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

该项目涉及非阿基米德动力学中的一些开放问题，这是一个连接数论和传统（阿基米德）动力学系统之间接口的领域。在过去的十年中，人们已经清楚地认识到，非阿基米德分析中的一类技术对象Berkovich空间对于非阿基米德理论是必不可少的。特别地，构造动力系统下的概率测度不变量需要使用Berkovich空间。同样在过去的十年中，我们对Berkovich空间上的函数分支的理解有了增长，并且随着它的增长，我们将不变测度的存在性应用于非阿基米德动力学中的某些问题的能力也有了增长。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中起到了很大的作用。利用所有这些新工具，PI计划研究几个尚未解决的问题，这些问题已经抵御了以前不那么复杂的攻击。该建议借鉴了复杂动力学和非阿基米德分析的工具，所研究的问题可以应用于全局领域的算术动力学理论，从而应用于丢芬图几何的某些问题。这个项目将动力系统和数论的不同领域结合在一起。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，即理解一组自然产生的多项式方程的有理数解的集合，一直是数论的一个主要主题，从古希腊到费马一直到今天。另一方面，动力系统的研究，特别是复杂动力学的研究，最近才兴起，不仅展示了纯粹的数学之美，而且还展示了分形和相关集合的壮观的计算机绘图。因此，该项目提出的非阿基米德动力学研究借鉴、建立并结合了古代和现代这两个领域。此外，与之前两个成功的项目一样，PI计划在REU的夏季研究项目中指导一些学生，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，以产生有趣的例子；如果是这样，任何产生的相关数据都将被发布或张贴在网络上，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，PI目前正在编写一本关于非阿基米德变量动力学的研究生水平教科书，因为该领域目前的说明性文本太少了。

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.