## 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.

### MSC:

 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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