Laminar currents and birational dynamics

Abstract

We study the dynamics of a bimeromorphic map $X\to X$, where $X$ is a compact complex Kähler surface. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic, and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of ${\mathbb{C}}^2$. This extends recent results by E. Bedford and J. Diller