Topological entropy for set valued maps

Any continuous map T on a compact metric space X induces in a natural way a continuous map T ¯ on the space K ( X ) of all non-empty compact subsets of X . Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map T ¯ is zero or infinity. Moreover, the topological entropy of T ¯ | C ( X ) is zero, where C ( X ) denotes the space of all non-empty compact and connected subsets of X . For general continuous maps on compact metric spaces these results are not valid.