Rational Periodic Sequences for the Lyness Recurrence

Consider the celebrated Lyness recurrence xn+2 = (a + xn+1)/xn with a ∈ Q. First we prove that there exist initial conditions and values of a for which it generates periodic sequences of rational numbers with prime periods 1, 2, 3, 5, 6, 7, 8, 9, 10 or 12 and that these are the only periods that rat...

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Autores: Gasull, Armengol|||0000-0002-1719-8231, Mañosa Fernández, Víctor|||0000-0002-5082-3334, Xarles Ribas, Francesc Xavier|||0000-0002-7053-2286
Tipo de recurso: artículo
Fecha de publicación:2012
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:150557
Acceso en línea:https://ddd.uab.cat/record/150557
https://dx.doi.org/urn:doi:10.3934/dcds.2012.32.587
Access Level:acceso abierto
Palabra clave:Lyness difference equations
Rational points over elliptic curves
Periodic points
Universal family of elliptic curves
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spelling Rational Periodic Sequences for the Lyness RecurrenceGasull, Armengol|||0000-0002-1719-8231Mañosa Fernández, Víctor|||0000-0002-5082-3334Xarles Ribas, Francesc Xavier|||0000-0002-7053-2286Lyness difference equationsRational points over elliptic curvesPeriodic pointsUniversal family of elliptic curvesConsider the celebrated Lyness recurrence xn+2 = (a + xn+1)/xn with a ∈ Q. First we prove that there exist initial conditions and values of a for which it generates periodic sequences of rational numbers with prime periods 1, 2, 3, 5, 6, 7, 8, 9, 10 or 12 and that these are the only periods that rational sequences {xn}n can have. It is known that if we restrict our attention to positive rational values of a and positive rational initial conditions the only possible periods are 1, 5 and 9. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of a, positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien & Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order n, n ≥ 5, including n infinity. This fact implies that the Lyness map is a universal normal form for most birational maps on elliptic curves. 22012-01-0120122012-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/150557https://dx.doi.org/urn:doi:10.3934/dcds.2012.32.587reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437open 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:1505572026-06-06T12:50:31Z
dc.title.none.fl_str_mv Rational Periodic Sequences for the Lyness Recurrence
title Rational Periodic Sequences for the Lyness Recurrence
spellingShingle Rational Periodic Sequences for the Lyness Recurrence
Gasull, Armengol|||0000-0002-1719-8231
Lyness difference equations
Rational points over elliptic curves
Periodic points
Universal family of elliptic curves
title_short Rational Periodic Sequences for the Lyness Recurrence
title_full Rational Periodic Sequences for the Lyness Recurrence
title_fullStr Rational Periodic Sequences for the Lyness Recurrence
title_full_unstemmed Rational Periodic Sequences for the Lyness Recurrence
title_sort Rational Periodic Sequences for the Lyness Recurrence
dc.creator.none.fl_str_mv Gasull, Armengol|||0000-0002-1719-8231
Mañosa Fernández, Víctor|||0000-0002-5082-3334
Xarles Ribas, Francesc Xavier|||0000-0002-7053-2286
author Gasull, Armengol|||0000-0002-1719-8231
author_facet Gasull, Armengol|||0000-0002-1719-8231
Mañosa Fernández, Víctor|||0000-0002-5082-3334
Xarles Ribas, Francesc Xavier|||0000-0002-7053-2286
author_role author
author2 Mañosa Fernández, Víctor|||0000-0002-5082-3334
Xarles Ribas, Francesc Xavier|||0000-0002-7053-2286
author2_role author
author
dc.subject.none.fl_str_mv Lyness difference equations
Rational points over elliptic curves
Periodic points
Universal family of elliptic curves
topic Lyness difference equations
Rational points over elliptic curves
Periodic points
Universal family of elliptic curves
description Consider the celebrated Lyness recurrence xn+2 = (a + xn+1)/xn with a ∈ Q. First we prove that there exist initial conditions and values of a for which it generates periodic sequences of rational numbers with prime periods 1, 2, 3, 5, 6, 7, 8, 9, 10 or 12 and that these are the only periods that rational sequences {xn}n can have. It is known that if we restrict our attention to positive rational values of a and positive rational initial conditions the only possible periods are 1, 5 and 9. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of a, positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien & Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order n, n ≥ 5, including n infinity. This fact implies that the Lyness map is a universal normal form for most birational maps on elliptic curves.
publishDate 2012
dc.date.none.fl_str_mv 2
2012-01-01
2012
2012-01-01
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
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dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/150557
https://dx.doi.org/urn:doi:10.3934/dcds.2012.32.587
url https://ddd.uab.cat/record/150557
https://dx.doi.org/urn:doi:10.3934/dcds.2012.32.587
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 MTM2008-03437
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
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instname:Universitat Autònoma de Barcelona
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