Abstract

The starting point of this talk is the well known Lyness map $F(x,y)=(y,(a+y)/x)$ with $a>0$, which is a diffeomorphism of $Q+$ has a unique fixed point in $Q+$, and has a first integral $V$, whose level curves are topological circles surrounding the unique critical point. In an unpublished paper, E.C. Zeeman, using results of algebraic geometry (in fact proving that these level curves have genus 1, and then $F$ can be defined through its internal product), it can be proved that the action of $F$ on each topological circle is conjugated to a rotation of the circle. Zeeman conjectured that the rotation number associated to each of these rotations, thought as a function of the energy levels of $V$ is a monotonic function. And left open the possibility of proving this by considering a hamiltonian vector field such that $F$ was its time-1 map. This vector field was found and the proof of Zeeman′s conjecture was given by Beukers and Cushman (J. Differential Equations. 143 (1998)). Taking this example as a motivation we are interested in the following problems.
(1) When a discrete dynamical system given by a map $F$ can be thought as an stroboscopic map of one flow? and
(2) how we can obtain the associated vector field?
(3) If a map can be thought as an stroboscopic map of a flow, how is its the dynamics? In fact if a map exhibits an analogous property as the one that posses the two dimensional Lyness map, then the rotation numbers can be studied analytically thought the properties of the associated vector field. Thus obtaining a possible alternative to some other techniques which only can be applied to maps such with invariant curves of genus 1. And even, opens possibilities for the numerical explorations only using standard numerical methods of integration of differential equations. We will focus our applications in the study of the third dimensional Lyness Map $F(x,y,z)=(y,z,(a+y+z)/x)$.

This is a joint work together with Anna Cima and Armengol Gasull (UAB)