Abstract

In this work we study the rational periodic orbits of the Lyness recurrence x_{n+2}=(a+x_{n+1})/x_{n}.
It is known that when a>0 the only possible rational orbits are 1,5 and 9-periodic. The first ones are easily obtained, and the last ones have resisted previous analysis. We prove that for infinitely many positive values of a, positive 9-period rational sequences occur. This result is our main result and gives a negative answer to a conjecture of E.C. Zeeman (1996).
In fact there exist initial conditions and values of the parameter a for which the Lyness recurrence generates rational periodic sequences with minimal periods 1,2,3,5,6,7,8,9,10 or 12 and that these are the only periods that rational sequences {x_n} can have.
The key of our study is the fact that the Lyness map preserves a foliation of algebraic curves which are mainly elliptic curves. That is, curves of genus 1 where it is possible to define an operation between their points giving rise to a group structure.
We will see that on each elliptic level, the Lyness map can be seen as a linear translation from P to P+Q, where + is the group operation. Since the set of rational points of an elliptic curve is an abelian finitely generated subgroup with a well known structure, the existence of rational periodic sequences is then related with this structure.

This is a joint work with Armengol Gasull and Xavier Xarles.