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The absorption theorem for affable equivalence relations

Published online by Cambridge University Press:  01 October 2008

THIERRY GIORDANO ,
HIROKI MATUI ,
IAN F. PUTNAM  and
CHRISTIAN F. SKAU
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, 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, B.C., Canada V8W 3P4
CHRISTIAN F. SKAU
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7034 Trondheim, Norway
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Abstract

We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being ‘small’ in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case—when Y is a finite set—this result is highly non-trivial. The result itself—called the absorption theorem—is a powerful and crucial tool for the study of the orbit structure of minimal ℤn-actions on the Cantor set, see Remark 4.8. The absorption theorem is a significant generalization of the main theorem proved in Giordano et al [Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys.24 (2004), 441–475] . However, we shall need a few key results from the above paper in order to prove the absorption theorem.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1509 - 1531
Copyright
Copyright © 2008 Cambridge University Press

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References

[1]Giordano, T., Matui, H., Putnam, I. F. and Skau, C. F.. Orbit equivalence for Cantor minimalℤ2-systems. J. Amer. Math. Soc. 21 (2008), 863–892.CrossRefGoogle Scholar
[2]Giordano, T., Putnam, I. F. and Skau, C. F.. Topological orbit equivalence and C *-crossed products. J. Reine Angew. Math. 469 (1995), 51–111.Google Scholar
[3]Giordano, T., Putnam, I. F. and Skau, C. F.. Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys. 24 (2004), 441–475.CrossRefGoogle Scholar
[4]Molberg, M.. AF-equivalence relations. Math. Scand. 99 (2006), 247–256.CrossRefGoogle Scholar
[5]Paterson, A. L. T.. Groupoids, Inverse Semigroups, and Their Operator Algebras (Progress in Mathematics, 170). Birkhäuser Boston, Boston, MA, 1999.CrossRefGoogle Scholar