RUI: Cocycles and rigidity for hyperbolic systems and actions

本文的主要研究对象是光滑动力系统，如微分同胚和流，以及高阶阿贝尔群的光滑作用。该项目的主要目标是研究双曲型、部分双曲型和非一致双曲型系统和作用的各种刚性性质。高阶作用具有不变测度的刚性和光滑刚性等显著性质。在第一种情况下，一些非典型系统会出现刚性，例如具有高不变叶光滑性或强收缩的系统。共循环在这两个领域都发挥着重要作用。研究双曲系统上非对易上循环的上同调是所提出的研究的主要部分。动力系统领域研究力学、物理和数学系统随时间的演化。它提供了数学的其他领域以及许多自然科学的应用，如物理、力学、计算机科学和生物学。这个领域起源于微分方程式和天体力学。它使用分析和拓扑学中的数学工具来研究“长期”系统的定性行为。该项目的一个主要部分是研究更复杂的系统，这些系统由几个相互通勤的独立系统组成。这样的复杂系统在代数、几何和物理中很自然地出现。PI为主要的地区性大学工作，该大学主要是本科生，并为当地大部分学校提供teachers.-----------------------------------------------------教育

The main objects of the proposed research are smooth dynamical systems such as diffeomorphisms and flows, as well as smooth actions of higher rank abelian groups. The main goal of the project is to study various rigidity properties for hyperbolic, partially hyperbolic, and non-uniformly hyperbolic systems and actions. Higher rank actions exhibit remarkable properties including rigidity of invariant measures and smooth rigidity. In rank one case, rigidity occurs for some non-typical systems, such as those with high smoothness of invariant foliations or strong pinching. An important role in both areas is played by cocycles. Investigating cohomology of non-commutative cocycles over hyperbolic systems is a major part of the proposed research.The field of Dynamical Systems studies evolution of mechanical, physical, and mathematical systems over time. It provides applications to other areas of mathematics as well as to many natural sciences such as physics, mechanics, computer science, and biology. This field originated from differential equations and celestial mechanics. It uses mathematical tools from analysis and topology to study qualitative behavior of systems "in the long run". A major part of the project is the study of more complex systems consisting of several individual ones that commute with each other. Such complex systems appear naturally in algebra, geometry, and physics. The PI works for the main regional university which is primarily undergraduate and educates most of the local school teachers.-----------------------------------------------------