Rigid Structures and Statistical Properties of Smooth Systems

光滑系统的刚性结构和统计特性

• 批准号: 2154796
• 负责人: Anne Wilkinson
• 金额: $ 37.57万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154796&HistoricalAwards=false
• 关键词: Rigid; Structures; Statistical; Properties; Smooth

The aim of this project is to discover new phenomena in the area of dynamical systems. Dynamical systems ("dynamics," for short) is the study of motion, and in particular motion that is dictated by an unchanging set of rules, such as the forces controlling planetary motion. Well-known experimental phenomena in dynamics such as chaotic trajectories combined with stable motion have been observed experimentally but are far from being fully understood from a theoretical perspective. The project will address several themes. The first is the stability of dynamical systems - that is, how small perturbations in initial conditions and even in the rules themselves affect the outcome of the evolution. Understanding robust mechanisms for stability is a fundamental pursuit, and the investigator has already discovered several novel such mechanisms. The second theme is genericity, or loosely to understand what dynamical features are present in a typical system. The third theme is rigidity, the study of symmetries of dynamical systems and those systems with optimal symmetries, the so-called ideal crystals of dynamics. An important aspect of the project is to further interaction between mathematical and adjacent scientific communities, such as physics. The PI has already collaborated on questions surrounding the design of particle accelerators and is currently collaborating with a physicist on studying the quantum dynamics behind the emergence of black holes. Furthermore, the PI has given several public lectures on dynamics and has written in the popular press about the work of mathematicians. The PI will expand these activities in the coming years. The project provides research training opportunities for undergraduate and graduate students.This project considers questions in smooth dynamical systems all the way from a general perspective, in particular those about genericity of certain foliation dynamics, to a local one, focused on the rigidity of specific families of group actions. These questions are motivated by well-known conjectures, but also by the desire to discover and explore new dynamical phenomena. The first circle of questions centers around Boltzmann’s original ergodic hypothesis as well as the modern and related conjectures of Pugh and Shub about stable ergodicity. The basic question they address is when one might expect a dynamical system to be ergodic. An important question that remains open is the symplectic version of the C1 Pugh-Shub conjecture, which the investigator will attack. The strategy to prove this conjecture involves interesting and timely aspects of the study of hyperbolic and partially hyperbolic dynamics and expanding foliations. A second project investigates the topological and statistical properties of the unstable foliations of partially hyperbolic systems, and in particular Anosov diffeomorphisms with a partially hyperbolic splitting. A third project concerns the rigidity properties of partially hyperbolic abelian actions. Here the action of the su-holonomy group plays an important role: in the actions considered, the joint action of the ambient, partially hyperbolic dynamics and the su-holonomy group are constrained by certain solvable groups for which known rigidity results described above hold.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的目的是发现动力系统领域的新现象动力系统（简称“动力学”）是对运动的研究，特别是由一组不变的规则决定的运动，例如。作为控制行星运动的众所周知的实验现象，例如混沌轨迹与稳定运动相结合，已经通过实验观察到，但从理论角度来看还远未得到充分理解。动力系统 - 也就是说，初始条件甚至规则本身的微小扰动如何影响演化的结果是一项基本追求，研究人员已经发现了几种新颖的此类机制。第三个主题是刚性，研究动力系统的对称性和具有最佳对称性的系统，即所谓的理想动力学晶体。数学和邻近的科学界， PI 已经就有关粒子加速器设计的问题进行了合作，目前正在与一位物理学家合作研究黑洞出现背后的量子动力学。 PI 将在未来几年扩大这些活动，为本科生和研究生提供研究培训机会。该项目从总体角度考虑平滑动力系统中的问题。特别是那些关于某些叶状动力学的通用性的问题，集中于特定群体行为的刚性。这些问题是由众所周知的猜想引起的，但也是出于发现和探索新的动力学现象的愿望。一系列问题围绕着玻尔兹曼最初的遍历假设以及 Pugh 和 Shub 关于稳定遍历性的现代相关猜想，他们解决的基本问题是人们何时期望动态系统存在。一个仍然悬而未决的重要问题是 C1 Pugh-Shub 猜想的辛版本，研究人员将对此猜想进行证明，该策略涉及双曲​​和部分双曲动力学以及扩展叶的研究的有趣且及时的方面。第二个项目研究部分双曲系统不稳定叶的拓扑和统计特性，特别是具有部分双曲分裂的阿诺索夫微分同胚。第三个项目涉及刚性特性。这里，su-完整群的作用起着重要作用：在所考虑的动作中，环境、部分双曲动力学和 su-完整群的联合作用受到某些已知可解群的约束。上述刚性结果成立。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

绝对连续性、李亚普诺夫指数和刚度 II：具有紧凑中心叶的系统

• DOI: 10.1017/etds.2021.42
• 发表时间: 2022
• 期刊: Ergodic Theory and Dynamical Systems
• 影响因子: 0.9
• 作者: AVILA, A.;VIANA, MARCELO;WILKINSON, A.
• 通讯作者: WILKINSON, A.