Ergodic Theory of Non-Amenable 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物理、数论和几何中的应用。然而，需要新的工具，特别是在对称群是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。该项目致力于通过将遍历理论推广到这一背景下，开发分析不可服从集体行动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熵理论，它是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。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氏链进行分类。2)对低维群的SOFIC近似进行分类(使用几何技巧)，并着眼于利用这些结果从低维群的汞齐构造第一个已知的非SOFIC群。3)通过主要研究者和NVO发展的度量等价技术和最近的L^1-陶氏反例背后的思想，探索几何定义的群(例如李群、CAT(0)立方群、相对双曲群)的逐点遍历定理的极限。还发展了新的乘法遍历定理，用于求迹von Neumann代数中的余循环值。4)将几何群论中的工具引入到度量等价关系(MERS)的研究中，用度量丛代替作用，用图代替Cayley图等。更具体的目标包括证明MERS的Tits替代方案的类比，并对低维MER的正常次等价关系进行分类。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 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.