-chain closing lemma for certain partially hyperbolic diffeomorphisms

Abstract

Abstract For every $r\in \mathbb {N}_{\geq 2}\cup \{\infty \}$ , we prove a $C^r$ -orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with one-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism f , if a point y is chain attainable from x through pseudo-orbits, then for any neighborhood U of x and any neighborhood V of y , there exist true orbits from U to V by arbitrarily $C^r$ -small perturbations. As a consequence, we prove that for $C^r$ -generic diffeomorphisms in this class, periodic points are dense in the chain recurrent set, and chain transitivity implies transitivity.