Entropy Theory for Non-Amenable Groups

最初，动力系统包括对随时间变化的系统的研究，如太阳系中行星的运动、天气的变化、交通模式和物种数量。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，而不是随着时间的推移而变化，而是由更一般的对称群转化。例如，这将适用于研究在一个大而高度规则的网络上传播的信息。在这种一般形式下，动力系统与数学和科学的其他分支有重要的联系，如数论、几何、组合学、统计物理、算子代数和数据科学。研究动力系统的一个基本工具是熵的概念，它是对系统混乱或不可预测程度的测量。这个概念最初是在1958年由Kolmogorov在时间变换的情况下引入的，但很快就被扩展到具有可服从变换组的系统，即边界现象大多无关紧要的系统。对于具有一组不可服从的变换的系统，熵的概念在十年前才被形式化。熵理论的这个新领域仍处于早期阶段，还没有像在经典环境中那样被充分理解或成为强大的工具。总的来说，这个项目的目标是进一步发展熵理论在非服从群体的背景下，使它可能成为一个广泛适用的工具，就像它的经典版本一样。这个项目的总体目标是提高我们对熵的理解，扩大它的应用范围。特别是，首席研究员打算解决sofic熵和Rokhlin熵是否重合于sofic群的主要代数作用，研究尾渗熵是否为sofic熵的下界，并确定完全正熵作用是否均匀混合。从历史上看，熵理论最有效的应用是对伯努利位移的研究。具体来说，熵在奥恩斯坦理论的发展中发挥了重要作用——奥恩斯坦理论完全描述和分类了伯努利位移在可数可服从群上的同构。在最近的工作中，首席研究员建立了西奈因子定理的推广到所有可数无限群。作为这个项目的一部分，PI计划利用和扩展这项最近的工作，向奥恩斯坦理论迈出第一步。作为这一目标的一部分，主要研究者计划进一步将Sinai的因子定理推广到f不变熵和朴素熵的设置中，研究有限确定过程的性质，研究2阶自由群上的高温Ising模型是否与伯努利位移同构，确定伯努利位移的Popa因子何时与伯努利位移同构。并研究自由群上伯努利位移的轨道等价类。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。

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.