Open Problems in the Theory of Mathematical Billiards

本项目致力于研究一类特殊的混沌动力系统，即双曲（和聚焦，或“椭圆”）数学台球。在双曲情况下，它们是具有奇点的非均匀双曲动力系统的原型。这类系统在统计物理严密的数学基础中起着重要的作用，因此对其强混合特性的研究和建立越来越具有物理意义。该提案的第一部分着重于关于这类动力系统的一个基本猜想，即所谓的“玻尔兹曼-西奈遍历假设”，该假设指出，在平坦环面上运动的（完全弹性）硬球的有限系统是完全双曲的和遍历的，当然，在其平凡的第一积分的水平集中。到目前为止，这一猜想的证明（在其全部普遍性方面）一直经受住了任何针对它的攻击。本提案的第一个主要部分直接针对这一猜想。第二部分是该方向进一步研究的蓝图，将原始Boltzmann-Sinai假设推广到圆柱形台球，以及物理上更相关的容器（即矩形盒，凸域等）中的台球。第三部分解决了M. Herman提出的一个问题，即如果运动发生在一个紧凑的凸域，那么硬球动力学的散射（双曲）效应是否最终胜过凸边界的聚焦效应。第四部分针对高维严格色散非均匀双曲台球n阶奇异集的基本复杂性问题。这些问题的答案对于进一步研究此类系统的精细统计特性至关重要。第五部分提出了一些关于高维台球的遍历性质的基本问题，其中硬核相互作用势被光滑的旋转对称相互作用势所取代。最后，结语部分针对平面台球理论中一些尚待解决的问题。动力系统理论研究复杂的多组分系统的时间演化，如统计物理中的粒子系统、化学中的反应动力学、大气的行为（因此与天气预报有关）、人口动力学、股票市场的发展等。从本质上讲，这一理论与微分方程和随机过程理论密切相关，并且部分源于微分方程和随机过程理论。动力系统理论的一个有趣的特点是，它帮助我们更好地理解上述系统在时间演化中的关键现象，如解对初始条件的高灵敏度，有时被称为混沌，或混沌行为。我目前的提议旨在调查和更好地理解一类流行的、重要的、大多表现混乱的数学模型，即所谓的数学台球。它们之所以得名，是因为它们模拟了球状粒子通过弹性碰撞相互作用的物理运动。

This project is devoted to a special class of chaotic dynamical systems, namely hyperbolic (and focusing, or `elliptic') mathematical billiards. In the hyperbolic case they are the prototype examples of non-uniformly hyperbolic dynamical systems with singularities. Such systems have been playing an important role in the rigorous mathematical foundation of statistical physics, so that the study and establishing their strong mixing properties is getting more and more physical relevance. The first part of the proposal focuses on a fundamental conjecture regarding this family of dynamical systems, namely the so called `Boltzmann-Sinai Ergodic Hypothesis', which states that any finite system of (totally elastic) hard spheres moving on a flat torus is fully hyperbolic and ergodic, of course, on the level set of its trivial first integrals. The proof of this conjecture (in its full generality) has been so far notoriously withstanding any attack against it. The first major part of the present proposal directly targets this conjecture. The second part is a blueprint for further research in this direction by generalizing the original Boltzmann-Sinai Hypothesis to cylindric billiards, and billiards in physically more relevant containers, i.e. rectangular boxes, convex domains, etc. The third part addresses a question posed by M. Herman, which asks if the scattering (hyperbolic) effect of the hard ball dynamics eventually prevails over the focusing effect of the convex boundary, if the motion takes place in a compact, convex domain. In the fourth part the fundamental complexity problems are targeted for the $n$-step singularity sets of higher dimensional, strictly dispersive, non-uniformly hyperbolic billiards. The answers to those questions are pivotal in further studies of the fine statistical properties of such systems. Part five poses some basic questions and problems concerning the ergodic properties of high-dimensional billiards, in which the hard core interaction potential is replaced by a smooth, rotational symmetric one. Finally, the closing part aims at some open problems in the theory of planar billiards.The theory of dynamical systems studies the time-evolution of complicated,multi-component systems, like particle systems in statistical physics, reaction kinetics from chemistry, the behavior of the atmosphere (hence the relevance in weather forecasting), population dynamics, developments on the stock market, etc. By nature, this theory is closely related to - and is partly arising from - the theory of differential equations and stochastic processes. An interesting feature of the theory of dynamical systems is that it helps us better understand such crucial phenomena in the time evolution of the above mentioned systems, as the high sensitivity of the solution to the initial conditions, sometimes referred to as chaos, or chaotic behavior. My present proposal targets the investigation and better understanding of a popular and important class of mostly chaotically behaving mathematical models, namely the so called mathematical billiards. They got their name after the fact that they model the physical motion of ball shaped particles interacting with each other via elastic collisions.