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

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  11 October 2023

YI SHI  and
XIAODONG WANG
YI SHI*
Affiliation:
School of Mathematics, Sichuan University, Chengdu 610065, P. R. China
XIAODONG WANG
Affiliation:
School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, P. R. China (e-mail: [email protected])
*
e-mail: [email protected]
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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.

Keywords

partially hyperbolic diffeomorphismchain closing lemmaperiodic orbitdynamical coherencechain recurrent set

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C20: Generic properties, structural stability 37C50: Approximate trajectories (pseudotrajectories, shadowing, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 7 , July 2024 , pp. 1923 - 1944
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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