Multidimensional continued fractions, dynamic renormalization and KAM theory
Autor(a) principal: | |
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Data de Publicação: | 2007 |
Outros Autores: | , |
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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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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1799133620632289280 |