Rigidity in Rank One Hyperbolic Dynamics and Related Topics

这个研究项目是在双曲动力系统领域。该领域包括各种真实世界的动态系统，或随时间变化的系统。例子可以简单到平面或三维空间中机械连杆的运动，也可以复杂到密闭容器中气体的行为或大气中对流的动力学。双曲动力学被通俗地称为“混沌理论”，研究这种动力系统的混沌（或“不可预测性”）和长期“行为”。该项目旨在在双曲动力学的某些基础问题上取得重大进展。这项研究可能会对物理，化学，工程和其他自然出现混沌动力系统的领域产生影响。该项目还具有教育部分，并支持研究生的培训。Anosov超同态，部分双曲超同态和负曲率测地流是混沌动力系统的主要例子。在过去的几十年里，对单个动力系统的统计和几何性质的研究蓬勃发展。然而，我们仍然有一个相当贫穷的了解空间的结构Anosov和部分双曲的超纯和流动。该项目旨在获得这样的理解，主要集中在光滑共轭类的结构。首席研究员计划利用动力学，分析和几何技术的武器库，以实现这一建议的目标。具体地说，该项目有三个主要部分：（1）获得高维Anosov同态和三维Anosov流的光滑共轭类的结构的基本理解：（2）推进负曲面和高维流形的刚性程序;（三）推进刚度计划从双曲线动力学到部分双曲线动力学。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估。

This research project is in the field of hyperbolic dynamical systems. This field encompasses a variety of real-world dynamical systems, or systems that change with time. Examples could be as simple as the motion of a mechanical linkage in the plane or in 3-dimensional space, and as complex as the behavior of gas in a closed vessel or the dynamics of convection currents in the atmosphere. Hyperbolic dynamics is colloquially known as “chaos theory” and studies chaotic (or “unpredictability” properties) and long-term ``behavior on the average” of such dynamical systems. The project seeks to make significant progress on certain foundational questions in hyperbolic dynamics. Potentially this research could have impact in physics, chemistry, engineering, and other areas where chaotic dynamical systems naturally appear. The project also has an educational component and supports the training of graduate students.Anosov diffeomorphisms, partially hyperbolic diffeomorphisms and geodesic flows in negative curvature are the prime examples of chaotic dynamical systems. In the past decades, the study of statistical and geometric properties of individual dynamical systems has flourished. However, we still have a rather poor understanding of the structure of the space of Anosov and partially hyperbolic diffeomorphisms and flows. The project seeks to gain such understanding primarily focusing on the structure of smooth conjugacy classes. The principal investigator plans to utilize an arsenal of dynamics, analytic and geometric techniques to achieve the goals of this proposal. More specifically the project has three major parts: (1) gaining fundamental understanding of the structure of smooth conjugacy classes of higher dimensional Anosov diffeomorphisms and 3-dimensional Anosov flows; (2) advancing the rigidity program for negatively curved surfaces and higher dimensional manifolds; (3) advancing the rigidity program from hyperbolic to partially hyperbolic dynamics.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.