José Manuel Rodríguez Sanjurjo

Morse equations and unstable manifolds of isolated invariant sets

We describe a new way of obtaining the Morse equations of a Morse decomposition of an isolated invariant set. This is achieved through a filtration of truncated unstable manifolds associated with the decomposition. The results in the paper make it possible to calculate the Morse equations (and also the Conley index) in many interesting situations without using index pairs. We also study the intrinsic topology of the unstable manifold and obtain new duality properties of the cohomological Conley index.

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.

A topology for the sets of shape morphisms

Abstract We introduce a topology on the set of shape morphisms between arbitrary topological spaces X , Y , Sn(X,Y) . These spaces allow us to extend, in a natural way, some classical concepts to the realm of topological spaces. Several applications are given to obtain relations between shape theory and N -compactness and shape-theoretic properties of the spaces of quasicomponents.

Shape and Conley Index of Attractors and Isolated Invariant Sets

This article is an exposition of several results concerning the theory of continuous dynamical systems, in which Topology plays a key role. We study homological and homotopical properties of attractors and isolated invariant compacta as well as properties of their unstable manifolds endowed with the intrinsic topology. We also provide a dynamical framework to express properties which are studied in Topology under the name of Hopf duality. Finally we see how the use of the intrinsic topology makes it possible to calculate the Conley-Zehnder equations of a Morse decomposition of an isolated invariant compactum, provided we have enough information about its unstable manifold.

Selections of multivalued maps and shape domination

Given an approximate mapping f − ={f k }:X→Y between compacta from the Hilbert cube [K. Borsuk, Fund. Math. 62 (1968), 223–254, the author associates with f − a (u.s.c.) multivalued mapping F:X→Y . If F is single-valued, F and f − induce the same shape morphism, S(F)=S(f − ) . If Y is calm [Z. Cerin, Pacific J. Math. 79 (1978), no. 1, 69–91 and all F(x) , x∈X , are sufficiently small sets, then the existence of a selection for F implies that S(f − ) is generated by some mapping X→Y . If F is associated with f − and admits a coselection (a mapping g:Y→X such that y∈F(g(y)) , for y∈Y ), then S(f − ) is a shape domination and therefore sh(Y)≤sh(X) . If Y is even an FANR, then every sufficiently small multivalued mapping F:X→Y , which admits a coselection, induces a shape domination S(F) .

The AR-property for Roberts’ example of a compact convex set with no extreme points Part 1: General result

We prove that the original compact convex set with no extreme points, constructed by Roberts (1977) is an absolute retract, therefore is homeomorphic to the Hilbert cube. Our proof consists of two parts. In this first part, we give a sufficient condition for a Roberts space to be an AR. In the second part of the paper, we shall apply this to show that the example of Roberts is an AR.

INTERNAL FUNDAMENTAL SEQUENCES AND APPROXIMATIVE RETRACTS

Abstract We introduce the notion of internal fundamental sequence and prove that any shape morphism from an arbitrary compactum X to an internally movable compactum Y is induced by an internal fundamental sequence. We use this special kind of fundamental sequences to give characterizations and some properties of AANR C -sets and AANR N -sets. The paper ends with a section devoted to internal FANRs.

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.

Shape invariance of N-compactifications

Abstract We use N -compactifications of 0-dimensional spaces to obtain a new shape invariant for the class of all topological spaces. We also point out that the shape and topological classifications are not the same in the realm of Tychonov spaces having a base of clopen sets. Finally we use the new shape invariant to obtain classes of spaces where the conditions “to have the same shape” and “to be homeomorphic” are equivalent.