Coarse non-wandering sets and their filtration

This paper investigates recurrence properties of dynamical systems under the restriction that control is available only through inputs and outputs. We introduce the concept of ``coarse non-wandering'', a generalization of the classical non-wandering concept, and construct an associated filtration based on levels that quantify the closeness of recurrence behavior under input/output-only control. The forward direction of this filtration describes how the level of control relates to recurrence properties, whereas the backward direction captures the robustness of such behaviors and, in particular, guarantees controllability through control applied only at the observation points when the observational noise is sufficiently small. Furthermore, we demonstrate that the existence of a wandering domain is equivalent to the presence of an orbit reachable within finite error but unable to return within any slightly enlarged error bound.