Minimal topological actions do not determine the measurable orbit equivalence class

Abstract

We construct an amenable action Φ of a non-amenable group Γ on a discrete space. This action extends to a minimal topological action Φ of Γ on a Cantor set C. We show that Φ is non-uniquely ergodic and furthermore there exist ergodic invariant measures μ1 and μ2 such that (Φ,C,μ1) and (Φ,C,μ2) are not orbit equivalent measurable equivalence relations. This also provides an instance of the failure of equivalence between the notions of “global” and “local” amenability for countable equivalence relations.