Translation Flows on Flat Surfaces and the Teichmueller Geodesic Flow

本研究是对平面上平动流的动力学性质的研究，特别强调与概率论的联系。在先前的资助工作中，提出了新的翻译流极限定理。这些结果给出了遍历理论中一类全新的极限定理，并打开了一系列广泛的问题。他将继续研究翻译流的极限分布。一个重要的特殊情况是沿伪anosov微分同胚的稳定叶理流动。本研究必不可少的一个具体工具是由作者开发和研究的符号编码，将翻译流作为Vershik自同构上的悬浮流，这是s.e to早期工作的发展。强混沌，双曲行为的Teichmueller流控制轻度混沌，抛物行为的平移流。在项目的第二部分，更几何的部分，提案人将继续他对Teichmueller测地线流的混沌特性的研究。我们在股票价格、天气模式、湍流和交通堵塞的演变中看到了混乱的行为。如何控制混沌行为？动力系统的遍历理论在我们对混沌的现代认知中起着关键作用。本文研究的是表面流动的慢混沌，这是几何和物理中的一类中心例子。该项目还有一个重要的教学组成部分。通过访问学者的讲座、工作研讨会和阅读与提案相关的项目，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。

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.