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

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.