Some chaotic and mixing properties of fuzzified dynamical systems

Let X be a compact metric space, let φ be a continuous self-map on X, and let F(X) denote the space of fuzzy sets on X equipped with the levelwise topology. In this paper we study relations between a given (crisp) dynamical system (X,φ) and its g-fuzzification (F(X),Φg). Among other things we study various (weak, strong, mild etc.) mixing properties and also several kinds of chaotic behaviors (Li-Yorke chaos, ω-chaos, distributional chaos, topological chaos etc.). We specified subspaces of fuzzy sets which makes sense to study further. Additionally, we proved that numerous local characteristics of the original dynamical system can be found in its fuzzy extension, while many global characteristics are preserved in the opposite direction. Thus, among other things, we obtained easy-to-check criteria for checking absence of complex behavior in induced fuzzy dynamical systems.