Multiorders in amenable group actions

Abstract

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group, we mean any probability measure $\nu$ on the collection $\tilde{\mathcal{O}}$ of linear orders of type $\mathbb{Z}$ on $G$, invariant under the natural action of $G$ on such orders. Multiorders exist on any countable amenable group (and only on such groups) and every multiorder has the Følner property, meaning that almost surely the order intervals starting at the unit form a Følner sequence. Every free measure-preserving $G$-action $(X,\mu ,G)$ has a multiorder $(\tilde{\mathcal{O}},\nu ,G)$ as a factor and has the same orbits as the $\mathbb{Z}$-action $(X,\mu ,S)$, where $S$ is the successor map determined by the multiorder factor. Moreover, the sub-sigma-algebra ${\Sigma }_{\tilde{\mathcal{O}}}$ associated with the multiorder factor is invariant under $S$, which makes the corresponding $\mathbb{Z}$-action $(\tilde{\mathcal{O}},\nu ,\tilde{S})$ a factor of $(X,\mu ,S)$. We prove that the entropy of any $G$-process generated by a finite partition of $X$, conditional with respect to ${\Sigma }_{\tilde{\mathcal{O}}}$, is preserved by the orbit equivalence with $(X,\mu ,S)$. Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to $h(\mu ,T,\mathcal{P})=H(\mu ,\mathcal{P}|{\mathcal{P}}^{-})$ known for $\mathbb{Z}$-actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss (2000). The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra $\Sigma$, as soon as the “orbit change” is measurable with respect to $\Sigma$. In our variant, we replace the measurability assumption by a simpler one: $\Sigma$ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion, we provide a characterization of the Pinsker sigma-algebra of any $G$-process in terms of an appropriately defined remote past arising from a multiorder. The paper has an appendix in which we present an explicit construction of a particularly regular (uniformly Følner) multiorder based on an ordered dynamical tiling system of $G$.