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.