Translation Flows on Flat Surfaces and the Teichmueller Geodesic Flow

平面上的平移流和 Teichmueller 测地线流

• 批准号: 1068735
• 负责人: Alexander Bufetov
• 金额: $ 12.99万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068735&HistoricalAwards=false
• 关键词: Translation; Flows; Flat; Surfaces; Teichmueller

The research is a study of dynamical properties of translation flows on flat surfaces, with a special emphasis on connections with probability theory. In previously funded work, the proposer has obtained new limit theorems for translation flows. These results give a completely new type of limit theorems in ergodic theory and open a wide array of questions. The proposer will continue his study of limit distributions for translation flows. An important particular case is that of flows along stable foliations of pseudo-Anosov diffeomorphisms.A specific tool essential for this study is the symbolic coding, developed and studied by the proposer, of translation flows as suspension flows over Vershik automorphisms, a construction developing earlier work of S.Ito. Strongly chaotic, hyperbolic behavior of the Teichmueller flow controls the mildly chaotic, parabolic behavior of translation flows.In the second, more geometric, part of the project, the proposer will continue his investigation of the chaotic properties of the Teichmueller geodesic flow.We see chaotic behavior in the evolution of stock prices, weather patterns, turbulent fluids and traffic jams. How to control chaotic behavior? A key role in our modern perception of chaos is played by the ergodic theory of dynamical systems. The proposed research studies the slow chaos for flows on surfaces, a central class of examples in geometry and physics. The project also has an important pedagogical component. Through lectures by visiting scholars, working seminars and reading projects related to the proposal, the proposer will continue involving Rice undergraduate and graduate students in Mathematical research.

该研究是对平面上平移流的动力学特性的研究，特别强调与概率论的联系。在之前资助的工作中，提议者获得了翻译流的新极限定理。这些结果给出了遍历理论中全新类型的极限定理，并提出了一系列广泛的问题。提议者将继续研究翻译流的极限分布。一个重要的特例是沿着伪阿诺索夫微分同胚的稳定叶流的流动。这项研究必不可少的一个特定工具是由提案者开发和研究的符号编码，将平移流作为 Vershik 自同构上的悬浮流，这是 S.Ito 早期工作的一个发展结构。 Teichmueller 流的强混沌双曲行为控制着平移流的轻度混沌抛物线行为。在该项目的第二个更具几何性的部分中，提议者将继续研究 Teichmueller 测地流的混沌特性。我们在股票价格、天气模式、湍流流体和交通拥堵的演变中看到了混沌行为。如何控制混乱的行为？动力系统的遍历理论在我们现代对混沌的认知中发挥着关键作用。拟议的研究研究了表面流动的缓慢混沌，这是几何和物理学的核心例子。该项目还有一个重要的教学部分。通过访问学者的讲座、工作研讨会和与该提案相关的阅读项目，提案者将继续让莱斯大学的本科生和研究生参与数学研究。