Aspects of Sofic Entropy and Algebraic Actions

索菲克熵和代数作用的方面

• 批准号: 1600802
• 负责人: Benjamin Hayes
• 金额: $ 10.9万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2018-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600802&HistoricalAwards=false
• 关键词: Aspects; Sofic; Entropy; Algebraic; Actions

The mathematical theory of dynamical systems concerns itself with states of certain physical systems (e.g., the amount of a certain gas inside a room, the population of a species, the temperature inside a building) as they evolve in time. It happens to be useful and natural to make "time" itself more abstract and replace it with any other discrete system of symmetries, say of some object (such symmetries form what is a called a discrete group). In the case of a very chaotic physical system, one cannot effectively predict its future state. This fact gives rise to an important quantity, the entropy of the system, that measures how unpredictable it is. Entropy was originally defined directly within an information theoretic context, but it turns out that, if one views it instead from the perspective of a statistical mechanics formalism, the scope of entropy theory is increased to encompass a very large class of groups known as "sofic groups." Seen through this new lens, entropy becomes a measurement of how many finitary approximations the physical system has. The general question of how well an infinitary process can be approximated by a finitary one is a fundamental question of scientific inquiry, and the class of sofic groups includes many natural examples coming from geometry and number theory. This project also sheds light on the connection to physics that is inherent in its statistical mechanical and information theoretic origins. Moreover, the subject has proven to have interesting links to diverse areas of mathematics, including the following: functional analysis, ergodic theory, combinatorics, and operator algebras. Many of the links are direct consequences of the principal investigator's research. The project is part of ongoing activity to strengthen and understand such connections.This project revolves around three main problems. The first is to understand the orbit equivalence consequences of positive or complete positive entropy for actions of nonamenable groups. For example, the principal investigator has shown that actions with complete positive entropy are strongly ergodic and that positive entropy actions are not weakly compact. He seeks to prove that actions with compete positive entropy are solidly ergodic, as defined by Chifan and Ioana, which will give further indication of the Bernoulli-like behavior that these actions exhibit. Part of the interest in this objective is that it places the theory of entropy for actions of nonamenable groups in stark contrast to the study of entropy for actions of amenable groups, where it is impossible to derive any orbit equivalence consequences of entropy. These consequences of complete positive entropy apply to algebraic actions (i.e., actions on compact groups by automorphisms), for the principal investigator can already demonstrate that algebraic actions related to invertible convolution operators have complete positive entropy. A second problem is concerned with investigating the entropy theory of algebraic actions of equivalence relations, which is joint work with Lewis Bowen and Dylan Airey. One aim of this part of the project is to generalize the principal investigator's results on entropy and Fuglede-Kadison determinants from the group case to the equivalence relation case. Lastly, the principal investigator will further the connections between dynamics and operator algebras by defining sofic entropy for actions on operator algebras.

动态系统的数学理论与某些物理系统的状态有关（例如，房间内某种气体的数量，物种的种群，建筑物内部的温度）随着时间的推移而发展。使“时间”本身更加抽象并用任何其他离散的对称系统（例如某些对象）（这种对称性形式形成一个所谓的离散群体），恰好是有用和自然的。在一个非常混乱的物理系统的情况下，人们无法有效地预测其未来状态。这一事实产生了重要数量，即系统的熵，从而衡量了它的不可预测性。熵最初是在信息理论背景下直接定义的，但事实证明，如果人们从统计力学形式主义的角度看待它，则增加了熵理论的范围，以包含一大批被称为“ Sofic群体”的群体。从这个新镜头中可以看出，熵成为了物理系统具有多少个临界近似值的测量。关于无限过程的一般问题，一个限制的过程是一个科学探究的基本问题，而索非人群体的类别包括许多来自几何学和数字理论的自然示例。该项目还阐明了与物理学的联系，这是其统计机械和信息理论起源中固有的。此外，事实证明，该主题与数学的不同领域具有有趣的联系，包括以下内容：功能分析，千古理论，组合学和操作员代数。许多链接都是主要研究者研究的直接后果。 该项目是加强和了解这种联系的持续活动的一部分。该项目围绕三个主要问题。首先是了解对不合同群体的作用的阳性或完全正熵的轨道等效后果。例如，主要研究者表明，具有完全正熵的作用是强烈的邪恶的，而正熵作用并不弱紧凑。他试图证明，正如Chifan和Ioana所定义的那样，具有竞争积极熵的行动是坚实的颈，这将进一步表明这些行动所表现出的类似Bernoulli的行为。这一目标的一部分是，它赋予了不可统的群体行为的熵理论，与对熵群体的熵的研究形成了鲜明的对比，在这种情况下，不可能得出任何熵的轨道等效后果。完全积极熵的这些后果适用于代数行动（即，通过自动形态对紧凑型组的行为），因为主要研究者已经可以证明，与可逆卷积操作员有关的代数行动具有完全积极的熵。第二个问题是研究等效关系代数行为的熵理论，这是与刘易斯·鲍恩（Lewis Bowen）和迪伦·艾尔（Dylan Airey）的联合工作。该项目的这一部分的目的是将主要研究者在熵和fuglede-kadison的决定因素上概括为从组案例到等效关系案例。最后，主要研究者将通过定义对操作员代数操作的SOFIC熵来进一步进一步的动力学和操作员代数之间的联系。