A dynamical proof of Matui's absorption theorem
Julien Melleray
TL;DR
This work provides an elementary dynamical proof of Matui's absorption theorem for minimal ample group actions on the Cantor set, clarifying a gap in prior approaches. It introduces malleable subsets and a strengthened Krieger-type result to upgrade local conjugacies to global ones via a back-and-forth construction, and then proves that adding countably many copies of a malleable piece (absorption) does not change orbit equivalence. The absorption result underpins a classification strategy for minimal ample groups by orbit equivalence, extending ideas known for $\mathbb Z$-actions. An appendix supplies a corrigendum that explains how to fix the related argument in Giordano–Putnam–Skau using the strengthened absorption theorem.
Abstract
We give a dynamical, relatively elementary proof of an "absorption theorem" which is closely related to a well-known result due to Matui. The construction is in the spirit of an earlier joint work of the author and S. Robert. In an appendix we explain how to use this result to correct the dynamical proof given by Melleray--Robert of a classification theorem for orbit equivalence of minimal ample groups due to Giordano, Putnam and Skau (the original argument had a gap).