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Frequently hypercyclic operators


Authors: Frédéric Bayart and Sophie Grivaux
Journal: Trans. Amer. Math. Soc. 358 (2006), 5083-5117
MSC (2000): Primary 47A16, 47A35, 37B05, 37A05
DOI: https://doi.org/10.1090/S0002-9947-06-04019-0
Published electronically: June 20, 2006
MathSciNet review: 2231886
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Abstract: We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators $ T$ on separable complex $ \mathcal{F}$-spaces: $ T$ is frequently hypercyclic if there exists a vector $ x$ such that for every nonempty open subset $ U$ of $ X$, the set of integers $ n$ such that $ T^{n}x$ belongs to $ U$ has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.


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Additional Information

Frédéric Bayart
Affiliation: Laboratoire Bordelais d’Analyse et de Géométrie, UMR 5467, Université Bordeaux 1, 351 Cours de la Libération, 33405 Talence Cedex, France
Email: bayart@math.u-bordeaux.fr

Sophie Grivaux
Affiliation: Laboratoire Paul Painlevé, UMR 8524, Université des Sciences et Technologies de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France
Email: grivaux@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0002-9947-06-04019-0
Keywords: Hypercyclic operators, frequently hypercyclic operators, unimodular point spectrum, ergodic and weak-mixing measure-preserving linear transformations, Gaussian measures on Hilbert spaces, Fock spaces
Received by editor(s): April 15, 2004
Received by editor(s) in revised form: November 25, 2004
Published electronically: June 20, 2006
Additional Notes: This work was supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2000-00116 (Analysis and Operators)
Article copyright: © Copyright 2006 American Mathematical Society

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