The measure of maximal entropy for the geodesic flow on closed convex projective manifolds

The seminar is virtual via Zoom. If you are interested in participating, please contact Stephan Stadler (stadler@mpim...).

The existence and uniqueness of a measure of maximal entropy for the geodesic flow on closed real hyperbolic manifolds, and more generally for Anosov flows, is a classical result that was obtained independently by Bowen and Margulis. There are many applications to the theory of dicrete subgroups of Lie groups, including to the recent and active topic of Anosov representations.

We will discuss the measure of maximal entropy in a different setting, which also strongly interacts with the theory of discrete linear groups: the geodesic flow on convex projective manifolds. Such a manifold is a quotient of a subset of a real projective space, which is open, convex and bounded in some affine chart, by a discrete group of projective transformations. Its geodesic flow is not uniformly hyperbolic in general, but it resembles the geodesic flow on non-positively curved Riemannian manifolds. As in the Riemannian setting, the main tool for constructing interesting invariant measures are Patterson--Sullivan densities, which we will also discuss.