Frequently hypercyclic operators

Bayart, Frédéric; Grivaux, Sophie

doi:10.1090/S0002-9947-06-04019-0

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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