Ergodic Theory of Nonamenable Group 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。将时间在动力系统中的作用推广到更复杂的对称组是很自然的，也是有实际意义的。这种广义的动力学概念导致了在统计力学、数论和几何学中的应用。然而，需要新的工具，特别是在对称群不可服从的特殊情况下，这意味着边界现象太重要了，不能安全地忽略(不像整数或实数中的区间)。不可服从群自然而然地出现在数学的许多部分，如几何和数论。这个项目致力于开发分析不可服从集体行动的遍历理论所需的工具。有五个具体目标。一是继续发展SOFIC熵理论。这是Kolmogorov-Sinai熵对SOFIC群作用的广泛推广，SOFIC群是一类包含所有从属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超越顺从群体的行为。第二个目标是利用PI和Amos Nevo最近发现的技巧，为几何定义的群(例如，Lie群、CAT(0)立方群、相对双曲群)建立逐点遍历定理。第三个目标是将几何群论的工具引入到可测等价关系的研究中。第四个目标是继续分析给定群的弱等价行动类空间的结构。这个空间是罗克林引理不适用于非顺从群的分类对象。由于罗克林引理在奥恩斯坦理论中是至关重要的，所以这个话题与第一个目标密切相关。第五个目标是继续阐明给定群的定态作用的空间是什么样子的。一个主要的焦点是定态作用的新构造(通过不变的随机子群或测量的等价关系)。另一个是定态作用的一般行为。还有一个是将新技术应用于调和函数和随机游动的研究，这些主题与定常作用密切相关。

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.