Cocycles over hyperbolic and partially hyperbolic systems

周期在经典动力学和群作用理论中都扮演着重要的角色。本文研究的主要内容是双曲和部分双曲微分同胚上的余循环。在这一背景下，通过微分同胚的微分及其对切丛的不变子丛的限制，给出了线性余循环的重要例子。如果丛是平凡的，则可以将线性上循环视为具有矩阵群中的值的上循环。PI还将研究具有其他组中的值的协循环，包括无限维组。PI计划研究余圈上同调中的各种问题。这一领域的问题部分是由双曲和部分双曲系统和群体行动的光滑动力学和刚性问题引起的。PI打算利用她最近的结果以及预测的结果来进一步推进这些领域的研究。动力学是研究系统随时间演变的数学领域，重点是描述它的长期行为。正在考虑的系统来自数学、物理和其他科学。双曲和部分双曲系统一直是可微动力学的主要研究对象之一。这些系统的指数收缩和膨胀会产生混沌行为，了解系统的复杂结构是一个重要的目标。导数在分析这些系统中起着至关重要的作用，它给出了一个系统上的协循环的例子。周期也自然地出现在联系两个系统的线性化方程中，因此在双曲系统的稳定性、刚性和分类的研究中很有用。

Cocycles play an important role in classical dynamics as well as in the theory of group actions. The main part of the proposed research is the study of cocycles over hyperbolic and partially hyperbolic diffeomorphisms. In this context important examples of linear cocycles are given by the differential of the diffeomorphism and by its restrictions to invariant sub-bundles of the tangent bundle. If the bundle is trivial the linear cocycle can be viewed as a cocycle with values in a matrix group. The PI will also study cocycles with values in other groups, including infinite dimensional ones. The PI plans to work on various problems in cohomology of cocycles. The questions in this area are motivated in part by problems in smooth dynamics and rigidity of hyperbolic and partially hyperbolic systems and group actions. The PI intends to use her recent results as well as the projected outcomes to further the research in these areas.Dynamics is the area of mathematics that studies evolution of a system over time with an emphasis on describing its long term behavior. The systems under consideration come from mathematics, physics, and other sciences. Hyperbolic and partially hyperbolic systems have been one of the main objects of study in differentiable dynamics. Exponential contraction and expansion in these systems produce chaotic behavior, and understanding the complex structure of the system is an important goal. The derivative plays a crucial role in analysing these systems, and it gives an example of a cocycle over the system. Cocycles also appear naturally in linearized equations that relate two systems and thus are useful in the study of stability, rigidity, and classification of hyperbolic systems.