Universally measure-preserving homeomorphisms of Cantor minimal systems. (English) Zbl 1233.37003

A Cantor minimal system \((X,S)\) is a compact, metrizable, totally disconnected space \(X\) with no isolated point, endowed with a minimal homeomorphism \(S: X\to X\). Such a system \((X,S)\) is known to be topologically conjugate to some symbolic dynamics, namely some “adic” transformation arising from some “simple ordered Bratteli diagram”, yielding a sort of coding of \((X,S)\) [R. H. Herman, I. F. Putnam and C. F. Skau, “Ordered Bratteli diagrams, dimension groups and topological dynamics”, Int. J. Math. 3, No. 6, 827–864 (1992; Zbl 0786.46053)]).
The concern of the authors is the following characterization of orbit equivalence between Cantor minimal systems \((X,S)\) and \((Y,T)\), first established in [“Full groups of Cantor minimal systems”, Isr. J. Math. 111, 285–320 (1999; Zbl 0942.46040)] by T. Giordano, I. F. Putnam and C. F. Skau: there exists a homeomorphism \(\phi:X\to Y\) such that (for any \(x\in X\)) \[ \{\phi\cdot S^i(x)\mid i\in \mathbb{Z}\}= \{T^i\cdot\phi(x)\mid i\in\mathbb{Z}\} \] if and only if there exists a homeomorphim \(\widetilde\phi: X\to Y\) such that for any probability measure \(\mu\) on \(X\), \(\mu\) is \(S\)-invariant if and only if \(\mu\circ\widetilde\phi^{- 1}\) is \(T\)-invariant.
The aim of this article is to provide a new proof of this theorem, using dynamical methods, which is hopefully also more elementary (i.e., involving neither \(K\)-theory nor hand-logical algebra).
The idea of the authors is, following Glasner and Weiss, to apply and improve a copying lemma by Katznelson and Weiss, and to use a finitary orbit equivalence technique on Bratteli diagrams.


37A05 Dynamical aspects of measure-preserving transformations
37A55 Dynamical systems and the theory of \(C^*\)-algebras
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
37B10 Symbolic dynamics
Full Text: DOI


[1] O. Bratteli, Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234. · Zbl 0264.46057
[2] M. Boyle and D. Handelman, Entropy versus orbit equivalence for minimal homeomorphisms, Pacific J. Math. 164 (1994), 1–13. · Zbl 0812.58025
[3] A. Connes, Une classification des facteurs de type III, Ann. Sci. École. Norm. Sup. 6 (1973), 133–252. · Zbl 0274.46050
[4] A. Connes, On hyperfinite factors of type III 0 and Krieger’s factors, J. Funct. Anal. 18 (1975), 318–327. · Zbl 0296.46067
[5] A. del Junco and A. Sahin, Dye’s theorem in the almost continuous category, Israel J. Math. 173 (2009), 235–251. · Zbl 1219.28016
[6] H. Dye, On groups of measure-preserving transformations 1, Amer. J. Math. 81 (1959), 119–159. · Zbl 0087.11501
[7] T. Giordano, I. Putnam, and C. Skau, Full groups of Cantor minimal systems, Israel J. Math. 111 (1999), 285–320. · Zbl 0942.46040
[8] T. Giordano, I. Putnam, and C. Skau Topological orbit equivalence and C*-crossed products, J. Reine Angew. Math. 469 (1995), 51–111. · Zbl 0834.46053
[9] E. Glasner and B. Weiss, Weak orbit equivalence of Cantor minimal systems, Internat. J. Math. 6 (1995), 559–579. · Zbl 0879.54046
[10] U. Haagerup Connes’s bicentralizer problem and uniqueness of the injective factor of type III 1, Acta. Math. 158 (1967), 95–148. · Zbl 0628.46061
[11] T. Hamachi, A measure theoretical proof of the Connes-Woods theorem on AT-flows, Pacific J. Math. 154 (1992), 67–85. · Zbl 0792.46048
[12] T. Hamachi, Canonical subrelations of ergodic equivalence relations-subrelations, J. Operator Theory 43 (2000), 3–34. · Zbl 0990.37001
[13] T. Hamachi and M. Keane, Finitary orbit equivalence of odometers, Bull. London Math. Soc. 38 (2006), 450–458. · Zbl 1108.37006
[14] T. Hamachi, M. Keane, and M. Roychowdhury, Finitary orbit equivalence and measured Bratteli diagrams, Colloq. Math. 110 (2008) 363–382. · Zbl 1172.37004
[15] R. Herman, I. Putnam, and C. Skau, Ordered Bratteli diagrams and topological dynamics, Internat. J. Math. 3 (1992) 827–864. · Zbl 0786.46053
[16] Y. Katznelson and B. Weiss, The classification of non-singular actions revisited, Ergodic Theory Dynam. Systems 11 (1991), 333–348. · Zbl 0759.58024
[17] W. Krieger, On non-singular transformations of a measure space I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1969), 83–97. · Zbl 0185.11901
[18] W. Krieger, On non-singular transformations of a measure space II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1969), 98–119. · Zbl 0185.11901
[19] W. Krieger, On ergodic flows and isomorphisms of factors, Math. Ann. 223 (1976), 19–70. · Zbl 0332.46045
[20] M. S. Keane and M. Smorodinsky, A class of finitary codes, Israel J.Math. 26 (1977), 352–371. · Zbl 0357.94012
[21] M. S. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitary isomorphic, Ann. of Math. (2) 109 (1979), 397–406. · Zbl 0405.28017
[22] N. S. Ormes, Strong orbit realization for minimal homeomorphisms, J. Anal. Math. 71 (1997), 103–133. · Zbl 0881.28013
[23] I. Putnam, Cantor minimal systems, a survey and a new proof, Expo. Math. 28 (2010), 101–131. · Zbl 1231.37006
[24] M. Roychowdhury, Irrational rotation of the circle and the binary odometer are finitary orbit equivalent, Publ. Res. Inst. Math. Sci. 43 (2007), 385–402. · Zbl 1129.37004
[25] C. Sutherland, Subfactors and ergodic theory, Current Topics in Operator Algebras, World Sci., Singapore, 1991, pp. 38–42. · Zbl 0817.46063
[26] F. Sugisaki, The relationship between entropy and strong orbit equivalence for the minimal homeomorphisms (I), Internat. J. Math. 14 (2003), 735–772. · Zbl 1055.37007
[27] F. Sugisaki, The relationship between entropy and strong orbit equivalence for the minimal homeomorphisms (II), Tokyo J. Math. 21 (1998), 311–351. · Zbl 1063.37500
[28] A. M. Vershik, A theorem on periodical Markov approximation in ergodic theory, Ergodic Theory and Related Topics, Akademie-Verlag, Berlin, 1982, pp. 195–206. · Zbl 0517.47005
[29] H. Yuasa, Not finitely but countably Hopf-equivalent clopen sets in a Cantor minimal system, Topol. Methods Nonlinear Anal. 33(2009), 355–371. · Zbl 1182.54048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.