Chaos for semigroups of unbounded operators

deLaubenfels, Ralph; Emamirad, Hassan; Grosse‐Erdmann, Karl‐Goswin

doi:10.1002/mana.200310112

Cited by 37 publications

(43 citation statements)

References 25 publications

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“…C 0 -semigroups generated by polynomials with constant coefficients of the derivative operator were treated on [12], under mild assumptions on the underlying space. These results were extended by deLaubenfels et al dealing with semigroups of unbounded operators [13,Section 5]. Using their Theorem 5.3 and Corollary 5.4 we can complete the one-dimensional case.…”

Section: One-dimensional Ornstein-uhlenbeck Operatorsmentioning

confidence: 59%

Hypercyclic Semigroups Generated by Ornstein-Uhlenbeck Operators

Conejero

Mangino

2010

Mediterr. J. Math.

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The chaotic and hypercyclic behavior of the C0-semigroups of operators generated by a perturbation of the Ornstein-Uhlenbeck operator with a multiple of the identity in L 2 (R N ) is investigated. Negative and positive results are presented, depending on the signs of the real parts of the eigenvalues of the matrix appearing in the drift of the operator.Mathematics Subject Classification (2010). Primary 47A16; Secondary 47D06, 47D07.

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Section: One-dimensional Ornstein-uhlenbeck Operatorsmentioning

confidence: 59%

Hypercyclic Semigroups Generated by Ornstein-Uhlenbeck Operators

Conejero

Mangino

2010

Mediterr. J. Math.

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“…As a generalization, deLaubenfels, Emamirad and Grosse-Erdmann studied the dynamics of semigroups of Cregularized semigroups of unbounded operators [140].…”

Section: Dynamics Of Semigroups On Herzog Spacesmentioning

confidence: 99%

Linear dynamics of semigroups generated by differential operators

et al. 2017

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During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C 0 -semigroups of linear and continuous operators.We will review some of these notions and we will discuss basic properties of the dynamics of C 0 -semigroups. We will also study in detail the dynamics of the translation C 0 -semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C 0 -semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models. . On the one hand, it is connected with the still unsolved Invariant Subspace Problem on Hilbert spaces, which asks for the existence of an operator with no nontrivial closed invariant subset. The answer was positive in Banach spaces, as it was seen by Read [4] in`1. On the other hand, there is a number of different connections of linear dynamics with different areas such as algebra, topology, real and complex analysis, functional analysis, approximation theory, number theory, and probability.The advances in the area were first compiled by Grosse-Erdmann [5,6]. The monographs of Bayart and Matheron [7] and Grosse-Erdmann and Peris [8] represent a good source of the state of the art in the area. The study of the size and the algebraic structure of the set of vectors with wild behaviour, provides an interesting field for continuing with the quest of surprising examples of sets of pathological elements that, however, are preserved by the elementary algebraic operators. In this line, a selection of topics was recently revisited by Aron et al. in [9, Ch. 4], see also [10].Given a family fT i g i 2I of linear and continuous operators on an infinite-dimensional separable Banach space X , we say that it is universal if there exists some element x 2 X such that, its orbit fT i x W i 2 I g is dense in X .

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“…The study will be carried out on Herzog type spaces [32]. Herzog's result was later improved in [24].…”

Section: Introductionmentioning

confidence: 99%

Chaotic semigroups from second order partial differential equations

Conejero

Lizama

Murillo‐Arcila

2017

Journal of Mathematical Analysis and Applications

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Abstract. We give general conditions on given parameters to ensure Devaney and distributional chaos for the solution C 0 -semigroup corresponding to a class of second-order partial differential equations. We also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups. In the case of chaos, we prove that the C 0 -semigroup admits a strongly mixing measure with full support. We also give concrete examples of partial differential equations, such as the telegraph equation, whose solutions satisfy these properties.

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Chaos for semigroups of unbounded operators

Cited by 37 publications

References 25 publications

Hypercyclic Semigroups Generated by Ornstein-Uhlenbeck Operators

Hypercyclic Semigroups Generated by Ornstein-Uhlenbeck Operators

Linear dynamics of semigroups generated by differential operators

Chaotic semigroups from second order partial differential equations

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