Strong orbit equivalence and residuality
type: Item
instance of: scholarly article
title: Strong orbit equivalence and residuality
zbMATH Open document ID: 1279.37017
author: Brett M. Werner
published in: Topological Methods in Nonlinear Analysis
publication date: 2013-07-12T00:00:00Z
reviewed by: Carlos Meniño
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 6187739
zbMATH Keywords: minimal Cantor system
zbMATH Keywords: orbit equivalence
MaRDI profile type: Publication
review text: In this paper, the author defines a space of minimal Cantor systems which contains (up to conjugacy) all minimal Cantor systems strongly orbit equivalent to a given one. A metric is defined in this space and it is showed that it becomes a (non-compact) separable, complete metric space. The author studies typical properties of this space; it is a remarkable fact that null entropy minimal Cantor systems form a residual set in this space.NEWLINENEWLINEThe paper is organized in three main blocks. In the first one, the author motivates the work and explains the background and prerequisites required for a good understanding of the results. In this way, the paper is nearly self-contained and the bibliography is accurate and well addressed. The key result is addressed to [\textit{T. Giordano}, \textit{I. F. Putnam} and \textit{C. Skau}, J. Reine Angew. Math. 469, 51--111 (1995; Zbl 0834.46053)], where it is shown that every minimal Cantor system is conjugate to a Bratteli-Vershik system and for each pair of strongly orbit equivalent Bratteli-Vershick systems there exists a strong orbit equivalence preserving maximal and minimal paths and preserving the cofinality of paths.NEWLINENEWLINEThis leads to the definition of the space \(\mathcal{S}(T,x_0)\) of minimal Cantor systems \(S\) which are related to a given minimal Cantor system \(T:X\to X\) by the following: {\parindent=6mm \begin{itemize}\item[-] \(S(x_0)=T(x_0)\) \item[-] \(\mathcal{O}^-_S(x_0)=\mathcal{O}^-_T(x_0)\) (past orbits relative to \(S\) and \(T\) of \(x_0\) agree) \item[-] \(\mathcal{O}^+_S(x_0)=\mathcal{O}^+_T(x_0)\) (future orbits relative to \(S\) and \(T\) of \(x_0\) agree) \item[-] \(S\) is orbit equivalent to \(T\) by the identity and the associated cocycles \(a\), \(b\) are continuous in \(X\setminus\{x_0\}\) NEWLINENEWLINE\end{itemize}} The previous results guarantee that any minimal Cantor system strongly orbit equivalent to \(T\) is conjugate to some \(S\in\mathcal{S}(T,x_0)\).NEWLINENEWLINEThe sec…
description: scientific article; zbMATH DE number 6187739