Ergodic Properties of Smooth Systems on Manifolds

这个项目主要研究光滑动力系统的混沌特性，这是当前数学分析研究的一个非常活跃的领域，与数学科学的联系和应用。动力学系统中的混沌行为具有对初始条件的敏感依赖性，当系统当前状态的微小变化可能导致其未来状态的巨大波动时，就会发生这种情况。这发生在许多自然系统中，例如人类的心脏、流体流动和全球天气模式。PI计划开发一个通用的框架和新的方法来理解一大类动力系统中的混沌行为。该项目的活动将导致我们对基本动力学现象的理解取得进展，并可能产生结果并应用于其他数学领域，如几何和数论，以及其他科学领域，如物理和经济学。该项目还将涉及对几名学生和博士后的培训。这个项目是研究流形上光滑系统的遍历和统计性质以及它们与几何和数论的相互作用的研究计划的一部分，主要有三个领域。其中一个领域涉及混沌系统的一些经典性质：K性质和伯努利性质、定量混合(和高阶混合)和极限定理。PI将研究部分双曲(或非一致部分双曲)系统的这些性质与它们的外观之间的关系，并继续为与定量混合和伯努利性质有关的问题开发几何框架。PI还计划构建具有新的遍历和统计行为的奇异动力系统的例子。第二个研究方向涉及抛物型系统的最新发展，不一定是代数起源的。尽管最近取得了进展，但这类系统的许多基本问题仍然悬而未决，例如，关于高阶混合的罗克林问题。PI将建立在他早期工作的技术基础上，并试图发展抛物型系统遍历性质的一般理论。该项目的第三部分将继续PI对稀疏均匀分布问题的研究，使用动力学(如定量均匀分布和混合)和分析数论(如筛选法和指数和)的方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project focuses on the chaotic properties of smooth dynamical systems, a very active area of current research in mathematical analysis, with connections and applications across the mathematical sciences. Chaotic behavior in dynamical systems is characterized by sensitive dependence on initial conditions, which occurs when small changes in a system’s present state may produce large fluctuations in its future state. This occurs in many natural systems, such as the human heart, fluid flows, and global weather patterns. The PI plans to develop a general framework and new approaches for understanding chaotic behavior in a large class of dynamical systems. Activities in the project will lead to progress in our understanding of fundamental dynamical phenomena, with possible consequences and applications in other mathematical fields, such as geometry and number theory, and other scientific areas, such as physics and economics. The project will also involve the training of several students and postdocs. This project is part of a program of research studying ergodic and statistical properties of smooth systems on manifolds and their interactions with geometry and number theory, with three main areas of focus. One area concerns some classical properties that are expected to hold for chaotic systems: K and Bernoulli properties, quantitative mixing (and higher order mixing) and limit theorems. The PI will investigate relations between these properties and their appearance for partially hyperbolic (or non-uniformly partially hyperbolic) systems, and continue developing a geometric framework for problems related to quantitative mixing and the Bernoulli property. The PI also plans to construct examples of exotic dynamical systems with new ergodic and statistical behavior. A second research direction relates to recent developments on parabolic systems, of not necessarily algebraic origin. Despite recent progress, many fundamental questions for such systems are still open, for instance, the Rokhlin problem on higher order mixing. The PI will build on techniques from his earlier work and try to develop a general theory for ergodic properties of parabolic systems. A third part of the project will continue the PI’s research on sparse equidistribution problems, using methods from dynamics (such as quantitative equidistribution and mixing) and analytic number theory (such as sieve methods and exponential sums).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.