Geometric and Functional Analysis

, Volume 20, Issue 1, pp 53–67

Ergodic Subequivalence Relations Induced by a Bernoulli Action

Authors

  • Ionut Chifan
    • Math. DeptUCLA
    • IMAR
    • Math. DeptUCLA
    • IMAR
Article

DOI: 10.1007/s00039-010-0058-7

Cite this article as:
Chifan, I. & Ioana, A. Geom. Funct. Anal. (2010) 20: 53. doi:10.1007/s00039-010-0058-7

Abstract

Let Γ be a countable group and denote by \({\mathcal{S}}\) the equivalence relation induced by the Bernoulli action \({\Gamma\curvearrowright [0, 1]^{\Gamma}}\), where [0, 1]Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation \({\mathcal{R}}\) of \({\mathcal{S}}\), there exists a partition {Xi}i≥0 of [0, 1]Γ into \({\mathcal{R}}\)-invariant measurable sets such that \({\mathcal{R}_{\vert X_{0}}}\) is hyperfinite and \({\mathcal{R}_{\vert X_{i}}}\) is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.

Keywords and phrases

Bernoulli actiondeformation/rigidityergodic subequivalence relationmalleablestrongly ergodic

2010 Mathematics Subject Classification

37A2037A15

Copyright information

© Birkhäuser / Springer Basel AG 2010