On the Julia Directions of the Value Distribution of Nondegenerate Transendental Holomorphic Maps of $\mathbf{C}^2$ to $\mathbf{C}^2$

Abstract

We prove that for a nondegenerate holomorphic map $F=(f(x,y),g(x,y))$ of $\mathbf{C}^2$ to $\mathbf{C}^2$ where $f$ and $g$ are entire functions and $f$ is a transendental one, there exists a ray $J(\theta) = \{(x,y); x = te^{i\theta},y = kte^{i\theta} (0 \leqq t < \infty)\}$ where $k$ is an arbitrarily fixed complex number except some Lebesgue measure zero set and $\theta$ is some real number depending on value $k$, such that $F(x,kx)$, in any open cone in $\mathbf{C}^2$ with vertex $(0,0)$ containing the ray $J(\theta)$, does not omit any algebraic curve with three irreducible components in a general position.