Finite order automorphisms and dimension groups of Cantor minimal systems. (English) Zbl 1029.37003

Let \((X,\phi)\) be a Cantor minimal system, i.e., \(X\) is the Cantor set and \(\phi\) is a minimal homeomorphism on \(X\). Denote by \(C(\phi)\) the group of all homeomorphisms on \(X\) commuting with \(\phi\) and call it the automorphism group or commutant group of \((X,\phi)\). Let \(B_{\phi} = \{f-f\circ \phi^{-1} : f\in C(X,\mathbb Z)\}\) be the coboundary subgroup of integer-valued continuous functions on \(X\). The dimension group \(K^0 (X, \phi)\) of \((X,\phi)\) is the quotient of \(C(X,\mathbb Z)\) by \(B_{\phi}\). Let \(T(\phi) = \{\psi \in C(\phi) : \psi ^* = \text{Id} \}\) where \(\psi ^*\) is the automorphism of the dimension group \(K^0 (X,\phi)\) induced by \(\psi\).
The author studies commutant groups by using dimension groups of Cantor minimal systems. He computes the dimension group of the skew product extension of a Cantor minimal system associated with a finite group-valued cocycle. Using it, he studies finite subgroups in the commutant group of a Cantor minimal system and proves that every finite subgroup of \(T(\phi)\) is cyclic. It is an open problem whether \(T(\phi)\) is always abelian. Further, in the paper it is given a certain obstruction for a finite subgroup \(G\) of \(C(\phi)\) which has nontrivial intersection with \(T(\phi)\). It is also given a necessary and sufficient condition for dimension groups so that \(T(\phi)\) can include an element of finite order.


37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
46L55 Noncommutative dynamical systems
54H20 Topological dynamics (MSC2010)
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