A Survey of Non-Archimedean Dynamics
A Survey of Non-Archimedean Dynamics
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非阿基米德动力学综述
DOI：
10.1090/noti2472
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发表时间：
2022
影响因子：
--
通讯作者：
Benedetto, Robert L
中科院分区：
文献类型：
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作者：
Benedetto, Robert L
在复动力学中，人们认为有理函数{∞}到其自身的映射。𝑧𝑓然后研究作用在球面上的迭代次数，由下式给出：𝑓𝑓𝑓𝑛𝑓一个丰富的理论如下-见，例如，在[CG 93，Mil 06]的论述-与著名的分形图片，展示美丽的定理。在过去的几十年里，一个年轻的，平行的理论已经发展的情况下，我们取代了阿基米德与adic，更一般地说，非阿基米德领域。三个主要动机推动了非阿基米德理论：寻找动力学现象与复杂理论进行比较和对比，如[HY 83，RL 03];将局部场结果应用于数论问题，如[BR 10，Sil 07];分析复杂动力系统族，特别是在退化点，如[DMF 14，Kiw 15]。
In complex dynamics, one considers a rational function 𝑓∈ ℂ (𝑧) as a map from the Riemann sphere ℙ1 (ℂ)∶= ℂ∪{∞} to itself. One then studies the iterates of 𝑓, given by 𝑓𝑛∶= 𝑓∘⋯∘ 𝑓⏟⎵⏟⎵⏟ 𝑛 times as they act on the sphere. A rich theory follows—see, for example, the expositions in [CG93, Mil06]—with famous fractal pictures that exemplify beautiful theorems. In the past few decades, a younger, parallel theory has developed for the case that we replace ℂ with 𝑝-adic and, more generally, non-archimedean fields. Three principal motivations have driven the non-archimedean theory: seeking dynamical phenomena for comparison and contrast with the complex theory, as in [HY83, RL03]; applying local field results to number-theoretic questions, as in [BR10, Sil07]; and analyzing families of complex dynamical systems, especially at degeneration points, as in [DMF14, Kiw15].