Mean Li-Yorke chaos in Banach spaces

Abstract

We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only if it has an absolutely mean irregular vector. As a consequence, absolutely Cesàro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke chaos is shown to be equivalent to the existence of a dense (or residual) set of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained. Moreover, we construct an example of an invertible hypercyclic operator T such that every nonzero vector is absolutely mean irregular for both T and T −1 . Several other examples are also presented. Finally, mean Li-Yorke chaos is also investigated for C 0 -semigroups of operators on Banach spaces. 1 1 2010 Mathematics Subject Classification: 47A16, 37D45.

Disjoint distributional chaos in Fréchet spaces

“…A. Conejero et al [20] (2016). Further information about distributional chaos in metric and Fréchet spaces can be obtained by consulting [8], [12]- [13], [18], [26]- [27], [30]- [31], [40], [42], [49]- [50], [55], [57] and references cited therein.…”

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Linear Dynamical Systems

“…To do this they modified the definition by varying the values of the distributional functions (4.1) and (4.2). Other interesting directions of research include the notion of mean Li-Yorke chaos that has been investigated by Bernardes et al [32].…”