We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ${\mathbb Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso-Zannier and Amoroso-Dvornicich, as well as properties of the Arakelov-Zhang pairing.

中文翻译：

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动力学伽罗瓦理论中的阿贝尔扩展

我们提出了当与多项式相关联的动态伽罗瓦群是阿贝尔群时的推测表征，并且我们在几种情况下证明了我们的猜想，包括 ${\mathbb Q}$ 上的稳定二次情况。在后临界无限的情况下，证明使用代数技术，包括关于循环 $p$-扩展塔中的分枝的结果。在后临界有限情况下，证明使用高度理论，结合 Amoroso-Zannier 和 Amoroso-Dvornicich 的结果，以及 Arakelov-Zhang 配对的性质。