Linear chaos and frequent hypercyclicity

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.

16 pages