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.

36 pages, 2 figures, Changes: slightly changed formulation and proof of Theorem 7.4, some remarks added

References in corpus (1)

• Symbolic extensions of amenable group actions and the comparison property

Cited by in corpus (1)

• Existence of asymptotic pairs in positive entropy group actions