Journal of Applied Mathematics

Volume 2013 (2013), Article ID 953082, 7 pages

http://dx.doi.org/10.1155/2013/953082

Research Article

Intersections of Translation of a Class of Self-Affine Sets

Jian Lu, Yuru Zou, and Lijing Wang

College of Mathematics and Computational Science, Shenzhen University, Shenzhen 518060, China

Received 20 September 2013; Revised 17 December 2013; Accepted 17 December 2013

Academic Editor: Naseer Shahzad

Copyright © 2013 Jian Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . Let be the expanding endomorphism of the -torus given by the matrix and let be the set of digits with , . 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 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 . 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 , 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 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, 20–22].

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 be a finite directed graph in which loops and multiple edges are allowed. Let be the set of symbols. Suppose that the edges of are labeled in symbols in 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 symbols. That is,

Suppose that is the resulting sofic system. We call

a T-invariant sofic set. An adjacency matrix , according to , is constructed, where, for two vertices , in , is the number of edges in from to .

Theorem 1. Suppose that . Then

To compute the Hausdorff dimension of , the following result in [6] is needed.

Kenyon-Peres’s Theorem (see [6, Theorem  3.2]). Let be a finite directed graph with edges labeled by and adjacency matrix . Let be the resulting sofic system. Then the Hausdorff dimension of is given by

where, for and vertices , in , denotes the number of edges from to in such that the second coordinate of their label is , .

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.

2. Equivalence Classes on

In this section we consider the geometrical structure of . First we define a suitable equivalence relation on the basic rectangles of and partition them into equivalence classes.

Let be defined as in (2); simple computation yields

Then define

such that can be viewed as the unique invariant nonempty compact set under mappings . Let , , obviously, satisfy the OSC. Set

In order to analyze the structure of , we get the following standard notations.

Let and be the set of all finite words, where is the set of all words of length , with containing only the empty word . . For , let denote the length of . For and , let be the concatenation of and . For and a positive integer , let denote the truncation of to the th place.

We define the iterated mappings using these notations. Let . Then

with , the identity map on . Thus we get

where and . We call , the basic rectangles of rank , respectively. For simplicity without confusion, we further call them basic -rectangles and we write and in this paper.

Suppose that there exist , , and such that

Then we connect a directed edge from to and denote it by . We call the parent of and an offspring of . Definitions of parent and offspring can be parallel extended to .

Following we investigate the relationships between the basic -rectangles of and .

Suppose that , , ; the neighborhood of with respect to the basic -rectangles of is defined as

Let . Note that, for some , , 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 we define

where , are the left-lower corner points of , , respectively. For and , we say that and are equivalent, if . Next we prove a recurrent relationship on .

Let and , where , . Suppose that and set

such that , . Let , be the offsprings of and connected by edge , respectively, and denote them by , . Similarly, , are the offsprings of and connected by edge , respectively. That is , .

Lemma 2. With above notations, let be an affine mapping,

where , are the left-lower corner points of and , respectively. Then . If , then , . Furthermore .

Proof. Note that

where . Then

We now prove that , . is the direct conclusion from , which yields , . can be easily obtained by the same argument as .

Proposition 3. With above notations, if , then, for , if and only if . If , then .

Proof. By Lemma 2, we have

so .

One can check that

By Lemma 2 and (21), (22) can be rewritten as

Observe that

where (defined as in (9)). By condition , 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 , we have

where .

Proposition 4.

Proof. It is easy to see that ; therefore

To prove the reverse inclusion, we let ; then for , large enough, such that for , and ( denotes the diameter of and the open -neighborhood of ). Let , then ( always exists; otherwise ). On the other hand , then . Hence

The desired result is then obtained by letting tend to 0.

For some , it is possible that, for all , ; namely, all offsprings of do not belong to . So set

Remark 5. We remark that, by Proposition 4, can also be expressed in radix expansions as follows:

where , .

3. Sofic Affine-Invariant Set

Suppose that is a finite set; let . Set and . We now define the directed edges on . Fix () and a vertex such that . Let in be an offspring of connected by edge . Consider

Note that if none of the offsprings of belongs to , then , 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 then is obtained. Let

Then 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 , . By the definition of in (30) and the construction of , one can claim that is an infinite path in and then belongs to . Thus . The inverse inclusion is left for the readers.

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

Proposition 7. Suppose that . Then the set is finite.

Proof. Suppose that , . , denote the left-lower corner points of , respectively; then

where ;

where . Consider

where ; is the least common multiple of and . , . So the coordinates of are integers. On the other hand implies that

Consequently,

Then, for any , . So the set is finite.

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

4. Examples

Example 1. Let and , where , , , and .

By Proposition 7, the set is finite. The types of the basic rectangles can be determined by (15). It is clear that , denoted by . For , among four offsprings of , only and have nonempty neighborhood, which are connected by edges and from their parent , respectively. By (15), they are of different types; denote them symbolically as

For , direct computation yields

They are of new types and , 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 belongs to ; then type and all paths going to are discarded.

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

By the same argument as above, we have

However none of the offsprings of , whose type is of , belongs to . Then type and all paths going to are deleted. By Proposition 3, the above process exhausts all possible types and yields the adjacency matrix

and by Kenyon-Peres’s Theorem, we have

Example 2. Let and , where , , , and . .

Again by Proposition 7, the set is finite. First, ; denote it by . For , , and , . They are of different types and then denoted by

For , , and , , . Their types are of , , and , respectively. Thus we have

Upon the third iteration, produces four offsprings in , which yields

Only one offspring of , whose type is of , belongs to ; then we get

None of the offsprings of , whose type is , belongs to ; then type and all paths going to are discarded. For , by the same argument as above, we have

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

Figure 2: Construction of . Red and blue rectangles denote the construction of and , respectively.

Also by Kenyon-Peres’s Theorem, we get

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).

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