  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.
