Structure of partially hyperbolic Hénon maps

Abstract

We describe the structure of “substantially dissipative” complex Hénon maps admitting a dominated splitting on the Julia set. We prove that the Fatou set consists of only finitely many components, each either attracting or parabolic periodic. In particular, there are no wandering components and no rotation domains. Moreover, we show that $J=J^{\star }$ and the dynamics on $J$ is hyperbolic away from parabolic cycles.