Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms
  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. 
Huyi Hu, Yunhua Zhou and Yujun Zhu
math.DS