RUI: Galois Action and Entropy in Non-archimedean Dynamics

这个项目涉及非阿基米德动力学中的一些公开问题，非阿基米德动力学是数论和传统(阿基米德)动力学系统之间的一个领域。这个项目结合了动力系统和数论中截然不同的领域。丢番图问题是关于多项式方程的集合有理数解的问题，从古至今一直是数论中的一个主要主题。另一方面，动力系统的研究是最近兴起的，不仅展示了纯粹的数学之美，而且还展示了壮观的计算机分形图和相关集合。该项目借鉴、构建并结合了这两个领域。此外，与之前三个成功的项目一样，研究人员计划指导REU暑期研究项目中的一些学生，以帮助他们进行数学训练。REU产生的任何计算数据都将发布或发布在网络上，以造福于更大的研究社区。自然，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目前正在编写一本研究生水平的关于一个非阿基米德变量的动力学教科书，因为该领域今天的说明性文本太少了。一类关键的问题涉及非阿基米德动力系统上的伽罗瓦作用，与理解有理数的绝对伽罗瓦群的中心数论问题有关。一方面，非阿基米德动力学提供了算术动力学所需的局部信息，而算术动力学又将其本身实现为一类特殊的丢番图问题。第二类问题涉及这种动力系统的遍历性质，特别是熵；熵是一个衡量系统中混沌和不可预测量的数字。这两个主题通过对Berkovich空间中Julia集的研究联系在一起，这些集是在过去十年中被证明在非阿基米德动力学研究中具有核心重要性的技术对象。该项目还利用了复杂动力学、遍历理论和非阿基米德分析的工具。要研究的问题是新的领域，但却是具有悠久历史的丰富理论的延续。特别地，Galois作用量问题有望为攻击绝对Galois群的研究提供新的(动力学)工具，而相关的熵问题的研究有望为动力系统遍历理论的研究提供新的例子。

This project concerns a number of open problems in non-archimedean dynamics, a field on the interface between number theory and traditional (archimedean) dynamical systems. This project joins together the very different realms of dynamical systems and of number theory. Diophantine problems, which ask about the set rational number solutions to polynomial equations, have been a major theme in number theory from ancient times to the present day. On the other hand, the study of dynamical systems has arisen far more recently, exhibiting not only a purely mathematical beauty but also spectacular computer drawings of fractals and related sets. This project draws on, builds on, and joins together both fields. In addition, as in three earlier successful projects, the investigator plans to supervise some students in an REU summer research project to aid in their mathematical training. Any computational data produced in the REU 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.One key class of problems concerns Galois actions on non-archimedean dynamical systems, related to the central number theory problem of understanding the absolute Galois group of the rational numbers. On the one hand, non-archimedean dynamics provides the local information needed in arithmetic dynamics, which in turn realizes itself as a particular kind of Diophantine problem. A second class of problems concerns the ergodic properties, especially the entropy, of such dynamical systems; the entropy is a number that measures the amount of chaos and unpredictability in the system. These two topics are tied together by the study of Julia sets in Berkovich spaces, which are technical objects that, in the past decade, have proven to be of central importance in the study of non-archimedean dynamics. The project also draws on tools from complex dynamics, ergodic theory, and non-archimedean analysis. The problems to be studied branch into new areas but are continuations of rich theories with long and storied histories. In particular, the Galois action problem promises to provide new (dynamical) tools for attacking the study of absolute Galois groups, while the study of the associated entropy issues promise to provide new examples in the study of the ergodic theory of dynamical systems.