Flexibility of entropies for surfaces of negative curvature

TL;DR

This paper investigates the range of possible entropy values for geodesic flows on negatively curved surfaces, showing that only the constant curvature case imposes restrictions on entropy values.

Contribution

It proves that the only restrictions on topological and metric entropies for negatively curved metrics are those already known from the constant curvature case.

Findings

Topological entropy is minimized by constant negative curvature.

Metric entropy with respect to Liouville measure is bounded below by the topological entropy.

No additional restrictions on entropy values beyond the constant curvature case.

Abstract

We consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a Riemannian metric $g$ of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for $g$ is greater than or equal to the topological entropy for the metric of constant negative curvature on $M$ with the same total area which is greater than or equal to the metric entropy with respect to the Liouville measure of geodesic flow for $g$. Equality holds only in the case of constant negative curvature. We prove that those are the only restrictions on the values of topological and metric entropies for metrics of negative curvature.