Partial Hyperbolicity and Rigidity

在过去的几十年里，部分双曲动力系统的主题已经成为一个主要方向，其中复杂（“混沌”）动力系统的理论已经扩展到双曲动力学的经典设置之外。这项研究的部分驱动力来自于这样一种认识，即动力系统在实验现象中的许多实际应用需要更广泛的理论。PI在过去十年中的研究集中在基本的超线性（fundamentalergodic）--即统计学--部分双曲系统一般可以预期显示的特性上。 PI和她的合作者的工作最近导致了对低维部分双曲系统的“Boltzmann遍历假设”的证明：一般的保守部分双曲系统是遍历的。拟议的研究有两个方面：* 扩展PI和她的合作者以前关于部分双曲同态遍历性的工作。*流形上可解群作用的刚性研究，特别是那些包含部分双曲元的作用。从这个研究计划中得到的材料将通过会议上的演讲，在线预印本服务器和学术期刊上的出版物广泛传播。 该补助金的一个重要组成部分是专门用于博士研究生的培训。水平

In the past few decades, the topic of partially hyperbolic dynamicalsystems has emerged as one main direction in which the theory ofcomplicated ("chaotic") dynamical systems has extended beyond theclassical setting of hyperbolic dynamics. This research has been driven inpart by the realization that many practical applications of dynamicalsystems to experimental phenomena require a much broader theory.The PI's research over the last decade has focused on the fundamentalergodic -- that is, statistical -- properties that partially hyperbolicsystems generically can be expected to display. The PI's work with hercollaborators has led recently to a proof of a "Boltzmann ErgodicHypothesis" for partially hyperbolic systems in low dimension: thegeneric conservative partially hyperbolic system is ergodic.The proposed research has two aspects:* Extending the previous work of the PI and her collaborators on theergodicity of partially hyperbolic diffeomorphisms.* A study of rigidity of solvable groups actions on manifolds, inparticular those actions that contain partially hyperbolic elements.The material resulting from this research proposal will be widelydisseminated, through talks at conferences, online preprint servers, andpublication in scholarly journals. A significant component of this grantis devoted to graduate-student training on the Ph.D. level.