For a quasiregular mapping \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n\), with \(n\ge 2\), a Julia limiting direction \(\theta \in S^{n-1}\) arises from a sequence \((x_n)_{n=1}^{\infty }\) contained in the Julia set of f, with \(|x_n| \rightarrow \infty \) and \(x_n/|x_n| \rightarrow \theta \). Julia limiting directions have been extensively studied for entire and meromorphic functions in the plane. In this paper, we focus on Julia limiting directions in higher dimensions. First, we give conditions under which every direction is a Julia limiting direction. Our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a sufficient, but not necessary, condition in \({\mathbb {R}}^3\) for a set \(E\subset S^2\) to be the set of Julia limiting directions for a quasiregular mapping. The methods here will require showing that certain sectorial domains in \({\mathbb {R}}^3\) are ambient quasiballs. This is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball \({\mathbb {B}}^3\) under an ambient quasiconformal mapping of \({\mathbb {R}}^3\) onto itself.

对于准规则映射\（f：{\ mathbb {R}} ^ n \ rightarrow {\ mathbb {R}} ^ n \），其中\（n \ ge 2 \）为Julia限制方向\（\ theta \ S ^ {n-1} \中的ins源自f的Julia集中包含的序列\（（x_n）_ {n = 1} ^ {\ infty} \），其中\（| x_n | \ rightarrow \ infty \）和\（x_n / | x_n | \ rightarrow \ theta \）。Julia极限方向已针对平面中的整体和亚纯函数进行了广泛研究。在本文中，我们重点关注朱莉娅在更高维度上的限制方向。首先，我们给出条件，其中每个方向都是Julia限制方向。我们的方法表明，如果准脂肪成分包含一个扇形域，则该扇形的增长存在多项式界。其次，我们在\（{\ mathbb {R}} ^ 3 \）中给一个\\ {E \ subset S ^ 2 \ }一个充分但非必要的条件，作为拟正则的Julia限制方向的集合映射。这里的方法将要求显示\（{\ mathbb {R}} ^ 3 \）中的某些扇区域是环境准球。这是一个众所周知的难题，它确定在环境拟形形\（{\ mathbb {R}} ^ 3下，哪个域是单位球\（{{mathbb {B}} ^ 3 \）的图像\）本身。