# Modular groups, Hurwitz classes and dynamic portraits of NET maps

## TL;DR

This paper develops invariants for nearly Euclidean Thurston (NET) maps, enabling their classification and enumeration, and provides a complete classification of their dynamic critical orbit portraits.

## Contribution

It introduces new invariants inspired by Hurwitz equivalence to classify and enumerate NET maps and their dynamic portraits.

## Key findings

- Complete classification of NET maps based on developed invariants.
- Enumeration method for NET maps using Hurwitz class concepts.
- Detailed classification of dynamic critical orbit portraits.

## Abstract

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