Geometry and entropies in a fixed conformal class on surfaces

TL;DR

This paper investigates the relationship between geometry and entropy in fixed conformal classes on surfaces, establishing restrictions on topological entropy and providing geometric decompositions and compactness results.

Contribution

It introduces new restrictions on topological entropy for non-positively curved metrics within a fixed conformal class and extends results to metrics with no focal points.

Findings

Restrictions on topological entropy in fixed conformal classes

A collar lemma and thick-thin decomposition for these metrics

Precompactness of the metric class under certain conditions

Abstract

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics. Also, we extend some of the results to metrics of fixed total area in a fixed conformal class with no focal points and with some integral bounds on the positive part of the Gaussian curvature.