Abstract

For any real $a>0$, consider the dynamical system generated by the map $F(x,y,z)=(y,z,(a+y+z)/x)$, defined in $D=\{(x,y,z):x>0,y>0,z>0\}$. The iterates of $F(x,y,z)$ give the behavior of the three order difference equation $x$_k+3 = (a+x_k+1+x_k+2) / x_k , also known as Lyness 3-D recurrence. For $a=1$ it is easy to see that all points in $D$ are 8-periodic by $F$. We define two subsets of $D$, $G$ of dimension 2 and $L$ of dimension 1 and prove that for any $a>0$:
1) The point $p$ intersection between $G$ and $L$ is a fix point of $F$ and and the set L\{p} is formed by two periodic points of $F$.
2) The set $G$ is invariant by $F$. Furthermore $G\backslash \{p\}$ is filled of invariant closed curves, each one of them being diffeomorphic to a circle and such that the dynamical system generated by $F$ over it is conjugated to a rotation.
3) The subset $D\backslash L$ is filled of closed curves, each one of them being invariant by $F2$, diffeomorphic to a circle and such that the dynamical system generated by $F2$ over it is also conjugated to a rotation.

This is a joint work with A. Gasull and V. Mañosa