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，物理或数学，随着时间的推移，根据一组确定性的规则演变。 双曲动力系统是那些在每一点上都表现出混乱、不可预测的特征的系统;它们是自然发生的，也是经过充分研究的。 还有其他的动力系统称为KAM系统（以Kolomogorov，Arnold和Moser命名），它们具有规则运动的稳定区域。 部分双曲系统提供了一个更一般的一类动力系统，比任何一个，并包括系统，联合收割机在某些方向上的双曲性与KAM行为在其他方向上。 部分双曲系统广泛存在于物理学中的动力学系统中;例如行星运动通常包含部分双曲子动力学，而粒子加速器（用于生物成像和理论物理）的有效构建需要对KAM和部分双曲动力学的详细理解。PI有一个超过15年的研究部分双曲系统的成熟研究计划，并准备将这些系统的理论提高到一个新的通用性和适用性水平。这项研究的影响将在未来应用于生物学，物理学和工程学系统中。 PI目前正在与费米实验室的粒子加速器小组合作，探索其中的一些潜在应用。这项资助支持的研究的深远目标是沿着过去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PI建议研究：遍历性的保守部分双曲同态;物理措施（耗散）部分双曲同态;刚性现象连接到部分双曲群行动;和遍历性的奇异部分双曲系统。 这些研究目标将通过各种方式进行，包括在同行评审期刊上发表论文，指导博士学位。学生，和公开演讲，无论是在研究会议和公众。

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.