Quasi-shadowing for partially hyperbolic diffeomorphisms

A partially hyperbolic diffeomorphism f has the quasi-shadowing property if for any pseudo orbit {x(k)}(k is an element of Z), there is a sequence of points {y(k)}(k is an element of 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 the quasi-shadowing property, and if f has a 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 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.