Finite order automorphisms and dimension groups of Cantor minimal systems ^{∗}^{†}

MATUI Hiroki

Abstract

We compute the dimension group of the skew product extension of a Cantor minimal system associated with a finite group valued cocycle. Using it, we study finite subgroups in the commutant group of a Cantor minimal system and prove that a finite subgroup of the kernel of the mod map must be cyclic. Moreover, we give a certain obstruction for finite subgroups of commutant groups to have nonzero intersection to the kernel of mod maps. We also give a necessary and sufficient condition for dimension groups so that the kernel of the mod map can include a finite order element.

1 Introduction

When X is the Cantor set and ϕ is a homeomorphism on X which has no non-trivial invariant closed subset, the topological dynamical system (X, ϕ) is called a Cantor minimal system. Cantor minimal systems are analogues of ergodic systems in the topological setting and several authors have been studied them in the various ways. Among many subjects about Cantor minimal systems, the study of automorphism groups is one of the most mysterious parts. We denote by C(ϕ) the set of all homeomorphisms on X commuting with ϕ for a Cantor minimal system (X, ϕ) and call it the automorphism group or commutant group of (X, ϕ). The symbol C(ϕ) usually means the set of all continuous maps γ : X → X which commute with ϕ, but we restrict our attention only on the set of commuting homeomorphisms in this paper. As ϕ is minimal, C(ϕ) acts on X freely. We denote by Zϕ the subgroup generated by ϕ in C(ϕ). In the present paper, we give a new direction in the study of commutant groups, by using dimension groups of Cantor minimal systems.

The notion of dimension group was introduced for Cantor minimal systems in [HPS] and this new invariant threw a new light on the study of Cantor minimal systems. Let (X, ϕ) be a Cantor minimal system and

B_{ϕ} = {f − f ∘ ϕ^{−1} ; f ∈ C(X, Z)}

be the coboundary subgroup of integer valued continuous functions C(X, Z). The dimension group K^{0}(X, ϕ) of a Cantor minimal system (X, ϕ) is the quotient of C(X, Z) by B_{ϕ}.

∗1991 Mathematics Subject Classification. 54H20.

†Key words and Phrases. Cantor minimal system, dimension group, automorphism

Define the positive cone

K^{0}(X, ϕ)^{+} = {[f] ∈ K^{0}(X, ϕ) ; f ∈ C(X, Z)^{+}},

where the bracket means the quotient map. The dimension group K^{0}(X, ϕ) is an ordered group with this positive cone and the equivalence class [1] of the constant function one is called the order unit of K^{0}(X, ϕ). It was proved in [GPS] that K^{0}(X, ϕ), as an ordered group with a distinguished order unit, characterizes the strong orbit equivalence class of (X, ϕ). The dimension group K^{0}(X, ϕ) is order isomorphic to the K_{0}-group of the C^{∗}-algebra C^{∗}(X, ϕ). In this paper, however, we don’t deal with C^{∗}-algebras.

One of purposes of this paper is to compute the dimension group of the Cantor minimal system (Y, ψ) arising from the skew product extension of a Cantor minimal system (X, ϕ) associated with a finite group valued cocycle. The dimension group of a Cantor minimal system (X, ϕ) is usually computed as the inductive limit system arising from the ordered Bratteli diagram of (X, ϕ) ([HPS]). Unfortunately, however, there is no explicit way to write down the ordered Bratteli diagram of the skew product system (Y, ψ) by means of the ordered Bratteli diagram of the original system (X, ϕ) and the cocycle. In Theorem 2.5, we will describe the dimension group K^{0}(Y, ψ) as the quotient of the restricted dimension group by the canonical infinitesimal subgroup.

Our main tool for the study of commutants is the mod map. Since γ ∈ C(ϕ) satisfies g ∘ γ^{−1} ∈ B_{ϕ} for all g ∈ B_{ϕ},

mod(γ)([f]) = [f ∘ γ^{−1}]

gives rise to an order automorphism of K^{0}(X, ϕ) preserving the order unit ([GPS2]). It can be easily checked that the mod map is a group homomorphism from C(ϕ) to Aut(K^{0}(X, ϕ)). We define T (ϕ) = C(ϕ) ∩ ker mod. In Section 3, we will prove that every finite subgroup of T (ϕ) is cyclic. Moreover, we will show that if a finite subgroup G of C(ϕ) includes an element of prime order p and it is in T (ϕ), then the p-Sylow group of G is cyclic. It should be remarked that every finite group can be embedded into the commutant group of a Cantor minimal system, which was shown in [LM]. Next, we will consider when the kernel of the mod map can contain a finite order automorphism, and give a necessary and sufficient condition for dimension groups. The invariant η defined in [M] will be computed for finite order elements. Several examples of finite subgroups of C(ϕ) and T (ϕ) are given in Section 4.

Acknowledgements.

The author is grateful to Professor Izumi for his helpful advice and encouragement.

2 Dimension groups of skew products

In this section, we define skew product extensions of Cantor minimal systems associated with finite group valued cocycles, and give an algorithm of computing the dimension group of skew product systems.

Definition 2.1 Let (X, ϕ) be a Cantor minimal system and G be a finite group. A continuous map c : X → G is called a G-valued cocycle. G-valued cocycles c and c′ are cohomologous if there is a continuous map b : X → G

such that b(x)c(x)b(ϕ(x))^{−1} = c′(x) holds for all x ∈ X.

Let c : X → G be a cocycle. We set Y = X × G and a homeomorphism ψ on Y such as ψ(x, g) = (ϕ(x), gc(ϕ(x))). This dynamical system (Y, ψ) is called the skew product extension or extension, simply, of (X, ϕ) associated with the G-valued cocycle c. Of cource, cohomologous cocycles determine isomorphic systems.

Let (Y, ψ) be the extension of (X, ϕ) associated with a cocycle c. Obviously, there is a factor map from (Y, ψ) to (X, ϕ). For each element h ∈ G, let γ_{h} be the homeomorphism on Y which sends (x, g) to (x, hg). Then {γ_{g}}_{g∈G} forms a subgroup of C(ψ) isomorphic to G. We call this finite subgroup a canonical commutant of the skew product extension.

Notice that the extension (Y, ψ) is not always minimal. For example, when (X, ϕ) is an odometer system and G is a finite group and not cyclic, the skew product extension (Y, ψ) is never minimal for any G-valued cocycle, because every cocycle c : X → G is cohomologous to a cocycle c′ whose range contains only the identity e and a single element g. The reader may refer to Section VIII.4 of [D] for odometer systems.

Lemma 2.2 Let (Y, ψ) be a Cantor minimal system and G ⊂ C(ψ) be a finite subgroup. Then, there exist a Cantor minimal system (X, ϕ) and a cocycle c : X → G, such that the extension of (X, ϕ) by the cocycle c is isomorphic to (Y, ψ) and the subgroup G coincides with the canonical commutant under this isomorphism.

Proof. Let {γ_{g}}_{g∈G} be the finite subgroup of C(ψ). We can consider the quotient system (X, ϕ) of (Y, ψ) by the action of {γ_{g}}_{g}. Denote the factor map by π. Since Y is the Cantor set, there exists a clopen subset Y_{0} such that the restriction π̃ of π on Y_{0} is a homeomorphism onto X. Set Y_{g} = γ_{g}(Y_{0}) for each g ∈ G. Then, {Y_{g}}_{g} is a clopen partition of Y and a map ρ sending (x, g) ∈ X × G to γ_{g}(π̃^{−1}(x)) is a homeomorphism. The minimal homeomorphism ρ^{−1} ∘ ψ ∘ ρ on X × G determines a cocycle c : X → G. It is clear that {γ_{g}}_{g} coincides with the canonical commutant. □

We fix a Cantor minimal system (X, ϕ) and a finite group valued cocycle c : X → G. Let (Y, ψ) be the extension. We would like to compute the dimension gruop K^{0}(Y, ψ). In order to do this, at first, we must represent the system (X, ϕ) by using an ordered Bratteli diagram ([HPS]) and fix the notation.

Let B = (V, E, ≤) be a simple ordered Bratteli diagram associated with (X, ϕ), where V = ∪_{n=0}^{∞}V_{n} and E = ∪_{n=1}^{∞}E_{n} are the sets of vertices and edges. We denote the range and source map by r and s. For every v ∈ V \ V_{0}, a linear order is defined on r^{−1}(v). Let (e_{1}, e_{2}, ⋯, e_{m}) be the ordered list of the edges in r^{−1}(v). We define a map θ from V \ V_{0} to the set of finite sequences consisting of vertices such as θ(v) = (s(e_{1}), s(e_{2}), ⋯, s(e_{m})). In this paper, we use the map θ to describe the partial order on E. We can identify (X, ϕ) with the infinite path space of B and the Bratteli-Vershik map on it. For each n ∈ N, the vertex set V_{n} corresponds to towers of Kakutani-Rohlin partitions of X. We denote this partition by

P_{n} = {X(n, v, k) ; v ∈ V_{n}, 1 ≤ k ≤ h(v)},

where each X(n, v, k) is the clopen set of level k in the tower corresponding to v and h : V → N is the height of the tower. Here, we have ϕ(X(n, v, k)) = X(n, v, k + 1) for v ∈ V_{n} and 0 ≤ k ≤ h(v) − 1. We set

X_{n} = ⋃_{v∈V_{n}} X(n, v, h(v))

and call it the top set of n-th step. Then, we also get

ϕ(X_{n}) = ⋃_{v∈V_{n}} X(n, v, 1).

The sequence of the top sets {X_{n}}_{n} is decreasing and shrinks to one point set {x_{max}}. We also have that P_{n+1} is a finer partition than P_{n} and {P_{n}}_{n} generates the topology of X.

Let us recall the way of computing the dimension group K^{0}(X, ϕ). We denote by C(P_{n}) the set of integer valued functions on X which are constant on each clopen set of P_{n}. The characteristic functions on the clopen sets contained in the same tower of P_{n} give the same element of K^{0}(X, ϕ). Therefore, we can view that C(P_{n}) forms a free abelian group Z^{V_{n}}. We denote the canonical basis of Z^{V_{n}} by the same symbols {v; v ∈ V_{n}} as vertices. For each n ∈ N, the edge set E_{n} determines the incidence matrix A_{n} from Z^{V_{n}} to Z^{V_{n+1}}, which is given by

A_{n}(v, w) = #{k ; 1 ≤ k ≤ h(w), X(n + 1, w, k) ⊂ X(n, v, h(v))}

for v ∈ V_{n} and w ∈ V_{n+1}. Hence, the dimension group is computed such as

K^{0}(X, ϕ) = lim A_{n} : Z^{V_{n}} → Z^{V_{n+1}}.

If we set u_{n} = ∑_{v∈V_{n}} h(v)v ∈ Z^{V_{n}} for all n ∈ N, we get u_{n}A_{n} = u_{n+1} and {u_{n}}_{n} is the order unit of K^{0}(X, ϕ).

We would like to consider the dimension group of the skew product (Y, ψ). We may assume that the Kakutani-Rohlin partition P_{1} is finer than the clopen partition determined by the cocycle c. We can define a map d from V \ V_{0} to G as follows;

d(v) = c(x)c(ϕ(x))c(ϕ^{2}(x)) ⋯ c(ϕ^{h(v)}(x)),

where x is an arbitrary point in X(n, v, 1). We call d(v) the label of the vertex v determined by the cocycle c. Let w be a vertex of V_{n+1}. If θ(w) = (v_{1}, v_{2}, ⋯, v_{m}), we get

d(w) = d(v_{1})d(v_{2}) ⋯ d(v_{m})

from the definition of d.

Lemma 2.3 In the above setting, the skew product (Y, ψ) is a Cantor minimal system if and only if {d(v) ; v ∈ V_{n}} generates G for every n ∈ N.

Proof. Assume {d(v) ; v ∈ V_{n}} does not generate G for some n. We can replace the cocycle c to c′, in the same cohomology class, whose range generates a proper subgroup of G, and so the extension (Y, ψ) is not minimal.

Let us prove the converse. We assume that d(V_{n}) generates G for all n. We can identify X and x_{max} ∈ X with the infinite path space of B = (V, E, ≤) and the unique maximal path. It suffices to show that the set

H = \{g ∈ G ; (x_{max}, g) ∈ \overline{Orb_{ψ}((x_{max}, e))}\}

coincides with G. Since H is a subgroup of G, it is enough to show that H includes a generating set of G. Let x_{max} = (e_{1}, e_{2}, ⋯) and ϕ(x_{max}) = (f_{1}, f_{2}, ⋯) be the unique maximal and minimal paths. We also assume that all maximal edges of E_{n} have the same source vertex as e_{n}. Take n ∈ N \ {1, 2} and set θ(r(f_{n})) = (v_{1}, v_{2}, ⋯, v_{m}). When we define

h_{i} = ∑_{j=1}^{i} h(v_{j}), g_{i} = d(v_{1})d(v_{2}) ⋯ d(v_{i})

for i = 1, 2, ⋯, m, we have ψ^{h_{i}}(x_{max}, e) = (ϕ^{h_{i}}(x_{max}), g_{i}) and the first n − 2 edges of ϕ^{h_{i}}(x_{max}) agree with (e_{1}, e_{2}, ⋯, e_{n−2}). We set K_{n} = {g_{i} ∈ G ; i = 1, 2, ⋯, m}. From the assumption, K_{n} is a generating set of G. Because G is a finite set, there exists a generating set K of G such that K_{n} = K holds for infinitely many n. Then we get (x_{max}, g) ∈ \overline{Orb_{ψ}((x_{max}, e))} for all g ∈ K, and so the proof is completed. □

From now on, we assume that the skew product (Y, ψ) is a Cantor minimal system. We define Kakutani-Rohlin partitions {Q_{n}}_{n} for (Y, ψ) as the following;

Y (n, v, k, g) = X(n, v, k) × {g},
Q_{n} = {Y (n, v, k, g) ; v ∈ V_{n}, 1 ≤ k ≤ h(v), g ∈ G},

and we set the top set as

Y_{n} = X_{n} × G = ⋃_{v∈V_{n},g∈G} Y (n, v, h(v), g).

The partition Q_{n} consists of #(V_{n} × G) towers, and {Q_{n}}_{n} generates the topology of Y. However, we should note that the intersection of the top sets {Y_{n}}_{n} is not one point. It is equal to {x_{max}} × G. Therefore, we cannot use the same method as the case of (X, ϕ). We need a proposition obtained by Putnam in order to compute the dimension group K^{0}(Y, ψ).

Proposition 2.4 ([P, Theorem 4.1]) Let (Y, ψ) be a Cantor minimal system and {y_{i}}_{i=1}^{m} be a finite subset of Y lying in distinct orbits. When we define

D = C(Y, Z)/{f − f ∘ ψ^{−1} ; f ∈ C(Y, Z), f(y_{i}) = 0 for all i = 1, 2, ⋯, m},

it is an ordered group. Moreover, the sequence

0 → Z ⟶^{j} Z^{m} ⟶^{δ} D ⟶^{q} K^{0}(Y, ψ) → 0

is exact, where the map j sends the generator of Z to (1, 1, ⋯, 1) and q is the natural quotient map. The map δ is given as follows; for u = (u_{i})_{i} ∈ Z^{m}, we take f ∈ C(Y, Z) such as f(y_{i}) = u_{i} and define δ(u) to be the equivalence class of f − f ∘ ψ^{−1}.

We would like to apply the proposition above for the extension (Y, ψ) and the finite subset {x_{max}} × G. Let us define

K^{0}(Y, ψ; G) = C(Y, Z)/{f − f ∘ ψ^{−1} ; f ∈ C(Y, ψ), f(x_{max}, g) = 0 for all g ∈ G},

and call it the restricted dimension group of (Y, ψ). The restricted dimension group K^{0}(Y, ψ; G) can be represented as an inductive limit sequence by means of the Kakutani-Rohlin partitions {Q_{n}}_{n}. At first, we give this description of K^{0}(Y, ψ; G).

Since the partition Q_{n} consists of #(V_{n}×G) towers, we can consider that C(Q_{n}) forms a free abelian group Z^{V_{n}×G} ≅ Z^{V_{n}} ⊗ Z[G]. We denote the canonical basis of Z^{V_{n}} ⊗ Z[G] by {v ⊗ g ; v ∈ V_{n}, g ∈ G}, and choose the characteristic function on Y (n, v, h(v), g) as the representative of v ⊗ g. Let us consider the incidence matrix B_{n} from n-th step to n + 1-th step. If v ∈ V_{n} and w ∈ V_{n+1} satisfy X(n + 1, w, k) ⊂ X(n, v, h(v)) for some k ∈ {1, 2, ⋯, h(w)}, then, for all g ∈ G, Y (n + 1, w, k, g) ⊂ Y (n, v, h(v), g) holds and there exists h ∈ G such that ψ^{h(w)−k}(Y (n+1, w, k, g)) = Y (n+1, w, h(w), gh). Of cource, if k = h(w), h is the identity. Otherwise, the element h is given by

h = c(x)c(ψ(x))c(ψ^{2}(x)) ⋯ c(ψ^{h(w)−k−1}(x)),

where x is an arbitrary point in X(n + 1, w, k + 1). We would like to write down the incidence matrix as the matrix the size of V_{n} × V_{n+1} which has entries in the group ring Z[G]. Let w be a vertex in V_{n+1} and assume θ(w) = (v_{1}, v_{2}, ⋯, v_{m}). For a vertex v ∈ V_{n}, we define non-negative integers {λ_{g}}_{g} as follows;

λ_{g} = #{i ; 1 ≤ i ≤ m, v = v_{i}, g = d(v_{i+1})d(v_{i+2}) ⋯ d(v_{m})},

and set B_{n}(v, w) = ∑_{g∈G} λ_{g}g. From the argument above, we see that the matrix B_{n} represents the connecting map from n-th step to n + 1-th step. Then, we get

K^{0}(Y, ψ; G) = lim B_{n} : Z^{V_{n}} ⊗ Z[G] → Z^{V_{n+1}} ⊗ Z[G],

where vectors are considered as row vectors of Z[G]^{V_{n}} and the product of vectors and matrices are computed in the obvious way. The positive cone is obtained by the inductive limit of the canonical positive cone of Z^{V_{n+1}}⊗Z[G], and the order unit is given by ∑_{g∈G} u_{n}⊗ g ∈ Z^{V_{n+1}} ⊗ Z[G] at n-th step.

[figure: V_{n}; E_{n}; V_{n+1}; v_{1}; v_{2}; w_{1}; w_{2}; w_{3}; Fig. 1; θ(w_{1}) = (v_{1}, v_{1}, v_{2}); θ(w_{2}) = (v_{1}, v_{2}, v_{2}, v_{1}, v_{2}); θ(w_{3}) = (v_{1}, v_{2}, v_{2})]

For example, consider the case of the diagram in Fig. 1, where the partial order in the edge set E_{n} is expressed by the map θ. If the label of v_{i} is g_{i} ∈ G for each i = 1, 2, then we have d(w_{1}) = g_{1}g_{1}g_{2}, d(w_{2}) = g_{1}g_{2}g_{2}g_{1}g_{2} and d(w_{3}) = g_{1}g_{2}g_{2}. The usual incidence matrix A_{n} : Z^{V_{n}} → Z^{V_{n+1}} and the incidence matrix B_{n} : Z[G]^{V_{n}} → Z[G]^{V_{n+1}} of the extension are given by

A_{n} = [ \begin{matrix}2 & 2 & 1 \\ 1 & 3 & 2 \end{matrix} ] , B_{n} = [ \begin{matrix}g_{1}g_{2} + g_{2} & g_{2}g_{2}g_{1}g_{2} + g_{2} & g_{2}g_{2} \\ e & g_{2}g_{1}g_{2} + g_{1}g_{2} + e & g_{2} + e \end{matrix} ] .

We can obtain A_{n} from B_{n}, sending each entry of B_{n} by the canonical ring homomorphism from Z[G] to Z.

In order to compute the dimension group K^{0}(Y, ψ), we must describe explicitly the image of the map δ in Proposition 2.4. We can realize that δ is the map from Z[G]. For g ∈ G and n ∈ N, let f be the characteristic function on the clopen set X_{n} × {g}. Then, we have f(x_{max}, g) = 1 and f(x_{max}, h) = 0 for h ≠ g, and so the image of g ∈ Z[G] by the map δ is given by the equivalence class of f − f ∘ ψ^{−1} in K^{0}(Y, ψ; G). When x_{v} ∈ X is a point in X(n, v, 1) for each v ∈ V_{n}, the function f ∘ψ^{−1} is the characteristic function on the clopen set ∪_{v∈V_{n}}Y (n, v, 1, gc(x_{v})). Since ψ^{h(v)−1}(Y (n, v, 1, gc(x_{v})) = Y (n, v, h(v), gd(v)) for every v ∈ V_{n}, we can conclude that δ(g) equals

e(n, g) = ∑_{v∈V_{n}} v ⊗ g − v ⊗ gd(v) ∈ Z^{V_{n}} ⊗ Z[G]

at n-th step. The equation e(n, g)B_{n} = e(n+1, g) can be checked easily, and so {e(n, g)}_{n} determines a subgroup Z in K^{0}(Y, ψ; G). We denote by Z_{n} the subgroup of Z^{V_{n}} ⊗ Z[G] generated by {e(n, g) ; g ∈ G}. Remark that ∑_{g} e(n, g) equals zero for all n.

Theorem 2.5 Keep the above notation. The dimension group K^{0}(Y, ψ) of the skew product (Y, ψ) is obtained as the quotient of

K^{0}(Y, ψ; G) = lim B_{n} : Z^{V_{n}} ⊗ Z[G] → Z^{V_{n+1}} ⊗ Z[G]

by the subgroup {Z_{n}}_{n} which is isomorphic to Z^{#G−1}.

Let us consider the action on K^{0}(Y, ψ) induced by the canonical commutant {γ_{g}}_{g}. In the following argument, we use the description of K^{0}(Y, ψ) in the above theorem. Because γ_{h} transfers Y (n, v, h(v), g) to Y (n, v, h(v), hg), the induced action on Z^{V_{n}} ⊗ Z[G] sends v ⊗ g to v ⊗ hg for all v ∈ V_{n} and g ∈ G. Assume γ_{h} ∈ T (ψ), that is, mod(γ_{h}) = id. Then, for all n ∈ N, v ∈ V_{n} and g ∈ G, there exists m such that

(v ⊗ (g − hg))B_{n}B_{n+1} ⋯ B_{n+m}

is contained in Z_{n+m+1}. The following lemma is needed to prove the main theorem in the next section.

Lemma 2.6 Let (Y, ψ) be the skew product as above and H be a normal subgroup of G such that mod(γ_{h}) = id for all h ∈ H. By telescoping the ordered Bratteli diagram B = (V, E, ≤), we may assume that (v ⊗ (g − hg))B_{n} ∈ Z_{n+1} for all v ∈ V_{n}, g ∈ G and h ∈ H.

For w ∈ V_{n+1}, let K be the cyclic subgroup of G generated by d(w). Then, we have A_{n}(v, w) ∈ lZ for all v ∈ V_{n}, where the natural number l is the index of the subgroup H ∩ K in H.

Proof. Let v be a vertex of V_{n}, and B_{n}(v, w) = ∑_{g∈G} λ_{g}g. We will prove ∑_{g∈G} λ_{g} ∈ lZ. From the assumption, there exists {μ_{g}}_{g∈G} ⊂ Z such that

(v ⊗ (e − h))B_{n} = ∑_{g∈G} μ_{g}e(n + 1, g),

and so we have

(e − h) ∑_{g∈G} λ_{g}g = ∑_{g∈G} μ_{g}(g − gd(w)),

which implies λ_{g} − λ_{h^{−1}g} = μ_{g} − μ_{gd(w)^{−1}} for all g ∈ G. When we write λ̃_{g} = ∑_{k∈K} λ_{gk}, we get λ̃_{g} = λ̃_{hg} and this equation holds for all h ∈ H. Let G = ∪_{i=1}^{m}Hg_{i}K be the decomposition into double cosets by H and K. Since H is a normal subgroup of G, each double coset consists of l left K-cosets. Hence, we get ∑_{g∈G} λ_{g} = l ∑_{i=1}^{m} λ̃_{g_{i}} ∈ lZ. □

3 Commutants and dimension groups

We will prove the main theorems about finite subgroups of the commutant group in this section. When a Cantor minimal system (Y, ψ) has a finite subgroup G in C(ψ), by Lemma 2.2, we can regard it as the skew product extension obtained by a Cantor minimal system (X, ϕ) and a cocycle c. Moreover, the dimension group K^{0}(Y, ψ) can be described as in Theorem 2.5 by means of the ordered Bratteli diagram of (X, ϕ). We often use this identification in the following argument.

Lemma 3.1 Let (Y, ψ) be a Cantor minimal system and τ ∈ C(ψ) be an element of finite order n. If γ ∈ T (ψ) satisfies γ ∘ τ ∘ γ^{−1} = τ^{k} for some k, then k must be one.

Proof. We may assume there exist a Cantor minimal system (X, ϕ) and a continuous function f : X → Z/nZ such that

Y = X × Z/nZ, ψ(x, l) = (ϕ(x), l + f(x)), τ(x, l) = (x, l + 1)

for all x ∈ X and l ∈ Z/nZ. Let π be the factor map from (Y, ψ) to (X, ϕ). From the assumption we have π ∘ γ ∘ τ = π ∘ γ, and so there exists γ̃ ∈ C(ϕ) such that γ̃ ∘ π = π ∘ γ. We denote by π^{∗} the injection from K^{0}(X, ϕ) to K^{0}(Y, ψ) induced by π ([GW, Proposition 3.1]). It can be easily seen that mod(γ) ∘ π^{∗} = π^{∗} ∘ mod(γ̃), hence we get mod(γ̃) = id, thus γ̃ ∈ T (ϕ). Therefore we can find a continuous function g : X → Z/nZ such as

f − f ∘ γ̃ = g − g ∘ ϕ.

We define a homeomorphism γ_{0} on Y = X × Z/nZ by γ_{0}(x, l) = (γ̃(x), l + g(x)) for (x, l) ∈ Y. It is clear that γ_{0} commutes with τ. We can also check that π ∘ γ^{−1} ∘ γ_{0} = π and

ψ ∘ γ_{0}(x, l) = (ϕ ∘ γ̃(x), l + g(x) + f(γ̃(x)))
= (γ̃ ∘ ϕ(x), l + f(x) + g(ϕ(x)))
= γ_{0} ∘ ψ(x, l)

for all (x, l) ∈ Y. Then we conclude that there exists m such that γ^{−1} ∘ γ_{0} = τ^{m} holds, which implies that γ commutes with τ. □

The following lemma shows that if T (ψ) contains a finite abelian subgroup H, then H is cyclic.

Lemma 3.2 Let (Y, ψ) be a Cantor minimal system and G be a finite p-subgroup of C(ψ) for a prime p. If a normal subgroup H of G is included in T (ψ) and the quotient of G by H is cyclic, then G is also cyclic.

Proof. Let ρ be the quotient map from G to G/H. We may consider that (Y, ψ) is the extension of (X, ϕ) by a G-valued cocycle c, and (X, ϕ) is represented by an ordered Bratteli diagram. We use the same notation as in Section 2. Because {γ_{h}}_{h} ∈ H is contained in T (ϕ), we may assume (v ⊗ (e − h))B_{1} ∈ Z_{2} for all v ∈ V_{1} and h ∈ H. Since (Y, ψ) is minimal, d(V_{2}) generates G, which implies that there exists a vertex w ∈ V_{2} such that ρ(d(w)) is a generator of the cyclic group G/H. Let K be the cyclic subgroup of G generated by d(w). By Lemma 2.6, we have A_{1}(v, w) ∈ lZ for every v ∈ V_{1}, where l is the index of H ∩ K in H. On the other hand, when θ(w) is equal to (v_{1}, v_{2}, ⋯, v_{m}), d(w) is the product of d(v_{i})’s, and so

ρ(d(w)) = ∑_{i=1}^{m} ρ(d(v_{i})) = ∑_{v∈V_{1}} A_{1}(v, w)ρ(d(v)).

If l is a multiple of p, ρ(d(w)) cannot be a generator of G/H. Hence, l is one and H is included in K, which means that G is a cyclic group generated by d(w). □

Using the above lemmas, we can prove the following.

Theorem 3.3 If (Y, ψ) is a Cantor minimal system and H is a finite subgroup of T (ψ), then H is a cyclic group.

Proof. We say that a group G has a property M if there exists a Cantor minimal system (Y, ψ) such that G is contained in T (ψ). The proof is by contradiction. Assume that H is a finite non-cyclic group of minimum order which has the property M. It is clear that all proper subgroups of H are cyclic. Take a Cantor minimal system (Y, ψ) which satisfies H ⊂ T (ψ). If K is a non-trivial normal subgroup of H, then we can consider the quotient system (X, ϕ) of (Y, ψ) by the action of K. By the same argument as in Lemma 3.1, we can deduce that H/K is contained in T (ϕ), that is, H/K has property M. From the assumption we infer that H/K is cyclic, and so we obtain that H is an abelian group by Lemma 3.1. However, we know that a finite abelian group having property M must be cyclic by Lemma 3.2, and it contradicts to the assumption of H. As a consequence, H is a simple group whose every proper subgroup is cyclic. In light of Exercise 7.(a) in Section 2.2 of [S], we get a contradiction. □

For substitution minimal subshifts of constant length, the automorphism groups were computed in [LM] and [HP], and it was also shown that there is a substitution minimal subshift (X, ϕ) for an arbitrary finite group G such that the commutant group C(ϕ) is isomorphic to G ⊕ Zϕ. In these papers, actually, measure-theoretic automorphims were investigated and we can deduce the same results directly in the topological dynamical setting. However, the above theorem says that the same statement never hold for T (ϕ). We will show that there exists a Cantor minimal system (Y, ψ) for every n ∈ N, which satisfies C(ψ) = T (ψ) = Z/nZ ⊕ Zψ, in the next sectinon.

Lemma 3.4 Let (Y, ψ) be a Cantor minimal system and Q be the quaternion group. If C(ψ) contains Q, then Q ∩ T (ψ) is trivial.

Proof. Assume Q is generated by a, b, c ∈ Q satisfying a = b^{2} = c^{2}, a^{2} = e and bc = acb. We may consider that (Y, ψ) is the extension of a Cantor minimal system (X, ϕ) associated with a Q-valued cocycle. We use the same notation as in Section 2. The proof is by contradiction. If Q ∩ T (ψ) is non-trivial, a ∈ Q must be in T (ψ). We may assume that (v ⊗ (e − a))B_{1} is included in Z_{2} for all v ∈ V_{1}. Since d(V_{2}) generates Q, we may also assume that there exist vertices w_{1}, w_{2} ∈ V_{2} such that d(w_{1}) = b and d(w_{2}) = c. Let v ∈ V_{1} be an arbitrary vertex. When we set B_{1}(v, w_{i}) = ∑_{g∈Q} λ_{g}^{i}g for i = 1, 2, there exists {ν_{g}}_{g} ⊂ Z such that

∑_{g∈Q}(e − a)λ_{g}^{i}g = ∑_{g∈Q} ν_{g}g(e − d(w_{i}))

holds for i = 1, 2. From these equations, we have

(λ_{e}^{1} − λ_{a}^{1}) + (λ_{b}^{1} − λ_{ab}^{1}) + (λ_{c}^{1} − λ_{ac}^{1}) + (λ_{abc}^{1} − λ_{bc}^{1})
= ν_{e} − ν_{ab} + ν_{b} − ν_{e} + ν_{c} − ν_{bc} + ν_{abc} − ν_{c}
= ν_{b} − ν_{abc} + ν_{abc} − ν_{ab} + ν_{abc} − ν_{ab} + ν_{ab} − ν_{bc}
= (λ_{b}^{2} − λ_{ab}^{2}) + (λ_{abc}^{2} − λ_{bc}^{2}) + (λ_{abc}^{2} − λ_{bc}^{2}) + (λ_{ab}^{2} − λ_{b}^{2}) ∈ 2Z.

Consequently we get A_{1}(v, w_{1}) = ∑_{g∈Q} λ_{g}^{1} ∈ 2Z. Because v is an arbitrary vertex of V_{1}, we obtain a contradiction as in Lemma 3.2. □

In the next section, we will give some examples of finite subgroups of C(ψ) and T (ψ). The following theorem, however, shows that there exists an obstruction for a finite group G which has non-trivial intersection with T (ψ).

Theorem 3.5 Let (Y, ψ) be a Cantor minimal system, G be a finite subgroup of C(ψ) and p be a prime. If a p-Sylow group of G has non-trivial intersection with T (ψ), then the p-Sylow group is cyclic.

Proof. We may assume G is a p-group which has non-zero intersection with T (ψ). From Lemma 3.2, we see that all the abelian subgroups of G are cyclic. Hence, by using of (4.4) of Section 4.4 in [S], we deduce that G is a cyclic group or a generalized quaternion group. If G is a generalized quaternion group, it contains the quaternion group Q and Q has non-trivial intersection with T (ψ). By means of Lemma 3.4, we get a contradiction.

□

We would like to consider a finite cyclic subgroup Z/mZ of T (ψ). In the rest of this section, we denote a generator of Z/mZ by a and the canonical ring homomorphism from Z[Z/mZ] to Z by ρ and set P = ∑_{j=0}^{m−1} a^{j}.

Let (Y, ψ) be the Cantor minimal system obtained as the extension of (X, ϕ) associated with a cocycle c : X → Z/mZ. We denote the generator of the canonical commutant by γ ∈ C(ψ). If a function f ∈ C(Y, Z) is fixed by mod(γ) in K^{0}(Y, ψ), there exists r ∈ Z[Z/mZ] such that

[f − f ∘ γ^{−1}] = δ(r),

where the bracket means the equivalence class in the restricted dimension group K^{0}(Y, ψ; G) and δ is as in Section 2. Then, we can define the map η̃(γ) from ker(id − mod(γ)) to Z/mZ by

η̃(γ)([f]) = ρ(r) + mZ.

The following is another obstruction for actions of commutants. Note that we can identify the dimension group K^{0}(X, ϕ) with Im ∑_{j=0}^{m−1} mod(γ)^{j}.

Lemma 3.6 When (Y, ψ) is a Cantor minimal system and γ ∈ C(ψ) is an element of order m, we have

ker(id − mod(γ))/ Im ∑_{j=0}^{m−1} mod(γ)^{j} ≅ Z/mZ,

and

ker ∑_{j=0}^{m−1} mod(γ)^{j} = Im(id − mod(γ)).

Proof. Let (Y, ψ) be the skew product extension of a Cantor minimal system (X, ϕ) associated with a Z/mZ-valued cocycle and η̃(γ) be as above. We use the same notation as in Section 2. Let λ_{v} ∈ {0, 1, ⋯, m − 1} be the natural number such that d(v) = a^{λ_{v}}. For every n, we set

s_{n} = ∑_{v∈V_{n}} v ⊗ (e + a + ⋯ + a^{λ_{v}})

in K^{0}(Y, ψ; G). It is not hard to check that the quotient image of s_{n} in K^{0}(Y, ψ) drops into ker(id−mod(γ)), and its value by η̃(γ) is one in Z/mZ. Therefore, η̃(γ) is surjective. We would like to show the injectivity. Let s = ∑_{v∈V_{n}} v ⊗ s_{v} be an arbitrary element of Z^{V_{n}} ⊗ Z[Z/mZ], and assume

∑_{v∈V_{n}} v ⊗ (e − a)s_{v} = ∑_{v∈V_{n}} v ⊗ (e − d(v))r

for some r ∈ Z[Z/mZ]. Then, s_{v} equals (e + a + ⋯ + a^{λ_{v}})r modulo a scalar multiple of P. If ρ(r) is zero modulo m, there exists r′ ∈ Z[Z/mZ] such that r is equal to (e − a)r′ modulo a scalar multiple of P. Because v ⊗ P is in Im ∑_{j=0}^{m−1} mod(γ)^{j}, we can see that s equals

∑_{v∈V_{n}} v ⊗ (e − d(v))r′

modulo Im ∑_{j=0}^{m−1} mod(γ)^{j}, which is zero in K^{0}(Y, ψ).

The other equation is proved in a similar fashion. □

Remark. The above lemma gives a quite strong restriction for the existence of finite order automorphisms. For example, we have the following.

(i) If K^{0}(Y, ψ) is isomorphic to Z[1/r] ⊕ Z for an odd number r as an abelian group, there does not exist an order two element in C(ψ).

(ii) If K^{0}(Y, ψ) is isomorphic to Z^{s+1} as an abelian group and s is not divisible by a prime p, then there does not exist an element of order p in C(ψ).

We would like to consider when there exists a finite order element in the kernel of the mod map. We say a triple (G, G^{+}, u) is a dimension group in an abstract sense, if (G, G^{+}) is an unperforated ordered group satisfying the Riesz interpolation property and u is a distinguished element of G^{+} \ {0} called the order unit ([GPS, Section 1]). For a Cantor minimal system (X, ϕ), of cource, (K^{0}(X, ϕ), K^{0}(X, ϕ)^{+}, [1]) becomes a dimension group in this meaning. Two dimension groups are said to be isomorphic, when there is an isomorphism preserving the positive cones and the order units.

Theorem 3.7 When (G, G^{+}, u) is a simple dimension group except Z, and m is a natural number, the following are equivalent.

(i) There exists a Cantor minimal system (Y, ψ) such that (K^{0}(Y, ψ), K^{0}(Y, ψ)^{+}, [1]) is isomorphic to (G, G^{+}, u) and T (ψ) has an element of order m.

(ii) G/mG is isomorphic to Z/mZ as an abelian group and the order unit u of G is divisible by m.

Proof. The implication (i) ⇒ (i) follows from Lemma 3.6.

Let us prove the other implication (ii) ⇒ (i). Take an element u_{0} ∈ G^{+} such as u = mu_{0}. For the simple dimension group (G, G^{+}, u_{0}), there exists a Bratteli diagram (V, E) and G is order isomorphic to the inductive limit

lim A_{n} : Z^{V_{n}} → Z^{V_{n+1}},

where A_{n} denotes the incidence matrix determined by E_{n}. Since G/mG is isomorphic to Z/mZ, by telescoping the diagram (V, E), we can choose a representative s_{n} = ∑_{v∈V_{n}} λ_{v}v ∈ Z^{V_{n}} of a generator of G/mG satisfying

1 ≤ λ_{v} ≤ m, s_{n}A_{n} ≡ s_{n+1} (mod m)

for every n ∈ N. We may further assume that there is c_{v} ∈ {0, 1, 2, ⋯, m − 1} such as

vA_{n} ≡ c_{v}s_{n+1} (mod m)

for every v ∈ V_{n} and n ∈ N, and it follows that

∑_{v∈V_{n}} λ_{v}c_{v} ≡ 1 (mod m).

Let X be the infinite path space of (V, E). Although a partial order on E has not yet been defined, we can find a cocycle c : X → Z/mZ which determines the label d : V_{n} → Z/mZ such that d(v) = a^{λ_{v}}.

We would like to define a partial order on the edge set E and construct a Cantor minimal system (X, ϕ) so that the canonical commutant of the skew product extension (Y, ψ) of (X, ϕ) associated with c is included in T (ψ). In this case, we must note that the dimension group of (Y, ψ) is automatically isomorphic to (G, G^{+}, u). By telescoping, we may assume that each entry of A_{n} is not less than m^{2} + m. Fix a natural number n and an arbitrary vertex v_{0} ∈ V_{n}. We will define a linear order on each r^{−1}(w) for w ∈ V_{n+1}, so that v_{0} is the source vertex of the minimum and maximum edges of r^{−1}(w) for all w and the incidence matrix B_{n} from Z[Z/mZ]^{V_{n}} to Z[Z/mZ]^{V_{n+1}} satisfies

(v ⊗ (e − a))B_{n} ∈ Z_{n+1}

for every v ∈ V_{n}.

We construct a finite set Γ_{w} of directed edges on the vertex set Z/mZ for each w ∈ V_{n+1}. Because ∑_{v∈V_{n}} λ_{v}c_{v} equals one modulo m, we can define a finite set Γ_{v} of directed edges over Z/mZ for every v ∈ V_{n}, as follows;

#Γ_{v} = c_{v}, s(x)r(x)^{−1} = a^{λ_{v}} for all x ∈ Γ_{v},

and for each b ∈ Z/mZ

#(r^{−1}(b) ∩ ⋃_{v∈V_{n}} Γ_{v}) − #(s^{−1}(b) ∩ ⋃_{v∈V_{n}} Γ_{v}) = \{ \begin{matrix}1 & b = e \\ −1 & b = a \\ 0 & otherwise \end{matrix} .

Moreover, we can associate an element r_{v} ∈ Z[Z/mZ] with v ∈ V_{n} by

r_{v} = ∑_{x∈Γ_{v}} r(x).

For each v ∈ V_{n} and w ∈ V_{n+1}, we set

Γ_{v,w} = Γ_{v} × {1, 2, ⋯, λ_{w}}

and

r(x, k) = r(x)a^{k−1}, s(x, k) = s(x)a^{k−1}

for all (x, k) ∈ Γ_{v,w}. Let Γ′_{v} be the set of directed edges over the vertex set Z/mZ such as

Γ′_{v} = {x_{0}, x_{1}, ⋯, x_{m−1}},
r(x_{k}) = a^{k}, s(x_{k}) = a^{k+λ_{v}}.

Since A_{n}(v, w) is equal to #Γ_{v,w} modulo m, by adding some disjoint copies of Γ′_{v} to Γ_{v,w}, we get the set Γ′_{v,w} consisting of A_{n}(v, w) directed edges. The element r_{v,w} ∈ Z[Z/mZ] associated with Γ_{v,w} is defined by

r_{v,w} = ∑_{y∈Γ′_{v,w}} r(y)

and it equals to

(e + a + ⋯ + a^{λ_{w}−1})r_{v}

modulo a scalar multiple of P and satisfies ρ(r_{v,w}) = A_{n}(v, w). Let Γ_{w} be the disjoint union of Γ′_{v,w} for all v ∈ V_{n}. For b ∈ Z/mZ, we have

#{x ∈ Γ_{w} ; r(x) = b} − #{x ∈ Γ_{w} ; s(x) = b}
= ∑_{k=1}^{λw} ∑_{v∈V_{n}} #(r^{−1}(ba^{−k+1}) ∩ Γ_{v}) − #(s^{−1}(ba^{−k+1}) ∩ Γ_{v})

= \{ \begin{matrix}1 & b = e, b ≠ d(w) \\ −1 & b = d(w), b ≠ e \\ 0 & otherwise \end{matrix} .

Since the directed graph Γ_{w} includes at least two copies of Γ′_{v} for every v ∈ V_{n}, thanks to the unicursal theorem, we can find a directed path y⃗ = (y_{1}, y_{2}, ⋯, y_{l}) which starts from d(w) and ends at e and in which each element of Γ_{w} appears exactly once. Moreover, we can choose the path so that y_{1} and y_{l} is contained in Γ′_{v_{0},w}. We can obtain from the directed path y⃗ the ordered list (e_{1}, e_{2}, ⋯, e_{l}) of edges in r^{−1}(w) such that the source vertex of e_{k} in the Bratteli diagram (V, E) is v ∈ V_{n} when y_{k} is in Γ′_{v,w}. In this way, the partial order on E_{n} is well defined and it gives a simple order on the Bratteli diagram (V, E). It is clear that B_{n}(v, w) is equal to r_{v,w}, and so we get

(v ⊗ (e − a))B_{n} = ∑_{w∈V_{n+1}} w ⊗ (e − d(w))r_{v} ∈ Z_{n},

thereby completing the proof. □

Let us consider the invariant η defined in [M]. The homomorphism η was defined on T (ψ) and takes its value in Ext(K^{0}(Y, ψ), Z). The Ext group is the cokernel of the natural map from Hom(K^{0}(Y, ψ), R) to Hom(K^{0}(Y, ψ), R/Z). In Section 5 of [M], we constructed a representative Φ(γ) of η(γ) in Hom(K^{0}(Y, ψ), R/Z) for γ ∈ T (ϕ), by fixing an invariant measure μ on Y.

Let (Y, ψ) be the Cantor minimal system obtained as the extension of (X, ϕ) associated with a cocycle c : X → Z/mZ. Assume the canonical commutant γ is in T (ψ). Then, the map η̃(γ) is identified with the quotient map from K^{0}(Y, ψ) to K^{0}(Y, ψ)/mK^{0}(Y, ψ) ≅ Z/mZ.

Lemma 3.8 In the above setting, η̃(γ) is a representative of η(γ).

Proof. Fix a ψ-invariant measure μ on Y. By taking the average of {μ ∘ γ^{j}}_{j=0}^{m}, we may assume that μ is also invariant under the action of the canonical commutant. We denote Z/mZ by G. Let Φ(γ) be the element of Hom(K^{0}(Y, ψ), R/Z) determined by μ (Section 5 of [M]). It suffices to show that η̃(γ)([f]) coincides with Φ(γ)([f]) for the characteristic function f on the clopen set Y (n, v, h(v), b) for all n ∈ N, v ∈ V_{n} and b ∈ G.

We can assume that there exists {ν_{k}}_{k} ⊂ Z such that (v ⊗ (b − ab))B_{n} is equal to ∑_{k∈G} ν_{k}e(n, k). Then, by definition, we have

η̃(γ)([f]) = \frac{1}{m} ∑_{k∈G} ν_{k} + Z.

Let (v ⊗ b)B_{n} = ∑_{w∈V_{n+1},k∈G} λ_{w,k}w ⊗ k. We define the function f̃ ∈ C(Y, Z) by

f̃ = ∑_{w∈V_{n+1},k∈G} λ_{w,k}χ_{Y (n+1,w,h(w),k)},

where χ means a characteristic function. Since we have [f] = [f̃] in K^{0}(Y, ψ), there exists a function F_{0} ∈ C(Y, Z) such that f − f̃ = F_{0} − F_{0} ∘ ψ^{−1}, which implies

f − f ∘ γ^{−1} = F_{0} − F_{0} ∘ γ^{−1} − (F_{0} − F_{0} ∘ γ^{−1}) ∘ ψ^{−1} + f̃− f̃∘ γ^{−1},

and μ(F_{0} − F_{0} ∘ γ^{−1}) = 0 by the definition of μ. Moreover, we have

f̃− f̃∘ γ^{−1} = ∑_{w∈V_{n+1},k∈G}(λ_{w,k} − λ_{w,a^{−1}k})χ_{Y (n+1,w,h(w),k)}
= ∑_{w∈V_{n+1},k∈G}(ν_{k} − ν_{kd(w)^{−1}})χ_{Y (n+1,w,h(w),k)},

and it is equal to F_{1} − F_{1} ∘ ψ^{−1}, where F_{1} is a function such as

F_{1} = ∑_{w∈V_{n+1},k∈G} ∑_{j=1}^{h(w)} ν_{k}χ_{Y (n+1,w,h(w),kd(w))} ∘ ψ^{j}.

Therefore, we get

Φ(γ)([f]) = μ(F_{1}) + Z = ∑_{w∈V_{n+1},k∈G} h(w)ν_{k}μ(Y (n + 1, w, h(w), kd(w))) + Z
= \frac{1}{m} ∑_{w∈V_{n+1},k∈G} h(w)ν_{k}μ(X(n + 1, w, h(w)) × G) + Z
= \frac{1}{m} ∑_{k∈G} ν_{k} + Z,

and so η̃(γ) is equal to Φ(γ). □

The following corollary says that η is either zero or injective on finite order elements of T (ψ).

Corollary 3.9 When (Y, ψ) is a Cantor minimal system and γ ∈ T (ψ) is an element of finite order, we have η(γ) = 0 if and only if Hom(K^{0}(Y, ψ), Z) ≠ 0, and in this case Hom(K^{0}(Y, ψ), Z) is isomorphic to Z.

Proof. Let γ be an element of order m. If η(γ) = 0, the representative η̃(γ) has a lifting to Hom(K^{0}(Y, ψ), R). Since η̃(γ) is a surjection to Z/mZ, the lifting gives a non-trivial element of Hom(K^{0}(Y, ψ), Z).

Conversely, if Hom(K^{0}(Y, ψ), Z) is non-zero, we get a surjection from K^{0}(Y, ψ) to Z/mZ. Because η̃(γ) coincides with the quotient map to K^{0}(Y, ψ)/mK^{0}(Y, ψ) ≅ Z/mZ, it should have a lifting to Hom(K^{0}(Y, ψ), Z). Therefore, η(γ) is zero in the Ext group. It is easy to check Hom(K^{0}(Y, ψ), Z) ≅ Z. □

4 Examples

(1) Let n, m be natural numbers and l = nm. We give a Cantor minimal system (Y, ψ) such that C(ψ) = Z/lZ ⊕ Zψ and T (ψ) = Z/mZ ⊕ Zψ, by using a bijective substitution ([LM]). Let ξ be a substitution of constant length l + m on the alphabet set L = {0, 1, 2, ⋯, l − 1} such as

ξ(0)_{i} = \{ \begin{matrix}i − k & (k − 1)(n + 1) + 1 ≤ i ≤ k(n + 1), k = 1, 2, ⋯, m − 1 \\ i − m & (m − 1)(n + 1) + 1 ≤ i ≤ l + m − 1 \\ 0 & i = l + m \end{matrix}

and ξ(j)_{i} = ξ(0)_{i} + j for j = 0, 1, ⋯, l − 1 and i = 1, 2, ⋯, l + m, where the addition is understood modulo l. Note that Morse substitution is obtained when we put n = 1 and m = 2. Let (Y, ψ) be the substitution subshift determined by ξ. We refer the reader to [DHS] for basic facts about substitution subshifts and Bratteli diagrams. We define a homeomorphism γ on L^{Z} such as γ(x)_{i} = x_{i} + 1 for x ∈ L^{Z} and i ∈ Z. It is easy to check that γ(Y) = Y and γ ∈ C(ψ). By Theorem 5 of [LM], C(ψ) is isomorphic to Z/lZ ⊕ Z and γ is a generator of Z/lZ. The substitution rule ξ can be extended to a continuous map ξ : Y → Y.

We define a map π : Y → {0, 1}^{Z} such as

π(x)_{i} = \{ \begin{matrix}0 & x_{i} − x_{i−1} = 0 \\ 1 & x_{i} − x_{i−1} = 1, 1 − l \end{matrix}

for x ∈ Y. It can be easily seen that π is a well-defined factor map from (Y, ψ) to the subshift on {0, 1}^{Z}. We denote by (X, ϕ) the image of (Y, ψ) by π. We can see that π(x) = π(x′) if and only if x and x′ have the same γ-orbit in Y, and so the system (X, ϕ) is the quotient system of (Y, ψ) by the action of γ. Let y ∈ Y be the fixed point of ξ such that y_{−1} = y_{0} = 0. By the definition of π, we get

π(y) = ⋯ 1| \underbrace{0 \overbrace{11 ⋯ 1}^{n} 0 \overbrace{11 ⋯ 1}^{n} ⋯ 0 \overbrace{11 ⋯ 1}^{n}}_{(n+1)×m} 1 \overbrace{11 ⋯ 1}^{n} ⋯,

where the vertical bar separates π(y)_{(−∞,−1]} from π(y)_{[0,∞)}. Define the substituion rule ζ of constant length l + m by

ζ(0) = \underbrace{0 \overbrace{11 ⋯ 1}^{n} 0 \overbrace{11 ⋯ 1}^{n} ⋯ 0 \overbrace{11 ⋯ 1}^{n}}_{(n+1)×(m−1)} 0 \overbrace{11 ⋯ 1}^{n}

ζ(1) = \underbrace{0 \overbrace{11 ⋯ 1}^{n} 0 \overbrace{11 ⋯ 1}^{n} ⋯ 0 \overbrace{11 ⋯ 1}^{n}}_{(n+1)×(m−1)} 1 \overbrace{11 ⋯ 1}^{n}

on the alphabet set {0, 1}. Let z ∈ {0, 1}^{Z} be the fixed point of ζ such that z_{−1} = 1 and z_{0} = 0, where ζ is extended to a continuous map on {0, 1}^{Z}. Then, we have

z = lim_{k→∞} ϕ^{1+n+n(l+m)+···+n(l+m)^{k}}(π(y)),

which implies that (X, ϕ) is the substitution subshift associated with ζ. When we put

c(x) = \{ \begin{matrix}e & x_{0} = 0 \\ a & x_{0} = 1 \end{matrix}

for x ∈ X, where a denotes the generator of Z/lZ, the system (Y, ψ) is the extension of (X, ϕ) associated with the cocycle c.

We wolud like to compute the dimension groups of (X, ϕ) and (Y, ψ). Since the substitution rule ζ is proper, we can easily write down the ordered Bratteli diagram (V, E, ≤) of (X, ϕ) ([DHS, Proposition 16]). Every vertex set V_{n} consists of two points, namely v_{n,0} and v_{n,1}, which correspond to the alphabets 0 and 1. The dimension group K^{0}(X, ϕ) is the inductive limit of Z^{2} with the incidence matrix

A = [ \begin{matrix}m & m − 1 \\ l & l + 1 \end{matrix} ] .

Therefore, we have

K^{0}(X, ϕ) ≅ Z[\frac{1}{l + m}] ⊕ Z,

(p, q) ∈ K^{0}(X, ϕ)^{+} if and only if p = q = 0 or p + \frac{(m − 1)q}{l + m − 1} > 0

and the order unit is (1, 0).

Let us consider the skew product extension (Y, ψ). The label of each vertex is determined by the cocycle c and we get d(v_{n,0}) = e, d(v_{n,1}) = a. Hence, we have

B = [ \begin{matrix}∑_{j=1}^{m} a^{jn} & ∑_{j=2}^{m} a^{jn+1} \\ ∑_{j=1}^{l} a^{j} & e + ∑_{j=1}^{l} a^{j} \end{matrix} ]

and the restricted dimension group K^{0}(Y, ψ; Z/lZ) is obtained as the inductive limit of Z^{2} ⊗ Z[Z/lZ] with the incidence matrix B. It is easy to check that (e − a^{n}, 0)B and (0, e − a^{n})B are contained in the subspace spanned by {(0, a^{k} − a^{k+1})}_{k}, and so we have γ^{n} ∈ T (ψ). Let (Z, τ) be the quotient system of (Y, ψ) by the action of γ^{n}. The dimension group K^{0}(Z, τ) is naturally isomorphic to the subgroup mK^{0}(Y, ψ) of K^{0}(Y, ψ), because γ^{n} acts identically on K^{0}(Y, ψ). Since the dimension group is torsion free, it suffices to compute K^{0}(Z, τ) instead of K^{0}(Y, ψ). We let c′ be the composition of c and the quotient map from Z/lZ to Z/nZ. The system (Z, τ) is isomorphic to the extension of (X, ϕ) associated with the cocycle c′. We denote Z/nZ by G. The restricted dimension group K^{0}(Z, τ; G) is obtained by the inductive limit of Z^{2} ⊗ Z[G] with the incidence matrix

C = [ \begin{matrix}me & (m − 1)b \\ m ∑_{j=1}^{n} b^{j} & e + m ∑_{j=1}^{n} b^{j} \end{matrix} ] ,

where b is the generator of G. By using Theorem 2.5, we can express the dimension groups K^{0}(Z, τ) and K^{0}(Y, ψ) within the abelian group Z[1/(l + m)] ⊕ Z ⊕ (Z[1/m] ⊗ Z[G]) such as

\{(\frac{p}{(l + m)^{k}} , q, \frac{1}{m^{k}} r(e − b)) ; \begin{matrix}p, q ∈ Z, k ∈ N, r ∈ Z[G], \\ p + lq = (l + m − 1)ρ(r) \end{matrix} \}

with the strict ordering from the first coordinate, where ρ is the canonical ring homomorphism from Z[G] to Z. The order unit of K^{0}(Z, τ) and K^{0}(Y, ψ) are given by (n(l+m−1), 0, 0) and (l(l+m−1), 0, 0) respectively. The canonical commutant γ ∈ C(ψ) acts on the dimension group K^{0}(Y, ψ) as the multiplication by b in the last coordinate, and so T (ψ) is isomorphic to Z/mZ ⊕ Zψ. By Corollary 3.9, we can see that η(γ^{n}) is zero.

When ξ is the Morse substitution, (Y, ψ) and (X, ϕ) are isomorphic to the systems described in Example (4) of [M].

(2) Let S_{3} be the symmetric group of degree three generated by a and b satisfying a^{3} = b^{2} = e and bab = a^{−1}. The element a forms the normal subgroup Z/3Z of S_{3}. We construct a Cantor minimal system (Y, ψ) such as C(ψ) = S_{3}⊕Zψ and T (ψ) = Z/3Z⊕Zψ.

Define a proper substitution ξ on the alphabet set L = {0, 1, 2} such that

ξ(0) = 010021112112
ξ(1) = 000002101212
ξ(2) = 010210121112,

and let (X, ϕ) be the associated substitution subshift. Thanks to Proposition 5 of [DHS], we obtain the stationary ordered Bratteli diagram (V, E, ≤) which represents (X, ϕ). The vertex set V_{n} is canonically identified with L, and so we put V_{n} = {v_{n,0}, v_{n,1}, v_{n,2}} for each n ∈ N. The incidence matrix from Z^{V_{n}} to Z^{V_{n+1}} is represented as

A = [ \begin{matrix}3 & 6 & 3 \\ 6 & 3 & 6 \\ 3 & 3 & 3 \end{matrix} ]

for all n ∈ N under the identification of V_{n} with L. The dimension group K^{0}(X, ϕ) is obtained as the inductive limit of Z^{3} with the incidence matrix A. Hence, we get

K^{0}(X, ϕ) ≅ \{(\frac{p}{12^{n}} , \frac{q}{(−3)^{n}}) ; p, q ∈ Z, p ≡ 2q (mod 5)\}

with the strict ordering from the first coordinate and the order unit is (20, 0).

Let Z = {0, 1, ⋯, 11}^{Z} and τ be the odometer system on Z. Because #r^{−1}(v_{n,i}) = 12 for all n ∈ N \ {1} and i = 0, 1, 2, we can construct a factor map ρ from (X, ϕ) to (Z, τ) in the same way of Section 2 of [GJ]. The same construction of factor maps can be found in Section 7 of [M], too. For every point z ∈ Z, the preimage ρ^{−1}(z) consists of at most three points. Because the substituion rule ξ satisfies

{ξ(0)_{i}, ξ(1)_{i}, ξ(2)_{i}} = {0, 1, 2} if and only if i = 5, 6, 8,

ρ^{−1}(z) includes three distinct points if and only if the tail of z ∈ Z consists of only 4, 5 and 7. When γ is in C(ϕ), there is σ ∈ C(τ) satisfying σ ∘ ρ = ρ ∘ γ and the subset

{z ∈ Z ; #ρ^{−1}(z) = 3}

must be preserved by σ. It follows that σ is a power of τ. On the other hand, if γ ∈ C(ϕ) satisfies ρ ∘ γ = ρ, γ is the identity because there exists a point z ∈ Z such that #ρ^{−1}(z) = 1. As a consequence, we obtain C(ϕ) = Zϕ.

We set a S_{3}-valued cocycle c on X as follows;

c(x) = \{ \begin{matrix}a & x_{0} = 0 \\ b & x_{0} = 1 \\ a^{−1} & x_{0} = 2 \end{matrix}

for all x ∈ X. Then, we have d(v_{n,0}) = a, d(v_{n,1}) = b and d(v_{n,2}) = a^{−1} for all n ∈ N, and so the extension (Y, ψ) associated with the cocycle c is a Cantor minimal system by Lemma 2.3. By the same argument in the last paragraph, we see that the automorphism group C(ψ) is isomorphic to S_{3} ⊕ Zψ.

The restricted dimension group K^{0}(Y, ψ; S_{3}) is computed as the inductive limit of Z^{3} ⊗ Z[S_{3}] by the incidence matrix

B = [ \begin{matrix}e + ab + a^{2}b & e + b + 2ab + 2a^{2}b & e + a + ab \\ 2a + a^{2} + b + ab + a^{2}b & a + a^{2} + b & a + 2a^{2} + b + ab + a^{2}b \\ e + a^{2} + a^{2}b & e + ab + a^{2}b & e + ab + a^{2}b \end{matrix} ] .

When we put

v(g) = (g − ga, g − gb, g − ga^{−1}) ∈ Z^{3} ⊗ Z[S_{3}]

for g ∈ S_{3}, we can check the equations

(e − a, 0, 0)B = v(e) + v(ab),
(0, e − a, 0)B = −v(e) + v(a)

and

(0, 0, e − a)B = −v(a) − v(b),

which imply γ_{a} ∈ T (ψ). In the same method as the exapmle (1), we have

K^{0}(Y, ψ) ≅ \{(\frac{p}{12^{l}} , \frac{q}{(−3)^{l}} , \frac{r}{(−2)^{l}}) ; p, q, r ∈ Z, , p ≡ 2q (mod 5)\}

with the strict ordering from the first coordinate and the order unit is (10, 0, 0). The element b ∈ S_{3} induces mod(γ_{b}) which changes the signal of the last coordinate.

Because there are no non-trivial homomorphisms from K^{0}(Y, ψ) to Z, by Corollary 3.9, η(γ_{a}) is not zero in Ext(K^{0}(Y, ψ), Z). Since (Y, ψ) is not strong orbit equivalent to odometer systems, this example gives a positive answer to the problem (1) raised in [M].

For the dihedral group D_{n} = ⟨a, b ; e = a^{n} = b^{2}, bab = a^{−1}⟩ of order 2n, we can construct a Cantor minimal system (Y, ψ) such that C(ψ) = D_{n} ⊕ Zψ and T (ψ) = Z/nZ ⊕ Zψ, whenever n is odd. Theorem 3.5 tells us that we can never construct it for even n.

(3) Let D_{6} be the dihedral group as above. We will show that there is a Cantor minimal system (Y, ψ) satisfying C(ψ) = D_{6} ⊕ Zψ and T (ψ) = Z/3Z ⊕ Zψ.

We define a proper substitution ξ of constant length such as

ξ(0) = 010121122102122
ξ(1) = 001220000122122
ξ(2) = 011001201001212

on the alphabet set {0, 1, 2}. We denote by (X, ϕ) the substitution minimal subshift determined by ξ. When the cocycle c is defined by

c(x) = \{ \begin{matrix}a & x_{0} = 0 \\ b & x_{0} = 1 \\ a^{−1} & x_{0} = 2 \end{matrix}

for all x ∈ X, we can show that the extension (Y, ψ) is minimal and C(ψ) is isomorphic to D_{3} ⊕ Zψ in the same way as the example (2). Moreover, we have γ_{a}^{2} ∈ T (ψ) and η(γ_{a}^{2}) is not zero. We omit the computation.

(4) We give a Cantor minimal system (Y, ψ) such that ker η is isomophic to Z ⊕ Zψ. Define an ordered Bratteli diagram (V, E, ≤) as follows; the vertex set V_{n} (n ≠ 0)

consists of two vertices, namely v_{n,0}, v_{n,1}, and the incidence matrix from n-th step to n + 1-th step is given by A^{n+1}, where A is

A = [ \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ] .

The partial order on the edge set E is defined by

θ(v_{n+1,0}) = (\overbrace{v_{n,0}, v_{n,0}, ⋯, v_{n,0}}^{3^{n}}, \overbrace{v_{n,0}, ⋯, v_{n,0}}^{m_{n}+1}, \overbrace{v_{n,1}, ⋯, v_{n,1}}^{m_{n}}, \overbrace{v_{n,1}, v_{n,1}, ⋯, v_{n,1}}^{3^{n}})

and

θ(v_{n+1,1}) = (\overbrace{v_{n,0}, v_{n,0}, ⋯, v_{n,0}}^{3^{n}}, \overbrace{v_{n,1}, ⋯, v_{n,1}}^{m_{n}+1}, \overbrace{v_{n,0}, ⋯, v_{n,0}}^{m_{n}}, \overbrace{v_{n,1}, v_{n,1}, ⋯, v_{n,1}}^{3^{n}}),

for n ∈ N, where m_{n} is (3^{n} − 1)/2. When we denote the infinite path space by X and the Bratteli-Vershik map by ϕ, we can see that (X, ϕ) is a Cantor minimal system, since (V, E, ≤) is simple. The dimension group K^{0}(X, ϕ) is isomorphic to Z[1/3] ⊕ Z, where the unique state is given by Z[1/3] ⊕ Z ∋ (p, q) ↦ p + q/2 and the order unit is (1, 0). By the same reason as the above examples, we have C(ψ) ≅ Zϕ.

Let G be the projective limit of Z/3^{n}Z and a be (1, 1, ⋯) ∈ G. The addition by a is the odometer system of type 3^{∞} on G. We denote by ρ_{n} the canonical projection from G to Z/3^{n}Z. Define a G-valued cocycle c on X such as c(x) = a if the infinite path x goes through v_{n,0} and c(x) = a^{−1} otherwise. The extension (Y, ψ) of (X, ϕ) associated with the G-valued cocycle c can be defined in the same way as the case of finite groups and we get the canonical commutant {γ_{g}}_{g∈G} ∈ C(ψ). Moreover, we have C(ψ) = G ⊕ Zψ. The system (Y, ψ) is the projective limit of the extension (Y_{n}, ψ_{n}) associated with the Z/3^{n}Zvalued cocycle ρ_{n} ∘ c. By Proposition 3.7 of [M], (Y, ψ) is a Cantor minimal system and the dimension group K^{0}(Y, ψ) is the inductive limit of K^{0}(Y_{n}, ψ_{n}).

We would like to show that the canonical commutant γ_{g} is in T (ψ) for every g ∈ G. Take an element g ∈ G. When we write γ_{n} as the generator of the canonical commutant of (Y_{n}, ψ_{n}), the restriction of mod(γ_{g}) on K^{0}(Y_{n}, ψ_{n}) is a power of mod(γ_{n}). Therefore, it suffices to show γ_{n} acts identically on K^{0}(Y_{n}, ψ_{n}) for all n ∈ N. The label of the vertices v_{m,0} and v_{m,1} determined by the cocycle ρ_{n} ∘ c are ρ_{n}(a) and ρ_{n}(a^{−1}) respectively. We write B_{m} ∈ M_{2}(Z[Z/3^{n}Z]) as the incidence matrix of the extension (Y_{n}, ψ_{n}) from m-th step to m + 1-th step. Then,

(e − ρ_{n}(a), 0)B_{m} ∈ Z_{m+1}

and

(0, e − ρ_{n}(a))B_{m} ∈ Z_{m+1}

are easily checked for all m > n, where Z_{m} is a subspace of Z^{2} ⊗ Z[Z/3^{n}Z] spanned by

(ρ_{n}(a^{j}) − ρ_{n}(a^{j+1}), ρ_{n}(a^{j}) − ρ_{n}(a^{j−1})) j = 1, ⋯, 3^{n}.

As a consequence, γ_{n} ∈ T (ψ_{n}) is derived, and so T (ψ) = C(ψ) = G ⊕ Zψ.

The dimension grop K^{0}(Y, ψ) is isomorphic to Z[1/3] ⊕ Z[1/3] with the unique state given by (p, q) ↦ p + q/2 and the order unit is (1, 0). The invariant η argued in Lemma 3.8 takes its value in

Ext(Z[\frac{1}{3}] ⊕ Z[\frac{1}{3}], Z) ≅ Ext(Z[\frac{1}{3}], Z) ⊕ Ext(Z[\frac{1}{3}], Z).

Let κ be the natural quotient map from G to Ext(Z[1/3], Z) whose kernel is generated by a ∈ G. For all g ∈ G, we have

η(γ_{g}) = (0, κ(g)),

which implies ker η is isomorphic to Z^{2} generated by γ_{a} and ψ. This example shows that the value of η can be zero for non-trivial infinite order elements, even if Hom(K^{0}(Y, ψ), Z) is zero.

We ramark that all Cantor minimal systems in the above examples are orbit equivalent to odometer systems by Theorem 2.2 of [GPS].

5 Problems

In the final section we present some open problems.

(1) We showed that every finite subgroup of T (ψ) is cyclic in Theorem 3.3, and so abelian, especially. In all examples investigated in last section, T (ψ) is an abelian group. Is the group T (ψ) abelian for every Cantor minimal system (Y, ψ)?

(2) For minimal free Z^{d}-actions on the Cantor set, dimension groups were defined analogously in [F] and we can also define the mod map in the same way. What can we say about the kernel of the mod map or its finite subgroups? When we consider the direct product of d copies of an odometer system, we see that the direct product of d copies of a finite cyclic group can be included in the kernel of the mod map.

References

[D] Davidson, K. R.; C^{∗}-algebras by examples, Fields Institute Monographs 6, Amer. Math. Soc., Providence, RI,1996.

[DHS] Durand, F.; Host, B.; Skau, C. F.; Substitution dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems, 19 (1999), 953-993.

[F] Forrest, A.; A Bratteli diagram for commuting homeomorphisms of the Cantor set, preprint.

[GPS] Giordano, T.; Putnam, I. F.; Skau, C. F.; Topological orbit equivalence and C^{∗}-crossed products, J. reine angew. Math., 469 (1995), 51-111.

[GPS2] Giordano, T.; Putnam, I. F.; Skau, C. F.; Full groups of Cantor minimal systems, Israel J. Math, 111 (1999), 285-320.

[GJ] Gjerde, R.; Johansen, Ø.; Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows, preprint.

[GW] Glasner, E.; Weiss, B.; Weak orbit equivalence of Cantor minimal systems, Internat. J. Math., 23 (1996), 737-769.

[HP] Host, B.; Parreau, F.; Homomorphismes entre systèmes dynamiques définis par substitutions, Ergodic Theory Dynam. Systems, 9 (1989), 469-477.

[HPS] Herman, R. H.; Putnam, I. F.; Skau, C. F.; Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math., 3 (1992), 827-864.

[LM] Lemańczyk, M.; Mentzen, M. K.; On metric properties of substitutions, Compositio Math., 65 (1988), 241-263.

[M] Matui, H.; Ext and OrderExt classes of certain automorphisms of C^{∗}-algebras arising from Cantor minimal systems, preprint.

[P] Putnam, I. F.; The C^{∗}-algebras associated with minimal homeomorphisms of the Cantor set, Pacific J. Math., 136 (1989), 329-353.

[S] Suzuki, M.; Group Theory I,II, Grundlehren der Mathematischen Wissenschaften, 247, 248. Springer-Verlag, New York-Berlin, 1986.

e-mail; matui@kusm.kyoto-u.ac.jp Department of Mathematics, University of Kyoto, Kitasirakawa-Oiwaketyô, Sakyôku, Kyôto, 606-8502, Japan.