Uniform hyperbolicity in nonflat billiards

TL;DR

This paper establishes a sufficient condition for nonflat billiards, modeled as Riemannian manifolds with boundary, to exhibit uniform hyperbolicity, extending understanding of chaotic geodesic flows on surfaces.

Contribution

It introduces a new criterion for uniform hyperbolicity in nonflat billiards, including a novel condition for Anosov geodesic flows on closed surfaces.

Findings

Provides a sufficient condition for uniform hyperbolicity in nonflat billiards

Derives a new criterion for Anosov geodesic flows on closed surfaces

Extends the class of billiard systems known to exhibit strong chaotic behavior

Abstract

Uniform hyperbolicity is a strong chaotic property which holds, in particular, for Sinai billiards. In this paper, we consider the case of a nonflat billiard, that is, a Riemannian manifold with boundary. Each trajectory follows the geodesic flow in the interior of the billiard, and bounces when it meets the boundary. We give a sufficient condition for a nonflat billiard to be uniformly hyperbolic. As a particular case, we obtain a new criterion to show that a closed surface has an Anosov geodesic flow.