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

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

1992-05-28

arxiv.org/abs/math/9205210v1

Mathematics

Dynamical Systems

Scientific paper

10.1007/BF01232426

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.

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