Two remarks on frequent hypercyclicity

Moothathu, T.K. Subrahmonian

doi:10.1016/j.jmaa.2013.06.034

Cited by 21 publications

(21 citation statements)

References 8 publications

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“…In [27], [41] the authors asked if the set F HC(T ) is always of first Baire category. The positive answer was recently given independently by Bayart-Ruzsa [10] (for Banach spaces) and by Moothathu [49] and Grivaux-Matheron [38].…”

Section: Problem 1 (A) Which (Strong) Dynamical Behaviour Does the Fmentioning

confidence: 95%

Frequently hypercyclic operators: Recent advances and open problems

Grosse‐Erdmann¹

2016

Advanced Courses of Mathematical Analysis VI

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We report on recent advances on one of the central notions in linear dynamics, that of a frequently hypercyclic operator. We also include a list of ten open problems. 1 Introduction Frequent hypercyclicity is one of the most fascinating notions in linear dynamics: it is a natural strengthening of the key concept in linear dynamics, that of hypercyclicity, and it is very close in spirit (though not, as we will see, in a strict sense) to that of linear chaos. Although initiated only ten years ago, the study of frequently hypercyclic operators has seen very deep and important developments, many of them quite recent. In this survey we will discuss some of these advances, and we will highlight several open problems. Let us first recall the two central concepts in linear dynamics. Throughout this paper, X will denote a separable F-space, that is, a topological vector space whose topology is induced by a complete translation-invariant metric. The reader will lose very little in assuming that X is a separable Banach space. Moreover, T : X → X will denote a (continuous and linear) operator on X.

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Section: Problem 1 (A) Which (Strong) Dynamical Behaviour Does the Fmentioning

confidence: 95%

Frequently hypercyclic operators: Recent advances and open problems

Grosse‐Erdmann¹

2016

Advanced Courses of Mathematical Analysis VI

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“…Theorem 15]. On the other hand, for the set F HC a pT q, if we look at the proof given by Moothathu [11,Theorem 1] in the case of frequent hypercyclicity, it is sufficient to show that d a pIq " d a pI`1q for every set I of non-negative integers.…”

Section: Weighted Densities Between D and Dmentioning

confidence: 99%

“…These two notions of hypercyclicity have interesting differences. For instance, the set U F HCpT q of U-frequently hypercyclic vectors is either empty or residual [4,7,11] but the set F HCpT q of frequently hypercyclic vectors is always meager [4]. For this reason, we try to create a bridge between these two notions in the hope to better understand their differences and their limits.…”

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confidence: 99%

A bridge between U-frequent hypercyclicity and frequent hypercyclicity

Menet

2020

Journal of Mathematical Analysis and Applications

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Given A the family of weights a " panqn decreasing to 0 such that the series ř 8 n"0 an diverges, we show that the supremum on A of lower weighted densities coincides with the unweighted upper density and that the infimum on A of upper weighted densities coincides with the unweighted lower density. We then investigate the notions of U -frequent hypercyclicity and frequent hypercyclicity associated to these weighted densities. We show that there exists an operator which is U -frequently hypercyclic for each weight in A but not frequently hypercyclic, although the set of frequently hypercyclic vectors always coincides with the intersection of sets of U -frequently hypercyclic vectors for each weight in A.

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“…9.11] yields that the Taylor shift in the setting above is also chaotic and mixing. Furthermore, in [10] the author showed that the set of frequently hypercyclic vectors is always meagre while the set of hypercyclic vectors is residual.…”

Section: Tools Of Linear Dynamicsmentioning

confidence: 99%

“…Again, [10] yields that the set of frequently hypercyclic vectors of T on H ( ) is of first Baire category.…”

Section: Remarkmentioning

confidence: 99%

Frequently Hypercyclic Taylor Shifts

Thelen

2016

Comput. Methods Funct. Theory

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It is known that unimodular eigenvalues of an operator can give information about its dynamical behaviour. Recently, some situations have been characterized in which the Taylor shift operator is hypercyclic. The aim of this article is to use an eigenvalue criterion to find assumptions that guarantee the frequent hypercyclicity of the Taylor shift operator. As a conclusion, we also obtain holomorphic functions with smooth boundary values.

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Two remarks on frequent hypercyclicity

Cited by 21 publications

References 8 publications

Frequently hypercyclic operators: Recent advances and open problems

Frequently hypercyclic operators: Recent advances and open problems

A bridge between U-frequent hypercyclicity and frequent hypercyclicity

Frequently Hypercyclic Taylor Shifts

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