Phragmén-Lindelöf Principles and Julia Limiting Directions of Quasiregular Mappings

Abstract

We show that the set of Julia limiting directions of a transcendental-type $K$-quasiregular mapping $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ must contain a component of a certain size, depending on the dimension $n$, the maximal dilatation $K$, and the order of growth of $f$. In particular, we show that if the order of growth is small enough, then every direction is a Julia limiting direction. We also show that if every component of the set of Julia limiting directions is a point, then $f$ has infinite order. The main tool in proving these results is a new version of a Phragmén-Lindelöf principle for sub-$F$-extremals in sectors, where we allow for boundary growth of the form $O(\log |x|)$ instead of the previously considered $O(1)$ bound.