Cr closing lemma is an important statement in the theory of dynamical systems, which implies that for a Cr generic system the union of periodic orbits is dense in the nonwondering domain. C1 closing lemma is proved in many classes of dynamical systems, however Cr closing lemma with r > 1 is proved only for few cases. In this talk, I’ll prove C∞ closing lemma for Reeb flows on closed contact three-manifolds. The proof uses recent developments in quantitative aspects of embedded contact homology (ECH). In particular, the key ingredient of the proof is a result by Cristofaro-Gardiner, Hutchings and Ramos, which claims that the asymptotics of ECH spectral invariants recover the volume of a contact manifold. Applications to closed geodesics on Riemannian two-manifolds and Hamiltonian diffeomorphisms of symplectic two-manifolds (joint work with M. Asaoka) will be also presented.
This video is part of the Institute for Advanced Study‘s Symplectic geometry seminar.
