Equivalent conditions for hyperbolicity on partially hyperbolic holomorphic map

Abstract

Let $f:\mathbb{C}\sp n\to\mathbb{C}\sp n$, $n\geq2$, be a biholomorphism and let $\Lambda\subseteq \mathbb{C}\sp n$ be a compact $f$-invariant set such that $f|\Lambda$ is partially hyperbolic. We give equivalent conditions to hyperbolicity on $\Lambda$. In the particular case of generalized H\'enon map with dominated splitting in the Julia set $J$, we characterize the hyperbolicity of $J$.