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Difference sets and frequently hypercyclic weighted shifts

Published online by Cambridge University Press:  13 November 2013

FRÉDÉRIC BAYART  and
IMRE Z. RUZSA
FRÉDÉRIC BAYART
Affiliation:
Clermont Université, Université Blaise Pascal, Laboratoire de Mathematiques, BP 10448, F-63000 Clermont-Ferrand, France email Frederic.Bayart@math.univ-bpclermont.fr CNRS, UMR 6620, Laboratoire de Mathématiques, F-63177 Aubiere, France email Frederic.Bayart@math.univ-bpclermont.fr
IMRE Z. RUZSA
Affiliation:
Alfréd Rényi Institute of Mathematics, Pf. 127, H-1364 Budapest, Hungary email ruzsa@renyi.hu
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Abstract

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 but not frequently hypercyclic, and that there exists an operator which is frequently hypercyclic but not distributionally chaotic. These (surprising) 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 3 , May 2015 , pp. 691 - 709
Copyright
© Cambridge University Press, 2013 

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