Mean Li-Yorke chaos in Banach spaces

TL;DR

This paper explores mean Li-Yorke chaos in Banach space operators, showing its distinction from distributional chaos, characterizing it via absolutely mean irregular vectors, and providing criteria and examples including hypercyclic operators.

Contribution

It introduces the concept of mean Li-Yorke chaos in Banach spaces, characterizes it through absolutely mean irregular vectors, and constructs examples including hypercyclic operators.

Findings

Mean Li-Yorke chaos differs from distributional chaos in Banach spaces.

Operators are mean Li-Yorke chaotic iff they have an absolutely mean irregular vector.

Every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some invariant subspace.

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 Ces\`aro 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.