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)K(X),H) be the space of all non-empty compact subsets of X endowed with the Hausdorff metric induced by d   and f¯:K(X)→K(X) be the map defined by f¯(A):{f(a):a∈A}. In this paper we investigate the relationships between the mixing property of (K(X),f¯) and the mixing property of (X, f  ). In addition, we discuss specification for the set-valued discrete dynamical system (K(X),f¯).