Mathematical Sciences: Topics in Ergodic Theory and Dynamical Systems.

数学科学：遍历理论和动力系统主题。

• 批准号: 8701584
• 负责人: Jacob Feldman
• 金额: $ 16.03万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1990-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701584&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Ergodic; Theory

Ergodic theory is an active, central area of Modern Analysis. Its origins lie in the theoretical formulation for classical statistical mechanics at the end of the nineteenth century. The modern point of view derives from the profound mathematical theory of recurrence and ergodicity, as developed by Poincare, Birkhoff, von Neumann and other giants earlier in this century. As the subject has evolved and sophistication continually increased, ergodic theory has acquired close relationships with other branches of mathematics, such as dynamical systems, probability theory, functional analysis, number theory, differential topology, and differential geometry, and with applications as far ranging as mathematical physics, information theory, and computer design. The principal objects of study in ergodic theory are transformations of an underlying set or space, and the ultimate goal is to determine long-term behavior. Professor Feldman is a leader in ergodic theory whose research has been highly influential. His recent work has involved rigidity of flows on negatively curved manifolds, positive entropy for certain flows on Euclidean space, non- measure-preserving transformations and random walks, and amenable group actions and consequences for von Neumann algebras of operators. The current proposal continues this ongoing research. In particular, Professor Feldman will study problems related to horocycle and geodesic flows, Kakutani equivalence, recurrence for nonstationary processes, and ordered pairs of ergodic equivalence relations.

遍历理论是现代数学的一个活跃的中心领域， 分析. 它的起源在于理论公式， 19世纪末的经典统计力学 世纪。现代的观点源于深刻的 递归和遍历的数学理论，由 庞加莱、伯克霍夫、冯·诺依曼和其他早期的巨人 世纪。 随着学科的发展和成熟 不断增加，遍历理论已获得接近 与其他数学分支的关系，如 动力系统，概率论，泛函分析， 数论、微分拓扑学和微分几何， 应用范围远至数学物理学， 信息论和计算机设计。 校长 遍历理论中的研究对象是 基本集合或空间，最终目标是确定 长期行为。 费尔德曼教授是遍历理论的领导者， 研究具有很大的影响力。 他最近的工作 包括负弯曲歧管上的流动刚性， 欧氏空间上某些流的正熵，非 保测度变换和随机游走，以及 冯·诺依曼的顺从群体作用和后果 算子代数 目前的建议继续这一点 正在进行的研究。 特别是费尔德曼教授将研究 角谷，与时周期和测地线流有关的问题 等价性，非平稳过程的递归，以及 遍历等价关系的有序对。