Abstract
Li–Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more useful characterizations of Li–Yorke chaos can be given in the special setting of composition operators on \(L^p\)-spaces. As a consequence we obtain a simple characterization of weighted shifts which are Li–Yorke chaotic. We give numerous examples to show that our results are sharp.
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References
Bayart, F., Darji, U.B., Pires, B.: Topological transitivity and mixing of composition operators. J. Math. Anal. Appl. 465(1), 125–139 (2018)
Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)
Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li–Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83–93 (2011)
Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Distributional chaos for linear operators. J. Funct. Anal. 265(9), 2143–2163 (2013)
Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Li–Yorke chaos in linear dynamics. Ergod. Theory Dyn. Syst. 35(6), 1723–1745 (2015)
Bernardes Jr., N.C., Cirilo, P.R., Darji, U.B., Messaoudi, A., Pujals, E.R.: Expansivity and shadowing in linear dynamics. J. Math. Anal. Appl. 461(1), 796–816 (2018)
Bès, J., Menet, Q., Peris, A., Puig, Y.: Recurrence properties of hypercyclic operators. Math. Ann. 366(1), 545–572 (2016)
Grivaux, S., Matheron, É., Menet, Q.: Linear dynamical systems on Hilbert spaces: typical properties and explicity examples. Preprint, arXiv:1703.01854
Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos. Springer, London (2011)
Hajian, A.B., Kakutani, S.: Weakly wandering sets and invariant measures. Trans. Am. Math. Soc. 110, 136–151 (1964)
Li, T.Y., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82(10), 985–992 (1975)
Menet, Q.: Linear chaos and frequent hypercyclicity. Trans. Am. Math. Soc. 369(7), 4977–4994 (2017)
Prǎjiturǎ, G.T.: Irregular vectors of Hilbert space operators. J. Math. Anal. Appl. 354(2), 689–697 (2009)
Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill Inc, New York (1991)
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The authors thank the referee whose valuable comments improved the presentation of the article.
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Communicated by H. Bruin.
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The first author was partially supported by Grant #2017/22588-0, São Paulo Research Foundation (FAPESP), and by CNPq. The second author was supported by Grant #2017/19360-8, São Paulo Research Foundation (FAPESP). The third author was partially supported by Grant #2018/06916-0, São Paulo Research Foundation (FAPESP), and by CNPq.
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Bernardes, N.C., Darji, U.B. & Pires, B. Li–Yorke chaos for composition operators on \(L^p\)-spaces. Monatsh Math 191, 13–35 (2020). https://doi.org/10.1007/s00605-019-01341-2
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DOI: https://doi.org/10.1007/s00605-019-01341-2