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Recurrence properties of hypercyclic operators

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Abstract

We generalize the notions of hypercyclic operators, \(\mathfrak {U}\)-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics, namely \(\mathcal {A}\)-hypercyclicity. We then state an \(\mathcal {A}\)-hypercyclicity criterion, inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion, and we show that this criterion characterizes the \(\mathcal {A}\)-hypercyclicity for weighted shifts. We also investigate which density properties can the sets \({N(x, U)=\{n\in \mathbb {N}\ ; \ T^nx\in U\}}\) have for a given hypercyclic operator, and we study the new notion of reiteratively hypercyclic operators.

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Acknowledgments

We would like to thank the referee, whose comments improved the presentation of the paper, and especially for the present version of Proposition 3.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH, 43403, USA

    J. Bès

  2. Laboratoire de Mathématiques de Lens, Université d’Artois, Rue Jean Souvraz S.P. 18, 62307, Lens, France

    Q. Menet

  3. Departament de Matemàtica Aplicada, IUMPA, Universitat Politècnica de València, Edifici 7A, 46022, Valencia, Spain

    A. Peris

  4. Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, 84105, Beer Sheva, Israel

    Y. Puig

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  1. J. Bès
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  2. Q. Menet
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  4. Y. Puig
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Corresponding author

Correspondence to A. Peris.

Additional information

Dedicated to Professor José Bonet on the occasion of his sixtieth birthday

This work is supported in part by MEC and FEDER, Project MTM2013-47093-P, and by GVA, Projects PROMETEOII/2013/013 and ACOMP/2015/005. The second author was a postdoctoral researcher of the Belgian FNRS.

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Bès, J., Menet, Q., Peris, A. et al. Recurrence properties of hypercyclic operators. Math. Ann. 366, 545–572 (2016). https://doi.org/10.1007/s00208-015-1336-3

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  • DOI: https://doi.org/10.1007/s00208-015-1336-3

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