Quasi-shadowing Property on Random Partially Hyperbolic Sets

In this paper we investigate the quasi-shadowing property for C1 random dynamical systems on their random partially hyperbolic sets. It is shown that for any pseudo orbit {xk} +... on a random partially hyperbolic set there exists a .center. pseudo orbit {yk} +... shadowing it in the sense that yk+1 is obtained from the image of yk by a motion along the center direction. Moreover, when the random partially hyperbolic set has a local product structure, the above .center. pseudo orbit {yk} +... can be chosen such that yk+1 and the image of yk lie in their common center leaf.