Asymptotic vectors of entire curves

Abstract

We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate a relationship between the asymptotic vectors and the Picard exceptional vectors. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called an asymptotic vector for the entire curve $\vec{G}(z)=(g_1(z),g_2(z),\ldots,g_p(z))$ if there exists a continuous curve $L: \mathbb{R}_+\to \mathbb{C}$ given by an equation $z=z\left(t\right)$, $0\le t<\infty $, $\left|z\left(t\right)\right|<\infty $, $z\left(t\right)\to \infty $ as $t\to \infty $ such that$$\lim\limits_{\stackrel{z\to\infty}{z\in L}} \frac{\vec{G}(z)\vec{a} }{\big\|\vec{G}(z)\big\|}=\lim\limits_{t\to\infty} \frac{\vec{G}(z(t))\vec{a} }{\big\|\vec{G}(z(t))\big\|} =0,$$ where $\big\|\vec{G}(z)\big\|=\big(|g_1(z)|^2+\ldots +|g_p(z)|^2\big)^{1/2}$, $\vec{G}(z)\vec{a}=g_1(z)\cdot\bar{a}_1+g_2(z)\cdot\bar{a}_2+\ldots+g_p(z)\cdot\bar{a}_p$. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called a Picard exceptional vector of an entire curve $\vec{G}(z)$ if the function $\vec{G}(z)\vec{a}$ has a finite number of zeros in $\left\{\left|z\right|<\infty \right\}$. We prove that any Picard exceptional vector of transcendental entire curve with linearly independent com\-po\-nents and without common zeros is an asymptotic vector.Here we de\-mon\-stra\-te that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which $\infty $ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph``Value Distribution of Meromorphic Functions''.Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve.At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve.