Geodesic
Orbit equivalence for Cantor minimal ℤ²-systems
Giordano, Thierry 1   ; Matui, Hiroki 2   ; Putnam, Ian 3   ; Skau, Christian 4
1 Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, Ontario, Canada K1N 6N5
2 Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan
3 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W 3P4
4 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway
Journal of the American Mathematical Society, Tome 21 (2008) no. 3, pp. 863-892 Cet article a éte moissonné depuis la source American Mathematical Society

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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.
DOI : 10.1090/S0894-0347-08-00595-X

Giordano, Thierry  1   ; Matui, Hiroki  2   ; Putnam, Ian  3   ; Skau, Christian  4

1 Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, Ontario, Canada K1N 6N5
2 Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan
3 Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W 3P4
4 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway 
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Giordano, Thierry; Matui, Hiroki; Putnam, Ian; Skau, Christian. Orbit equivalence for Cantor minimal ℤ²-systems. Journal of the American Mathematical Society, Tome 21 (2008) no. 3, pp. 863-892. doi: 10.1090/S0894-0347-08-00595-X

[1] Connes, A., Feldman, J., Weiss, B. An amenable equivalence relation is generated by a single transformation Ergodic Theory Dynam. Systems 1981

[2] Dye, H. A. On groups of measure preserving transformations. I Amer. J. Math. 1959 119 159

[3] Forrest, Alan A Bratteli diagram for commuting homeomorphisms of the Cantor set Internat. J. Math. 2000 177 200

[4] Giordano, Thierry, Putnam, Ian F., Skau, Christian F. Topological orbit equivalence and 𝐶*-crossed products J. Reine Angew. Math. 1995 51 111

[5] Giordano, Thierry, Putnam, Ian, Skau, Christian Affable equivalence relations and orbit structure of Cantor dynamical systems Ergodic Theory Dynam. Systems 2004 441 475

[6] Giordano, Thierry, Putnam, Ian F., Skau, Christian F. The orbit structure of Cantor minimal ℤ²-systems 2006 145 160

[7] Herman, Richard H., Putnam, Ian F., Skau, Christian F. Ordered Bratteli diagrams, dimension groups and topological dynamics Internat. J. Math. 1992 827 864

[8] Jackson, S., Kechris, A. S., Louveau, A. Countable Borel equivalence relations J. Math. Log. 2002 1 80

[9] Lightwood, Samuel J., Ormes, Nicholas S. Bounded orbit injections and suspension equivalence for minimal ℤ² actions Ergodic Theory Dynam. Systems 2007 153 182

[10] Matui, Hiroki A short proof of affability for certain Cantor minimal ℤ²-systems Canad. Math. Bull. 2007 418 426

[11] Matui, Hiroki Affability of equivalence relations arising from two-dimensional substitution tilings Ergodic Theory Dynam. Systems 2006 467 480

[12] Ornstein, Donald S., Weiss, Benjamin Ergodic theory of amenable group actions. I. The Rohlin lemma Bull. Amer. Math. Soc. (N.S.) 1980 161 164

[13] Ornstein, Donald S., Weiss, Benjamin Entropy and isomorphism theorems for actions of amenable groups J. Analyse Math. 1987 1 141

[14] Phillips, N. Christopher Crossed products of the Cantor set by free minimal actions of ℤ^{𝕕} Comm. Math. Phys. 2005 1 42

[15] Renault, Jean A groupoid approach to 𝐶*-algebras 1980

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