Aspects of Sofic Entropy and Algebraic Actions

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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.