Partial Hyperbolicity and Rigidity

部分双曲性和刚性

• 批准号: 0401326
• 负责人: Anne Wilkinson
• 金额: $ 21.63万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401326&HistoricalAwards=false
• 关键词: Partial; Hyperbolicity; Rigidity

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.

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”)动力系统理论超越双曲动力学经典背景的一个主要方向。这项研究的部分原因是认识到，动力学系统对实验现象的许多实际应用需要更广泛的理论。PI在过去十年的研究集中在部分双曲系统通常可以预期展示的基本警示性质--即统计性质上。PI与合作者的工作最近证明了低维部分双曲系统的“Boltzmann遍历假设”：一般保守的部分双曲系统是遍历的。拟议的研究有两个方面：*扩展了PI及其合作者先前关于部分双曲微分同胚的遍历性的工作。*研究流形上可解群作用的刚性，特别是那些包含部分双曲元素的作用。这项研究建议的材料将通过在会议上的演讲、在线预印服务器和在学术期刊上发表来广泛传播。该助学金的一个重要组成部分致力于博士研究生的培训。