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

Abstract

<p style="text-indent:20px;">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-<inline-formula><tex-math id="M1">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> 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.