Linear Fractional Recurrences: Periodicities and Integrability

Bedford, Eric; Kim, Kyounghee

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. S2, pp. 33-56.

Les récurrences fractionnaires linéaires sont données par $z_{n+k}=A\left(z\right)/B\left(z\right)$, où $A\left(z\right)$ et $B\left(z\right)$ sont des fonctions linéaires de $z_n,z_{n+1},$ $\cdots ,z_{n+k-1}$. Dans cet article nous considérons deux questions concernant ces récurrences : (1) Trouver $A\left(z\right)$ et $B\left(z\right)$ telles que la récurrence soit périodique ; (2) Trouver des intégrales invariantes dans le cas où le degré de l’application birationnelle induite a une croissance quadratique. L’approche de ces questions se fait en considérant l’application birationnelle induite et en déterminant ses degrés dynamiques. Le premier théorème montre que pour tout $k$ il y a des récurrences fractionnaires linéaires à $k$ pas qui sont périodiques de période $4k$. Le second théorème montre que le degré du procédé de Lyness $A\left(z\right)=a+z_{n+1}+z_{n+2}+\cdots +z_{n+k-1}$ et $B\left(z\right)=z_{n+1}$ est à croissance quadratique. Le procédé de Lyness est intégrable, et nous en discuterons les intégrales connues.

Linear fractional recurrences are given as $z_{n+k}=A\left(z\right)/B\left(z\right)$, where $A\left(z\right)$ and $B\left(z\right)$ are linear functions of $z_n,z_{n+1},\cdots ,z_{n+k-1}$. In this article we consider two questions about these recurrences: (1) Find $A\left(z\right)$ and $B\left(z\right)$ such that the recurrence is periodic; and (2) Find (invariant) integrals in case the induced birational map has quadratic degree growth. We approach these questions by considering the induced birational map and determining its dynamical degree. The first theorem shows that for each $k$ there are $k$-step linear fractional recurrences which are periodic of period $4k$. The second theorem shows that the Lyness process, $A\left(z\right)=a+z_{n+1}+z_{n+2}+\cdots +z_{n+k-1}$ and $B\left(z\right)=z_{n+1}$ has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.

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