Generalized Bebutov systems: a dynamical interpretation of shape

Dedicated to Professor Joaquin Arregui on his 70th Birthday

Abstract

We define a semidynamical system--inspired by some classical dynamical systems studied by Bebutov in function spaces--in the space of approximative maps A(X, Y) between two metric compacta, with a suitable metric. Shape and strong shape morphisms are characterized as invariant subsets of this system. We study their structure and asymptotic properties and use the obtained results to give dynamical characterizations of basic notions in shape theory, like trivial shape, shape domination by polyhedra and internal FANRs.