Skip to main content

\(\Gamma \)-Supercyclicity for Strongly Continuous Semigroups


We characterize the subsets \(\Gamma \) of the complex plane \(\mathbb {C}\) for which the notion of \(\Gamma \)-supercyclicity for strongly continuous semigroups coincides with that of hypercyclicity. In addition, we characterize the sets \(\Gamma \subset {\mathbb {C}}\) such that for every strongly continuous semigroup of operators \({\mathcal {T}}=(T_t)_{t\geqslant 0}\) on a Banach space X, a vector \(x\in X\) is hypercyclic for \({\mathcal {T}}\) if and only if \({\mathrm {Orb}}(\Gamma x,\mathcal {T})\) is somewhere dense in X. We derive a characterization of \(C_0\)-semigroup versions of 1-dimensional hypercyclic subsets and Bourdon–Feldman subsets. We finally prove a multi-\(\Gamma \)-supercyclicity result for \(C_0\)-semigroups.

This is a preview of subscription content, access via your institution.


  1. 1.

    Bayart, F., Matheron, E.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  2. 2.

    Bermúdez, T., Bonilla, A., Peris, A.: \(\mathbb{C}\)-supercyclic versus \(\mathbb{R}^{+}\)-supercyclic operators. Arch. Math. 79, 125–130 (2002)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bourdon, P.S., Feldman, N.S.: Somewhere dense orbits are everywhere dense. Indiana Univ. Math. J. 52(3), 811–819 (2003)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Charpentier, S., Ernst, R.: Hypercyclic subsets. J. Anal. Math. arXiv:1711.10932 (to appear)

  5. 5.

    Charpentier, S., Ernst, R., Menet, Q.: \(\Gamma \)-supercyclicity. J. Funct. Anal. 270, 4443–4465 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Conejero, J.A., Peris, A.: Linear transitivity criteria. Topol. Appl. 153, 767–773 (2005)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Costakis, G.: On a conjecture of D. Herrero concerning hypercyclic operators. C. R. Acad. Sci. Paris Série I 330, 179–182 (2000)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Costakis, G., Peris, A.: Hypercyclic semigroups and somewhere dense orbits. C. R. Acad. Sci. Paris Ser. I 335, 895–898 (2002)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and Chaotic Semigroups of Linear Operators, vol. 17, pp. 793–819. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  10. 10.

    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin (1991)

    MATH  Google Scholar 

  11. 11.

    Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos. Springer, London (2011)

    Book  Google Scholar 

  12. 12.

    Léon-Saavedra, F., Müller, V.: Rotations of hypercyclic and supercyclic operators. Integral Equ. Oper. Theory 50, 385–391 (2004)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Matsui, M., Yamada, M., Takeo, F.: Supercyclic and chaotic translation semigroups. Proc. Am. Math. Soc. 131(11), 3535–3546 (2003)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Moothathu, T.K.S.: Linear independence of a hypercyclic orbit for semigroups. J. Math. Anal. Appl. 467, 704–710 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Oxtoby, J.C., Ulam, S.M.: Measure-preserving homeomorphisms and metrical transitivity. Ann. Math. 42(4), 874–920 (1941)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)

    Book  Google Scholar 

  17. 17.

    Peris, A.: Multi-hypercyclic operators are hypercyclic. Math. Z. 236, 779–786 (2001)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Shkarin, S.: Universal elements for non-linear operators and their applications. J. Math. Anal. Appl. 348, 193–210 (2008)

    MathSciNet  Article  Google Scholar 

Download references


I gratefully acknowledge many helpful suggestions, comments and remarks of Stéphane Charpentier and Evgeny Abakumov during the preparation of the paper. I also would like to thank Romuald Ernst for reading this paper and for his support. I am grateful to the referee for helpful comments and suggestions.

Author information



Corresponding author

Correspondence to Arafat Abbar.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Isabelle Chalendar.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Abbar, A. \(\Gamma \)-Supercyclicity for Strongly Continuous Semigroups. Complex Anal. Oper. Theory 13, 3923–3942 (2019).

Download citation


  • \(C_0\)-semigroup
  • Hypercyclicity
  • Supercyclicity

Mathematics Subject Classification

  • 47A16