Feng Zhu

Equidistribution and counting in Hilbert geometry

Following the seminal work of Margulis, many counting problems in negatively curved geometries have been fruitfully studied using dynamical methods, often involving the mixing of appropriate flows (e.g. the geodesic flow) and equidistribution to distinguished probability measures. We use this approach to asymptotically count orbit points and primitive closed geodesics in the setting of rank-one properly convex projective structures equipped with their Hilbert metrics. These are geometric structures which are not in general even CAT(0), but which nevertheless retain many qualitative features of negative curvature. This is joint work with Pierre-Louis Blayac.