Ergodic Subequivalence Relations Induced by a Bernoulli Action

TL;DR

This paper proves that any subequivalence relation of the Bernoulli action equivalence relation can be decomposed into hyperfinite and strongly ergodic parts, revealing a structured ergodic decomposition.

Contribution

It establishes a novel ergodic decomposition for subequivalence relations induced by Bernoulli actions of countable groups.

Findings

Existence of a measurable partition with hyperfinite and strongly ergodic components.

Any subequivalence relation admits a decomposition into hyperfinite and strongly ergodic parts.

The result applies to Bernoulli actions of arbitrary countable groups.

Abstract

Let $\Gamma$ be a countable group and denote by $\Cal S$ the equivalence relation induced by the Bernoulli action $\Gamma\curvearrowright [0,1]^{\Gamma}$, where $[0,1]^{\Gamma}$ is endowed with the product Lebesgue measure. We prove that for any subequivalence relation $\Cal R$ of $\Cal S$, there exists a partition $\{X_i\}_{i\geq 0}$ of $[0,1]^{\Gamma}$ with $\Cal R$-invariant measurable sets such that $\Cal R_{|X_0}$ is hyperfinite and $\Cal R_{|X_i}$ is strongly ergodic (hence ergodic), for every $i\geq 1$.