Cohomology of Dynamical Systems, Rigidity of Smooth Group Actions, and Partially Hyperbolic Diffeomorphisms

动力系统的上同调、光滑群作用的刚性和部分双曲微分同胚

• 批准号: 9971826
• 负责人: Viorel Nitica
• 金额: $ 6.68万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2001-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971826&HistoricalAwards=false
• 关键词: Cohomology; Dynamical; Systems; Rigidity; Smooth

AbstractNiticaThe proposed research has three goals. One goal is to continue the study of cocycles with values in Lie groups and in diffeomorphism groups over hyperbolic actions. In particular we would like to generalize Livsic's cohomological result, as well as the cohomological results of Katok-Spatzier and Katok-Nitica-Torok, about higher-rank abelian actions to various classes of non-abelian cocycles. The second goal is to use cohomological results as a tool in the rigidity theory of hyperbolic or partially hyperbolic abelian group actions, and higher rank lattice actions on compact manifolds. This goal is part of the rigidity program initiated by Zimmer, and developed by Hurder, Katok, Lewis, Margulis, Nitica, Qian, Spatzier, Torok, Zimmer, and others. The third goal is to find generic classes of partially hyperbolic transformations with interesting dynamic and stochastic properties. This is part of a program initiated by Pugh and Shub. The aim of the program is to show that stably ergodicity occurs much more frequently than expected, and that some partial hyperbolicity is sufficient to prove stably ergodicity. We are mostly interested in skew-products and perturbations of skew-products. This project targets the field of dynamical systems. The theory of dynamical systems is a major mathematical discipline that is intertwined to all the other areas of mathematics. Concepts coming from dynamical systems have inspired researchers from other sciences (e.g., statistics, mathematical physics, biology, engineering). They also stimulated the emergence of a new area called nonlinear dynamics, or chaos theory. An important feature of the well understood examples with chaotic dynamics is the presence of hyperbolicity (i.e., expansion and contraction). Many natural examples of dynamical systems that appear from practical applications exhibit only partial hyperbolicity. One of the goals of the proposed research is to show that one can find chaotic behavior for these examples as well. Another goal is to find obstructions to chaotic behavior for partially hyperbolic systems.

AbstractNitica提出的研究有三个目标。 目标之一是继续研究具有李群和双曲作用微分同胚群中的值的余循环。 特别是，我们希望将关于高阶阿贝尔作用的 Livsic 上同调结果以及 Katok-Spatzier 和 Katok-Nitica-Torok 的上同调结果推广到各类非阿贝尔余循环。第二个目标是使用上同调结果作为双曲或部分双曲阿贝尔群作用以及紧流形上的高阶晶格作用的刚性理论的工具。该目标是 Zimmer 发起的刚性计划的一部分，并由 Hurder、Katok、Lewis、Margulis、Nitica、Qian、Spatzier、Torok、Zimmer 等人开发。 第三个目标是找到具有有趣的动态和随机属性的部分双曲变换的通用类。 这是 Pugh 和 Shub 发起的计划的一部分。该程序的目的是表明稳定遍历性发生的频率比预期高得多，并且某些部分双曲性足以证明稳定遍历性。我们最感兴趣的是偏斜产品和偏斜产品的扰动。 该项目针对动力系统领域。动力系统理论是一门主要的数学学科，与数学的所有其他领域交织在一起。来自动力系统的概念启发了其他学科（例如统计学、数学物理、生物学、工程学）的研究人员。 他们还刺激了一个称为非线性动力学或混沌理论的新领域的出现。众所周知的混沌动力学例子的一个重要特征是双曲性（即膨胀和收缩）的存在。实际应用中出现的许多动力系统的自然例子仅表现出部分双曲性。拟议研究的目标之一是表明人们也可以找到这些示例的混沌行为。另一个目标是找到部分双曲系统混沌行为的障碍。