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Centralizers of derived-from-Anosov systems on ${\mathbb T}^3$: rigidity versus triviality

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

Published online by Cambridge University Press:  02 August 2021

SHAOBO GAN ,
YI SHI ,
DISHENG XU  and
JINHUA ZHANG
SHAOBO GAN
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China (e-mail: gansb@pku.edu.cn, shiyi@math.pku.edu.cn)
YI SHI
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China (e-mail: gansb@pku.edu.cn, shiyi@math.pku.edu.cn)
DISHENG XU
Affiliation:
Beijing International Center for Mathematical Research, Peking University, Beijing 100871, P. R. China (e-mail: xudisheng@bicmr.pku.edu.cn)
JINHUA ZHANG*
Affiliation:
School of Mathematical Sciences, Beihang University, Beijing 100191, P. R. China (e-mail: jinhua_zhang@buaa.edu.cn)
*
e-mail: zjh200889@gmail.com
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Abstract

In this paper, we study the centralizer of a partially hyperbolic diffeomorphism on ${\mathbb T}^3$ which is homotopic to an Anosov automorphism, and we show that either its centralizer is virtually trivial or such diffeomorphism is smoothly conjugate to its linear part.

Keywords

centralizerpartial hyperbolicityAnosov diffeomorphismrigidity

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C05: Smooth mappings and diffeomorphisms 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2841 - 2865
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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