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.