Nikita Shekutkovski

Asymptotically stable attractors are only a particular case of a large family of invariant compacta whose global topological structure is regular. We devote this paper to investigating the shape properties of this class of compacta, the nonsaddle sets. Stable attractors and unstable attractors having only internal explosions are examples of nonsaddle sets. The main aim of this paper is to generalize the well-known theorem for the shape of attractors to nonsaddle sets using the intrinsic approach to shape which combines continuity up to a covering and the corresponding homotopies of first order.

For flows defined on a compact manifold with or without boundary, it is shown that the connectivity components of a chain recurrent set possess a stronger connectivity known as joinability (or pointed 1-movability in the sense of Borsuk). As a consequence, the Vietoris–van Dantzig solenoid cannot be a component of a chain recurrent set, although the solenoid appears as a minimal set of a flow.