Effective Ergodic Theory: Parabolic and Hyperbolic

该项目致力于研究一类动力系统的长期行为，称为抛物系统，它显示附近轨道的次指数（多项式）发散。与显示轨道指数发散的双曲动力系统相比，抛物线系统的理解要少得多。双曲系统也将被研究，只要它们作为研究抛物系统的辅助工具出现，抛物系统是主要的焦点。研究将强调与应用相关的定量方面。该项目旨在提高我们对动力系统的基础知识，并开发数学研究的新思想和新方法，这些新思想和新方法可能应用于几何和数论，从而间接应用于其他科学学科。此外，由于抛物线行为出现在哈密顿力学中，该项目可以直接推进我们对经典物理系统（多边形台球、统计力学、行星运动）的简单数学模型的理解。研究培训和指导学生和博士后是该项目的一个重要目标，特别关注代表性不足的群体的成员。研究者将在几个方向上进行研究（平移流的有效弱混合，Ruelle渐近，特征变化的动力学），并将继续研究长期开放的问题，如非全球单幂流的有效遍历性（有效Ratner理论），高次多项式Weyl和的最优界，平面上测地流的遍历理论和非理性多边形上台球的遍历理论，以及光滑抛物流的遍历和谱理论。抛物系统的一种重要研究方法，通常称为重整化方法，用辅助双曲系统的动力学研究取代了对抛物系统动力学的直接研究，双曲系统的动力学研究更容易用双曲理论（不变流形，李雅普诺夫指数）的方法来理解。由于最近对抛物动力学和双曲动力学之间通过重正化的关系的兴趣，研究者从不稳定叶的有效（多项式）等价分布的角度扩展了他的研究，从相关的指数衰减和双曲系统的Ruelle渐近性的角度，并将进一步研究抛物动力学中的有效等价分布与双曲动力学中的有效混合之间的相互作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project is devoted to the study of long-term behavior of a class of dynamical systems, called parabolic, which display sub-exponential (polynomial) divergence of nearby orbits. In contrast with hyperbolic dynamical systems, which display exponential divergence of orbits, parabolic systems are much less understood. Hyperbolic systems will also be studied insofar they appear as auxiliary tools in the study of parabolic ones, which are the main focus. The research will emphasize quantitative aspects that are relevant for applications. The project aims to advance our fundamental knowledge of dynamical systems and to develop new ideas and new methods of mathematical investigation with potential applications to geometry and number theory and thus indirectly to other scientific subjects. Also, since parabolic behavior appears in Hamiltonian mechanics, the project can directly advance our understanding of simple mathematical models of classical physical systems (billiards in polygons, statistical mechanics, planetary motions). Research training and mentoring of students and postdocs is an important goal of the project, with particular attention to members of underrepresented groups.The investigator will carry out research in several directions (effective weak mixing of translation flows, Ruelle asymptotics, dynamics on character varieties) and will continue work on longstanding open questions such as on effective ergodicity for non-horospherical unipotent flows (effective Ratner theory), optimal bounds on Weyl sums for higher degree polynomials, the ergodic theory of geodesic flows on flat surfaces and of billiards in non-rational polygons, as well as the ergodic and spectral theory of smooth parabolic flows. An important approach to parabolic systems, often called renormalization approach, replaces the direct studies of the parabolic dynamics with that of an auxiliary hyperbolic system, which is easier to understand by methods of hyperbolic theory (invariant manifolds, Lyapunov exponents). Motivated by recent interest in the relation between parabolic and hyperbolic dynamics via renormalization, the investigator has broadened his research with work on exponential decay of correlations and Ruelle asymptotics for hyperbolic systems from the point of view of the effective (polynomial) equidistribution of unstable foliations, and will further pursue research on the interplay between effective equidistribution in parabolic dynamics and effective mixing in hyperbolic dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.