AI 中文总结
本文在全局函数域上证明了Andrews--Petsche猜想的一个类比：当多项式次数小于域特征时，阿贝尔动力伽罗瓦群恰为等平凡的，并给出了完整分类猜想。
AI 中文摘要
我们建立了Andrews--Petsche最近一个猜想的函数域类比。我们的主要结果刻画了阿贝尔动力伽罗瓦群恰好是等平凡的那些，只要多项式的次数小于域的特征$p$。这一特征的证明通过以下四个独立步骤完成：$$\text{阿贝尔} \implies \text{有限分歧} \implies \text{PCF映射} \implies \text{等平凡映射} \implies \text{等平凡对},$$并且更一般地适用于具有超吸引不动点的映射。我们观察到这一蕴含链在次数上是尖锐的：一旦达到$p$，就会出现来自Drinfeld模的新的非等平凡例子。我们提出了所有次数下的完整猜想性分类，考虑了来自Drinfeld模及其相关Lattès映射的所有新的奇特例子。
英文摘要
We establish a function field analogue of a recent conjecture of Andrews--Petsche. Our main result characterizes abelian dynamical Galois groups to be precisely the isotrivial ones, whenever the degree of the polynomial is smaller than $p$, the characteristic of the field. The proof of this characterization is achieved in four independent steps as follows: $$\text{Abelian} \implies \text{Finite ramification} \implies \text{PCF map} \implies \text{Isotrivial map} \implies \text{Isotrivial pair}, $$ and works more generally for maps with a superattracting fixed point. We observe that this chain of implications is sharp in the degree: as soon as one reaches $p$ there are new non-isotrivial examples coming from Drinfeld modules. We propose a full conjectural classification in all degrees, taking into account all of the new exotic examples coming from Drinfeld modules and their associated Lattès maps.
