Chaos for induced hyperspace maps

For (X, d) be a metric space, f : X → X a continuous map and the space of non-empty compact subsets of X with the Hausdorff metric, one may study the dynamical properties of the induced map H. Román-Flores [A note on in set-valued discrete systems. Chaos, Solitons & Fractals 2003;17:99–104] has shown that if is topologically transitive then so is f, but that the reverse implication does not hold. This paper shows that the topological transitivity of is in fact equivalent to weak topological mixing on the part of f. This is proved in the more general context of an induced map on some suitable hyperspace of X with the Vietoris topology (which agrees with the topology of the Hausdorff metric in the case discussed by Román-Flores.