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Orbit equivalence for Cantor minimal $ \mathbb{Z}^{2}$-systems

Author(s): Thierry Giordano; Hiroki Matui; Ian F. Putnam; Christian F. Skau
Journal: J. Amer. Math. Soc. 21 (2008), 863-892.
MSC (2000): Primary 37B99; Secondary 37B50, 37A20
Posted: 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.

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Additional Information:

Thierry Giordano
Affiliation: Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, Ontario, Canada K1N 6N5

Hiroki Matui
Affiliation: Graduate School of Science and Technology, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan

Ian F. Putnam
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W 3P4

Christian F. Skau
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway

DOI: 10.1090/S0894-0347-08-00595-X
PII: S 0894-0347(08)00595-X
Received by editor(s): September 22, 2006
Posted: January 22, 2008
Additional Notes: The first author was supported in part by a grant from NSERC, Canada
The second author was supported in part by a grant from the Japan Society for the Promotion of Science
The third author was supported in part by a grant from NSERC, Canada
The last author was supported in part by the Norwegian Research Council
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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