Topological entropy and chaos for maps induced on hyperspaces

If f is a continuous selfmap of a compact metric space X then by the induced map we mean the map f defined on the space of all nonempty closed subsets of X by f (K) = f (K). The paper mainly deals with the topological entropy of induced maps. We show that under some nonrecurrence assumption the induced map f is always topologically chaotic, that is it has positive topological entropy. Additionally we characterize topological weak and strong mixing off in terms of the omega limit set of induced map. This allows the description of the dynamics of the map f induced by a transitive graph map f on the space of all subcontinua of a given graph G. It follows that in this case f has the same topological entropy as f. (c) 2006 Elsevier Ltd. All rights reserved.