Abstract
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).
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