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

Matui, Hiroki

J. Math. Soc. Japan 54, No. 1, 135-160 (2002).

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.

Reviewer: Ľubomír Snoha (Banská Bystrica)

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)

Keywords:

Cantor minimal system; dimension group; automorphism; skew product