Interval Exchange Maps and Dynamics in Moduli Space

本文研究了紧Riemann曲面的模空间上Teichmueller测地流的动力学性质，特别强调了它在几何上的应用。主要的工具是由区间交换映射的重整化理论给出的流的符号编码。PI将这一技术应用于周期Teichmueller测地线及其相关计数问题的研究。第二个更具分析性的部分是对Teichmueller测地线流的混沌行为的定量研究，推广了PI得到的中心极限定理。其主要目的是证明Teichmueller测地线表现为布朗运动的轨迹。利用区间交换映射的重整化，PI旨在将紧致负曲流形上的经典结果推广到Teichmueller空间的背景下，而Teichmueller空间既不是紧致的，也不是Gromov双曲的。动力混沌理论在现代科学中起着至关重要的作用。这一理论起源于1890年彭加莱的S关于行星运动稳定性的回忆录，现已应用于气候预测和海洋学、统计物理学和计算机科学中的复杂性理论。

AbstractThe proposed research is a study of dynamical properties of the Teichmueller geodesic flow on the moduli space of compact Riemann surfaces, with a special emphasis on applications in geometry. The main tool is a symbolic coding for the flow, given by the theory of renormalization for interval exchange maps. This technique is applied by the PI to the study of periodic Teichmueller geodesics and related counting problems. The second, more analytic, part of the project is a quantitative study of chaotic behavior of the Teichmueller geodesic flow,extending the Central Limit Theorem obtained by the PI. The broad aim is to show that Teichmueller geodesics behave as trajectories of Brownian motion. Using renormalization for interval exchange maps, the PI aims to carry over classical results on compact negatively curved manifolds to the context of the Teichmueller space, which is neither compact nor Gromov hyperbolic.The theory of dynamical chaos plays a crucial role in modern science. Originating in 1890 in Poincare''s memoir on the stability of planetary motion, today the theory has applications in climate prediction and oceanography, statistical physics and complexity theory in computer science. The proposed research is a study of dynamical chaos in the moduli space of compact Riemann surfaces.