Precise asymptotics for periodic orbits of the geodesic flow in nonpositive curvature

We establish the most precise asymptotic formula ever found for the number of periodic orbits for the geodesic flow, counted by homotopy. We prove it for every compact manifold of nonpositive curvature with rank one.

This extends a celebrated result of Fields medalist G.A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper.

While proving this result, we also manage to carry out Margulis' construction of the measure of maximal entropy without requiring strong hyperbolicity.