Lie Symmetries of Birational Maps Preserving Genus 0 Fibrations

TL;DR

This paper demonstrates that planar birational integrable maps preserving genus 0 fibrations possess Lie symmetries and invariant measures, enabling systematic analysis of their global dynamics and explicit computation of rotation numbers and periods.

Contribution

It establishes the existence of Lie symmetries and invariant measures for such maps, providing a new framework for studying their global dynamics.

Findings

Existence of Lie symmetries for these maps

Explicit expressions for rotation numbers in examples

Determination of the set of periods for specific maps

Abstract

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.