Quelques propri\'et\'es des transformations birationnelles du plan projectif complexe

TL;DR

This paper explores algebraic and dynamical properties of the Cremona group, including generators, relations, subgroups, automorphisms, and dynamics, with a focus on automorphisms with entropy on rational surfaces.

Contribution

It compiles known properties of the Cremona group and discusses the construction of entropic automorphisms on rational surfaces, linking algebraic and dynamical aspects.

Findings

Properties of the Cremona group are summarized, including generators and relations.

Connections with polynomial automorphisms of the affine plane are discussed.

Construction methods for entropic automorphisms on rational surfaces are presented.

Abstract

We present some (unfortunately not all) known properties on the Cremona group; when it's possible we mentioned links with the most known group of polynomial automorphisms of the affine plane. The mentioned properties are essentially algebraic properties: generators, relations, finite subgroups, subgroups of finite type, automorphisms of the Cremona group, Tits alternative... but also dynamical properties: classification of birational maps, centralizer, dynamic of an Heisenberg subgroup... We deal with the construction of entropic automorphisms on rational surfaces.