Difference sets and frequently hypercyclic weighted shifts

By Fr'ed'eric Bayart, Imre Z. Ruzsa. Ergodic Theory and Dynamical Systems, 2013

129 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/968d94e023296001940a96bd1127b23ebcde1141

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In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Mixing operators and small subsets of the circle (2011, background): “2) is satisfied as soon as the weighted shift Bw is frequently hypercyclic ([11]).”
Linear chaos and frequent hypercyclicity (2014, background): “In particular, we know thanks to Bayart and Ruzsa [9] that a weighted shift on p is chaotic if and only if it is frequently hypercyclic.”
Upper frequent hypercyclicity and related notions (2016, methodology): “In view of the results in [6] and [7] it is not surprising that our conditions involve a sequence of sets Ap that interact.”
Shadowing and structural stability for operators (2020, background): “Frequent hypercyclic weighted shifts on `p(Z) and c0(Z) were characterized by Bayart and Ruzsa [6] and their characterization shows that condition (C) implies the frequent hypercyclicity criterion.”
Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “Frequent hypercyclicity was introduced in [5] and rather extensively studied since then (for instance in [10], [14], [28], [30], [40], [49] .”