Dimension groups of topological joinings and non-coalescence of Cantor minimal systems ^{∗}

MATUI Hiroki

Abstract

When we have two extensions of a Cantor minimal system which are both oneto-one on at least one orbit, we can construct new Cantor minimal systems called topological joinings. We compute the dimension group of the joining in a special case. As an application, we show that a non-invertible endomorphism can induce the identity map on the dimension group of a Cantor minimal system.

1 Introduction

By a topological dynamical system (Y, ψ), we mean a compact Hausdorff space Y endowed with a homeomorphism ψ. When (Y_{i}, ψ_{i}), i = 0, 1 are two topological dynamical systems, ψ_{0} × ψ_{1}-invariant closed subsets of Y_{0} × Y_{1} are called (topological) joinings, and when (Y_{0}, ψ_{0}) equals (Y_{1}, ψ_{1}), they are called self-joinings. In the measure-theoretical setting, the notion of self-joinings were introduced by D. Rudolph in [R], and it was proved that the minimal self-joining property implies coalescence and zero entropy. In this paper, we will compute the dimension group of joinings of Cantor minimal systems.

When Y is the Cantor set and a homeomorphism ψ on Y has no non-trivial invariant closed set, (Y, ψ) is called a Cantor minimal system. We define the dimension group K^{0}(Y, ψ) of (Y, ψ) as the quotient of C(Y, Z) by the coboundary subgroup

B_{ψ} = {f − f ∘ ψ^{−1} ; f ∈ C(Y, Z)}.

In [GPS], it was proved that the dimension group K^{0}(Y, ψ), as an ordered group with a distinguished order unit, characterizes the strong orbit equivalence class of (Y, ψ).

We would like to consider the case that a joining (Z, τ) of Cantor minimal systems (Y_{0}, ψ_{0}) and (Y_{1}, ψ_{1}) is also a Cantor minimal system. (We must distinguish the property of minimal self-joinings and minimal systems in the joinings.) We don’t know a necessary and sufficient condition so that the joining is minimal. In a special case, however, we will prove that the joining becomes a Cantor minimal system, and compute its dimension group. Our main result is Theorem 5, in which we will show that the dimension group of the joining is order isomorphic to the relative direct sum of K^{0}(Y_{i}, ψ_{i})’s.

In the last section, we will consider the non-coalescence of Cantor minimal systems. We denote by C(ψ) the set of continuous maps on Y which commute with ψ and call it the centralizer of (Y, ψ). If C(ψ) consists of homeomorphisms, we say that the system (Y, ψ) is topologically coalescent, and say that (X, ϕ) is non-coalescent, if C(ψ) contains a non-invertible endomorphism. For an element γ ∈ C(ψ), we can define an order homomorphism mod(γ) on K^{0}(Y, ψ) such as mod(γ)([f]) = [f ∘ γ] for [f] ∈ K^{0}(Y, ψ). The map mod gives a homomorphism from the automorphism group of (Y, ψ) to the automorphism group of the dimension group. Finite subgroups of the kernel of the mod map were studied by the author in [M2]. In the present paper, we will prove that there exist a lot of non-coalescent Cantor minimal systems whose non-invertible endomorphisms induce automorphisms on the dimension groups. Especially, we construct a non-coalescent Cantor minimal system which is strong orbit equivalent to an odometer system. For detailed information on odometer groups, we refer to [HR].

∗1991 Mathematics Subject Classification. 54H20

Acknowledgments.

The author is grateful to Professor Izumi for his constant encouragement.

2 Dimension groups of joinings

Let {(Y_{i}, ψ_{i})}_{i∈I} be a family of topological dynamical systems. For (y_{i})_{i∈I} in Y^{I}, the orbit closure of (y_{i})_{i} by the diagonal action is called the joining generated by (y_{i})_{i}.

Lemma 1 Let {(Y_{i}, ψ_{i})}_{i∈I} and (X, ϕ) be topological dynamical systems and π_{i} : (Y_{i}, ψ_{i}) → (X, ϕ) be factor maps. If y = (y_{i})_{i∈I} satisfies π_{i}(y_{i}) = π_{j}(y_{j}) and π_{i}^{−1}(π_{i}(y_{i})) = {y_{i}} for all i, j ∈ I, and (X, ϕ) is minimal, then the joining generated by (y_{i})_{i∈I} ∈ Y^{I} is minimal.

Proof. It suffices to prove the case I = {0, 1}. Let ψ be the diagonal action ψ_{0} × ψ_{1}. For every open neighborhood O of y = (y_{0}, y_{1}), there exists an open set U of X such that V = π_{0}^{−1}(U) × π_{1}^{−1}(U) is a neighborhood of y contained in O. To prove the minimality, it is enough to show the almost periodicity of y for V, that is,

{k ∈ Z ; ψ^{k}(y) ∈ V }

has a bounded gap in Z. But, we have ψ^{k}(y) ∈ V if and only if ϕ^{k}(π_{0}(y_{0})) ∈ U, and so the assertion comes from the minimality of (X, ϕ). □

When an unperforated ordered group G satisfies the Riesz interpolation property, G is called a dimension group in the abstract sense ([GPS, Section 1]). Let π : H → G be an order homomorphism between dimension groups. We say that π is an order embedding, if π is injective and π(h) ≥ 0 if and only if h ≥ 0. We also say that π(H) ⊂ G is a pure subgroup, if ng ∈ π(H) implies g ∈ π(H) for g ∈ G and n ∈ N.

Definition 2 Let π_{i} : H → G_{i}, i = 0, 1, be order embeddings between simple dimension groups except for Z. When π_{i}(H) is a pure subgroup of G_{i} for i = 0, 1, we call the quotient D of G_{0} ⊕ G_{1} by

{(π_{0}(h), −π_{1}(h)) ; h ∈ H}

the relative direct sum of G_{i} with respect to π_{i}(H). Define the positive cone D^{+} such as

D^{+} = {[g_{0}, g_{1}] ∈ D ; g_{i} ∈ G_{i}^{+}, i = 0, 1},

where [·, ·] means the quotient map. When π_{0} and π_{1} preserve the distinguished order units, the order unit of (D, D^{+}) can be defined in the obvious way.

We can define the relative direct sum of more than two dimension groups in a similar fashion.

Lemma 3 Let (D, D^{+}) be the relative direct sum of G_{i} with respect to π_{i}(H). Then, (D, D^{+}) is an unperforated ordered group, and for x ∈ D^{+} \ {0} and y ∈ D^{+}, there exists a natural number n ∈ N such that 0 ≤ y ≤ nx.

Proof. It is easy to see that (D, D^{+}) is an ordered group. Suppose n[g_{0}, g_{1}] ≥ 0 for [g_{0}, g_{1}] ∈ D and n ∈ N. By the definition, there exists an element h ∈ H such that ng_{0} + π_{0}(h) ≥ 0 and ng_{1} − π_{1}(h) ≥ 0. If ng_{0} + π_{0}(h) is equal to zero, h is divisible by n in H, because π_{0}(H) is a pure subgroup of G. Hence, we get [g_{0}, g_{1}] ≥ 0 in D. We have the same conclusion, if ng_{1} − π_{1}(h) is zero. Therefore, we may assume the strict inequality. Since H is a simple dimension group except for Z, we can find ϵ_{i} ∈ H for i = 0, 1 such as 0 < π_{0}(ϵ_{0}) < ng_{0} + π_{0}(h) and 0 < π_{1}(ϵ_{1}) < ng_{1} − π_{1}(h), by Corollary 4.10 of [GH]. By using 4.10 of [GH] again, we get an element h′ such as

h − ϵ_{0} < nh′ < h + ϵ_{1}.

Then, g_{0} + π_{0}(h′) and g_{1} − π_{1}(h′) are positive, which implies [g_{0}, g_{1}] are positive in D.

To prove the second statement, let x = [g_{0}, g_{1}] be in D^{+} \ {0}. We may assume g_{0} > 0 and g_{1} ≥ 0. Because we can choose h ∈ H^{+} \ {0} such that g_{0} − π_{0}(h) > 0, the assertion has been proved. □

The relative direct sum D may not satisfy the Riesz interpolation property. For example, let G be Q^{2} and define the positive cone by

G^{+} = {(x, y) ; x > |y|} ∪ {(0, 0)}.

If H = {(x, 0)} ≅ Q is a subgroup of G, it can be seen that the relative direct sum of two copies of G with respect to H doesn’t satisfy the Riesz interpolation property.

We say π(H) is order dense in G, if there is h ∈ H for every g < g′ such that g < π(h) < g′. By [GPS2, Proposition 1.1], π(H) is order dense in G, if and only if π induces an isomorphism between the state space S(G) and S(H).

Lemma 4 The relative direct sum (D, D^{+}) of G_{i} with respect to π_{i}(H) satisfies the Riesz interpolation property, if π_{1}(H) is order dense in G_{1}.

Proof. We check the Riesz decomposition property. Assume the inequality 0 ≤ [a, a′] < [b, b′] + [c, c′] in D for a, b, c ∈ G_{0}^{+} and a′, b′, c′ ∈ G_{1}^{+}. We may also assume [b, b′] ≠ 0 and [c, c′] ≠ 0. Because [b + c, b′ + c′] is strictly larger than [a, a′], we can find v, ϵ ∈ H such that

0 ≤ (b + c) − a + π_{0}(v), 0 < 2π_{1}(ϵ) < (b′ + c′) − a′ − π_{1}(v).

Since π_{1}(H) is order dense in G_{1}, we also get t, u ∈ H satisfying,

0 ≤ b + π_{0}(t), 0 < b′ − π_{1}(t) < π_{1}(ϵ)

and

0 ≤ c + π_{0}(u), 0 < c′ − π_{1}(u) < π_{1}(ϵ).

Take ϵ′ ∈ H such as 0 < π_{1}(ϵ′) < b′ − π_{1}(t) and 0 < π_{1}(ϵ′) < c′ − π_{1}(u). By using the order density again, we have s ∈ H satisfying

0 ≤ a + π_{0}(s), 0 < a′ − π_{1}(s) < π_{1}(ϵ′).

Then, we obtain

π_{1}(ϵ′) < (b′ + c′) − a′ − π_{1}(t + u − s) < 2π_{1}(ϵ),

and so t + u − s is greater than v. If we replace [a, a′], [b, b′] and [c, c′] with [a + π_{0}(s), a′ − π_{1}(s)], [b+π_{0}(t), b′ −π_{1}(t)] and [c+π_{0}(u), c′ −π_{1}(u)], the assertion is followed by the Riesz decomposition property of G_{i}. □

We would like to state the main theorem. Let π_{i} : (Y_{i}, ψ_{i}) → (X, ϕ) be factor maps between Cantor minimal systems for i = 0, 1. Define

E_{i} = {x ∈ X ; #π_{i}^{−1}(x) ≠ 1}

for each i and assume that y_{i,max} ∈ Y_{i} and x_{max} ∈ X satisfy

π_{0}(y_{0,max}) = π_{1}(y_{1,max}) = x_{max} ∈ (E_{0} ∪ E_{1})^{c}.

By Lemma 1, the joining (Z, τ) generated by (y_{0,max}, y_{1,max}) is a Cantor minimal system. In general, a factor map π_{i} between Cantor minimal systems induces the order embedding π_{i}^{∗} : K^{0}(X, ϕ) → K^{0}(Y_{i}, ψ_{i}). In Lemma 6, we will prove that π_{i}^{∗}(K^{0}(X, ϕ)) is a pure subgroup of K^{0}(Y_{i}, ψ_{i}). We denote by p_{i} the canonical factor maps from (Z, τ) to (Y_{i}, ψ_{i}) for i = 0, 1. The direct sum of p_{0}^{∗} and p_{1}^{∗} induces the homomorphism from K^{0}(Y_{0}, ψ_{0}) ⊕ K^{0}(Y_{1}, ψ_{1}) to K^{0}(Z, τ).

Theorem 5 In the above setting, we further assume that E_{0} and E_{1} are disjoint. Then, the dimension group K^{0}(Z, τ) is isomorphic to the relative direct sum of K^{0}(Y_{i}, ψ_{i}) with respect to π_{i}(K^{0}(X, ϕ)) as an unital ordered group. Moreover, the state space of K^{0}(Z, τ) is equal to

{(μ_{0}, μ_{1}) ; μ_{i} ∈ S(K^{0}(Y_{i}, ψ_{i})), μ_{0} ∘ π_{0}^{∗} = μ_{1} ∘ π_{1}^{∗}},

where S(K^{0}(Y_{i}, ψ_{i})) is the state space of K^{0}(Y_{i}, ψ_{i}).

If E_{0} and E_{1} are disjoint, the topological joining Z coincides with the pull-back associated with π_{0} and π_{1}, that is,

Z = {(y_{0}, y_{1}) ∈ Y_{0} × Y_{1} ; π_{0}(y_{0}) = π_{1}(y_{1})}.

Therefore, the theorem asserts that the dimension group of the pull-back system is isomorphic to the push-out of the dimension group.

For an ergodic measure μ on (X, ϕ), either of μ(E_{0}) and μ(E_{1}) must be zero when E_{0} and E_{1} are disjoint. If μ(E_{0}) = 0 for every ergodic measure μ, then π_{0}^{∗}(K^{0}(X, ϕ)) is order dense in K^{0}(Y_{0}, ψ_{0}). But, we must remark that even if E_{0} and E_{1} are disjoint, π_{i}^{∗}(K^{0}(X, ϕ)) may not be order dense in K^{0}(Y_{i}, ψ_{i}) for each i = 0, 1. See the example at the last of this section.

In order to prove the above theorem, at first, we need to fix Kakutani-Rohlin partitions for (X, ϕ) and (Y_{i}, ψ_{i}).

Let {X_{n}}_{n∈N} be a decreasing sequence of clopen sets of X such that the intersection of all X_{n}’s is {x_{max}}. By Theorem 4.2 of [HPS], for each X_{n}, we can find a finite set V_{n}, a map h : V_{n} → N and a clopen partition

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

of X, which satisfy the following properties.

• For every v ∈ V_{n} and k ≠ h(v), we have ϕ(X(n, v, k)) = X(n, v, k + 1).

• The clopen set X_{n} equals the disjoint union of X(n, v, h(v)) for all v ∈ V_{n}.

• The partition P_{n+1} is finer than P_{n} for every n ∈ N, and the family of partitions {P_{n}}_{n} generates the topology of X.

The clopen set X_{n} is called the top set and the map h is called the height function. In order to compute the dimension group K^{0}(X, ϕ), we must consider the free abelian group Z^{V_{n}}. We denote the canonical basis of Z^{V_{n}} by the same symbol v ∈ V_{n}. When we set the rectangular matrix A_{n} ∈ M_{V_{n}×V_{n+1}}(Z) such as

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

for every v ∈ V_{n} and v′ ∈ V_{n+1}, the dimension group K^{0}(X, ϕ) is computed by the inductive limit of Z^{V_{n}} with the connecting map A_{n}.

The partition P_{n} induces the clopen partition of Y_{i} for each i = 0, 1 and n ∈ N. This family of partitions, however, does not generate the topology of Y_{i}. When we divide every clopen set π_{i}^{−1}(X(n, v, k)) so that the topology is generated, we obtain a finite set W_{i,n}, a map ρ_{i} : W_{i,n} → V_{n} and a clopen partition

Q_{i,n} = {Y_{i}(n, w, k) ; w ∈ W_{i,n}, 1 ≤ k ≤ h(ρ_{i}(w))}

of Y_{i} and the following holds for each i = 0, 1.

• For every w ∈ W_{i,n} and k ≠ h(ρ_{i}(w)), we have ψ_{i}(Y_{i}(n, w, k)) = Y_{i}(n, w, k + 1).

• For every v ∈ V_{n} and 1 ≤ k ≤ h(v), π_{i}^{−1}(X(n, v, k)) is the disjoint union of Y_{i}(n, w, k) for all w ∈ ρ_{i}^{−1}(v).

• As in the case of (X, ϕ), Y_{i,n} = π_{i}^{−1}(X_{n}) is the disjoint union of Y_{i}(n, w, h(ρ_{i}(w))) for all w ∈ W_{i,n}.

• The partition Q_{i,n+1} is finer than Q_{i,n} and the family of partitions {Q_{i,n}}_{n} generates the topology of Y_{i}.

Notice that the intersection of all Y_{i,n}’s equals {y_{i,max}} because x_{max} is not contained in E_{i}. We denote the canonical basis of Z^{W_{i,n}} by the same symbol w ∈ W_{i.n}. We can compute the dimension group K^{0}(Y_{i}, ψ_{i}), using the rectangular matrix B_{i,n} ∈ M_{W_{i,n}×W_{i,n+1}}(Z) determined in a similar fashion to the case of A_{n}. The homomorphisms sending v ∈ Z^{V_{n}} to ∑_{w∈ρ_{i}^{−1}(v)} w ∈ Z^{W_{i,n}} induce the order embedding π_{i}^{∗} : K^{0}(X, ϕ) → K^{0}(Y_{i}, ψ_{i}).

Lemma 6 Let π : (Y, ψ) → (X, ϕ) be a factor map between Cantor minimal systems. If there is a point y ∈ Y such that π^{−1}(π(y)) = {y}, then π^{∗}(K^{0}(X, ϕ)) is a pure subgroup of K^{0}(Y, ψ).

Proof. We will prove the lemma for π_{0} : (Y_{0}, ψ_{0}) → (X, ϕ) above. The embedding of K^{0}(X, ϕ) to K^{0}(Y_{0}, ψ_{0}) is obtained by the natural injection from Z^{V_{n}} to Z^{W_{i,n}} induced by ρ_{0}. Since Z^{V_{n}} is a pure subgroup of Z^{W_{i,n}}, we get the conclusion. □

We return to the setting of Theorem 5. We would like to describe Kakutani-Rohlin partitions of (Z, τ). When w_{0} ∈ W_{0,n} and w_{1} ∈ W_{1,n} are preimages of the same v ∈ V_{n}, we set

Z(n, w_{0}, w_{1}, k) = Z ∩ (Y_{0}(n, w_{0}, k) × Y_{1}(n, w_{1}, k))

for all 1 ≤ k ≤ h(v). Let W_{n} be the set of pairs (w_{0}, w_{1}) such that the clopen set Z(n, w_{0}, w_{1}, h(v)) is not empty. Then, we get the partition

R_{n} = {Z(n, w_{0}, w_{1}, k) ; (w_{0}, w_{1}) ∈ W_{n}, 1 ≤ k ≤ h(ρ_{i}(w_{i}))}

of Z. We can check that the sequence of clopen partitions {R_{n}}_{n} satisfies the similar properties as in the case of {P_{n}}_{n} or {Q_{i,n}}. Hence, we can compute the dimension group K^{0}(Z, τ) by the inductive limit of Z^{W_{n}}. We write the canonical basis of Z^{W_{n}} by the same symbol (w_{0}, w_{1}) ∈ W_{n}. The connecting map from Z^{W_{n}} to Z^{W_{n+1}} is given by a matrix C_{n} and the entry of C_{n} corresponding to (w_{0}, w_{1}) ∈ W_{n} and (w′_{0}, w′_{1}) ∈ W_{n+1} is equal to the cardinality of the set

{1 ≤ k ≤ h(ρ_{i}(w′_{i})) ; Z(n, w′_{0}, w′_{1}, k) ⊂ Z(n, w_{0}, w_{1}, h(ρ_{i}(w_{i})))}.

The order embedding p_{0}^{∗} is induced from the homomorphism sending w_{0} ∈ Z^{W_{0,n}} to ∑_{(w_{0},w)∈W_{n}}(w_{0}, w) ∈ Z^{W_{n}}. The order embedding p_{1}^{∗} is described in the same way.

Define

E_{i}(m, Q) = {P ∈ P_{m} ; π_{i}^{−1}(P) ∩ Q and π_{i}^{−1}(P) ∩ Q^{c} are both non-empty}

for Q ∈ Q_{i,n}, n ≤ m and i = 0, 1. If P′ ∈ P_{m+1} is a subset of P ∈ P_{m} and P′ ∈ E_{i}(m + 1, Q), then we have P ∈ E_{i}(m, Q).

Lemma 7 For every Q_{0} ∈ Q_{0,n} and Q_{1} ∈ Q_{1,n}, there exists a natural number m greater than n such that E_{0}(m, Q_{0}) and E_{1}(m, Q_{1}) are disjoint.

Proof. The proof is by contradiction. Let U_{i,m} be the disjoint union of P ∈ E_{i}(m, Q_{i}), for every m ≥ n and i = 0, 1. The clopen set U_{m} = U_{0,m} ∩ U_{1,m} is non-empty and forms a decreasing sequence. Therefore, there exists a point x ∈ X contained in all U_{m}. Then, π_{i}^{−1}(x) contains at least two distinct points for each i = 0, 1, which contradicts to the disjointness of E_{0} and E_{1}. □

We are now ready to prove the maim theorem. Proof of Theorem 5. By virtue of Lemma 7, we may assume that E_{0}(n + 1, Q_{0}) and E_{1}(n + 1, Q_{1}) are disjoint for all Q_{i} ∈ Q_{i,n} and n ∈ N. We denote the factor map π_{0} ∘ p_{0} = π_{1} ∘ p_{1} by π.

At first, we prove the surjectivity of the map p_{0}^{∗} ⊕ p_{1}^{∗} from K^{0}(Y_{0}, ψ_{0}) ⊕ K^{0}(Y_{1}, ψ_{1}) to K^{0}(Z, τ). Fix n ∈ N and (w_{0}, w_{1}) ∈ W_{n} arbitrarily. Let v = ρ_{0}(w_{0}) = ρ_{1}(w_{1}) ∈ V_{n}. Set

S = {R ∈ R_{n+1} ; R ∩ Z(n, w_{0}, w_{1}, h(v)) is not empty},
S_{i} = {R ∈ S ; R ⊂ π^{−1}(P) for some P ∈ E_{i}(n + 1, Y_{i}(n, w_{i}, h(v)))},

for each i = 0, 1, and

S_{2} = {R ∈ S ; R ∩ π^{−1}(P) is empty for all P ∈ E_{i}(n + 1, Y_{i}(n, w_{i}, h(v))), i = 0, 1}.

Then, we have

Z(n, w_{0}, w_{1}, h(v)) = ⋃_{R∈S} R
= ⋃_{R∈S_{0}} R ∪ ⋃_{R∈S_{1}} R ∪ ⋃_{R∈S_{2}} R.

The first summand is equal to

⋃ Z ∩ (Y_{0}(n + 1, w′_{0}, k) × Y_{1}),

where the union runs over w′_{0} and k satisfying Z(n + 1, w′_{0}, w′_{1}, k) ∈ S_{0} for some w′_{1} ∈ W_{1,n+1}. The second summand can be written in the same way. The last summand is equal to π^{−1}(U) for some clopen set U of X. Hence, the surjectivity has been proved.

Let us consider the kernel of p_{0}^{∗} ⊕ p_{1}^{∗}. Suppose a_{0} = ∑_{w_{0}∈W_{0,n}} λ_{0,w_{0}}w_{0} and a_{1} = ∑_{w_{1}∈W_{1,n}} λ_{1,w_{1}}w_{1} satisfy

p_{0}^{∗}(a_{0}) = p_{1}^{∗}(a_{1})

in K^{0}(Z, τ). We may assume that

∑_{(w_{0},w_{1})∈W_{n}} λ_{0,w_{0}}(w_{0}, w_{1}) = ∑_{(w_{0},w_{1})∈W_{n}} λ_{1,w_{1}}(w_{0}, w_{1}),

and so we have λ_{0,w_{0}} = λ_{1,w_{1}}, if (w_{0}, w_{1}) exists in W_{n}. Therefore, we may further assume that there exists a subset F_{i} ⊂ W_{i,n} for each i = 0, 1, such that

• If w_{0} ∈ F_{0} and (w_{0}, w_{1}) ∈ W_{n}, then w_{1} ∈ F_{1}.

• If w_{1} ∈ F_{1} and (w_{0}, w_{1}) ∈ W_{n}, then w_{0} ∈ F_{0}.

• The element a_{i} is equal to ∑_{w_{i}∈F_{i}}(w_{0}, w_{1}) for each i = 0, 1.

Define the clopen set

W = ⋃_{w_{0}∈F_{0}} Z(n, w_{0}, w_{1}, h(ρ_{i}(w_{i}))) = ⋃_{w_{1}∈F_{1}} Z(n, w_{0}, w_{1}, h(ρ_{i}(w_{i}))).

Let Z(n + 1, w′_{0}, w′_{1}, k) ∈ R_{n+1} be a clopen set which is contained in W. Then, there exist w_{0} ∈ F_{0} and w_{1} ∈ F_{1} such that Z(n + 1, w′_{0}, w′_{1}, k) is contained in Z(n, w_{0}, w_{1}, h(ρ_{i}(w_{i}))). Without loss of generality, we may assume that X(n + 1, ρ_{0}(w′_{0}), k) is not contained in E_{0}(n + 1, Y_{0}(n, w_{0}, h(ρ_{0}(w_{0}))). Take a pair (w′′_{0}, w′′_{1}) ∈ W_{n+1} satisfying ρ_{i}(w′′_{i}) = ρ_{i}(w′_{i}), arbitrarily. There exists a pair (w̃_{0}, w̃_{1}) ∈ W_{n} satisfying Z(n + 1, w′′_{0}, w′′_{1}, k) ⊂ Z(n, w̃_{0}, w̃_{1}, h(ρ_{i}(w̃_{i}))). By the assumption, we have w̃_{0} = w_{0}, and so w̃_{1} must be in F_{1}. Hence, π^{−1}(X(n + 1, ρ_{0}(w′_{0}), k)) is contained in W. By repeating this argument, we see that p_{0}^{∗}(a_{0}) = p_{1}^{∗}(a_{1}) comes from K^{0}(X, ϕ).

We have proved that p_{0}^{∗} ⊕ p_{1}^{∗} induces an isomorphism from the relative direct sum to K^{0}(Z, τ). It is obvious from the above argument that the isomorphism preserves the positive cones and the distinguished order units, and so the proof is completed. □

Corollary 8 Let {(Y_{i}, ψ_{i})}_{i∈I} be a family of at most countable Cantor minimal systems and π_{i} : (Y_{i}, ψ_{i}) → (X, ϕ) be factor maps to a Cantor minimal system (X, ϕ). Define

E_{i} = {x ∈ X ; #π_{i}^{−1}(x) ≠ 1}

and assume y_{i} ∈ Y_{i} and x ∈ X satisfies π_{i}(y_{i}) = x and x ∈ E_{i}^{c} for all i ∈ I. When we further assume that E_{i}’s are disjoint for all i, the dimension group of the joining (Z, τ) generated by (y_{i})_{i} is isomorphic to the relative direct sum of K^{0}(Y_{i}, ψ_{i})’s with respect to π_{i}^{∗}(K^{0}(X, ϕ)).

Proof. Use Theorem 5 repeatedly. □

Example. Let {q_{n}}_{n∈N} be a sequence of natural numbers satisfying

α = lim_{n→∞} ∏_{k=1}^{n} \frac{q_{k} + 1}{q_{k} − 1} < ∞.

For convenience, we assume that, for every p ∈ N, there exists n ∈ N such that ∏_{k=1}^{n}(q_{k} + 1),∏_{k=1}^{n}(q_{k} − 1) and ∏_{k=1}^{n}(q_{k} − 2) are divisible by p.

We would like to define a simple ordered Bratteli diagram (V, E, ≤) ([HPS]). Set the vertex set V_{n} be {v_{n}, v′_{n}} and define the edge set E_{n} so that the connecting matrix from Z^{V_{n}} to Z^{V_{n+1}} equals

A_{n} = [ \begin{matrix}q_{n} & 1 \\ 1 & q_{n} \end{matrix} ] ,

where the 1-1 entry of A_{n} corresponds with the edges from v_{n} to v_{n+1}. Let the source vertex of the initial edge in r^{−1}(v_{n+1}) and r^{−1}(v′_{n+1}) be v_{n}, and the source vertex of the final edge be v′_{n}. Then, we obtain a simple ordered Bratteli diagram. Let (X, ϕ) be the Bratteli-Vershik system of (V, E, ≤). The dimension group K^{0}(X, ϕ) is isomorphic to Q^{2} and the positive cone is

K^{0}(X, ϕ)^{+} = {(x, y) ; αx > |y|} ∪ {(0, 0)},

and so there are two ergodic measures. The distinguished order unit [1] is equal to (2, 0).

Let us construct two extensions of (X, ϕ), namely (Y_{0}, ψ_{0}) and (Y_{1}, ψ_{1}). We would like to define two simple ordered Bratteli diagrams (W_{0}, F_{0}, ≤) and (W_{1}, F_{1}, ≤). Set the vertex set W_{0,n} be {w_{0,n}, w′_{0,n}, w′′_{0,n}} and the vertex set W_{1,n} be {w_{1,n}, w′_{1,n}, w′′_{1,n}}. To describe the partial order on the edge set, for each vertex w, we denote by θ(w) the ordered list of the source vertices of edges in r^{−1}(w). Define the partial order on F_{0} such as

θ(w_{0,n+1}) = (w_{0,n}, \overbrace{w_{0,n}, ⋯, w_{0,n}}^{q_{n}−2}, w′′_{0,n}, w′_{0,n}),

θ(w′_{0,n+1}) = (w_{0,n}, \overbrace{w′_{0,n}, ⋯, w′_{0,n}}^{q_{n}−2}, w′_{0,n}, w′_{0,n}),

and

θ(w′′_{0,n+1}) = (w_{0,n}, \overbrace{w′′_{0,n}, ⋯, w′′_{0,n}}^{q_{n}−2}, w′′_{0,n}, w′_{0,n}).

Similarly, define the partial order on F_{1} such as

θ(w_{1,n+1}) = (w_{1,n}, \overbrace{w_{1,n}, ⋯, w_{1,n}}^{q_{n}−2}, w_{1,n}, w′_{1,n}),

θ(w′_{1,n+1}) = (w_{1,n}, \overbrace{w′_{1,n}, ⋯, w′_{1,n}}^{q_{n}−2}, w′′_{1,n}, w′_{1,n}),

and

θ(w′′_{1,n+1}) = (w_{1,n}, \overbrace{w′′_{1,n}, ⋯, w′′_{1,n}}^{q_{n}−2}, w′′_{1,n}, w′_{1,n}),

for each n ∈ N. Then, we get two Cantor minimal systems (Y_{0}, ψ_{0}) and (Y_{1}, ψ_{1}) associated with simple ordered Bratteli diagrams (W_{0}, F_{0}, ≤) and (W_{1}, F_{1}, ≤). There exists a factor map π_{0} : (Y_{0}, ψ_{0}) → (X, ϕ) which sends the tower w_{0,n} and w′′_{0,n} to v_{n}, and the tower w′_{0,n} to v′_{n}. In the same way, there exists a factor map π_{1} : (Y_{1}, ψ_{1}) → (X, ϕ). For i = 0, 1, we set

E_{i} = {x ∈ X ; #π_{i}^{−1}(x) ≠ 1}.

It is not hard to see that E_{0} and E_{1} are disjoint.

Let β be the limit of ∏_{k=1}^{n}(q_{k} − 2)/(q_{k} − 1). The dimension group K^{0}(Y_{i}, ψ_{i}) is isomorphic to Q^{3} and the positive cone is

K^{0}(Y_{i}, ψ_{i})^{+} = \{(a, b, c) ; \begin{matrix}αa − b − βc > 0 \\ αa + b + 2βc > 0 \\ αa + b − βc > 0 \end{matrix} \} ∪ \{(0, 0, 0)\},

for both i = 0, 1, hence, (Y_{i}, ψ_{i}) has three ergodic measures. The order embeddings π_{0}^{∗} and π_{1}^{∗} send (x, y) ∈ Q^{2} to (x, y, 0) and (x, −y, 0), respectively.

Choose x ∈ X \ (E_{0} ∪ E_{1}) and y_{i} ∈ Y_{i} such as π_{i}(y_{i}) = x. The topological joining (Z, τ) generated by (y_{0}, y_{1}) is a Cantor minimal system. The dimension group K^{0}(Z, τ) is isomorphic to the relative direct sum of K^{0}(Y_{0}, ψ_{0}) and K^{0}(Y_{1}, ψ_{1}) with respect to K^{0}(X, ϕ). Therefore, K^{0}(Z, τ) equals Q^{4} and the positive cone is determined by four linear inequalities. In this case, K^{0}(Z, τ) satisfies the Riesz interpolation property, although neither π_{0}(K^{0}(X, ϕ)) nor π_{1}(K^{0}(X, ϕ)) is order dense.

3 Non-coalescence and dimension groups

In this section, we construct non-invertible endomorphisms of Cantor minimal systems which induce automorphisms on the dimension groups. We say an automorphism γ of a Cantor minimal system (X, ϕ) has essentially infinite order, if γ^{i} does not equal ϕ^{j} for all i ∈ N and j ∈ Z.

Lemma 9 Let (X, ϕ) be a Cantor minimal system, γ ∈ C(ϕ) be an automorphism which has essentially infinite order and x_{max} ∈ X be a point. Then, there exist a Cantor minimal system (Y, ψ) and a factor map π : (Y, ψ) → (X, ϕ) such that the following properties are satisfied;

(i) π^{∗} is an order isomorphism between K^{0}(X, ϕ) and K^{0}(Y, ψ).

(ii) When we set E = {x ∈ X ; #π^{−1}(x) ≠ 1}, γ^{i}(x_{max}) and γ^{j}(E) are all disjoint for i, j ∈ Z.

Proof. We may assume that X is the infinite path space of an ordered Bratteli diagram (V, E, ≤), x_{max} is the unique maximal path and ϕ is the Bratteli-Vershik map. Let x_{min} be the unique minimal path. We denote the source map by s : E_{n} → V_{n} and the range map by r : E_{n} → V_{n+1}. The symbol E_{n,m} means the set of finite paths from V_{n} to V_{m} and [e] for e ∈ E_{n,m} means the corresponding clopen set of X. Let A_{n} be the incidence matrix from Z^{V_{n}} to Z^{V_{n+1}}, and h(v) be the number of edges connecting v_{0} ∈ V_{0} to v ∈ V_{n}.

By telescoping the diagram, we would like to choose v_{n} ∈ V_{n} and f_{n}, f′_{n} ∈ E_{n} for each n ∈ N which satisfy the following.

• The source and range vertex of f_{n} and f′_{n} are v_{n} and v_{n+1}.

• The clopen sets γ^{j}([f_{n}]) and γ^{k}([f′_{n}]) for |j|, |k| ≤ n are all disjoint.

• The clopen set [f_{n}] ∪ [f′_{n}] doesn’t contain γ^{k}(x_{max}) and γ^{k}(x_{min}) for |k| ≤ n.

The construction of v_{n}, f_{n} and f′_{n} is by induction. At first, we choose v_{1} ∈ V_{1} arbitrarily. Let us assume that v_{n}, f_{n−1} and f′_{n−1} have been already chosen. Since γ is essentially infinite order, we can find m > n and v′ ∈ V_{m} satisfying

[e] ∩ γ^{k}[e] is empty for all e ∈ E_{v_{n},v′}, 0 < |k| ≤ n

and

#E_{v_{n},v′} > 2(2n + 1) + h(v_{n})(4n + 1) + 1,

where E_{v_{n},v′} means the set of edges from v_{n} to v′. Choose f ∈ E_{v_{n},v′} so that [f] does not contain γ^{k}(x_{max}) and γ^{k}(x_{min}) for |k| ≤ n. For a large number l > m, we may assume that

∀e ∈ E_{0,l}, ∀|k| ≤ 2n, ∃e′ ∈ E_{n,m}, s.t. γ^{k}([e]) ⊂ [e′].

Then, let f_{n} ∈ E_{n,l} be an arbitrary extension of f and we can choose f′_{n} ∈ E as an extension of an edge of E_{v_{n},v′} so that the above property is satisfied. Hence, the induction is completed.

Let F be the set of infinite paths consisting of f_{n}’s and f′_{n}’s and E be the union of all ϕ^{k}(F) for k ∈ Z. We construct a new ordered Bratteli diagram (W, E′, ≤). Define the vertex set W_{n} such as the disjoint union of V_{n} and v′_{n}, and let the map ρ : W_{n} → V_{n} be ρ(v′_{n}) = v_{n} and ρ(v) = v for v ∈ V_{n}. Set the edge set E′_{n} so that the incidence matrix B_{n} satisfies

B_{n}(w, w′) = \{ \begin{matrix}0 & w = v′_{n}, w′ ≠ v_{n+1}, v′_{n+1} \\ 1 & w = v′_{n}, w′ = v_{n+1}, v′_{n+1} \\ A_{n}(v_{n}, v_{n+1}) − 1 & w = v_{n}, w′ = v_{n+1}, v′_{n+1} \\ A_{n}(ρ(w), ρ(w′)) & otherwise \end{matrix}

for w ∈ W_{n} and w′ ∈ W_{n+1}. It is not hard to check that the inductive limit of Z^{W_{n}} with the connecting map B_{n} is order isomorphic to K^{0}(X, ϕ). Let us define the linear order on the set r^{−1}(w′) for each w′ ∈ W_{n+1}. When w′ is not equal to v_{n+1} nor v′_{n+1}, we define the linear order by exactly the same way as in r^{−1}(ρ(w′)). If w′ is v_{n+1} or v′_{n+1}, we change the source vertex of the edge f_{n} or f′_{n}, respectively, to v′_{n}. Then, we get a well-defined ordered Bratteli diagram.

Let (Y, ψ) be the Cantor minimal system determined by (W, E′, ≤). Obviously, there is a factor map π : (Y, ψ) → (X, ϕ), and π is not one-to-one exactly on the subset E. By the construction of E, we see that E satisfies the condition (ii) above. □

For an automorphism γ which induces the identity on the dimension group, we introduced a new invariant η(γ) in the Ext group ([M1]). When γ is a non-invertible endomorphism, we can define η(γ) in the same way, by replacing γ^{−1} with γ in [M1].

Theorem 10 Let (X, ϕ) be a Cantor minimal system, γ ∈ C(ϕ) be an automorphism which has essentially infinite order. Then, there exists a Cantor minimal system (Z, τ), a factor map Φ : (Z, τ) → (X, ϕ) and a non-invertible endomorphism γ̃ ∈ C(τ) such that γ ∘ Φ = Φ ∘ γ̃ and Φ induces the isomorphism between the dimension groups K^{0}(X, ϕ) and K^{0}(Z, τ). Moreover, η(γ̃) equals η(γ) in the Ext group.

Proof. Take x_{max} ∈ X arbitrarily and let (Y, ψ) and π as in Lemma 9. For each i ∈ N∪{0}, let y_{i} ∈ Y be the unique point such that π(y_{i}) = γ^{i}(x_{max}) and (Z, τ) be the topological joining generated by (y_{i})_{i}. By applying Lemma 1 to γ^{−i} ∘ π, we see that (Z, τ) is a Cantor minimal system. Let Φ be the composition of π and the projection from Z to the first summand. By Corollary 8, the factor map Φ induces the isomorphism between K^{0}(X, ϕ) and K^{0}(Z, τ). It is easily seen that the one-sided subshift on Y^{N∪{0}} gives the well-defined centralizer γ̃ ∈ C(τ) and γ̃ is a non-invertible endomorphism satisfying γ ∘ Φ = Φ ∘ γ̃. □

By applying the above theorem to an odometer system (X, ϕ), we get a Cantor minimal system (Z, τ) which is strong orbit equivalent to (X, ϕ) and has a non-invertible endomorphism γ. Of course, γ induces the identity map on the dimension group K^{0}(Z, τ). Actually, in the proof of Lemma 9, we can find the Cantor minimal system (Y, ψ) in the class of Toeplitz subshifts, if (X, ϕ) is an odometer system. Therefore, the Cantor minimal system (Z, τ) is the projective limit of Toeplitz minimal subshifts. But, the system (Z, τ) itself is not expansive. In [D], T. Downarowicz constructed an example of Toeplitz minimal subshift which admits a non-invertible endomorphism. We can check that the endomorphism in his example does not induce the identity on the dimension group. The author doesn’t know non-coalescent minimal subshifts except for Downarowicz’s example.

References

[D] Downarowicz, T.; The royal couple conceals their mutual relationship: a noncoalescent Toeplitz flow, Israel J. Math., 97 (1997), 239-251.

[GH] Goodearl, K. R.; Handelman, D. E.; Metric completions of partially ordered abelian groups, Indiana. Univ. Math. J., 29 (1980), 861-895.

[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.; K-theory and asymptotic index for certain almost one-to-one factors, to appear in Math. Scand.

[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.

[HR] Hewitt, E.; Ross, A. R. ; Abstract Harmonic Analysis, Volume I, Grundlehren der matematischen Wissenschaften 115, Springer-Verlag, 1979.

[M1] Matui, H.; Ext and OrderExt classes of certain automorphisms of C^{∗}-algebras arising from Cantor minimal systems, to appear in Canad. J. Math.

[M2] Matui, H.; Finite order automorphisms and dimension groups of Cantor minimal systems, preprint.

[PSS] Putnam, I. F.; Schmidt, K.; Skau, C. F.; C^{∗}-algebras associated with Denjoy homeomorphisms of the circle, J. Operator Theory, 16 (1986), 99-126.

[R] Rudolph, D.; An example of a measure-preserving map with minimal selfjoinings, J. Analyse Math., 35 (1979), 97-122.

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