JournalsggdVol. 2 , No. 2DOI 10.4171/ggd/34

Minimal topological actions do not determine the measurable orbit equivalence class

  • Tullio Ceccherini-Silberstein

    Università del Sannio, Benevento, Italy
  • Gábor Elek

    Hungarian Academy of Sciences, Budapest, Hungary
Minimal topological actions do not determine the measurable orbit equivalence class cover

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.