Perron-Frobenius operators and the Klein-Gordon equation

For a smooth curve Γ and a set Λ in the plane R2, let AC(Γ; Λ) be the space of finite Borel measures in the plane supported on Γ, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ. Following [12], we say that (Γ, Λ) is a Heisenberg uniqueness pair if AC(Γ...

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Autores: Canto Martín, Francisco Manuel, Hedenmalm, Håkan, Montes Rodríguez, Alfonso
Formato: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2014
País:España
Recursos:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/45142
Acesso em linha:http://hdl.handle.net/11441/45142
https://doi.org/10.4171/JEMS/427
Access Level:acceso abierto
Palavra-chave:Trigonometric system
Inversion
Composition operator
Klein-Gordon equation
Ergodic theory
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spelling Perron-Frobenius operators and the Klein-Gordon equationCanto Martín, Francisco ManuelHedenmalm, HåkanMontes Rodríguez, AlfonsoTrigonometric systemInversionComposition operatorKlein-Gordon equationErgodic theoryFor a smooth curve Γ and a set Λ in the plane R2, let AC(Γ; Λ) be the space of finite Borel measures in the plane supported on Γ, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ. Following [12], we say that (Γ, Λ) is a Heisenberg uniqueness pair if AC(Γ; Λ) = {0}. In the context of a hyperbola Γ, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Γ; Λ) when it is nonzero. We will fix the curve Γ to be the hyperbola x1x2 = 1, and the set Λ = Λα,β to be the lattice-cross Λα,β = (αZ × {0}) ∪ ({0} × βZ), where α, β are positive reals. We will also consider Γ+, the branch of x1x2 = 1 where x1 > 0. In [12], it is shown that AC(Γ; Λα,β) = {0} if and only if αβ ≤ 1. Here, we show that for αβ > 1, we get a rather drastic “phase transition”: AC(Γ; Λα,β) is infinite-dimensional whenever αβ > 1. It is shown in [13] that AC(Γ+; Λα,β) = {0} if and only if αβ < 4. Moreover, at the edge αβ = 4, the behavior is more exotic: the space AC(Γ+; Λα,β) is one-dimensional. Here, we show that the dimension of AC(Γ+; Λα,β) is infinite whenever αβ > 4. Dynamical systems, and more specifically Perron-Frobenius operators, will play a prominent role in the presentation.Ministerio de Ciencia e InnovaciónGöran Gustafsson FoundationJunta de AndalucíaEuropean Mathematical SocietyAnálisis MatemáticoFQM260: Variable Compleja y Teoria de Operadores2014info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/45142https://doi.org/10.4171/JEMS/427reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésJournal of the European Mathematical Society, 16 (1), 31-66.MTM2009-09501FQM260http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=16&iss=1&rank=2info:eu-repo/semantics/openAccessoai:idus.us.es:11441/451422026-06-17T12:51:07Z
dc.title.none.fl_str_mv Perron-Frobenius operators and the Klein-Gordon equation
title Perron-Frobenius operators and the Klein-Gordon equation
spellingShingle Perron-Frobenius operators and the Klein-Gordon equation
Canto Martín, Francisco Manuel
Trigonometric system
Inversion
Composition operator
Klein-Gordon equation
Ergodic theory
title_short Perron-Frobenius operators and the Klein-Gordon equation
title_full Perron-Frobenius operators and the Klein-Gordon equation
title_fullStr Perron-Frobenius operators and the Klein-Gordon equation
title_full_unstemmed Perron-Frobenius operators and the Klein-Gordon equation
title_sort Perron-Frobenius operators and the Klein-Gordon equation
dc.creator.none.fl_str_mv Canto Martín, Francisco Manuel
Hedenmalm, Håkan
Montes Rodríguez, Alfonso
author Canto Martín, Francisco Manuel
author_facet Canto Martín, Francisco Manuel
Hedenmalm, Håkan
Montes Rodríguez, Alfonso
author_role author
author2 Hedenmalm, Håkan
Montes Rodríguez, Alfonso
author2_role author
author
dc.contributor.none.fl_str_mv Análisis Matemático
FQM260: Variable Compleja y Teoria de Operadores
dc.subject.none.fl_str_mv Trigonometric system
Inversion
Composition operator
Klein-Gordon equation
Ergodic theory
topic Trigonometric system
Inversion
Composition operator
Klein-Gordon equation
Ergodic theory
description For a smooth curve Γ and a set Λ in the plane R2, let AC(Γ; Λ) be the space of finite Borel measures in the plane supported on Γ, absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ. Following [12], we say that (Γ, Λ) is a Heisenberg uniqueness pair if AC(Γ; Λ) = {0}. In the context of a hyperbola Γ, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the issue of finding the dimension of AC(Γ; Λ) when it is nonzero. We will fix the curve Γ to be the hyperbola x1x2 = 1, and the set Λ = Λα,β to be the lattice-cross Λα,β = (αZ × {0}) ∪ ({0} × βZ), where α, β are positive reals. We will also consider Γ+, the branch of x1x2 = 1 where x1 > 0. In [12], it is shown that AC(Γ; Λα,β) = {0} if and only if αβ ≤ 1. Here, we show that for αβ > 1, we get a rather drastic “phase transition”: AC(Γ; Λα,β) is infinite-dimensional whenever αβ > 1. It is shown in [13] that AC(Γ+; Λα,β) = {0} if and only if αβ < 4. Moreover, at the edge αβ = 4, the behavior is more exotic: the space AC(Γ+; Λα,β) is one-dimensional. Here, we show that the dimension of AC(Γ+; Λα,β) is infinite whenever αβ > 4. Dynamical systems, and more specifically Perron-Frobenius operators, will play a prominent role in the presentation.
publishDate 2014
dc.date.none.fl_str_mv 2014
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/45142
https://doi.org/10.4171/JEMS/427
url http://hdl.handle.net/11441/45142
https://doi.org/10.4171/JEMS/427
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of the European Mathematical Society, 16 (1), 31-66.
MTM2009-09501
FQM260
http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=16&iss=1&rank=2
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv European Mathematical Society
publisher.none.fl_str_mv European Mathematical Society
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.301603

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