Dynamics Seminar

We exhibit a large class of Riemannian metrics on the two-dimensional torus whose geodesic flows admit a (non-split) separatrix associated with a hyperbolic closed geodesic. Building on earlier results by Katok and Krikorian, we prove that, in the near integrable regime, such non-split separatrices are accumulated by KAM curves. On the other hand, we also construct examples of metrics whose non-split separatrices are isolated from KAM curves. I will motivate these results from the point of view of the rigidity of integrable metrics on the torus and I will explain a few key ideas in the proof.