Quasi-Shadowing and Quasi-Stability for Dynamically Coherent Partially Hyperbolic Diffeomorphisms

Let $f$ be a partially hyperbolic diffeomorphism. $f$ is called has the quasi-shadowing property if for any pseudo orbit ${x_k}_{kin mathbb{Z}}$, there is a sequence ${y_k}_{kin mathbb{Z}}$ tracing it in which $y_{k+1}$ lies in the local center leaf of $f(y_k)$ for any $kin mathbb{Z}$. $f$ is called topologically quasi-stable if for any homeomorphism $g$ $C^0$-close to $f$, there exist a continuous map $pi$ and a motion $tau$ along the center foliation such that $picirc g=taucirc fcircpi$. In this paper we prove that if $f$ is dynamically coherent then it has quasi-shadowing and topological quasi-stability properties.