Nearly Euclidean Thurston maps

Abstract

We take an in-depth look at Thurston’s combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston maps. These are orientation-preserving branched maps f : S 2 → S 2 f\colon S^2\to S^2 whose local degree at every critical point is 2 2 and which have exactly four postcritical points. These maps are simple enough to be tractable, but are complicated enough to have interesting dynamics.