Origami, affine maps, and complex dynamics

TL;DR

This paper explores the properties of nearly Euclidean Thurston maps (NET maps), which are affine perturbations of folding maps, revealing their diverse behaviors and computationally accessible invariants.

Contribution

It introduces a detailed analysis of NET maps, highlighting their diversity, and demonstrates how their relationship with affine maps facilitates invariant computation.

Findings

NET maps exhibit diverse dynamical behaviors

Affine relationships enable tractable invariant calculations

New phenomena in the dynamics of NET maps

Abstract

We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena.