On Julia limiting directions in higher dimensions

In this paper we study, for the first time, Julia limiting directions of quasiregular mappings in $\mathbb{R}^n$ of transcendental-type. First, we give conditions under which every direction is a Julia limiting direction. Along the way, 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 contribution to the inverse problem in $\mathbb{R}^3$ of determining which compact subsets of $S^2$ can give rise to Julia limiting directions. The methods here will require showing that certain sectorial domains in $\mathbb{R}^3$ are ambient quasiballs, which 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 map of $\mathbb{R}^3$ to itself.