Ergodic Theory of Parabolic Flows

抛物线流的遍历理论

• 批准号: 1201534
• 负责人: Giovanni Forni
• 金额: $ 33.5万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201534&HistoricalAwards=false
• 关键词: Ergodic; Theory; Parabolic; Flows

Dynamical systems can roughly be classified according to the speed of divergence of nearby trajectories. Systems with sub-exponential, polynomial divergence of nearby orbits are often called parabolic. Parabolic dynamical systems arise in many mathematical models of scientific phenomena and in applications of dynamical systems to other branches of mathematics, in particular to number theory and geometry. The main long term goal of the research is to explore how far the ideas and methods developed in the study of the above-mentioned systems, based on renormalization and harmonic analysis, can be generalized towards a theory of parabolic dynamics. In the near future, Forni plans to organize his research around three main themes: ergodic theory on finite-area translation surfaces, compact and non-compact, and of billiards in polygons; ergodic theory of smooth time-changes of nilflows and horocycle flows; renormalization and quantitative equidistribution of (higher-step) nilflows and Weyl sums. Among the questions considered are long-standing open problems such as ergodicity of billiards in polygons, the spectral type of smooth time-changes of horocycle flows and optimal bounds on Weyl sums.The proponent's research is in the field of dynamical systems, that is, the study of motion of a deterministic system with time. The most classical example is the motion of the planets, but dynamical systems arise in all area of sciences, for instance in physics, biology, economics. In addition, the methods of dynamical systems can be applied to study problems in other fields of mathematics, in particular in geometry and number theory, as it has been done very successfully in recent decades as dynamical systems has moved closer and closer to the core of pure mathematical research (without abandoning its original strong connection to natural sciences and to applied mathematics). In dynamical systems there is a well-established theory of chaos which applies to systems whose nearby trajectories diverge exponentially fast with time (the weather is perhaps the most famous example.) At the other end of the spectrum, there is regular motion, characterized by trajectories that move all together. The proponent's goal is to advance fundamental research on the intermediate case of weakly chaotic systems, that is, systems that have some measure of chaotic behavior, but whose nearby trajectories diverge at most polynomially fast with time. The motion of an idealized billiard ball on a polygonal table is an example. Systems of this kind, called parabolic, are especially important in applications to geometry, number theory and to mathematical models coming from several branches of physics: solid-state physics, celestial mechanics, statistical mechanics. For instance, in recent years in number theory and, to a lesser extent, in geometry many questions have been reformulated (sometimes solved) as questions on the dynamics of certain parabolic flows. In physics, the motion of an electron on the the so-called Fermi surface of an atom (solid-state physics), the motion of planets near a singularity (celestial mechanics), the motion of an atom in a box (statistical mechanics) are related to parabolic systems. This project also has an important training component with the goal of forming researchers with wide mathematical knowledge.

动力系统可以根据附近轨迹的散度速度粗略地分类。具有次指数、多项式散度的系统常被称为抛物线系统。抛物动力系统出现在许多科学现象的数学模型中，也出现在动力系统在其他数学分支，特别是数论和几何中的应用中。研究的主要长期目标是探索在上述系统的研究中发展起来的思想和方法，基于重整化和谐波分析，可以推广到抛物动力学理论的程度。在不久的将来，Forni计划围绕三个主要主题组织他的研究：有限面积平移曲面的遍历理论，紧实和非紧实，以及多边形台球；零流和环流光滑时变的遍历理论（高阶）nilflow和Weyl和的重整化和定量均分。其中考虑的问题是长期悬而未决的问题，如多边形台球的遍历性，环流的光滑时变的谱型和Weyl和的最优界。倡议者的研究领域是动力系统，即研究确定性系统随时间的运动。最经典的例子是行星的运动，但动力系统出现在所有科学领域，例如物理学、生物学、经济学。此外，动力系统的方法可以应用于研究其他数学领域的问题，特别是几何和数论，因为近几十年来，随着动力系统越来越接近纯数学研究的核心（而不放弃其与自然科学和应用数学的原始强烈联系），它已经做得非常成功。在动力系统中，有一个完善的混沌理论，适用于其附近轨迹随时间呈指数级快速发散的系统（天气可能是最著名的例子）。在光谱的另一端，有规律的运动，其特征是轨迹一起运动。支持者的目标是推进弱混沌系统中间情况的基础研究，即系统具有一定程度的混沌行为，但其附近轨迹最多随时间以多项式速度发散。一个理想的台球在多边形桌上的运动就是一个例子。这种系统被称为抛物线，在几何学、数论和来自物理学几个分支的数学模型的应用中尤为重要：固体物理学、天体力学、统计力学。例如，近年来，在数论和较小程度的几何中，许多问题被重新表述（有时被解决）为某些抛物线流动的动力学问题。在物理学中，电子在所谓的原子费米表面上的运动（固态物理学），行星在奇点附近的运动（天体力学），盒子里原子的运动（统计力学）都与抛物线系统有关。这个项目还有一个重要的训练组成部分，目标是培养具有广泛数学知识的研究人员。