Orbit equivalence for Cantor minimal $\mathbb {Z}^{2}$-systems
HTML articles powered by AMS MathViewer
- 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
- PDF | Request permission
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 [GMPS]GMPS:affable T. Giordano, H. Matui, I.F. Putnam and C.F. Skau, The absorption theorem for affable equivalence relations, Ergodic Theory Dynam. Systems, to appear.
- 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 [J]J:thesis Ø. Johansen, Ordered K-theory and Bratteli diagrams:Implications for Cantor minimal systems, Ph.D. thesis, NTNU, 1998.
- 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 [PPZ]PPZ:survey J. Peebles, I.F. Putnam and I.F. Zwiers, A survey of orbit equivalence for Cantor minimal dynamics, in preparation.
- 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
Bibliographic 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
- MR Author ID: 142845
- Christian F. Skau
- Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway
- MR Author ID: 233522
- 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 - © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc. 21 (2008), 863-892
- MSC (2000): Primary 37B99; Secondary 37B50, 37A20
- DOI: https://doi.org/10.1090/S0894-0347-08-00595-X
- MathSciNet review: 2393431