Chaos on hyperspaces

By Juan Luis García Guirao, Dominik Kwietniak, Marek Lampart, Piotr Oprocha, Alfredo Peris. 2009

81 citations recorded by the source corpus.

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 ¯ 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 ¯ and vice versa. We consider distributional chaos, Li–Yorke chaos, ω -chaos, Devaney chaos, topological chaos (positive topological entropy), specification property and their variants. This answers questions stated independently by Roman-Flores and Banks.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ab25613d36f1f52f01a4b6e0a582b23b08ea61d5

Citations in the corpus

• Recurrence properties and disjointness on the induced spaces (2013, unknown): “weak mixing property of (K(X),TK)(resp. (M(X),TM)), see [2] and [24]. Since then the connection of dynamical properties among (X,T), (K(X),TK)and (M(X),TM)has been studied by many authors, see, e.g., [2, 8, 9, 13, 14, 15, 18, 23]. A remarkable result by Glasner and Weiss [8] stated that the topologi”