Generalized H\'enon mappings and foliation by injective Brody curves

TL;DR

This paper proves that leaves of certain level sets of Green functions for generalized Hénon mappings are injective Brody curves in projective space, revealing new geometric properties of these complex dynamical systems.

Contribution

It establishes that each leaf in the foliation by level sets of the Green function is an injective Brody curve, a novel geometric insight into the structure of these dynamical systems.

Findings

Leaves are injective Brody curves in ^2

Level sets are foliated by biholomorphic images of 

Behavior of level sets near infinity analyzed

Abstract

We consider a finite composition of generalized H\'{e}non mappings $\mathfrak{f}:\mathbb{C}^2\to\mathbb{C}^2$ and its Green function $\mathfrak{g}^+:\mathbb{C}^2\to\mathbb{R}_{\ge 0}$ (see Section 2). It is well known that each level set $\{\mathfrak{g}^+=c\}$ for $c>0$ is foliated by biholomorphic images of $\mathbb{C}$ and each leaf is dense. In this paper, we prove that each leaf is actually an injective Brody curve in $\mathbb{P}^2$ (see Section 4). Namely, for any injective holomorphic parametrization of any leaf, its derivative is bounded over $\mathbb{C}$ with respect to the Fubini-Study metric of $\mathbb{P}^2$. We also study the behavior of the level sets of $\mathfrak{g}^+$ near infinity.