Teichmüller theory and critically finite endomorphisms

We present a systematic way to generate critically finite endomorphisms of PnPn. These maps arise in the context of Teichmüller theory, specifically in Thurstonʼs topological characterization of rational maps. The dynamical objects for the endomorphisms correspond to central objects from Thurstonʼs theorem. Our theorems build infinitely many of these endomorphisms; in fact, a large number of examples of critically finite endomorphisms of PnPn found in the literature arise from this construction.