Automorphismes d'entropie positive, le cas des surfaces rationnelles

TL;DR

This paper explores automorphisms with positive entropy on rational surfaces, reviewing key results and recent advances in the classification and dynamics of such complex algebraic surfaces.

Contribution

It provides an overview of recent developments and specific results concerning automorphisms of rational surfaces with positive entropy, building on prior foundational work.

Findings

Classification of surfaces with positive entropy automorphisms

Recent results by Bedford, Kim, McMullen, and Grivaux

Connections between birational maps and dynamical properties

Abstract

A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be a torus, a K3 surface, an Enriques surface or a non-minimal rational surface. We deal with results obtained in this last case. After some recalls on birational maps, algebraic geometry and dynamic, we will speak about Bedford and Kim's works, but also about McMullen's work and more recently about Grivaux and the author's work. These notes have been written for four talks given at IMPA in february/march 2010.