Generalized Bebutov systems: A dynamical interpretation of shape. (English) Zbl 1033.55007

Summary: We establish a new connection between shape and dynamics by adopting a different point of view. The theory of dynamical systems is used here to give a new interpretation of shape. We define a structure of semidynamical system in the space \(A(X,Y)\) of approximative maps between two metric compacta \(X\) and \(Y\). The metric structure of \(A(X,Y)\) is inspired by topologies introduced in previous papers. The dynamical structure is inspired by some classical dynamical systems studied by Bebutov in function spaces. According to this interpretation, shape morphisms and strong shape morphisms are invariant subspaces of the Bebutov space \(A(X,Y)\). This means that shape and strong shape morphisms can be viewed as semidynamical systems themselves. The paper is devoted to the study of the structure of the Bebutov system \(A(X,Y)\) and in particular to the recognition of Lagrange and Poisson stable orbits and non-wandering points, the determination of the limit sets and the properties of attractors and Lyapunov stable motions. These results are used to give dynamical characterizations of some basic notions in shape theory. For instance: 1. A shape morphism is generated by a map if and only if it is not dispersive (when viewed as a semidynamical system). 2. For a metric compactum \(X\) the following are equivalent: 1. \(X\) has trivial shape 2. The set of Lagrange stable motions of the Bebutov system \(A(X,X)\) is contained in a connected component of \(A(X,X)\). 3. There exists a connected attractor in \(A(X,X)\) containing a periodic orbit, and every attractor containing a periodic orbit contains all periodic orbits. 3. A metric compactum \(X\) is shape dominated by a polyhedron if and only if its Bebutov system is prolongable.


55P55 Shape theory
37B25 Stability of topological dynamical systems
54C56 Shape theory in general topology
54H20 Topological dynamics (MSC2010)
Full Text: DOI