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Northern Illinois University（北伊利诺伊大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明单位球上一致拟正则自映射的Denjoy-Wolff定理，利用Schwarz-Pick引理和链度量，并应用于稳定Fatou分量的分类，构造了非单射例子。
AI 中文摘要
我们证明了$\u005cmathbb{R}^n$（$n\geq 2$）中单位球上一致拟正则自映射的Denjoy-Wolff定理：要么所有轨道相对紧致，要么迭代在局部一致意义下收敛到球面上的单一点。我们不需要假设映射是正常、满射或具有边界延拓。关键工具是一个适用于所有维数的Schwarz-Pick引理：每个保持有界可测共形结构的拟正则映射对于由该结构构造的模度量都是非扩张的，而一致拟正则映射通过Iwaniec和Martin的定理保持这样的结构。由于该度量不是测地线度量，我们转而考虑一个相关的链度量，我们证明该度量是Gromov双曲的，其边界为球面，并应用Karlsson的一个定理。作为应用，$\overline{\mathbb{R}^n}$上一致拟正则映射的一个稳定Fatou分量，若它是具有一致完美边界的均匀域，则要么是吸引域，要么是旋转域，要么是具有唯一边界不动点的抛物型域。在每一维数中，我们构造了两种逃逸类型的非单射例子。
英文摘要
We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $\mathbb{R}^n$, $n\geq 2$: either all orbits are relatively compact, or the iterates converge locally uniformly to a single point of the sphere. No properness, surjectivity, or boundary extension is assumed. The key tool is a Schwarz-Pick lemma valid in all dimensions: every quasiregular map preserving a bounded measurable conformal structure is non-expanding for a modulus metric built from that structure, and uniformly quasiregular maps preserve such a structure by a theorem of Iwaniec and Martin. Since this metric is not geodesic, we pass to an associated chain metric, which we show is Gromov hyperbolic with boundary the sphere, and apply a theorem of Karlsson. As an application, a stable Fatou component of a uniformly quasiregular map of $\overline{\mathbb{R}^n}$ that is a uniform domain with uniformly perfect boundary is an attracting basin, a rotation domain, or a parabolic basin with a unique boundary fixed point. Non-injective examples of both escaping types are constructed in every dimension.
