RUI: Galois Action and Entropy in Non-archimedean Dynamics

RUI：非阿基米德动力学中的伽罗瓦作用和熵

• 批准号: 1501766
• 负责人: Robert Benedetto
• 金额: $ 17.87万
• 依托单位: Amherst College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501766&HistoricalAwards=false
• 关键词: RUI; Galois; Action; Entropy; Non

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.

这个项目涉及非阿基米德动力学中的一些开放问题，这是数论和传统（阿基米德）动力系统之间的接口领域。这个项目将动力系统和数论的不同领域结合在一起。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重要课题。另一方面，动力系统的研究是最近才兴起的，它不仅展示了纯粹的数学之美，而且展示了壮观的分形和相关集合的计算机绘图。这个项目借鉴，建立在，并结合在一起这两个领域。此外，在三个较早的成功的项目，调查员计划监督一些学生在REU夏季研究项目，以帮助他们的数学训练。在REU产生的任何计算数据将被发布或张贴在网络上，为更大的研究社区的利益。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教科书，因为该领域目前的临时教科书太少了。其中一类关键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，这反过来又实现了自己作为一种特殊的丢番图问题。 第二类问题是关于这种动力系统的遍历性，特别是熵;熵是一个衡量系统中混沌和不可预测性的数字。这两个主题是联系在一起的研究朱莉娅集在Berkovich空间，这是技术对象，在过去的十年中，已被证明是至关重要的非阿基米德动力学的研究。该项目还借鉴了复杂动力学，遍历理论和非阿基米德分析的工具。 所要研究的问题是新领域的分支，但又是历史悠久的丰富理论的延续。特别是，伽罗瓦作用问题有望为攻击绝对伽罗瓦群的研究提供新的（动力学）工具，而相关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。