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Quasi-stability of partially hyperbolic diffeomorphisms

Authors: Huyi Hu and Yujun Zhu
Journal: Trans. Amer. Math. Soc. 366 (2014), 3787-3804
MSC (2010): Primary 37D30; Secondary 37C20, 37B40
Published electronically: March 14, 2014
MathSciNet review: 3192618
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Abstract: A partially hyperbolic diffeomorphism $ f$ is structurally quasi-
stable if for any diffeomorphism $ g$ $ C^1$-close to $ f$, there is a homeomorphism $ \pi $ of $ M$ such that $ \pi \circ g$ and $ f\circ \pi $ differ only by a motion $ \tau $ along center directions. $ f$ is topologically quasi-stable if for any homeomorphism $ g$ $ C^0$-close to $ f$, the above holds for a continuous map $ \pi $ instead of a homeomorphism. We show that any partially hyperbolic diffeomorphism $ f$ is topologically quasi-stable, and if $ f$ has $ C^1$ center foliation $ W^c_f$, then $ f$ is structurally quasi-stable. As applications we obtain continuity of topological entropy for certain partially hyperbolic diffeomorphisms with one or two dimensional center foliation.

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Additional Information

Huyi Hu
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Yujun Zhu
Affiliation: College of Mathematics and Information Science, and Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

Keywords: Partial hyperbolicity, quasi-stability, entropy
Received by editor(s): April 4, 2012
Received by editor(s) in revised form: November 1, 2012
Published electronically: March 14, 2014
Additional Notes: The second author was supported by NSFC (No: 11371120), NSFC (No: 11071054), NCET (No: 11-0935), the Key Project of Chinese Ministry of Education (No: 211020), the Plan of Prominent Personnel Selection and Training for the Higher Education Disciplines in Hebei Province (BR2-219) and the SRF for ROCS, SEM
Article copyright: © Copyright 2014 American Mathematical Society
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