A. Giraldo

On the global structure of invariant regions of flows with asymptotically stable attractors

Several authors have pointed out the usefulness of shape theory, which was introduced by K.Borsuk in 1968, as a tool in the study of dynamical systems. In particular, Bogatyi and Gutsu [4], Garay [10], Gunther and Segal [11], Hastings [12], Robbin and Salamon [16], Rogers [17] and Tezer [24] have obtained interesting results in dynamical systems using shape-theoretical techniques. One of the authors of the present article has given another connection between shape and topological dynamics [20], based on the approach to shape developed in [18] and [19]. See also [21] for applications of shape theory to the study of uniform attractors. Most of the mentioned papers study properties of flows, although [16] and [24] are devoted to discrete dynamical systems. In particular, a shape Conley index is constructed in [16] for discrete dynamical systems and some shape-theoretical ideas are used in [24] to study properties of topological conjugacy of shifts and homotopical shift equivalence for maps of polyhedra. In the papers [4,10,11,12,20,21] some topological properties of the attractors of flows are studied. For instance, Hastings established in [12] a higher dimensional Poincare-Bendixson theorem, proving the existence of attractors of certain flows in the interior of submanifolds of the Euclidean space and comparing the shape of the attractor with that of the manifold. Other properties studied in the above papers are related to strong cellularity of the global attractors [10] and polyhedral shape of asymptotically stable compacta [4,11,20]. A remarkable fact proved by

Some duality properties of non-saddle sets

Abstract 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.

Singular Continuations of Attractors

We study dynamical and topological properties of the singularities of continuations of attractors of flows on manifolds. Despite the fact that these singularities are not isolated invariant sets, they share many of the properties of attractors; in particular, they have finitely generated Cech homology and cohomology, and they have the Cech homotopy type of attractors. This means that, from a global point of view, the singularities of continuations are topological objects closely related to finite polyhedra. The global structure is preserved even for weaker forms of continuation. An interesting case occurs with the Lorenz system for parameter values close to the situation of preturbulence. A general result, motivated by this particular case, is presented.

Finite approximations to & Cech homology

We show in this paper how to represent intrinsically Cech homology of compacta, in terms of inverse limits of discrete approximations. We establish some relations between inverse limits and non-continuous homotopies and, as a consequence, we get a strong form of the classical continuity property of Cech homology.

Density and finiteness. A discrete approach to shape

Abstract We show in this paper that the category of shape can be modelled in discrete terms using maps defined in dense subsets of compacta. This approach to shape provides in addition a characterization of the shape of compacta which does not require external elements and uses only continuous single-valued functions, in contrast with the existing internal characterizations of shape. As an application, we prove a connection between the notion of shape image, due to Lisica, and the basic notion of omega limit of a dynamical system.

APPROXIMATE POLYHEDRA, DENSITY AND DISCRETE MAPS

Some extension properties of maps defined on dense subsets are studied for approximate polyhedra. The latter are characterised as approximate extensors for finite maps with small oscillation.

Multifibrations. A class of shape fibrations with the path lifting property

In this paper we introduce a class of maps possessing a multivalued homotopy lifting property with respect to every topological space. We call these maps multifibrations and they represent a formally stronger concept than that of shape fibration. Multifibrations have the interesting property of being characterized in a completely intrinsic way by a path lifting property involving only the total and the base space of the fibration. We also show that multifibrations (and also, with some restrictions, shape fibrations) have a lifting property for homotopies of fine multivalued maps. This implies, when the spaces considered are metric compacta, that the possibility of lifting a fine multivalued map is a property of the corresponding strong shape morphism and not of the particular map considered.