the Results: First Part 5. Proof of the Results: Second Part Appendix: Matings Seen in Parameter Space and Some Numerical Observations Acknowledgements ReferencesWe study a family of cubic branched coverings and matings of cubic polynomials of the form g ? ? f, with g = g a : z 7 ! z 3 + a and f = P i for i = 1, 2, 3 or 4. We give criteria for matability or not of critically finite g a with each P i . The maps g a ? ?P 1 illustrate features that do not occur for matings of quadratic polynomials: they never have Levy cycles but do sometimes have Thurston obstructions.

We apply Thurston's characterization of postcritically finite rational maps as branched coverings of the sphere to give new classes of combination theorems for postcritically finite rational maps. Our constructions increase the degree of the map but always yield branched coverings which are equivalent to rational maps, independent of the combinatorics of the original map. The main tool is a general theorem based on the intersection number of arcs and curves which controls the region in the sphere in which an obstruction may reside.

In this paper we prove a priori bounds for infinitely renormalizable quadratic polynomials satisfying a "molecule condition". Roughly speaking, this condition ensures that the renormalization combinatorics stay away from the satellite types. These a priori bounds imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values. Stony Brook IMS Preprint