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；动力学是研究系统，无论是物理的还是数学的，根据一套确定性的规则随着时间的推移而演变。双曲动力系统是那些在每个点上都表现出混沌的、不可预测的特征的系统；它们既是自然发生的，也是经过充分研究的。还有其他称为KAM系统的动力系统(以Kolomogorov、Arnold和Moser命名)，它们有规则运动的稳定区域。部分双曲系统提供了比任何一个更一般的动力系统类别，并且包括在某些方向上结合双曲性与在其他方向上的KAM行为相结合的系统。部分双曲系统广泛存在于物理产生的动力学系统中；例如，行星运动通常包含部分双曲子动力学，而有效构建粒子加速器(用于生物成像和理论物理)需要对KAM和部分双曲动力学有详细的了解。PI有一个完善的研究计划，研究部分双曲系统超过15年，并准备将这些系统的理论提高到一个新的普适性和适用性的水平。这项研究的影响将在未来生物、物理和工程系统的应用中看到。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。这项拨款支持的研究是以一个深远的目标为指导的，即沿着过去40年发展的双曲理论发展部分双曲系统的一般理论。特别是，PI建议研究：保守的部分双曲微分同胚的遍历性质；(耗散的)部分双曲微分同胚的物理度量；与部分双曲群作用有关的刚性现象；以及奇异部分双曲系统的遍历性。这些研究目标将通过各种方式实现，包括在同行评议的期刊上发表论文，指导博士生，以及在研究会议上和对公众发表公开演讲。