An absorption theorem for minimal AF equivalence relations on Cantor sets

TL;DR

This paper proves that small extensions of minimal AF equivalence relations on Cantor sets remain orbit equivalent to AF relations, generalizing previous results and expanding understanding of their structure.

Contribution

It introduces a new class of extensions of minimal AF relations on Cantor sets and shows they are orbit equivalent to AF relations, broadening the scope of existing theorems.

Findings

Small extensions of minimal AF relations are orbit equivalent to AF relations.

The result generalizes previous theorems to broader classes of equivalence relations.

Provides a framework for understanding extensions 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].