On Net Maps: Examples and Nonexistence Results

TL;DR

The paper studies nearly Euclidean Thurston maps, characterizing when their associated pullback maps on Teichmuller space are constant, and proves that degree 2 Thurston maps cannot have this property.

Contribution

It provides new criteria for when NET maps have constant pullback maps and establishes a nonexistence result for degree 2 Thurston maps with this property.

Findings

Characterization of NET maps with constant pullback maps

Proof that no degree 2 Thurston map has a constant pullback map

Insights into the structure of Thurston maps and their Teichmuller dynamics

Abstract

A Thurston map is called nearly Euclidean if its local degree at each critical point is 2 and it has exactly four postcritical points. Nearly Euclidean Thurston (NET) maps are simple generalizations of rational Lattes maps. We investigate when such a map has the property that the associated pullback map on Teichmuller space is constant. We also show that no Thurston map of degree 2 has constant pullback map