Partial hyperbolicity and rigidity

部分双曲性和刚性

• 批准号: 1316534
• 负责人: Anne Wilkinson
• 金额: $ 17.25万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1316534&HistoricalAwards=false
• 关键词: Partial; hyperbolicity; rigidity

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.

该项目的指导下是沿着过去四十年发展的双曲线理论的路线发展的一般双曲系统的一般理论的指导。特别是，首席研究者将研究以下主题：保守的部分双曲线差异性的千古特性； （耗散）部分双曲线差异的物理措施；刚度与部分双曲组作用有关的僵化现象；和奇异的部分双曲系统。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的论文，这是一个协作书籍项目，负责博士学位。学生以及在会议上发表讲话，包括即将举行的印度海得拉巴国际数学家大会。动态系统是一个系统（例如，物理过程的状态空间），根据确定性规则集随时间发展。精心研究的动态系统包括所谓的双曲线系统，这些系统在每个点都显示混乱，不可预测的功能，以及具有稳定的常规运动区域的KAM系统。部分双曲线系统是比双曲线类更一般的动力系统类别，并且包括将双曲线与某些方向上的双曲线与KAM行为相结合的系统。 部分双曲系统广泛发生在物理中出现的动态系统中。例如，行星运动通常包含部分双曲线亚动力学。首席研究者已经制定了一项完善的研究计划，已有15年来研究部分双曲线系统，并准备将这些系统的理论提高到新的一般性和适用性。