Topological entropy of fuzzified dynamical systems

Let X be a compact metric space and let f : X → X be continuous. It is well known that a given discrete dynamical system (X, f) induces a map Φ on the space F(X) of fuzzy sets. In this paper we study relations between sizes of topological entropies of a given crisp dynamical system (X, f) and its fuzzy counterpart (F(X),Φ). Among another things, we present a constructive proof of the fact that even very weak assumptions acquired by the crisp discrete dynamical system ensures infinite topological entropy of the fuzzy one. However, we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system.