Periodic orbits of billiards and closed geodesics on flat surfaces

本项目致力于研究两个相互关联的动力系统的周期轨道：台球流和平面上的测地线流。台球是哈密顿动力学中一个非常流行和非常有趣的例子，它说明了各种动态行为。例如，经典的Birkhoff台球被证明是一个近可积的哈密顿系统，而星形区域中的台球推动了非一致双曲动力学的发展。该项目针对台球在多边形和多面体中的周期轨道的一系列问题，它们构成了台球的一个边界类。问题的中心是关于在任意多边形或多面体中存在周期台球轨道的猜想。对多边形中台球的研究自然会导致利用泰希穆勒理论中的技术来研究某些平面。该项目的另一部分解决了关于一般台球流的测度论非周期性的长期猜想。台球流是近一个世纪前由G.D.Birkhoff引入的动力学系统。这是一个有趣现象的丰富来源，其研究不会受到令人望而生畏的技术困难的阻碍。由不同类型的台球桌确定的不同类型的台球需要许多研究方法，因此，对动力学、几何学、数学物理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了解系统内的周期性运动。这个项目解决了关于多边形和多面体中的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动。在过去的二十年里，多边形台球已经发展成为一个成功地将动力系统和复杂分析这两个传统数学领域结合在一起的领域。一方面，由于研究对象与平面的联系，以及泰希穆勒理论的方法，在理解多边形中的台球方面取得了重大进展。另一方面，台球使人们对平面的研究产生了新的兴趣，这导致了泰希穆勒理论中某些问题的解决。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.