Translation Flows on Flat Surfaces and the Teichmueller Geodesic Flow

这项研究是对平板上平移流的动力学性质的研究，特别强调与概率论的联系。在以前的工作中，作者得到了关于平移流的新的极限定理。这些结果给出了遍历理论中一种全新的极限定理，并提出了一系列新的问题。提出者将继续研究平移流的极限分布。一个重要的特例是沿着伪Anosov微分同构的稳定叶的流，对此研究的一个重要工具是由作者开发和研究的对Vershik自同构上的悬浮流的平移流的符号编码，该结构发展了S.Ito的早期工作。Teichmueller流的强混沌、双曲行为控制了平移流的轻度混沌、抛物线行为。在项目的第二部分，更几何的部分，提出者将继续他对Teichmueller测地线流的混沌特性的研究。我们在股票价格、天气模式、湍流和交通拥堵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对混沌的现代认知中起着关键作用。这项拟议的研究研究了表面流动的慢混沌，这是几何学和物理学中的一类核心例子。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，提名者将继续让赖斯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.