Chaos on hyperspaces

Let ff be a continuous self-map of a compact metric space XX. The transformation ff induces in a natural way a self-map f¯ defined on the hyperspace K(X)K(X) of all nonempty closed subsets of XX. We study which of the most usual notions of chaos for dynamical systems induced by ff are inherited by f¯ and vice versa. We consider distributional chaos, Li–Yorke chaos, ωω-chaos, Devaney chaos, topological chaos (positive topological entropy), specification property and their variants. This answers questions stated independently by Roman-Flores and Banks.