Invariant measures for frequently hypercyclic operators

By S. Grivaux, É. Matheron. 2014

54 citations recorded by the source corpus.

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 c_0 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 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.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/78bb5f1644e709908c4ce456d9817fe9432006bb

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In Applied Mathematics

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Citations in the corpus

Frequently recurrent operators (2020, background): “The notion of upper frequent recurrence was ﬁrst introduced by Costakis and Parissis [20], while Grivaux and Matheron [26] have introduced a concept of frequent recurrence that is (at least formally) weaker than ours.”
Upper frequent hypercyclicity and related notions (2016, background): “Recently, Grivaux and Matheron [18] introduced a new type of A-hypercyclicity that is even stronger than frequent hypercyclicity.”
Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “By a result of [30], every ergodic operator on H is densely distributionally chaotic.”
Expansivity and Shadowing in Linear Dynamics (2016, background): “We refer the reader to the books [2, 19] and to the more recent papers [3, 6, 7, 8, 17, 18], where many additional references can be found.”