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~{\em 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. 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 \emph{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. 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.