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Set-Valued Chaos in Linear Dynamics

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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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Acknowledgements

This work was partially done on a visit of the first author to the Departament de Matemàtica Aplicada at Universitat Politècnica de València (Spain). The first author is very grateful for the hospitality and support. We thank the referee for valuable suggestions that produced a better presentation of the paper.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530 , Rio de Janeiro, RJ , 21945-970, Brazil

    N. C. Bernardes Jr.

  2. Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Edifici 8E, Cub F, 4a Planta, 46022, València, Spain

    A. Peris & F. Rodenas

Authors
  1. N. C. Bernardes Jr.
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  2. A. Peris
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  3. F. Rodenas
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Correspondence to A. Peris.

Additional information

The first author was partially supported by CNPq (Brazil) and by the EBW+ Project (Erasmus Mundus Programme). The second and third authors were supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P. The second author was partially supported by GVA, Project PROMETEOII/2013/013.

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Bernardes, N.C., Peris, A. & Rodenas, F. Set-Valued Chaos in Linear Dynamics. Integr. Equ. Oper. Theory 88, 451–463 (2017). https://doi.org/10.1007/s00020-017-2394-6

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  • DOI: https://doi.org/10.1007/s00020-017-2394-6

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