Entropy Theory for Non-Amenable Groups

不服从群体的熵理论

• 批准号: 1955090
• 负责人: Brandon Seward
• 金额: $ 15.72万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1955090&HistoricalAwards=false
• 关键词: Entropy; Theory; Non; Amenable; Groups

Originally, dynamical systems comprised the study of systems that change over time, such as the motion of planets in the solar system, variations in the weather, traffic patterns, and populations of species. As this field of study has grown, it has expanded to include more general types of systems that, rather than changing with time, are transformed by more general groups of symmetries. This would apply, for instance, to the study of information spread over a large and highly regular network. In this general form, dynamical systems has significant connections to other branches of mathematics and science, such as number theory, geometry, combinatorics, statistical physics, operator algebras, and data science. One fundamental tool in the study of dynamical systems is the concept of entropy, which is a measurement of how chaotic or unpredictable a system is. This notion was first introduced in 1958 by Kolmogorov in the case of time transformations but was soon after extended to systems with an amenable group of transformations, i.e. systems where boundary phenomena are mostly inconsequential. For systems having a non-amenable group of transformations, the concept of entropy was formalized only a decade ago. This new terrain of entropy theory is still in its early stages and has not yet become as well understood or as powerful a tool as in classical settings. The goal of this project, broadly speaking, is to further the development of entropy theory in the context of non-amenable groups so that it may become a widely applicable tool like its classical version.The overall goal of this project is to improve our understanding of entropy and widen its scope of applications. In particular, the principal investigator intends to address whether sofic entropy and Rokhlin entropy coincide for principal algebraic actions of sofic groups, to investigate whether tailed-percolation entropy is a lower bound to sofic entropy, and to determine whether completely-positive-entropy actions are uniformly mixing. Historically, the most potent applications of entropy theory have been to the study of Bernoulli shifts. Specifically, entropy played a significant role in the development of Ornstein theory - a theory that completely characterized and classified Bernoulli shifts over countable amenable groups up to isomorphism. In recent work, the principal investigator established a generalization of Sinai's factor theorem to all countably infinite groups. As part of this project, the PI plans to utilize and expand upon this recent work to make initial steps towards an Ornstein theory. As part of this goal, the principal investigator plans to further generalize Sinai's factor theorem to the settings of f-invariant entropy and naive entropy, to study the properties of finitely determined processes, to investigate whether the high temperature Ising model over the rank 2 free group is isomorphic to a Bernoulli shift, to determine when Popa factors of Bernoulli shifts are isomorphic to Bernoulli shifts, and to investigate the orbit equivalence class of Bernoulli shifts over free groups.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.

最初，动力系统包括研究随时间变化的系统，如太阳系中行星的运动、天气、交通模式和物种数量的变化。随着这一研究领域的发展，它已经扩展到包括更一般类型的系统，这些系统不是随着时间而变化，而是被更一般的对称性群所改变。例如，这将适用于对分布在大型且高度规则的网络上的信息的研究。在这种一般形式下，动力系统与数学和科学的其他分支，如数论、几何学、组合学、统计物理学、算子代数和数据科学有着重要的联系。研究动力系统的一个基本工具是熵的概念，它是衡量一个系统有多混乱或不可预测的指标。这一概念最早是由Kolmogorov于1958年在时间变换的情况下提出的，但不久之后被扩展到具有一组服从变换的系统，即边界现象大多无关紧要的系统。对于具有不可服从的一组变换的系统，熵的概念仅仅在十年前才被形式化。熵理论的这一新领域仍处于早期阶段，还没有像经典环境中那样被很好地理解或成为一种强大的工具。从广义上讲，这个项目的目的是为了进一步发展不可服从群背景下的熵理论，使其成为一个像经典版本一样广泛适用的工具。本项目的总体目标是增进我们对熵的理解，扩大其应用范围。特别地，主要研究者打算讨论对于SOFIC群的主要代数作用，SOFIC熵和Rokhlin熵是否重合，尾部渗流熵是否是SOFIC熵的下界，以及完全正熵作用是否一致混合。从历史上看，熵理论最有效的应用是研究伯努利位移。具体地说，熵在Ornstein理论的发展中发挥了重要作用--Ornstein理论完全刻画并分类了可数服从群上的Bernoulli变换，直到同构。在最近的工作中，主要研究者建立了西奈因子定理在所有可数无限群上的推广。作为这个项目的一部分，PI计划利用和扩展这项最新的工作，向Ornstein理论迈出第一步。作为这一目标的一部分，首席研究员计划将Sinai的因子定理进一步推广到f-不变熵和朴素熵的设置下，研究有限确定过程的性质，调查秩2自由群上的高温Ising模型是否同构于Bernoulli移位，确定Bernoulli移位的Popa因子何时与Bernoulli移位同构，并研究自由群上Bernoulli移位的轨道等价类别。这一裁决反映了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。