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

这个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，特别是那些由一组不变的规则所决定的运动，比如控制机械运动的牛顿力。众所周知的动力学实验现象，如混沌轨迹与稳定运动相结合，已经在实验中观察到，但还远远没有从理论的角度完全理解。该研究将解决在广泛的动力系统中混沌运动背后的理论机制，包括物理和几何性质的系统。基于首席研究员和她的合作者先前的工作，称为熵和混沌运动的数值不变量之间的相互作用将得到进一步的理解。另一方面，首席研究员提出了几个问题，将熵和相关不变量与一种称为刚性的现象联系起来。当指导系统的规则在数量上的微小变化迫使最终动态发生根本的质变时，刚性就会发生。确定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和相邻科学团体（如物理学）之间的互动。首席研究员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，首席研究员还做了几次关于动力学的公开演讲，并在大众媒体上发表了关于数学家工作的文章。她建议在未来几年扩大这些活动。动力学是对系统（例如，物理过程的状态空间）的研究，这些系统根据一组确定的规则随时间而演变。这类动力系统的研究很充分，包括所谓的双曲系统，它在每个点都显示混沌，不可预测的特征，以及KAM系统，它有规则运动的稳定区域。部分双曲型系统是比双曲型系统更一般的一类动力系统，它包括在某些方向上双曲性与在其他方向上KAM行为相结合的系统。部分双曲系统广泛地出现在物理学中的动力系统中；例如，行星运动通常包含部分双曲次动力学，有效构建粒子加速器（用于生物成像和理论物理）需要对KAM和部分双曲动力学都有详细的了解。首席研究员有一个超过15年的研究计划，研究部分双曲系统，并准备将这些系统的理论提高到一个新的水平，通用性和适用性。这项研究的影响将在未来的应用中看到一个具体的起源系统，在生物学，物理学和工程学。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。该奖项支持的研究是由一个深远的目标指导的，即沿着过去40年来发展的双曲理论的路线发展部分双曲系统的一般理论。特别地，主要研究者建议研究：保守部分双曲微分同态的遍历性质；流形上大量微分同态集合和嵌入的作用群体行为中的刚性现象。本研究的重点是探讨双曲度与KAM现象之间的相互作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.