Algorithmic Randomness For Amenable Groups

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.