On turbulent, erratic and other dynamical properties of Zadeh's extensions

Let (X, d) be a compact metric space and f: X → X a continuous function. Consider the hyperspace (K(X),H) of all nonempty compact subsets of X endowed with the Hausdorff metric induced by d, and let (F(X),d ^∞) be the metric space of all nonempty compact fuzzy set on X equipped with the supremum metric d_∞ which is calculated as the supremum of the Hausdorff distances of the corresponding level sets. If f̄ is the natural extension of f to (K(X),H) and f̂ is the Zadeh's extension of f to (F(X),d^∞), then the aim of this paper is to study the dynamics of f̄ and f̂ when f is turbulent (erratic, respectively).