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An orientation-preserving branched covering $f: S^2 \to S^2$ is a nearlyEuclidean Thurston (NET) map if each critical point is simple and itspostcritical set has exactly four points. Inspired by classical, non-dynamicalnotions such as Hurwitz equivalence of branched covers of surfaces, we developinvariants for such maps. We then apply these notions to the classification andenumeration of NET maps. As an application, we obtain a complete classificationof the dynamic critical orbit portraits of NET maps.

Modular groups, Hurwitz classes and dynamic portraits of NET maps