Partial hyperbolicity and rigidity

这个项目的指导目标是沿着过去40年发展的双曲型理论发展部分双曲型系统的一般理论这一深远的目标。特别是，首席研究者将研究以下主题：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，合作出版项目，指导博士生，以及在会议上发言，包括即将在印度海得拉巴举行的国际数学家大会。首席研究员是部分双曲动力学领域的领先专家。动态系统是根据一组确定性规则随时间演变的系统(例如，物理过程的状态空间)。研究得很好的动力系统包括所谓的双曲系统和KAM系统，双曲系统在每一点都表现出混沌的、不可预测的特征，KAM系统具有稳定的规则运动区域。部分双曲系统是一类比双曲系统更一般的动力系统，它包括在一些方向上结合双曲性和在另一个方向上的KAM行为的系统。部分双曲系统广泛存在于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。首席研究员已经制定了15年多的研究计划来研究部分双曲型系统，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的水平。

This project is guided by the far-reaching goal of developing a general theory of partially hyperbolic systems along the lines of the hyperbolic theory developed over the past forty years. In particular, the principal investigator will study the following topics: 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, a collaborative book project, supervising Ph.D. students, and speaking at conferences, including the upcoming International Congress of Mathematicians in Hyderabad, India.The principal investigator is a leading expert in the field of partially hyperbolic dynamics. A dynamical system is a system (for example, a state space for a physical process) that evolves over time according to a deterministic set of rules. Well-studied classes of 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 that arise in physics. For example, planetary motion usually contains partially hyperbolic subdynamics. The principal investigator has had a well-developed research plan for over fifteen years to study partially hyperbolic systems and is poised to raise the theory of these systems to a new level of generality and applicability.