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复杂（“混沌”）动力系统理论的一个主要方向，它已经超越了双曲动力学的经典设置。推动这项研究的部分原因是认识到动力学系统在实验现象中的许多实际应用需要更广泛的理论。在过去的十年里，PI的研究主要集中在基本性质上，即统计性质，部分双曲系统通常可以表现出来。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“玻尔兹曼遍历假设”的证明：一般保守部分双曲系统是遍历的。提出的研究有两个方面：*扩展了PI及其合作者先前关于部分双曲微分同态的遍演性的工作。流形上可解群作用的刚性研究，特别是那些含有部分双曲元素的作用。本研究计划所产生的材料将通过会议演讲、在线预印本服务器和在学术期刊上发表而广泛传播。这笔拨款的一个重要组成部分是专门用于博士水平的研究生培训。

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.