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arXiv · 1305.2325

Difference sets and frequently hypercyclic weighted shifts

Frédéric Bayart·Imre Ruzsa
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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, 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.

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Frédéric Bayart, Imre Ruzsa. 2013-05-10. Difference sets and frequently hypercyclic weighted shifts. https://doi.org/10.1017/etds.2013.77

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