A characterization of hyperbolic rational maps

Abstract

In the early 1980's Thurston gave a topological characterization of rational maps whose critical points have finite iterated orbits (\cite{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.