Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings

A theorem of Ritt states the Poincaré linearizer L of a rational map f at a repelling fixed point is periodic only if f is conjugate to a power of z, a Chebyshev polynomial or a Lattes map. The converse, except for the case where the fixed pointis an endpoint of the interval Julia set for a Chebyshev polynomial, is also true. In this paper, we prove the analogous statement in the setting of strongly automorphic quasiregular mappings and uniformly quasiregular mappings in ℝn. Along the way, we characterize the possible automorphy groups that can arise via crystallographic orbifolds and a use of the Poincaré conjecture. We further give a classification of the behaviour of uniformly quasiregular mappings on their Julia set when the Julia set is a quasisphere, quasidisk or all of ℝn and the Julia set coincides with the set of conical points. Finally, we prove an analogue of the Denjoy-Wolff Theorem for uniformly quasiregular mappings in B3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{B}^3}$$\end{document}, the first such generalization of the Denjoy—Wolff Theorem where there is no guarantee of non-expansiveness with respect to a metric.