An absorption theorem for minimal AF equivalence relations on Cantor sets

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].