On Li–Yorke and distributionally chaotic direct sum operators

Yin, Zongbin; He, Shengnan; Yu, Huan

doi:10.1016/j.topol.2018.02.012

“…Regarding distributionally chaotic backward shift operators, mention should be also made of papers [14]- [15], [34] and [37]. For the sake of brevity, we will not reconsider the related problematic for m n -distributional chaos here.…”

Section: Conclusion Final Remarks and Open Problemsmentioning

confidence: 99%

Reiterative $m_{n}$-distributional chaos of type $s$ in Fr\' echet spaces

Kostić¹

2019

Preprint

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The main aim of this paper is to consider various notions of (dense) mn-distributional chaos of type s and (dense) reiterative mn-distributional chaos of type s for general sequences of linear not necessarily continuous operators in Fréchet spaces. Here, (mn) is an increasing sequence in [1, ∞) satisfying lim infn→∞ mn n > 0 and s could be 0, 1, 2, 2+, 2 1 2 , 3, 1+, 2−, 2 Bd , 2 Bd + . We investigate mn-distributionally chaotic properties and reiteratively mndistributionally chaotic properties of some special classes of operators like weighted forward shift operators and weighted backward shift operators in Fréchet sequence spaces, considering also continuous analogues of introduced notions and some applications to abstract partial differential equations.2010 Mathematics Subject Classification. 47A06, 47A16. Key words and phrases. mn-distributional chaos of type s, reiterative mn-distributional chaos of type s, λ-distributional chaos of type s, reiterative λ-distributional chaos of type s, Fréchet spaces.The author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.

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“…The notion of a Li-Yorke irregular vector in Hilbert space has been defined for the first time by B. Beauzamy in [4]. After that, Li-Yorke linear dynamics in Hilbert, Banach and Frechet function spaces has been analyzed by a great number of other authors including G. T. Prǎjiturǎ [35], T. Bermúdez et al [5], N. C. Bernardes Jr et al [8], X. Wu [38] and Z. Yin et al [39] (see also [23], [28]). It is well known that any linear hypercyclic operator needs to be Li-Yorke chaotic as well as that the converse statement does not hold in general.…”

Section: Introductionmentioning

confidence: 99%

Disjoint Li-Yorke Chaos in Frechet Spaces ´ M. Kostic

2020

Electronic Journal of Mathematical Analysis and Applications

0

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“…The notion of a Li-Yorke irregular vector in Hilbert space has been defined for the first time by B. Beauzamy in [4]. After that, Li-Yorke linear dynamics in Hilbert, Banach and Frechet function spaces has been analyzed by a great number of other authors including G. T. Prǎjiturǎ [35], T. Bermúdez et al [5], N. C. Bernardes Jr et al [8], X. Wu [38] and Z. Yin et al [39] (see also [23], [28]). It is well known that any linear hypercyclic operator needs to be Li-Yorke chaotic as well as that the converse statement does not hold in general.…”

Section: Introductionmentioning

confidence: 99%