Orbit equivalence for Cantor minimal systems
Authors:
Thierry Giordano, Hiroki Matui, Ian F. Putnam and Christian F. Skau
Journal:
J. Amer. Math. Soc. 21 (2008), 863892
MSC (2000):
Primary 37B99; Secondary 37B50, 37A20
Published electronically:
January 22, 2008
MathSciNet review:
2393431
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We show that every minimal, free action of the group on the Cantor set is orbit equivalent to an AFrelation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AFrelations, actions and actions.
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 [D]
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 A. Forrest, A Bratteli diagram for commuting homeomorphisms of the Cantor set, Internat. J. Math. 11 (2000), 177200. MR 1754619 (2001d:37008)
 [GMPS]
 T. Giordano, H. Matui, I.F. Putnam and C.F. Skau, The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, to appear.
 [GPS1]
 T. Giordano, I.F. Putnam and C.F. Skau, Topological orbit equivalence and crossed products, J. Reine Angew. Math. 469 (1995), 51111. MR 1363826 (97g:46085)
 [GPS2]
 T. Giordano, I.F. Putnam and C.F. Skau, Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergodic Theory Dynam. Systems 23 (2004), 441475. MR 2054051 (2005d:37017)
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 T. Giordano, I.F. Putnam and C.F. Skau, The orbit structure of Cantor minimal systems, Proceedings of the first Abel Symposium, O. Bratteli, S. Neshveyev and C. Skau, Eds., SpringerVerlag, Berlin, 2006. MR 2265047 (2007i:37018)
 [HPS]
 R.H. Herman, I.F. Putnam and C.F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), 827864. MR 1194074 (94f:46096)
 [JKL]
 A. Jackson, A.S. Kechris and A. Louveau, Countable Borel equivalence relations, J. Math. Log. 2 (2002), 180. MR 1900547 (2003f:03066)
 [J]
 Ø. Johansen, Ordered Ktheory and Bratteli diagrams:Implications for Cantor minimal systems, Ph.D. thesis, NTNU, 1998.
 [LO]
 S. Lightwood and N. Ormes, Bounded orbit injections and suspension equiavlence for minimal actions, Ergodic Theory Dynam. Systems 27 (2007), 153182. MR 2297092
 [M1]
 H. Matui, A short proof of affability for certain Cantor minimal systems, Canad. Math. Bull. 50 (2007), 418426. MR 2344176
 [M2]
 H. Matui, Affability of equivalence relations arising from twodimensional substitution tilings, Ergodic Theory Dynam. Systems 26 (2006), 467480. MR 2218771
 [OW1]
 D.S. Ornstein and B. Weiss, Ergodic theory of amenable group actions I: The Rohlin lemma, Bull. Amer. Math. Soc. 2 (1980), 161164. MR 551753 (80j:28031)
 [OW2]
 D.S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1141. MR 910005 (88j:28014)
 [PPZ]
 J. Peebles, I.F. Putnam and I.F. Zwiers, A survey of orbit equivalence for Cantor minimal dynamics, in preparation.
 [Ph]
 N.C. Phillips, Crossed products of the Cantor set by free, minimal actions of , Comm. Math. Phys. 256 (2005), 142. MR 2134336 (2006g:46107)
 [R]
 J. Renault, A Groupoid Approach to algebras, Lecture Notes in Mathematics 793, Springer, Berlin, 1980. MR 584266 (82h:46075)
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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, 133 Yayoicho, Inageku, Chiba 2638522, 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), N7491 Trondheim, Norway
DOI:
http://dx.doi.org/10.1090/S089403470800595X
PII:
S 08940347(08)00595X
Received by editor(s):
September 22, 2006
Published electronically:
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
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
