ON THE INTERSECTION OF TRANSLATION OF MIDDLE-α CANTOR SETS

WENXIA LI & DONGMEI XIAO Department of Mathematics, Central China Normal University Wuhan 430079, P.R. China

Abstract

Let Γ_{α} be the middle-α Cantor set and let Γ_{α} + t = {x + t : x ∈ Γ_{α}} for −1 ≤ t ≤ 1. Let H_{p} = {t ∈ Γ_{α} : dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = dim_{B}(Γ_{α} ⋂(Γ_{α} + t)) = −p \frac{log 2}{log β} \}, 0 < p < 1 and H_{p}^{∗} = \{t ∈ Γ_{α} − Γ_{α} : dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = dim_{B}(Γ_{α} ⋂(Γ_{α} + t)) = −p \frac{log 2}{log β} \}, 0 < p < 1. We give the dimensions of Γ_{α} ⋂(Γ_{α} + t), H_{p} and H_{p}^{∗}. In addition the characteristic of H_{p} and H_{p}^{∗} are described, respectively, by means of some measures μ and μ^{∗} supported by Γ_{α} and Γ_{α} − Γ_{α} respectively.

1 Introduction

The middle-α Cantor sets are a straightforward generalization of the classical middle third Cantor set. To define a middle-α Cantor set in the interval [0,1], first choose α ∈ (0, 1), let β = (1 − α)/2 and then define the similarity maps

F_{i}(x) = βx + i(1 − β), i = −1, 0, 1.

The middle-α Cantor set Γ_{α} in the interval [0,1] is defined as the unique invariant nonempty compact set under maps F_{0} and F_{1}, i.e.

Γ_{α} = F_{0}(Γ_{α}) ⋃ F_{1}(Γ_{α}).  (1)

As we know, in dynamical systems there exists much more subtle and complicated dynamical behavior associated with a homoclinic tangency. For example, the unfolding of homoclinic tangencies provide examples on how one can pass directly from a simple nonchaotic system to the chaotic region, without going through cascades of period doubling bifurcations (see [2],[10]). A homoclinic tangency is associated with a saddle fixed or periodic point of a diffeomorphism. This saddle may or not be part of a larger invariant set, say a horseshoe,i.e. a Cantor set. Generally the Hausdorff dimension of such a Cantor set is a number strictly between zero and one. Consider a zero-dimensional hyperbolic invariant set Λ in the plane. The stable and unstable manifolds, W^{s}(Λ) and W^{u}(Λ), are each locally the product of a Cantor set with an interval. The question is what conditions guarantee that there will be a point of tangency between two of the segments in these sets. If we regard W^{u}(Λ) as a Cantor set of horizontal lines, then the question is whether some curve in W^{s}(Λ) has a point with a horizontal tangency which is contained in W^{u}(Λ). Naively, it would appear that if this is the case, then a slight vertical shift of W^{s}(Λ) would separate the Cantor set at which W^{s}(Λ) has a horizontal tangent from W^{u}(Λ), as happens when Λ is a point or a periodic orbit. However, for general hyperbolic sets, Newhouse observed that this is not always possible. And the creation of horseshoes have led to the problem of analyzing the difference sets {t : Γ ⋂(Γ′ + t) ≠ ∅} of Cantor sets Γ and Γ′ embedded in the real line (see [6][8][9]).

Now taking Γ = Γ′ = Γ_{α}, it seems very interesting to analyze the difference set Γ_{α} − Γ_{α}. Kraft[5] applyed cardinality and Lebesgue measure to the difference set Γ_{α} − Γ_{α} to distinguish between different middle-α Cantor sets. The name and notation for the difference set come from the definition,

Γ_{α} − Γ_{α} := {x − y : x, y ∈ Γ_{α}}.  (2)

Here is a more ”dynamical” way of defining the difference set,

Γ_{α} − Γ_{α} = {t : Γ_{α} ⋂(Γ_{α} + t) ≠ ∅}  (3)

where Γ_{α} + t := {x + t : x ∈ Γ_{α}}. Notice that for any choice of α ∈ (0, 1), Γ_{α} ⊂ Γ_{α} − Γ_{α} ⊆ [−1, 1]. Now we hope to apply the fractal dimensions to Γ_{α} − Γ_{α} and the related sets. We shall try to answer the following questions:

Question 1. What are the fractal dimensions of Γ_{α} − Γ_{α} for α ∈ (0, 1) ?. Question 2. What are the fractal dimensions of Γ_{α} ⋂(Γ_{α} + t) for t ∈ Γ_{α} − Γ_{α} ?. Question 3. Since Γ_{α} ⋂(Γ_{α} + t) ⊆ Γ_{α}, we have 0 ≤ dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) ≤ dim_{H} Γ_{α} = −\frac{log 2}{log β}. Then for any given 0 ≤ p ≤ 1, is there t ∈ Γ_{α} − Γ_{α} such that dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = −p\frac{log 2}{log β} ?. Question 4. For 0 < p < 1 define

H_{p} := \{t ∈ Γ_{α} : dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = dim_{B}(Γ_{α} ⋂(Γ_{α} + t)) = −p\frac{log 2}{log β} \} ,

H_{p}^{∗} := \{t ∈ Γ_{α} − Γ_{α} : dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = dim_{B}(Γ_{α} ⋂(Γ_{α} + t)) = −p\frac{log 2}{log β} \} .

What are the fractal dimensions of H_{p} and H_{p}^{∗} ?. Can we characterize H_{p} and H_{p}^{∗} by means of some measures μ and μ^{∗} supported by Γ_{α} and Γ_{α} − Γ_{α} respectively ?.

Note that the sets Γ_{α} − Γ_{α}, Γ_{α} ⋂(Γ_{α} + t), H_{p} and H_{p}^{∗} can be described geometrically. Consider the set Γ_{α} × Γ_{α} in the xoy-plane. Let L_{t} denote the straight line y = x + t. Then Γ_{α} − Γ_{α} consists of those t which make the A_{t} := L_{t} ⋂(Γ_{α} × Γ_{α}) ≠ ∅. The set Γ_{α} ⋂(Γ_{α} + t) is the orthogonal projection of A_{t} on the y-axis and dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = dim_{H} A_{t}, dim_{B}(Γ_{α} ⋂(Γ_{α} + t)) = dim_{B} A_{t}. H_{p}^{∗} consists of those t such that dim_{H} A_{t} = dim_{B} A_{t} = −p\frac{log 2}{log β}, while H_{p} consists of those t such that the intersection points of L_{t} with y-axis lie in Γ_{α} × Γ_{α} and dim_{H} A_{t} = dim_{B} A_{t} = −p\frac{log 2}{log β}.

It is not difficult to verify that

Γ_{α} − Γ_{α} = F_{0}(Γ_{α} − Γ_{α}) ⋃ F_{−1}(Γ_{α} − Γ_{α}) ⋃ F_{1}(Γ_{α} − Γ_{α}),  (4)

i.e. Γ_{α} − Γ_{α} is a self-similar set generated by similarity maps F_{0}, F_{−1} and F_{1}. Now Let us introduce the following notations:

(i) Ω := ∏_{1}^{∞}{0, 1}, Ω^{(n)} := ∏_{1}^{n}{0, 1}, Ω^{∗} := ⋃_{n=1}^{∞} Ω^{(n)}; Ω_{1} := ∏_{1}^{∞}{−1, 0, 1},

Ω_{1}^{(n)} := ∏_{1}^{n}{−1, 0, 1}, Ω_{1}^{∗} := ⋃_{n=1}^{∞} Ω_{1}^{(n)};

(ii) For σ = (σ(1), σ(2), ⋯) ∈ Ω or Ω_{1}, k ∈ N let σ|k := (σ(1), ⋯, σ(k));

(iii) Denote F_{σ} := F_{σ(1)} ∘ ⋯ ∘ F_{σ(n)} for σ ∈ Ω^{(n)} or Ω_{1}^{(n)}.

Thus we have

Γ_{α} = ⋂_{n=1}^{∞}( ⋃_{σ∈Ω^{(n)}} F_{σ}([0, 1])).  (5)

Γ_{α} − Γ_{α} = ⋂_{n=1}^{∞}( ⋃_{σ∈Ω_{1}^{(n)}} F_{σ}([−1, 1])).  (6)

Two natural continuous correspondences ϕ : Ω ⟼ Γ_{α} and ϕ_{1} : Ω_{1} ⟼ Γ_{α} − Γ_{α} are given as follows respectively:

ϕ(σ) : = ⋂_{n=1}^{∞} F_{σ|n}([0, 1]) = ∑_{i=0}^{∞} σ(i + 1)β^{i}(1 − β)

= lim_{n→∞} ∑_{i=0}^{n−1} σ(i + 1)β^{i}(1 − β) = lim_{n→∞}F_{σ|n}(0);  (7)

ϕ_{1}(σ) : = ⋂_{n=1}^{∞} F_{σ|n}([−1, 1]) = ∑_{i=0}^{∞} σ(i + 1)β^{i}(1 − β)

= lim_{n→∞} ∑_{i=0}^{n−1} σ(i + 1)β^{i}(1 − β) = lim_{n→∞}F_{σ|n}(0).  (8)

We call the sequence {σ(1), σ(2), ⋯} as the address of ϕ(σ) ∈ Γ_{α}, respectively, of ϕ_{1}(σ) ∈ Γ_{α} − Γ_{α}. Clearly for each x ∈ Γ_{α} it has unique address in Ω. But for each x ∈ Γ_{α} −Γ_{α} it may have multiple addresses in Ω_{1} unless 0 < β < 1/3 when the maps F_{i}, i = −1, 0, 1, satisfy the open set condition related to the open set (−1, 1).

It is known that dim_{H} Γ_{α} = dim_{B} Γ_{α} = −\frac{log 2}{log β} since the maps F_{0}, F_{1} satisfy the open set condition related to the open set (0, 1).

The rest of this present paper will be arranged as follows. In §2 we first list some basic properties of sets Γ_{α} − Γ_{α} and Γ_{α} ⋂(Γ_{α} + t) which can be found in [5] but are proved in simpler way here. The dimensions of Γ_{α} − Γ_{α}, Γ_{α} ⋂(Γ_{α} + t), H_{p} and H_{p}^{∗} are studied in §3 and §4 respectively.

2 Basic properties of Γ_{α} − Γ_{α} and Γ_{α} ⋂(Γ_{α} + t)

In this section we discuss some basic properties of Γ_{α} − Γ_{α} and Γ_{α} ⋂(Γ_{α} + t) which can be found in [5] but are proved in simpler way here.

Theorem 2.1 If β < \frac{1}{3}, then Γ_{α} − Γ_{α} is a Cantor set of Lebesque measure zero. If β ≥ \frac{1}{3}, then Γ_{α} − Γ_{α} = [−1, 1].

Proof When β < 1/3 the maps F_{−1}, F_{0} F_{1} satisfy the open set condition . Then dim_{H}(Γ_{α} − Γ_{α}) = −\frac{log 3}{log β} < 1. So Γ_{α} − Γ_{α} is a Cantor set of Lebesgue measure zero. When β ≥ 1/3 we have [−1, 1] = F_{0}([−1, 1]) ⋃ F_{−1}([−1, 1]) ⋃ F_{1}([−1, 1]) which indicates Γ_{α} − Γ_{α} = [−1, 1] by (6).

Lemma 2.2 Choose β ∈ (0, 1/2). If Γ_{α} ⋂(Γ_{α} + t) is a single point, then t ∈ ∧_{α} := {∑_{i=0}^{∞} u_{i}β^{i}(1 − β) : u_{i} ∈ {−1, 1} for all i}.

Proof Let σ ∈ Ω_{1} be such that ϕ_{1}(σ) = t ∈ Γ_{α} − Γ_{α}. Thus {σ(1), σ(2), ⋯} is a address in Ω_{1} of t and

t = ∑_{i=0}^{∞} σ(i + 1)β^{i}(1 − β).

Suppose that there exits some σ(k) = 0. Take τ, ω ∈ Ω such that

(τ(i), ω(i)) = \{ \begin{matrix}(0, 0) & if σ(i) = −1, 0 and i ≠ k; \\ (1, 1) & if σ(i) = 1 ; \\ (0, 1) & if i = k. \end{matrix}

Thus we will get two different points ϕ(τ), ϕ(ω) ∈ Γ_{α} ⋂(Γ_{α} + t).

Theorem 2.3 If β ≥ 1/3, then Γ_{α} ⋂(Γ_{α} + t) is a Cantor set for almost all t ∈ [−1, 1].

Proof Let Δ := {σ ∈ Ω_{1} : there is some k ≥ 0 such that σ(i) ≠ 0 for i ≥ k}. Thus

ϕ_{1}(Δ) = ∧_{α} ⋃( ⋃_{k=1}^{∞} ⋃_{τ∈Ω_{1}^{(k)}}(β^{k} ∧_{α} + ∑_{i=0}^{k−1} τ(i + 1)β^{i}(1 − β))).

Therefore ϕ_{1}(Δ) ⊂ Γ_{α} − Γ_{α} = [−1, 1] has Lebesgue measure zero since ∧_{α} has Lebesgue measure zero. So we only need to prove for any t ∈ [−1, 1] \ ϕ_{1}(Δ), Γ_{α} ⋂(Γ_{α} + t) has not any isolated point. Now for any fixed t ∈ [−1, 1] \ ϕ_{1}(Δ) we have that for each σ = (σ(1), σ(2), ⋯) ∈ ϕ_{1}^{−1}(t) there are n_{i} ⟶ ∞ with σ(n_{i}) = 0. Thus for any x ∈ Γ_{α} ⋂(Γ_{α} + t) with address {τ(1), τ(2), ⋯} there exists some σ ∈ ϕ_{1}^{−1}(t) such that

x = ∑_{i=1}^{∞} τ(i)β^{i−1}(1 − β), τ(i) ∈ A(σ(i))

by Lemma 3.3. Let n_{i} ⟶ ∞ with σ(n_{i}) = 0. Therefore the points x_{i} := x + (1 − 2τ(n_{i}))β^{(n_{i}−1)}(1 − β) ∈ Γ_{α} ⋂(Γ_{α} + t) will tend to x.

Theorem 2.4 If β ∈ (\sqrt{2} − 1, 1/2), then Γ_{α} ⋂(Γ_{α} + t) contains a Cantor set for all t ∈ (−1, 1).

Proof First we will prove that for t ∈ (−1, 1) it has a address {u_{1}, u_{2}, ⋯} such that there are infinitely many u_{i} = 0. This address {u_{1}, u_{2}, ⋯} can be obtained as follows. If t ∈ [−β, β] then take u_{1} = 0. If t ∈ (−1, −β) since F_{−1}(β) > −β and

F_{−1}^{n}(F_{0}(−1)) = −β^{n+1} − ∑_{i=0}^{n−1} β^{i}(1 − β) ⟶ − ∑_{i=0}^{∞} β^{i}(1 − β) = −1 (n → ∞)

where F_{−1}^{n} denotes the n times iterations of F_{−1}, there exists some n such that t ∈ F_{−1}^{n} ∘ F_{0}([−1, 1])\ ⋃_{i=1}^{n−1} F_{−1}^{i} ∘ F_{0}([−1, 1]). At this case we take u_{1} = ⋯ = u_{n} = −1 and u_{n+1} = 0. If t ∈ (β, 1) the similar argument as above shows that there exits some n such that t ∈ F_{1}^{n} ∘ F_{0}([−1, 1])\ ⋃_{i=1}^{n−1} F_{1}^{i} ∘ F_{0}([−1, 1]). And at this case we take u_{1} = ⋯ = u_{n} = 1 and u_{n+1} = 0. Consequently according to the above for t ∈ (−1, 1) we can get n ∈ {0} ⋃ N such that u_{1} = ⋯ = u_{n} = −1 or 1 and u_{n+1} = 0. Now denote t_{1} = β^{−(n+1)}(t−∑_{i=0}^{n} u_{i+1}β^{i}(1−β)). Then t_{1} ∈ (−1, 1). For t_{1} repeating the above process we can get some n_{1} ∈ {0} ⋃ N such that t_{1} ∈ F_{1}^{n_{1}} ∘ F_{0}([−1, 1])\ ⋃_{i=1}^{n_{1}−1} F_{1}^{i} ∘F_{0}([−1, 1]) or t_{1} ∈ F_{−1}^{n_{1}} ∘F_{0}([−1, 1])\ ⋃_{i=1}^{n_{1}−1} F_{−1}^{i} ∘F_{0}([−1, 1]). Now we take u_{n+2} = ⋯ = u_{n+n_{1}+1} = 1 or −1, u_{n+n_{1}+2} = 0. We can repeat the above process for t_{2} := β^{−(n+n_{1}+2)}(t−∑_{i=0}^{n+n_{1}+1} u_{i+1}β^{i}(1−β)) to further determine consequent u_{i}’s.

For the address {u_{1}, u_{2}, ⋯} chosen above of t let

Q := {x = ∑_{i=0}^{∞} σ(i + 1)β^{i}(1 − β) : σ(i) = 0  if u_{i} = −1, σ(i) = 1 if u_{i} = 1,
σ(i) = 0 or 1 if u_{i} = 0}.

Thus Q ⊂ Γ_{α} ⋂(Γ_{α} + t) and Q is a generalized Moran set which is a Cantor set (see §3).

3 The dimensions of Γ_{α} − Γ_{α} and Γ_{α} ⋂(Γ_{α} + t)

In this section we will apply the Hausdorff, box and packing dimensions to the middle-α Cantor sets, the difference sets Γ_{α} − Γ_{α} and Γ_{α} ⋂(Γ_{α} + t) to distinguish between different middle-α Cantor sets. To do this let us recall the generalized Moran set and related results.

A class of generalized Moran sets is introduced [3,4,7] and studied which is defined as follows:

Let {n_{k}}_{k≥1} be sequence of integers and let T_{k} = {0 < t_{j}^{(k)} < 1 : 1 ≤ j ≤ n_{k}} for k ∈ N. Besides, the following notations are used:

(i) S = {σ = (σ(1), σ(2), ⋯) : 1 ≤ σ(j) ≤ n_{j}};

(ii) S^{k} = {σ = (σ(1), σ(2), ⋯, σ(k)) : 1 ≤ σ(j) ≤ n_{j}} for k ∈ N and S^{∗} = ⋃_{k=1}^{∞}S^{k};

(iii) t(σ) = t_{σ(1)}^{(1)}t_{σ(2)}^{(2)} ⋯ t_{σ(k)}^{(k)} for σ ∈ S^{k};

(iv) σ ∗ i = (σ(1), σ(2), ⋯, σ(k), i) where σ = (σ(1), σ(2), ⋯, σ(k)) ∈ S^{k} and 1 ≤ i ≤ n_{k+1}; (v) σ|k = (σ(1), σ(2), ⋯, σ(k)) for σ ∈ S and k ∈ N.

Now for each σ ∈ S^{∗} we construct compact set J_{σ} ⊂ R^{d}. Given nonempty compact set J ⊂ R^{d} with \overline{intJ} = J, we take a family of nonoverlapping nonempty compact subset \{J_{i} : 1 ≤ j ≤ n_{1}\} of J such that \overline{intJ_{j}} = J_{j} and |J_{j}| = t_{j}^{(1)}|J|. Suppose that J_{σ} is given for some σ ∈ S^{k}. Take a family of nonoverlapping nonempty compact subsets \{J_{σ∗i} : 1 ≤ i ≤ n_{k+1}\} of J_{σ} such that \overline{intJ_{σ∗i}} = J_{σ∗i} and |J_{σ∗i}| = t_{i}^{(k+1)}|J_{σ}|. The nonempty compact set

E = ⋂_{k=1}^{∞}⋃_{σ∈S^{k}}J_{σ}  (9)

is called generalized Moran set. The following results have been proved in [3,4].

Theorem 3.1 Assume (1) there is a positive number c such that for all σ ∈ S^{∗}, J_{σ} contains a ball with diameter c|J_{σ}|; (2) inf_{k,j}t_{j}^{(k)} > 0 and (3)lim_{k→∞}sup_{σ∈S^{k}}t(σ) = 0. Then we have

dim_{H}E = \underline{s}, dim_{P} E = \overline{dim}_{B}E = \overline{s}

where \underline{s} = \underline{lim}_{k→∞}s_{k}, \overline{s} = \overline{lim}_{k→∞}s_{k} and s_{k} satisfy ∏_{i=1}^{k}[∑_{j=1}^{n_{i}} (t_{j}^{(i)})^{s_{k}}] = 1.

Example Let {Φ_{k}}_{k≥1} = {ϕ_{1}^{(k)}, ⋯, ϕ_{n_{k}}^{(k)}}_{k≥1} be a family of similar maps in R^{d}

and the similar ratio for ϕ_{j}^{(k)} be t_{j}^{(k)}. Let {Φ_{k}}_{k≥1} satisfy the uniformly open set condition,i.e. there is a bounded nonempty open set V ⊂ R^{d} such that for each k ∈ N and 1 ≤ j ≤ n_{k}, ϕ_{j}^{(k)}(V) ⊂ V and ϕ_{i}^{(k)}(V) ⋂ ϕ_{j}^{(k)}(V) = ∅ for 1 ≤ i ≠ j ≤ n_{k}. Now take J = \overline{V} and J_{σ} = ϕ_{σ(1)}^{(1)} ∘ ϕ_{σ(2)}^{(2)} ∘ ⋯ ∘ ϕ_{σ(k)}^{(k)}(J) for each k ∈ N and σ ∈ S^{k}. Thus by (9) we can get a class of generalized Moran set generated by the similarity contraction maps family {Φ_{k}}_{k≥1}. It is easy to see that this kind of generalized Moran sets satisfy the condition (1) of Theorem 3.1.

On the sets Γ_{α} − Γ_{α}, from the Theorem 2.1 it follows

Theorem 3.2 When 0 < β < 1/3, dim_{H}(Γ_{α} − Γ_{α}) = dim_{B}(Γ_{α} − Γ_{α}) = −\frac{log 3}{log β}. When 1/3 ≤ β < 1/2, dim_{H}(Γ_{α} − Γ_{α}) = dim_{B}(Γ_{α} − Γ_{α}) = 1.

Let A := {0, 1} and A + i := {i + j : j ∈ A}, i = −1, 0, 1. Let A(i) := A ⋂(A + i), i = −1, 0, 1. To get the dimensions of sets Γ_{α} ⋂(Γ_{α} + t) we need the following

Lemma 3.3 For t ∈ Γ_{α} − Γ_{α}

Γ_{α} ⋂(Γ_{α} + t) = ⋃_{σ∈ϕ_{1}^{−1}(t)} {∑_{i=1}^{∞} τ(i)β^{i−1}(1 − β) : τ(i) ∈ A(σ(i))} .

Proof For any x ∈ Γ_{α} ⋂(Γ_{α} + t) there exists u ∈ Ω such that ϕ(τ) − ϕ(u) = t where τ = ϕ^{−1}(x), i.e.

t = ϕ(τ) − ϕ(u) = ∑_{i=1}^{∞}(τ(i) − u(i))β^{i−1}(1 − β).

Taking σ(i) = τ(i) − u(i) then σ ∈ ϕ_{1}^{−1}(t) and τ(i) ∈ A(σ(i)).

On the other hand for any x ∈ ⋃_{σ∈ϕ_{1}^{−1}(t)}{∑_{i=1}^{∞} τ(i)β^{i−1}(1−β) : τ(i) ∈ A(σ(i))} there is some σ ∈ ϕ_{1}^{−1}(t) and τ ∈ Ω with τ(i) ∈ A(σ(i)) such that

x = ∑_{i=1}^{∞} τ(i)β^{i−1}(1 − β),

which means x ∈ Γ_{α}. Taking u(i) = τ(i) − σ(i) then we have u ∈ Ω and x = ϕ(u) + t ∈ (Γ_{α} + t).

Theorem 3.4 Let 0 < β ≤ 1/3. Then for any t ∈ Γ_{α} − Γ_{α}

dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = \underline{dim}_{B}(Γ_{α} ⋂(Γ_{α} + t))
= \underline{lim}_{k→∞} − \frac{log(∏_{j=1}^{k} m(σ(j)))}{k} · \frac{1}{log β} = −\frac{log 2}{log β} \underline{t};

dim_{P} (Γ_{α} ⋂(Γ_{α} + t)) = \overline{dim}_{B}(Γ_{α} ⋂(Γ_{α} + t))
= \overline{lim}_{k→∞} − \frac{log(∏_{j=1}^{k} m(σ(j)))}{k} · \frac{1}{log β} = −\frac{log 2}{log β} \overline{t},

where σ = (σ(1), σ(2), ⋯) ∈ ϕ_{1}^{−1}(t), m(σ(j)) := CardA(σ(j)) and \overline{t} (resp. \underline{t}) is the upper (resp. lower) density of 0 occuring in ϕ_{1}^{−1}(t).

Proof Note that maps F_{i}, i = −1, 0, 1, satisfy the open set condition with the open set (−1, 1) and the set Γ_{α} − Γ_{α} is invariant under F_{i}, i = −1, 0, 1. Thus for any t ∈ Γ_{α} − Γ_{α} the set ϕ_{1}^{−1}(t) is a singleton when 0 < β < 1/3, contains at most two elements when β = 1/3. In addition when ϕ_{1}^{−1}(t) contains exactly two elements, one has its all components equal to 1 with finite exception and the other has its all components equal to −1 with finite exception. Therefore when ϕ_{1}^{−1}(t) contains exactly two elements, Γ_{α} ⋂(Γ_{α} + t) is a finite set by Lemma 3.3. In this case we have lim_{k→∞}[log(∏_{j=1}^{k} m(σ(j)))]/k = 0 since m(−1) = m(1) = 1 which proves our results. When ϕ_{1}^{−1}(t) is a singleton the Lemma 3.3 and the Example tell us that Γ_{α} ⋂(Γ_{α} + t) is a generalized Moran set generated by the maps family {Φ_{k}}_{k≥1} where {Φ_{k}} = {F_{i} : i ∈ A(σ(k)}. Let s_{k} be such that

∏_{j=1}^{k}[ ∑_{i∈A(σ(j))} β^{s_{k}}] = 1,

i.e. s_{k} = −[log(∏_{j=1}^{k} m(σ(j)))]/[k log β]. Therefore by Theorem 3.1

dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = \underline{lim}_{k→∞} − \frac{log(∏_{j=1}^{k} m(σ(j)))}{k} · \frac{1}{log β} = −\frac{log 2}{log β} \underline{t};

dim_{P} (Γ_{α} ⋂(Γ_{α} + t)) = \overline{dim}_{B}(Γ_{α} ⋂(Γ_{α} + t))
= \overline{lim}_{k→∞} − \frac{log(∏_{j=1}^{k} m(σ(j)))}{k} · \frac{1}{log β} = −\frac{log 2}{log β} \overline{t}.

Since all the maps F_{i}, i = 0, 1 have the same similarity ratio we have dim_{H}(Γ_{α} ⋂(Γ_{α} +t)) = \underline{dim}_{B}(Γ_{α} ⋂(Γ_{α} + t)) by Theorem 2.4 in [7].

Corollary 3.5 Let 0 < β ≤ 1/3. For any 0 ≤ p ≤ 1 there exists t ∈ Γ_{α} − Γ_{α} such that

dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = dim_{B}(Γ_{α} ⋂(Γ_{α} + t)) = −p\frac{log 2}{log β} .

In addition the above t can be chosen from both Γ_{α} and (Γ_{α} − Γ_{α}) \ Γ_{α}.

Proof When p = 0 we can take t = 0 ∈ Γ_{α} or t = (1 − β)^{2} ∈ (Γ_{α} − Γ_{α})\Γ_{α} with address {1, −1, 0, 0, ⋯}. When p ≠ 0 let n_{k} be the maximal natural number such that [n_{k}p] = k where [n_{k}p] is the integer part of n_{k}p. So n_{k+1} ≤ n_{k} + n_{1}. Let σ(n) = 0 if n = n_{k} for some k ∈ N, otherwise |σ(n)| = 1. Taking σ = (σ(1), σ(2), ⋯) we have

lim_{k→∞} \frac{log(∏_{j=1}^{n_{k}} m(σ(j)))}{n_{k}} = lim_{k→∞} \frac{(n_{k} − ∑_{j=1}^{n_{k}} |σ(j)|) log 2}{n_{k}} = lim_{k→∞} \frac{k log 2}{n_{k}} = p log 2.

On the other hand for any n ∈ N there exists k ∈ N such that n_{k} ≤ n < n_{k+1} ≤ n_{k} + n_{1}. Thus

\frac{n_{k}}{n} · \frac{[n_{k}p]}{n_{k}} ≤ \frac{n − ∑_{i=1}^{n} |σ(i)|}{n} ≤ \frac{n_{k}}{n} · \frac{[n_{k}p] + 1}{n_{k}}.

Therefore lim_{n→∞}[log(∏_{j=1}^{n} m(σ(j)))]/n = p log 2 which means

dim_{H}(Γ_{α} ⋂(Γ_{α} + t)) = dim_{B}(Γ_{α} ⋂(Γ_{α} + t)) = −p\frac{log2}{logβ}

by Theorem 3.4 if taking t = ϕ_{1}(σ). In addition letting σ(n) = 1 for all n ≠ n_{k} leads t ∈ Γ_{α} and letting σ(n) = 1 for infinitely many n ≠ n_{k} and σ(n) = −1 for infinitely many n ≠ n_{k} will lead t ∈ (Γ_{α} − Γ_{α})\Γ_{α}.

Open problem. What are the dimensions of Γ_{α} ⋂(Γ_{α} + t) for 1/3 < β < 1/2 ?.

4 The Dimensions of H_{p} and H_{p}^{∗}

We will apply the multifractal decomposition results for Moran sets to Γ_{α} and Γ_{α} − Γ_{α} to get the dimension results of H_{p} and H_{p}^{∗} respectively. Note that, as a special case, the middle-α Cantor set Γ_{α} is a pairwise disjoint, map specified Moran construction and so is the set Γ_{α} − Γ_{α} when 0 < β < 1/3. First let us recall the related results on the multifractal decomposition results of Moran sets applied to the case of middle-α Cantor sets.

In the definition of generalized Moran set in §3, if taking n_{k} ≡ n and t_{j}^{(k)} ≡ t_{j}, the nonempty compact set E determined by (9) is called as Moran set. In [1] the authors gave an important results on the multifractal decompositions of these Moran sets. In the following we will list some results on the multifractal decompositions of the middle-α Cantor set Γ_{α} according to [1]. In this case we prefer to take S = Ω = ∏_{i=1}^{∞}{0, 1} with using index 0, 1 instead of index 1, 2 just for convenience.

Fix a probability vector (p_{0}, p_{1}) with each p_{i} positive, p_{0} ≠ p_{1} and p_{0} + p_{1} = 1. Let μ̂ be the corresponding infinite product measure on S. Let μ be the image measure on Γ_{α} induced by ϕ defined by (7). So μ(E) = μ̂(ϕ^{−1}(E)), for E ⊂ R. Let t_{0} = t_{1} = β. For σ ∈ S write

p(σ|k) = p_{σ(1)}p_{σ(2)} ⋯ p_{σ(k)}; t(σ|k) = t_{σ(1)}t_{σ(2)} ⋯ t_{σ(k)}

For each q ∈ R there is a unique number β(q) such that

p_{0}^{q}β^{β(q)} + p_{1}^{q}β^{β(q)} = 1,

i.e. β(q) = −(log(p_{0}^{q} + p_{1}^{q}))/ log β. Define

α(q) = −β′(q) = \frac{p_{0}^{q} log p_{0} + p_{1}^{q} log p_{1}}{p_{0}^{q} log β + p_{1}^{q} log β}; f(q) = qα(q) + β(q),

Ê_{q} = \{σ ∈ S : lim_{k→∞} \frac{log p(σ|k)}{log t(σ|k)} = α(q)\}; E_{q} = ϕ(Ê_{q}).

Note that α(q) is strictly decreasing because of p_{0} ≠ p_{1} and satisfies

lim_{q→∞} α(q) = min \{\frac{log p_{0}}{log β} , \frac{log p_{1}}{log β} \}; lim_{q→−∞} α(q) = max \{\frac{log p_{0}}{log β} , \frac{log p_{1}}{log β} \} .

According to [1], for middle-α Cantor set Γ_{α} the following multifractal decompositions results hold.

Theorem 4.1 (1) For each q ∈ R, dim_{H} E_{q} = dim_{P} E_{q} = f(q);

(2) x ∈ E_{q} iff the pointwise dimension of μ at x is α(q):

lim_{ε→0} \frac{log μ(B(x, ε))}{log ε} = α(q),

where B(x, ε) denotes the ball with center at x and radius ε.

Let G(λ) = log p_{1}/ log β + (log p_{0}/ log β − log p_{1}/ log β)λ which is strictly monotonic because of p_{0} ≠ p_{1}. Thus we can determine a function q = Q(λ) from (0, 1) onto R by letting G(λ) = α(q).

It follows immediately from Theorem 3.4 and Corollary 3.5

Lemma 4.2 Let 0 < β ≤ 1/3. For 0 < p < 1

H_{p} = \{t ∈ Γ_{α} : lim_{k→∞} \frac{k − ∑_{i=1}^{k} σ(i)}{k} = p, σ = (σ(1), σ(2), ⋯) ∈ ϕ^{−1}(t)\} .

Note that for σ = (σ(1), σ(2), ⋯) ∈ Ω

\frac{log p(σ|k)}{log t(σ|k)} = \frac{log p_{1}}{log β} + (\frac{log p_{0}}{log β} − \frac{log p_{1}}{log β} ) \frac{k − ∑_{i=1}^{k} σ(i)}{k}

Thus by the definitions of E_{q} and Q(λ), Theorem 4.1 and Lemma 4.2 we can get

Theorem 4.3 Let 0 < β ≤ 1/3. We have (1) H_{p} = E_{Q(p)} = {t ∈ Γ_{α} : lim_{ε→0} log μ(B(t, ε))/ log ε = α(Q(p))};

(2) dim_{H} H_{p} = dim_{P} H_{p} = f(Q(p)).

Similar to H_{p} we can study the dimensions of H_{p}^{∗} by means of the multifractal decomposition results on Γ_{α} − Γ_{α}. Note that Γ_{α} − Γ_{α} is an invariant set under F_{i}, i = −1, 0, 1 which satisfy the open set condition if 0 < β ≤ 1/3. To do this we take a probability vector {p_{−1}, p_{0}, p_{1}} with each positive, p_{0} ≠ p_{−1} = p_{1} and p_{0}+2p_{1} = 1. Let μ̂^{∗} be the corresponding infinite product measure on Ω_{1}. Let μ^{∗} be the image measure on Γ_{α}−Γ_{α} induced by ϕ_{1} defined by (8). So μ^{∗}(E) = μ̂^{∗}(ϕ_{1}^{−1}(E)), for E ⊂ R.

Here we take S = Ω_{1} = ∏_{i=1}^{∞}{−1, 0, 1} and t_{−1} = t_{0} = t_{1} = β. For σ ∈ Ω_{1} the p(σ|k) and t(σ|k) are defined in the same method as above.

For each q ∈ R there is a unique number β^{∗}(q) such that

p_{0}^{q}β^{β^{∗}(q)} + 2p_{1}^{q}β^{β^{∗}(q)} = 1,

i.e. β^{∗}(q) = −(log(p_{0}^{q} + 2p_{1}^{q}))/ log β. Define

α^{∗}(q) = −β^{∗}′(q) = \frac{p_{0}^{q} log p_{0} + 2p_{1}^{q} log p_{1}}{(p_{0}^{q} + 2p_{1}^{q}) log β}; f^{∗}(q) = qα^{∗}(q) + β^{∗}(q),

Ê_{q}^{∗} = \{σ ∈ Ω_{1} : lim_{k→∞} \frac{log p(σ|k)}{log t(σ|k)} = α^{∗}(q)\}; E_{q}^{∗} = ϕ_{1}(Ê_{q}^{∗}).

According to [1] for set Γ_{α} −Γ_{α} the following multifractal decompositions results hold.

Theorem 4.4 (1) Let 0 < β ≤ 1/3. For each q ∈ R, dim_{H}(E_{q}^{∗}) = dim_{P} (E_{q}^{∗}) = f^{∗}(q).

(2) Let 0 < β < 1/3. x ∈ E_{q}^{∗} iff the pointwise dimension of μ^{∗} at x is α^{∗}(q):

lim_{ε→0} \frac{log μ^{∗}(B(x, ε))}{log ε} = α^{∗}(q).

Let G^{∗}(λ) = log p_{1}/ log β + (log p_{0}/ log β − log p_{1}/ log β)λ which is strictly monotonic because of p_{0} ≠ p_{1}. Note that when p_{0} ≠ p_{1} the function α^{∗}(q) defined on R is strictly decreasing and satisfies

lim_{q→∞} α^{∗}(q) = min \{\frac{log p_{0}}{log β} , \frac{log p_{1}}{log β} \}; lim_{q→−∞} α^{∗}(q) = max \{\frac{log p_{0}}{log β} , \frac{log p_{1}}{log β} \} .

Thus we can determine a function q = Q^{∗}(λ) from (0, 1) onto R by letting G^{∗}(λ) = α^{∗}(q). Therefore Theorem 3.4 and Corollary 3.5 mean

Lemma 4.5 Let 0 < β ≤ 1/3. For 0 < p < 1

H_{p}^{∗} = \{t ∈ Γ_{α} − Γ_{α} : lim_{k→∞} \frac{k − ∑_{i=1}^{k} |σ(i)|}{k} = p, σ = (σ(1), σ(2), ⋯) = ϕ_{1}^{−1}(t)\} .

Note that for σ = (σ(1), σ(2), ⋯) ∈ Ω_{1}

\frac{log p(σ|k)}{log t(σ|k)} = \frac{log p_{1}}{log β} + (\frac{log p_{0}}{log β} − \frac{log p_{1}}{log β} ) \frac{k − ∑_{i=1}^{k} |σ(i)|}{k}.

Thus by the definitions of E_{q}^{∗} and Q^{∗}(λ), Theorem 4.1 and Lemma 4.5 we can get

Theorem 4.6 (1) Let 0 < β < 1/3. We have

H_{p}^{∗} = E_{Q^{∗}(p)}^{∗} = \{t ∈ Γ_{α} − Γ_{α} : lim_{ε→0} \frac{log μ^{∗}(B(t, ε))}{log ε} = α^{∗}(Q^{∗}(p))\};

(2) Let 0 < β ≤ 1/3. We have dim_{H} H_{p}^{∗} = dim_{P} H_{p}^{∗} = f^{∗}(Q^{∗}(p)).

Acknowledgment

This work is supported by the National Science Foundation of China. The authors would like to thank the referee for helpful comments and suggestions and Dr. Miroslav M. Novak for careful reading and corrections.

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