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 of f   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.