Skip to main content
SpringerLink
Log in

\(\mathcal{U}\)-frequent hypercyclicity notions and related weighted densities

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study dynamical notions lying between \(\mathcal{U}\)-frequent hypercyclicity and reiterative hypercyclicity by investigating weighted upper densities between the unweighted upper density and the upper Banach density. While chaos implies reiterative hypercyclicity, we show that chaos does not imply \(\mathcal{U}\)-frequent hypercyclicity with respect to any weighted upper density. Moreover, we show that if T is \(\mathcal{U}\)-frequently hypercyclic (resp. reiteratively hypercyclic) then the n-fold product of T is still U-frequently hypercyclic (resp. reiteratively hypercyclic) and that this implication is also satisfied for each of the considered \(\mathcal{U}\)-frequent hypercyclicity notions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. I. Ansari, Existence of hypercyclic operators on topological vector spaces, Journal of Functional Analysis 148 (1997), 384–390.

    Article  MathSciNet  Google Scholar 

  2. F. Bayart and S. Grivaux, Hypercyclicité: le rôle du spectre ponctuel unimodulaire, Comptes Rendus Mathématique. Académie des Sciences. Paris 338 (2004), 703–708.

    Article  MathSciNet  Google Scholar 

  3. F. Bayart and S. Grivaux, Frequently hypercyclic operators, Transactions of the American Mathematical Society 358 (2006), 5083–5117.

    Article  MathSciNet  Google Scholar 

  4. F. Bayart and É. Matheron, Hypercyclic operators failing the hypercyclicity criterion on classical banach spaces, Journal of Functional Analysis 250 (2007), 426–441.

    Article  MathSciNet  Google Scholar 

  5. F. Bayart and I. Z. Ruzsa, Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory and Dynamical Systems 35 (2015), 691–709.

    Article  MathSciNet  Google Scholar 

  6. J. Bès, Q. Menet, A. Peris and Y. Puig, Recurrence properties of hypercyclic operators, Mathematische Annalen 366 (2016), 545–572.

    Article  MathSciNet  Google Scholar 

  7. G. D. Birkhoff, Démonstration d’un théorème élémentaire sur les fonctions entières, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 189 (1929), 473–475.

    MATH  Google Scholar 

  8. A. Bonilla and K.-G. Grosse-Erdmann, Upper frequent hypercyclicity and related notions, Revista Matemática Complutense 31 (2018), 673–711.

    Article  MathSciNet  Google Scholar 

  9. M. De la Rosa and C. Read, A hypercyclic operator whose direct sum T ⊕ T is not hypercyclic, Journal of Operator Theory 61 (2009), 369–380.

    MathSciNet  MATH  Google Scholar 

  10. R. Ernst and A. Mouze, Frequent universality criterion and densities, Ergodic Theory and Dynamical Systems, https://doi.org/10.1017/etds.2019.103.

  11. R. Ernst and A. Mouze, A quantitative interpretation of the frequent hypercyclicity criterion, Ergodic Theory and Dynamical Systems 39 (2019), 898–924.

    Article  MathSciNet  Google Scholar 

  12. A. R. Freedman and J. J. Sember, Densities and summability, Pacific Journal of Mathematics 95 (1981), 293–305.

    Article  MathSciNet  Google Scholar 

  13. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, Journal of Functional Analysis 98 (1991), 229–269.

    Article  MathSciNet  Google Scholar 

  14. S. Grivaux, E. Matheron and Q. Menet, Linear dynamical systems on hilbert spaces: typical properties and explicit examples, Memoirs of the American Mathematical Society, to appear.

  15. K.-G. Grosse-Erdmann and A. Peris, Frequently dense orbits, Comptes Rendus Mathématique. Académie des Sciences. Paris 341 (2005), 123–128.

    Article  MathSciNet  Google Scholar 

  16. K.-G. Grosse-Erdmann and A. Peris, Weakly mixing operators on topological vector spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matematicas 104 (2010), 413–426.

    Article  MathSciNet  Google Scholar 

  17. C. Kitai, Invariant closed sets for linear operators, Ph.D. thesis, University of Toronto, Canada, 1982.

  18. Q. Menet, Linear chaos and frequent hypercyclicity, Transactions of the American Mathematical Society 369 (2017), 4977–4994.

    Article  MathSciNet  Google Scholar 

  19. Q. Menet, A bridge between U-frequent hypercyclicity and frequent hypercyclicity, Journal of Mathematical Analysis and Applications 482 (2020), Article no. 123569.

  20. S. Shkarin, On the spectrum of frequently hypercyclic operators, Proceedings of the American Mathematical Society 137 (2009), 123–134.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LMPA, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, Université du Littoral Côte d’Opale, UR 2597, F-62100, Calais, France

    Romuald Ernst

  2. Institut de Mathématique, Universitéde Liège, Allée de la Découverte 12, 4000, Liège, Belgique

    Céline Esser

  3. Département de Mathématique, Université de Mons, 20 Place du Parc, 7000, Mons, Belgique

    Quentin Menet

Authors
  1. Romuald Ernst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Céline Esser
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Quentin Menet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Romuald Ernst.

Additional information

The first and the third author were supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front) and by Programme PEPS JC 2018 INSMI.

The third author is a Research Associate of the Fonds de la Recherche Scientifique — FNRS.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ernst, R., Esser, C. & Menet, Q. \(\mathcal{U}\)-frequent hypercyclicity notions and related weighted densities. Isr. J. Math. 241, 817–848 (2021). https://doi.org/10.1007/s11856-021-2115-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2115-3

Search

Navigation