Universally measure-preserving homeomorphisms of cantor minimal systems

Abstract

In 1995, T. Giordano, I. Putnam, and C. Skau [GPS] made a significant breakthrough in Cantor minimal system theory. They completely classified Cantor minimal systems in the sense of topological orbit equivalence by using C*-algebra and homological algebra techniques. Since then, a dynamical proof of their theorem has been conjectured. Such a proof is presented in this paper. We establish orbit equivalence theory based on a finite coordinate change relation arising from an ordered Bratteli diagram, which is known from [HK] in the finitary case of ergodic probability measure-preserving transformations. We obtain the Orbital Extension Theorem. This theorem is considered a topological version of the Copying Lemma of Y. Katznelson and B. Weiss [KW], which has played an important role in measured orbit equivalence theory.