Centralisateurs dans le groupe de Jonqui\`eres

TL;DR

This paper provides a criterion for linear degree growth of certain birational maps of the complex projective plane fixing a rational fibration and computes their centralizers, addressing classical problems in difference equations.

Contribution

It introduces a new criterion for analyzing degree growth and explicitly computes the centralizer of birational maps preserving rational fibrations.

Findings

Criterion for linear degree growth of birational maps

Explicit description of centralizers of such maps

Connection to classical difference equations problems

Abstract

We give a criterion to determine when the degree growth of a birational map of the complex projective plane which fixes (the action on the basis of the fibration is trivial) a rational fibration is linear up to conjugacy. We also compute the centraliser of such maps. It allows us to describe the centraliser of the birational maps of the complex projective plane which preserve a rational fibration (the action on the basis of the fibration being not necessarily trivial); this question is related to some classical problems of difference equations.