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Rational periodic sequences for the Lyness recurrence
Gasull Embid, Armengol; Mañosa Fernández, Víctor; Xarles Ribas, Xavier
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III; Universitat Politècnica de Catalunya. CODALAB - Control, Dinàmica i Aplicacions
Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. 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 $\{x_n\}_n$ 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 $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.
Preprint
2012-05-10
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en diferències
Recurrent sequences (Mathematics)
Lyness difference equations
Rational points over elliptic curves
Periodic points
Universal family of elliptic curves
Seqüències recurrents
Classificació AMS::39 Difference and functional equations::39A Difference equations
Classificació AMS::14 Algebraic geometry::14H Curves
Open Access
Other
         

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