On mixing property in set-valued discrete systems

Let (X, d) be a compact metric space and f : X -> X be a continuous map. Let (K(X),H) be the space of all nonempty compact subsets of X endowed with the Hausdorff metric induced by d and (f) over bar : K(X) -> K(X) be the map defined by f (A) : {f (a) : a E A}. In this paper we investigate the relationships between the mixing property of (K(X), (f) over bar) and the mixing property of (X,f). In addition, we discuss specification for the set-valued discrete dynamical system (K(X), (f) over bar). (c) 2005 Elsevier Ltd. All rights reserved.