Mathematical Sciences: Ergodic Theory, Differential Dynamicsand Combinatorial Number Theory

小行星9501383 部分研究涉及一个全新的方法来稳定性结果的KAM类型。这种方法填补了早期工作中的空白，圆的同构，是最佳的。该方法给出了一个新的证明的Moser扭曲定理;给出了最佳光滑不变曲线的旋转数属于一个“好”集的充分措施;给出不变曲线的扭曲地图不一定接近刚性扭曲。该项目将涉及完成这项工作，并扩展到更高的维度。在其他方向的调查人员将继续他们的工作统计版本的结构稳定性和应用遍历理论的组合和数论。 一些研究集中在这样一个问题上：如果我们只有关于控制动力系统的方程的近似信息，我们能对动力系统的长期行为说些什么？本研究将集中在两个简单的例子，涉及地图的圆或环。 还将对混沌系统进行研究。遍历理论和动力系统的应用组合和数论问题，以及统计和数据压缩将进行调查。 ***

9501383 Ornstein Part of the research involves a completely new approach to stability results of the KAM type. This approach fills the gaps in earlier work on diffeomorphisms of the circle and are optimal. The method gives a new proof of the Moser twist theorem; gives optimal smoothness for invariant curves whose rotation numbers belong to a "good" set of full measure; gives invariant curves for twist maps which are not necessarily close to a rigid twist. The project will involve completing this work and extending to higher dimensions. In other directions the investigators will continue their work on a statistical version of structural stability and applications of ergodic theory to combinatorics and number theory. Some of the research is focused on the question: What can we say about the long-term behavior of a dynamical system, if we have only approximate information about the equations that govern them? This research will focus on two simple examples involving maps of the circle or an annulus. A study will also be made of chaotic systems. Applications of ergodic theory and dynamical systems to combinatorial and number theoretic problems as well as statistics and data compression will be investigated. ***