Ergodic Theory Beyond Amenability

经典动力学研究系统如何随时间变化。遍历理论关注于动力系统的统计行为。遍历理论的应用非常广泛：从交通建模到航空航天工程和人口动力学。将动力系统中时间的作用推广到更复杂的对称群是很自然的，也是很有实际意义的。这种广义的动力学概念在统计物理学、数论和几何学中得到了应用。然而，新的工具是需要的，特别是在特定的情况下，当组的对称性是不服从的，这意味着边界现象是太重要了，不能安全地忽略。非顺从群自然地出现在数学的许多部分，如几何和数论。本计画的目的是将遍历理论推广到非服从性群体行为的统计行为分析，并发展相关的工具。本计画为研究生提供研究训练的机会，其研究目标为：（1）群体行为所产生的测量等价关系。MER既是一个工具，也是一个有趣的例子来源，用于将经典遍历理论扩展到不服从的群作用。本计画的目的是透过分类低维MER的正常次等价关系，以及从几何群论中寻找对象的MER-类比，来发展MER的结构。(2)Sofic熵理论是经典熵对sofic群作用的推广，sofic群是一类包含顺从群和剩余有限群的群。它是相对较新的。具体目标包括：确定条件下熵是不变的轨道等价，发展一个局部紧凑版本的sofic熵理论，分类混合马尔可夫链自由群和确定如何sofic熵取决于sofic近似的选择。(3)将遍历理论扩展到非顺从群的一个主要工具是sofic近似，其中群对自身的作用平均地局部近似为有限集合上的部分作用序列。没有已知的情况下，该工具不能使用。这个项目的一个主要目标是通过修改最近用于解决Connes嵌入猜想的技术来找到非sofic群和低维群的作用。(4)Sofic近似适合于Benjamini-Schramm（BS）收敛的更广泛的框架，其中考虑有限图或紧度量空间的序列，并且极限对象是随机点图或空间。自然发生的序列包括随机平移表面、二次微分和测量叠层。该项目将确定这些序列的BS极限，并将其与已知对象（如Curien-Werner Markovian三角测量和高斯分析函数）联系起来。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Classical dynamics studies how systems change in time. Ergodic theory focusses on the statistical behavior of dynamical systems. Applications of ergodic theory are widespread: from traffic modeling to aerospace engineering and population dynamics. It is natural and of practical importance to generalize the role of time in a dynamical system to more complicated groups of symmetries. This generalized notion of dynamics leads to applications in statistical physics, number theory and geometry. However, new tools are needed especially in the particular case when the group of symmetries is non-amenable, which means that boundary phenomena are too significant to be safely ignored. Non-amenable groups naturally arise in many parts of mathematics such as geometry and number theory. This project is concerned with developing the tools needed to analyze the statistical behavior of non-amenable group actions by generalizing ergodic theory to this context. The project provides research training opportunities for graduate students.The research goals of this projects are: (1) Measured equivalence relations (MERs) arise from actions of groups. MERs are both a tool and a source of interesting examples for extending classical ergodic theory to non-amenable group actions. This project aims to develop the structure of MERs by classifying the normal subequivalence relations of low-dimensional MERs and finding MER-analogs of objects from geometric group theory. (2) Sofic entropy theory is a generalization of classical entropy to actions of sofic groups, a class of groups including amenable and residually finite groups. It is relatively new. Specific goals include: determine conditions under which entropy is invariant under orbit-equivalence, develop a locally compact version of sofic entropy theory, classify mixing Markov chains over free groups and determine how sofic entropy depends on the choice of sofic approximation. (3) A major tool for extending ergodic theory to non-amenable groups is sofic approximation, in which the action of the group on itself is approximated locally on average by a sequence of partial actions on finite sets. There are no known cases when this tool cannot be used. A major goal of this project is to find non-sofic groups and actions of low-dimensional groups by modifying recent techniques used to solve Connes’ Embedding Conjecture. (4) Sofic approximation fits into the broader framework of Benjamini-Schramm (BS) convergence, in which one considers sequence of finite graphs or compact measured metric spaces and the limit object is a random pointed graph or space. Naturally occurring sequences include random translation surfaces, quadratic differentials and measured laminations. This project will determine the BS-limit of these sequences and relate them to known objects such as the Curien-Werner Markovian triangulation and Gaussian Analytic Functions.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.