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。有一个完善的混沌理论，适用于其附近轨迹随时间呈指数快速发散的动力系统（天气可能是最著名的例子）。在光谱的另一端，有规律的运动，以缓慢发展的轨迹族为特征。本研究项目旨在推进弱混沌系统（通常称为抛物型）中间情况的基础研究。抛物型系统的特征是其附近轨迹最多以多项式速度随时间发散。这类系统在几何学、数论和在物理学的几个分支中产生的数学模型中尤其重要，这些分支包括固态物理学、天体力学和统计力学。例如，近年来，在数论中，在较小程度上，在几何学中，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，电子在费米表面的行为（固态物理学），行星在奇点附近的运动（天体力学），以及受限原子的运动（统计力学）都与抛物线系统有关。该项目旨在开发解决这些重要问题的新方法，并将研究生纳入基础数学研究。根据附近轨迹的散度速度可以粗略地区分动力系统。具有次指数、多项式散度的系统常被称为抛物线系统。对于一些抛物线系统，一个非常成功的方法是基于重整化，这是一个有效的工具，每当系统显示自己至少近似自相似时，这意味着它在越来越小的尺度上几乎是相同的，可能在改变坐标之后。然而，许多基本抛物型系统，包括大多数均匀流，似乎不具有自相似性质，并且没有为此类系统开发的重整化方法。因此，人们对这些系统的动力学性质了解甚少，很少进行研究。该项目旨在建立在主要研究者和合作者最近的研究结果的基础上，以研究不可重整的系统。指导原则是发展一种标度方法来推广在缺乏自相似的重整化。不可重整抛物型系统包括高阶零流和高阶零流形上的高阶阿贝尔作用。这包括半单李群商和非有理多边形上的台球上的大多数无效阿贝尔作用。

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.