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A Thurston map is a branched covering map from  to  with a finite postcritical set. We associate a natural Gromov hyperbolic graph  with an expanding Thurston map  and a Jordan curve  on  containing . The boundary at infinity of  with associated visual metrics can be identified with  equipped with the visual metric induced by the expanding Thurston map . We define asymptotic upper curvature of an expanding Thurston map  to be the asymptotic upper curvature of the associated Gromov hyperbolic graph, and establish a connection between the asymptotic upper curvature of  and the entropy of .

Thurston maps and asymptotic upper curvature