Chaos for multivalued maps and induced hyperspace maps

Let (X, d) be a compact metric space and φ: X⊸X be a multivalued map. At first, we will extend for these maps the notions of a topological entropy and Robinson’s chaos from a single-valued into a multivalued setting and show their basic properties. Then, for a subclass of multivalued continuous maps with compact values, we will clarify their relationship to the induced (hyper)maps φ*:K(X)→K(X) in the hyperspace (K(X),dH), endowed with the Hausdorff metric dH, where K(X) consists of all compact subsets of X. Concretely, we will show that a positive topological entropy h(φ) of φ implies a positive topological entropy h(φ*) of φ*. On the other hand, Robinson’s chaos to φ* implies in a reverse way Robinson’s chaos to φ.