Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings

Abstract

A theorem of Ritt states the a linearizer of a holomorphic function at a repelling fixed point is periodic only if the holomorphic map is conjugate to a power of $z$, a Chebyshev polynomial or a Lattès map. The converse, except for some exceptions, is also true. In this paper, we prove the analogous statement in the setting of strongly automorphic quasiregular mappings and uniformly quasiregular mappings in $\mathbb{R}^n$. 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 $\mathbb{R}^n$ 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 $\mathbb{B}^3$, the first such generalization of the Denjoy-Wolff Theorem where there is no guarantee of non-expansiveness with respect to a metric.