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

该计划将推进动力系统现代结构理论主要领域的研究。一个经典的动力系统是一个空间，连同空间本身的地图或流。有时也有一组对称的空间。如果对称组足够复杂，则系统通常是“刚性”的。这意味着在不破坏整个系统的情况下，不可能改变对称性的重要动力学特性。相比之下，具有单一地图或流程的经典系统通常非常灵活。本研究项目旨在了解这种灵活性。在双曲动力学中，经典光滑系统（微分同态和流动）的行为与高阶阿贝尔群的行为之间存在着对比。后者在过去十年中一直是PI的研究中心，表现出显著的行为僵化。另一方面，经典系统是相当灵活的。柔性程序的主要挑战是，数值动态不变量只能在极少数情况下精确计算，大多数是代数起源。大多数已知的构造都是微扰的，因此最多只能覆盖模型允许的值的一个小邻域，或者更常见的是，甚至不能覆盖，因为均匀系统通常是“极端的”。因此，建立灵活性需要在大族中使用非摄动或大摄动结构来覆盖不变量的可能值。这需要李亚普诺夫特征指数理论、光滑遍历理论和几何方法的结合。对于熟悉现代动力系统理论的关键概念的广大数学家和科学家来说，主要问题相对容易解释，但这种观点是相当新的，并且在过去几年中由PI明确地表述出来。第二个研究方向是进一步发展高阶阿贝尔群行动的刚性程序。“非均匀测量刚性”下的技术和见解的组合导致了对最大等级行动的几乎确定的描述。从测度论的角度来看，这些行为具有算术性质，在适当的条件下，从几何的角度来看，也具有算术性质。对于可以执行这种操作的流形的拓扑结构有重要的含义。提出的研究包括进一步研究最大秩行动的拓扑结构，以及将算法结果推广到秩与环境流形维数无关的更广泛的行动类别。其他研究方向涉及零熵系统，包括抛物线（多项式复杂度）和椭圆（低复杂度）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.