On the cost of generating an equivalence relation. (English) Zbl 0843.28010

Levitt, Gilbert

Ergodic Theory Dyn. Syst. 15, No. 6, 1173-1181 (1995).

In this paper a generating system for a measurable equivalence relation \(R\) is a countable family \(\Phi\) of partially defined \(\mu\)-preserving isomorphisms \(\varphi: A_j\to B_j\) \((A_j, B_j\subset X)\) on a standard Borel space \(X\), so that \(R\) is the smallest equivalence relation generated by the \(\varphi_j\)’s. It is shown that \[ \sum_j \mu(A_j)+ \int {1\over \# R(x)} \mu(dx)\geq 1\quad (\text{where}\quad {1\over \infty}= 0) \] and equality holds if and only if \(R\) is amenable and the generators are independent. Applications to pseudogroups of measure-preserving homeomorphisms are also given.

Reviewer: M.Denker (Göttingen)

MSC:

28D15	General groups of measure-preserving transformations
37A99	Ergodic theory

Keywords:

orbit equivalence; measurable equivalence relation; pseudogroups; measure-preserving homeomorphisms

References:

[1]	Gaboriau, Dynamique des systèmes d’isométries et actions de groupes sur les arbres réels (1993)
[2]	Feldman, Trans. Amer. Math. Soc. 234 pp 289– (1977) · doi:10.1090/S0002-9947-1977-0578656-4
[3]	DOI: 10.1017/S014338570000136X · Zbl 0491.28018 · doi:10.1017/S014338570000136X
[4]	DOI: 10.1017/S0143385700005368 · Zbl 0667.28003 · doi:10.1017/S0143385700005368
[5]	Salem, Riemannian Foliations, Progress in Math. 73 (1988)
[6]	DOI: 10.2307/2118526 · Zbl 0843.57026 · doi:10.2307/2118526
[7]	Haefliger, Structures Transverses des Feuilletages 116 pp 70– (1984)
[8]	DOI: 10.1007/BF01391835 · Zbl 0594.57014 · doi:10.1007/BF01391835
[9]	DOI: 10.1017/S0143385700008580 · Zbl 0839.58022 · doi:10.1017/S0143385700008580
[10]	DOI: 10.1007/BF02773004 · Zbl 0824.57001 · doi:10.1007/BF02773004
[11]	DOI: 10.1007/BF01244321 · Zbl 0791.58055 · doi:10.1007/BF01244321