Parabolic Dynamics

我们将研究定量等分布和其他遍历性质，如弱混合/混合，抛物流的几个例子，特别是台球流多边形和重新参数化的幂零流。在过去的几年中，我们已经开发出一种方法来调查的遍历性的理论性质的抛物流的基础上的研究不变的分布（分布障碍存在的解决方案的上同调方程），Gottschalk-Hedlund定理和建设的重整化动力学。我们已经成功地应用我们的想法在几篇论文中，我们已经证明了在一些基本情况下的遍历速度的界限。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，主要是因为没有重整化方案。我们打算攻击的问题，包括长期存在的开放问题，如问题上的弱混合不变曲面的合理多边形的流动和速度的问题（唯一）遍历nilflows. We的长期目标是有助于发展一类弱混沌动力系统的理论，称为抛物，其中，尽管最近取得了一些进展，尚未得到充分的理解。抛物线运动的特征是幂律发散（例如线性、二次等）。与时间的关系它代表了一种介于强烈混沌运动（指数快速发散）和规则运动（无发散或极慢发散）之间的中间状态。在光谱的极端末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具体问题，选择他们的基本性质和他们的相关性在应用物理学，几何和数论。例如，某些抛物系统与天体力学的研究有关，或者作为经典力学和量子力学（量子混沌）之间关系的实验场，其他系统与数论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有关的动力学现象的基本知识。

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.