Partial Hyperbolicity and the Structure of Diffeomorphism Groups

拟议的研究扩展了两个方面的主要研究者以前的研究计划在光滑动力学。在第一，她的工作与合作者最近导致了一个证明的“玻尔兹曼遍历假设”的部分双曲系统：典型的保守部分双曲系统是遍历的。 在其他最近的工作中，主要研究者和她的合作者已经证明了具有最小可微性的典型保守非同态具有平凡中心化子;也就是说，它们没有光滑对称性。该项目有两个主要组成部分：（1）将首席研究员及其合作者先前关于部分双曲型同构遍历性的工作扩展到更广泛的背景，包括耗散系统和非一致部分双曲系统;（2）从泛型和刚性的角度研究流形上光滑群作用的动力学性质。例如，主要研究者建议在保守环境下扩展她与合作者的先前工作，以证明具有最小可微性的典型双曲同态具有平凡的中心化子。在过去的几十年中，部分双曲动力系统的主题已经成为复杂（“混沌”）动力系统理论扩展到经典双曲动力学环境之外的一个主要方向。这项研究的部分驱动力来自于这样一种认识，即动力系统在实验现象中的许多实际应用需要更广泛的理论。在一个平行的发展，光滑系统的典型（或通用）属性已经开始出现，作为日益复杂的扰动技术已经开发。首席研究员在过去十年的研究集中在基本的定性特征上--例如，遍历性（统计“混沌”），缺乏对称性，以及轨道复杂性的其他标志--这可能由一个典型的光滑动力系统所表现出来。拟议中的研究将在理解特定系统家族中何时出现混乱方面取得基础性进展，例如在预测全球天气模式或外星物体轨迹的实践中出现的混乱。从这项研究计划中产生的材料将通过会议上的演讲，在线预印本服务器和学术期刊上的出版物广泛传播。 这笔赠款的一个重要组成部分是专门用于博士研究生培训。水平

The proposed research extends two aspects of the principal investigator's previous research program in smooth dynamics. In the first, her work with collaborators has led recently to a proof of a "Boltzmann Ergodic Hypothesis" for partially hyperbolic systems: the typical conservative partially hyperbolic system is ergodic. In other recent work, the principal investigator and her collaborators have shown that typical conservative diffeomorphisms with minimal differentiability have trivial centralizers; that is, they have no smooth symmetries. The project has two main components: (1) extending the previous work of the principal investigator and her collaborators on the ergodicity of partially hyperbolic diffeomorphisms to a broader context, including dissipative systems and nonuniformly partially hyperbolic systems; (2) a study of the dynamical properties of smooth group actions on manifolds from the perspectives of genericity and rigidity. For one example, the principal investigator proposes to extend previous work with her collaborators in the conservative setting to show that the typical diffeomorphisms with minimal differentiability have trivial centralizers.In the past few decades, the topic of partially hyperbolic dynamical systems has emerged as one main direction in which the theory of complicated ("chaotic") dynamical systems has extended beyond the classical setting of hyperbolic dynamics. This research has been driven in part by the realization that many practical applications of dynamical systems to experimental phenomena require a much broader theory. In a parallel development, the typical (or generic) properties of smooth systems have begun to emerge, as increasingly sophisticated perturbation techniques have been developed. The principal investigator's research over the last decade has focused on the fundamental qualitative features -- e.g., ergodicity (statistical "chaos"), lack of symmetries, and other hallmarks of orbit complexity -- that might be displayed by a typical smooth dynamical system. The proposed research will make foundational advances in understanding when to expect chaos in a given family of systems, such as those that arise in practice in predicting global weather patterns or the trajectories of extraterrestrial bodies. The material resulting from this research proposal will be widely disseminated, through talks at conferences, online preprint servers, and publication in scholarly journals. A significant component of this grant is devoted to graduate-student training on the Ph.D. level.