Thurston maps and asymptotic upper curvature

A Thurston map is a branched covering map from $\S^2$ to $\S^2$ with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\G=\G(f,\mathcal C)$ with an expanding Thurston map $f$ and a Jordan curve $\mathcal C$ on $\S^2$ containing $\post(f)$. The boundary at infinity of $\G$ with associated visual metrics can be identified with $\S^2$ equipped with the visual metric induced by the expanding Thurston map $f$. We define asymptotic upper curvature of an expanding Thurston map $f$ to be the asymptotic upper curvature of the associated Gromov hyperbolic graph, and establish a connection between the asymptotic upper curvature of $f$ and the entropy of $f$.