Ergodic Properties of Mathematical Billiards

动力系统理论研究复杂的多组分系统的时间演化，如统计物理学中的粒子系统，化学反应动力学，大气行为（因此在天气预报的相关性），人口动态，股票市场的发展，等等。这一理论与微分方程和随机过程理论密切相关，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时间演化中的关键现象，如解对初始条件的高敏感性，有时称为混沌或混沌行为。我目前的建议的目标是调查和更好地了解一个流行的和重要的一类主要是混沌行为的数学模型，即所谓的数学台球。他们的名字是因为他们模拟了球形粒子通过弹性碰撞相互作用的物理运动。 由于上述原因，本项目主要致力于一类特殊的具有奇异性的非一致双曲动力系统，即双曲或至少部分双曲数学台球。这类系统在统计物理的严格数学基础中起着重要的作用，因此研究它们并建立它们的遍历性和统计性质越来越具有物理意义。该提案的主要部分集中在证明关于硬球系统的基本猜想的最后步骤，即所谓的“Boltzmann-Sinai遍历假设”，该假设指出任何在平坦环面上移动的（全弹性）硬球的有限系统是完全双曲和遍历的，当然，在其平凡第一积分的水平集上。这个猜想的证明（在其充分的一般性）到现在为止已经臭名昭著地经受住了任何攻击。在提出一个归纳证明（关于相互作用的球的数目的归纳）中最困难的部分到目前为止已经证明了所谓的“Anglomerate”，一个全局动力几何条件，声称在所有奇异轨道的范围内奇异轨道的几乎必然双曲性。 因为到现在为止，我已经非常接近完成这个跨越了几个人几十年艰苦研究工作的大项目的最后一步，我现在可以自信地将完成这个假设的证明作为最近提案的第一个主要部分。此外，这个计划还包含了我和我的同事N共同努力的承诺。I. 2004年，他出版了一本关于玻尔兹曼-西奈遍历假说的完整证明的综合性书籍。出版这样一本书的目的是双重的：首先，它将提供一个全面的阐述这一理论与一个统一的系统的符号，参考资料等，更容易阅读。其次，它将更彻底和清楚地解释理论的许多技术细节，以便使更广泛的动力系统研究人员能够理解。 该提案的后续部分包含了在这个方向上进一步研究的蓝图，通过将原始的玻尔兹曼-西奈假说推广到圆柱形台球，以及物理上更相关的容器中的台球，即。e.矩形框，凸域等，这些部分还解决了一些进一步的开放问题，在理论的数学台球仍然开放后，我attampts证明他们在我以前的研究期间。其清单可在本提案的详细说明中找到。

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 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. Due to the above said, this project is primarily devoted to a special class of non-uniformly hyperbolic dynamical systems with singularities, namely hyperbolic, or at least partially hyperbolic mathematical billiards. Such systems have been playing an important role in the rigorous mathematical foundation of statistical physics, so that studying them and establishing their ergodic and statistical properties is getting more and more physical relevance. The main part of the proposal focuses on a the final steps in proving a fundamental conjecture regarding hard ball 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) up until now has been notoriously withstanding any attack against it. The hardest part in presenting an inductive proof (induction on the number of interacting spheres) has been so far proving the so called "Ansatz", a global dynamical-geometric condition, claiming the almost sure hyperbolicity of singular orbits inside the realm of all singular trajectories. Since by now I got extremely close to make the closing steps in this big project that spanned over a couple decades of hard research work of several people, I can now confidently include finishing the proof of this hypothesis as the first major part of the recent proposal. Furthermore, this plan also contains my promise to join efforts with my colleague N. I. Chernov to publish a comprehensive book on the complete proof of the Boltzmann-Sinai Ergodic Hypothesis. The goal of publishing such a volume is two-fold: First, it would provide a comprehensive exposition of this theory with a unified system of notations, references, etc. for easier reading. Secondly, it would more thoroughly and clearly explain many involved technicalities of the theory, in order to make it accessible for a wider group of researchers of dynamical systems. The subsequent parts of the proposal contain blueprints 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. These sections also address some further open problems in the theory of mathematical billiards that still remain open after my attampts to prove them in my previous research period. Their list can be found in the detailed description of the current proposal.