Intersections of Translation of a Class of Self-Affine Sets

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.


Introduction
Fix integers 2 ≤  ≤ .Let  be the expanding endomorphism of the 2-torus T 2 = R 2 /Z 2 given by the matrix diag(, ) and let  ⊆  ×  be the set of digits with  = {0, 1, . . .,  − 1},  = {0, 1, . . .,  − 1}.The set  , is defined as follows: which can be viewed as the unique invariant set of a family of (= #) affine contractions as follows: If no confusion arises about , , we will write  instead of  , .Throughout this paper, we let # denote the cardinality of .The set , 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 .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 selfaffine 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][10][11][12][13][14][15][16][17][18][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 ln 2/ ln 3. 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- 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  ∩ ( + ) for  ∈  − , which extends our previous results in [18] and is different from the listed papers.It is easy to see that  ∩ ( + ) ̸ = 0 if and only if  ∈  − ; so throughout this paper, we always suppose that  ∈  − .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  ∩ ( + ) and partitioning them into equivalence classes, a sofic affine-invariant set coinciding with  ∩ ( + ) 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][21][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  = (V, E) be a finite directed graph in which loops and multiple edges are allowed.Let  0 = {0, 1, . . .,  − 1} × {0, 1, . . .,  − 1} be the set of symbols.Suppose that the edges of  are labeled in symbols in  0 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  form a sofic system on # 0 symbols.That is, is an infinite path in , where   ∈  0 } .
(3) Suppose that Θ ⊂  N 0 is the resulting sofic system.We call a T-invariant sofic set.An adjacency matrix , according to  = (V, E), is constructed, where, for two vertices To compute the Hausdorff dimension of  ∩ ( + ), the following result in [6] is needed.
Kenyon-Peres's Theorem (see [6,Theorem 3.2]).Let  = (V, E) be a finite directed graph with edges labeled by  0 and adjacency matrix .Let Θ ⊂  N 0 be the resulting sofic system.Then the Hausdorff dimension of Γ  (Θ) is given by where, for 0 ≤  <  and vertices This paper is arranged as follows.A description for equivalence classes on  ∩ ( + ) is arranged in Section 2. A sofic affine-invariant set coinciding with  ∩ ( + ) 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.

Equivalence Classes on 𝐾 ∩ (𝐾 + 𝑡)
In this section we consider the geometrical structure of  ∩ ( + )( ̸ = 0).First we define a suitable equivalence relation on the basic rectangles of  ∩ ( + ) and partition them into equivalence classes.
Following we investigate the relationships between the basic -rectangles of  and  + .
Suppose that Ω  ,  ∈ Σ  ,  ≥ 0; the neighborhood of Ω  with respect to the basic -rectangles of  +  is defined as Let Λ  = { ∈ Σ  : (Ω  ) ̸ = 0,  ≥ 0}.Note that, for some Ω  ,  ∈ Σ  , (Ω  ) may be empty; namely, these basic -rectangles do not meet any basic -rectangles of +, and then they have nothing to do with the intersection  ∩ ( + ), which will be deleted.So let The basic -rectangles of  and  +  are completely determined by their left-lower corner points.For this point, if  ∈ M  we define where δ, τ are the left-lower corner points of , , respectively.

Proposition 3. With above notations, if
Proof.By Lemma 2, we have so By Lemma 2 and ( 21), ( 22) can be rewritten as Observe that where  k ∈  * (defined as in ( 9)).By condition   1 ( 1 ) =   2 ( 2 ), we obtain Let  be defined as in (16); if  is a finite set, we then call that the types of the basic rectangles of  ∩ ( + ) is finite.
We remark that, for  ∈  − , if the types of the basic rectangles of  ∩ ( + ) is recursive, such as type  produces types , and ,  produces ,  produces  and finally  produces , then by Proposition 3, one can claim that the types of the basic rectangles of  ∩ ( + ) are finite.
From (11), and the definition of Λ  , we have where Proposition 4.

Sofic Affine-Invariant Set
Suppose that  is a finite set; let V =  = {  () :  ∈ M   ≥ 0}.Set V = {,  1 , . . .,   } and  0 (Ω) = .We now define the directed edges Note that if none of the offsprings of  belongs to M +1 , then  , = 0, for all .We discard those   '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 , A finite directed graph  = (V, E) then is obtained.Let is an infinite path in , where   ∈ } . (34) Then Θ ⊂  ℓ is the resulting sofic system and is a sofic invariant set.Theorem 6.If  is a finite set, then Proof.By Remark 5,  ∩ ( + ) can be expressed as where    ∈ ,  = ( 1 , . . .,   , . ..) ∈ Λ.By the definition of Λ in (30) and the construction of  = (V, E), one can claim that   1 ⋅ ⋅ ⋅    ⋅ ⋅ ⋅ is an infinite path in  and then belongs to Θ. Thus  ∩ ( + ) ⊆ Γ  (Θ).The inverse inclusion is left for the readers.
Following we give some sufficient condition for the set  (defined as in ( 16)) to be finite.
By Proposition 7, the set  is finite.The types of the basic rectangles can be determined by (15).It is clear that  0 (Ω) = , denoted by .For  = 1, among four offsprings of Ω, only Ω 3 and Ω 4 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 → .
For  = 2, direct computation yields They are of new types and Ω 32 , Ω 43 are of the same type.So we have → , Above computation process can also be seen from Figure 1.Upon the third iteration, none of the offsprings of Ω 41 belongs to M 3 ; then type  and all paths going to  are discarded.By the same argument as above, we have → .
However none of the offsprings of Ω 321 , whose type is of , belongs to M 4 .Then type  and all paths going to  are deleted.By Proposition 3, the above process exhausts all possible types and yields the adjacency matrix

2 j
j be the offsprings of  1 and  2 connected by edge j ∈ Σ 1 , respectively, and denote them by  1 .Similarly, V  k ,   k are the offsprings of V  and   connected by edge k ∈ Σ 1 , respectively.

Figure 1 :For
Figure 1: Construction of  ∩  + .Red and blue rectangles denote the construction of  and  + , respectively.