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

On Net Maps: Examples and Nonexistence Results