Some remarks on topological orbit equivalence of Cantor minimal systems

MATUI Hiroki

August 2002

Abstract

We will study the number of discontinuities of the orbit cocycles associated with orbit equivalence between Cantor minimal systems.

1 Introduction

In [GPS] the following was stated as Theorem 2.5.

Theorem (original version) . Let (X_{i}, φ_{i}) be Cantor systems (i = 1, 2). The following are equivalent:

(i) (X_{1}, φ_{1}) is orbit equivalent to (X_{2}, φ_{2}), and there exists an orbit map F : X_{1} → X_{2} so that the associated orbit cocycles m, n : X_{1} → ℤ each have finitely many points of discontinuity in (necessarily the same) disjoint φ_{1}-orbits. Assume k + 1 is the least possible number of discontinuity points by considering all such maps F. (If there are no discontinuity points we set k = 0.)

(ii) There exist subgroups, both isomorphic to ℤ^{k}, of

Inf(K^{0}(X_{1}, φ_{1})) and Inf(K^{0}(X_{2}, φ_{2})),

respectively, so that the quotient groups

K^{0}(X_{i}, φ_{i})/ℤ^{k} (i = 1, 2)

(with the induced order) are order isomorphic by a map preserving the distinguished order units (more precisely, the quotient image of the distinguished order units), where k is the least natural number with this property.

In the above statement, while the implication (i)⇒(ii) is valid, the other implication (ii)⇒(i) is not correct. We will show it by constructing a concrete counter example in this paper.

The original proof contains a gap in the final step. In page 99 of [GPS] three orbit equivalent maps G_{1}, G_{2} and H are considered and the associated orbit cocycles each have finitely many points of discontinuity. The composition map F, however, does not have such a nice property. Thus, the orbit cocycles associated with F may have infinitely many discontinuities. What they actually proved is the following.

Theorem (corrected version) . Let G_{1} and G_{2} be two simple (acyclic) dimension groups. The following are equivalent.

(i) There exist subgroups, both isomorphic to ℤ^{k}, of Inf(G_{1}) and Inf(G_{2}), respectively, so

that the quotient groups

G_{i}/ℤ^{k} (i = 1, 2)

(with the induced order) are order isomorphic by a map preserving the distinguished order units (more precisely, the quotient image of the distinguished order units), where k is the least natural number with this property.

(ii) There exist Cantor minimal systems (X_{i}, φ_{i}), i = 1, 2, so that K^{0}(X_{1}, φ_{1}) ≅ G_{1} and K^{0}(X_{2}, φ_{2}) ≅ G_{2} (where ≅ denotes order isomorphism by a map preserving the distinguished order units), and an orbit map F : X_{1} → X_{2} so that the associated orbit cocycles m, n : X_{1} → ℤ each have k + 1 discontinuity points in (necessarily the same) disjoint φ_{1}- orbits. Furthermore for k ≥ 1, k + 1 is the least possible number of discontinuity points by considering all orbit maps and all Cantor minimal systems satisfying the above condition. If k = 0, the least possible number of discontinuity points is zero.

We will collect some basic facts concerning orbit equivalence of Cantor minimal systems in Section 2. The counter example will be constructed in Section 3. In the last section we will restrict our attention to 2-strong orbit equivalence and discuss some further problems.

Acknowledgments. The author would like to express his sincerest thanks to Professor Christian Skau for the numerous insightful conversations.

2 Preliminaries

We call a compact metrizable totally disconnected and perfect space the Cantor set. The Cantor set is homeomorphic to {0, 1}^{ℕ} with the product topology. A homeomorphism ϕ ∈ Homeo(X) on a topological space is said to be minimal if every ϕ-orbit is dense in X. If ϕ is a minimal homeomorphism on the Cantor set X, the pair (X, ϕ) is called a Cantor minimal system. In [GPS] it was proved that the K^{0}-group of (X, ϕ) is a complete invariant for the orbit equivalence class of (X, ϕ). We have to recall this fact at first.

Let (X, ϕ) and (Y, ψ) be Cantor minimal systems. When there exists a homeomorphism F : X → Y such that F ({ϕ^{n}(x) : n ∈ ℤ}) = {ψ^{n}(F (x)); n ∈ ℤ} holds for every x ∈ X, two systems (X, ϕ) and (Y, ψ) are said to be orbit equivalent. Since ϕ and ψ have no periodic points, the orbit cocycles n : X → ℤ and m : Y → ℤ are uniquely determined by

F (ϕ(x)) = ψ^{n(x)}(F (x)), F^{−1}(ψ(y)) = ϕ^{m(y)}(F^{−1}(y)).

If each of n and m has exactly k discontinuities and these k points have distinct orbits, then we say that F gives an orbit equivalence with k discontinuities.

Definition 1. Two Cantor minimal systems (X, ϕ) and (Y, ψ) are said to be k-strong orbit equivalent, if there exists a homeomorphism F : X → Y which gives an orbit equivalence with l discontinuities for some l ≤ k. 1-strong orbit equivalence is called strong orbit equivalence simply.

We should notice that k-strong orbit equivalence is not an equivalence relation if k ≥ 2. For a Cantor minimal system (X, ϕ), we set

K^{0}(X, ϕ) = C(X, ℤ)/{f − fϕ^{−1} ; f ∈ C(X, ℤ)}.

We denote the equivalence class of f ∈ C(X, ℤ) in K^{0}(X, ϕ) by [f]. The K^{0}-group is a unital ordered group with the positive cone

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

and the order unit [1_{X}]. Moreover K^{0}(X, ϕ) is unperforated and satisfies the Riesz interpolation property, and so it is so called a unital dimension group.

Main results of [GPS] are the following.

Theorem 2 ([GPS, Theorem 2.1]). When (X, ϕ) and (Y, ψ) are Cantor minimal systems, the following are equivalent.

(i) (X, ϕ) and (Y, ψ) are strong orbit equivalent.

(ii) K^{0}(X, ϕ) and K^{0}(Y, ψ) are order isomorphic by a map preserving the distinguished order units.

Theorem 3 ([GPS, Theorem 2.2]). When (X, ϕ) and (Y, ψ) are Cantor minimal systems, the following are equivalent.

(i) (X, ϕ) and (Y, ψ) are orbit equivalent.

(ii) K^{0}(X, ϕ)/ Inf(K^{0}(X, ϕ)) and K^{0}(Y, ψ)/ Inf(K^{0}(Y, ψ)) are order isomorphic by a map preserving the distinguished order units.

Let (X, ϕ) be a Cantor minimal system and x_{0}, x_{1}, …, x_{k−1} ∈ X be k distinct points. We denote the K_{0}-group of the AF subalgebra A_{{x_{0},...,x_{k−1}}} by E(x_{0}, x_{1}, …, x_{k−1}) (see Section 3 of [P]). That is,

E(x_{0}, x_{1}, …, x_{k−1}) = C(X, ℤ)/{f − fϕ^{−1} ; f(x_{i}) = 0 for all i = 0, 1, …, k − 1},
E(x_{0}, x_{1}, …, x_{k−1})^{+} = {[f] ∈ E(x_{0}, x_{1}, …, x_{k−1}) ; f ≥ 0}.

For i = 1, 2, …, k − 1, take a clopen neighborhood U_{i} of x_{i} which does not contain the other x_{j}’s. Define a homomorphism ι from ℤ^{k−1} to E(x_{0}, x_{1}, …, x_{k−1}) by sending the i-th canonical basis to the representative class of 1_{U_{i}} − 1_{ϕ(U_{i})}. Then, from [P],

0 → ℤ^{k−1} →^{ι} E(x_{0}, x_{1}, …, x_{k−1}) →^{q} K^{0}(X, ϕ) → 0

is exact, where q is the natural quotient map. We denote the Ext class of this exact sequence by ζ(x_{0}, x_{1}, …, x_{k−1}) ∈ Ext(K^{0}(X, ϕ), ℤ^{k−1}).

Proposition 4. In the above setting, we have the following.

(i) ζ(x_{0}, x_{1}, …, x_{k−1}) = ⨁_{i=1}^{k−1} ζ(x_{0}, x_{i})

(ii) ζ(x_{0}, x_{1}, …, x_{k−1}) depends only on the orbits of x_{i}’s.

(iii) ζ(x_{0}, ϕ(x_{0})) = 0

(iv) ζ(x_{0}, x_{1}) + ζ(x_{1}, x_{2}) = ζ(x_{0}, x_{2})

(v) If x_{i}’s have distinct orbits, E(x_{0}, x_{1}, …, x_{k−1}) is a simple dimension group and the range of the map ι is contained in the infinitesimal subgroup.

Proof. Although every assertion is obvious from the argument in [P] or [GPS], we would like to give a proof for the reader’s convenience.

(i) Let q_{i} be the canonical quotient map from E(x_{0}, x_{1}, …, x_{k−1}) to E(x_{0}, x_{i}). Then,

0 ⟶ ℤ^{k−1} ⟶^{ι} E(x_{0}, x_{1}, …, x_{k−1}) ⟶^{q} K^{0}(X, ϕ) ⟶ 0
↓  ^{q_{i}}↓  ∥
0 ⟶ ℤ ⟶^{ι}  E(x_{0}, x_{i})  ⟶^{q} K^{0}(X, ϕ) ⟶ 0

is commutative and q_{i} sends the i-th basis of ℤ^{k−1} to the basis of ℤ. Hence we get the conclusion.

(ii) It suffices to show ζ(x_{0}, x_{1}) = ζ(x_{0}, ϕ(x_{1})). Let U be a clopen neighborhood of ϕ(x_{1}) which does not contain x_{0}. The map sending f to f − f(ϕ(x_{1}))(1_{U} − 1_{ϕ(U)}) gives rise to a homomorphism π from E(x_{0}, x_{1}) to E(x_{0}, ϕ(x_{1})). It is easy to see that π is an isomorphism and ζ(x_{0}, x_{1}) equals ζ(x_{0}, ϕ(x_{1})) via π.

(iii) The map f ↦ f(ϕ(x_{0})) gives rise to a homomorphism from E(x_{0}, ϕ(x_{0})) to ℤ and this is a left inverse of ι.

(iv) Let U be a clopen neighborhood of x_{2} which does not contain x_{0} and x_{1}. Suppose [f] ∈ E(x_{0}, x_{1}), [g] ∈ E(x_{1}, x_{2}) and q([f]) = q([g]) in K^{0}(X, ϕ). There exists h ∈ C(X, ℤ) such that f = g + h − hϕ^{−1}. By sending ([f], [g]) to [f − (h(x_{1}) − h(x_{2}))(1_{U} − 1_{U}ϕ^{−1})], we obtain a homomorphism from

{([f], [g]) ∈ E(x_{0}, x_{1}) ⊕ E(x_{1}, x_{2}); q([f]) = q([g])}

to E(x_{0}, x_{2}). Since its kernel is

{(ι(n), −ι(n)) ; n ∈ ℤ} ≅ ℤ,

we can conclude ζ(x_{0}, x_{1}) + ζ(x_{1}, x_{2}) = ζ(x_{0}, x_{2}).

(v) This is exactly Corollary 2 of [GPS, Theorem 1.17].

The following proposition is clear from the proof of the implication (i)⇒(ii) of [GPS, Theorem 2.5].

Proposition 5 ([GPS]). Let (X, ϕ) and (Y, ψ) be Cantor minimal systems and F : X → Y be a homeomorphism which gives an orbit equivalence with k discontinuities. Suppose x_{0}, x_{1}, …, x_{k−1} ∈ X and y_{0}, y_{1}, …, y_{k−1} ∈ Y are discontinuities of the orbit cocycles. Then C(X, ℤ) ∋ f ↦ fF^{−1} ∈ C(Y, ℤ) induces a unital order isomorphism from E(x_{0}, x_{1}, …, x_{k−1}) to E(y_{0}, y_{1}, …, y_{k−1}).

In the next section we need the following lemma. The proof is obvious.

Lemma 6. Let π : (X, ϕ) → (Y, ψ) be a factor map between Cantor minimal systems. Suppose x_{0}, x_{1}, …, x_{k−1} ∈ X drop to distinct k points in Y. Then, ζ(x_{0}, x_{1}, …, x_{k−1}) is sent to ζ(π(x_{0}), π(x_{1}), …, π(x_{k−1})) by the canonical homomorphism from Ext(K^{0}(X, ϕ), ℤ^{k−1}) to Ext(K^{0}(Y, ψ), ℤ^{k−1}) induced by π^{∗} : K^{0}(Y, ψ) → K^{0}(X, ϕ).

3 A counter example

At first we construct a Cantor minimal system (X_{0}, ϕ_{0}) whose K^{0}-group is isomorphic to ℚ ⊕ ℚ ⊕ ℚ. Let {a_{n}}_{n∈ℕ} be a sequence of natural numbers which satisfies

α = lim_{N→∞} ∏_{n=1}^{N} \frac{a_{n} + 2}{a_{n} − 1} < ∞.

Moreover, we assume that for every m ∈ ℕ there exists N ∈ ℕ such that ∏_{n=1}^{N}(a_{n} + 2) and ∏_{n=1}^{N}(a_{n} − 1) are both divisible by m. We define a properly ordered simple Bratteli diagram B = (V, E) as follows. Put V_{0} = {v_{0}} and V_{n} = {u_{n}, v_{n}, w_{n}} for all n ∈ ℕ. Connect v_{0} to each vertex of V_{1} by a single edge. The edge set E_{n+1} is defined so that the incidence matrix of n-th step is

[ \begin{matrix}a_{n} & 1 & 1 \\ 1 & a_{n} & 1 \\ 1 & 1 & a_{n} \end{matrix} ] .

It is clear that B = (V, E) is a simple Bratteli diagram. Since

[ \begin{matrix}1 & 1 & 1 \\ 2 & −1 & −1 \\ 0 & 1 & −1 \end{matrix} ] = [ \begin{matrix}a_{n} + 2 \\ & a_{n} − 1 \\ & & a_{n} − 1 \end{matrix} ]^{−1} [ \begin{matrix}1 & 1 & 1 \\ 2 & −1 & −1 \\ 0 & 1 & −1 \end{matrix} ] [ \begin{matrix}a_{n} & 1 & 1 \\ 1 & a_{n} & 1 \\ 1 & 1 & a_{n} \end{matrix} ] ,

by sending xu_{n} + yv_{n} + zw_{n} ∈ ℤ^{V_{n}} to

[ \begin{matrix}3 ∏_{k=1}^{n−1}(a_{k} + 2) \\ & 6 ∏_{k=1}^{n−1}(a_{n} − 1) \\ & & 2 ∏_{k=1}^{n−1}(a_{n} − 1) \end{matrix} ]^{−1} [ \begin{matrix}1 & 1 & 1 \\ 2 & −1 & −1 \\ 0 & 1 & −1 \end{matrix} ] [ \begin{matrix}x \\ y \\ z \end{matrix} ] ,

we have

K_{0}(V, E) ≅ ℚ ⊕ ℚ ⊕ ℚ,

K_{0}(V, E)^{+} = \{(p, q, r) ∈ ℚ^{3} ; \begin{matrix}αp − q + r > 0 \\ αp − q − r > 0 \\ 2αp + q > 0 \end{matrix} \} ∪ \{0\}

and the order unit 1_{B} = (1, 0, 0). Note that 2u_{1} − v_{1} − w_{1} corresponds to (0, 1, 0) and v_{1} − w_{1} corresponds to (0, 0, 1). We write this dimension group by (G, G^{+}, 1_{G}). Notice that the dimension group G has no non-trivial automorphisms except for the flip which changes the signal of the third coordinate.

We define a linear order on each of r^{−1}(u_{n+1}), r^{−1}(v_{n+1}) and r^{−1}(w_{n+1}) so that the first edge has the source vertex u_{n}, the second edge has the source vertex v_{n} and the last edge has the source vertex w_{n}. Then B = (V, E) is obviously properly ordered. Let (X_{0}, ϕ_{0}) be the Cantor minimal system determined by B = (V, E).

Let ψ be the adding machine on

Y = ∏_{n=1}^{∞}{0, 1, 2, …, a_{n} + 1}.

Then the dimension group of (Y, ψ) is (ℚ, ℚ^{+}, 1). It is easily seen that there exists an almost one-to-one factor map π : (X_{0}, ϕ_{0}) → (Y, ψ) and π^{∗} sends p ∈ ℚ to (p, 0, 0) ∈ ℚ^{3}. The following lemma is also clear.

Lemma 7. The factor map π is three-to-one on ⋃_{n} ψ^{n}(E), where

E = ∏_{n=1}^{∞}{2, 3, …, a_{n}} ⊂ Y,

and one-to-one on the other orbits.

Take y = (y_{n})_{n} ∈ E. Suppose x_{0} ∈ X_{0} and x_{1} ∈ X_{1} are distinct preimages of y. We would like to consider ζ(x_{0}, x_{1}) ∈ Ext(K^{0}(X_{0}, ϕ_{0}), ℤ). We identify Ext(K^{0}(X_{0}, ϕ_{0}), ℤ) with Ext(ℚ, ℤ)^{3}. Then it is not hard to see that the first summand of ζ(x_{0}, x_{1}) is zero. Let η_{1} and η_{2} be the second and third summands of ζ(x_{0}, x_{1}). We will compute them. Note that Ext(ℚ, ℤ) is divisible and torsion-free, thus Ext(ℚ, ℤ) is a vector space over ℚ.

Let F (V) be the free abelian group over V and ∂ : F (V) → F (V) be the homomorphism defined by ∂(v) = v − ∑_{s(e)=v} r(e). Then

0 → F (V) ⟶^{∂} F (V) ⟶ K_{0}(V, E) → 0

gives a projective resolution of K_{0}(V, E), and Ext(K^{0}(X_{0}, ϕ_{0}), ℤ) is the quotient of Hom(F (V), ℤ) by the image of ∂^{∗}. By the same computation as in [GPS], we see that ζ(x_{0}, x_{1}) has a representative ρ : F (V) → ℤ given by

ρ(v) = #{e ∈ E_{n+1}; s(e) = v and x_{1}(n + 1) < e} − #{e ∈ E_{n+1}; s(e) = v and x_{0}(n + 1) < e}

for v ∈ V_{n} = {u_{n}, v_{n}, w_{n}}. Suppose x_{0} goes through u_{n}’s and x_{1} goes through v_{n}’s. Then we get

(ρ(u_{n}), ρ(v_{n}), ρ(w_{n})) = (y_{n} − a_{n}, a_{n} − y_{n}, 0).

We denote the basis of the free abelian group F (ℕ) by {e_{n}}_{n∈ℕ}. Put ∂(e_{n}) = e_{n} − (a_{n} − 1)e_{n+1}. Then

0 → F (ℕ) ⟶^{∂} F (ℕ) ⟶ ℚ → 0

gives a projective resolution of ℚ. Let ι_{1} : F (ℕ) → F (V) be the homomorphism defined by ι_{1}(e_{n}) = 2u_{n} − v_{n} − w_{n}. Then we have ∂ι_{1} = ι_{1}∂, and so a representative of η_{1} is given by e_{n} ↦ 3(y_{n} − a_{n}). Similarly, by considering ι_{2}(e_{n}) = v_{n} − w_{n}, we know that a representative of η_{2} is given by e_{n} ↦ a_{n} − y_{n}. Hence we obtain −3η_{2} = η_{1}. Similar computation can be done, when x_{0} or x_{1} goes through w_{n}’s.

These observations give us the following.

Lemma 8. Suppose that x_{0}, x_{1} ∈ X_{0} have distinct orbits. If ζ(x_{0}, x_{1}) = (0, η_{1}, η_{2}) ∈ Ext(ℚ, ℤ)^{3}, then η_{1} and η_{2} are linearly dependent over ℚ.

By [GPS2], we get a Cantor minimal system (X_{1}, ϕ_{1}) and a factor map π_{1} : (X_{1}, ϕ_{1}) → (X_{0}, ϕ_{0}) which satisfy the following:

• The dimension group K^{0}(X_{1}, ϕ_{1}) is isomorphic to G ⊕ ℤ and the positive cone is G^{+}.

• The factor map π_{1} induces the embedding G ∋ p ↦ (p, 0) ∈ G ⊕ ℤ.

• The factor map π_{1} is at most two-to-one and the factor map ππ_{1} is at most three-to-one.

Since π^{∗} induces the canonical isomorphism from Ext(K^{0}(X_{1}, ϕ_{1}), ℤ) to Ext(K^{0}(X_{0}, ϕ_{0}), ℤ), we get the exactly same statement as Lemma 8 for (X_{1}, ϕ_{1}).

Take two elements ξ_{1} and ξ_{2} in Ext(ℚ, ℤ) which are linearly independent over ℚ. Let

0 → ℤ → D → G → 0

be an exact sequence corresponding to (0, ξ_{1}, ξ_{2}) ∈ Ext(G, ℤ). Let D^{+} be the union of zero and the inverse image of G^{+} \ {0}, and let 1_{D} be a preimage of 1_{G}. Then (D, D^{+}, 1_{D}) is a unital simple dimension group, and so there exists a Cantor minimal system (X_{2}, ϕ_{2}) whose dimension group is isomorphic to (D, D^{+}, 1_{D}).

Clearly, Inf(K^{0}(X_{1}, ϕ_{1})) ≅ Inf(K^{0}(X_{2}, ϕ_{2})) ≅ ℤ, and K^{0}(X_{1}, ϕ_{1})/ Inf(K^{0}(X_{1}, ϕ_{1})) is isomorphic to K^{0}(X_{2}, ϕ_{2})/ Inf(K^{0}(X_{2}, ϕ_{2})) as unital simple dimension group. However, we have the following.

Theorem 9. In the above setting, (X_{1}, ϕ_{1}) and (X_{2}, ϕ_{2}) are not 2-strong orbit equivalent.

Proof. Because K^{0}(X_{1}, ϕ_{1}) and K^{0}(X_{2}, ϕ_{2}) are not isomorphic, these two systems are not strong orbit equivalent. Suppose there exists a homeomorphism F : X_{1} → X_{2} which gives an orbit equivalence with two discontinuities. Let x_{0}, x_{1} ∈ X_{1} and y_{0}, y_{1} ∈ X_{2} be the discontinuities of the orbit cocycles. From Proposition 5, the unital dimension groups E(x_{0}, x_{1}) and E(y_{0}, y_{1}) are isomorphic. Let θ be the isomorphism.

Both of E(x_{0}, x_{1}) and E(y_{0}, y_{1}) have the infinitesimal subgroup isomorphic to ℤ^{2}. Hence θ on ℤ^{2} gives a matrix S ∈ GL(2, ℤ), and the following diagram is obtained:

0 ⟶ ℤ^{2} ⟶ E(x_{0}, x_{1}) ⟶ G ⟶ 0
≅↓^{S}  ≅↓_{θ}  ≅↓
0 ⟶ ℤ^{2} ⟶ E(y_{0}, y_{1}) ⟶ G ⟶ 0

Since G has no non-trivial automorphisms except for the flip on the third coordinate, we may assume that θ induces the identity map on G.

Suppose the above two exact sequences are given by (0, η), (ξ, ζ) ∈ Ext(G, ℤ) ⊕ Ext(G, ℤ) respectively, where ξ = (0, ξ_{1}, ξ_{2}) ∈ Ext(ℚ, ℤ)^{3}. Then we get

[ \begin{matrix}ξ \\ ζ \end{matrix} ] = S [ \begin{matrix}0 \\ η \end{matrix} ] .

Therefore

S_{12}η = S_{12}(η_{0}, η_{1}, η_{2}) = (0, ξ_{1}, ξ_{2}),

which shows η_{0} = 0. From Lemma 8, however, η_{1} and η_{2} are linearly dependent over ℚ and this contradicts the linear independence of ξ_{1} and ξ_{2}.

By starting from linearly independent three elements ξ_{1}, ξ_{2}, ξ_{3} ∈ Ext(ℚ, ℤ) and the corresponding exact sequence, we get a counter example for 3-strong orbit equivalence in a similar fashion. For k ≥ 4, we do not need the linear independence and we can get a contradiction much easily, because the factor map π is at most three-to-one.

In Theorem 9 we have shown that (X_{1}, ϕ_{1}) and (X_{2}, ϕ_{2}) are not 2-strong orbit equivalent. But they may be k-strong orbit equivalent for some k ≥ 3. Therefore we may have a chance to show the following statement.

Conjecture 10. When (X_{1}, ϕ_{1}) and (X_{2}, ϕ_{2}) are Cantor minimal systems, the following are equivalent.

(i) (X_{1}, ϕ_{1}) is k-strong orbit equivalent to (X_{2}, ϕ_{2}) for some k ≥ 0.

(ii) For some l ≥ 0, there exist subgroups, both isomorphic to ℤ^{l}, of

Inf(K^{0}(X_{1}, ϕ_{1})) and Inf(K^{0}(X_{2}, ϕ_{2})),

respectively, so that the quotient groups

K^{0}(X_{i}, ϕ_{i})/ℤ^{l} (i = 1, 2)

(with the induced order) are order isomorphic by a map preserving the distinguished order units (more precisely, the quotient image of the distinguished order units).

It seems rather hard to find a counter example for the above conjecture.

4 More on 2-strong orbit equivalence

The 2-strong orbit equivalence closely relates to the surjectivity of the map X^{2} ∋ (x_{0}, x_{1}) ↦ ζ(x_{0}, x_{1}) ∈ Ext(K^{0}(X, ϕ), ℤ). Of course, this map is not surjective in general, which was shown in Lemma 8. But we can show the surjectivity in some cases. Let us denote the real analogue of K^{0}-groups by K_{ℝ}^{0}(X, ϕ) (see [O]). That is,

K_{ℝ}^{0}(X, ϕ) = C(X, ℝ)/{f − fϕ^{−1} ; f ∈ C(X, ℝ)}.

Notice that K_{ℝ}^{0}(X, ϕ) is a real vector space.

Theorem 11. When (X, ϕ) is a Cantor minimal system and K^{0}(X, ϕ) is isomorphic to ℚ or ℚ^{2}, the map (x_{0}, x_{1}) ↦ ζ(x_{0}, x_{1}) is surjective. Moreover we can make x_{0} and x_{1} have distinct orbits, unless (X, ϕ) is an odometer system.

Proof. Suppose K^{0}(X, ϕ) is isomorphic to ℚ. There exists a factor map π from (X, ϕ) to the odometer system (Y, ψ) whose K^{0}-group is ℚ. For every ξ ∈ Ext(ℚ, ℤ), there exist distinct points y_{0}, y_{1} ∈ Y such that ζ(y_{0}, y_{1}) = ξ. Then, Lemma 4 leads us to the conclusion. When ξ = 0 and (X, ϕ) is not an odometer system, take distinct points x_{0} and x_{1} with π(x_{0}) = π(x_{1}). Then ζ(x_{0}, x_{1}) = 0.

Next, let (X, ϕ) be a Cantor minimal system whose K^{0}-group is ℚ⊕ℚ. Take a clopen set U so that [1_{X}] and [1_{U}] are linearly independent over ℚ in K^{0}(X, ϕ). If they are linearly independent over ℝ in K_{ℝ}^{0}(X, ϕ), then f = 1_{U} satisfies the condition (i) of Lemma 13. If [1_{U}] = s[1_{X}] in K_{ℝ}^{0}(X, ϕ) for some irrational number s ∈ (0, 1), there exist f ∈ C(X, ℤ) and h ∈ C(X, ℝ) such that 0 ≤ h ≤ 4/3, [f] = [1_{U}] and f = s1_{X} + h − hϕ^{−1}. That is, the condition (ii) of Lemma 13 is satisfied. We put

G = {s ∈ K^{0}(X, ϕ) ; ns = m[1_{X}] for some n ∈ ℕ, m ∈ ℤ}

and

H = {t ∈ K^{0}(X, ϕ) ; nt = m[f] for some n ∈ ℕ, m ∈ ℤ}.

Then they are both isomorphic to ℚ and G⊕H ∋ (s, t) ↦ s+t ∈ K^{0}(X, ϕ) is an isomorphism. Let π_{G} and π_{H} be the natural homomorphism from Ext(K^{0}(X, ϕ), ℤ) to Ext(G, ℤ) and Ext(H, ℤ). Of course π_{G} ⊕ π_{H} is an isomorphism.

To prove the surjectivity, assume ξ ∈ Ext(K^{0}(X, ϕ), ℤ) is given and ξ ≠ 0. Let η_{G} ∈ Hom(G, ℝ/ℤ) and η_{H} ∈ Hom(H, ℝ/ℤ) be representatives of π_{G}(ξ) and π_{H}(ξ). We may assume that η_{G}([1_{X}]) = η_{H}([f]) = 0. We use the notation of Lemma 12. For every natural number n we set

F_{n} = \{(x, y) ∈ X^{2} ; η_{G}(\frac{1}{n!}[1_{X}]) = ρ_{1_{X}}^{x,y}(\frac{1}{n!}), η_{H}(\frac{1}{n!}[f]) = ρ_{f}^{x,y}(\frac{1}{n!})\} .

From the definition of ρ_{f}^{x,y} it is easy to see that F_{n} is a closed set and F_{n+1} ⊂ F_{n}. Thanks to Lemma 13 each F_{n} is not empty. Hence we obtain (x_{0}, x_{1}) ∈ ⋂ F_{n}. It is clear that ζ(x_{0}, x_{1}) is equal to ξ.

In the case of ξ = 0, we get the conclusion by Lemma 18.

Lemma 12. Let (X, ϕ) be a Cantor minimal system and x, y ∈ X be distinct points. Suppose τ : ℚ → K^{0}(X, ϕ) is an injection and τ(1) = [f]. If a sequence of natural numbers {a_{n}}_{n} satisfies lim_{n→∞} ϕ^{a_{n}}(y) = x, then

ℚ ∋ r ↦ ( lim_{n→∞} r ∑_{k=1}^{a_{n}} f(ϕ^{k}(y))) + ℤ

gives a well-defined homomorphism ρ_{f}^{x,y} from ℚ to ℝ/ℤ and it is a representative of τ^{∗}(ζ(x, y)).

Proof. For any natural number N, there exists g, h ∈ C(X, ℤ) with f − Ng = h − hϕ^{−1}. Then

∑_{k=a_{l}+1}^{a_{m}} f(ϕ^{k}(y)) = N ∑_{k=a_{l}+1}^{a_{m}} g(ϕ^{k}(y)) + h(ϕ^{a_{m}}(y)) − h(ϕ^{a_{l}}(y))

is zero modulo N for sufficiently large l < m, since the last two terms are canceled. Hence ρ_{f}^{x,y} : ℚ → ℝ/ℤ is a well-defined homomorphism.

Let us show that ρ_{f}^{x,y} is a representative of τ^{∗}(ζ(x, y)). Put

F = {s ∈ E(x, y) ; q(s) ∈ τ(ℚ)}.

Suppose [g] ∈ F and q([g]) = τ(r). There exists a locally constant function h : X → ℚ such that rf − g = h − hϕ^{−1}. By sending [g] ∈ F to

lim_{n→∞} ∑_{k=1}^{a_{n}}(rf(ϕ^{k}(y)) − g(ϕ^{k}(y))) = lim_{n→∞} h(ϕ^{a_{n}}(y)) − h(y) = h(x) − h(y),

we get a well-defined homomorphism ρ : F → ℚ. If U is a clopen neighborhood of y which does not contain x, we see ρ([1_{U} − 1_{ϕ(U)}]) = 1. The proof is completed because ρ induces ρ_{f}^{x,y}.

Lemma 13. Let (X, ϕ) be a Cantor minimal system and suppose a function f ∈ C(X, ℤ) satisfies either of the following:

(i) [f] and [1_{X}] are linearly independent over ℝ in K_{ℝ}^{0}(X, ϕ) and f is a characteristic function on a clopen set.

(ii) There exist an irrational number s ∈ (0, 1) and h ∈ C(X, ℝ) such that 0 ≤ h ≤ 4/3 and f = s1_{X} + h − hϕ^{−1}.

Then, for any natural numbers N, m and l, there exist x ∈ X and k ∈ ℕ such that

k ≡ m, f(ϕ(x)) + f(ϕ^{2}(x)) + ⋯ + f(ϕ^{k}(x)) ≡ l (mod N).

Proof. Put

g_{n} = fϕ + fϕ^{2} + ⋯ + fϕ^{nN+m}

for every n ∈ ℕ.

(i) Take an invariant measure μ. Since f −μ(f)1_{X} is not a coboundary, we get lim_{n→∞} ||g_{n} − (nN + m)μ(f)1_{X}|| = ∞. Besides, we have

max{g_{n}(x) − (nN + m)μ(f) ; x ∈ X} ≥ 0

and

min{g_{n}(x) − (nN + m)μ(f) ; x ∈ X} ≤ 0

because μ(g_{n} − (nN + m)μ(f)1_{X}) = 0. Hence we obtain

max_{x∈X} g_{n}(x) − min_{x∈X} g_{n}(x) ≥ N

for some n ∈ ℕ. As f is a characteristic function, g_{n}(x) − g_{n}(ϕ(x)) ∈ {−1, 0, 1} for every x ∈ X. Therefore we can find x ∈ X with g_{n}(x) ≡ l (mod N).

(ii) Since s is irrational, we can find n ∈ ℕ and t ∈ (1/3, 2/3) with (nN + m)s − t ∈ Nℤ + l. Then we have

g_{n} = (nN + m)s1_{X} + hϕ^{nN+m} − h

and hϕ^{nN+m}(x) − h(x) equals −t or 1 − t because of 0 ≤ h ≤ 4/3. As a coboundary cannot be positive, there exists x ∈ X with hϕ^{nN+m}(x) − h(x) = −t. Consequently we get g_{n}(x) ≡ l (mod N).

In [GPS] the following was stated as a corollary.

Conjecture 14. Let (X, ϕ) and (Y, ψ) be Cantor minimal systems. If there exists an order isomorphism from K^{0}(X, ϕ) to K^{0}(Y, ψ) preserving the order units modulo infinitesimal subgroups, then (X, ϕ) and (Y, ψ) are 2-strong orbit equivalent.

If the discontinuities of the orbit cocycle are allowed to lie in the same orbit, the above conjecture is solved positively. This is because we can prove that E(x_{0}, ϕ(x_{0})) is unital order isomorphic to E(y_{0}, ψ(y_{0})) for every x_{0} ∈ X and y_{0} ∈ Y. But, 2-strong orbit equivalence requires the discontinuities to lie in distinct orbits. Thus, we can show the above conjecture if the following is true.

Conjecture 15. Let (X, ϕ) be a Cantor minimal system which is not conjugate to an odometer system. Then there exist x_{0}, x_{1} ∈ X lying in distinct orbits such that ζ(x_{0}, x_{1}) is equal to zero.

We do not know whether the above conjectures are correct or not.

Definition 16. Let (X, ϕ) be a Cantor minimal system. Two distinct points x_{0}, x_{1} ∈ X are said to be positively asymptotic, if d(ϕ^{n}(x_{0}), ϕ^{n}(x_{1})) converges to zero as n → ∞.

Proposition 17. When (X, ϕ) is a Cantor minimal system and two distinct points x_{0}, x_{1} ∈ X are positively asymptotic, ζ(x_{0}, x_{1}) is zero in Ext(K^{0}(X, ϕ), ℤ).

Proof. For f ∈ C(X, ℤ), put

ρ(f) = ∑_{n=0}^{∞} f(ϕ^{n}(x_{0})) − f(ϕ^{n}(x_{1})),

which is well-defined because f(ϕ^{n}(x_{0})) = f(ϕ^{n}(x_{1})) for sufficiently large n. When f(x_{0}) = f(x_{1}) = 0, we can see ρ(f − fϕ^{−1}) = 0. Hence ρ gives rise to a homomorphism from E(x_{0}, x_{1}) to ℤ. If a clopen neighborhood U of x_{1} does not contain x_{0}, then ρ(1_{U} − 1_{ϕ(U)}) = 1. Therefore ζ(x_{0}, x_{1}) is equal to zero.

Thus, if (X, ϕ) and (Y, ψ) have positively asymptotic pairs, Conjecture 14 is true. It is well known that every minimal subshift has positively asymptotic pairs. In a recent paper [BHR], systems of positive entropy also have positively asymptotic pairs. Hence Conjecture 14 is true for these kinds of Cantor minimal systems. In general, however, it is not known whether or not a Cantor minimal system always has positively asymptotic pairs unless it is not an odometer system.

We would like to conclude this section with the following lemma, which implies Conjecture 14 is also true when K^{0}-groups are of finite rank.

Lemma 18. When (X, ϕ) is a Cantor minimal system except for odometer systems and K^{0}(X, ϕ) is of finite rank, there exist x_{0}, x_{1} ∈ X lying in distinct orbits and ζ(x_{0}, x_{1}) = 0.

Proof. Let p_{1}, p_{2}, …, p_{m} ∈ C(X, ℤ) be a maximal family of independent basis of K^{0}(X, ϕ). We can define a map π from X to (ℤ^{m})^{ℤ} by

π(x)_{k} = (p_{1}(x), p_{2}(x), …, p_{m}(x))

for k ∈ ℤ. The infinite sequence π(x) actually consists of finite alphabets, and so π is regarded as a factor map to a subshift. Thus, we can find a factor map π : (X, ϕ) → (Y, ψ) so that (Y, ψ) is a minimal subshift and π^{∗}(K^{0}(Y, ψ)) contains [p_{1}], [p_{2}], …, [p_{m}].

Let Γ = K^{0}(X, ϕ)/π^{∗}(K^{0}(Y, ψ)). Note that Γ is a countable torsion group. Suppose [f] ∈ K^{0}(X, ϕ) is of order n in Γ. Then there exist g ∈ C(Y, ℤ) and h ∈ C(X, ℤ) such that nf = gπ + h − hϕ^{−1}. Let ψ̃ be a homeomorphism on Y × ℤ/nℤ determined by

ψ̃(y, k) = (ψ(y), k + g(ψ(x))),

where the addition is understood modulo n. The dynamical system (Y × ℤ/nℤ, ψ̃) is called the skew product extension of (Y, ψ) associated with the ℤ/nℤ-valued cocycle g. Lemma 3.6 of [M] tells us that (Y × ℤ/nℤ, ψ̃) is a Cantor minimal system. When we define a map π_{1} from X to Y × ℤ/nℤ by π_{1}(x) = (π(x), h(x)), it is easy to see that π_{1} is a factor map from (X, ϕ) to (Y × ℤ/nℤ, ψ̃) and π = π_{0}π_{1} where π_{0} is the canonical projection from Y × ℤ/nℤ to Y. Let γ ∈ Homeo(Y × ℤ/nℤ) be the canonical centralizer determined by γ(y, k) = (y, k + 1). Take s ∈ K^{0}(Y ×ℤ/nℤ, ψ̃). We have ms ∈ π_{0}^{∗}(K^{0}(Y, ψ)) for some m ∈ ℕ because Γ is a torsion group. Hence ms = mod(γ)(ms) = m mod(γ)(s), which implies s = mod(γ)(s). That is, mod(γ) is the identity map on K^{0}(Y × ℤ/nℤ, ψ̃). Thanks to Theorem 3.7 of [M2], we can conclude that K^{0}(Y, ψ)/nK^{0}(Y, ψ) is isomorphic to ℤ/nℤ and [g] is a generator. We also remark that [gπ_{0}] is n-divisible in K^{0}(Y × ℤ/nℤ, ψ̃), and so [f] ∈ π_{1}^{∗}(K^{0}(Y × ℤ/nℤ, ψ̃).

If [f], [f′] ∈ K^{0}(X, ϕ) are both of order n, the above argument implies that both of n[f] = [gπ] and n[f′] = [g′π] are generators of K^{0}(Y, ψ)/nK^{0}(Y, ψ) ≅ ℤ/nℤ. Therefore we get [g′] − k[g] ∈ nK^{0}(Y, ψ) for some k ∈ ℕ, and so [f′] − k[f] ∈ π^{∗}(K^{0}(Y, ψ)). Consequently Γ has only one n-cyclic component, which implies that Γ is a subgroup of ℚ/ℤ.

We can find a sequence of natural numbers {a_{n}}_{n} such that a_{1}|a_{2}|a_{3}| … and

Γ ≅ ⋃_{n=1}^{∞} \frac{1}{a_{n}} ℤ + ℤ.

It is well-known that the dual group of the discrete abelian group Γ is

Γ̂ = proj lim ℤ/a_{n}ℤ

which is a compact zero-dimensional abelian group. Suppose f_{1} ∈ C(X, ℤ) is of order a_{1} in Γ and {f_{n}}_{n=1}^{∞} ⊂ C(X, ℤ) satisfies f_{1} − a_{1}^{−1}a_{n}f_{n} ∈ π^{∗}(K^{0}(Y, ψ)). Let h_{n} ∈ C(X, ℤ) and g_{n} ∈ C(Y, ℤ) be functions satisfying

a_{n}f_{n} = g_{1}π + ∑_{k=1}^{n−1} a_{k}g_{k+1}π + h_{n} − h_{n}ϕ^{−1}.

For some fixed x_{0} ∈ X we may assume that h_{n}(x_{0}) = 0 for all n ∈ ℕ. Then we have h_{n+1}(x) − h_{n}(x) is a_{n}-divisible for all n ∈ ℕ and x ∈ X. Hence

H : X ∋ x ↦ (h_{1}(x), h_{2}(x), h_{3}(x), …)

is a well-defined continuous map from X to Γ̂. Similarly

G : Y ∋ y ↦ (g_{1}(y), g_{1}(y) + a_{1}g_{2}(y), g_{1}(y) + a_{1}g_{2}(y) + a_{2}g_{3}(y), …)

is well-defined as a continuous map from Y to Γ̂. In the same way as the case of the cyclic group valued cocycle, we can define a homeomorphism ψ̃ on Y × Γ̂ by

ψ̃(y, k) = (ψ(y), k + G(ψ(y))).

Moreover π_{1} : X ∋ x ↦ (π(x), H(x)) gives a factor map from (X, ϕ) to (Y × Γ̂, ψ̃) and π = π_{0}π_{1} where π_{0} is the canonical projection from Y × Γ̂ to Y. For every f ∈ C(X, ℤ), there exists a_{n} such that a_{n}[f] ∈ π^{∗}(K^{0}(Y, ψ)). Since every element of K^{0}(Y, ψ) is a_{n}-divisible in K^{0}(Y ×Γ̂, ψ̃), we can conclude that π_{1}^{∗} is an isomorphism.

There exists a positively asymptotic pair (y_{0}, y_{1}) in Y, as (Y, ψ) is a minimal subshift. Then

k = lim_{n→∞} ∑_{i=1}^{n}(G(ψ^{i}(y_{0})) − G(ψ^{i}(y_{1})))

exists in Γ̂. Therefore ((y_{0}, 0), (y_{1}, k)) is a positively asymptotic pair in (Y ×Γ̂, ψ̃) and, by virtue of Proposition 17, ζ((y_{0}, 0), (y_{1}, k)) is zero. Because π_{1}^{∗} is an isomorphism, preimages of (y_{0}, 0) and (y_{1}, k) by π_{1} do the work.

References

[BHR] Blanchard, F.; Host, B.; Ruette, S.; Asymptotic pairs in positive-entropy systems, Ergodic Theory Dynam. Systems 22 (2002), 671–686.

[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, Math. Scand. 89 (2001), 297–319.

[M] Matui, H.; Ext and OrderExt classes of certain automorphisms of C^{∗}-algebras arising from Cantor minimal systems, Canad. J. Math. 53 (2001), 325–354.

[M2] Matui, H.; Finite order automorphisms and dimension groups of Cantor minimal systems, J. Math. Soc. Japan 54 (2002), 135–160.

[O] Ormes, N.; Real coboundaries for minimal Cantor systems, Pacific J. Math. 195 (2000), 453–476.

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

e-mail; matui@math.s.chiba-u.ac.jp Department of Mathematics and Informatics, Faculty of Science, Chiba University, Yayoityô 1-33, Inageku, Chiba, 263-8522, Japan.