Topological mixing of the geodesic flow on convex projective manifolds

TL;DR

This paper proves that the geodesic flow on certain convex projective manifolds exhibits topological mixing behavior, especially on the biproximal unit tangent bundle and in higher-rank cases, indicating complex dynamical properties.

Contribution

It introduces the biproximal unit tangent bundle and establishes topological mixing of the geodesic flow on it for irreducible convex projective manifolds, extending understanding of their dynamics.

Findings

Geodesic flow is topologically mixing on the biproximal unit tangent bundle for irreducible manifolds.

Flow is topologically mixing on each component of the non-wandering set in higher-rank cases.

The biproximal unit tangent bundle is closed and invariant under the geodesic flow.

Abstract

We introduce a natural subset of the unit tangent bundle of a convex projective manifold, the biproximal unit tangent bundle; it is closed and invariant under the geodesic flow, and we prove that the geodesic flow is topologically mixing on it whenever the manifold is irreducible. We also show that, for higher-rank, irreducible, compact convex projective manifolds, the geodesic flow is topologically mixing on each connected component of the non-wandering set.