Non-Commutative Cocycles and Dynamics of Systems with Hyperbolic Behavior

动力系统理论是现代数学的一个分支，起源于对物理和机械问题的研究。它描述了各种抽象和现实系统如何随着时间的推移而演变，因此它具有广泛的适用性。这个项目的重点是表现出双曲行为的系统，即在某些方向上呈指数膨胀，在其他方向上呈指数收缩。膨胀和收缩产生了系统丰富而复杂的行为，通常被描述为混沌，单个轨迹对初始条件的微小变化高度敏感。然而，如果一个系统在强烈的意义上是双曲的，那么它作为一个整体是稳定的，也就是说，在质量上类似于任何小的扰动。环是动力系统研究中的一个基本工具。对于由可微函数给出的系统，其导数和相关对象是环的素数例子。另一类重要的例子是矩阵或映射的随机序列。在研究两个动力系统何时相似，在何种程度上相似，特别是当一个系统与扰动或标准模型相似时，共环是有用的。循环自然地出现在动力学中，更普遍地出现在群体行为中。首席研究员将研究双曲、部分双曲和非均匀双曲动力系统上的共环。本文将重点研究非交换非紧群中的带值环，如一般线性群、Hilbert和Banach空间上的线性算子群、紧流形的微分同态群、非正曲率空间的等距群及其推广。主要研究者将研究环的上同调，两个环间共轭的正则性，以及与更简单环共轭的存在性。首席研究员还将研究估算周期增长及其Lyapunov指数的相关问题，重点是使用周期数据。这些问题的部分动机是由于系统和动作的平滑动力学和刚性问题，表现出一些双曲性。主要研究者将利用共环的上同性和非平稳范式的结果来推进这一领域的发展。主要研究单个双曲系统的拓扑刚性和光滑刚性，以及任意流形上高阶双曲阿贝尔作用的全局刚性。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

The theory of dynamical systems is a modern branch of mathematics which originated from the study of physical and mechanical problems. It describes how various abstract and real-life systems evolve over time, and so it has a wide range of applicability. This project is focused on systems that exhibit hyperbolic behavior, that is, exponential expansion in some directions and exponential contraction in other directions. The expansion and contraction produce a rich and complex behavior of the system, often described as chaotic, with individual trajectories highly sensitive to small changes in the initial conditions. Nonetheless, if a system is hyperbolic in a strong sense it is stable as a whole, that is, qualitatively similar to any small perturbation. Cocycles are a fundamental tool in the study of dynamical systems. For systems given by differentiable functions, the derivative and related objects are the prime examples of cocycles. Another important class of examples is given by random sequences of matrices or maps. Cocycles are useful in studying when two dynamical systems are similar and to what extent and, in particular, when a system is similar to a perturbation or to a standard model.Cocycles appear naturally in dynamics and, more generally, in group actions. The principal investigator will study cocycles over hyperbolic, partially hyperbolic, and non-uniformly hyperbolic dynamical systems. The research will be focused on cocycles with values in non-commutative non-compact groups, such as the general linear group, groups of linear operators on Hilbert and Banach spaces, groups of diffeomorphisms of compact manifolds, and groups of isometries of spaces of non-positive curvature and their generalizations. The principal investigator will investigate cohomology of cocycles, regularity of a conjugacy between two cocycles, and existence of conjugacy to simpler cocycles. The principal investigator will also work on related problems of estimating growth of a cocycle and its Lyapunov exponents, with a focus on using the periodic data. These questions are motivated in part by problems in smooth dynamics and rigidity of systems and actions exhibiting some hyperbolicity. The principal investigator will use the results on cohomology of cocycles and on non-stationary normal forms to advance the development of this area. The principal investigator will focus on topological and smooth rigidity of a single hyperbolic system and on global rigidity of higher rank hyperbolic abelian actions on an arbitrary manifold.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.