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Minimal topological actions do not determine the measurable orbit equivalence class
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 …