We prove that any planar birational integrable map, which preserves
a fibration given by genus $0$ curves has a Lie symmetry and some
associated invariant measures. The obtained results allow to study
in a systematic way the global dynamics of these maps. Using this
approach, the dynamics of several maps is described. In particular
we are able to give, for particular examples, the explicit
expression of the rotation number function, and the set of periods
of the considered maps.
Llorens, M.; Mañosa, V. "Lie symmetries of birational maps preserving genus 0 fibrations". 2015.