## Institute for Mathematical Sciences

## Preprint ims92-8

** E. Bedford, M. Lyubich, and J. Smillie**
* 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 $\mu^\pm$ 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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