Growth rate of closed geodesics on surfaces without conjugate points

Abstract: Let (M,g) be a closed Riemannian surface of of genus at least 2 and no conjugate points. By the uniformization theorem such a surface admits a metric of negative curvature and therefore the topological entropy h of the geodesic flow is positive. Denote by P(t) the number of free homotopy classes containing a closed geodesic of period $\le t $. We will show: P(t) is asymptotically equivalent to e^(ht)/(ht) =F(t), i.e. the ratio of P and F converges to 1 as t tends to infinity. An important ingredient in the proof is a mixing flow invariant measure given by the unique measure of maximal entropy. Under suitable hyperbolicity assumptions this result carries over to closed Riemannian manifolds without conjugate and higher dimension.

For closed manifolds of negative curvature the above estimate is well known and has been originally obtained by Margulis. In a recent preprint the estimate has been also obtained by Ricks for certain closed manifolds (rank 1 mflds) of non-positive curvature. This is a joint work with Vaughn Climenhaga and Khadim War.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

Series comments: On the week of the seminar, an announcement with the Zoom link is mailed to the seminar mailing list. To receive these e-mails, please sign up by writing to Lev Buhovsky (http://www.math.tau.ac.il/~levbuh/).