Chaos induced by turbulent and erratic functions

Abstract

Let ( X, d) be a compact metric space and f : X -> X a continuous function and consider the hyperspace (K(X), H) of all nonempty compact subsets of X endowed with the Hausdorff metric induced by d. Let (f) over bar : K(X) -> K(X) be defined by (f) over bar (A) = {f(a)/a is an element of A} the natural extension of f to K(X), then the aim of this work is to study the dynamics of (f) over bar when f is turbulent (erratic, respectively) and its relationships.