Preprint C99/2010
Dominated Splitting and Critical Sets for Polynomials Automorphisms on $\bold C^2$
DDIC
Keywords: dominated splitting | pseudo hyperbolicty | generalized Hénon maps | critical points | linear and projective cocyles
Generalized H�non maps are the polynomial automorphisms of $\bold C^2$ that represent the first step for a global understanding of holomorphic dynamics in higher dimension. As in the one dimensional context (polynomials on $\bold C$), it can be defined the Julia set, which is the set that concentrate the interesting dynamical behavior of these type of systems. One of the goals of the present work, is to describe the dynamics of this maps under the hypotheses of dominated splitting in the Julia set, and to find sufficient conditions (forward expansivity in the center-unstable leaves and some types of uniformly hyperbolicity in the set of all periodic point), to guarantee hyperbolicity in the Julia set. The second main goal is to understand the dynamical obstruction for domination. This allow us to introduce a notion of critical point (and critical set) for polynomial automorphisms that capture many of the dynamical properties of their one-dimensional counterpart.