Partial Hyperbolicity and the Structure of Diffeomorphism Groups

这项拟议的研究扩展了首席研究员先前在平滑动力学方面的研究计划的两个方面。首先，她与合作者的工作最近证明了部分双曲系统的“玻尔兹曼遍历假设”：典型的保守部分双曲系统是遍历的。在最近的另一项工作中，主要研究者和她的合作者证明了典型的具有最小可微性的保守微分同胚有平凡的中心化子，即它们没有光滑的对称性。该项目有两个主要部分：(1)将主要研究者和她的合作者关于部分双曲微分同态的遍历性的工作推广到更广泛的背景下，包括耗散系统和非一致部分双曲系统；(2)从泛性和刚性的角度研究流形上光滑群作用的动力学性质。例如，主要研究人员建议在保守环境下扩展以前与她的合作者的工作，以证明具有最小可微性的典型可微同胚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”)动力系统理论超越双曲动力学经典背景的一个主要方向。这项研究的部分原因是认识到，动力系统对实验现象的许多实际应用需要更广泛的理论。在一个平行的发展中，随着越来越复杂的摄动技术的发展，光滑系统的典型(或一般)性质已经开始显现。在过去十年中，首席研究人员的研究重点放在基本的定性特征上--例如，遍历性(统计“混沌”)、缺乏对称性和轨道复杂性的其他特征--这些特征可能由典型的光滑动力系统表现出来。拟议中的研究将在理解何时预期给定系统家族中出现混乱方面取得基础性进展，例如预测全球天气模式或外星天体轨迹的实践中出现的系统。这项研究提案产生的材料将通过在会议上的演讲、在线预印服务器和在学术期刊上发表来广泛传播。这笔助学金的很大一部分用于博士水平的研究生培训。

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.