Orbit equivalence for Cantor minimal ℤ²-systems

Giordano, Thierry; Matui, Hiroki; Putnam, Ian; Skau, Christian

doi:10.1090/S0894-0347-08-00595-X

Orbit equivalence for Cantor minimal $\mathbb {Z}^{2}$-systems
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by Thierry Giordano, Hiroki Matui, Ian F. Putnam and Christian F. Skau

J. Amer. Math. Soc. 21 (2008), 863-892

DOI: https://doi.org/10.1090/S0894-0347-08-00595-X

Published electronically: January 22, 2008

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Abstract:

We show that every minimal, free action of the group $\mathbb {Z}^{2}$ on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, $\mathbb {Z}$-actions and $\mathbb {Z}^{2}$-actions.

References

A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by a single transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 431–450 (1982). MR 662736, DOI 10.1017/s014338570000136x
H. A. Dye, On groups of measure preserving transformations. I, Amer. J. Math. 81 (1959), 119–159. MR 131516, DOI 10.2307/2372852
Alan Forrest, A Bratteli diagram for commuting homeomorphisms of the Cantor set, Internat. J. Math. 11 (2000), no. 2, 177–200. MR 1754619, DOI 10.1142/S0129167X00000106

The absorption theorem for affable equivalence relations

Thierry Giordano, Ian F. Putnam, and Christian F. Skau, Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math. 469 (1995), 51–111. MR 1363826
Thierry Giordano, Ian Putnam, and Christian Skau, Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 441–475. MR 2054051, DOI 10.1017/S014338570300066X
Thierry Giordano, Ian F. Putnam, and Christian F. Skau, The orbit structure of Cantor minimal $\Bbb Z^2$-systems, Operator Algebras: The Abel Symposium 2004, Abel Symp., vol. 1, Springer, Berlin, 2006, pp. 145–160. MR 2265047, DOI 10.1007/978-3-540-34197-0_{7}
Richard H. Herman, Ian F. Putnam, and Christian F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. MR 1194074, DOI 10.1142/S0129167X92000382
S. Jackson, A. S. Kechris, and A. Louveau, Countable Borel equivalence relations, J. Math. Log. 2 (2002), no. 1, 1–80. MR 1900547, DOI 10.1142/S0219061302000138

Ordered K-theory and Bratteli diagrams:Implications for Cantor minimal systems

Samuel J. Lightwood and Nicholas S. Ormes, Bounded orbit injections and suspension equivalence for minimal $\Bbb Z^2$ actions, Ergodic Theory Dynam. Systems 27 (2007), no. 1, 153–182. MR 2297092, DOI 10.1017/S014338570600068X
Hiroki Matui, A short proof of affability for certain Cantor minimal $\Bbb Z^2$-systems, Canad. Math. Bull. 50 (2007), no. 3, 418–426. MR 2344176, DOI 10.4153/CMB-2007-040-3
Hiroki Matui, Affability of equivalence relations arising from two-dimensional substitution tilings, Ergodic Theory Dynam. Systems 26 (2006), no. 2, 467–480. MR 2218771, DOI 10.1017/S0143385705000611
Donald S. Ornstein and Benjamin Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164. MR 551753, DOI 10.1090/S0273-0979-1980-14702-3
Donald S. Ornstein and Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141. MR 910005, DOI 10.1007/BF02790325

A survey of orbit equivalence for Cantor minimal dynamics

N. Christopher Phillips, Crossed products of the Cantor set by free minimal actions of $\Bbb Z^d$, Comm. Math. Phys. 256 (2005), no. 1, 1–42. MR 2134336, DOI 10.1007/s00220-004-1171-y
Jean Renault, A groupoid approach to $C^{\ast }$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR 584266, DOI 10.1007/BFb0091072