Ergodicity and equidistribution in strictly convex Hilbert geometry
arXiv:2008.00328
Abstract
In this paper we show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT(-1) or rank-one CAT(0) spaces, also hold for geometrically-finite strictly convex projective structures equipped with their Hilbert metric. More specifically, such structures admit a finite Sullivan measure; with respect to this measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects.
32 pages. Title edited to more accurately reflect scope of results; minor changes to formatting and terminology; typos corrected