Ergodic Theory, Differential Dynamics and Combinatorial Number Theory

Katznelson/Ornstein Furstenberg、Katznelson、Ornstein和Weiss将继续和扩展几个关于动力系统的稳定性以及极限集的性质和其他极限行为的长期项目。在有关稳定的项目中：1)我们开发了一种替代方法来解决似乎只有通过KAM战略才能解决的问题。在一维和二维空间中，我们得到了圆微分同胚共轭的光滑性(完善了Herman和Yoccoz的开创性结果)和某些不变圆的扭转映射在扰动下的持久性(Moser定理和推广)，以及曲面微分同胚下不变曲线的光滑性。这些方法有望成为研究高维不变环面的存在性和光滑性的工具。2)证明一个流是Bernoulli流不仅完全是抽象测度论的流，而且是证明统计稳定性的主要步骤。我们正在正式确定流动的伯努利性质的一般标准。在关于极限行为的项目中，我们提到了多项式次数的多重递推的研究，以及处理分形图和其他极限集的维度的问题。动力系统是自然科学和社会科学中进化现象的数学模型。它们还提供了统计学、计算机科学、组合学等方面的重要工具。在这个广泛而广泛的学科中，我们将主要精力集中在处理稳定性的问题上：系统的可感知属性在多大程度上是健壮的，即，如果系统受到某种扰动(修改)，系统将持续存在。在任何类型的实际应用中，当数据和参数仅为近似已知时，人们很难依赖不稳定的性质。我们的主要工作之一是找到保证系统下存在曲面不变的条件。在正确的维度中，这样的(超)表面将空间分成不变的区域，或轨道无法逃脱的区域，如果表面在扰动下持续存在--区域也是如此。换句话说，我们可以保证某些情况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.