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 Lattes 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 Poincare 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.

中文翻译：

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强自守映射和 Julia 一致拟正则映射集

Ritt 定理指出，只有当全纯映射与 $z$ 的幂、切比雪夫多项式或 Lattes 映射共轭时，在排斥不动点的全纯函数的线性化器才是周期性的。反过来，除了一些例外，也是如此。在本文中，我们证明了 $\mathbb{R}^n$ 中强自守拟正则映射和一致拟正则映射的设置中的类似陈述。在此过程中，我们表征了可能通过晶体学轨道和使用庞加莱猜想产生的自同构群。我们进一步给出了当 Julia 集是一个准球面、quasidisk 或 $\mathbb{R}^n$ 并且 Julia 集与圆锥点集重合时一致拟正则映射在其 Julia 集上的行为的分类。最后，