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$C^r$-chain closing lemma for certain partially hyperbolic diffeomorphisms
Published online by Cambridge University Press: 11 October 2023
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.
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