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arXiv · 1210.4988

Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms

Huyi Hu·Yunhua Zhou·Yujun Zhu
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Abstract

A partially hyperbolic diffeomorphism $f$ has quasi-shadowing property if for any pseudo orbit ${x_k}_{k\in \mathbb{Z}}$, there is a sequence of points ${y_k}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_k)$ by a motion $\tau$ along the center direction. We show that any partially hyperbolic diffeomorphism has quasi-shadowing property, and if $f$ has $C^1$ center foliation then we can require $\tau$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^0$-perturbation. When $f$ has uniformly compact $C^1$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems holden for uniformly hyperbolic systems, such as Anosov closing lemma, cloud lemma and spectral decomposition theorem.

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Huyi Hu, Yunhua Zhou, Yujun Zhu. 2012-10-17. Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms. https://doi.org/10.1017/etds.2014.126

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