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A theorem of Ritt states the a linearizer of a holomorphic function at arepelling fixed point is periodic only if the holomorphic map is conjugate to apower of $z$, a Chebyshev polynomial or a Latt\`es map. The converse, exceptfor some exceptions, is also true. In this paper, we prove the analogousstatement in the setting of strongly automorphic quasiregular mappings anduniformly quasiregular mappings in $\mathbb{R}^n$. Along the way, wecharacterize the possible automorphy groups that can arise via crystallographicorbifolds and a use of the Poincar\'e conjecture. We further give aclassification of the behaviour of uniformly quasiregular mappings on theirJulia 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 uniformlyquasiregular mappings in $\mathbb{B}^3$, the first such generalization of theDenjoy-Wolff Theorem where there is no guarantee of non-expansiveness withrespect to a metric.

Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings