Intersections of Translation of a Class of Self-Affine Sets

Abstract

Intersections of translation of a class of self-affine sets are investigated by constructing sofic affine-invariant sets which coincide with these sets and then a scheme for computing the Hausdorff dimensions of these intersections is given.

1. Introduction

Fix integers 2≤𝑚≤𝑛. Let 𝑇 be the expanding endomorphism of the 2-torus 𝕋²=ℝ²/ℤ² given by the matrix diag(𝑛,𝑚) and let 𝐷⊆𝐼×𝐽 be the set of digits with 𝐼={0,1,…,𝑛−1}, 𝐽={0,1,…,𝑚−1}. The set 𝐾_𝑇,𝐷 is defined as follows: 𝐾_𝑇,𝐷=^∞_𝑘=1𝑇^−𝑘𝑑_𝑘∶𝑑_𝑘∈𝐷∀𝑘≥1, (1) which can be viewed as the unique invariant set of a family of 𝑁(=#𝐷) affine contractions as follows: 𝐹_𝑗𝑥𝑦=𝑇⁻¹𝑥𝑦+𝑑_𝑗,𝑑_𝑗∈𝐷,𝑗=1,2,…,𝑁. (2) If no confusion arises about 𝑇, 𝐷, we will write 𝐾 instead of 𝐾_𝑇,𝐷. Throughout this paper, we let #𝐷 denote the cardinality of 𝐷. The set 𝐾, called McMullen’s self-affine carpet, was first studied by McMullen [1] and Bedford [2], independently, to determine its Hausdorff and box-counting dimensions. From then on, there has been a fast growth in general interest in the study of McMullen’s self-affine carpet 𝐾. Its packing and Hausdorff measures were explored in [3, 4]. The results of [1, 2] were extended to the compact subsets of the 2-torus corresponding to shifts of finite type or sofic shifts and to the Sierpinski sponges by Kenyon and Peres [5, 6]. Gui and Li [7] studied a class of subset of Sierpinski carpets for which the allowed digits in the expansions fall into each fiber set with a prescribed frequency. Hua [8] investigated a class of self-affine sets with overlaps and showed that they can be viewed as sofic affine invariant sets without overlaps.

Intersection of Cantor sets has been the subject of several studies [9–19], the context and motivation being numerous. Davis and Hu [12] explored the Hausdorff dimension of the intersection of two middle third Cantor sets and showed that the Hausdorff dimension can take any value from 0 to ln2/ln3. Williams [10] produced examples to show that the intersection of two Cantor sets can vary from being one point to containing another Cantor set. Li and Xiao [13] showed that intersection resulting from two middle-𝛼 sets being translated across one another is a generalized Moran set if the contraction lies in [0,1/3], whose Hausdorff, packing, upper Box dimension then can be obtained by the results of the Moran set. Nekka and Li [14, 15] studied the properties of intersection of Cantor sets with their translations. Dai and Tian [16, 17] got the related properties of a class of self-similar sets in the plane.

In this paper, we develop a new algorithm to explore the structure of 𝐾∩(𝐾+𝑡) for 𝑡∈𝐾−𝐾, which extends our previous results in [18] and is different from the listed papers. It is easy to see that 𝐾∩(𝐾+𝑡)≠∅ if and only if 𝑡∈𝐾−𝐾; so throughout this paper, we always suppose that 𝑡∈𝐾−𝐾. The intersection, generally, has a complicated structure which leads the set quite different from McMullen’s self-affine carpet. The main difficulty in the study of the structure of the intersection comes from the fact that the number of affine contractions used in the construction may vary from step to step. However, by defining a suitable equivalence relation on the basic rectangles of 𝐾∩(𝐾+𝑡) and partitioning them into equivalence classes, a sofic affine-invariant set coinciding with 𝐾∩(𝐾+𝑡) then can be constructed, if the number of equivalence classes is finite. Our idea of the construction of the sofic affine-invariant set is enlightened by [8, 20–22].

To state our main result, we need to recall some definitions and results about the sofic affine-invariant set. For details, one can refer to [6, 9].

Let 𝐺=(𝒱,ℰ) be a finite directed graph in which loops and multiple edges are allowed. Let 𝐷₀={0,1,…,𝑛−1}×{0,1,…,𝑚−1} be the set of symbols. Suppose that the edges of 𝐺 are labeled in symbols in 𝐷₀ in a right resolving fashion: no two edges emanating from the same vertex have the same symbol. Then the symbol sequences which arise from infinite paths in 𝐺 form a sofic system on #𝐷₀ symbols. That is, Θ=𝑑_𝑘𝑘≥1∶𝑑_𝑘𝑘≥1isaninfinitepathin𝐺,where𝑑_𝑘∈𝐷₀. (3) Suppose that Θ⊂𝐷^ℕ₀ is the resulting sofic system. We call Γ_𝑇(Θ)=^∞_𝑘=1𝑇^−𝑘𝑑_𝑘∶𝑑_𝑘𝑘≥1∈Θ (4) a T-invariant sofic set. An adjacency matrix 𝐴, according to 𝐺=(𝒱,ℰ), is constructed, where, for two vertices 𝑣₁, 𝑣₂ in 𝒱, 𝐴(𝑣₁,𝑣₂) is the number of edges in ℰ from 𝑣₁ to 𝑣₂.

Theorem 1. Suppose that 𝑡=(𝑝₁/𝑞₁,𝑝₂/𝑞₂)∈ℚ². Then 𝐾∩(𝐾+𝑡)=Γ_𝑇(Θ). (5)

To compute the Hausdorff dimension of 𝐾∩(𝐾+𝑡), the following result in [6] is needed.

Kenyon-Peres’s Theorem (see [6, Theorem  3.2]). Let 𝐺=(𝒱,ℰ) be a finite directed graph with edges labeled by 𝐷₀ and adjacency matrix 𝐴. Let Θ⊂𝐷^ℕ₀ be the resulting sofic system. Then the Hausdorff dimension of Γ_𝑇(Θ) is given bydim_𝐻Γ_𝑇(Θ)=lim_𝑘→∞log_{𝑚0≤𝑖1,…,𝑖𝑘≤𝑚−1}𝐴_𝑖𝑘⋅𝐴_𝑖𝑘−1⋅…⋅𝐴_𝑖1^𝛼, (6)where, for 0≤𝑗<𝑚 and vertices 𝑣₁, 𝑣₂ in 𝐺, 𝐴_𝑗(𝑣₁,𝑣₂) denotes the number of edges from 𝑣₁ to 𝑣₂ in 𝐺 such that the second coordinate of their label is 𝑗, 𝛼=log𝑚/log𝑛.

This paper is arranged as follows. A description for equivalence classes on 𝐾∩(𝐾+𝑡) is arranged in Section 2. A sofic affine-invariant set coinciding with 𝐾∩(𝐾+𝑡) is then constructed in Section 3, if the number of equivalence classes is finite (see Theorem 6). A sufficient condition for the number of equivalence classes to be finite is also established in Section 3 (see Proposition 7). Finally, the proof of the Theorem 1 is completed in this section. Some examples are included in Section 4.

2. Equivalence Classes on 𝐾∩(𝐾+𝑡)

In this section we consider the geometrical structure of 𝐾∩(𝐾+𝑡)(≠∅). First we define a suitable equivalence relation on the basic rectangles of 𝐾∩(𝐾+𝑡) and partition them into equivalence classes.

Let {𝐹_𝑗}^𝑁_𝑗=1 be defined as in (2); simple computation yields 𝐾+𝑡=^𝑁_𝑗=1𝑇⁻¹(𝐾+𝑡)+𝑇⁻¹𝑑_𝑗+𝑡−𝑇⁻¹𝑡. (7) Then define 𝑆_𝑗𝑥𝑦=𝑇⁻¹𝑥𝑦+𝑑_𝑗+𝑇𝑡−𝑡,𝑗=1,…,𝑁, (8) such that 𝐾+𝑡 can be viewed as the unique invariant nonempty compact set under mappings 𝑆_𝑗,𝑗=1,…,𝑁. Let 𝑂=𝑡+(0,1)², {𝑆_𝑗}^𝑁_𝑗=1, obviously, satisfy the OSC. Set 𝐷^∗=𝑑_𝑗+𝑇𝑡−𝑡,𝑑_𝑗∈𝐷. (9)

In order to analyze the structure of 𝐾∩(𝐾+𝑡), we get the following standard notations.

Let Σ={1,…,𝑁} and Σ^∗=∪_𝑝≥0Σ^𝑝 be the set of all finite words, where Σ^𝑝 is the set of all words of length 𝑝, with Σ⁰ containing only the empty word ∅. Σ^ℕ={𝑖₁𝑖₂⋯∶𝑖_𝑛∈Σforall1≤𝑛<∞}. For 𝐽∈Σ^𝑝, let |𝐽|=𝑝 denote the length of 𝐽. For 𝐼∈Σ^𝑝 and 𝐽∈Σ^𝑞, let 𝐼𝐽 be the concatenation of 𝐼 and 𝐽. For 𝐼=𝑖₁𝑖₂⋯∈Σ^ℕ and a positive integer 𝑘, let 𝐼∣𝑘=𝑖₁⋯𝑖_𝑘 denote the truncation of 𝐼 to the 𝑘th place.

We define the iterated mappings using these notations. Let 𝐽=(𝑗₁,…,𝑗_𝑘)∈Σ^𝑘. Then 𝐹_𝐽=𝐹_𝑗1∘𝐹_𝑗2∘⋯∘𝐹_𝑗𝑘,𝑆_𝐽=𝑆_𝑗1∘𝑆_𝑗2∘⋯∘𝑆_𝑗𝑘 (10) with 𝐹_∅=𝑆_∅=𝐼, the identity map on ℝ². Thus we get 𝐾=^∞_{𝑘=1𝐽∈Σ𝑘}𝐹_𝐽(Ω),𝐾+𝑡=^∞_{𝑘=1𝐽∈Σ𝑘}𝑆_𝐽Ω^𝑡, (11) where Ω=[0,1]² and Ω^𝑡=𝑡+[0,1]². We call 𝐹_𝐽(Ω), 𝑆_𝐽(Ω^𝑡) the basic rectangles of rank 𝑘, respectively. For simplicity without confusion, we further call them basic 𝑘-rectangles and we write 𝐹_𝐽(Ω)=Ω_𝐽 and 𝑆_𝐽(Ω^𝑡)=Ω^𝑡_𝐽 in this paper.

Suppose that there exist 𝐼∈Σ^𝑘, 𝐽∈Σ^𝑘+1, and ℓ∈Σ¹ such that 𝜏₁=Ω_𝐼,𝜏₂=Ω_𝐽,𝐽=𝐼ℓ. (12) Then we connect a directed edge ℓ from 𝜏₁ to 𝜏₂ and denote it by 𝜏→−₁^ℓ𝜏₂. We call 𝜏₁ the parent of 𝜏₂ and 𝜏₂ an offspring of 𝜏₁. Definitions of parent and offspring can be parallel extended to 𝐾+𝑡.

Following we investigate the relationships between the basic 𝑘-rectangles of 𝐾 and 𝐾+𝑡.

Suppose that Ω_𝐽, 𝐽∈Σ^𝑘, 𝑘≥0; the neighborhood of Ω_𝐽 with respect to the basic 𝑘-rectangles of 𝐾+𝑡 is defined as 𝑁Ω_𝐽=Ω^𝑡_𝐼∶𝐼∈Σ^𝑘,Ω_𝐽∩Ω^𝑡_𝐼≠∅. (13) Let Λ_𝑘={𝐽∈Σ^𝑘∶𝑁(Ω_𝐽)≠∅,𝑘≥0}. Note that, for some Ω_𝐽, 𝐽∈Σ^𝑘, 𝑁(Ω_𝐽) may be empty; namely, these basic 𝑘-rectangles do not meet any basic 𝑘-rectangles of 𝐾+𝑡, and then they have nothing to do with the intersection 𝐾∩(𝐾+𝑡), which will be deleted. So let ℳ_𝑘=Ω_𝐽,𝐽∈Λ_𝑘. (14)

The basic 𝑘-rectangles of 𝐾 and 𝐾+𝑡 are completely determined by their left-lower corner points. For this point, if 𝜏∈ℳ_𝑘 we define 𝑅_𝑘(𝜏)=𝑇^𝑘𝜏(𝜏,𝑅𝛿−∶𝛿∈𝑁)𝑅=_𝑘(𝜏)∶𝜏∈ℳ_𝑘,𝑘≥0, (15)(16) where 𝛿, 𝜏 are the left-lower corner points of 𝛿, 𝜏, respectively. For 𝜏₁∈ℳ_𝑘1 and 𝜏₂∈ℳ_𝑘2, we say that 𝜏₁ and 𝜏₂ are equivalent, if 𝑅_𝑘1(𝜏₁)=𝑅_𝑘2(𝜏₂). Next we prove a recurrent relationship on 𝑅.

Let 𝜏₁=Ω_𝐼 and 𝜏₂=Ω_𝐽, where 𝐼∈Σ^𝑘1, 𝐽∈Σ^𝑘2. Suppose that 𝑅_𝑘1(𝜏₁)=𝑅_𝑘2(𝜏₂) and set 𝑁𝜏₁=𝑣₁,…,𝑣_𝑡,𝑁𝜏₂=𝑢₁,…,𝑢_𝑡, (17) such that 𝑇^𝑘1(𝑣_𝑖̂−𝜏₁)=𝑇^𝑘2(𝑢_𝑖−𝜏₂), 1≤𝑖≤𝑡. Let 𝜎¹_𝐣, 𝜎²_𝐣 be the offsprings of 𝜏₁ and 𝜏₂ connected by edge 𝐣∈Σ¹, respectively, and denote them by 𝜏→−₁^𝐣𝜎¹_𝐣, 𝜏→−₂^𝐣𝜎²_𝐣. Similarly, 𝑣^𝑖_𝐤, 𝑢^𝑖_𝐤 are the offsprings of 𝑣_𝑖 and 𝑢_𝑖 connected by edge 𝐤∈Σ¹, respectively. That is 𝑣→−_𝑖^𝐤𝑣^𝑖_𝐤, 𝑢→−_𝑖^𝐤𝑢^𝑖_𝐤.

Lemma 2. With above notations, let 𝑔∶ℝ²→ℝ² be an affine mapping, 𝑔(𝑥)=𝑇^{𝑘1−𝑘2}𝑥+𝜏₂−𝑇^{𝑘1−𝑘2}𝜏₁, (18) where 𝜏₁, 𝜏₂ are the left-lower corner points of 𝜏₁ and 𝜏₂, respectively. Then 𝜎²_𝐣=𝑔(𝜎¹_𝐣). If 𝑅_𝑘1(𝜏₁)=𝑅_𝑘2(𝜏₂), then 𝑢_𝑖=𝑔(𝑣_𝑖), 𝑖=1,…,𝑡. Furthermore 𝑢^𝑖_𝐤=𝑔(𝑣^𝑖_𝐤).

Proof. Note that 𝜎¹_𝐣=𝑇^−𝑘1−1(Ω)+𝜏₁+𝑑_𝐣𝑇^−𝑘1−1,𝜎²_𝐣=𝑇^−𝑘2−1(Ω)+𝜏₂+𝑑_𝐣𝑇^−𝑘2−1, (19) where 𝑑_𝐣∈𝐷. Then 𝑔𝜎¹_𝐣=𝑇^−𝑘2−1(Ω)+𝑇^{𝑘1−𝑘2}𝜏₁+𝑑_𝐣𝑇^−𝑘2−1+𝜏₂−𝑇^{𝑘1−𝑘2}𝜏₁=𝜎²_𝐣. (20) We now prove that 𝑢_𝑖=𝑔(𝑣_𝑖), 𝑖=1,…,𝑡. 𝜏₂−𝑇^{𝑘1−𝑘2}𝜏₁=𝑢_𝑖−𝑇^{𝑘1−𝑘2}𝑣_𝑖̂ is the direct conclusion from 𝑅_𝑘1(𝜏₁)=𝑅_𝑘2(𝜏₂), which yields 𝑢_𝑖=𝑔(𝑣_𝑖), 𝑖=1,…,𝑚. 𝑢^𝑖_𝐤=𝑔(𝑣^𝑖_𝐤) can be easily obtained by the same argument as 𝜎²_𝐣=𝑔(𝜎¹_𝐣).

Proposition 3. With above notations, if 𝑅_𝑘1(𝜏₁)=𝑅_𝑘2(𝜏₂), then, for 𝐣∈Σ¹, 𝜎¹_𝐣∈ℳ_𝑘1+1 if and only if 𝜎²_𝐣∈ℳ_𝑘2+1. If 𝜎¹_𝐣∈ℳ_𝑘1+1, then 𝑅_𝑘1+1(𝜎¹_𝐣)=𝑅_𝑘2+1(𝜎²_𝐣).

Proof. By Lemma 2, we have 𝑣^𝑖_𝐤∩𝜎¹_𝐣≠∅⟺𝑔𝑣^𝑖_𝐤∩𝑔𝜎¹_𝐣=𝑢^𝑖_𝐤∩𝜎²_𝐣≠∅, (21) so 𝜎¹_𝐣∈ℳ_𝑘1+1⇔𝜎²_𝐣∈ℳ_𝑘2+1.
One can check that 𝑁𝜎¹_𝐣=𝑣^𝑖_𝐤∶𝑣_𝑖∈𝑁𝜏₁,𝑣^𝑖_𝐤∩𝜎¹_𝐣,𝑁𝜎≠∅_2𝐣=𝑢^𝑖_𝐤∶𝑢_𝑖∈𝑁𝜏₂,𝑢^𝑖_𝐤∩𝜎²_𝐣≠∅. (22) By Lemma 2 and (21), (22) can be rewritten as 𝑁𝜎²_𝐣=𝑔𝑣^𝑖_𝐤∶𝑢_𝑖∈𝑁𝜏₂,𝑢^𝑖_𝐤∩𝜎²_𝐣=𝑔𝑣≠∅_𝑖𝐤∶𝑣_𝑖∈𝑁𝜏₁,𝑣^𝑖_𝐤∩𝜎¹_𝐣=𝑔𝑣≠∅_𝑖𝐤∶𝑣^𝑖_𝐤∈𝑁𝜎¹_𝐣. (23) Observe that 𝜎¹_𝐣=𝜏₁+𝑑_𝐣𝑇^−𝑘1−1,𝜎²_𝐣=𝜏₂+𝑑_𝐣𝑇^−𝑘2−1,𝑣^𝑖_𝐤̂=𝑣_𝑖̂+𝑐_𝐤𝑇^−𝑘1−1+𝑇^−𝑘1𝑡𝑇⁻¹,𝑢−1_𝑖𝐤=𝑢_𝑖+𝑐_𝐤𝑇^−𝑘2−1+𝑇^−𝑘2𝑡𝑇⁻¹−1, (24) where 𝑐_𝐤∈𝐷^∗ (defined as in (9)). By condition 𝑅_𝑘1(𝜏₁)=𝑅_𝑘2(𝜏₂), we obtain 𝑇^𝑘2+1𝑢^𝑖_𝐤−𝜎²_𝐣=𝑇^𝑘1+1𝑣^𝑖_𝐤̂−𝜎¹_𝐣. (25)

Let 𝑅 be defined as in (16); if 𝑅 is a finite set, we then call that the types of the basic rectangles of 𝐾∩(𝐾+𝑡) is finite.

We remark that, for 𝑡∈𝐾−𝐾, if the types of the basic rectangles of 𝐾∩(𝐾+𝑡) is recursive, such as type 𝐴 produces types 𝐵, and 𝐶, 𝐵 produces 𝐷, 𝐶 produces 𝐵 and finally 𝐷 produces 𝐴, then by Proposition 3, one can claim that the types of the basic rectangles of 𝐾∩(𝐾+𝑡) are finite.

From (11), and the definition of Λ_𝑘, we have 𝐾∩(𝐾+𝑡)=^∞_{𝑘=1𝐽∈Σ𝑘}Ω_𝐽^∞_{𝑘=1𝐼∈Σ𝑘}Ω^𝑡_𝐼=^∞_𝑘=1Π_𝑘, (26) where Π_𝑘=(⋃_{𝐽∈Λ𝑘}Ω_𝐽)⋂(⋃_{𝐼∈Σ𝑘}Ω^𝑡_𝐼).

Proposition 4. 𝐾∩(𝐾+𝑡)=^∞_{𝑘=1𝐽∈Λ𝑘}Ω_𝐽. (27)

Proof. It is easy to see that Π_𝑘⊂⋃_{𝐽∈Λ𝑘}Ω_𝐽; therefore 𝐾∩(𝐾+𝑡)⊆^∞_{𝑘=1𝐽∈Λ𝑘}Ω_𝐽. (28) To prove the reverse inclusion, we let 𝜀/2>0; then for 𝑘₁, 𝑘₂ large enough, such that |Ω_𝐽|≤𝜀/2 for 𝐽∈Λ_𝑘1, and Π_𝑘2⊆[𝐾∩(𝐾+𝑡)]_𝜀/2 (|𝐴| denotes the diameter of 𝐴 and [𝐴]_𝜀 the open 𝜀-neighborhood of 𝐴). Let 𝑘=max{𝑘₁,𝑘₂}, then max{|Ω_𝐽′|,𝐽^′∈Λ_𝑘}≤𝜀/2 (𝐽^′ always exists; otherwise 𝐾∩(𝐾+𝑡)=∅). On the other hand Ω_𝐽′∩Π_𝑘≠∅, then Ω_𝐽′⊆[𝐾∩(𝐾+𝑡)]_𝜀. Hence ^∞_{𝑘=1𝐽′∈Λ𝑘}Ω_𝐽′⊆𝐾(𝐾+𝑡)_𝜀. (29) The desired result is then obtained by letting 𝜀 tend to 0.

For some 𝐽∈Λ_𝑘, it is possible that, for all 𝑖∈Σ¹, 𝐽𝑖∉Λ_𝑘+1; namely, all offsprings of Ω_𝐽 do not belong to ℳ_𝑘+1. So set Λ=𝐽=𝑗₁,𝑗₂,…,𝑗_𝑘∶𝑗,…forany𝑘≥0,𝐽∣𝑘=₁,𝑗₂,…,𝑗_𝑘∈Λ_𝑘. (30)

Remark 5. We remark that, by Proposition 4, 𝐾∩(𝐾+𝑡) can also be expressed in radix expansions as follows: 𝐾∩(𝐾+𝑡)=lim^𝑘_{𝑘→∞𝑖=1}𝑇^−𝑖𝑑_𝑙𝑖, (31) where 𝑑_𝑙𝑖∈𝐷, 𝐼=(𝑙₁,…,𝑙_𝑘,…)∈Λ.

3. Sofic Affine-Invariant Set

Suppose that 𝑅 is a finite set; let 𝒱=𝑅={𝑅_𝑘(𝜏)∶𝜏∈ℳ_𝑘𝑘≥0}. Set 𝒱={𝜔,𝜔₁,…,𝜔_𝑄} and 𝑅₀(Ω)=𝜔. We now define the directed edges ℰ={𝐸_𝑖,𝑗}^𝑄_𝑖,𝑗=1 on 𝒱. Fix 𝑖 (1≤𝑖≤𝑄) and a vertex 𝜏=Ω_𝐼∈ℳ_𝑘 such that 𝑅_𝑘(𝜏)=𝜔_𝑖. Let 𝛿_ℓ=Ω_𝐼ℓ in ℳ_𝑘+1 be an offspring of 𝜏 connected by edge ℓ∈Σ¹. Consider 𝐸_𝑖,𝑗=𝑑_ℓ∶𝜏⟶^ℓ𝛿_ℓ,𝑅_𝑘+1𝛿_ℓ=𝜔_𝑗,𝑑_ℓ∈𝐷. (32) Note that if none of the offsprings of 𝜏 belongs to ℳ_𝑘+1, then 𝐸_𝑖,𝑗=∅, for all 𝑗. We discard those 𝜔_𝑖’s that there is no path going out and discard all the paths going to such discarded vertices. Hence, without loss of generality, we assume, that for each 𝑖, ^𝑄_𝑗=1#𝐸_𝑖,𝑗≥1. (33) A finite directed graph 𝐺=(𝒱,ℰ) then is obtained. Let Θ=𝑑_𝑘𝑘≥1∶𝑑_𝑘𝑘≥1isaninfinitepathin𝐺,where𝑑_𝑘∈𝐷. (34) Then Θ⊂𝐷^ℓ is the resulting sofic system and Γ_𝑇(Θ)=^∞_𝑘=1𝑇^−𝑘𝑑_𝑘∶𝑑_𝑘𝑘≥1∈Θ (35) is a sofic invariant set.

Theorem 6. If 𝑅 is a finite set, then 𝐾∩(𝐾+𝑡)=Γ_𝑇(Θ). (36)

Proof. By Remark 5, 𝐾∩(𝐾+𝑡) can be expressed as 𝐾∩(𝐾+𝑡)=lim^𝑘_{𝑘→∞𝑖=1}𝑑_𝑙𝑖𝑇^−𝑖, (37) where 𝑑_𝑙𝑖∈𝐷, 𝐼=(𝑙₁,…,𝑙_𝑘,…)∈Λ. By the definition of Λ in (30) and the construction of 𝐺=(𝒱,ℰ), one can claim that 𝑑_𝑙1⋯𝑑_𝑙𝑘⋯ is an infinite path in 𝐺 and then belongs to Θ. Thus 𝐾∩(𝐾+𝑡)⊆Γ_𝑇(Θ). The inverse inclusion is left for the readers.

Following we give some sufficient condition for the set 𝑅 (defined as in (16)) to be finite.

Proposition 7. Suppose that 𝑡=(𝑝₁/𝑞₁,𝑝₂/𝑞₂)∈ℚ². Then the set 𝑅 is finite.

Proof. Suppose that 𝜏∈𝑀_𝑘, 𝛿∈𝑁(𝜏). 𝜏, 𝛿 denote the left-lower corner points of 𝜏, 𝛿 respectively; then 𝜏=^𝑘_𝑖=1𝑑_ℓ𝑖𝑇^−𝑖, (38) where 𝑑_ℓ𝑖∈𝐷; 𝛿=^𝑘_𝑖=1𝑐_ℓ𝑖𝑇^−𝑖+𝑇^−𝑘𝑡, (39) where 𝑐_ℓ𝑖∈𝐷^∗. Consider 𝑇^𝑘𝛿−𝜏=𝑇^𝑘𝑘_𝑖=1𝑐_ℓ𝑖−𝑑_ℓ𝑖𝑇^−𝑖+𝑇^−𝑘𝑡=1𝑞^𝑘_𝑖=1𝑒_ℓ𝑖+𝑝𝑇−𝑝𝑇^𝑘−𝑖+𝑝, (40) where 𝑝=(𝑞𝑝₁/𝑞₁,𝑞𝑝₂/𝑞₂); 𝑞 is the least common multiple of 𝑞₁ and 𝑞₂. 𝑒_ℓ𝑖∈𝑞(𝐷−𝐷), 𝑞𝐷∶={𝑞𝑑_𝑗,𝑑_𝑗∈𝐷}. So the coordinates of 𝑞𝑇^𝑘(𝛿−𝜏) are integers. On the other hand 𝜏∩𝛿≠∅ implies that 𝑇^𝑘𝛿−𝜏∈[−𝑛,𝑛]×[−𝑚,𝑚]. (41) Consequently, 𝑞𝑇^𝑘𝛿−𝜏∈−𝑞𝑛,𝑞𝑛×−𝑞𝑚,𝑞𝑚. (42) Then, for any 𝜏∈𝑀_𝑘, 𝑅_𝑘(𝜏)⊂(([−𝑞𝑛,𝑞𝑛]×[−𝑞𝑚,𝑞𝑚])∩ℤ²)/𝑞. So the set 𝑅 is finite.

Proof of Theorem 1. It is a direct conclusion from Theorem 6 and Proposition 7.

4. Examples

Example 1. Let 𝑇=⁴⁰₀₃ and 𝐷={𝑑₁,𝑑₂,𝑑₃,𝑑₄}, where 𝑑₁=⁰₀, 𝑑₂=³₀, 𝑑₃=¹₁, and 𝑑₄=³₂.𝑡=(1/3,1/4).

By Proposition 7, the set 𝑅 is finite. The types of the basic rectangles can be determined by (15). It is clear that 𝑅₀(Ω)=𝑡, denoted by 𝐴. For 𝑘=1, among four offsprings of Ω, only Ω₃ and Ω₄ have nonempty neighborhood, which are connected by edges (1,1) and (3,2) from their parent Ω, respectively. By (15), they are of different types; denote them symbolically as 𝐴⟶^(1,1)𝐵,𝐴⟶^(3,2)𝐶. (43) For 𝑘=2, direct computation yields Ω₃₂,Ω₄₁,Ω₄₃∈ℳ₂. (44) They are of new types and Ω₃₂, Ω₄₃ are of the same type. So we have 𝐵⟶^(3,0)𝐷,𝐶⟶^(1,1)𝐷,𝐶⟶^(0,0)𝐸. (45) Above computation process can also be seen from Figure 1. Upon the third iteration, none of the offsprings of Ω₄₁ belongs to ℳ₃; then type 𝐸 and all paths going to 𝐸 are discarded.

Figure 1 

Construction of 𝐾∩𝐾+𝑡. Red and blue rectangles denote the construction of 𝐾 and 𝐾+𝑡, respectively.

By the same argument as above, we have 𝐷⟶^(0,0)𝐹,𝐷⟶^(1,1)𝐶. (46) However none of the offsprings of Ω₃₂₁, whose type is of 𝐹, belongs to ℳ₄. Then type 𝐹 and all paths going to 𝐹 are deleted. By Proposition 3, the above process exhausts all possible types and yields the adjacency matrix 𝑆=0110000100010010, (47) and by Kenyon-Peres’s Theorem, we have 𝑆₀=,𝑆0000000100000000₁=,𝑆0100000000010010₂=.0010000000000000 (48)

Example 2. Let 𝑇=⁵⁰₀₄ and 𝐷={𝑑₁,𝑑₂,𝑑₃,𝑑₄}, where 𝑑₁=⁰₀, 𝑑₂=⁴₀, 𝑑₃=²₁, and 𝑑₄=³₂. 𝑡=(12/25,5/16).

Again by Proposition 7, the set 𝑅 is finite. First, 𝑅₀(Ω)=𝑡; denote it by 𝐴. For 𝑘=1, Ω₃,Ω₄∈ℳ₁, and 𝑁(Ω₃)={Ω^𝑡₁}, 𝑁(Ω₄)={Ω^𝑡₁}. They are of different types and then denoted by 𝐴⟶^(2,1)𝐵,𝐴⟶^(3,2)𝐶. (49) For 𝑘=2, Ω₃₃,Ω₃₄,Ω₄₁∈ℳ₂, and 𝑁(Ω₃₃)={Ω^𝑡₁₁}, 𝑁(Ω₃₄)={Ω^𝑡₁₁,Ω^𝑡₁₃}, 𝑁(Ω₄₁)={Ω^𝑡₁₄}. Their types are of 𝐷, 𝐸, and 𝐹, respectively. Thus we have 𝐵⟶^(2,1)𝐷,𝐵⟶^(3,2)𝐸,𝐶⟶^(0,0)𝐹. (50) Upon the third iteration, Ω₃₃ produces four offsprings in ℳ₃, which yields 𝐷⟶^(0,0)𝐷,𝐷⟶^(2,1)𝐺,𝐷⟶^(3,2)𝐻,𝐷⟶^(4,0)𝐷. (51) Only one offspring of Ω₃₄, whose type is of 𝐸, belongs to ℳ₃; then we get 𝐸⟶^(4,0)𝐼. (52) None of the offsprings of Ω₄₁, whose type is 𝐹, belongs to ℳ₃; then type 𝐹 and all paths going to 𝐹 are discarded. For 𝑘=4, by the same argument as above, we have 𝐺⟶^(0,0)𝐷,𝐺⟶^(2,1)𝐺,𝐺⟶^(3,2)𝐻,𝐺^(4,0)𝐻⟶𝐷,_(0,0)⟶𝐷,𝐻⟶^(2,1)𝐷,𝐻⟶^(3,2)𝐻,𝐻^(4,0)𝐼⟶𝐷,_(4,0)⟶𝐼. (53) Figure 2 can avail to understand the above computation process. By Proposition 3, these are all the types of the intersection and the adjacency matrix is 𝑆=0100000001100000201100000001002011000201100000001. (54)

Figure 2 

Construction of 𝐾∩𝐾+𝑡. Red and blue rectangles denote the construction of 𝐾 and 𝐾+𝑡, respectively.

Also by Kenyon-Peres’s Theorem, we get 𝑆₀=,𝑆0000000000000000200000000001002000000200000000001₁=,𝑆0100000001000000001000000000000010000001000000000₂=,𝑆0000000000100000000100000000000001000000100000000₃=.0000000000000000000000000000000000000000000000000 (55)

Acknowledgments

The authors would like to thank the anonymous referees for their helpful comments and suggestions that led to a significant improvement of the paper. This work was supported by the National Natural Science Foundation of China (nos. 61003178, 11201312, 61070087, 61373087, 11071150, and 61272252) and by the Municipal Science and Technology Plan of Shenzhen in China (JC201105170615A, JC201005280508A).