Coarse non-wandering sets and their filtration

This paper develops a new topological framework for the global analysis of dynamical systems, focusing on wandering behavior. More precisely, we introduce the concept of “coarse non-wandering”, a generalization of the classical non-wandering concept, and construct an associated filtration that quantifies the persistence of non-wandering behavior under noise localized near the observation points. The forward direction of this filtration clarifies the persistence of wandering behavior, while the backward direction captures the resistance to wandering and, in particular, ensures the robustness of the non-wandering property when the localized noise is sufficiently small. Furthermore, we demonstrate that the existence of a wandering domain is equivalent to the presence of an orbit that is reachable within finite noise but cannot return within any slightly enlarged noise bound.