Periodic orbits of planar integrable birational maps, part II: the genus 0 case and other approaches [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Víctor Mañosa
Universitat Politècnica de Catalunya


We will consider birational planar maps possessing a rational first integral so that they preserve a foliation of the plane given by algebraic curves which have generically genus 0. We present a systematic way to study this case. It is shown that on each invariant curve any birational map is conjugate to a Moebius transformation, and that any of these maps possesses a Lie symmetry with an associated invariant measure (which is vector field such that the map sends any orbit of the differential system determined by the vector field, to another orbit of this system).

The results allow to give a global analysis of the dynamics of the maps under consideration, and in particular, to obtain the explicit expression of the rotation number function associated to the maps defined on an open set foliated by closed invariant curves, which facilitates the full characterization of the set of periods of the maps as well as the explicit identification of the level sets where this periodic orbits are located.

If there is enough time we will se how to deal with periodic points of planar integrable birational maps without taking care of the genus of the preserved foliation.

We will focus on the results obtained with M. Llorens, but also we will also refer some results obtained with A. Cima and A. Gasull.