Partial hyperbolicity and rigidity

这个项目的目标是沿着过去四十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。特别是，主要研究者将研究以下主题：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，包括在同行评议期刊上发表论文、合作出书项目、指导博士生以及在会议上发言，包括即将在印度海德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权威专家。动态系统是一个系统（例如，一个物理过程的状态空间），它根据一组确定的规则随时间演变。研究得很好的动力系统包括所谓的双曲系统，它在每个点都显示混沌，不可预测的特征，以及KAM系统，它有稳定的规则运动区域。部分双曲型系统是比双曲型系统更一般的一类动力系统，它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。首席研究员已经制定了一个完善的研究计划，研究部分双曲系统超过十五年，并准备将这些系统的理论提高到一个新的水平，通用性和适用性。

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.