Measure theory and geometric topology in dynamics

动力系统理论的目标之一是了解具有某些特殊特征的系统的轨道结构。在这个项目中，首席研究员将专注于显示某种双曲线的动力学。该项目将在两个主要方向上发展，具有多维时间的动力系统（即高阶群的作用）和正熵和经典动力系统（即一维时间）显示一些均匀（尽管可能是部分）双曲。目的是开发技术来理解测量系统的理论和拓扑性质之间的相互作用。这种相互作用是双向的。例如，在拓扑测量理论的方向上，观察到在三维零流形上部分双曲系统总是k系统。在拓扑测量理论的方向上，已知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公认的观点，即一些系统表现出某种混沌行为。虽然系统的起源可能非常多样化，例如来自机械、生物、股票市场等，但它们的抽象数学模型有望被分类，然后用各种数学技术进行研究。这个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，从而将纯数学和非常强大的技术引入到其他复杂问题的研究中。虽然混沌行为根据定义是不可能被确定性预测的，但代数系统呈现的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、调查报告、课堂讲稿等），并与迷你课程的教学合作，组织讲习班、研讨会等。

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.