Hereditarily frequently hypercyclic operators and disjoint frequent hypercyclicity

Bayart, Frédéric; Grivaux, Sophie; Matheron, Étienne; Menet, Quentin

doi:10.1017/etds.2025.13

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Hereditarily frequently hypercyclic operators and disjoint frequent hypercyclicity

Bayart, Frédéric; Grivaux, Sophie; Matheron, Étienne et al.

2025 • In Ergodic Theory and Dynamical Systems, p. 52

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https://hdl.handle.net/20.500.12907/52509

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10.1017/etds.2025.13

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Keywords :

countable Lebesgue spectrum; disjointness; frequent hypercyclicity; Furstenberg families; Mathematics (all); Applied Mathematics

Abstract :

[en] We introduce and study the notion of hereditary frequent hypercyclicity, which is a reinforcement of the well-known concept of frequent hypercyclicity. This notion is useful for the study of the dynamical properties of direct sums of operators; in particular, a basic observation is that the direct sum of a hereditarily frequently hypercyclic operator with any frequently hypercyclic operator is frequently hypercyclic. Among other results, we show that operators satisfying the frequent hypercyclicity criterion are hereditarily frequently hypercyclic, as well as a large class of operators whose unimodular eigenvectors are spanning with respect to the Lebesgue measure. However, we exhibit two frequently hypercyclic weighted shifts Bw,Bw on c0(Z+) whose direct sum {BwBw' is not \mathcal {U} -frequently hypercyclic (so that neither of them is hereditarily frequently hypercyclic), and we construct a C-type operator on ℓp(Z+, 1≤ p<∞, which is frequently hypercyclic but not hereditarily frequently hypercyclic. We also solve several problems concerning disjoint frequent hypercyclicity: we show that for every N\in N, any disjoint frequently hypercyclic N-tuple of operators (T1,⋯,TN) can be extended to a disjoint frequently hypercyclic (N+1) -tuple (T1,\⋯,TN, TN+1) as soon as the underlying space supports a hereditarily frequently hypercyclic operator; we construct a disjoint frequently hypercyclic pair which is not densely disjoint hypercyclic; and we show that the pair (D,τa) is disjoint frequently hypercyclic, where D is the derivation operator acting on the space of entire functions and τa is the operator of translation by a ϵ C 0. Part of our results are in fact obtained in the general setting of Furstenberg families.

Disciplines :

Mathematics

Author, co-author :

Bayart, Frédéric ; Laboratoire de Mathématiques Blaise Pascal Umr 6620 Cnrs, Université Clermont Auvergne, Aubière, France

Grivaux, Sophie ; Univ. Lille, Cnrs, Umr 8524, Laboratoire Paul Painlevé, Lille, France

Matheron, Étienne ; Univ. Artois, Ur 2462 - Laboratoire de Mathématiques de Lens (LML), Lens, France

Menet, Quentin ; Université de Mons - UMONS > Faculté des Sciences > Service de Probabilité et statistique

Language :

English

Title :

Hereditarily frequently hypercyclic operators and disjoint frequent hypercyclicity

Publication date :

2025

Journal title :

Ergodic Theory and Dynamical Systems

ISSN :

0143-3857

eISSN :

1469-4417

Publisher :

Cambridge University Press

Pages :

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0143385725000136

Research unit :

S844 - Probabilité et statistique

Research institute :

R150 - Institut de Recherche sur les Systèmes Complexes

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