Rigidity in Rank One Hyperbolic Dynamics and Related Topics

本研究项目是关于双曲动力系统的研究。这一领域涵盖了各种真实世界的动态系统，或随时间变化的系统。例子可以简单到机械连杆在平面或三维空间中的运动，也可以复杂到封闭容器中气体的行为或大气中对流的动力学。双曲动力学通俗地称为“混沌理论”，研究这种动力系统的混沌(或“不可预测性”)和长期的“平均行为”。该项目寻求在双曲动力学的某些基本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。该项目也有一个教育部分，并支持研究生的培训。Anosov微分同胚、部分双曲微分同胚和负曲率测地线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，我们对Anosov空间的结构以及部分双曲微分同胚和流的了解仍然很少。该项目寻求获得这样的理解，主要集中在光滑共轭类的结构上。首席研究人员计划利用动力学、解析和几何技术来实现这项提案的目标。更具体地说，该项目有三个主要部分：(1)对高维Anosov微分同胚和三维Anosov流的光滑共轭类的结构有了基本的了解；(2)推进了负曲面和高维流形的刚性计划；(3)将刚性计划从双曲线推进到部分双曲动力学。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 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.