Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-25T19:35:44.018Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ernst.r@math.cnrs.fr
AUGUSTIN MOUZE
Affiliation:
Laboratoire Paul Painlevé, UMR 8524, Cité Scientifique, 59650 Villeneuve d’Ascq, France email Augustin.Mouze@math.univ-lille1.fr
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

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

References

Bayart, F. and Grivaux, S.. Hypercyclicité: le rôle du spectre ponctuel unimodulaire. C. R. Math. Acad. Sci. Paris 338(9) (2004), 703–708.Google Scholar
Bayart, F. and Grivaux, S.. Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358(11) (2006), 5083–5117.Google Scholar
Bayart, F. and Grivaux, S.. Invariant Gaussian measures for operators on Banach spaces and linear dynamics. Proc. Lond. Math. Soc. (3) 94(1) (2007), 181–210.Google Scholar
Bayart, F. and Matheron, É.. Dynamics of Linear Operators (Cambridge Tracts in Mathematics) . Cambridge University Press, Cambridge, 2009.Google Scholar
Bayart, F. and Ruzsa, I. Z.. Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys. 35(3) (2015), 691–709.Google Scholar
Bès, J., Menet, Q., Peris, A. and Puig, Y.. Recurrence properties of hypercyclic operators. Math. Ann. 366(1) (2016), 545–572.Google Scholar
Bonilla, A. and Grosse-Erdmann, K.-G.. Upper frequent hypercyclicity and related notions. Preprint, 2016, arXiv:1601.07276.Google Scholar
Freedman, A. R. and Sember, J. J.. Densities and summability. Pacific J. Math. 95(2) (1981), 293–305.Google Scholar
Grosse-Erdmann, K.-G. and Peris, A.. Linear Chaos (Universitext Series) . Springer, London, 2011.Google Scholar
Mouze, A. and Munnier, V.. Polynomial inequalities and universal Taylor series. Math. Z. 284(3–4) (2016), 919–946.Google Scholar
Zygmund, A.. Trigonometric Series (Cambridge Mathematical Library, 1) . Cambridge University Press, Cambridge, 2002.Google Scholar