A $C^{r}$ Closing Lemma for a Class of Symplectic Diffeomorphisms
  We prove a $C^r$ closing lemma for a class of partially hyperbolic symplectic diffeomorphisms. We show that for a generic $C^r$ symplectic diffeomorphism, $r =1, 2, ...,$, with two dimensional center and close to a product map, the set of all periodic points is dense. 
Zhihong Xia, Hua Zhang