An absorption theorem for minimal AF equivalence relations on Cantor sets. (English) Zbl 1170.37009

Matui, Hiroki

J. Math. Soc. Japan 60, No. 4, 1171-1185 (2008).

Minimal \(\mathbb Z\)-actions and \(\mathbb Z^2\)-actions on Cantor sets have been classified up to orbit equivalence [T. Giordano, I. F. Putnam and C. F. Skau, J. Reine Angew. Math. 469, 51–111 (1995; Zbl 0834.46053); T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Ergodic Theory Dyn. Syst. 28, No. 5, 1509–1531 (2008; Zbl 1166.37004)]. The strategy is to prove that the equivalence relation associated with the given minimal action is orbit equivalent to an AF relation. Recall that an étale equivalence relation \(R\) is called an AF relation, if there exists an increasing sequence \(R_1\subset R_2\subset\cdots\) of compact open subrelations of \(R\) such that \(R=\cup_{n\in{\mathbb N}}R_n\). To prove this, a key step is to prove the absorption theorem [T. Giordano, I. F. Putnam and C. F. Skau, Ergodic Theory Dyn. Syst. 24, No. 2, 441–475 (2004; Zbl 1074.37010)]. However, the proof method of the absorption theorem in the case of minimal \(\mathbb Z^2\)-actions cannot be generalized to the case of minimal \(\mathbb Z^d\)-actions directly for \(d\geq 3\). The aim of the present paper is to generalize the absorption theorem which is needed for the study of \(\mathbb Z^d\)-actions.

Reviewer: Enhui Shi (Suzhou)

MSC:

37B05

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

Keywords:

Cantor sets; orbit equivalence; minimal dynamical systems

Citations:

Zbl 0834.46053; Zbl 1074.37010; Zbl 1166.37004

References:

[1]	T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal \(\Z^2\)-systems, J. Amer. Math. Soc., 21 (2008), 863-892. · Zbl 1254.37012
[2]	T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, The absorption theorem for affable equivalence relations, to appear in Ergodic Theory Dynam. Systems, · Zbl 1166.37004
[3]	T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal \(\Z^d\)-systems, in preparation. · Zbl 1293.37004
[4]	T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and \(C^*\)-crossed products, J. Reine Angew. Math., 469 (1995), 51-111. · Zbl 0834.46053
[5]	T. Giordano, I. F. Putnam and C. F. Skau, Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergodic Theory Dynam. Systems, 24 (2004), 441-475. · Zbl 1074.37010