Thurston maps and asymptotic upper curvature

Abstract

A Thurston map is a branched covering map from $\mathbb {S}^2$ to $\mathbb {S}^2$ with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\mathcal {G}=\mathcal {G}(f,{\mathcal {C}})$ with an expanding Thurston map $f$ and a Jordan curve ${\mathcal {C}}$ on $\mathbb {S}^2$ containing ${{\mathrm{post}}}(f)$. The boundary at infinity of $\mathcal {G}$ with associated visual metrics can be identified with $\mathbb {S}^2$ 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 $f$.