  We study the dynamics of a bimeromorphic selfmap of a compact complex
K\"ahler surface $X$. 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.
