On topological properties of closed attractors

Abstract

The notion of an attractor has various definitions in the theory of dynamical systems. Under compactness assumptions, several of those definitions coincide and the theory is rather complete. However, without compactness, the picture becomes blurry. To improve our understanding, we characterize in this work when a closed, not necessarily compact, asymptotically stable attractor on a locally compact metric space is homotopy equivalent to its domain of attraction. This enables a further structural study of the corresponding feedback stabilization problem.