Quasi-shadowing for partially hyperbolic diffeomorphisms
Quasi-shadowing for partially hyperbolic diffeomorphisms
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部分双曲微分同胚的准阴影
DOI：
10.1017/etds.2014.126
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发表时间：
2015
影响因子：
0.9
通讯作者：
Zhu Yujun
中科院分区：
文献类型：
--
作者：
Hu Huyi;Zhou Yunhua;Zhu Yujun
摘要：部分双曲微分同态$f$具有拟阴影性质，如果对于\mathbb{Z} $中的任意伪轨道$\{x_{k}\}_{k\ \} $，在\mathbb{Z} $中存在跟踪它的点序列$\{y_{k}\}_{k\ \，其中$y_{k+1}$由$f(y_{k})$沿中心方向运动${\it\tau}$得到。我们证明了任何部分双曲微分同构具有拟阴影性质，如果$f$具有$C^{1}$中心叶理，则我们可以要求${\it\tau}$沿中心叶理移动点。作为应用，我们证明了任何部分双曲微分同态在C^{0}$-扰动下是拓扑拟稳定的。当$f$具有一致紧化$C^{1}$中心叶形时，我们也给出了一些对一致双曲系统成立的定理的部分双曲微分同态版本，如Anosov闭引理、云引理和谱分解定理。
Abstract A partially hyperbolic diffeomorphism $f$ has the 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 ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ 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 a uniformly compact $C^{1}$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems which hold for uniformly hyperbolic systems, such as the Anosov closing lemma, the cloud lemma and the spectral decomposition theorem.