Some Dimensional Results of a Class of Homogeneous Moran Sets

Abstract

In this paper, we construct a class of special homogeneous Moran sets: -quasi-homogeneous perfect sets, and obtain the Hausdorff dimension of the sets under some conditions. We also prove that the upper box dimension and the packing dimension of the sets can get the maximum value of the homogeneous Moran sets under the condition and estimate the upper box dimension of the sets in two cases.

1. Introduction

The fractal dimensions of the Moran sets have been studied by many authors and close connected with many subjects, such as the multifractals (see [1–3]), the quasi-conformal mappings (see [4, 5]), the ergodic theory (see [6]), and the number theory (see [7]). In these applications, the homogeneous Moran sets play an important role. There are many classical results about the fractal dimensions of the homogeneous Moran sets (see [8, 9]), but Wen et al. [8, 9] only showed the maximal values and the minimal values of the Hausdorff dimension, the upper box dimension, and the packing dimension (see [10]) of the family of the homogeneous Moran sets, and we need to get some more exact expressions of the fractal dimensions for some special homogeneous Moran sets. Wen and Wu [11] constructed a class of homogeneous Moran sets by the gaps, which are called homogeneous perfect sets, and obtained the Hausdorff dimension of the homogeneous perfect sets under some conditions. Later, Wang and Wu [12] obtained the upper box dimension and the packing dimension of the homogeneous perfect sets under some conditions. Hu [13, 14] constructed a special kind of homogeneous Moran sets which is called -quasi-homogeneous Cantor sets by the connected components and the gaps and obtained its Hausdorff dimension, upper box dimension, and packing dimension.

In this paper, a class of special homogeneous Moran sets, which is called -quasi-homogeneous perfect sets, is constructed by the connected components and the gaps, and several results about the fractal dimensions of the sets are obtained, which extend the results in [11, 12, 14] under some conditions.

This paper is organized as follows. In Section 2, we give the definitions of the homogeneous Moran sets, the homogeneous perfect sets, and the -quasi-homogeneous perfect sets. Our main results are stated in Section 3. Section 4 gives the proofs of our theorems.

2. Preliminaries

First, we recall the definition of homogeneous Moran sets.

Let be a sequence of positive integers and be a sequence of positive real numbers, such that , and , for any . Let , for any , and let and . If and , let .

Definition 1. (homogeneous Moran sets, (see [8]). Let be a nonempty closed interval and be a collection of closed subintervals of . We say that has homogeneous Moran structure if it satisfies(i)(ii)For any , , are subintervals of , and the interiors of and are disjoint for any (iii)For any , , and , we have , where denotes the diameter of the set LetThen, we call a homogeneous Moran set satisfying and denote by the class of all homogeneous Moran sets associated with , , and .
Let ; then, and any in is called a -order basic interval of . Without loss of generality, we set and assume that are located from left to right for any and any .
Based on the basic intervals and the gaps, Wen and Wu [11] constructed a special kind of homogeneous Moran sets: homogeneous perfect sets.

Definition 2. (homogeneous perfect sets, (see [11]). Let , and we say that satisfies homogeneous perfect structure if it satisfies the following: there exists a sequence of real numbers such that, for any , we have , and for any , , and , we have , , and . Then, we call a homogeneous perfect set satisfying and denote by the class of all homogeneous perfect sets associated with .
Based on the connected components of the basic intervals and their gaps, we can construct a special class of homogeneous Moran sets: -quasi-homogeneous perfect sets.

Definition 3. (-quasi-homogeneous perfect sets). Let , if for all and , the -order basic intervals arbitrarily connect forming connected components (denote by , which are called -order connected components). We say that satisfies homogeneous perfect structure if it satisfies(i)(ii)There exists a sequence of real numbers such that, for any , we have and and for any , , and , we have , , and Let ; then, we call a -quasi-homogeneous perfect set satisfying and denote by the class of all -quasi-homogeneous perfect sets associated with . For any , , denote by the number of the -order basic intervals contained in the -order connected components .

Remark 1. Let ; then, we have(i)If , for any , then is a homogeneous perfect set and , where , for any , (ii)If , , and , for any , then is a homogeneous Cantor set [8]

3. Main Results

Let

Our main results are stated as follows.

Theorem 1. Let with . For any , , and , there exists a positive integer such that for the -order connected component . If there exist positive constants such that at least one of the following two conditions is satisfied for any :(A).(B); then,

Example 1. Let be a -quasi-homogeneous perfect set satisfying the conditions of Theorem 1, then we have , which is equivalent to Theorem 1.2 (1) in [11]. Notice that if is a homogeneous perfect set, then and satisfies (for any ) if . Thus, Theorem 1 generalizes Theorem 1.2 (A) and (B) in [11] under the condition ().

Theorem 2. Suppose . If , then

Example 2. Let be a -quasi-homogeneous Cantor set (refer to [14]); then, and satisfies and , for any . Since the conclusion of Theorem 2 holds under the condition , Theorem 2 generalizes Theorem 2.3 in [14].

Theorem 3. Let with and , for any . If there exists a positive constant such that, for any , there exists a sequence of nonnegative integers with and , satisfying that, for any -order connected component is one of , and the following condition is satisfied:Then,

Remark 2. The conditions of and in Theorem 3 imply that there exist , such that, for any , we have .

Theorem 4. Let with , for any . Suppose that there exist positive constants such that, for any , at least one of the following three conditions is satisfied:(C), and .(D).(E) and ; then,

Example 3. We obtain the following conclusions about the relationships between theorems in this paper and Theorem 1.4 in [12]:(i)If Theorem 1.4 (A) in [12] holds for which satisfies (for any ) and there exists such that (for any ) or Theorem 1.4 (B) or (C) holds for which satisfies then we have and satisfies the conditions of Theorem 4, then (7) holds for . Since and , (7) is equivalent to (1.4) in [12] for the upper box dimension.(ii)If Theorem 1.4 (D) in [12] holds for which satisfies (for any ), then . Thus, and satisfies the conditions of Theorem 2; then, (4) holds for . Since , we have and (4) is equivalent to (1.4) in [12].By the argument above, Theorems 2 and 4 generalize Theorem 1.4 in [12] for the upper box dimension under the conditions and (for any ) (notice that if satisfies (B) or (C) or (D) of Theorem 1.4 in [12], we only need the condition ).

Remark 3. The conditions of Theorem 3 and the conditions of Theorem 4 (C) do not contain each other.
If satisfies the conditions of Theorem 3, such that , , and , then as , and does not satisfy the conditions of Theorem 4 (C).
If satisfies the conditions of Theorem 4 (C), such that and for any (it means ) and there exists a positive integer such that, for any and any , the numbers of the -order basic intervals contained in are , then does not satisfy the conditions of Theorem 3.

Remark 4. The upper box dimension in Theorems 3 or 4 is different from and .
If satisfies the conditions of Theorem 3, such that , (for any ), (for any ), , (for any ), , and (for any ) or if satisfies the conditions of Theorem 4 (D) such that , (for any ), (for any ), and (for any ), thenIt can be concluded that is different from and .

4. Proofs of the Theorems

We shall use the following lemmas to prove the theorems.

Lemma 1. (mass distribution principle, (see [10]). Let be a Borel set and be a Borel measure supported on such that . Suppose that, for some , there are numbers and such that , for all sets U with . Then, .

Lemma 2. (see [13]). Let ; then,

Lemma 3. (see [8]). Let , then

Lemma 4. (see [15]) Let ; then,

4.1. Proof of Theorem 1

Since the conditions of Theorem 1 imply that , for all , we have . We only need to prove .

By Lemma 2, it is suffices to prove that if satisfies the conditions of Theorem 1, then .

For convenience, let

For any , there exists a positive integer such that, for any , we have

Let be a Borel probability measure supported on , such that, for each -order basic intervals , . Then, for each -order connected component , .

Let be an interval with ; then, there exists a positive integer such that . Thus, the number of the -order basic intervals that intersect is at most . Denote by the number of the -order connected components that intersect . In order to estimate , we divide our proof into the following two cases: Case (i): . It is obvious that . Then, Case (ii):

We discuss separately according to the two conditions (A) and (B) of Theorem 1. Condition (A): . Then, we have On the one hand, since the number of the -order basic intervals that intersect is at most , we have . On the other hand, the number of the -order connected components that intersect is at most . Then, by (13) and (15), we have Condition (B): . In this case, we have

On the one hand, . On the other hand, by the definition of , we have

Then,

Combining (13), (17), and (19), we have

By (14), (16), (20), and Lemma 1, we have . By the arbitrariness of , we obtain and complete the proof of Theorem 1.

4.2. Proof of Theorem 2

By Lemma 3, it is suffices to prove . The conclusion is obvious if . Now, we assume . For any , there exist , such that, for any ,

Let ; then, . By the definitions of and , we have . Since , for any , we have and . By the convexity of , we obtain that

Let ; by (21) and (22), we have

By Lemma 4, we have . By the arbitrariness of , we have and complete the proof of Theorem 1.

4.3. Proof of Theorem 3

First, we use Lemma 4 to estimate the upper bound for .

Proposition 1. Suppose ; then,

Proof. For any , there exist and such that, for any ,and it follows thatwhich impliesBy , for any , and the convexity of , we haveSince , we have for any . Then, by (27) and (28), we haveBy Lemma 4, we have . By the arbitrariness of , we have .
Now, we estimate the lower bound for .

Proposition 2. Suppose satisfies the conditions of Theorem 3; then,

Proof. Let and (for any ). We estimate for a fixed large enough.
By the condition of Theorem 3, we have . Let be the maximum number of the -order connected components intersecting a ball of diameter ; then, we havewhich impliesLet , then we have . Then, . Thus,Since as , then, by (33), we haveFrom Propositions 1 and 2, we have Theorem 3.

4.4. Proof of Theorem 4

It is obvious that the conclusion of Proposition 1 also holds if satisfies the conditions of Theorem 4, then it is suffices to estimate the lower bound of under the conditions of Theorem 4.

Proposition 3. Suppose satisfies the conditions of Theorem 4; then,

Proof. We discuss separately according to the three conditions (C), (D), and (E) of Theorem 4 to estimate for a fixed large enough, and we use the definitions of and in the proof of Theorem 3. Condition (C): in this case, and . Then, we have it follows which implies . Thus, Condition (D): in this case, . Then, which implies that the number of the -order connected components intersecting a ball of diameter is at most 2; then, the number of the -order connected components intersecting a ball of diameter is at most . Thus, Condition (E): in this case, . We distinguish two cases. Case 1: ; then, . Similar to the discussions in condition (C), we have Case 2: . In this case, we haveand it follows thatSimilarly to the discussions in Condition (D), we haveNotice that as . Then, by (38), (40), (41), and (44), we haveFrom Propositions 1 and 3, we have Theorem 4.

Data Availability

The data used to support the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank Professor Wu Min of South China University of Technology for her helpful discussions and suggestions. This work was supported by the National Natural Science Foundation of China (no. 11901121), Guangxi Natural Science Foundation (2020GXNSFAA297040), special foundation for Guangxi Ba Gui Scholars, Guangdong Natural Science Foundation (2018A030313971), and The Education Department Foundation of Guangdong Province (2017KTSCX112).