Abstract: A topologically expansive system on X is one where the diagonal is repelling. Such systems always have an expansive metric, one where, sufficiently close points a moved apart by a definite ratio. By the expansivity, we mean the maximum ratio one can achieve by varying the metric. This is related but distinct from the topological entropy; in some sense expansivity measures the minimal amount nearby points are stretched apart, while topological entropy measures the maximal amount.

We propose to study expansivity, and give ways to get upper and lower bounds on it using graphs; for instance, we can determine it exactly for many rational maps of the Riemann sphere, considered as acting on their Julia sets. It is related to the p=\infty case of a family of asymptotic energies E^p of a rational map, where p=2 determines whether or not the map is rational and the value of p where E^p = 1 gives a quasi-symmetry invariant of the Julia set, the Ahlfors regular conformal dimension.