## Journal of the Mathematical Society of Japan

### An absorption theorem for minimal AF equivalence relations on Cantor sets

Hiroki MATUI

#### Abstract

We prove that a ‘small’ extension of a minimal AF equivalence relation on a Cantor set is orbit equivalent to the AF relation. By a ‘small’ extension we mean an equivalence relation generated by the minimal AF equivalence relation and another AF equivalence relation which is defined on a closed thin subset. The result we obtain is a generalization of the main theorem in [GMPS2]. It is needed for the study of orbit equivalence of minimal $\bm{Z}^d$ -systems for $d>2$ [GMPS3], in a similar way as the result in [GMPS2] was needed (and sufficient) for the study of minimal $\bm{Z}^2$ -systems [GMPS1].

#### Article information

Source
J. Math. Soc. Japan, Volume 60, Number 4 (2008), 1171-1185.

Dates
First available in Project Euclid: 5 November 2008

https://projecteuclid.org/euclid.jmsj/1225894037

Digital Object Identifier
doi:10.2969/jmsj/06041171

Mathematical Reviews number (MathSciNet)
MR2467874

Zentralblatt MATH identifier
1170.37009

#### Citation

MATUI, Hiroki. An absorption theorem for minimal AF equivalence relations on Cantor sets. J. Math. Soc. Japan 60 (2008), no. 4, 1171--1185. doi:10.2969/jmsj/06041171. https://projecteuclid.org/euclid.jmsj/1225894037

#### References

• T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal $\Z^2$-systems, J. Amer. Math. Soc., 21 (2008), 863–892.
• T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, The absorption theorem for affable equivalence relations, to appear in Ergodic Theory Dynam. Systems, arXiv:0705.3270.
• T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal $\Z^d$-systems, in preparation.
• T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and $C^*$-crossed products, J. Reine Angew. Math., 469 (1995), 51–111.
• T. Giordano, I. F. Putnam and C. F. Skau, Affable equivalence relations and orbit structure of Cantor dynamical systems, Ergodic Theory Dynam. Systems, 24 (2004), 441–475.