  This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$.
One can associate to such an automorphism two currents $\mu^\pm$ and the
equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some
geometric and dynamical properties of these objects. First, we characterize
$\mu$ as the unique measure of maximal entropy. Then we show that the measure
$\mu$ has a local product structure and that the currents $\mu^\pm$ have a
laminar structure. This allows us to deduce information about periodic points
and heteroclinic intersections. For example, we prove that the support of $\mu$
coincides with the closure of the set of saddle points. The methods used
combine the pluripotential theory with the theory of non-uniformly hyperbolic
dynamical systems.
