Li-Yorke and distributionally chaotic operators

TL;DR

This paper investigates Li-Yorke and distributional chaos in Banach space operators, providing characterizations, criteria, spectral properties, and examples of chaotic operators, including hypercyclic ones, in infinite-dimensional spaces.

Contribution

It offers new characterizations and computable criteria for chaos, links chaos to spectral properties, and constructs examples of distributionally chaotic hypercyclic operators.

Findings

Li-Yorke chaos characterized by irregular vectors

Criteria for distributional and Li-Yorke chaos established

Existence of distributionally chaotic hypercyclic operators in all infinite-dimensional separable Banach spaces

Abstract

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.