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态系统领域的新现象。动态系统（简称“动力学”）是运动的研究，特别是由一组不变的规则（例如控制行星运动的力）决定的。已经在实验中观察到了众所周知的实验现象，例如混沌轨迹和稳定运动的混合轨迹，但从理论的角度来看远未完全理解。项目首先是动态系统的稳定性 - 即，在初始条件甚至规则本身中的扰动程度很小，也会影响进化的结果。了解稳定的强大机制是一种基本追求，研究者已经发现了几种新颖的机制。第二个主题是通用，或者松散地了解典型系统中存在哪些动态功能。第三个主题是刚性，对动态系统对称性的研究以及具有最佳对称性的系统，即所谓的理想动力学晶体。该项目的一个重要方面是在数学和相邻科学界（例如物理学）之间进一步相互作用。 PI已经在围绕粒子加速器设计的问题上进行了合作，目前正在与物理学家合作研究黑洞出现背后的量子动态。此外，PI还为动态提供了几次公开演讲，并在流行媒体上写了有关数学家的工作。 PI将在未来几年扩大这些活动。该项目为本科生和研究生提供了研究培训机会。该项目从一般的角度，尤其是关于某些叶面动力的一般性的人，尤其是那些侧重于本地群体的问题，都关注集体行动的特定家庭的刚性，从一般的角度，尤其是关于某些叶面动力的一般性的问题，尤其是那些涉及一般性的培训机会。这些问题是出于众所周知的猜想的动机，也是出于发现和探索新的动态现象的愿望。第一个问题的圈子围绕着鲍尔茨曼的原始奇异假说，以及关于稳定的真诚性的现代和相关的猜想。他们解决的基本问题是，何时可能期望动态系统是千古的。仍然开放的一个重要问题是C1 PUGH-SHUB猜想的符合性版本，研究人员将攻击。证明这一概念的策略涉及双曲​​线和部分双曲动力学研究以及扩大叶子的有趣及时的方面。第二个项目调查了部分双曲系统不稳定叶子的拓扑和统计特性，尤其是Anosov的差异性，并具有部分双曲线分裂。第三个项目涉及部分双曲线阿贝尔行动的僵化特性。在这里，SU-SOLONOMAY小组的行动起着重要的作用：在所考虑的行动中，环境，部分双曲动力和SU-So-Solononomy组的共同行动受到某些可解决的群体的约束，上面描述的已知刚性结果持有。该奖项反映了NSF的法定任务，反映了通过使用基金会的智力综述和宽阔的影响来评估支持的珍贵的支​​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.