Periodic orbits of billiards and closed geodesics on flat surfaces

本计画致力于研究两个相关动力系统的周期轨道，即平面结构曲面上的台球流与测地线流。台球是哈密顿动力学中一个非常流行和有趣的例子，它说明了各种动力学行为。例如，经典的伯克霍夫台球被证明是一个近似可积的哈密顿系统，而星形域中的台球则推动了非一致双曲动力学的发展。该项目的目标是一系列问题的周期性轨道的台球在多边形和多面体，这构成了一个边界类的台球。这些问题的中心是关于任意多边形或多面体中周期台球轨道存在性的一个猜想。在多边形台球的研究自然导致研究某些平面利用技术从泰希穆勒理论。该项目的另一部分解决了一个长期存在的关于一般台球流的测度论非周期性的猜想。D.伯克霍夫 它是有趣现象的丰富来源，其研究不会受到令人生畏的技术困难的阻碍。由不同类型的台球桌决定的不同类型的台球需要大量的研究方法，因此，对动力学，几何学，数学物理学和光谱理论的发展产生了强烈的影响。 根据庞加莱的方法，研究任何动力系统的首要任务是理解系统内的周期运动。本计画针对多边形与多面体中周期性台球轨道的一些问题。它需要研究密切相关的动力系统，即平面上的测地线流。 在过去的二十年里，多边形台球已经发展成为一个领域，成功地结合了两个传统领域的数学，动力系统和复杂的分析。一方面，在多边形台球的理解取得了重大进展，因为该主题的联系，以平面，从而对泰希穆勒理论的方法。另一方面，这是台球，创造了一个新的兴趣研究的平面，然后导致解决某些问题的Teichmuller理论。该项目旨在进一步扩大这种互动。

This project is devoted to the study of periodic orbits of two related dynamical systems, billiard flows and geodesic flows on surfaces with flat structure. Billiards is a popular and extremely interesting example in Hamiltonian dynamics that illustrates a variety of dynamic behaviors. For example, classical Birkhoff billiards was shown to be a near-integrable Hamiltonian system, while billiards in star-shaped domains powered the development of nonuniformly hyperbolic dynamics. The project targets a series of problems concerning periodic orbits of billiards in polygons and polyhedra, which constitute a borderline class of billiards. The problems are centered on a conjecture about the existence of a periodic billiard orbit in an arbitrary polygon or polyhedron. The study of billiards in polygons naturally leads to the study of certain flat surfaces by exploiting techniques from Teichmuller theory. Another part of the project addresses a long-standing conjecture on the measure-theoretic aperiodicity of general billiard flows.Billiard flow is a dynamical system introduced almost a century ago by G. D. Birkhoff. It is a rich source of interesting phenomena whose study is not obstructed by daunting technical difficulties. Different kinds of billiards determined by varying types of billiard tables have required a host of methods of study and, as a result, have had a strong impact on developments in dynamics, geometry, mathematical physics, and spectral theory. According to Poincare's approach, the primary task in the study of any dynamical system is to understand the periodic motions within the system. This project addresses a number of problems concerning periodic billiard orbits in polygons and polyhedra. It requires the study of closely related dynamical systems, namely, geodesic flows on flat surfaces. In the last twenty years, billiards in polygons has evolved into a field that successfully weds two traditional areas of mathematics, dynamical systems and complex analysis. On the one hand, significant progress in understanding billiards in polygons has been made because of the linkage of the subject to flat surfaces, and thus to the methods of Teichmuller theory. On the other hand, it was billiards that created a renewed interest in the study of flat surfaces, which then led to the solution certain problems in Teichmuller theory. This project aims to expand this interaction even further.