Beyond Renormalization in Parabolic Dynamics

动力系统理论研究确定性系统的长期行为。一个经典的例子是行星的运动。动力系统出现在所有科学领域，例如物理学、生物学和经济学。此外，动力系统的方法还可以应用于研究其他数学领域的问题，包括几何和数论。有一个久负盛名的混沌理论适用于其附近轨迹随时间以指数级速度偏离的动力系统(天气可能是最著名的例子)。在光谱的另一端，有规则的运动，其特征是缓慢演变的轨迹家族。这一研究项目旨在推进弱混沌系统(通常称为抛物线)的中间情形的基础研究。抛物线系统的特征是附近的轨迹随时间最多以多项式的速度发散。这类系统在几何学、数论和物理学的几个分支中产生的数学模型中尤其重要，包括固体物理、天体力学和统计力学。例如，近年来，在数论中，在较小程度上，在几何中，许多问题被重新表述(有时被解决)为关于某些抛物线流动的动力学的问题。在物理学中，电子在费米表面的行为(固态物理)、行星在奇点附近的运动(天体力学)和受限原子的运动(统计力学)都与抛物系统有关。这个项目旨在开发解决这些重要问题的新方法，并将让研究生参与基础数学研究。动力系统可以根据附近轨迹的发散速度大致区分开来。附近轨道具有次指数、多项式发散的系统通常被称为抛物线。对一些抛物系统的一种非常成功的方法是基于重整化，这是一个有效的工具，只要系统显示自己至少是近似自相似的，这意味着当以越来越小的尺度观察时，可能在改变坐标之后，它几乎是相同的。然而，许多基本抛物系统，包括大多数均匀流，似乎不具有自相似性质，并且还没有为这类系统发展重整化方法。因此，人们对这些系统的动力学性质知之甚少，也很少成为研究的对象。该项目旨在以主要研究人员和合作者的最新结果为基础，研究不可重整化系统。其指导原则是发展一种标度方法，在没有自相似的情况下推广重整化。不可重整化抛物系统包括高阶零流和高阶零流形上的高阶Abelian作用。这包括非有理多边形中的半单李群和台球的商上的大多数单幂Abel作用。

The theory of dynamical systems studies the long term behavior of deterministic systems. A classical example is the motion of the planets. Dynamical systems arise in all areas of sciences, for instance in physics, biology, and economics. In addition, the methods of dynamical systems can be applied to study problems in other fields of mathematics, including geometry and number theory. There is a well-established theory of chaos that applies to dynamical systems whose nearby trajectories diverge exponentially rapidly with time (the weather is perhaps the most famous example). At the other end of the spectrum, there is regular motion, characterized by slowly-evolving families of trajectories. This research project aims to advance fundamental research on the intermediate case of weakly chaotic systems, often called parabolic. Parabolic systems are characterized by the property that nearby trajectories diverge at most polynomially rapidly with time. Systems of this kind are especially important in geometry, number theory, and mathematical models arising in several branches of physics, including solid-state physics, celestial mechanics, and 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 behavior of electrons at the Fermi surface (solid-state physics), the motion of planets near a singularity (celestial mechanics), and the motion of a confined atom (statistical mechanics) are related to parabolic systems. This project aims to develop new approaches to these important questions and will involve graduate students in the fundamental mathematical research.Dynamical systems can be roughly distinguished according to the speed of divergence of nearby trajectories. Systems with sub-exponential, polynomial divergence of nearby orbits are often called parabolic. An extremely successful approach to some parabolic systems is based on renormalization, which is an effective tool whenever the system reveals itself to be at least approximately self-similar, meaning it is nearly the same when viewed at smaller and smaller scales, possibly after changing of coordinates. However, many fundamental parabolic systems, including most homogeneous flows, do not seem to possess self-similarity properties, and no renormalization method has been developed for such systems. As a consequence, the dynamical properties of these systems are very poorly understood and are rarely the subject of investigation. This project aims to build on recent results by the principal investigator and collaborators to investigate non-renormalizable systems. The guiding principle is to develop a scaling method to generalize renormalization in the absence of self-similarity. Non-renormalizable parabolic systems include higher-step nilflows and higher rank Abelian actions on higher step nilmanifolds. This includes most unipotent Abelian actions on quotients of semisimple Lie groups and billiards in non-rational polygons.