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 (T(X), d(infinity)) be the metric space of all nonempty compact fuzzy set on X equipped with the supremum metric d(infinity) which is calculated as the supremum of the Hausdorff distances of the corresponding level sets. If (f) over bar is the natural extension off to (K(X), H) and f is the Zadeh's extension of f to (F(X), d(infinity)), then the aim of this paper is to study the dynamics of (f) over bar and (f) over cap when! is turbulent (erratic, respectively). (C) 2011 Elsevier Ltd. All rights reserved.