Frequent universality criterion and densities

Ernst, Romuald; Mouze, Augustin

doi:10.1017/etds.2019.103

Cited by 6 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us recall that any operator satisfying the Frequent Universality Criterion is automatically α-universal whenever α D s for some s ≥ 1 [21]. Since each of the criteria given in Section 2 are natural strengthenings of the Frequent Hypercyclicity Criterion, we could expect a positive answer to this question for any such α.…”

Section: Common Frequent Hypercyclicity With Respect To Densitiesmentioning

confidence: 99%

“…Furthermore, Ernst and Mouze recently proved [20,21] that any operator satisfying the usual Frequent Universality Criterion in fact enjoys a stronger form of frequent universality. Let α = (α k ) k≥1 be a sequence of non-negative real numbers with k≥1 α k = +∞.…”

Section: Introductionmentioning

confidence: 99%

“…For s ∈ N ∪ {∞}, D s := (exp(k/ log (s) (k))) k≥1 where log (s) = log • • • • • log, log appearing s times, with the conventions log (0) (x) = x and log (∞) (x) = 1 for anyx > 0. One can check that ϕ Ds (n) ∼ log (s) (n) exp(x/ log (s) (n)) for s ∈ N (see[21, Remark 3.10]) and ϕ D∞ (n) ∼ e e−1 exp(n). For r ≥ 1 we shall also write P r := (k r ) k≥1 .…”

mentioning

confidence: 99%

“…A similar calculation as that of [20,Lemma 4.10] allows to conclude that for all s ≥ 1 d Ds ((n k (f ))) > 0. Therefore this sequence (n k (f )) allows to construct a hypercyclic vector for T which will be D s -frequently hypercyclic for all s ≥ 1 (we refer the reader to the beginning of Section 2 of [21]).…”

mentioning

confidence: 99%

See 3 more Smart Citations

Common frequent hypercyclicity

Charpentier¹,

Ernst²,

Mestiri³

et al. 2020

Preprint

View full text Add to dashboard Cite

We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are variants of the classical Frequent Hypercyclicity Criterion and of a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset Λ of the complex plane for the family {λT : λ ∈ Λ} to have a common frequently hypercyclic vector. In passing, this permits us to easily exhibit frequent hypercyclic weighted shifts which do not possess common frequent hypercyclic vectors. We also provide with criteria for families of the recently introduced operators of C-type to share a common frequently hypercyclic vector. Further, we prove that the same problem of common α-frequent hypercyclicity may be vacuous, where the notion of α-frequent hypercyclicity extends that of frequent hypercyclicity replacing the natural density by more general weighted densities. Finally, it is already known that any operator satisfying the classical Frequent Universality Criterion is α-frequently universal for any sequence α satisfying a suitable condition. We complement this result by showing that for any such operator, there exists a vector x which is α-frequently universal for T , with respect to all such α.

show abstract

Section: Common Frequent Hypercyclicity With Respect To Densitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Common frequent hypercyclicity

Charpentier¹,

Ernst²,

Mestiri³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…We recall that the family S is intended to give rise to dynamical notions between reiterative hypercyclicity and U-frequent hypercyclicity. In [10,11], the authors focused on the links between frequent hypercyclicity and such dynamical notions given by lower densities. Moreover, the implication between U-frequent hypercyclicity with respect to a ∈ S and U-frequent hypercyclicity is given to us by the following lemma.…”

mentioning

confidence: 99%

$\mathcal{U}$-Frequent hypercyclicity notions and related weighted densities

Ernst¹,

Esser²,

Menet³

2019

Preprint

View full text Add to dashboard Cite

We study dynamical notions lying between U-frequent hypercyclicity and reiterative hypercyclicity by investigating weighted upper densities between the unweighted upper density and the upper Banach density. While chaos implies reiterative hypercyclicity, we show that chaos does not imply U-frequent hypercyclicity with respect to any weighted upper density. Moreover, we show that if T is U-frequently hypercyclic (resp. reiteratively hypercyclic) then the n-fold product of T is still U-frequently hypercyclic (resp. reiteratively hypercyclic) and that this implication is also satisfied for each of the considered U-frequent hypercyclicity notions.

show abstract