Functional envelope of a dynamical system

If (X, f) is a dynamical system given by a compact metric space X and a continuous map f : X -> X then by the functional envelope of (X, f) we mean the dynamical system (S(X), F-f) whose phase space S(X) is the space of all continuous selfmaps of X and the map F-f : S(X) -> S(X) is defined by F-f (phi) = f circle phi for any phi epsilon S(X). The functional envelope of a system always contains a copy of the original system. Our motivation for the study of dynamics in functional envelopes comes from semigroup theory, from the theory of functional difference equations and from dynamical systems theory. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope. Special attention is paid to orbit closures, omega-limit sets, (non) existence of dense orbits and topological entropy.