Lyapunov exponents in Hilbert geometry

TL;DR

This paper investigates the behavior of Hilbert geometries at infinity along geodesics, linking boundary shape and Lyapunov exponents through a dynamical systems perspective and convex function regularity.

Contribution

It introduces a regularity property of convex functions to connect boundary geometry with Lyapunov exponents in Hilbert geometries.

Findings

Boundary shape determines behavior at infinity.

Lyapunov exponents are linked to boundary regularity.

A new convex function regularity property is proposed.

Abstract

We study the behaviour of a Hilbert geometry when going to infinity along a geodesic line. We prove that all the information is contained in the shape of the boundary at the endpoint of this geodesic line and have to introduce a regularity property of convex functions to make this link precise. The point of view is a dynamical one and the main interest of this article is in Lyapunov exponents of the geodesic flow.