$C^0$-Robustness of topological entropy for geodesic flows

TL;DR

This paper investigates the stability of topological entropy under $C^0$ perturbations of Riemannian metrics, demonstrating robustness results especially on the 2-torus and in hyperbolic metric spaces.

Contribution

It establishes the $C^0$-robustness of topological entropy for geodesic flows, particularly on the 2-torus and in Teichmüller space, extending previous results and providing new quantitative bounds.

Findings

Metrics with contractible closed geodesics have robust entropy.

Metrics with robust positive entropy are $C^{ ext{infty}}$ generic on the torus.

The set of metrics with high positive entropy is dense and large in the $C^0$ topology.

Abstract

In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the $C^0$ topology. We establish several instances of entropy robustness (persistence of entropy non-vanishing after small $C^0$ perturbations). A large part of this paper is dedicated to metrics on the 2-dimensional torus, for which our main results are that metrics with a contractible closed geodesic have robust entropy (thus generalizing and quantifying a result of Denvir-Mackay) and that metrics with robust positive entropy on the torus are $C^{\infty}$ generic. Moreover, we quantify the asymptotic behavior of volume entropy in the Teichm\~A{\OE}ller space of hyperbolic metrics on a punctured torus, which bounds from below the topological entropy for these metrics. For general closed manifolds of dimension at least 2 we prove that the set of metrics with robust and high positive entropy is $C^0$-large in the sense that it is dense, contains cones and arbitrarily large balls.