Some duality properties of non-saddle sets

We show in this paper that the class of compacta that can be isolated non-saddle sets of flows in ANRs is precisely the class of compacta with polyhedral shape. We also prove—reinforcing the essential role played by shape theory in this setting—that the Conley index of a regular isolated non-saddle set is determined, in certain cases, by its shape. We finally introduce and study the notion of dual of a non-saddle set. Examples of compacta related by duality are attractor–repeller pairs. We use the complement theorems in shape theory to prove that the shape of the dual set is determined by the shape of the original non-saddle set.