Ergodic Theory, Differential Dynamics and Combinatorial Number Theory

摘要Katznelson/Ornstein Furstenberg，Katznelson，Ornstein和韦斯将继续和扩展几个长期项目，处理稳定性，以及极限集和其他极限行为的性质，动力系统。在关于稳定性的项目中：1）我们已经开发了一种替代方法来解决似乎只能通过KAM策略才能解决的问题。在一维和二维中，我们得到了更好的结果，无论是在光滑的共轭的圆仿射（完成开创性的结果赫尔曼和Yoccoz）和持久性扰动下的扭曲映射的某些不变的圆（Moser定理和扩展），以及光滑的曲线不变下的表面仿射。 这些方法 作为研究高维不变环面的存在性和光滑性的工具是很有前途的。 2)建立一个流是伯努利流，不仅将它完全刻画为一个抽象测度流，而且也是证明统计稳定性的主要步骤。我们正在正式的过程中的一般标准的伯努利流。在极限行为的项目中，我们提到了多项式时间的多重递归的研究和分形维数和其他极限集的处理问题。 动力系统是自然科学和社会科学中进化现象的数学模型。他们还提供了重要的工具，统计，计算机科学，组合学等，在这个广泛的和不断扩大的主题，我们直接我们的主要努力的问题处理稳定性：在何种程度上是一个系统的感知属性强大，也就是说，将持续如果系统受到扰动（修改）有点。 在任何一种实际应用中，当数据和参数只是近似地已知时，人们很难依赖不稳定的性质。我们的主要项目之一是找到保证系统下不变曲面存在的条件。在正确的维度上，这样的（超）表面将空间分隔成不变的区域，或者说是轨道无法逃脱的区域，如果表面在微扰下持续存在，那么区域也是如此。换句话说，我们可以保证某些条件永远不会演变成其他（特定）类型。

Abstract Katznelson/Ornstein Furstenberg, Katznelson, Ornstein, and Weiss will continue and expand several long-term projects dealing with stability, as well as properties of limit sets and other limit behavior, of dynamical systems. Among the projects on stability: 1) We have devloped an alternative approach to problems that seemed accessible only via the KAM strategy. In dimensions one and two we obtained finer results both in the smoothness of the conjugation of circle diffeomorphism (completing the pioneering results of Herman and Yoccoz) and persistence under perturbation of twist maps of certain invariant circles (Moser's theorem and extensions), as well as smoothness of curves invariant under surface diffeomorphism. These methods look promising as tools for the study of existence and smoothness of invariant tori in higher dimensions. 2) Establishing that a flow is Bernoulli not only characterizes it completely as an abstract measure theoretic flow, but also serves as the main step in proving statistical stability. We are in the process of formalizing general criteria for Bernoullicity of flows. Among the projects on limit behavior we mention the study of multiple recurrence with polynomial times and problems dealing with dimensions of fractals and other limit sets. Dynamical systems are mathematical models of evolutionary phenomena in both natural and social sciences. They also provide important tools in statistics, computer sciences, combinatorics, etc. In this broad and broadening subject, we direct our main effort to problems dealing with stability: to what extent are the perceived properties of a system robust, that is, will persist if the system is perturbed (modified) somewhat. In any sort of practical application, where the data and the parameters are known only approximately, one can hardly rely on properties which are not stable. One of our main projects is to find conditions which guarantee the existence of surfaces invariant under the system. In the right dimension, such (hyper) surfaces separate the space into invariant regions, or domains from which orbits cannot escape, and if the surface persists under perturbation-so do the domains. In other words, we can guarantee that some conditions will never evolve into some other (specific) type.