Measure theory and geometric topology in dynamics

动力学测量理论和几何拓扑

• 批准号: 1201326
• 负责人: Federico Rodriguez Hertz
• 金额: $ 22.7万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-05-01 至 2016-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201326&HistoricalAwards=false
• 关键词: Measure; theory; geometric; topology; dynamics

One of the goals of dynamical system theory is to understand the orbit structure of systems displaying some particular feature. In this project the principal investigator will focus on dynamics displaying some sort of hyperbolicity. The project will develop in two main directions, dynamical systems with multidimensional time (i.e. actions of higher rank groups) and positive entropy and classical dynamical systems (i.e. one dimensional time) displaying some uniform (though possibly partial) hyperbolicity. The aim is to develop techniques to understand the interactions between measure theoretical and topological properties of a system. This interaction happens to go both ways. For instance, in the direction of topology to measure theory it is observed that on 3-dimensional nilmanifolds partially hyperbolic systems are always K-systems. And in the direction of measure theory to topology it is known that a non-uniformly hyperbolic higher rank action on a 3-dimensional irreducible manifold can only exist on the torus. Boltzmann's study of gas particles produced the nowadays accepted idea that several systems display some chaotic behavior. While the system may be of very diverse origin, e.g. from mechanical, biological, stock-market, etc., their abstract mathematical model can hopefully be classified and then studied with various mathematical technologies. The aim of this project is to classify systems displaying some hyperbolic behavior, the hallmark of chaotic dynamics, and prove that such systems should have some algebraic origin, thus introducing a purely mathematical and very powerful technology into the study of an otherwise intricate problem. While chaotic behavior is by definition impossible to be deterministically predicted, algebraic systems present symmetries that allows the researcher to study the long run behavior, and be able to describe the orbit structure of such systems. The PI plans to continue with the formation of graduate students, the writing of preparatory material on the subject (e.g. books, survey papers, lecture notes, etc.) and collaborate with the teaching of mini-courses, organization of workshop, seminars, etc.

动力系统理论的目标之一是了解系统的轨道结构显示一些特定的功能。在这个项目中，主要研究人员将集中在动力学显示某种双曲线。该项目将在两个主要方向发展，多维时间（即高阶群的作用）和正熵的动力系统和经典动力系统（即一维时间）显示出一些统一的（虽然可能是部分的）双曲性。其目的是开发技术，以了解系统的测量理论和拓扑性质之间的相互作用。这种互动是双向的。例如，在拓扑学到测度理论的方向上，可以观察到在3维零流形上，部分双曲系统总是K-系统。从测度论到拓扑学，我们知道三维不可约流形上的非一致双曲高阶作用只能存在于环面上。玻尔兹曼对气体粒子的研究产生了现在被接受的观点，即一些系统显示出某种混沌行为。虽然该系统可以具有非常不同的来源，例如来自机械、生物、股票市场等，其抽象的数学模型有望被分类，然后用各种数学技术加以研究。该项目的目的是对显示一些双曲行为的系统进行分类，这是混沌动力学的标志，并证明这些系统应该有一些代数起源，从而将一种纯粹的数学和非常强大的技术引入到一个复杂问题的研究中。虽然混沌行为根据定义是不可能被确定性预测的，但代数系统提供了对称性，使研究人员能够研究长期行为，并能够描述此类系统的轨道结构。PI计划继续培养研究生，编写有关该主题的准备材料（例如书籍，调查论文，演讲稿等）。并与微型课程的教学，讲习班，研讨会等的组织合作。