Rational Periodic Sequences for the Lyness Recurrence

TL;DR

This paper characterizes all possible rational periodic sequences generated by the Lyness recurrence, identifying specific prime periods, and demonstrates the existence of infinitely many positive rational sequences with period 9, solving an open problem.

Contribution

It proves the existence of rational sequences with prime periods up to 12, especially infinitely many with period 9, and shows the Lyness map's universality for elliptic curves.

Findings

Rational sequences with prime periods 1,2,3,5,6,7,8,9,10,12 exist.

Only periods 1,5,9 occur for positive rational initial conditions and parameter.

Lyness map's invariant level sets form a universal family of elliptic curves.

Abstract

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.