Universally measure-preserving homeomorphisms of Cantor minimal systems
type: Item
instance of: scholarly article
title: Universally measure-preserving homeomorphisms of Cantor minimal systems
zbMATH Open document ID: 1233.37003
author: Toshihiro Hamachi
author: Michael Keane
author: Hisatoshi Yuasa
published in: Journal d'Analyse Mathématique
publication date: 2011-06-08T00:00:00Z
review text: 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.
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 37A55
Mathematics Subject Classification ID: 37A20
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 5905291
zbMATH Keywords: measure preserving homeomorphisms
zbMATH Keywords: Cantor minimal system
zbMATH Keywords: Bratteli diagram
zbMATH Keywords: symbolic dynamics
zbMATH Keywords: edge sequence space
zbMATH Keywords: topological orbit equivalence
MaRDI profile type: Publication
full work available at URL: s11854-011-0001-3
OpenAlex ID: W1989184176
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DOI: 10.1007/S11854-011-0001-3
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description: scientific article; zbMATH DE number 5905291