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力系统领域的新现象。 动力系统（简称“动力学”）是对运动的研究，特别是由一组不变的规则决定的运动，例如控制机械运动的牛顿力。 众所周知的动力学实验现象，例如混沌轨迹与稳定运动相结合，已经通过实验观察到，但从理论角度来看还远未得到充分理解。该研究将解决广泛的动力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。 根据首席研究员和她的合作者之前的工作，我们将进一步了解称为熵的数值不变量与混沌运动之间的相互作用。另一方面，首席研究员提出了将熵和相关不变量与刚性现象联系起来的几个问题。 当指导系统的规则在数量上发生微小变化，迫使所产生的动态发生根本性的质变时，就会出现刚性。识别刚性系统是对物理和几何系统中观察到的某些特殊动力学行为进行分类的第一步。该项目的一个重要方面是进一步促进数学与相邻科学界（例如物理学界）之间的互动。 首席研究员已经就有关粒子加速器设计的问题进行了合作，目前正在与一位物理学家合作研究黑洞出现背后的量子动力学。 此外，首席研究员还做过几次关于动力学的公开讲座，并在大众媒体上撰写了有关数学家工作的文章。她建议在未来几年扩大这些活动。动力学是对根据一组确定性规则随时间演变的系统（例如，物理过程的状态空间）的研究。 经过深入研究的此类动力系统包括所谓的双曲系统（其在每个点都表现出混乱、不可预测的特征）和 KAM 系统（其具有规则运动的稳定区域）。 部分双曲系统是比双曲类更通用的动力系统，包括将某些方向上的双曲性与其他方向上的 KAM 行为相结合的系统。 部分双曲系统广泛出现在物理学中出现的动力系统中；例如，行星运动通常包含部分双曲子动力学，而粒子加速器（用于生物成像以及理论物理）的有效构造需要对 KAM 和部分双曲动力学有详细的了解。首席研究员制定了超过 15 年的完善研究计划，研究部分双曲系统，并准备将这些系统的理论提升到通用性和适用性的新水平。 这项研究的影响将在未来对生物学、物理学和工程学等具体系统的应用中体现出来。 首席研究员目前正在与费米实验室的粒子加速器小组合作探索其中一些潜在的应用。该奖项支持的研究以一个深远的目标为指导，即沿着过去 40 年发展的双曲理论的路线发展部分双曲系统的一般理论。 特别是，主要研究者建议研究： 保守部分双曲微分同胚的遍历特性；大量微分同胚和流形嵌入的行为；以及群体行动中的僵化现象。 拟议研究的一个亮点是调查双曲性和 KAM 现象之间的相互作用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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