Translation Flows on Flat Surfaces and the Teichmueller Geodesic Flow

这项研究是一项研究的动力学性质的平移流在平面上，特别强调与概率论的连接。在以前资助的工作中，提议者获得了新的极限定理的翻译流。这些结果给出了遍历理论中一个全新类型的极限定理，并提出了一系列广泛的问题。提议者将继续他的研究极限分布的翻译流。一个重要的特例是沿着沿着伪Anosov自同构的稳定叶理的流，这个研究的一个重要工具是由提出者发展和研究的符号编码，作为Vershik自同构上的悬浮流的平移流，一个发展S.Ito早期工作的构造。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。在项目的第二部分，更多的几何部分，提议者将继续他对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识中起着关键作用。拟议的研究研究表面上的流动，在几何和物理学的一个中心类的例子缓慢的混乱。该项目还有一个重要的教学内容。通过访问学者的讲座，工作研讨会和阅读项目相关的建议，提案人将继续参与水稻本科生和研究生的数学研究。

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.