Relationship Between Non-Wandering Point and Stability in Random Dynamical Systems

Abstract

This paper aims to study the concepts of random dynamical systems (RDSs) and investigates the relationships between several fundamental concepts, including recurrent points, Poisson stability, Lyapunov stability, Lipschitz stability, and non-wandering points. The main results show that any positively or negatively Poisson stable point is necessarily a non-wandering point and that recurrent points in Lyapunov or orbitally Lipschitz stable systems are also non-wandering. In addition, it is proven that the set of Poisson stable points is dense in a complete metric space in which every point is non-wandering. Overall, the study enhances the understanding of the interplay between stability and recurrence in random dynamical systems and clarifies their qualitative behavior in stochastic settings.