Ergodic Theory Beyond Amenability

经典动力学研究系统如何随时间变化。遍历理论关注的是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。将时间在动力系统中的作用推广到更复杂的对称群是很自然的，而且具有实际意义。这种广义的动力学概念在统计物理、数论和几何中都有应用。然而，新的工具是必要的，特别是在特殊情况下，当组的对称性是不可服从的，这意味着边界现象太重要，不能安全地忽略。不可服从群自然出现在许多数学领域，如几何和数论。该项目关注的是通过将遍历理论推广到这一背景，开发分析不可服从群体行为的统计行为所需的工具。本项目为研究生提供研究训练机会。本课题的研究目标是：(1)可测量等价关系（MERs）是由群的作用产生的。对于将经典的遍历理论扩展到不可服从的群体行为，MERs既是一个工具，也是一个有趣例子的来源。本项目旨在通过对低维MERs的正态亚等价关系进行分类，并从几何群论中寻找对象的mer -类似物，来发展MERs的结构。(2) Sofic熵理论是将经典熵推广到Sofic群的行为，Sofic群是一类包括可服从群和剩余有限群的群。它相对较新。具体目标包括：确定在轨道等价下熵不变的条件，发展局部紧化的sofic熵理论，对自由群上的混合马尔可夫链进行分类，并确定sofic熵如何依赖于sofic近似的选择。(3)将遍历理论推广到不可服从群的一个主要工具是局部逼近，其中群对自身的作用是由有限集合上的部分作用序列局部平均逼近的。没有已知的不能使用此工具的情况。该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群和低维群的行为。(4) Sofic近似符合广义的Benjamini-Schramm （BS）收敛框架，即考虑有限图序列或紧致度量空间，极限对象为随机点图或空间。自然序列包括随机平移面、二次微分和测量层积。该项目将确定这些序列的bs极限，并将它们与已知对象（如curen - werner马尔可夫三角剖分和高斯解析函数）联系起来。该奖项反映了美国国家科学基金会的法定使命，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.