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Ergodic Theory of Non-Amenable Group Actions

不服从群体行为的历经理论

基本信息

批准号：
1900386
负责人：
Lewis Bowen
金额：
$ 32.97万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-06-01 至 2022-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900386&HistoricalAwards=false
关键词：
Ergodic Theory Non Amenable Group

项目摘要

Classical dynamics studies how systems change in time. Ergodic theory focuses 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 research goals of this project are: 1) Continue developing sofic entropy theory, which is a vast generalization of Kolmogorov-Sinai entropy to actions of sofic groups, a class of groups that contains all amenable groups and residually finite groups. One ambitious goal is to classify the mixing Markov chains over free groups using asymptotic geometric invariants of associated model spaces. 2) Classify sofic approximations of low-dimensional groups (using geometric techniques), with an eye towards using the results to construct the first known non-sofic groups from amalgams of low-dimensional groups. 3) Explore the limits of pointwise ergodic theorems for geometrically defined groups (e.g. Lie groups, CAT(0) cubulated groups, relatively hyperbolic groups) via the measured-equivalence techniques developed by the principal investigator and Nevo and the ideas behind the recent L^1-counterexample of Tao. Also develop new multiplicative ergodic theorems for cocycles taking values in tracial von Neumann algebras. 4) Import tools from geometric group theory into the study of measured equivalence relations (MERs) through the use of metric bundles in place of actions, graphings in place of Cayley graphs and so on. More specific goals include proving analogs of the Tits alternative for MERs and classifying the normal sub-equivalence relations of low-dimensional MERs.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.

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。本课题的研究目标是：1)继续发展sofic entropy theory，这是将Kolmogorov-Sinai熵推广到sofic groups的行为，sofic groups是一类包含所有可服从群和剩余有限群的群。一个雄心勃勃的目标是利用相关模型空间的渐近几何不变量对自由群上的混合马尔可夫链进行分类。2)对低维群的sofic近似进行分类（使用几何技术），着眼于利用结果从低维群的混合物中构造已知的第一个非sofic群。3)探索几何定义群（如李群，CAT(0)计算群，相对双曲群）的点遍历定理的极限，通过由首席研究员和Nevo开发的测量等效技术以及最近Tao的L^1反例背后的思想。也发展了在迹迹冯·诺伊曼代数中取值的环的新的乘法遍历定理。4)将几何群论中的工具引入到可测量等价关系（MERs）的研究中，方法是用度量束代替作用，用图解代替凯利图等。更具体的目标包括证明MERs的Tits替代品的类似物和分类低维MERs的正常亚等价关系。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。