Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents

Abstract

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $μ^\pm$ and the equilibrium measure $μ=μ^+\wedgeμ^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $μ$ as the unique measure of maximal entropy. Then we show that the measure $μ$ has a local product structure and that the currents $μ^\pm$ have a laminar structure. This allows us to deduce information about periodic points and heteroclinic intersections. For example, we prove that the support of $μ$ 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.