Linear chaos and frequent hypercyclicity

By Quentin Menet. 2014

82 citations recorded by the source corpus.

We answer one of the main current questions in Linear Dynamics by constructing a chaotic operator on $\ell^1$ which is not $\mathcal{U}$-frequently hypercyclic and thus not frequently hypercyclic. This operator also gives us an example of a chaotic operator which is not distributionally chaotic. We complement this result by showing that every chaotic operator is reiteratively hypercyclic.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a2be3b57fb8faab82605fa80a7f421581311a347

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Upper frequent hypercyclicity and related notions (2016, background): “The corollary easily implies Menet’s result [28] that every chaotic mapping on a complete metric space is reiteratively hypercyclic; but the proof is essentially the same as Menet’s direct proof.”
Frequently recurrent operators (2020, background): “In particular, since periodic points are reiteratively recurrent, we have the following result of Menet [35].”
Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “Moreover, our constructions will also enable us to improve the main result of [40] by showing that there exist - operators on H which are chaotic and topologically mixing but not U-frequently hypercyclic.”