RUI: Families, Ramification, and Berkovich Spaces in Non-archimedean Dynamics

这个项目涉及非阿基米德动力学中的一些开放问题，非阿基米德动力学是一个连接数论和传统(阿基米德)动力学系统之间的桥梁。在过去的十年里，作为非阿基米德分析中的一类技术对象的Berkovich空间，对于非阿基米德理论来说是必不可少的。特别地，在动力系统下构造概率度量不变量需要使用Berkovich空间。也是在过去的十年里，我们对Berkovich空间上的函数分支的理解有所增长，并随之增长了我们将不变测度的存在性应用于非阿基米德动力学中某些问题的能力。此外，单参数族的使用在构造非阿基米德动力系统的病理例子方面也取得了很大的效果。利用所有这些新可用的工具，PI计划研究几个悬而未决的问题，这些问题已经抵抗了以前不那么复杂的攻击。该方案利用了复动力学和非阿基米德分析的工具，所要研究的问题应用于全局域上的算术动力学理论，从而应用于丢番图几何中的某些问题。该项目结合了动力系统和数论的非常不同的领域。一方面，非阿基米德动力学是算术动力学的一个子领域，它涉及一类特殊的丢番图几何问题。这样的问题，即理解自然产生的一组多项式方程的有理数解的集合，从古希腊人费马到今天一直是数论的一个主要主题。另一方面，对动力系统的研究，特别是对复杂动力学的研究，最近兴起得多，不仅表现出纯粹的数学之美，而且还展示了壮观的计算机分形图及其相关集合。因此，该项目提出的对非阿基米德动力学的研究借鉴、建立并结合了古代和现代两个领域。此外，与之前两个成功的项目一样，PI计划指导REU暑期研究项目中的一些学生，以帮助他们进行数学训练。根据学生的兴趣，REU可能会涉及一些密集的计算机计算来生成有趣的例子；如果是这样的话，任何产生的相关数据都将被公布或发布在网络上，以造福于更大的研究社区。当然，任何结果也将通过Arxiv等网站传播，并在数学期刊上发表。此外，由于该领域目前说明性文本太少，PI目前正在编写一本关于动力学的研究生水平的教科书，其中包含一个非阿基米德变量。

This project concerns a number of open questions in non-archimedean dynamics, a field bridging the interface between number theory and traditional (archimedean) dynamical systems. In the past ten years, it has become clear that Berkovich spaces, a certain class of technical objects in non-archimedean analysis, are essential for the non-archimedean theory. In particular, the construction of a probability measure invariant under the dynamical system requires the use of Berkovich spaces. Also in the past decade, our understanding of ramification of functions on Berkovich spaces has grown, and with it has grown our ability to apply the existence of invariant measures to certain problems in non-archimedean dynamics. In addition, the use of one-parameter families has been used to great effect in constructing pathological examples of non-archimedean dynamical systems. Using all these newly available tools in tandem, the PI plans to study several open questions that have resisted previous less sophisticated attacks. The proposal draws on tools from both complex dynamics and non-archimedean analysis, and the problems to be studied have applications to the theory of arithmetic dynamics over global fields and hence to certain problems in Diophantine geometry.This project joins together the very different realms of dynamical systems and of number theory. On the one hand, non-archimedean dynamics is a subfield of arithmetic dynamics, which concerns a particular class of Diophantine geometry problems. Such problems, i.e., understanding the set of rational number solutions to a naturally arising set of polynomial equations, have been a major theme in number theory from the ancient Greeks through Fermat and into the present day. On the other hand, the study of dynamical systems, and especially of complex dynamics, has arisen far more recently, exhibiting not only a purely mathematical beauty but also spectacular computer drawings of fractals and related sets. This project's proposed study of non-archimedean dynamics thus draws on, builds on, and joins together both fields, the ancient and modern alike. In addition, as in two earlier successful projects, the PI plans to supervise some students in an REU summer research project to aid in their mathematical training. Depending on the interest of the students, the REU may involve some intensive computer computations to generate interesting examples; if so, any relevant data generated will be published or posted on the web, for the benefit of the larger research community. Naturally, any results will also be disseminated via websites such as ArXiv and publication in mathematical journals.In addition, the PI is currently writing a graduate-level textbook on dynamics in one non-archimedean variable, as the field has too few expository texts today.