A comprehensive program in modern dynamics

本项目旨在跨越现代动力系统结构理论的几个主要领域：双曲（均匀和非均匀）、部分双曲、抛物线和椭圆的研究方向取得进展。它将特别强调各种刚性现象以及在这些领域中几种强大方法的适用性。在过去的几年中，首席研究员和他的合作者引入了新的方法和见解，这些方法和见解在理解高阶阿贝尔群作用的不变测度的刚性和轨道结构的可微刚性方面取得了重大进展。基于这些方法所取得的进展，在数论中的丢芬图近似问题上产生了富有成效的应用，并首次提供了大类别动作的不变几何结构存在的例子。项目的研究方向包括：高阶阿贝尔群的部分双曲代数作用的可微刚性规划的完成，酉群表示理论和KAM方法在单幂齐次作用刚性中的应用。开始研究非齐次抛物型系统、非标准kam型不变曲线定理和零拓扑熵的低维系统。动力系统是自然科学和社会科学范围内过程的时间演化的数学模型。它们在核心数学学科中也有着惊人的广泛应用，尤其是在几何和数论的各个领域。在许多情况下，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，或者是更一般的性质，这是由关键的数学概念“群”所捕获的。阻碍对重要模型进行全面理解的关键困难可以描述如下：虽然建立产生混沌行为的某些初始条件的存在性通常相对容易，但证明混沌行为存在于大多数或许多动力系统中（在所谓的相空间中的体积意义上）超出了现有甚至预期的数学方法的范围。首席研究员和他的合作者发现，对于具有多维时间的系统，在某些非常普遍的条件下，这个困难是可以克服的：乍一看似乎只保证某些混沌轨道存在的拓扑或动力学性质的全局条件，实际上意味着存在一组这样的轨道，它们在相空间中占据正体积，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产生重要影响。

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.