Topological spectrum of locally compact Cantor minimal systems ^{∗}

MATUI Hiroki

March 2002

Abstract

We show that there exists a locally compact Cantor minimal system whose topological spectrum has a given Hausdorff dimension.

1 Introduction

The purpose of this paper is to introduce the topological spectrum for LCCM systems and to discuss the relation to the dimension groups. In the measurable setting, the group of L^{∞}- eigenvalues of a non-singular ergodic transformations were studied by several authors ([A], [AN], [IKS], [HMP]), and it was known that the group may be uncountable and can have arbitrary Hausdorff dimension. We would like to consider its topological analogue in this paper.

Our object of study is an LCCM or CM system. Let us introduce them. A homeomorphism ϕ from a topological space X to itself is called minimal, if every ϕ-orbit is dense in X. This is obviously equivalent to non-existence of non-trivial closed invariant sets. When X is the Cantor set and ϕ ∈ Homeo(X) is minimal, we say (X, ϕ) is a Cantor minimal system, or CM system shortly. A non-closed open subset of the Cantor set is called a locally compact Cantor set. When ϕ is a minimal homeomorphism on the locally compact Cantor set X, the pair (X, ϕ) is called a locally compact Cantor minimal system, or LCCM system shortly. CM and LCCM systems were studied in many papers ([GPS], [D], [M]), and it was proved that their dimension groups are the complete invariant for topological orbit equivalence. We view these topological dynamical systems as an analogue of ergodic systems of type II_{1} or II_{∞}.

At first we give a definition of eigenvalues and eigenfunctions. Of course the eigenfunctions have to be continuous, and we will see that this restriction implies the richness of continuous characters. In Section 2, after introducing the Polish topology on the topological spectrum group, we will consider topology of the dual group and the duality. In Section 3, we will construct an LCCM system whose topological spectrum has a given Hausdorff dimension and which is strong orbit equivalent to a given LCCM system. The method of construction is similar to that of measurable cases, given in [A].

2 Topological spectrum

We begin with the definition of topological spectrum.

Definition 1. Let (X, ϕ) be a CM or LCCM system. If a continuous function f : X → ℝ/ℤ and λ ∈ ℝ/ℤ satisfy fϕ = f + λ, we call f an eigenfunction and λ an eigenvalue of (X, ϕ). We denote the set of eigenfunctions by E(ϕ) and the set of eigenvalues by e(ϕ). Each of these sets clearly forms an abelian group. We call e(ϕ) the topological spectrum of (X, ϕ).

∗2000 Mathematics Subject Classification. 37B05.

We denote by P the canonical homomorphism from E(ϕ) to e(ϕ). Equip E(ϕ) with the topology of uniform convergence on compact subsets on X and e(ϕ) with the quotient topology. For λ ∈ ℝ/ℤ we write the distance from zero by |λ|.

Lemma 2. Let (X, ϕ) be a CM or LCCM systems and U ⊂ X be a compact open set with U ∩ ϕ(U) ≠ ∅. Then, d(f, g) = max{|f(x) − g(x)|; x ∈ U} gives a complete metric on E(ϕ), and E(ϕ) is a Polish group.

Proof. For every compact set K ⊂ X, there exists N such that K ⊂ ⋃_{|n|≤N} ϕ^{n}(U). Therefore d(·, ·) gives a metric. Separability and completeness follow from those of C(U).

The following lemma is obvious.

Lemma 3. For a CM or LCCM system (X, ϕ), we have the following.

(i) The canonical inclusion e(ϕ) → ℝ/ℤ is continuous.

(ii) For every x ∈ X there exists a continuous homomorphism P_{x} : e(ϕ) → E(ϕ) such that P P_{x} = id and P_{x}(λ)(x) = 0 for all λ ∈ e(ϕ). Especially, E(ϕ) ≅ e(ϕ) × ℝ/ℤ as topological groups.

(iii) The topological spectrum e(ϕ) is a Polish group.

Since e(ϕ) is Polish, a Borel set of e(ϕ) is Borel in ℝ/ℤ. In the same fashion as in the measurable case [AN], we can show that a Polish topology on e(ϕ) under which the injection map e(ϕ) → ℝ/ℤ is continuous is unique. Note that, however, (ii) of the above lemma is rather different from the measurable case [AN, Theorem 2.5]. Fix x_{0} ∈ X and P_{x_{0}} : e(ϕ) → E(ϕ). For every x ∈ X,

e(ϕ) ∋ λ ↦ P_{x_{0}}(λ)(x) ∈ ℝ/ℤ

gives a (continuous) character of e(ϕ). We denote it by τ_{x} and the set of all characters of e(ϕ) by ε(ϕ). In contrast to the measurable case, the splitting homomorphism P_{x_{0}} ensures the rich structure of ε(ϕ), unless e(ϕ) is trivial.

Lemma 4. The spectrum e(ϕ) of a CM or LCCM system (X, ϕ) is a proper subset of ℝ/ℤ. Moreover e(ϕ) is a weak Dirichlet set of ℝ/ℤ.

Proof. Take a probability measure μ on e(ϕ). For x_{0} ∈ X, there exists {a_{n}}_{n} ⊂ ℤ \ {0} such that x_{n} = ϕ^{a_{n}}(x_{0}) → x_{0}. By the Lebesgue convergence theorem, we have

lim_{n→∞} ∫_{e(ϕ)} |e^{2πia_{n}λ} − 1|^{2} dμ = lim_{n→∞} ∫_{e(ϕ)} |e^{2πiτ_{xn}(λ)} − e^{2πiτ_{x_{0}}(λ)}|^{2} dμ = 0,

that is, e(ϕ) is a weak Dirichlet set.

If (X, ϕ) has a finite invariant measure, it is not hard to see that e(ϕ) is at most countable and endowed with the discrete topology. Hence we will mainly focus our attention on the case that (X, ϕ) is an LCCM system without finite invariant measures.

Let B = (V, E) be a properly ordered almost simple Bratteli diagram and ∞_{B} = (e_{n})_{n}. For convenience we assume that r(s^{−1}(V_{n} \ {r(e_{n})})) equals V_{n+1} \ {r(e_{n+1})} for all n ∈ ℕ (see [M, Remark 2.5]). We denote the ideal part of B = (V, E) by B̃ = (Ṽ , Ẽ). For v ∈ Ṽ_{n} and λ ∈ ℝ/ℤ we let Var(v, λ) be the supremum of |mλ|, where m runs over all integers such that there exists x ∈ X_{B} which goes through v and whose initial n edges agree with those of ϕ_{B}^{m}(x). Then we have the following.

Proposition 5. In the above setting,

e(ϕ_{B}) = {λ ∈ ℝ/ℤ ; Var(v, λ) ≤ 3^{−1} for some v ∈ Ṽ }
= {λ ∈ ℝ/ℤ ; Var(v, λ) → 0 as v → ∞}.

Proof. Take λ ∈ e(ϕ_{B}). We will show that there exists a vertex v such that Var(v, λ) ≤ 3^{−1}. For x_{0} = (e_{1}, e_{2}, …) ∈ X_{B} \ {∞_{B}}, let U_{n} be a set of infinite paths whose initial n edges agree with x_{0}. Then U_{n} is a clopen neighborhood of x_{0} and ⋂_{n} U_{n} = {x_{0}}. Because an eigenfunction f corresponding to λ is continuous, there exists n such that |f(x)−f(y)| < 3^{−1} for all x, y ∈ U_{n}. Then the vertex r^{(}e_{n}) does the job.

Next, suppose Var(w_{0}, λ) ≤ 3^{−1} holds for some λ ∈ ℝ/ℤ and w_{0} ∈ Ṽ_{n}. We would like to show lim Var(v, λ) = 0. By telescoping B = (V, E), we may assume that every vertices v, v′ lying in successive levels are connected. The proof is by contradiction. Suppose Var(v, λ) > ε holds for infinitely many v ∈ Ṽ. At first, we can find n_{1} > n and w_{1} ∈ Ṽ_{n_{1}} such that Var(w_{1}, λ) > ε. Hence there exist an infinite path x_{1} and m_{1} ∈ ℤ such that x_{1} and ϕ^{m_{1}}(x_{1}) have the same initial n_{1} edges and |m_{1}λ| > ε. We may assume that x_{0} goes through w_{0} and m_{1}λ ∈ (ε, 3^{−1}] in ℝ/ℤ. Since x_{0} and ϕ^{m_{1}}(x_{0}) have the same tail, we can find n_{2} > n_{1} and w_{2} ∈ Ṽ_{n_{2}} such that Var(w_{2}, λ) is greater than ε and k-th edges of x_{0} and ϕ^{m_{1}}(x_{0}) agree for every k > n_{2}. Therefore there exist an infinite path x_{2} and m_{2} ∈ ℤ, such that x_{2} and ϕ^{m_{2}}(x_{2}) have the same initial n_{2} edges and |m_{2}λ| > ε. In a similar fashion to the first step, we may assume that x_{2} and x_{1} have the same initial n_{2} − 1 edges and m_{2}λ ∈ (ε, 3^{−1}] in ℝ/ℤ. By repeating this argument, we get x_{k} and m_{1}, m_{2}, …, m_{k} ∈ ℤ. Then we have

|(m_{1} + m_{2} + ⋯ + m_{k})λ| ≤ Var(w_{0}, λ) ≤ 3^{−1}

and m_{i}λ ∈ (ε, 3^{−1}], which shows a contradiction.

At last, suppose lim Var(v, λ) = 0. Fix an infinite path x_{0} ∈ X_{B} \ {∞_{B}}. We define a function f : X_{B} \ {∞_{B}} → ℝ/ℤ by

f(x) = lim_{n→∞} a_{n}λ,

when ϕ^{a_{n}}(x_{0}) goes to x. If ϕ^{i}(x_{0}) and ϕ^{j}(x_{0}) have the same initial n edges (e_{1}, e_{2}, …, e_{n}) and r(e_{n}) = v ∈ Ṽ_{n}, then |(i − j)λ| ≤ Var(v, λ). Hence the function f is well-defined. By the same reason we have the continuity of f, and so f is in E(ϕ) and λ ∈ e(ϕ).

We can say the following regarding the topology of e(ϕ).

Lemma 6. Let B = (V, E) be a properly ordered almost simple Bratteli diagram as above.

(i) Suppose {λ_{n}}_{n∈ℕ} ⊂ e(ϕ) and λ ∈ e(ϕ). If λ_{n} goes to λ in e(ϕ), then Var(v, λ_{n}) converges to zero uniformly on n as v → ∞.

(ii) Suppose {λ_{n}}_{n∈ℕ} ⊂ e(ϕ) and λ_{n} goes to λ in ℝ/ℤ. If Var(v, λ_{n}) converges to zero uniformly on n as v → ∞, then λ is an eigenvalue of ϕ and λ_{n} → λ in e(ϕ).

(iii) The spectrum e(ϕ) is σ-compact in ℝ/ℤ.

Proof. We have λ_{n} → 0 in e(ϕ) if and only if λ_{n} → 0 in ℝ/ℤ and sup_{v∈Ṽ} Var(v, λ_{n}) → 0 as n → ∞. Thus (i) and (ii) are clear.

We prove (iii). It suffices to show that

{λ ∈ ℝ/ℤ ; Var(v, λ) ≤ 3^{−1}}

is compact for every v ∈ Ṽ. But it is obvious.

Note that e(ϕ) itself is not σ-compact unless it is countable.

Example 7. We define an almost simple Bratteli diagram B = (V, E) by the following figure.

[figure: v_{0}; v_{1}; v_{2}; v_{3}; v_{4}; v_{5}; u_{1}; u_{2}; u_{3}; u_{4}; u_{5}; a_{1}; a_{2}; a_{3}; a_{4}]

The oblique thick lines show that there exist a_{n} edges from v_{n} to u_{n} for each n ∈ ℕ, where a_{n} is a natural number greater than one. We introduce a linear order on each r^{−1}(u_{n}) for n ≥ 2 so that the source vertex of the maximum edge and the minimum edge is v_{n−1} and the two edges from u_{n−1} appear consecutively. Then B = (V, E) is properly ordered.

Let b_{n} be the number of paths from v_{0} to u_{n}. Thus {b_{n}}_{n} satisfies b_{n+1} = 2b_{n} + a_{n}. Then, from Proposition 5, we have

e(ϕ_{B}) = {λ ∈ ℝ/ℤ ; ∑_{n∈ℕ}|a_{n}λ| < ∞}.

This kind of groups is called H_{1}-groups ([HMP]). The polish topology of e(ϕ_{B}) is given by the norm λ ↦ |λ| + ∑ |a_{n}λ|.

We would like to discuss the topology of ε(ϕ) and the duality. We have two candidates of the topology on ε(ϕ). One is the topology of pointwise convergence and the other is of uniform convergence on compact sets of e(ϕ). We denote ε(ϕ) equipped with the former weak topology by ε(ϕ)_{w}, and denote ε(ϕ) equipped with the latter strong topology by ε(ϕ)_{s}. They are obviously topological groups.

Lemma 8. For a CM or LCCM system (X, ϕ), the natural map X ∋ x ↦ τ_{x} ∈ ε(ϕ)_{s} is continuous. Hence X ∋ x ↦ τ_{x} ∈ ε(ϕ)_{w} is also continuous.

Proof. Let K ⊂ e(ϕ) be a compact set. Since P_{x_{0}}(K) is a compact subset of E(ϕ), it is equicontinuous on compact sets of X. Thus we have the conclusion.

For λ ∈ e(ϕ), we denote the homomorphism ε(ϕ) ∋ τ ↦ τ(λ) ∈ ℝ/ℤ by λ̂. Thus, λ̂ is an element of the dual group of ε(ϕ)_{w}, which is clearly a subgroup of the dual group of ε(ϕ)_{s}. We write these dual groups by ε̂(ϕ)_{w} and ε̂(ϕ)_{s}. The character group ε(ϕ) clearly contains ℤ generated by the canonical inclusion map e(ϕ) → ℝ/ℤ unless e(ϕ) is finite. One can show that ℤ is dense in ε(ϕ)_{w}, and so e(ϕ) is isomorphic to ε̂(ϕ)_{w} via the map λ ↦ λ̂ as abelian group.

Lemma 9. In the above setting, λ ↦ λ̂ gives an isomorphism of topological groups e(ϕ) and ε̂(ϕ)_{w}, where ε̂(ϕ)_{w} is equipped with the topology of uniform convergence on compact sets of ε(ϕ)_{s}.

Proof. Suppose λ_{n} → 0 in e(ϕ). Let K be a compact subset of ε(ϕ)_{s}. We would like to prove sup_{τ∈K} τ(λ_{n}) goes to zero. Assume the assertion were false. Then we would have δ > 0 and τ_{n} ∈ K with |τ_{n}(λ_{n})| > δ. Because K is compact, we can find τ ∈ K whose every neighborhood contains infinitely many τ_{n}. There exists a natural number M such that |τ(λ_{m})| < δ/4 for all m ≥ M. Since {λ_{m}}_{m≥M} ∪{0} is compact, there exists a neighborhood U of τ such that |τ′(λ_{m})| is less than δ/2 for all τ′ ∈ U. This is a contradiction.

Conversely, suppose λ̂_{n} goes to zero in ε̂(ϕ)_{w}. Let K be a compact subset of X. Then,

sup_{x∈K} P_{x_{0}}(λ_{n})(x) = sup_{x∈K} λ̂_{n}(τ_{x}) → 0,

because {τ_{x} ; x ∈ K} is compact in ε(ϕ)_{s}.

It is unclear whether or not the canonical subgroup ℤ is dense in ε(ϕ)_{s}. If it is dense, one can see that ε̂(ϕ)_{w} equals ε̂(ϕ)_{s} and the duality theorem holds.

In the above lemma, we equipped ε̂(ϕ)_{w} with the topology of uniform convergence on compact sets of ε(ϕ)_{s}. It is easy to see that this topology does not coincide with the topology of pointwise convergence in general. But we know the following.

Lemma 10. Let {f_{n}}_{n∈ℕ} ⊂ E(ϕ) be a sequence of eigenfunctions. Suppose there exists an eigenfunction f ∈ E(ϕ) such that f_{n}(x) converges to f(x) for all x ∈ X. Then, f_{n} converges to f in E(ϕ).

Proof. It suffices to show that f_{n} converges to f uniformly on compact sets of X. We may suppose f = 0. Let λ_{n} ∈ e(ϕ) be the eigenvalue corresponding to f_{n}. It is clear that λ_{n} goes to zero in ℝ/ℤ.

Assume the assertion were false. Then we would have δ > 0 and {x_{n}}_{n∈ℕ} ⊂ X such that |f_{n}(x_{n})| is greater than δ and x_{n} goes to x_{0} ∈ X. At first, we can find a compact open subset V_{1} such that |f_{1}(y)| > δ/2 for all y ∈ V_{1}. Put g_{1} = f_{1}. The minimality implies ϕ^{k}(x_{0}) ∈ V_{1} for some k ∈ ℤ. By taking sufficiently large m, we may assume |kλ_{m}| < δ/2 and ϕ^{k}(x_{m}) ∈ V_{1}. Since |f_{m}(ϕ^{k}(x_{m}))| > δ − δ/2 = δ/2, there exists a clopen subset V_{2} ⊂ V_{1} such that |f_{m}(y)| > δ/2 for all y ∈ V_{2}. Put g_{2} = f_{m}. By repeating this procedure, we will find a subsequence {g_{n}}_{n} and a decreasing sequence of clopen sets V_{n} such that |g_{n}(y)| > δ/2 for all y ∈ V_{n}, which implies that g_{n}(x) does not converge to zero for x ∈ ⋂ V_{n}. This is a contradiction.

3 Hausdorff dimension

We will show Theorem 12 in this section. Let D = (W, F) be a simple Bratteli diagram. We may assume each vertex of W_{1} connects to the top vertex by a single edge. We will construct an almost simple Bratteli diagram B = (V, E) containing D as its ideal part. Put

q_{n} = max{#r^{−1}(w) ; w ∈ W_{n+1}}.

By telescoping we may assume 2^{n^{2}} ≤ q_{n}. Let b_{n} be the integer satisfying 2^{b_{n}} ≤ q_{n} < 2^{b_{n}+1}. Take w_{n} ∈ W_{n} with q_{n−1} = r^{−1}(w_{n}). We can define a proper order on D = (W, F) so that the unique minimum infinite path and unique maximum infinite path go through w_{n} ∈ W_{n}.

We would like to construct an almost simple Bratteli diagram B = (V, E) by adding vertices and edges to D = (W, F). Let V_{0} = {v_{0}} and V_{n} be the disjoint union of a copy of W_{n} and {v_{n}} for every n ∈ ℕ. Connect v_{n−1} to v_{n} by a single edge for n ∈ ℕ. These edges form the infinite path ∞_{B}. Take a sequence {a_{n}}_{n∈ℕ} of natural numbers satisfying a_{1} = 0 and a_{n+1} > a_{n}+b_{n}+n. We define E_{n} by adding to F_{n} some edges going from v_{n−1} to W_{n}, so that the number of paths from the top vertex to w in B = (V, E) is p_{n} = 2^{a_{n}} for all w ∈ W_{n}.

Take n ≥ 2 and w ∈ W_{n}. Let (f_{1}, f_{2}, …, f_{m}) be the ordered list of edges of r^{−1}(w) ∩ F_{n} in D = (W, F). We define a linear order on each r^{−1}(w) ⊂ E_{n}, so that the ordered list of edges in B = (V, E) is

(e_{1}, e_{2}, …, e_{p_{n−1}}, f_{1}, f_{2}, …, f_{m}, e′_{1}, e′_{2}, …, e′_{l}),

where e_{i}’s and e′_{j}’s have the source vertex v_{n−1} and 2^{a_{n−1}} + m2^{a_{n−1}} + l = 2^{a_{n}}. Then, B = (V, E) is clearly a properly ordered almost simple Bratteli diagram. Let (X, ϕ) be the associated LCCM system.

Lemma 11. In the above setting, we have

e(ϕ) = {λ ∈ ℝ/ℤ ; ∑_{n=1}^{∞}(q_{n} − 1)|p_{n}λ| < ∞}.

Proof. Suppose λ ∈ e(ϕ). By Proposition 5, there exists a natural number N such that Var(w_{N}, λ) ≤ 3^{−1}. Let (f_{1}, f_{2}, …) be the unique minimum path in X_{D}. We can identify x_{0} = (f_{1}, f_{2}, …) with a point in X = X_{B} \ {∞_{B}}. Put

L^{+} = {n ∈ ℕ ; n ≥ N + 1 and p_{n}λ ∈ [0, 2^{−1}) in ℝ/ℤ}

and

L^{−} = {n ∈ ℕ ; n ≥ N + 1 and p_{n}λ ∈ [−2^{−1}, 0) in ℝ/ℤ}.

Let n_{1} < n_{2} < ⋯ < n_{k} and n_{i} ∈ L^{+}. It is easy to see that ϕ^{jp_{n_{k}}} (x_{0}) has the same initial N edges with x_{0} for j = 1, 2, …, q_{n_{k}} − 1. Hence we have (q_{n_{k}} − 1)|p_{n_{k}}λ| ≤ Var(w_{N}, λ) ≤ 3^{−1}. Moreover, the initial n_{k} edges of ϕ^{(q_{n_{k}} −1)p_{n_{k}}} (x_{0}) agree with those of x_{0}. Therefore ϕ^{(q_{n_{k}} −1)p_{n_{k}} +jp_{n_{k−1}}} (x_{0}) also has the same initial N edges with x_{0} for j = 1, 2, …, q_{n_{k−1}} − 1, and so we get

(q_{n_{k}} − 1)|p_{n_{k}}λ| + (q_{n_{k−1}} − 1)|p_{n_{k−1}}λ| ≤ 3^{−1}

since n_{k}, n_{k−1} ∈ L^{+}. By repeating this argument,

∑_{i=1}^{k}(q_{n_{i}} − 1)|p_{n_{i}}λ| ≤ 3^{−1}

is obtained. Hence

∑_{n∈L^{+}}(q_{n} − 1)|p_{n}λ| + ∑_{n∈L^{−}}(q_{n} − 1)|p_{n}λ| ≤ 3^{−1} + 3^{−1} = \frac{2}{3},

which means that ∑_{n}(q_{n} − 1)|p_{n}λ| is finite.

Let us prove the other inclusion. We may suppose that ∑_{n=N}^{∞}(q_{n} −1)|p_{n}λ| ≤ 6^{−1}. We would like to prove Var(w_{N}, λ) ≤ 3^{−1}. To do that, it suffices to show that |mλ| ≤ 6^{−1} whenever x_{0} and ϕ^{m}(x_{0}) have the same initial N edges. There exists a natural number M greater than N such that the nth edges of x_{0} and ϕ^{m}(x_{0}) coincide for every n > M. We can find k_{M−1} with 0 ≤ k_{M−1} < q_{M−1} so that the M-th edges of ϕ^{m}(x_{0}) and ϕ^{k_{M−1}p_{M−1}}(x_{0}) coincide. Next, we can find k_{M−2} with 0 ≤ k_{M−2} < q_{M−2} so that the M −1-st edges of ϕ^{m}(x_{0}) and ϕ^{k_{M−2}p_{M−2}+k_{M−1}p_{M−1}}(x_{0}) coincide. By repeating this procedure, we get

m = k_{N}p_{N} + k_{N+1}p_{N+1} + ⋯ + k_{M−1}p_{M−1}

for some 0 ≤ k_{i} < q_{i}. Therefore |mλ| ≤ 6^{−1}.

Since 2^{b_{n}} ≤ q_{n} < 2^{b_{n}+1},

e(ϕ) = {λ ∈ ℝ/ℤ ; ∑_{n=1}^{∞} 2^{b_{n}}|2^{a_{n}}λ| < ∞},

where {a_{n}}_{n} and {b_{n}}_{n} satisfy a_{n+1} > a_{n} + b_{n} + n and b_{n} ≥ 2^{n}. Put

K = ⋃_{n=1}^{∞}[a_{n}, a_{n} + b_{n}], L = ⋃_{n=1}^{∞}[a_{n}, a_{n} + b_{n} + n].

Then, from [IKS], we can conclude that the Hausdorff dimension of e(ϕ) equals

1 − lim sup_{n→∞} \frac{1}{n}#(K ∩ [0, n]) = 1 − lim sup_{n→∞} \frac{1}{n}#(L ∩ [0, n]).

Hence we obtain the following theorem.

Theorem 12. Let (D, D^{+}) be a simple dimension group and α ∈ [0, 1] a real number. Then, there exists an LCCM system (X, ϕ) such that the Hausdorff dimension of e(ϕ) is equal to α and K^{0}(X, ϕ) is isomorphic to (D, D^{+}, D^{+}).

Proof. If α ≠ 1, let a_{n} be the largest integer which is not greater than

\frac{∑_{i=1}^{n−1} b_{i} + i}{1 − α} + n.

If α = 1, let a_{n} be a sequence increasing rapidly.

References

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[D] Danilenko, A. I.; Strong orbit equivalence of locally compact Cantor minimal systems, Internat. J. Math. 12 (2001), 113–123.

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

[HMP] Host, B.; Mela, J.-F.; Parreau, F.; Nonsingular transformations and spectral analysis of measures, Bull. Soc. Math. France 119 (1991), 33–90.

[IKS] Ito, Y.; Kamae, T.; Shiokawa, I.; Point spectrum and Hausdorff dimension, Number theory and combinatorics (Japan 1984), 209–227, World Sci. Publishing, Singapore, 1985.

[M] Matui, H.; Topological orbit equivalence of locally compact Cantor minimal systems, to appear in Ergodic Theory Dynam. Systems.

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.