Journal of the Mathematical Society of Japan

An absorption theorem for minimal AF equivalence relations on Cantor sets

Hiroki MATUI

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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 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 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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1225894037

Digital Object Identifier
doi:10.2969/jmsj/06041171

Mathematical Reviews number (MathSciNet)
MR2467874

Zentralblatt MATH identifier
1170.37009

Subjects
Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)

Keywords
Cantor sets orbit equivalence minimal dynamical systems

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


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References

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  • T. Giordano, H. Matui, I. F. Putnam and C. F. Skau, Orbit equivalence for Cantor minimal $\Z^d$-systems, in preparation.
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