A comprehensive program in modern dynamics: flexibility, rigidity and low complexity systems

拟议的计划将推进动力系统现代结构理论的主要领域的研究。一个经典的动力系统是一个空间，以及该空间本身的映射或流动。有时也有一组空间的对称性。如果这组对称足够复杂，那么这个系统往往是“僵化的”。这意味着在不破坏整个系统的情况下，对称性的重要动力学性质是不可能改变的。相比之下，只有一个地图或流程的经典系统通常是相当灵活的。这项研究计划旨在了解这种灵活性。在双曲动力学中，经典光滑系统的行为(微分同胚和流)与高阶阿贝尔群的作用之间存在着反差。后者在过去十年中一直是PI研究的中心，表现出显著的行为僵化。另一方面，古典系统是相当灵活的。灵活性计划的主要挑战是，数值动力学不变量只能在极少数情况下精确计算，大多数情况下是代数起源的。大多数已知的结构都是微扰的，因此充其量只能覆盖模型允许的值的一小部分，或者更经常地，甚至不是这样，因为齐次系统通常是“极端的”。因此，建立灵活性需要在大族中使用非微扰或大微扰结构，以涵盖不变量的可能值。这需要结合李亚普诺夫特征指数理论、光滑遍历理论和几何学的方法。虽然主要问题相对容易向熟悉现代动力系统理论的关键概念的数学家和科学家的相当广泛的受众解释，但观点是相当新的，并已由PI在过去几年中明确提出。第二个研究方向是进一步发展高阶阿贝尔群作用的刚性程序。在“非统一度量刚性”的名称下，技术和洞察力的结合导致了对最大排序动作的几乎确定的描述。从测度论的角度来看，这些行为被证明是算术性质的，如果有适当的限制，几何观点也是如此。对于能够承载这种作用的流形的拓扑来说，存在着不平凡的含义。所提出的研究包括进一步研究最大秩作用的拓扑学，以及将算术性结果推广到更广泛的作用类，其秩与环境流形的维度无关。其他研究方向涉及零熵系统，包括抛物型(多项式复杂性)和椭圆型(低复杂性)系统的几个研究领域。

The proposed program will advance research across the principal areas of the modern structural theory of dynamical systems. A classical dynamical system is a space together with a map or flow of the space into itself. Sometimes there is also a group of symmetries of the space. If the group of symmetries is complicated enough, the system is often "rigid". This means the important dynamical properties of the symmetries cannot be changed without destroying the entire system. In contrast, the classical systems, with a single map or flow, are often quite flexible. This research program is aimed at understanding this flexibility. In hyperbolic dynamics there is a contrast between behavior of classical smooth systems (diffeomorphisms and flows) and actions of higher rank abelian groups. The latter, which have been at the center of the PI's research during the last decade, exhibit remarkable rigidity of behavior. Classical systems, on the other hand, are quite flexible. The principal challenge of the flexibility program is that numerical dynamical invariants can only be precisely calculated in very few cases, mostly of algebraic origin. Most known constructions are perturbative and hence at best would allow to cover a small neighborhood of the values allowed by the model, or more often, not even that, since homogeneous systems are often "extremal". So establishing flexibility calls for non-perturbative or large perturbation constructions in large families to cover possible values of invariants. This calls for a combination of methods from the theory of Lyapunov characteristic exponents, smooth ergodic theory, and geometry. While the principal problems are relatively easy to explain to a fairly broad audience of mathematicians and scientists familiar with the key notions of the modern theory of dynamical systems, the point of view is quite new and has been explicitly formulated by the PI within the last few years. A second direction of research is the further development of the rigidity program for actions of higher rank abelian groups. The combinations of techniques and insights that goes under the name "non-uniform measure rigidity" resulted in the almost definitive description of maximal rank actions. Those actions turn out to have an arithmetic nature from measure-theoretic and, with proper qualifications, also geometric point of view. There are non-trivial implications for the topology of manifolds that can carry such actions. The proposed research includes further study of the topology of maximal rank actions, as well as an extension of arithmeticity results to broader classes of actions whose rank is not related to the dimension of the ambient manifold. Additional directions of research deal with zero entropy systems and include several areas of study in both parabolic (polynomial complexity) and elliptic (low complexity) systems.