Minimal topological actions do not determine the measurable orbit equivalence class

Abstract

We construct a minimal topological action $\wp$ of a non-amenable group on a Cantor set $C$, which is non-uniquely ergodic and furthermore there exist ergodic invariant measures ${\mu }_1$ and ${\mu }_2$ such that $(\wp ,C,{\mu }_1)$ and $(\wp ,C,{\mu }_2)$ are not orbit equivalent measurable equivalence relations.