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

Bernardes, Nilson C.; Darji, Udayan B.; Pires, Benito

doi:10.1007/s00605-019-01341-2

Cited by 18 publications

(20 citation statements)

References 14 publications

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“…Chaotic operators are well studied in linear dynamics and well understood. For example, in [8], the authors showed a characterization of Li-Yorke chaos for bilateral weighted backward shifts. Moreover, Grosse-Erdmann in [14] gave a complete characterization of backward weighted shifts which are chaotic.…”

Section: Types Of Chaosmentioning

confidence: 99%

“…A systematic study of composition operators on L p (X) in the setting of linear dynamics was initiated in [2] and [8]. The motivation for the study of composition operators is to have a concrete but large class of operators which can be utilized as examples and counterexamples in linear dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…For general (X, B, µ, f, T f ) as above, characterizations of when T f is hypercyclic, when T f is mixing [2], and when T f is Li-Yorke chaotic [8] were given. These are very general characterizations and, as a specific case, when X = Z, f is +1-map we obtain the characterizations of hypercyclicity, mixing and Li-Yorke chaos for weighted backward shifts.…”

Section: Introductionmentioning

confidence: 99%

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Shift-like Operators on $L^p(X)$

D’Aniello¹,

Darji²,

Maiuriello³

2021

Preprint

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In this article we develop a general technique which takes a known characterization of a property for weighted backward shifts and lifts it up to a characterization of that property for a large class of operators on L p (X) which we call shift-like. The properties of interest include chaotic properties such as Li-Yorke chaos, hypercyclicity, frequent hypercyclicity as well as tame properties such as shadowing, expansivity and generalized hyperbolicity. Shift-like operators appear naturally as composition operators on L p (X) when the underlying space is a dissipative measure system. In the process of proving the main theorem, we prove some results concerning when a property is shared by a linear dynamical system and its factors.

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Section: Types Of Chaosmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Shift-like Operators on $L^p(X)$

D’Aniello¹,

Darji²,

Maiuriello³

2021

Preprint

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“…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]. Recently, generalized hyperbolicity among composition operators was characterized by the first author, D'Aniello and Maiuriello in [10].…”

Section: Introductionmentioning

confidence: 99%

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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“…At this point we like to point out that a systematic study of composition operators in the setting of linear dynamics was initiated in [3] and [6]. In [3], necessary and sufficient conditions were given for a composition operator to be topologically transitive and mixing.…”

Section: Introductionmentioning

confidence: 99%

Generalized Hyperbolicity and Shadowing in $L^p$ spaces

D'Aniello,

Darji,

Maiuriello

2020

Preprint

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It is rather well-known that hyperbolic operators have the shadowing property. In the setting of finite dimensional Banach spaces, having the shadowing property is equivalent to being hyperbolic. In 2018, Bernardes et al. constructed an operator with the shadowing property which is not hyperbolic, settling an open question. In the process, they introduced a class of operators which has come to be known as generalized hyperbolic operators. This class of operators seems to be an important bridge between hyperbolicity and the shadowing property. In this article, we show that for a large natural class of operators on L p (X) the notion of generalized hyperbolicity and the shadowing property coincide. We do this by giving sufficient and necessary conditions for a certain class of operators to have the shadowing property. We also introduce computational tools which allow construction of operators with and without the shadowing property. Utilizing these tools, we show how some natural probability distributions, such as the Laplace distribution and the Cauchy distribution, lead to operators with and without the shadowing property on L p (X).

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

Cited by 18 publications

References 14 publications

Shift-like Operators on $L^p(X)$

Shift-like Operators on $L^p(X)$

Chaos and frequent hypercyclicity for composition operators

Generalized Hyperbolicity and Shadowing in $L^p$ spaces

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