Aspects of Sofic Entropy and Algebraic Actions

动力系统的数学理论涉及某些物理系统的状态(例如，房间内的某种气体的数量、物种的数量、建筑物内的温度)，因为它们随着时间的演变而变化。将“时间”本身变得更抽象，并用任何其他离散的对称系统来取代它，比如某些对象(这样的对称形成了一个所谓的离散群)，这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它的未来状态。这一事实产生了一个重要的量，即系统的熵，它衡量了系统的不可预测性。熵最初是在信息论的背景下直接定义的，但事实证明，如果人们从统计力学形式主义的角度来看待它，熵理论的范围就扩大到包括一类非常大的被称为“SOFIC群”的群。通过这个新的透镜，熵变成了对物理系统有多少有限近似的度量。有限过程逼近无穷过程的一般问题是科学研究的一个基本问题，而SOFIC群的类包括许多来自几何和数论的自然例子。该项目还阐明了统计、机械和信息理论起源所固有的与物理学的联系。此外，这门学科已被证明与数学的不同领域有有趣的联系，包括以下内容：泛函分析、遍历理论、组合学和算子代数。许多联系都是首席研究人员研究的直接结果。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展开。第一个是理解正的或完全的正熵对于不可服从群的作用的轨道等价结果。例如，主要研究人员已经证明，具有完全正熵的行为是强遍历的，并且正熵行为不是弱紧的。他试图证明具有竞争正熵的行为是坚实的遍历行为，正如Chifan和Ioana所定义的那样，这将进一步表明这些行为所展示的类Bernoulli行为。这一目标的部分有趣之处在于，它将非顺从群作用的熵理论与顺从群作用的熵研究形成了鲜明对比，在后者中，不可能推导出任何轨道等价的结果。完全正熵的这些结果适用于代数作用(即紧群上的自同构作用)，因为主要研究者已经证明了与可逆卷积算子有关的代数作用具有完全正熵。第二个问题是与Lewis Bowen和Dylan Airey共同研究等价关系的代数作用的熵理论。该项目这一部分的目的之一是将主要研究者关于熵和Fuglede-Kadison行列式的结果从群情形推广到等价关系情形。最后，主要研究者将通过定义算子代数上的作用的SOFIC熵来进一步研究动力学和算子代数之间的联系。

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.