Geodesic flow of nonstrictly convex Hilbert geometries

TL;DR

This paper studies the complex topological dynamics of geodesic flows on certain 3-manifolds modeled by nonstrictly convex Hilbert geometries, revealing mixing properties and entropy characteristics.

Contribution

It provides the first detailed analysis of geodesic flow behavior, including mixing and entropy properties, for nonstrictly convex Hilbert geometries on 3-manifolds.

Findings

Geodesic flow is topologically mixing.

Flow satisfies a nonuniform Anosov closing lemma.

Entropy-expansiveness holds for these flows.

Abstract

In this paper we describe the topological behavior of the geodesic flow for a class of closed 3-manifolds realized as quotients of nonstrictly convex Hilbert geometries, constructed and described explicitly by Benoist. These manifolds are Finsler geometries which have isometrically embedded flats, but also some hyperbolicity and an explicit geometric structure. We prove the geodesic flow of the quotient is topologically mixing and satisfies a nonuniform Anosov closing lemma, with applications to entropy and orbit counting. We also prove entropy-expansiveness for the geodesic flow of any compact quotient of a Hilbert geometry.