Fundamental group of uniquely ergodic Cantor minimal systems

Abstract

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