Chaos for multivalued maps and induced hyperspace maps

Abstract

• It is the first paper dealing really with chaos for multivalued maps in the relatioship with the induced hypermaps in hyperspaces. • Topological entropy for multivalued maps forces the one for induced hypermaps, but not vice versa. • Robinson’s chaos for multivalued maps is reversely forced by the on efor thr induced hypermaps 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 ) , 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 (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.