Multidimensional continued fractions, dynamic renormalization and KAM theory

Detalhes bibliográficos
Autor(a) principal: Khanin, Kostya
Data de Publicação: 2007
Outros Autores: Dias, João Lopes, Marklof, Jens
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10400.5/29044
Resumo: The disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SL(d,Z)\ SL(d, R) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension..
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spelling Multidimensional continued fractions, dynamic renormalization and KAM theoryFraction AlgorithmsDynamics of FlowsGeneric VectorsKAM TheoryThe disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SL(d,Z)\ SL(d, R) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension..SpringerRepositório da Universidade de LisboaKhanin, KostyaDias, João LopesMarklof, Jens2023-10-13T14:25:14Z20072007-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.5/29044engKhanin, Kostya; João Lopes Dias and Jens Marklof . (2007). “Multidimensional continued fractions, dynamic renormalization and KAM theory”. Communications in Mathematical Physics , Volume 270: pp.197-231 . (Search PDF in 2023)Doi: 10.1007/s00220-006-0125-y1432-0916info:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-12T01:31:32Zoai:www.repository.utl.pt:10400.5/29044Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T20:35:46.744347Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Multidimensional continued fractions, dynamic renormalization and KAM theory
title Multidimensional continued fractions, dynamic renormalization and KAM theory
spellingShingle Multidimensional continued fractions, dynamic renormalization and KAM theory
Khanin, Kostya
Fraction Algorithms
Dynamics of Flows
Generic Vectors
KAM Theory
title_short Multidimensional continued fractions, dynamic renormalization and KAM theory
title_full Multidimensional continued fractions, dynamic renormalization and KAM theory
title_fullStr Multidimensional continued fractions, dynamic renormalization and KAM theory
title_full_unstemmed Multidimensional continued fractions, dynamic renormalization and KAM theory
title_sort Multidimensional continued fractions, dynamic renormalization and KAM theory
author Khanin, Kostya
author_facet Khanin, Kostya
Dias, João Lopes
Marklof, Jens
author_role author
author2 Dias, João Lopes
Marklof, Jens
author2_role author
author
dc.contributor.none.fl_str_mv Repositório da Universidade de Lisboa
dc.contributor.author.fl_str_mv Khanin, Kostya
Dias, João Lopes
Marklof, Jens
dc.subject.por.fl_str_mv Fraction Algorithms
Dynamics of Flows
Generic Vectors
KAM Theory
topic Fraction Algorithms
Dynamics of Flows
Generic Vectors
KAM Theory
description The disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SL(d,Z)\ SL(d, R) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension..
publishDate 2007
dc.date.none.fl_str_mv 2007
2007-01-01T00:00:00Z
2023-10-13T14:25:14Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10400.5/29044
url http://hdl.handle.net/10400.5/29044
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Khanin, Kostya; João Lopes Dias and Jens Marklof . (2007). “Multidimensional continued fractions, dynamic renormalization and KAM theory”. Communications in Mathematical Physics , Volume 270: pp.197-231 . (Search PDF in 2023)
Doi: 10.1007/s00220-006-0125-y
1432-0916
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron:RCAAP
instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron_str RCAAP
institution RCAAP
reponame_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
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