# Ergodicity of Bowen-Margulis measure for the Benoist 3-manifolds

## TL;DR

This paper investigates the ergodic properties of the Bowen-Margulis measure for a class of non-Riemannian, non-CAT(0), hyperbolic 3-manifolds with Hilbert geometries, establishing its ergodicity and constructing the measure explicitly.

## Contribution

It proves the ergodicity of the Bowen-Margulis measure for Benoist 3-manifolds and explicitly constructs this measure in a non-Riemannian hyperbolic setting.

## Key findings

- Proves ergodicity of Bowen-Margulis measure.
- Constructs the measure explicitly.
- Shows the Patterson-Sullivan density is canonical.

## Abstract

We study the geodesic flow of a class of 3-manifolds introduced by Benoist which have some hyperbolicity but are non-Riemannian, not CAT(0), and with non-C^1 geodesic flow. The geometries are nonstrictly convex Hilbert geometries in dimension three which admit compact quotient manifolds by discrete groups of projective transformations. We prove the Patterson-Sullivan density is canonical, with applications to counting, and construct explicitly the Bowen-Margulis measure of maximal entropy. The main result of this work is ergodicity of the Bowen-Margulis measure.