Thurston maps and asymptotic upper curvature

TL;DR

This paper introduces a new way to analyze expanding Thurston maps by associating them with Gromov hyperbolic graphs and defining their asymptotic upper curvature, linking it to the map's entropy.

Contribution

It defines asymptotic upper curvature for expanding Thurston maps via associated hyperbolic graphs and explores its relationship with the entropy of the map.

Findings

Asymptotic upper curvature is well-defined for expanding Thurston maps.

The boundary at infinity of the associated hyperbolic graph corresponds to the sphere with a visual metric.

A relationship between asymptotic upper curvature and the entropy of the map is established.

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 $f$.