Operators with dense, invariant, cyclic vector manifolds

Godefroy, Gilles; Shapiro, Joel H.

doi:10.1016/0022-1236(91)90078-j

Cited by 506 publications

(541 citation statements)

References 22 publications

Supporting

Mentioning

509

Contrasting

Unclassified

Order By: Relevance

“…It is shown in [3] that if φ is topologically transitive and has dense periodic points, then φ has sensitive dependence on initial conditions. In the special case where φ is a bounded operator on a separable Banach space X, it is shown in [9] that φ is topologically transitive if and only if it is hypercyclic. Thus φ is chaotic if and only if it is hypercyclic and has a dense set of periodic points.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Hypercyclic operators with an in¯nite dimensional closed subspace of periodic points

Grivaux¹

2003

Rev. Mat. Complut.

View full text Add to dashboard Cite

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Hypercyclic operators with an in¯nite dimensional closed subspace of periodic points

Grivaux¹

2003

Rev. Mat. Complut.

View full text Add to dashboard Cite

show abstract

“…Interesting constructions and counterexamples have been given in the framework of certain subspaces of analytic functions, see for instance [10,40,14]. Godefroy & Shapiro also considered Hardy and Bergman spaces for studying the dynamics of shift operators, see [28]. Our results show an interesting duality between chaos and stability, which is distinguished by means of a critical parameter depending on the given equation.…”

Section: Introductionmentioning

confidence: 80%

Chaotic semigroups from second order partial differential equations

Conejero

Lizama

Murillo‐Arcila

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

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.

show abstract

“…It is a variation of the (DSW) criterion [DSW97] which depends on verifying that the point spectrum of the infinitesimal generator of the C 0 -semigroup contains "enough" eigenvalues. A first criterion stated in these terms was given for operators by Godefroy and Shapiro in [GS91]. We will use the following version of the (DSW) Criterion, see [GEPM11,Th.…”

Section: Preliminariesmentioning

confidence: 99%

On the existence of chaos for the viscous van Wijngaarden–Eringen equation

Conejero

Lizama

Murillo‐Arcila

2016

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

Abstract. We study the viscous van WijngaardenEringen equation:∂ 4 u ∂t 2 ∂x 2 which corresponds to the linearized version of the equation that models the acoustic planar propagation in bubbly liquids. We show the existence of an explicit range, solely in terms of the constants a 0 and Re d , in which we can ensure that this equation admits a uniformly continuous, Devaney chaotic and topologically mixing semigroup on Herzog's type Banach spaces.

show abstract

Operators with dense, invariant, cyclic vector manifolds

Cited by 506 publications

References 22 publications

Hypercyclic operators with an in¯nite dimensional closed subspace of periodic points

Hypercyclic operators with an in¯nite dimensional closed subspace of periodic points

Chaotic semigroups from second order partial differential equations

On the existence of chaos for the viscous van Wijngaarden–Eringen equation

Contact Info

Product

Resources

About