Robust and generic mechanisms in smooth dynamics

平稳动力学中稳健且通用的机制

• 批准号: 1402852
• 负责人: Anne Wilkinson
• 金额: $ 60万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1402852&HistoricalAwards=false
• 关键词: Robust; generic; mechanisms; smooth; dynamics

The PI is a leading expert in the field of partially hyperbolic dynamics; dynamics is the study of systems, physical or mathematical, that evolve over time according to a deterministic set of rules. Hyperbolic dynamical systems are those which display chaotic, unpredictable features at every point; they are both naturally occurring and well-studied. There are other dynamical systems called KAM systems (named after Kolomogorov, Arnold, and Moser), which have stable regions of regular motion. Partially hyperbolic systems provide a more general class of dynamical systems than either, 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 subdynamics, 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 PI 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 in biology, physics and engineering. The PI is currently collaborating with the particle accelerator group at Fermilab to explore some of these potential applications.The research supported by this grant 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 PI proposes to study: ergodic properties of conservative partially hyperbolic diffeomorphisms; physical measures for (dissipative) partially hyperbolic diffeomorphisms; rigidity phenomena connected to partially hyperbolic group actions; and ergodicty of singular partially hyperbolic systems. These research goals will be carried out through a variety of modalities, including published papers in peer-reviewed journals, supervising Ph.D. students, and public speaking, both at research conferences and to the general public.

PI是部分双曲动力学领域的领先专家；动力学是对物理或数学系统的研究，这些系统根据一组确定性规则随时间演变。 双曲动力系统是指在每个点都表现出混乱、不可预测特征的系统。 它们都是自然发生的并且经过充分研究。 还有其他称为 KAM 系统（以 Kolomogorov、Arnold 和 Moser 命名）的动力系统，它们具有稳定的规则运动区域。 部分双曲系统提供了比这两种系统更通用的动力系统，并且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。 PI 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、物理学和工程系统的应用中体现出来。 PI 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。这笔资助支持的研究以沿着双曲理论的思路开发部分双曲系统的一般理论为指导。过去40年的发展。特别是，PI 建议研究： 保守部分双曲微分同胚的遍历特性； （耗散）部分双曲微分同胚的物理测量；与部分双曲群作用相关的刚性现象；和奇异部分双曲系统的遍历性。 这些研究目标将通过多种方式实现，包括在同行评审期刊上发表论文、指导博士学位。学生，以及在研究会议和公众面前的公开演讲。