Teichmüller Theory and Critically Finite Endomorphisms

Abstract. We present a systematic way to generate critically finite endomorphisms of Pn. These maps arise in the context of Teichmüller theory, specifically in Thurston’s topological characterization of rational maps. The dynamical objects for the endomorphisms correspond to central objects from Thurston’s theorem. Our theorems build infinitely many of these endomorphisms; in fact, a large number of examples of critically finite endomorphisms of Pn found in the literature arise from this construction. Introduction Let S be an oriented topological 2-sphere. We begin with an orientation-preserving branched cover f : (S, P ) ! (S, P ), where the domain and range spheres are identified, and P is the postcritical set of f . If |P | < 1 then f is called a Thurston map. Each Thurston map induces a holomorphic endomorphism