Strong orbit equivalence and residuality
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 second block is devoted to the topological properties of the metric space \((\mathcal{S}(T,x_0),m_T)\), where \(m_T\) is a suitable metric defined as NEWLINE\[NEWLINE m_T(S,S')=\inf_{\varepsilon>0}\{a(x)=a'(x)\,,\;b(x)=b'(x)\;\text{ for all }x\in X\setminus B(x_0,\varepsilon)\} + \sup_{x\in X}d(Sx,S'x), NEWLINE\]NEWLINE where \(a\), \(b\), \(a'\), \(b'\) are the respective cocycles of \(S\) and \(S'\) relative to \(T\) and \(d\) is a fixed metric in \(X\) compatible with its topology (and \(B(x,r)\) is the respective ball with center \(x\) and radius \(r\)). The author shows that this metric space is separable and complete but non-compact. In fact, for every \(S\in\mathcal{S}(T,x_0)\), there exists a dense and countable family of minimal Cantor systems conjugate to \(S\), and these systems are constructed explicitly.NEWLINENEWLINEThe final block deals with questions about typical properties of systems in \(\mathcal{S}(T,x_0)\), here ``typical'' means ``in a residual set''.NEWLINENEWLINEFirst, if \(T\) is of finite rank \(K\) then the set of finite rank systems is residual in \(\mathcal{S}(T,x_0)\) (finite rank \(K\) means that the Bratteli-Vershick system representing \(T\) has at most \(K\) vertices on each level). At this point, I would like to address a question in this review about the non-finite rank case: is the set of finite rank systems dense/residual?. In the special case where \(T\) has rank \(1\) (an odometer), the rank \(1\) systems are residual.NEWLINENEWLINEFinally, it is shown that null entropy systems are also residual. The author address to [\textit{P. Walters}, An introduction to ergodic theory. New York etc.: Springer (1982; Zbl 0475.28009)] for the basic topics on topological entropy. The main step is the remarkable fact that any minimal Cantor system is strongly orbit equivalent to a minimal Cantor system with null entropy (see [\textit{M. Boyle} and \textit{D. Handelman}, Pac. J. Math. 164, No. 1, 1--13 (1994; Zbl 0812.58025)]). Thus its conjugacy class in \(\mathcal{S}(T,x_0)\) is formed by null entropy systems and it is dense by previous results. Finally, considering a sequence of coverings \(\{\mathcal{P}_l\}\) generating the topology of \(X\), the author shows that the set of systems \(S\in\mathcal{S}(T,x_0)\) with \(h(S,\mathcal{P}_l)<1/n\) is open in \(\mathcal{S}(T,x_0)\) (where \(h(S,\mathcal{P}_l)\) is the entropy respective to the covering \(P_l\), and recall that \(\lim_l h(S,\mathcal{P}_l)=h(S)\)) finishing the proof.
scientific article; zbMATH DE number 6187739