Abstract

We prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions. Subsequently, we consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples in which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition of the smallest compression ratio .

1. Introduction

Let us begin with the definition of the Assouad dimension and the lower dimension. For , and denotes the smallest number of open sets required for an r-cover of a bounded set E.

Definition 1. The Assouad dimension of a nonempty set is defined by There exists a constant such that, for any , and , .
If the Hausdorff dimension provides fine, but global, geometric information, then the Assouad dimension which was introduced by Assouad [1] provides coarse, but local, geometric information. The Assouad dimension is a fundamental notion of dimension used to study fractal objects in a wide variety of contexts. An important theme in dimension theory is that dimensions often come in pairs. The natural partner of the Assouad dimension is the lower dimension, which was introduced by Larman [2], where it was called the minimal dimensional number.

Definition 2. The lower dimension of F is defined by There exists a constant such that, for any , and , .
The lower dimension is not monotonic, and the modified lower dimension is defined byThe Assouad dimension has recently received an enormous interest in the mathematical literature due to its connections with the doubling property. This lead Larman to introduce the dual notion of dimension, namely, the lower Assouad dimension, often simply called the lower dimension. Just like the Assouad dimension, the lower dimension has also received an enormous interest in the mathematical literature due to its connections with the uniform property of metric spaces. As a result of this, a large number of papers have investigated the Assouad dimension and the lower dimension of different classes of fractal sets. Olsen [3] gave a simple and direct proof that the Assouad dimension of a graph-directed Moran fractal satisfying the open-set condition which is Ahlfors regular coinciding with its Hausdorff and box dimensions. However, in general, it is difficult to obtain the Assouad dimensions of sets which are not Ahlfors regular. Mackay [4] calculated the Assouad dimension of the self-affine carpets of Bedford and McMullen and his main result solved the problem posed by Olsen [3]. For the Moran sets introduced by Wen [5] which are not Ahlfors regular, Li et al. [6] obtained the Assouad dimensions of Moran sets under suitable condition and studied the Assouad dimensions of Cantor-like sets. Jinjun Li [7] also show that the Assouad dimensions of some Moran sets coincide with their packing and upper box dimensions. However, Li [7] did not compute the lower dimension of this class of fractals, the main conclusions of the paper [6, 7] must satisfy the condition that the smallest compression ratio and the paper [7] conjecture that the conclusion remains true if the condition is removed (see Remark 1 [7]). In this paper, we prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions. Subsequently, we consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition that the smallest compression ratio , and we give a negative answer to the conjecture in the paper [7].

2. Lower Dimensions of Some Moran Sets

Firstly, let us recall the definition of Moran sets introduced by Wen [5]. Let be a sequence of positive integers. Define , and for any , set , , and . If , let . And if , remark for .

Definition 3. Suppose that is a compact set with . Let be a sequence of positive real vectors with and We say the collection of closed subsets of fulfills the Moran structure if it satisfies the following Moran structure conditions (MSC):(1)For is geometrically similar to , i.e., there exists a similarity such that . For convenience, we write (2)For all and are subsets of and satisfy that (3)For any and ,where denotes the diameter of A.
Suppose that F is a collection of closed subsets of J fulfilling the Moran structure, setIt is ready to see that E is a nonempty compact set. The set is called the Moran set associated with the collection F.
Let and . The elements of are called kth-level basic sets of E and the elements of F are called the basic sets of E. Suppose that the set and the sequences and are given. We denote by the class of the Moran sets satisfying the MSC. We call the Moran class associated with the triplet .

Remark 1. From the above definition, we see that if the Moran sets and , then the relative positions of k^th-level basic sets of E₁ and those of E₂ may be different, although they satisfy the same MSC.
Under some mild conditions, Hua et al. [8] gave the Hausdorff packing and upper box dimensions of Moran sets. To state their result, we need some notations. Let be a Moran class. Let . Letwhere satisfies the following equation:SetWe can now present the main result of Hua et al. [8].

Theorem 1. (see [8]). Suppose that is a Moran class satisfying . Then, for any ,

Li [7] computed the Assouad dimension of a fairly general (and important) class of Moran fractals.

Theorem 2. (see [7]). Suppose that is a Moran class satisfying . Then, for any ,

The natural partner of the Assouad dimension is the lower dimension, and we prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions.

Theorem 3. Suppose that is a Moran class satisfying and Then, for any ,

Lemma 1. There exists a probability measure supported by the Moran E such thatfor any and .

Proof. Take a sequence of probability measures supported by E such thatfor any .
More precisely, we can construct as follows.
First, we distribute the unit mass among the m^th-level basic elements according to (11). Inductively, suppose that we have already distributed the mass of proportion to a k^th-level basic set ; then, we distribute the mass concentrated on evenly to each of its th-level basic subsets, i.e.,for .
Repeating the above procedure, we get the desired measure.
Now, fix some ; for any and , we obtainCombining it with (11), we haveFor any , by the definitions of E,and thus by (14),This givesObserving thatone obtainsTo summarize, we obtain a sequence of probability measures supported by E and satisfy (10) for any and .
Now, Hellys theorem [9] enables us to extract a subsequence converging weakly to a limit measure .
To verify that fulfills the desired requirements, we fix some and . Then, by the properties of the weak convergence,Combining with (19), this impliesOn the other hand, take an small enough so that the -neighborhood of is separated from the other m^th-level basic set; then, . By the properties of weak convergence, the following holds:Combining with (19) yieldsWe have for any and ,Finally, for any which is not in E, since E is a closed set, there exists an open set U containing and separated from E, and thus, , which asserts that is supported by E.

Lemma 2. (see [1, 2]). If is closed, then

For , we denote by - the word obtained by deleting the last letter of . For , we define by .

Lemma 3. (see [7], Lemma 3.1). If , there exists a constant such that for all and .

Remark 2. Some subtly different definitions of the low dimension are given as follows: There exist two constant and such that, for any , and , .
It is easy to check this definition and Definition 2 coincides.

Proof of Theorem 3. Fix and ; there exists such that and . By Lemma 1, there exists a probability measure supported by E, such thatwhere denotes the rank of , i.e., is the ^th-level basic set. This implies thatBy Remark 2, let R be small enough such that and Therefore,Using Lemma 3 and (28), we attainwhich implies and by Lemma 2 and Theorem 1, the proof of Theorem 3 is completed.

3. Assouad Dimensions and Lower Dimensions of Homogeneous Cantor Set

Definition 4. Suppose that is the interval [0, 1] and for any , , in Definition 3. For all , , , and the left endpoint of is the left endpoint of and the right endpoint of is the right endpoint of . The set is called the homogeneous Cantor set. Write .

Theorem 4. (see [10]). Suppose . Then,

Theorem 5. Suppose . If , then .

Proof. Take x as the left endpoint of some basic elements of order k and ; then, is the length of basic elements of order k. It is obvious that . Notice thatHere, is the length of the basic elements of order k + 1 of the Moran set, and is the length of the interval among the basic elements of order k + 1 of the Moran set E. Take ; then,Note that when ; therefore, .

Theorem 6. Suppose . If , then .

Proof. Take x as the left endpoint of some basic elements of order k + 1 andHere, is the length of the basic elements of order k + 1 of the Moran set and is the length of the interval among the basic elements of order k + 1 of the Moran set E. Take . It is obvious thatNote that when ; therefore, .