Mathematical Sciences: Ergodic Theory, Differential Dynamicsand Combinatorial Number Theory

9501383 Ornstein部分研究涉及一种全新的方法来获得KAM类型的稳定性结果。这种方法填补了早期圆的微分同态研究的空白，是最优的。该方法给出了莫泽扭定理的一个新的证明；给出了旋转数属于满测度集的不变曲线的最优平滑性；给出不一定接近刚性扭转的扭转映射的不变曲线。该项目将包括完成这项工作并扩展到更高的维度。在其他方向上，研究人员将继续研究结构稳定性的统计版本以及遍历理论在组合学和数论中的应用。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？本研究将集中于两个简单的例子，涉及到圆或环的地图。还将对混沌系统进行研究。将研究遍历理论和动力系统在组合和数论问题以及统计和数据压缩中的应用。＊＊＊

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. ***