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A quantitative interpretation of the frequent hypercyclicity criterion

Published online by Cambridge University Press:  27 June 2017

ROMUALD ERNST  and
AUGUSTIN MOUZE
ROMUALD ERNST
Affiliation:
LMPA, Centre Universitaire de la Mi-Voix, Maison de la Recherche Blaise-Pascal, 50 rue Ferdinand Buisson, BP 699, 62228 Calais Cedex, France email [email protected]
AUGUSTIN MOUZE
Affiliation:
Laboratoire Paul Painlevé, UMR 8524, Cité Scientifique, 59650 Villeneuve d’Ascq, France email [email protected]
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Abstract

We give a quantitative interpretation of the frequent hypercyclicity criterion. Actually we show that an operator which satisfies the frequent hypercyclicity criterion is necessarily $A$-frequently hypercyclic, where $A$ refers to some weighted densities sharper than the natural lower density. In that order, we exhibit different scales of weighted densities that are of interest to quantify the ‘frequency’ measured by the frequent hypercyclicity criterion. Moreover, we construct an example of a unilateral weighted shift which is frequently hypercyclic but not $A$-frequently hypercyclic on a particular scale.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 4 , April 2019 , pp. 898 - 924
Copyright
© Cambridge University Press, 2017 

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Footnotes

†

Current address: École Centrale de Lille, Cité Scientifique, CS20048, 59651 Villeneuve d’Ascq cedex, France.

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