Partial hyperbolicity and rigidity

这个项目是由发展的一般理论的部分双曲系统沿着线的双曲理论在过去的四十年发展的深远的目标。特别是，首席研究员将研究以下主题：遍历性的保守部分双曲同态;（耗散）部分双曲同态的物理措施;刚性现象连接到部分双曲群作用;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，合作图书项目，监督博士学位。学生，并在会议上发言，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。动态系统是一个系统（例如，物理过程的状态空间），它根据一组确定性规则随时间演化。研究得很好的动力系统包括所谓的双曲系统，它在每一点上都显示出混沌的、不可预测的特征，以及KAM系统，它具有规则运动的稳定区域。部分双曲系统是比双曲类更一般的一类动力系统，并且包括在某些方向上将双曲性与在其他方向上的KAM行为结合的联合收割机系统。 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。首席研究员已经有了一个完善的研究计划超过十五年来研究部分双曲系统，并准备将这些系统的理论提高到一个新的水平的一般性和适用性。

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.