Fundamental group of uniquely ergodic Cantor minimal systems

TL;DR

This paper introduces and computes the fundamental group for uniquely ergodic Cantor minimal systems, revealing algebraic structures linked to the acting groups and connections with crossed product C*-algebras.

Contribution

It defines the fundamental group for these systems, computes it for several cases, and explores its algebraic structure and relation to crossed product C*-algebras.

Findings

Fundamental groups can be characterized as positive units of certain subrings of real numbers.

For free actions of finitely generated abelian groups, the fundamental group is explicitly described.

Connections between fundamental groups of dynamical systems and C*-algebra structures are established.

Abstract

We introduce the fundamental group ${\mathcal F}(\mathcal{R}_{G, \phi})$ of a uniquely ergodic Cantor minimal $G$-system $\mathcal{R}_{G, \phi}$ where $G$ is a countable discrete group. We compute fundamental groups of several uniquely ergodic Cantor minimal $G$-systems. We show that if $\mathcal{R}_{G, \phi}$ arises from a free action $\phi$ of a finitely generated abelian group, then there exists a unital countable subring $R$ of $\mathbb{R}$ such that $\mathcal{F}(\mathcal{R}_{G, \phi})=R_{+}^\times$. We also consider the relation between fundamental groups of uniquely ergodic Cantor minimal $\mathbb{Z}^n$-systems and fundamental groups of crossed product $C^*$-algebras $C(X)\rtimes_{\phi} \mathbb{Z}^n$.