Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder

We extend the results and techniques from [7] to study the combinatorial dynamics (forcing) and entropy of quasiperiodically forced skewproducts on the cylinder. For these maps we prove that a cyclic permutation τ forces a cyclic permutation ν as interval patterns if and only if τ forces ν as cylind...

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Detalhes bibliográficos
Autores: Alsedà, Lluís|||0000-0001-9908-1063, Mañosas, Francesc|||0000-0003-2535-0501, Morales, Leopoldo
Tipo de documento: artigo
Data de publicação:2015
País:España
Recursos:Universitat Autònoma de Barcelona
Repositório:Dipòsit Digital de Documents de la UAB
Idioma:inglês
OAI Identifier:oai:ddd.uab.cat:145307
Acesso em linha:https://ddd.uab.cat/record/145307
https://dx.doi.org/urn:doi:10.1016/j.jmaa.2015.03.038
Access Level:Acceso aberto
Palavra-chave:Combinatorial dynamics
Forcing entropy
Irrational rotation
Quasiperiodically forced systems on the cylinder
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spelling Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinderAlsedà, Lluís|||0000-0001-9908-1063Mañosas, Francesc|||0000-0003-2535-0501Morales, LeopoldoCombinatorial dynamicsForcing entropyIrrational rotationQuasiperiodically forced systems on the cylinderWe extend the results and techniques from [7] to study the combinatorial dynamics (forcing) and entropy of quasiperiodically forced skewproducts on the cylinder. For these maps we prove that a cyclic permutation τ forces a cyclic permutation ν as interval patterns if and only if τ forces ν as cylinder patterns. This result gives as a corollary the Sharkovski˘ı Theorem for quasiperiodically forced skew-products on the cylinder proved in [7]. Next, the notion of s-horseshoe is defined for quasiperiodically forced skewproducts on the cylinder and it is proved, as in the interval case, that if a quasiperiodically forced skew-product on the cylinder has an s-horseshoe then its topological entropy is larger than or equals to log(s). Finally, if a quasiperiodically forced skew-product on the cylinder has a periodic orbit with pattern τ, then h(F) ≥ h(fτ ), where fτ denotes the connectthe-dots interval map over a periodic orbit with pattern τ. This implies that if the period of τ is 2nq with n ≥ 0 and q ≥ 1 odd, then h(F) ≥ log(λq) 2n , where λ1 = 1 and, for each q ≥ 3, λq is the largest root of the polynomial x q - 2x q-2 - 1. Moreover, for every m = 2nq with n ≥ 0 and q ≥ 1 odd, there exists a quasiperiodically forced skew-product on the cylinder Fm with a periodic orbit of period m such that h(Fm) = log(λq) 2n . This extends the analogous result for interval maps to quasiperiodically forced skew-products on the cylinder. 22015-01-0120152015-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/145307https://dx.doi.org/urn:doi:10.1016/j.jmaa.2015.03.038reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-01486Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2011-26995-C02-0open accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:1453072026-06-06T12:50:31Z
dc.title.none.fl_str_mv Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
title Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
spellingShingle Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
Alsedà, Lluís|||0000-0001-9908-1063
Combinatorial dynamics
Forcing entropy
Irrational rotation
Quasiperiodically forced systems on the cylinder
title_short Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
title_full Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
title_fullStr Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
title_full_unstemmed Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
title_sort Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder
dc.creator.none.fl_str_mv Alsedà, Lluís|||0000-0001-9908-1063
Mañosas, Francesc|||0000-0003-2535-0501
Morales, Leopoldo
author Alsedà, Lluís|||0000-0001-9908-1063
author_facet Alsedà, Lluís|||0000-0001-9908-1063
Mañosas, Francesc|||0000-0003-2535-0501
Morales, Leopoldo
author_role author
author2 Mañosas, Francesc|||0000-0003-2535-0501
Morales, Leopoldo
author2_role author
author
dc.subject.none.fl_str_mv Combinatorial dynamics
Forcing entropy
Irrational rotation
Quasiperiodically forced systems on the cylinder
topic Combinatorial dynamics
Forcing entropy
Irrational rotation
Quasiperiodically forced systems on the cylinder
description We extend the results and techniques from [7] to study the combinatorial dynamics (forcing) and entropy of quasiperiodically forced skewproducts on the cylinder. For these maps we prove that a cyclic permutation τ forces a cyclic permutation ν as interval patterns if and only if τ forces ν as cylinder patterns. This result gives as a corollary the Sharkovski˘ı Theorem for quasiperiodically forced skew-products on the cylinder proved in [7]. Next, the notion of s-horseshoe is defined for quasiperiodically forced skewproducts on the cylinder and it is proved, as in the interval case, that if a quasiperiodically forced skew-product on the cylinder has an s-horseshoe then its topological entropy is larger than or equals to log(s). Finally, if a quasiperiodically forced skew-product on the cylinder has a periodic orbit with pattern τ, then h(F) ≥ h(fτ ), where fτ denotes the connectthe-dots interval map over a periodic orbit with pattern τ. This implies that if the period of τ is 2nq with n ≥ 0 and q ≥ 1 odd, then h(F) ≥ log(λq) 2n , where λ1 = 1 and, for each q ≥ 3, λq is the largest root of the polynomial x q - 2x q-2 - 1. Moreover, for every m = 2nq with n ≥ 0 and q ≥ 1 odd, there exists a quasiperiodically forced skew-product on the cylinder Fm with a periodic orbit of period m such that h(Fm) = log(λq) 2n . This extends the analogous result for interval maps to quasiperiodically forced skew-products on the cylinder.
publishDate 2015
dc.date.none.fl_str_mv 2
2015-01-01
2015
2015-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/145307
https://dx.doi.org/urn:doi:10.1016/j.jmaa.2015.03.038
url https://ddd.uab.cat/record/145307
https://dx.doi.org/urn:doi:10.1016/j.jmaa.2015.03.038
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-01486
Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM2011-26995-C02-0
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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