Chaos for multivalued maps and induced hyperspace maps

Let (X, d) be a compact metric space and phi: X(sic)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 phi* : K(X) -> K(X) in the hyperspace (K(X), d(H)), endowed with the Hausdorff metric d(H), where K(X) consists of all compact subsets of X. Concretely, we will show that a positive topological entropy h(phi) of phi implies a positive topological entropy h(phi*) of phi*. On the other hand, Robinson's chaos to phi* implies in a reverse way Robinson's chaos to phi. (C) 2020 Elsevier Ltd. All rights reserved.