$C^r$-Closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle

Abstract: For every $r\in\mathbb{N}_{\geq2}\cup{\infty}$, we prove the $C^r$-closing lemma for general and conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle. In particular, it implies periodic points are dense for $C^r$-generic conservative partially hyperbolic diffeomorphisms with one-dimensional center bundle.