Universally measure-preserving homeomorphisms of Cantor minimal systems
A Cantor minimal system \((X,S)\) is a compact, metrizable, totally disconnected space \(X\) with no isolated point, endowed with a minimal homeomorphism \(S: X\to X\). Such a system \((X,S)\) is known to be topologically conjugate to some symbolic dynamics, namely some ``adic'' transformation arising from some ``simple ordered Bratteli diagram'', yielding a sort of coding of \((X,S)\) [\textit{R. H. Herman}, \textit{I. F. Putnam} and \textit{C. F. Skau}, ``Ordered Bratteli diagrams, dimension groups and topological dynamics'', Int. J. Math. 3, No. 6, 827--864 (1992; Zbl 0786.46053)]). The concern of the authors is the following characterization of orbit equivalence between Cantor minimal systems \((X,S)\) and \((Y,T)\), first established in [``Full groups of Cantor minimal systems'', Isr. J. Math. 111, 285--320 (1999; Zbl 0942.46040)] by \textit{T. Giordano}, \textit{I. F. Putnam} and \textit{C. F. Skau}: there exists a homeomorphism \(\phi:X\to Y\) such that (for any \(x\in X\)) \[ \{\phi\cdot S^i(x)\mid i\in \mathbb{Z}\}= \{T^i\cdot\phi(x)\mid i\in\mathbb{Z}\} \] if and only if there exists a homeomorphim \(\widetilde\phi: X\to Y\) such that for any probability measure \(\mu\) on \(X\), \(\mu\) is \(S\)-invariant if and only if \(\mu\circ\widetilde\phi^{- 1}\) is \(T\)-invariant. The aim of this article is to provide a new proof of this theorem, using dynamical methods, which is hopefully also more elementary (i.e., involving neither \(K\)-theory nor hand-logical algebra). The idea of the authors is, following Glasner and Weiss, to apply and improve a copying lemma by Katznelson and Weiss, and to use a finitary orbit equivalence technique on Bratteli diagrams.
scientific article; zbMATH DE number 5905291
scientific article