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

摘要：本研究有三个目标。一个目标是继续研究双曲作用上李群和微分同胚群中的带值环。特别地，我们将Livsic关于高阶阿贝尔动作的上同结果，以及Katok-Spatzier和Katok-Nitica-Torok关于高阶阿贝尔动作的上同结果推广到各种非阿贝尔环上。第二个目标是利用上同调结果作为紧流形上双曲或部分双曲阿贝尔群作用和高阶格作用的刚性理论的工具。这个目标是齐默发起的刚性计划的一部分，并由赫德、卡托克、刘易斯、马古利斯、尼蒂卡、钱、斯帕齐尔、托洛克、齐默等人发展。第三个目标是找到具有有趣的动态和随机性质的部分双曲变换的一般类。这是Pugh和Shub发起的一个项目的一部分。该程序的目的是表明稳定遍历性比预期发生的频率要高得多，并且某些部分双曲性足以证明稳定遍历性。我们最感兴趣的是斜积和斜积的微扰。这个项目的目标是动力系统领域。动力系统理论是一个主要的数学学科，它与所有其他数学领域交织在一起。来自动力系统的概念启发了其他科学（如统计学、数学物理、生物学、工程学）的研究人员。它们还刺激了一个叫做非线性动力学或混沌理论的新领域的出现。很好理解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。在实际应用中出现的许多动力系统的自然例子只表现出部分双曲性。所提出的研究的目标之一是表明人们也可以在这些例子中找到混沌行为。另一个目标是找到部分双曲系统混沌行为的障碍。

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.