Topological entropy and chaos for maps induced on hyperspaces

Abstract 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.

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