Limit Theorems in Dynamical Systems

本项目开发方法，以进行精确计算与确定性动力系统的混沌行为相关的统计量。具有不稳定行为的确定性系统出现在从原子到行星尺度的广泛应用中。在许多情况下，不稳定性是由附近轨迹的指数发散引起的。除非精确地知道初始条件，否则不能精确地预测这类系统的长时间行为，因此有必要使用轨道的统计描述。该项目还涉及混合动力学行为的简单模型，其中系统表现出与规则和随机行为相关的特征。这些模型作为解释更复杂现象的范例出现。此外，该项目还为研究生的研究培训和指导提供了机会。主要研究者还将通过讲座和论文写作来传播研究结果。该项目侧重于动力系统的统计特性。全文共分四个部分。首先，将考虑动力系统极限定理的精确渐近性。尖锐极限定理估计最近已被证明的主要研究者和其他人有应用动力学。本文将研究非自治系统的一个局部极限定理。主要研究者计划对经典渐近展开式的障碍进行分类，并开发适用于经典展开式失败的情况的新工具。要考虑的第二个主题是指数混合。指数混合系统的锐极限定理将通过探索相空间中的几何结构来研究。接下来，首席研究员将通过构建具有奇异特性的动力系统来研究统计特性的灵活性。考虑的最后一个主题是平均理论。首席研究员以前作出了几项贡献的研究平均系统的双曲快速运动。这些技术将进一步调查，着眼于扩展到准周期快速dynamics.This奖项的设置反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This project develops methods to carry out precise computations of statistical quantities associated with chaotic behavior of deterministic dynamical systems. Deterministic systems with unstable behavior appear in a wide range of applications, ranging from atomic to planetary scales. In many cases the instability is caused by exponential divergence of nearby trajectories. Long time behavior of such systems cannot be predicted exactly unless the initial conditions are known precisely, and consequently it is necessary to use statistical descriptions of the orbits. The project also addresses simple models of mixed dynamical behavior where systems exhibit features associated with both regular and stochastic behaviors. Such models occur as paradigmatic examples explaining more complicated phenomena. In addition, the project provides opportunities for research training and mentoring of graduate students. The principal investigator will also disseminate results through lectures and expository writing.The project focuses on the statistical properties of dynamical systems. It consists of four parts. First, precise asymptotics in limit theorems for dynamical systems will be considered. Sharp limit theorem estimates have recently been shown by the principal investigator and others to have applications in dynamics. A local limit theorem for non-autonomous systems will be studied. The principal investigator plans to classify obstructions to classical asymptotic expansions, and to develop new tools applicable in cases where the classical expansions fail. A second topic to be considered is exponential mixing. Sharp limit theorems for exponentially mixing systems will be studied through an exploration of geometric structures in the associated phase space. Next, the principal investigator will study flexibility of statistical properties through the construction of dynamical systems with exotic properties. A final topic under consideration is averaging theory. The principal investigator has previously made several contributions to the study of averaging for systems with hyperbolic fast motion. These techniques will be investigated further with an eye towards extensions to the setting of quasi-periodic fast 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.