Mathematical Sciences: Ergodic Theory of Unipotent Translations on Homogeneous Spaces

数学科学：齐次空间上的单能平移遍历理论

• 批准号: 8701840
• 负责人: Marina Ratner
• 金额: $ 11万
• 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: 1987-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8701840&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Ergodic; Theory; Unipotent

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. If for example, the transformation results from a differential equation, is time dependent, or more generally evolves according to a more extensive process than time, one speaks of a flow on the underlying geometry. This is the context of the modern theory of dynamical systems, in which a fundamentalcritical question is the classification of flows on homogenous spaces. Professor Ratner is one of the world leaders in the study of certain geometric flows and their dynamics. Her work on "rigidity" -- the classification of flows in terms of minimal data -- of horocycle flows is one of the top recent achievements in ergodic theory and dynamical systems. It comprises the most significant work to date in the classification of flows on homogeneous manifolds. Her work has produced examples and counterexamples, and it has brought an important geometrical point of view to the subject. In the current proposal she plans to study more general, so-called unipotent, flows. She hopes to classify all Borel probability measures preserved by unipotent tranformations on finite volume homogeneous spaces; to investigate the rigidity properties of such transformations; and to establish exponential mixing for ergodic flows on unit tangent bundles of compact surfaces of variable negative curvature.

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。 被代理人不同意 遍历理论的研究是一个基本的变换 设置或空间。 例如，如果转换是由 微分方程，是时间相关的，或者更一般地说， 根据一个比时间更广泛的过程演变，一个 谈到了底层几何的流动。 这就是背景 现代动力系统理论的一部分， 基本的关键问题是流的分类， 同质空间 拉特纳教授是世界上研究 某些几何流及其动力学。 她的工作 “刚性”-根据最小 最近的最大成就之一是 在遍历理论和动力系统中。 它包含了世界上 迄今为止在对流动进行分类方面所做的重要工作 齐次流形 她的工作产生了一些例子， 反例，它带来了重要的几何 对主题的观点。 在目前的提案中，她计划 来研究更一般的，所谓的幂幺流 她希望 分类所有保幂幺的Borel概率测度 有限体积齐性空间上的变换;到 研究此类变换的刚性特性;以及 建立单位切线上遍历流的指数混合 具有可变负曲率的紧致曲面的丛。