Li–Yorke and distributionally chaotic operators

We study Li–Yorke chaos and distributional chaos for operators on Banach spaces. More precisely, we characterize Li–Yorke chaos in terms of the existence of irregular vectors. Sufficient “computable” criteria for distributional and Li–Yorke chaos are given, together with the existence of dense scram...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2011, Vol.373 (1), p.83-93
Main Authors: Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.
Format: Article
Language:English
Subjects:
Distributional chaos
Distributionally irregular vectors
Exact sciences and technology
Functional analysis
Irregular vector
Li–Yorke chaos
Mathematical analysis
Mathematics
Operator theory
Sciences and techniques of general use
Weighted shift operators
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Summary:We study Li–Yorke chaos and distributional chaos for operators on Banach spaces. More precisely, we characterize Li–Yorke chaos in terms of the existence of irregular vectors. Sufficient “computable” criteria for distributional and Li–Yorke chaos are given, together with the existence of dense scrambled sets under some additional conditions. We also obtain certain spectral properties. Finally, we show that every infinite dimensional separable Banach space admits a distributionally chaotic operator which is also hypercyclic.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2010.06.011