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

TL;DR

This paper proves a $C^r$-closing lemma for partially hyperbolic diffeomorphisms with one-dimensional center, showing that periodic points are dense in generic conservative cases, advancing understanding of dynamical stability.

Contribution

It establishes the $C^r$-closing lemma for a broad class of partially hyperbolic diffeomorphisms with one-dimensional center, including conservative cases.

Findings

Periodic points are dense for $C^r$-generic conservative partially hyperbolic diffeomorphisms.

The $C^r$-closing lemma holds for all $r eq 1$, including infinity.

The result applies to general and conservative systems with one-dimensional center bundles.

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.