Generalized Bebutov systems: a dynamical interpretation of shape

Abstract. We define a semidynamical system—inspired by some classical dynamical systems studied by Bebutov in function spaces—in the space of approximative maps AðX;Y Þ 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. Introduction. The theory of shape, introduced by K. Borsuk in 1968 [7], has proved to be a successful instrument for the study of the global topological properties of dynamical systems. Shape theory is especially useful when applied to spaces of locally complicate