Partial Hyperbolicity and the Structure of Diffeomorphism Groups

偏双曲性和微分同胚群的结构

• 批准号: 0701018
• 负责人: Anne Wilkinson
• 金额: $ 26.04万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-15 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701018&HistoricalAwards=false
• 关键词: Partial; Hyperbolicity; Structure; Diffeomorphism; Groups

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.

拟议的研究扩展了主要研究者先前研究计划方面的两个方面。首先，她与合作者的合作最近导致了部分双曲线系统的“鲍尔茨曼·奇异假说”的证明：典型的保守性部分双曲线系统是千古的。 在最近的其他工作中，首席调查员及其合作者表明，典型的保守性差异性具有最小的不同性质具有微不足道的中心化。也就是说，它们没有光滑的对称性。该项目有两个主要组成部分：（1）将主要研究人员及其合作者的先前工作扩展到对部分双曲线差异性的怪异性，包括更广泛的环境，包括耗散系统和部分夸张的双曲线系统； （2）从一般性和刚性的角度研究平滑群体作用在流形上的动力学特性的研究。例如，主要研究人员建议在保守环境中与她的合作者进行以前的工作，以表明具有最小可差权的典型差异性具有微不足道的中心化。在过去的几十年中，部分双曲动力学系统的主题已成为一个主要方向，其中一个动态学理论是“ chaotic dynancity”的一个主要方向。这项研究部分是由于意识到动态系统在实验现象上的许多实际应用需要更广泛的理论。在平行的开发中，随着越来越复杂的扰动技术的发展，光滑系统的典型（或通用）特性已经开始出现。主要研究者在过去十年中的研究集中在基本的定性特征上，例如ergodicity（统计“混乱”），缺乏对称性和其他轨道复杂性的标志 - 可能由典型的平滑动态系统显示出来。拟议的研究将在理解何时在给定的系统家族中期待混乱的基础进步，例如在预测全球天气模式或外星物体的轨迹中实践中出现的混乱。该研究提案所产生的材料将通过会议，在线预印式服务器和学术期刊上的出版物进行广泛传播。 该赠款的重要组成部分致力于博士学位的研究生培训。等级。