Difference sets and frequently hypercyclic weighted shifts

Frédéric Bayart (1) , Imre Ruzsa (2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 Alfréd Rényi Institute of Mathematics

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on $\ell^p(\mathbb Z)$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is $\mathcal U$-frequently hypercyclic, yet not frequently hypercyclic and that there exists an operator which is frequently hypercyclic, yet not distributionally chaotic. These (surprizing) counterexamples are given by weighted shifts on $c_0$. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.

Domaines: Analyse fonctionnelle [math.FA]

Frédéric Bayart, Imre Ruzsa. Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory and Dynamical Systems, 2013, 35 (3), pp.691-709. ⟨hal-00821458⟩