Robust and generic mechanisms in smooth dynamics

PI是部分双曲动力学领域的领先专家；动力学是对系统的研究，物理的或数学的，根据一组确定的规则随着时间的推移而发展。双曲动力系统在每一点都表现出混沌的、不可预测的特征；它们都是自然发生的，并且经过了充分的研究。还有其他动力系统称为KAM系统（以Kolomogorov， Arnold和Moser命名），它具有规则运动的稳定区域。部分双曲系统提供了一种比两者都更一般的动力系统，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。PI已经制定了一个超过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一般性和适用性水平。这项研究的影响将在未来应用于生物、物理和工程系统中。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。该基金支持的研究工作的长远目标是沿着过去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，PI提出研究：保守部分双曲微分同态的遍历性质；（耗散的）部分双曲微分同态的物理测度部分双曲群作用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多种方式实现，包括在同行评审期刊上发表论文，指导博士生，以及在研究会议和公众面前发表演讲。

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.