$C^r$-Chain closing lemma for certain partially hyperbolic diffeomorphisms

TL;DR

This paper establishes a $C^r$-orbit connecting lemma for certain partially hyperbolic diffeomorphisms, showing that chain attainability can be realized by true orbits through small perturbations, with implications for generic dynamics.

Contribution

It proves a $C^r$-orbit connecting lemma for dynamically coherent, plaque expansive partially hyperbolic diffeomorphisms with 1D center, extending chain recurrence results.

Findings

Periodic points are dense in the chain recurrent set for generic diffeomorphisms.

Chain transitivity implies transitivity in this class.

The lemma holds for all $r eq 1$, including $C^\infty$.

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 1-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.