Ergodic Theory, Differential Dynamics and Combinatorial Number Theory

Furstenberg、Katznelson、Ornstein和Weiss将继续和扩展几个长期项目，研究动力系统的稳定性、极限集的性质和其他极限行为。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有通过KAM策略才能解决的问题。在维一和维二中，我们在圆的微分同态共轭的平滑性（完成Herman和Yoccoz的开创性成果）和某些不变圆的扭转映射摄动下的持久性（Moser定理和推广）以及曲面微分同态下曲线的不变性平滑性方面都获得了更精细的结果。这些方法有望成为研究高维不变环面存在性和光滑性的工具。2)确定流是伯努利流，不仅使其完全成为抽象的测度理论流，而且是证明统计稳定性的主要步骤。我们正在对流动的伯努利性的一般标准进行形式化。在关于极限行为的项目中，我们提到了多项式次的多次递归的研究，以及处理分形维数和其他极限集的问题。动力系统是自然科学和社会科学中进化现象的数学模型。它们也为统计学、计算机科学、组合学等提供了重要的工具。在这个广泛和不断扩大的主题中，我们将主要精力放在处理稳定性的问题上：系统的感知属性在多大程度上是鲁棒的，也就是说，如果系统受到某种干扰（修改），它将持续存在。在任何一种实际应用中，如果数据和参数只是近似地已知，人们就很难依赖不稳定的性质。我们的主要课题之一是寻找保证系统下曲面不变存在的条件。在正确的维度中，这样的（超）曲面将空间分割成不变的区域，或者轨道无法逃离的区域，如果表面在扰动下持续存在，那么这些区域也会持续存在。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.