Topological transitivity and mixing of composition operators

Bayart, Frédéric; Darji, Udayan B.; Pires, Benito

doi:10.1016/j.jmaa.2018.04.063

Cited by 21 publications

(16 citation statements)

References 14 publications

(29 reference statements)

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“…Proof. We use an example from our earlier work [2]. In particular, our X is an odometer and f is the +1-map.…”

Section: Example 43mentioning

confidence: 99%

“…The continuous linear operator T f : L p (X) → L p (X) defined by ϕ → ϕ • f is called the composition operator induced by f . In such a general setting, hypercyclic composition operators and topologically mixing composition operators on L p -spaces were completely characterized in a joint work of the authors with Bayart in [2]. Li-Yorke chaotic composition operators on L p -spaces were completely characterized in a joint work of the authors with Bernardes Jr. in [6].…”

Section: Introductionmentioning

confidence: 99%

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Chaos and frequent hypercyclicity for composition operators

Darji¹,

Pires²

2021

Proceedings of the Edinburgh Mathematical Society

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The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Ruzsa in 2015 that for backward weighted shifts on $\ell _p(\mathbb {Z})$ , the notions of chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^{p}$ -spaces, the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class, an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible operator frequently hypercyclic on $\ell _1$ whose inverse is not frequently hypercyclic is constructed.

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“…Proof. We use an example from our earlier work [2]. In particular, our X is an odometer and f is the +1-map.…”

Section: Example 43mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Chaos and frequent hypercyclicity for composition operators

Darji¹,

Pires²

2021

Proceedings of the Edinburgh Mathematical Society

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“…Throughout this note, (X, B, µ) will denote a measure space with µ(X) = 0 and f : X → X will be a bimeasurable map (that is, f (B) ∈ B and f −1 (B) ∈ B for every B ∈ B) for which there exists a constant c > 0 such that (1) µ f (B) ≥ cµ(B) for every B ∈ B.…”

Section: Introductionmentioning

confidence: 99%

“…Condition (1) ensures that the composition operator T f : ϕ → ϕ • f is a continuous linear operator acting on L p (X, B, µ) (1 ≤ p < ∞). This constitutes a natural class of operators.…”

Section: Introductionmentioning

confidence: 99%

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Li–Yorke chaos for composition operators on $$L^p$$-spaces

2019

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Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more useful characterizations of Li-Yorke chaos can be given in the special setting of composition operators on L p spaces. As a consequence we obtain a simple characterization of weighted shifts which are Li-Yorke chaotic. We give numerous examples to show that our results are sharp.2010 Mathematics Subject Classification. Primary 47A16, 47B33; Secondary 37D45.

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Hypercyclicity of Weighted Composition Operators on $$L^p$$-Spaces

Azimi

Jabbarzadeh

2022

Mediterr. J. Math.

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Topological transitivity and mixing of composition operators

Cited by 21 publications

References 14 publications

Chaos and frequent hypercyclicity for composition operators

Chaos and frequent hypercyclicity for composition operators

Li–Yorke chaos for composition operators on $$L^p$$-spaces

Hypercyclicity of Weighted Composition Operators on $$L^p$$-Spaces

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