Research Article | Open Access

# Multifractal Structure of the Divergence Points of Some Homogeneous Moran Measures

**Academic Editor:**Christian Maes

#### Abstract

The point for which the limit does not exist is called divergence point. Recently, multifractal structure of the divergence points of self-similar measures has been investigated by many authors. This paper is devoted to the study of some Moran measures with the support on the homogeneous Moran fractals associated with the sequences of which the frequency of the letter exists; the Moran measures associated with this kind of structure are neither Gibbs nor self-similar and than complex. Such measures possess singular features because of the existence of so-called divergence points. By the box-counting principle, we analyze multifractal structure of the divergence points of some homogeneous Moran measures and show that the Hausdorff dimension of the set of divergence points is the same as the dimension of the whole Moran set.

#### 1. Introduction and Statement of Results

##### 1.1. Moran Set

Let be a sequence of positive integers and let be a sequence of positive real number with for any . Define , and for any , set , , and If , , let . And for , remark .

*Definition 1. *Suppose is a closed interval of length 1. The collection of closed subintervals of is said to have a homogeneous Moran structure, if it satisfies the following conditions (MSC):(i);
(ii)for all and , are subintervals of and satisfy that , where denotes the interior of ;(iii)for any and , ,
where denotes the diameter of .

Suppose that is a collection of closed subintervals of having homogeneous Moran structure, and set It is ready to see that is a nonempty compact set. The set is called the homogeneous Moran set associated with the collection .

Let , and let . The elements of are called the basic elements of order of the homogeneous Moran set and the elements of are called the basic elements of the homogeneous Moran set .

*Remark 2. *If , then contains interior points. Thus, the measure and dimension properties will be trivial. We assume therefore that

Proposition 3 (see [1, Proposition 3.1]). *For a homogeneous Moran set defined as above, suppose furthermore that
**
Then we have
**
where satisfies the equation for each .*

Let , and let be a sequence over , . For , write ; then . We denote by the number of occurrences of the letter in . If for any , , then we say that the sequence has the frequency vector . It is easy to see that and . For , let For , let and let be a positive real number with . For , in the homogeneous Moran construction above, for any if take , . Then we construct the homogeneous Moran set relating to and denote it by .

*Remark 4. *In this paper, we assume that , let be the basic intervals of order contained in arranged from left to right. For all , let , where is a sequence of positive real number. Let . In this paper we suppose .

##### 1.2. Moran Measure

Let be probability vectors; that is, and . For any , , from Section 1.1, we know where , if . For , define as follows: let , be the occurrences of the letter in ; then . For convenience we will write , where . In fact, is a rearrangement of . We make the convention that if .

Now define It is obvious that for any . We make the convention that if .

Let be a mass distribution on , such that for any , , and . Since is related to , we denote it by . Here is a homogeneous Moran measure on , and it is an extension of the self-similar measure by Hutchinson [2].

##### 1.3. Main Results

From now on, we assume that is a homogeneous Moran fractal defined in Section 1.1, and is a probability measure introduced in Section 1.2. The notationa , , , , are as above; in the following, if , , let , and , . Now we define an auxiliary function as follows. For each and , there is a unique number such that By simple calculation, we get

Proposition 5 (see [3, Proposition 2.3]). *For all , defined by (10) satisfies the following:*(i)*;*(ii)* is strictly decreasing, and **;*(iii)* is convex in **, and ** is strictly convex if and only if ** is not the same for all **, **.*

Let be a Borel probability measure on ; let where the supremum is taken over all families of disjoint closed balls with . If exists, we call that the spectrum of exists; that is, Peres and Solomyak [4] give alternative definition of spectrum. Let be a Borel probability measure on . Let be the partition of into grid boxes with . For , denote . If exists, we call that the spectrum of exists; that is, Peres and Solomyak [4] prove that .

Proposition 6 (see [5]). *Let be a Moran measure supported on the homogeneous Moran fractals ; then
*

*The Legendre transform* of is the function defined by
where , .

Let be a complete separable metric space and a finite Borel measure on . In the multifractal analysis one is interested in the size of the following level sets: The space has the following natural decomposition: where The set is called the set of divergence points and the point for which the limit does not exist is called divergence point. Recently, multifractal structure of the divergence points of self-similar measures has been investigated by a large number of authors. Barreira and Schmeling [6] and Chen and Xiong [7] have shown that for self-similar measures satisfying the SSC the set of divergence points typically has the Hausdorff dimension as the support . Furthermore, Olsen and Winter [8] analyse its structure and give a decomposition of this set for the case that the SSC satisfies. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of , which makes things more difficult. By the box-counting principle we (2011) [9] show that the set of divergence point has still the same Hausdorff dimension as the support for self-similar measures satisfying the OSC. Li et al. [10] further analyse its structure and give a decomposition of this set for the case that the OSC satisfies. This paper is devoted to the study of some Moran measures with the support on the homogeneous Moran fractals associated with the sequences of which the frequency of the letter exists; such measures possess singular features because of the existence of so-called divergence points. By the box-counting principle, we analyze Multifractal structure of the divergence points of some homogeneous Moran measures and show that the Hausdorff dimension of the set of divergence points is the same as the dimension of the whole Moran set. It should be pointed out that the Moran measures associated with this kind of structure are neither Gibbs nor self-similar and more than complex.

Theorem 7. *Let be a Moran measure supported on the homogeneous Moran fractals associated with the sequences of which the frequency of the letter exists as above. Set
**
Then
*

By Theorem 7 and Proposition 5, we easily obtain that the Hausdorff dimension of the set of divergence points is the same as the dimension of the whole Moran set.

#### 2. Several Lemma

Lemma 8. *Suppose that is such that for some . Then for any , , *(I)*there exist , , an integral number , and satisfying the following properties:(a) for all ,(b) for all ;*(II)

*there exist an integral number such that for any integer , there exist , , an integral number , and satisfying the following properties:(a)*

*for all ;*(b)*for all .**Proof. *For the given , we choose a small such that
Using (10), we can pick and an integral number such that for any integer ,
Set
Then
Similarly, we have
These two inequalities together with (22)–(26) imply
Note that for each , . Hence,
which combining with (30) yields
We suppose is an integral number satisfying and . This completes the proof of (I).

Next we prove (II). Note that ; we get
Note that and for and using (33), we can choose an integral such that for any integral , the following properties are satisfied; that is,
Combining (34) we get
Therefore
Using the same method, we can choose an integral such that for any integral ,
Set , and (36) and (37) satisfy simultaneously. We suppose is an integral number satisfying and
Set . Replace (25) and (26) with (36) and (37); we can prove (II) by the same method with (I).

Lemma 9. *Let be a Moran measure supported on the homogeneous Moran fractals associated with the sequences of which the frequency of the letter exists as above. Set
**
Then
*

*Proof. *Let , then for any , ; Using (39), there exist and a number sequence such that
for any . We can choose such that . For any and , we consider grid boxes ; there exist adjacent grid boxes such that , and there and are neighbours with . Therefore there exists such that (by ). On the other hand, notice that (by (41)). Thus
Thus
Therefore
Using (14) and (15), notice that ; we have
Thus

#### 3. Proof of Theorem 7

Lemma 10. *Suppose ( is from Remark 4), for all , and , and choose such that
**
where denote the basic elements of order that contains the point . Then
*

*Proof. *It is obvious that . Now let , but ; then there exists such that , and , and , . Therefore,
which is a contradiction since ; therefore .

Since and . (48) imply that
which yields

*Proof of Theorem 7. *For any , define a number sequence in the following manner:
We choose a positive sequence . For and , there exist an integral number sequence ( corresponds to in Lemma 8) and a real number such that the follow properties are satisfied.

For and , using Lemma 8 we can pick satisfying , for all and .

For , remark .

By the same step, for and , we also construct such that (a);
(b);
(c) for , ;(d) for , . Also we let be a sequence of integers large enough such that(e);
(f) for each ;(g).

Now we define a sequence of subsets of symbol set in the following manner:
and relabel them as . Let
It is easy to check that is the homogeneous* Moran set* which is a subset of the homogeneous* Moran set *.

Next we show that

Let , then there exist such that
Let ; there exists such that
Using Lemma 10, we attain
Thus to calculate , we need to estimate and for . By (c) and (d), we obtain
By (54), (60)–(62), and (g), we obtain
Let ; there exists such that
By the same method, we obtain

Next we show . For small enough, there is a unique large integer such that
Using the same method as above, we attain
Now we estimate and for , . For large , write with . In the case that , by (c) and (d), we obtain
In the other case , we have the similar inequalities where the lower bounds for and in (68) and (69) are replaced, respectively, by
By (54), using the inequalities (67)–(71), (f), and (g), we obtain

Combining (63), (65), and (72), we obtain

To prove , recall that is the homogeneous* Moran set* which is a subset of the homogeneous* Moran set *. For large , write