Dense chaos and densely chaotic operators

Abstract

The aim of this paper is to study dense chaos and densely chaotic operators on Banach spaces. First, we prove that a dynamical system is densely δ-chaotic for some δ > 0 if and only if it is densely chaotic and sensitive. Meanwhile, we also show that for general dynamical systems, Devaney chaos and dense chaos do not imply each other. Then, by using these results, we have that for a operator defined on a Banach space, dense chaos, dense δ-chaos, generic chaos and generic δ-chaos are equivalent and they are all strictly stronger than Li-Yorke chaos.