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$C^r$-chain closing lemma for certain partially hyperbolic diffeomorphisms
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 7 / July 2024
- Published online by Cambridge University Press:
- 11 October 2023, pp. 1923-1944
- Print publication:
- July 2024
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- Article
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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.
