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

Abstract

This paper concerns the dynamics of polynomial automorphisms of ${\mathbf{C}}^2$. 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.

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