Linear chaos and frequent hypercyclicity

TL;DR

This paper constructs a specific chaotic operator on lex^1 that is not requently hypercyclic, providing new insights into the distinctions among various forms of hypercyclicity in linear dynamics.

Contribution

It introduces a novel chaotic operator on lex^1 that lacks  frequent hypercyclicity and distributional chaos, and proves all chaotic operators are reiteratively hypercyclic.

Findings

Constructed a chaotic operator on lex^1 not  frequently hypercyclic.

Provided an example of a chaotic operator that is not distributionally chaotic.

Proved that every chaotic operator is reiteratively hypercyclic.

Abstract

We answer one of the main current questions in Linear Dynamics by constructing a chaotic operator on $\ell^1$ which is not $\mathcal{U}$-frequently hypercyclic and thus not frequently hypercyclic. This operator also gives us an example of a chaotic operator which is not distributionally chaotic. We complement this result by showing that every chaotic operator is reiteratively hypercyclic.