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We consider the structure of substantially dissipative complex H\'enon mapsadmitting a dominated splitting on the Julia set. The dominated splittingassumption corresponds to the one-dimensional assumption that there are nocritical points on the Julia set. Indeed, we prove the correspondingdescription of the Fatou set, namely that it consists of only finitely manycomponents, each either attracting or parabolic periodic. In particular thereare no rotation domains, and no wandering components. Moreover, we show that $J= J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycles.

Structure of partially hyperbolic H\'enon maps