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，如微分同态和流动，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、部分双曲和非均匀双曲系统和动作的各种刚性性质。高阶作用具有显著的性质，包括不变测度的刚性和光滑刚性。在第一种情况下，刚性出现在一些非典型系统中，例如那些具有高度平滑的不变叶理或强挤压的系统。自行车在这两个领域都发挥着重要作用。研究双曲系统上非交换共环的上同调是本研究的主要内容。动力系统研究机械、物理和数学系统随时间的演变。它提供了应用到数学的其他领域，以及许多自然科学，如物理学，力学，计算机科学和生物学。这个领域起源于微分方程和天体力学。它使用来自分析和拓扑学的数学工具来研究系统“长期”的定性行为。该项目的一个主要部分是研究由几个相互通勤的独立系统组成的更复杂的系统。这种复杂的系统自然地出现在代数、几何和物理学中。π为主要工作区域主要是大学生和大学教育的大部分当地学校教师 .-----------------------------------------------------

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.-----------------------------------------------------