抛物自映射的同时线性化与中心化子II：正双曲步长
Simultaneous linearization and centralizers of parabolic self-maps II: positive hyperbolic step
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中文总结 AI 辅助
研究单位圆盘正双曲步长抛物自映射的交换性、同时线性化和全纯模型关系，得到交换对同时线性化的存在唯一性结果，还表明特定族可与\(\varphi\)同时线性化当且仅当族中元素两两交换，扩展了Cowen相关结果。
中文摘要 AI 辅助
单位圆盘的全纯自映射在复合运算下的交换性研究可追溯到1964年、1970年、1973年和1984年的相关文献。在许多情况下，全纯自映射\(\varphi:\mathbb D \to \mathbb D\)的中心化子，即半群\(\mathcal Z_\forall(\varphi):=\{\psi\in\mathsf{Hol}(\mathbb D):\psi\circ\varphi=\varphi\circ\psi\}\)是可交换的。然而，对于正双曲步长的抛物自映射\(\varphi\)情况并非如此，本文对此进行了详细分析。我们研究了交换性、同时线性化和全纯模型之间的关系。特别地，我们得到了交换对\(\varphi\)，\(\psi\in \mathcal Z_\forall(\varphi)\)同时线性化的存在性和唯一性结果。此外，将此概念扩展到任意全纯自映射族，我们表明当且仅当\(\Delta\)中的任意两个元素相互交换时，中心化子\({\Delta\subset\mathcal Z_\forall(\varphi)}\)中的给定（有限或无限）族可以与\(\varphi\)同时线性化。这给出了Cowen关于\(\mathsf{Hol}(\mathbb D)\)中交换对结果的深远扩展。
英文摘要
The study of holomorphic self-mappings of the unit disc commuting under the composition goes back to A.L. Shields (1964), W.A. Pranger (1970), D.F. Behan (1973), and C.C. Cowen (1984). In many situations, the centralizer of a holomorphic self-map ${φ:\mathbb D \to \mathbb D}$, i.e. the semigroup $\mathcal Z_\forall(φ):=\{ψ\in\mathsf{Hol}(\mathbb D):ψ\circφ=φ\circψ\}$ turns out to be commutative. However, this does not hold for the case of a parabolic self-map $φ$ of positive hyperbolic step, which is analyzed in detail in this paper. We investigate the relationships among commutativity, simultaneous linearization, and holomorphic models. In particular, we obtain existence and uniqueness results for the simultaneous linearization of commuting pairs $φ$, $ψ\in \mathcal Z_\forall(φ)$. Furthermore, extending this notion to arbitrary families of holomorphic self-mappings, we show that a given (finite or infinite) family in the centralizer ${Δ\subset\mathcal Z_\forall(φ)}$ can be simultaneously linearized together with$~φ$ if and only if any two elements of$~Δ$ commute with each other. This gives a far reaching extension of Cowen's result concerning commuting pairs in$~\mathsf{Hol}(\mathbb D)$.
