A note on transitivity in set-valued discrete systems

Let (X, d) be a metric space and f : X --> X is a continuous function. If we consider the space (K (X), H) of all non-empty compact subsets of X endowed with the Hausdorff metric induced by d and (f) over bar : K(X) --> K(X), (f) over bar (A) = {f(a)/a is an element of A}, then the aim of this work is to show that (f) over bar transitive implies f transitive. Also, we give an example showing that f transitive does not implies (f) over bar transitive. (C) 2002 Elsevier Science Ltd. All rights reserved.