Mean Li-Yorke chaos in Banach spaces

Author(s)

Bernardes, Jr., N. C. ^[1] ; Bonilla, A. ^[2] ; Peris, A. ^[3]

Affiliation

^[1] Univ Fed Rio de Janeiro, Inst Matemat, Dept Matemat Aplicada, Caixa Postal 68530, BR-21945970 Rio De Janeiro, RJ - Brazil

^[2] Univ La Laguna, Dept Anal Matemat, San Cristobal la Laguna 38271, Tenerife - Spain

^[3] Univ Politecn Valencia, Inst Univ Matemat Pura & Aplicada, Edifici 8E, Acces F, 4a Planta, E-46022 Valencia - Spain

• Document type: Journal article
• Source: JOURNAL OF FUNCTIONAL ANALYSIS; v. 278, n. 3 FEB 1 2020.
• Web of Science Citations: 0

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 Cesaro 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. (C) 2019 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 17/22588-0 - Dynamics, operators and ergodicity
• Grantee: Ali Messaoudi
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil