Abstract
Let f be a continuous self-map of a compact metric space X. The transformation f induces in a natural way a self-map (f) over bar defined on the hyperspace K(X) of all nonempty closed subsets of X. We study which of the most usual notions of chaos for dynamical systems induced by f are inherited by (f) over bar and vice versa. We consider distributional chaos, Li-Yorke chaos, omega-chaos, Devaney chaos, topological chaos (positive topological entropy), specification property and their variants. This answers questions stated independently by Roman-Flores and Banks. (c) 2008 Elsevier Ltd. All rights reserved.