Multiorders in amenable group actions

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 $\mathcal O$ 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 $(\mathcal O,\nu,G)$ 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. The sub-sigma-algebra $\Sigma_{\mathcal O}$ associated with the multiorder factor is invariant under $S$, which makes the corresponding $\mathbb Z$-action $(\mathcal O,\nu,\tilde S)$ 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 $\Sigma_{\mathcal O}$, 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 the one known for $\mathbb Z$-actions. This fact is 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 $\Sigma$-measurable. 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 prove that the Pinsker sigma-algebra of any $G$-process can be identified (with probability 1) using the following algorithm: (1) fix an arbitrary multiorder on $G$, (2) select any order from the support of that multiorder, (3) in the process, find the "remote past" along the selected order.