Thurston maps and asymptotic upper curvature

Abstract

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 $f$.

26 pages. arXiv admin note: text overlap with arXiv:1109.2664; and with arXiv:1009.3647 by other authors

Cited by in corpus (1)

• Periodic Points and the Measure of Maximal Entropy of an Expanding Thurston Map