Full groups of Cantor minimal systems. (English) Zbl 0942.46040

For a dynamical system \((X,\phi)\) on the compact Hausdorff space \(X\), the associated full group \([\phi]\) is the subgroup of all homeomorphisms \(\gamma\) of \(X\) such that \(\gamma(x)\in \text{Orb}_\phi(x)\) (the \(\phi\)-orbit of \(x\)) for all \(x\in X\). In the particular case of a Cantor minimal system (i.e. \(\phi\) is a minimal homeomorphism of a Cantor set \(X\)), the authors obtain an analogue of Krieger’s theorem [W. Krieger, Math. Ann. 252, 87-95 (1980; Zbl 0472.54028)] which characterizes the topological orbit equivalence of two minimal homeomorphisms on a Cantor set \(X\).
In this paper, the full group \([\phi]\) of a Cantor minimal system \((X,\phi)\) is defined; namely a homeomorphism \(\psi\) of \(X\) belongs to \([\phi]\) if \(\psi(x)= \phi(x)^{n(x)}\), \(n(x)\in \mathbb{Z}\) for all \(x\in X\). The topological full group \(\tau[\phi]\) of \((X,\phi)\) is the subgroup of \([\phi]\) consisting in the homeomorphisms whose associated orbit cocycle \(n(x)\) is continuous. Following Dye’s definition [H. A. Dye, Am. J. Math. 85, 551-576 (1963; Zbl 0191.42803)], for an open set \({\mathcal O}\in X\) and for \(\Gamma\) the full group (i), or the topological full group (ii), or a minimal AF-system (iii), the local subgroup \(\Gamma_{\mathcal O}\) of \(\Gamma\) is defined by \(\Gamma_{\mathcal O}= \{\gamma\in\Gamma\); \(\gamma(x)= x\), for all \(x\in{\mathcal O}^c\}\).
In Section 3, the local subgroups \(\Gamma_U\) for \(U\) a clopen subset of \(X\) is algebraically characterized. In Section 4, the results of the preceeding section are used to prove the following result:
Theorem. Let \((X_1,\phi_1)\) and \((X_2,\phi_2)\) be Cantor minimal systems.
(i) \((X_1,\phi_1)\) and \((X_2,\phi_2)\) are orbit if and only if \([\phi_1]\) and \([\phi_2]\) are isomorphic.
(ii) \((X_1,\phi_1)\) and \((X_2,\phi_2)\) are flip-conjugate if and only if \(\tau[\phi_1]\) and \(\tau[\phi_2]\) are isomorphic.
(iii) \((X_1,\phi_1)\) and \((X_2,\phi_2)\) are strong orbit equivalent if and only if \(\tau[\phi_1]_{y_1}\) and \(\tau[\phi_2]_{y_2}\) are isomorphic for any \(y_i\in X_i\), \(i= 1,2\).
(For \(y\in X\), \(\tau[\phi]_y\) denotes the subgroup of \(\gamma\in \tau[\phi]\) such that \(\gamma(\text{orb}^+_\phi(y))= \text{Orb}^+_\phi(y)\), where \(\text{Orb}^+_\phi(y)\) is the forward \(\phi\)-orbit of \(y\).)
In Section 5, the authors show that, up to normalization, there exists only one non-trivial homomorphism from \(\tau[\phi]\) to \(\mathbb{Z}\) called the index map, the kernel of which is a complete algebraic invariant of flip-conjugacy of a Cantor minimal system \((X,\phi)\).


46L80 \(K\)-theory and operator algebras (including cyclic theory)
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