Let f be a meromorphic function in the complex plane. A value θ ∈ [0, 2 π ) is called a Julia limiting direction of f if there is an unbounded sequence { z n } in the Julia set J ( f ) satisfying lim n→∞ arg z n = θ (mod 2 π ). We denote by L ( f ) the set of all Julia limiting directions of f . Our main result is that, for any non-empty compact set E ⊆ [0, 2 π ) and ρ ∈ [0, ∞], there are an entire function f of infinite lower order and a transcendental meromorphic function g of order ρ such that L ( f ) = L ( g ) = E. In addition, we have also constructed some transcendental entire functions whose lower order is ρ ∈ (1/2, ∞) and whose L ( f ) coincides with a certain kind of compact set. To prove our results, we have established a criterion for a direction θ to be a Julia limiting direction of a function by utilizing the growth rate of the function in the direction θ. The criterion may be of independent interest.

中文翻译：

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关于亚纯函数的 Julia 极限方向

设 f 是复平面中的亚纯函数。如果 Julia 集合 J ( f ) 中存在无界序列 { zn } 满足 lim n→∞ arg zn = θ (mod 2 π )，则值 θ ∈ [0, 2 π ) 被称为 f 的 Julia 极限方向。我们用 L ( f ) 表示 f 的所有 Julia 极限方向的集合。我们的主要结果是，对于任何非空紧致集合 E ⊆ [0, 2 π ) 和 ρ ∈ [0, ∞]，都有一个无穷低阶的完整函数 f 和阶 ρ 的超越亚纯函数 g，例如L ( f ) = L ( g ) = E. 此外，我们还构造了一些超越整函数，其低阶为 ρ ∈ (1/2, ∞) 且其 L ( f ) 符合某种紧致放。为了证明我们的结果，我们通过利用函数在 θ 方向的增长率，建立了一个标准，使 θ 方向成为函数的 Julia 极限方向。该标准可能具有独立的利益。