A comprehensive program in modern dynamics

现代动力学综合课程

• 批准号: 1002554
• 负责人: Anatole Katok
• 金额: $ 30.02万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1002554&HistoricalAwards=false
• 关键词: comprehensive; program; modern; dynamics

This project aims to make progress in several directions of research across the principal areas of the modern structural theory of dynamical systems: hyperbolic (uniform and nonuniform), partially hyperbolic, parabolic, and elliptic. It will put particular emphasis on various kinds of rigidity phenomena and on applicability of several powerful methods across those areas. New methods and insights have been introduced by the principal investigator and his collaborators over the past years that have led to significant advances in the understanding of rigidity of invariant measures and the differentiable rigidity of orbit structure for actions of higher rank Abelian groups. The progress achieved based on these methods has engendered fruitful applications to Diophantine approximation problems in number theory and provided the first examples of the existence of invariant geometric structure for large classes of actions. Avenues of pursuit in the project include the following: tame and wild behavior in the classification of Anosov systems up to a differentiable conjugacy, global rigidity of hyperbolic measures for actions of higher rank Abelian groups and applications to the Zimmer program, completing the program of differentiable rigidity of partially hyperbolic algebraic actions of higher rank Abelian groups, applications of the theory of unitary group representations and the KAM method to rigidity of unipotent homogeneous actions, commencement of a comprehensive program of investigation of nonhomogeneous parabolic systems, nonstandard KAM-type invariant curve theorems, and low-dimensional systems with zero topological entropy.Dynamical systems serve as mathematical models for the time-evolution of processes that range across the spectrum of the natural and social sciences. They also have a surprisingly broad range of applications in core mathematical disciplines, most particularly in various areas of geometry and number theory. Within many of these contexts the term "time" does not necessarily connote the usual one-dimensional time but can be multidimensional or of an even more general nature that is captured by the key mathematical concept of a "group." The crucial difficulty that impedes efforts in obtain a comprehensive understanding of important models can be described as follows: while it is often relatively easy to establish existence of some initial conditions that produce chaotic behavior, proving that chaotic behavior exists in most or many dynamical systems (in the sense of so-called volume in the phase-space) is beyond the reach of present or even anticipated mathematical methods. The principal investigator and his collaborators have discovered that, for systems with multidimensional time and under certain very general conditions, this difficulty can be overcome: global conditions of a topological or dynamical nature that at first glance would seem to guarantee only the existence of some chaotic orbits actually imply the existence of a set of such orbits that fill a positive volume in phase-space or, in certain cases, even provide a complete description of the orbit structure. This seemingly technical fact could have important implications for physics and engineering.

该项目旨在在动力系统的现代结构理论的主要领域的几个研究方向取得进展：双曲（均匀和非均匀），部分双曲，抛物和椭圆。它将特别强调各种刚性现象和在这些领域的几个强大的方法的适用性。在过去的几年里，主要研究者和他的合作者引入了新的方法和见解，这使得对高阶阿贝尔群作用的不变测度刚性和轨道结构的可微刚性的理解取得了重大进展。基于这些方法所取得的进展已经产生了富有成效的应用丢番图逼近问题在数论中，并提供了第一个例子的存在不变的几何结构的大型类的行动。该项目的执行途径包括：Anosov系统分类中的驯服和狂野行为直到可微共轭，高阶Abel群作用的双曲测度的整体刚性及其在Zimmer程序中的应用，完成高阶Abel群部分双曲代数作用的可微刚性程序，酉群表示理论和KAM方法在幂单齐次作用刚性上的应用，开始研究非齐次抛物系统的综合方案，非标准KAM型不变曲线定理，动力学系统是自然科学和社会科学中各种过程时间演化的数学模型。他们也有一个令人惊讶的广泛的应用在核心数学学科，特别是在几何和数论的各个领域。在许多这样的背景下，术语“时间”并不一定意味着通常的一维时间，而是可以是多维的，或者具有更一般的性质，这是由“群”的关键数学概念所捕获的。“妨碍全面了解重要模式的努力的关键困难可描述如下：虽然建立产生混沌行为的某些初始条件的存在通常相对容易，证明大多数或许多动力系统中存在混沌行为（在所谓的相空间体积的意义上）是目前甚至预期的数学方法所无法达到的。首席研究员和他的合作者发现，对于具有多维时间的系统，在某些非常一般的条件下，这个困难是可以克服的：拓扑或动力学性质的全局条件乍一看似乎只保证某些混沌轨道的存在，实际上暗示存在一组这样的轨道，它们填充相空间中的正体积，或者，在某些情况下，甚至提供了轨道结构的完整描述。这个看似技术性的事实可能对物理学和工程学有重要的影响。