Parabolic Dynamics

抛物线动力学

• 批准号: 0800673
• 负责人: Giovanni Forni
• 金额: $ 35.98万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800673&HistoricalAwards=false
• 关键词: Parabolic; Dynamics

We will study quantitative equidistribution and other ergodic properties, such as weak mixing/mixing, for several examples of parabolic flows, in particular billiard flows in polygons and reparametrizations of nilpotent flows. In the past several years, we have developed a method to investigate the ergodic theoretical properties of parabolic flows based on the study of invariant distributions (distributional obstructions to the existence of solutions of cohomological equations), on the Gottschalk-Hedlund theorem and on the construction of a renormalization dynamics. We have succesfully applied our ideas in several papers where we have proved bounds on the speed of ergodicity in a few fundamental cases. We intend to test our method further in other more challenging cases, which so far have been out of reach mainly beacuse no renormalization scheme is available. The problems that we intend to attack include longstanding open questions such as the question on weak mixing on invariant surfaces for flows in rational polygons and the question on the speed of (unique) ergodicity for nilflows.Our long term goal is to contribute to develop a theory on a class of weakly chaotic dynamical systems, called parabolic, which, despite some recent progress, are not yet sufficiently well understood. Parabolic motion is characterized by a power-law divergence (for instance linear, quadratic, etc.) of nearby trajectories with time. It represents an intermediate situation between strongly chaotic motion (exponentially fast divergence) and regular motion (no or extremely slow divergence). Motions at the extreme ends of the spectrum are comparatevely much better understood than parabolic motion. We will study specific questions on the dynamics of specific classes of examples, chosen for their fundamental nature and for their relevance in applications to physics, to geometry and to number theory. For instance, certain parabolic systems are relevant in the study of celestial mechanics, or as a testing ground for conjectures on the relation between classical and quantum mechanics (quantum chaos), other systems have deep connections to questions in number theory. Advances in our understanding of these systems will improve our fundamental knowledge of dynamical phenomena which are relevant for the natural sciences and for technological applications.

我们将研究一些抛物流的定量均匀分布和其他遍历性质，如弱混合/混合，特别是多边形的台球流和幂零流的再参数化。在过去的几年里，我们发展了一种研究抛物流遍历理论性质的方法，该方法基于不变分布（对上同调方程解存在性的分布障碍）、Gottschalk-Hedlund定理和重整化动力学的构造。我们已经在几篇论文中成功地应用了我们的想法，在这些论文中，我们证明了一些基本情况下遍历速度的界限。我们打算在其他更具挑战性的情况下进一步测试我们的方法，这些情况到目前为止还无法实现，主要是因为没有可用的重整化方案。我们打算解决的问题包括长期存在的开放问题，如在合理多边形中流动的不变表面上的弱混合问题和零流的（唯一）遍历速度问题。我们的长期目标是为发展一类弱混沌动力系统的理论做出贡献，称为抛物线系统，尽管最近取得了一些进展，但尚未得到充分的理解。抛物运动的特征是附近轨迹随时间的幂律发散（例如线性、二次等）。它代表了一种介于强混沌运动（指数快速散度）和规则运动（无散度或极慢散度）之间的中间状态。相对而言，光谱极端端的运动比抛物线运动更容易理解。我们将研究特定类别的例子的动力学的特定问题，选择它们的基本性质以及它们在物理，几何和数论应用中的相关性。例如，某些抛物系统与天体力学的研究有关，或者作为对经典力学和量子力学（量子混沌）之间关系的猜想的试验场，其他系统与数论中的问题有着深刻的联系。我们对这些系统的理解的进步将提高我们对与自然科学和技术应用相关的动力学现象的基本知识。