Equivalent conditions for hyperbolicity on partially hyperbolic holomorphic map

TL;DR

This paper establishes equivalent conditions for hyperbolicity in partially hyperbolic holomorphic maps and characterizes hyperbolicity in Julia sets of generalized Hénon maps with dominated splitting.

Contribution

It provides new criteria for hyperbolicity in partially hyperbolic holomorphic maps and applies these to characterize hyperbolicity in Julia sets of generalized Hénon maps.

Findings

Equivalent conditions for hyperbolicity on invariant sets.

Characterization of hyperbolicity in Julia sets of generalized Hénon maps.

Insights into the structure of partially hyperbolic holomorphic dynamics.

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 $J$.