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**Affable equivalence relations and orbit structure of Cantor dynamical systems.**
*(English)*
Zbl 1074.37010

The important result by A. Connes, J. Feldman and B. Weiss [Ergodic Theory Dyn. Syst. 1, 431–450 (1981; Zbl 0491.28018)] states that every free ergodic action of an amenable group by measure class preserving transformations is orbit equivalent to an action of the group of integers. Whether or not a similar result holds in the context of topological dynamics, is far from being clear, even for the actions of the group \({\mathbb Z}^2\). For purely topological reasons, it only makes sense to consider the fundamental question above for zero-dimensional compact sets \(X\).

Let a countable, amenable group \(G\) act freely and minimally on the Cantor set \(C\); does there exist an action of \(\mathbb Z\) on \(C\) generating the same orbit structure? With this purpose, the authors consider the following concepts. All equivalence relations under consideration are étale, that is, the relation \(R\subseteq X\times X\) is given a topology of its own under which the projection \(r\) of \(R\) to \(X\) is a local homeomorphism. An étale equivalence relation \(R\) on a locally compact zero-dimensional Hausdorff space \(X\) is CEER (compact étale equivalence relation) if \(R\) minus the diagonal is compact. (In this case, the topology on \(R\) is easily seen to be the relative topology induced from \(X\times X\).) An equivalence relation \(R\) is an \(AF\)-relation (from “approximately finite”), if \(R\) is the union of an increasing sequence of open subequivalence relations which are CEER, and \(R\) is equipped with the inductive limit topology. For instance, actions of locally finite countable groups on the Cantor set give rise to AF-equivalence relations.

Finally, the central new notion of the paper is that of an affable equivalence relation: so is called a countable equivalence relation \(R\) which can be given a topology making it into an \(AF\)-relation. Equivalently, \(R\) is orbit equivalent to an \(AF\)-relation. The name “affable” is based on a word play: AF-able. The authors develop a spectacular machinery of dealing with AF-relations in the language of Bratteli diagrams.

As proved in the paper, the main problem motivating the present article can be now stated as follows: is the orbit equivalence relation determined by a free minimal action of a countable amenable group on the Cantor set affable? Furthermore, a number of tests for affability of concrete equivalence relations are established, largely with a view of forthcoming investigations in the area by the same authors.

Let a countable, amenable group \(G\) act freely and minimally on the Cantor set \(C\); does there exist an action of \(\mathbb Z\) on \(C\) generating the same orbit structure? With this purpose, the authors consider the following concepts. All equivalence relations under consideration are étale, that is, the relation \(R\subseteq X\times X\) is given a topology of its own under which the projection \(r\) of \(R\) to \(X\) is a local homeomorphism. An étale equivalence relation \(R\) on a locally compact zero-dimensional Hausdorff space \(X\) is CEER (compact étale equivalence relation) if \(R\) minus the diagonal is compact. (In this case, the topology on \(R\) is easily seen to be the relative topology induced from \(X\times X\).) An equivalence relation \(R\) is an \(AF\)-relation (from “approximately finite”), if \(R\) is the union of an increasing sequence of open subequivalence relations which are CEER, and \(R\) is equipped with the inductive limit topology. For instance, actions of locally finite countable groups on the Cantor set give rise to AF-equivalence relations.

Finally, the central new notion of the paper is that of an affable equivalence relation: so is called a countable equivalence relation \(R\) which can be given a topology making it into an \(AF\)-relation. Equivalently, \(R\) is orbit equivalent to an \(AF\)-relation. The name “affable” is based on a word play: AF-able. The authors develop a spectacular machinery of dealing with AF-relations in the language of Bratteli diagrams.

As proved in the paper, the main problem motivating the present article can be now stated as follows: is the orbit equivalence relation determined by a free minimal action of a countable amenable group on the Cantor set affable? Furthermore, a number of tests for affability of concrete equivalence relations are established, largely with a view of forthcoming investigations in the area by the same authors.

Reviewer: Vladimir Pestov (Ottawa, Ontario)

### MSC:

37B10 | Symbolic dynamics |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |

37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |