Ergodic Theory, Differential Dynamics and Combinatorial Number Theory

遍历理论、微分动力学和组合数论

• 批准号: 0100581
• 负责人: Donald Ornstein
• 金额: $ 29.42万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100581&HistoricalAwards=false
• 关键词: Ergodic; Theory; Differential; Dynamics; Combinatorial

This proposal contains a number of different problems whose common thread is a core approach combining dynamics, probability theory, and a basic set of techniques that were mainly developed in the classical ergodic theoretic setting. These techniques have been very successfully adapted to problems from hard analysis and combinatorics, in areas such as: KAM theory; Ramsey theory (in particular density versions of results previously known for partitions, e.g., the density version of the Hales-Jewett theorem); Fractal geometry and Haussdorff dimension of sets in Euclidean space; the theory of amenable groups and their actions. These are the main areas that we propose to develop further.Physical systems satisfying the same set of laws exhibit a wide variety of behaviors going from the extreme of completely chaotic systems which appear to be completely random to systems that exhibit a surprising level of stability (for instance, a particle moving in a cyclotron). Mathematical dynamics and ergodic theory explain these different types of behaviors. The same methods turn out to be extremely useful in other areas, in particular in combinatorics, information theory, data compression, etc. Our work is developing new methods that give better explanations of these phenomena and extend the applicability of the field to a broader class of problems.

这个建议包含了一些不同的问题，其共同的线程是一个核心的方法相结合的动力学，概率论，和一套基本的技术，主要是在经典遍历理论的设置。 这些技术已经非常成功地适用于来自硬分析和组合学的问题，例如：KAM理论; Ramsey理论（特别是以前已知的分区结果的密度版本，例如，密度版本的黑尔斯-朱厄特定理）;分形几何和Haussdorff维数集在欧几里德空间;理论的顺从群体和他们的行动。 这些是我们建议进一步发展的主要领域。满足同一组定律的物理系统表现出各种各样的行为，从完全混沌的极端系统（看起来完全随机）到表现出惊人稳定性的系统（例如，粒子在回旋加速器中运动）。 数学动力学和遍历理论解释了这些不同类型的行为。 同样的方法在其他领域也非常有用，特别是在组合学、信息论、数据压缩等领域。我们的工作是开发新的方法，更好地解释这些现象，并将该领域的适用性扩展到更广泛的问题。