Algorithmic Randomness For Amenable Groups

TL;DR

This paper extends algorithmic randomness theory to computable amenable groups, providing an effective Shannon-McMillan-Breiman theorem and linking topological entropy with Hausdorff dimension.

Contribution

It introduces a new framework for algorithmic randomness in amenable groups and proves an effective Shannon-McMillan-Breiman theorem within this context.

Findings

Effective Shannon-McMillan-Breiman theorem for amenable groups

Equivalence of topological entropy and Hausdorff dimension in this setting

Utilization of Ornstein-Weiss work to support proofs

Abstract

We develop the theory of algorithmic randomness for the space $A^G$ where $A$ is a finite alphabet and $G$ is a computable amenable group. We give an effective version of the Shannon-McMillan-Breiman theorem in this setting. We also extend a result of Simpson equating topological entropy and Hausdorff dimension. This proof makes use of work of Ornstein and Weiss which we also present.