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高校应用数学学报
Applied Mathematics A Journal of Chinese Universities  2018, Vol. 33 Issue (2): 243-252    DOI:
    
Quasi-shadowing near the partially hyperbolic set
WANG Lin$^{1}$,  WANG Xinsheng$^{2}$
1. College of Applied Mathematics,  Shanxi University of Finance and Economics,   Taiyuan 030006,  China; 2. College of Mathematics and Information Science,  Hebei Normal University,   Shijiazhuang 050024,  China
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Abstract   In this paper, the concept of the so called partial hyperbolic set is introduced, and
it is proved that a diffeomorphism on a compact Riemannian manifold
has the quasi-shadowing property in a neighborhood of the partially
hyperbolic set in the following sense: Let $f$ be a partially
hyperbolic diffeomorphism on a compact Riemannian manifold $M$,  and
$\mathit\Lambda\subset M$ be a partially hyperbolic set of $f$.
There exists the neighborhood $O(\mathit\Lambda)$ of
$\mathit\Lambda$ such that for any $\varepsilon>0$ there exists
$\delta>0$ such that for any $\delta$-pseudo orbit
$\{x_{k}\}_{k\in\mathbb{Z}}\subset O(\mathit\Lambda)$ of $f$, there
exist a sequence of points $\{y_{k}\}_{k\in\mathbb{Z}}$ and a
sequence of center vectors $\{u_{k}\in
E_{x_{k}}^{c}\}_{k\in\mathbb{Z}}$ such that $d(x_{k},
y_{k})<\varepsilon$, where
$y_{k}=\exp_{x_{k}}(\exp^{-1}_{x_{k}}(f(y_{k-1}))+u_{k})$. As an
application, it is show that any diffeomorphism is topologically
quasi-stable under $C^{0}$-perturbation, if there exists an
invariant set in a neighborhood of the partially hyperbolic set.

Key words： partially hyperbolicity      quasi-shadowing      quasi-stability     
Published: 26 July 2018
CLC:  O193  
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WANG Lin
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Cite this article:

WANG Lin, WANG Xinsheng. Quasi-shadowing near the partially hyperbolic set. Applied Mathematics A Journal of Chinese Universities, 2018, 33(2): 243-252.

URL:

http://www.zjujournals.com/amjcua/     OR     http://www.zjujournals.com/amjcua/Y2018/V33/I2/243


部分双曲集附近的拟跟踪性

引入部分双曲集的概念,证明了紧黎曼流形上的微分同胚在其部分双曲集的小邻域内具有如下形式的拟跟踪性:
设$f$为紧黎曼流形$M$上的一个微分同胚,
$\mathit\Lambda$为$f$的部分双曲集.
则存在$\mathit\Lambda$的邻域$O(\mathit\Lambda)$,
使得对于任意$\varepsilon>0$, 存在$\delta>0,
$使得$f$在$O(\mathit\Lambda)$中的任意$\delta$-伪轨$\{x_{k}\}_{k\in\mathbb{Z}},
$ 存在点列$\{y_{k}\}_{k\in\mathbb{Z}}, $和中心向量列$\{u_{k}\in
E_{x_{k}}^{c}\}_{k\in\mathbb{Z}}$满足$d(x_{k}, y_{k})<\varepsilon$,
其中$y_{k}=\exp_{x_{k}}(\exp^{-1}_{x_{k}}(f(y_{k-1}))+u_{k})$.
作为一个应用,  给出任意微分同胚在$C^{0}$扰动下,
如果在双曲集邻域内存在不变集,  则其是拓扑拟稳定的.

关键词： 部分双曲,  拟跟踪,  拟稳定 
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