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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.frCNRS, 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

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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.

Information

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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