Entropy Theory for Non-Amenable Groups

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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.