2020/08/05 by Fletcher, Alastair
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2008.02201
In this paper we study, for the first time, Julia limiting directions of quasiregular mappings in ℝn 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 ℝ3 of determining which compact subsets of S2 can give rise to Julia limiting directions. The methods here will require showing that certain sectorial domains in ℝ3 are ambient quasiballs, which is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball \mathbbB3 under an ambient quasiconformal map of ℝ3 to itself.