Set-Valued Chaos in Linear Dynamics

We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator \(T : X \rightarrow X\) on a topological vector space X, and the natural hyperspace extensions \(\overline{T}\) and \(\widetilde{T}\) of T to the spaces \(\mathcal {K}(X)\) of compact subsets of X and \(\mathcal {C}(X)\) of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T, \(\overline{T}\) and \(\widetilde{T}\). Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in Banks (Chaos Solitons Fractals 25(3):681–685, 2005) and Peris (Chaos Solitons Fractals 26(1):19–23, 2005).