Orbit equivalence for Cantor minimal Z^d-systems

TL;DR

This paper proves that all minimal actions of finitely generated abelian groups on the Cantor set are orbit equivalent to AF relations, expanding the classification of such dynamical systems.

Contribution

It establishes that any minimal abelian group action on the Cantor set is orbit equivalent to an AF relation, broadening the scope of classification.

Findings

All minimal abelian group actions on the Cantor set are orbit equivalent to AF relations.

Extends classification of minimal Cantor systems to include Z^d-actions.

Shows orbit equivalence for a broad class of dynamical systems.

Abstract

We show that every minimal action of any finitely generated abelian group on the Cantor set is (topologically) orbit equivalent to an AF relation. As a consequence, this extends the classification up to orbit equivalence of minimal dynamical systems on the Cantor set to include AF relations and Z^d-actions.