On the $C^{\infty}$ closing lemma for Hamiltonian flows on symplectic $4$-manifolds

The main result in this paper is the $C^{\infty}$ closing lemma for a large family of Hamiltonian flows on $4$-dimensional symplectic manifolds, which includes classical Hamiltonian systems. First we prove the $C^{\infty}$ closing lemma and the $C^r$ general density theorem for geodesic flows on closed Finsler surfaces by combining a result of Asaoka-Irie with the dual lens map technique. Then we extend our results to Hamitonian flows with certain restriction. We also list some applications of our results in differential geometry and contact topology.