Invariant measures for frequently hypercyclic operators |
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Authors: | Sophie Grivaux Étienne Matheron |
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Institution: | 1. CNRS, Laboratoire Paul Painlevé, UMR 8524, Université Lille 1, Cité Scientifique, 59655 Villeneuve d''Ascq Cedex, France;2. Laboratoire de Mathématiques de Lens, Université d''Artois, rue Jean Souvraz S. P. 18, 62307 Lens Cedex, France |
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Abstract: | We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T ; that there exist frequently hypercyclic operators on the sequence space c0 admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are satisfied by all the known chaotic operators) for an operator T to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support. |
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Keywords: | 47A16 47A35 37A05 |
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