The absorption theorem for affable equivalence relations

We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being `small' in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case--when Y is a finite set--this result is highly non-trivial. The result itself--called the absorption theorem--is a powerful and crucial tool for the study of the orbit structure of minimal Z^n-actions on the Cantor set [GMPS]. The absorption theorem is a significant generalization of the main theorem proved in [GPS2]. However, we shall need a few key results from [GPS2] in order to prove the absorption theorem.