Rational Periodic Sequences for the Lyness Recurrence
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mañosa Fernández, Víctor 1971- (Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III)
Xarles Ribas, Francesc Xavier (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2012
Abstract:	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.
Grants:	Ministerio de Ciencia y Tecnología MTM2008-03437
Note:	Agraïments: The authors are partially supported by MCYT through grant DPI2008-06699-C02-02 (second author) and MTM2009-10359 (third author). The authors are also supported by the Government of Catalonia through the SGR program.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Lyness difference equations ; Rational points over elliptic curves ; Periodic points ; Universal family of elliptic curves
Published in:	Discrete and continuous dynamical systems. Series A, Vol. 32 Núm. 2 (2012) , p. 587-604, ISSN 1553-5231

DOI: 10.3934/dcds.2012.32.587

Postprint 18 p, 426.2 KB

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