Ergodicity, Rigidity, and the Interplay Between Chaotic and Regular Dynamics

遍历性、刚性以及混沌动力学和规则动力学之间的相互作用

• 批准号: 1900411
• 负责人: Anne Wilkinson
• 金额: $ 33万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900411&HistoricalAwards=false
• 关键词: Ergodicity; Rigidity; Interplay; Between; Chaotic

The aim of this project is to discover new phenomena in the area of dynamical systems. Dynamical systems ("dynamics," for short) is the study of motion, and in particular motion that is dictated by an unchanging set of rules, such as the Newtonian forces controlling mechanical motion. Well-known experimental phenomena in dynamics such as chaotic trajectories combined with stable motion have been observed experimentally but are far from being fully understood from a theoretical perspective. The research will address the theoretical mechanisms behind chaotic motion in broad classes of dynamical systems, which include systems of both a physical and geometric nature. Based on previous work of the principal investigator and her collaborators, the interplay between a numerical invariant called entropy and chaotic motion will be further understood. On the flip side, the principal investigator proposes several problems connecting entropy and related invariants with a phenomenon called rigidity. Rigidity occurs when quantitatively small changes to the rules guiding a system force fundamental qualitative changes in the resulting dynamics. Identifying the rigid systems is a first step toward classification of certain peculiar dynamical behaviors observed in physical and geometric systems. An important aspect of the project is to further interaction between mathematical and adjacent scientific communities, such as physics. The principal investigator has already collaborated in questions surrounding the design of particle accelerators and is currently collaborating with a physicist studying the quantum dynamics behind the emergence of black holes. Furthermore, the principal investigator has given several public lectures on dynamics and has written in the popular press about the work of mathematicians. She proposes to expand these activities in the coming years.Dynamics is the study of systems (for example, a state space for a physical process) that evolve over time according to a deterministic set of rules. Well-studied classes of such dynamical systems include the so-called hyperbolic systems, which display chaotic, unpredictable features at every point, and KAM systems, which have stable regions of regular motion. The partially hyperbolic systems are a more general class of dynamical systems than the hyperbolic class and include systems that combine hyperbolicity in some directions with KAM behavior in other directions. Partially hyperbolic systems occur widely in dynamical systems arising in physics; for example, planetary motion usually contains partially hyperbolic sub-dynamics, and the effective construction of particle accelerators (used in biological imaging, as well as theoretical physics) requires a detailed understanding of both KAM and partially hyperbolic dynamics. The principal investigator has a well-developed research plan of over 15 years studying partially hyperbolic systems and is poised to raise the theory of these systems to a new level of generality and applicability. The impacts of this research will be seen in future applications to systems of a concrete origin, in biology, physics and engineering. The principal investigator is currently collaborating with the particle accelerator group at Fermilab to explore some of these potential applications. The research supported by this award is guided by the far-reaching goal of developing a general theory of partially hyperbolic systems along the lines of the hyperbolic theory developed in the past 40 years. In particular the principal investigator proposes to study: ergodic properties of conservative partially hyperbolic diffeomorphisms; actions of large collections of diffeomorphisms and embeddings on manifolds; and rigidity phenomena in actions of groups. A highlight of the proposed research is to investigate the interaction between hyperbolicity and KAM phenomena.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的目的是发现动态系统领域的新现象。 动力系统（简称“动力学”）是运动的研究，特别是由一组不变的规则（例如控制机械运动的牛顿力量）决定的。 已经在实验中观察到了众所周知的实验现象，例如混沌轨迹和稳定运动的混合轨迹，但从理论的角度来看远未完全理解。该研究将解决广泛的动态系统中混沌运动背后的理论机制，其中包括物理和几何性质的系统。 基于首席研究员及其合作者的先前工作，将进一步了解称为熵和混乱运动的数值不变性之间的相互作用。另一方面，主要研究人员提出了几个问题，将熵和相关的不变性与称为刚度的现象相关联。 当定量变化对指导系统力的基本定性变化的规则变化时，就会发生刚性。识别刚性系统是在物理和几何系统中观察到的某些特殊动力学行为的分类的第一步。该项目的一个重要方面是在数学和相邻科学界（例如物理学）之间进一步相互作用。 首席研究者已经在围绕粒子加速器设计的问题上进行了合作，目前正在与研究黑洞出现背后的量子动态的物理学家合作。 此外，首席研究员已经为动态提供了几次公开演讲，并在流行媒体上写了有关数学家的工作。她建议在未来几年内扩展这些活动。贫民学是对系统的研究（例如，用于物理过程的状态空间），根据确定性规则集随时间发展。 经过充分研究的这种动态系统的类别包括所谓的双曲线系统，这些系统在每个点都显示混乱，不可预测的功能，以及具有稳定的常规运动区域的KAM系统。 部分双曲线系统是比双曲线类更一般的动力系统类别，并且包括将双曲线与某些方向上的双曲线与KAM行为相结合的系统。 部分双曲系统广泛存在于物理中引起的动力系统中。例如，行星运动通常包含部分双曲线亚动力学，并且有效的粒子加速器（用于生物成像以及理论物理学）需要详细了解KAM和部分双曲动力学。首席研究者的研究计划已有15年以上，研究了部分双曲线系统，并准备将这些系统的理论提升到新的一般性和适用性水平。 这项研究的影响将在未来对具体起源系统，生物学，物理和工程系统的系统中看到。 主要研究者目前正在Fermilab的粒子加速器组合作，以探索其中一些潜在的应用。该奖项支持的研究以深远的目的为指导，即沿着过去40年中开发的双曲线理论的路线开发部分双曲线系统的一般理论。 特别是主要研究者提出的研究：保守的部分双曲线差异；大量的差异性和嵌入流形的作用；和群体行动中的僵化现象。 拟议的研究的亮点是调查双曲线和KAM现象之间的相互作用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准通过评估来支持的。

项目成果

Ratner's work on unipotent flows and its impact.

拉特纳关于单能流及其影响的工作。

• DOI: 10.1090/noti/1829
• 发表时间: 2019
• 期刊: Notices of the American Mathematical Society
• 作者: Lindenstrauss, Elon;Sarnak, Peter;Wilkinson, Amie
• 通讯作者: Wilkinson, Amie

Symplectomorphisms with positive metric entropy

• DOI: 10.1112/plms.12437
• 发表时间: 2019-04
• 期刊: Proceedings of the London Mathematical Society
• 影响因子: 1.8
• 作者: A. Avila;S. Crovisier;A. Wilkinson

Pathology and asymmetry: Centralizer rigidity for partially hyperbolic diffeomorphisms

• DOI: 10.1215/00127094-2021-0053
• 发表时间: 2019-02
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Danijela Damjanović;A. Wilkinson;Disheng Xu