On NET maps: Examples and nonexistence results

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 Lattès maps. We investigate when such a map has the property that the associated pullback map on Teichmüller space is constant. We also show that no Thurston map of degree 2 has constant pullback map.