权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Ergodic Theory of Nonamenable Group Actions

无名群体行为的历经理论

基本信息

批准号：
1500389
负责人：
Lewis Bowen
金额：
$ 37.34万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-05-15 至 2019-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500389&HistoricalAwards=false
关键词：
Ergodic Theory Nonamenable Group Actions

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 modelling 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 mechanics, number theory, and geometry. However, new tools are needed, especially in the particular case when the group of symmetries is non nonamenable, which means that boundary phenomena are too significant to be safely ignored (unlike intervals in the integers or real numbers). Nonamenable 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 ergodic theory of nonamenable group actions. There are five specific goals. The first is to continue developing sofic entropy theory. This 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 major aim of this research is to determine to what extent Ornstein theory can be extended beyond actions of amenable groups. The second goal is to establish pointwise ergodic theorems for geometrically defined groups (e.g. Lie groups, CAT(0) cubulated groups, relatively hyperbolic groups) using techniques recently discovered by the PI and Amos Nevo. The third goal is to import tools from geometric group theory into the study of measured equivalence relations. The fourth goal is to continue to analyzing the structure of the space of weak equivalence classes of actions of a given group. This space serves as a classifying object for the ways in which the Rokhlin Lemma fails for nonamenable groups. Because the Rokhlin Lemma is of crucial importance in Ornstein theory, this topic is intimately connected with the first goal. The fifth goal is to continue clarifying what the space of stationary actions of a given group "looks like.'' One major focus is on new constructions of stationary actions (via invariant random subgroups or measured equivalence relations). Another is on generic behavior of stationary actions. Yet another is to apply the new techniques to the study of harmonic functions and random walks; topics deeply connected with stationary actions.

经典动力学研究系统如何随时间变化。遍历理论主要研究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力学、数论和几何学中得到了应用。然而，需要新的工具，特别是在对称群是不可靠的特殊情况下，这意味着边界现象太重要而不能被安全地忽略（不像整数或真实的数中的区间）。非顺从群自然地出现在数学的许多部分，如几何和数论。这个项目是关于发展分析不可服从的群体行为的遍历理论所需的工具。有五个具体目标。一是继续发展热力学熵理论。这是柯尔莫哥洛夫-西奈熵对sofic群作用的一个广泛推广，sofic群是一类包含所有顺从群和剩余有限群的群。这项研究的一个主要目的是确定奥恩斯坦理论可以在多大程度上扩展到顺从群体的行动之外。第二个目标是使用PI和Amos Nevo最近发现的技术为几何定义的群（例如李群，CAT（0）cubulated群，相对双曲群）建立点态遍历定理。第三个目标是从几何群论导入工具到测量等价关系的研究中。第四个目标是继续分析给定群的弱等价作用类空间的结构。这个空间可以作为一个分类对象，说明罗克林引理在不服从群的情况下是如何失效的。因为罗克林引理在奥恩斯坦理论中至关重要，所以这个主题与第一个目标密切相关。第五个目标是继续澄清一个给定群体的静止行动空间是什么样子的。一个主要的焦点是新的静态动作的构造（通过不变的随机子群或测量等价关系）。另一个是关于静态动作的一般行为。另一个是将新技术应用于调和函数和随机游动的研究;这些主题与静态行为密切相关。