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和一个映射或空间流到它自身。 有时也有一组空间对称。 如果对称群足够复杂，系统往往是“刚性的”。 这意味着对称性的重要动力学性质不能在不破坏整个系统的情况下改变。 相比之下，具有单个映射或流的经典系统通常非常灵活。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（微分同胚和流）的行为与更高秩阿贝尔群的作用之间存在对比。后者在过去十年中一直是PI研究的中心，表现出显着的行为刚性。另一方面，经典系统非常灵活。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，主要是代数起源。大多数已知的构造是微扰的，因此最多允许覆盖模型所允许的值的一个小邻域，或者更经常地，甚至不是这样，因为齐次系统通常是“极值”的。 因此，建立灵活性要求在大的家庭中的非微扰或大扰动结构，以涵盖可能的不变量值。这就要求从理论上的李雅普诺夫特征指数，光滑遍历理论，和几何的方法相结合。虽然主要的问题是相对容易解释一个相当广泛的观众数学家和科学家熟悉的关键概念的现代理论的动力系统，观点是相当新的，并已明确制定了PI在过去几年。第二个研究方向是进一步发展的刚性计划行动的高阶阿贝尔集团。在“非均匀测度刚性”的名义下，技术和见解的结合导致了对最大秩动作的几乎确定的描述。从测量理论来看，这些行为具有算术性质，并且在适当的条件下，也具有几何观点。对于可以承载这种作用的流形的拓扑，存在着非平凡的含义。建议的研究包括进一步研究的拓扑结构的最大秩的行动，以及扩展的算术结果，以更广泛的行动类，其排名是不相关的尺寸的环境流形。其他研究方向涉及零熵系统，包括抛物（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.