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

Authors: JiaQing Xiao; YouMing He
Journal: Advances in Mathematical Physics
 (2014)

---

## Abstract

The pointx for which the limit lim
r
→
0
⁡(
log
⁡
μ
B
x
,
r
/
log
⁡
r
) 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.


---

## Body

## 1. Introduction and Statement of Results

### 1.1. Moran Set

Let{
n
k
}
k
≥
1 be a sequence of positive integers and let {
r
k
} be a sequence of positive real number with n
k
r
k
<
1 for any k
∈
ℕ. Define D
o
=
ϕ, and for any k
≥
1, set D
m
,
k
=
{
(
i
m
,
i
m
+
1
,
…
,
i
k
)
;
1
≤
i
j
≤
n
j
,
m
≤
j
≤
k
}, D
k
=
D
1
,
k, and
(1)
D
=
⋃
k
≥
0
‍
D
k
.
If σ
=
(
σ
1
,
σ
2
,
…
,
σ
k
)
∈
D
k, τ
=
(
τ
1
,
…
,
τ
m
-
k
)
∈
D
k
+
1
,
m, let σ
*
τ
=
(
σ
1
,
…
,
σ
k
,
τ
1
,
…
,
τ
m
-
k
). And for 1
≤
l
≤
k, remark σ
∣
l
=
(
σ
1
,
…
,
σ
l
).Definition 1.
SupposeJ is a closed interval of length 1. The collection ℱ
=
{
J
σ
;
σ
∈
D
} of closed subintervals of J is said to have a homogeneous Moran structure, if it satisfies the following conditions (MSC):(i)
J
ϕ
=
J;
(ii)
for allk
≥
0 and σ
∈
D
k, J
σ
*
1
,
J
σ
*
2
,
…
,
J
σ
*
n
k
+
1 are subintervals of J
σ and satisfy that J
τ
*
i
∘
⋂
‍
J
τ
*
j
∘
=
ϕ
(
i
≠
j
), where A
∘ denotes the interior of A;
(iii)
for anyk
≥
1 and σ
∈
D
k
-
1, 1
≤
j
≤
n
k,
(2)
|
J
σ
*
j
|
|
J
σ
|
=
r
k
,where |
A
| denotes the diameter of A.Suppose thatℱ is a collection of closed subintervals of J having homogeneous Moran structure, and set
(3)
E
k
=
⋃
σ
∈
D
k
‍
J
σ
,
E
=
⋂
k
≥
0
‍
E
k
.
It is ready to see that E is a nonempty compact set. The set E
:
=
E
(
ℱ
) is called the homogeneous Moran set associated with the collection ℱ.Letℱ
k
=
{
J
σ
;
σ
∈
D
k
}, and let ℱ
=
⋃
k
≥
0
ℱ
k. The elements of ℱ
k are called the basic elements of order k of the homogeneous Moran set E and the elements of ℱ are called the basic elements of the homogeneous Moran set E.Remark 2.
Iflim
⁡
n
→
∞
⁡
sup
⁡
σ
∈
D
n
⁡
|
J
σ
|
>
0, then E contains interior points. Thus, the measure and dimension properties will be trivial. We assume therefore that
(4)
lim
⁡
n
→
∞
sup
⁡
σ
∈
D
n
|
J
σ
|
=
0
.Proposition 3 (see [1, Proposition 3.1]).
For a homogeneous Moran setE defined as above, suppose furthermore that
(5)
lim
⁡
k
→
∞
log
⁡
r
k
log
⁡
r
1
r
2
⋯
r
k
-
1
=
0
.
Then we have
(6)
dim
⁡
H
E
=
liminf
⁡
k
→
∞
s
k
,
where s
k satisfies the equation ∑
σ
∈
D
k
‍
r
σ
s
k
=
1 for each k.LetA
=
{
a
1
,
a
2
,
…
,
a
m
}, and let ω
=
s
1
s
2
⋯
s
k
⋯ be a sequence over A, s
i
∈
A. For k
≥
1, write ω
k
=
ω
∣
k
=
s
1
s
2
⋯
s
k; then |
ω
k
|
=
k. We denote by |
ω
k
|
a
i the number of occurrences of the letter a
i in ω
k. If for any a
i
∈
A, lim
⁡
k
→
∞
⁡
(
|
ω
k
|
a
i
/
k
)
=
η
i
>
0, then we say that the sequence ω has the frequency vector η
=
(
η
1
,
η
2
,
…
,
η
m
). It is easy to see that ∑
i
=
1
m
|
ω
k
|
a
i
=
k and ∑
j
=
1
m
η
j
=
1. For η
=
(
η
1
,
η
2
,
…
,
η
m
), let
(7)
A
η
ℕ
=
{
ω
=
{
s
k
}
k
≥
1
;
s
k
∈
A
,
lim
⁡
k
→
∞
|
ω
k
|
a
i
k
=
η
i
,
1
≤
i
≤
m
}
.
For 1
≤
i
≤
m, let m
i
∈
ℕ and let c
i be a positive real number with m
i
c
i
≤
1. For ω
∈
A
ℕ, in the homogeneous Moran construction above, for any k
≥
1 if s
k
=
a
i take n
k
=
m
i, r
k
=
c
i. Then we construct the homogeneous Moran set relating to ω
∈
A
η
ℕ and denote it by E
(
ω
)
=
{
J
,
{
n
k
}
,
{
r
k
}
}.Remark 4.
In this paper, we assume thatJ
σ
∈
ℱ
k
(
k
≥
1
), let J
σ
*
1
,
J
σ
*
2
,
…
,
J
σ
*
n
k
+
1 be the n
k
+
1 basic intervals of order k
+
1 contained in J
σ arranged from left to right. For all 1
≤
j
≤
n
k
+
1
-
1, let d
(
J
σ
*
j
,
J
σ
*
(
j
+
1
)
)
≥
Δ
k
|
J
σ
|, where {
Δ
k
} is a sequence of positive real number. Let Δ
=
inf
⁡
k
≥
1
⁡
Δ
k. In this paper we suppose Δ
>
0.


### 1.2. Moran Measure

LetP
a
i
=
(
p
i
1
p
i
2
⋯
p
i
m
i
)
(
1
≤
i
≤
m
) be probability vectors; that is, p
i
j
>
0 and ∑
j
=
1
m
i
p
i
j
=
1
(
1
≤
i
≤
m
). For any k
≥
1, σ
∈
D
k, from Section 1.1, we know σ
=
σ
1
σ
2
⋯
σ
k
∈
D
k where σ
k
∈
{
1,2
,
…
,
m
i
}, if s
k
=
a
i. For σ
=
σ
1
σ
2
⋯
σ
k, define σ
(
a
i
) as follows: let ω
k
=
s
1
⋯
s
k, e
1
<
e
2
<
⋯
e
|
ω
k
|
a
i be the occurrences of the letter a
i in ω
k; then σ
(
a
i
)
=
σ
e
1
σ
e
2
⋯
σ
e
|
ω
k
|
a
i. For convenience we will write σ
(
a
i
)
=
σ
i
1
σ
i
2
⋯
σ
i
|
ω
k
|
a
i, where σ
i
j
∈
{
1,2
,
…
,
m
i
}
(
1
≤
i
≤
m
). In fact, σ
(
a
1
)
*
σ
(
a
2
)
*
⋯
*
σ
(
a
m
) is a rearrangement of σ
=
σ
1
⋯
σ
k. We make the convention that σ
(
a
i
)
=
ϕ if |
ω
k
|
a
i
=
0.Now define(8)
p
σ
(
a
i
)
=
p
i
σ
i
1
⋯
p
i
σ
i
|
ω
k
|
a
i
,
1
≤
i
≤
m
.
It is obvious that ∑
σ
∈
D
k
∏
i
=
1
m
p
σ
(
a
i
)
=
1 for any k
≥
1. We make the convention that p
σ
(
a
i
)
=
1 if σ
(
a
i
)
=
ϕ.Letμ be a mass distribution on E
(
ω
), such that for any J
σ
∈
ℱ
k, σ
∈
D
k,
(9)
μ
(
J
σ
)
=
p
σ
(
a
1
)
p
σ
(
a
2
)
⋯
p
σ
(
a
m
)
,
and μ
(
∑
σ
∈
D
k
J
σ
)
=
1. Since μ is related to ω, we denote it by μ
(
ω
). Here μ
(
ω
) is a homogeneous Moran measure on E
(
ω
), and it is an extension of the self-similar measure by Hutchinson [2].


### 1.3. Main Results

From now on, we assume thatE
(
ω
) is a homogeneous Moran fractal defined in Section 1.1, and μ
(
ω
) is a probability measure introduced in Section 1.2. The notationa D, D
k, J
σ, μ
(
J
σ
), P
σ
(
a
i
)
(
1
≤
i
≤
m
) are as above; in the following, if σ
∈
D
k, τ
∈
D
k
+
1
,
k
+
n, let r
σ
=
|
J
σ
|, p
σ
=
μ
(
J
σ
) and r
τ
=
|
J
σ
*
τ
|
/
|
J
σ
|, p
τ
=
μ
(
J
σ
*
τ
)
/
μ
(
J
σ
). Now we define an auxiliary function β
(
q
) as follows. For each q
∈
ℝ and k
≥
1, there is a unique number β
k
(
q
) such that
(10)
∑
σ
∈
D
k
‍
p
σ
q
r
σ
β
k
(
q
)
=
1
.
By simple calculation, we get
(11)
lim
⁡
k
→
∞
β
k
(
q
)
=
β
(
q
)
=
∑
i
=
1
m
η
i
log
⁡
(
∑
j
=
1
m
i
p
i
j
q
)
∑
i
=
1
m
η
i
log
⁡
c
i
.Proposition 5 (see [3, Proposition 2.3]).
For allq
∈
ℝ, β
(
q
) defined by (10) satisfies the following:(i)
β
(
0
)
=
dim
⁡
E
;
(ii)
β
(
q
)
is strictly decreasing, and
lim
⁡
q
→
∓
∞
⁡
β
(
q
)
=
±
∞
;
(iii)
β
(
q
)
is convex in
q
, and
β
(
q
)
is strictly convex if and only if
log
⁡
p
i
j
/
log
⁡
c
i
is not the same for all
1
≤
j
≤
m
i
,
i
=
1,2
,
…
,
m
.Letμ be a Borel probability measure on ℝ
d; let
(12)
Θ
(
q
;
r
)
=
sup
⁡
∑
i
‍
μ
(
B
(
x
i
,
r
)
)
q
,
r
>
0
,
q
∈
ℝ
,
where the supremum is taken over all families of disjoint closed balls {
B
(
x
i
,
r
)
}
i with x
i
∈
supp
⁡
μ. If lim
⁡
r
→
0
(
log
⁡
Θ
(
q
;
r
)
/
log
⁡
r
) exists, we call that the L
q
-spectrum τ
(
q
) of μ exists; that is,
(13)
τ
(
q
)
=
lim
⁡
r
→
0
log
⁡
Θ
(
q
;
r
)
-
log
⁡
r
.
Peres and Solomyak [4] give alternative definition of L
q
-spectrum. Let μ be a Borel probability measure on ℝ
d. Let D
n be the partition of ℝ
d into grid boxes ∏
i
=
1
d
‍
[
k
i
2
-
n
,
(
k
i
+
1
)
2
-
n
) with k
i
∈
ℤ. For q
>
0, denote τ
n
(
q
)
(
μ
)
=
∑
Q
∈
D
n
‍
(
μ
Q
)
q. If lim
⁡
n
→
0
(
log
⁡
τ
n
(
q
)
(
μ
)
/
-
n
log
⁡
2
) exists, we call that the L
q
-spectrum τ
0
(
q
) of μ exists; that is,
(14)
τ
0
(
q
)
=
lim
⁡
n
→
0
log
⁡
τ
n
(
q
)
(
μ
)
-
n
log
⁡
2
.
Peres and Solomyak [4] prove that τ
(
q
)
=
τ
0
(
q
).Proposition 6 (see [5]).
Letμ be a Moran measure supported on the homogeneous Moran fractals E; then
(15)
τ
(
q
)
=
τ
0
(
q
)
=
β
(
q
)
.The Legendre transform of β is the function β
*
:
(
α
min
⁡
,
α
max
⁡
)
→
ℝ defined by
(16)
β
*
(
α
)
=
inf
⁡
q
∈
ℝ
{
β
(
q
)
+
α
q
}
,
where α
min
⁡
=
inf
⁡
q
∈
ℝ
β
′
(
q
), α
max
⁡
=
sup
⁡
q
∈
ℝ
β
′
(
q
).LetX be a complete separable metric space and μ a finite Borel measure on X. In the multifractal analysis one is interested in the size of the following level sets:
(17)
X
α
=
{
x
∈
X
:
lim
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
α
}
,
X
α
=
Tl
=
x
∈
X
:
lim
⁡
r
→
0
-
∞
≤
α
≤
+
∞
.
The space X has the following natural decomposition:
(18)
X
=
⋃
-
∞
≤
α
≤
+
∞
‍
X
α
∪
X
o
,
where
(19)
X
o
=
{
x
∈
X
:
lim
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
does
not
exist
}
.
The set X
o is called the set of divergence points and the point x for which the limit lim
⁡
r
→
0
⁡
(
log
⁡
μ
(
B
(
x
,
r
)
)
/
log
⁡
r
) 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 K. 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 ℝ
d, 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 K 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.
Letμ be a Moran measure supported on the homogeneous Moran fractals E
(
Ω
) associated with the sequences Ω of which the frequency of the letter exists as above. Set
(20)
E
q
1
,
q
2
=
{
limsup
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
2
)
x
∈
E
(
Ω
)
aaa
∣
β
′
(
q
1
)
=
liminf
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
a
a
a
a
a
a
a
a
a
a
<
limsup
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
2
)
}
.
Then
(21)
dim
⁡
E
q
1
,
q
2
=
β
*
(
β
′
(
q
1
)
)
.By Theorem7 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.


## 1.1. Moran Set

Let{
n
k
}
k
≥
1 be a sequence of positive integers and let {
r
k
} be a sequence of positive real number with n
k
r
k
<
1 for any k
∈
ℕ. Define D
o
=
ϕ, and for any k
≥
1, set D
m
,
k
=
{
(
i
m
,
i
m
+
1
,
…
,
i
k
)
;
1
≤
i
j
≤
n
j
,
m
≤
j
≤
k
}, D
k
=
D
1
,
k, and
(1)
D
=
⋃
k
≥
0
‍
D
k
.
If σ
=
(
σ
1
,
σ
2
,
…
,
σ
k
)
∈
D
k, τ
=
(
τ
1
,
…
,
τ
m
-
k
)
∈
D
k
+
1
,
m, let σ
*
τ
=
(
σ
1
,
…
,
σ
k
,
τ
1
,
…
,
τ
m
-
k
). And for 1
≤
l
≤
k, remark σ
∣
l
=
(
σ
1
,
…
,
σ
l
).Definition 1.
SupposeJ is a closed interval of length 1. The collection ℱ
=
{
J
σ
;
σ
∈
D
} of closed subintervals of J is said to have a homogeneous Moran structure, if it satisfies the following conditions (MSC):(i)
J
ϕ
=
J;
(ii)
for allk
≥
0 and σ
∈
D
k, J
σ
*
1
,
J
σ
*
2
,
…
,
J
σ
*
n
k
+
1 are subintervals of J
σ and satisfy that J
τ
*
i
∘
⋂
‍
J
τ
*
j
∘
=
ϕ
(
i
≠
j
), where A
∘ denotes the interior of A;
(iii)
for anyk
≥
1 and σ
∈
D
k
-
1, 1
≤
j
≤
n
k,
(2)
|
J
σ
*
j
|
|
J
σ
|
=
r
k
,where |
A
| denotes the diameter of A.Suppose thatℱ is a collection of closed subintervals of J having homogeneous Moran structure, and set
(3)
E
k
=
⋃
σ
∈
D
k
‍
J
σ
,
E
=
⋂
k
≥
0
‍
E
k
.
It is ready to see that E is a nonempty compact set. The set E
:
=
E
(
ℱ
) is called the homogeneous Moran set associated with the collection ℱ.Letℱ
k
=
{
J
σ
;
σ
∈
D
k
}, and let ℱ
=
⋃
k
≥
0
ℱ
k. The elements of ℱ
k are called the basic elements of order k of the homogeneous Moran set E and the elements of ℱ are called the basic elements of the homogeneous Moran set E.Remark 2.
Iflim
⁡
n
→
∞
⁡
sup
⁡
σ
∈
D
n
⁡
|
J
σ
|
>
0, then E contains interior points. Thus, the measure and dimension properties will be trivial. We assume therefore that
(4)
lim
⁡
n
→
∞
sup
⁡
σ
∈
D
n
|
J
σ
|
=
0
.Proposition 3 (see [1, Proposition 3.1]).
For a homogeneous Moran setE defined as above, suppose furthermore that
(5)
lim
⁡
k
→
∞
log
⁡
r
k
log
⁡
r
1
r
2
⋯
r
k
-
1
=
0
.
Then we have
(6)
dim
⁡
H
E
=
liminf
⁡
k
→
∞
s
k
,
where s
k satisfies the equation ∑
σ
∈
D
k
‍
r
σ
s
k
=
1 for each k.LetA
=
{
a
1
,
a
2
,
…
,
a
m
}, and let ω
=
s
1
s
2
⋯
s
k
⋯ be a sequence over A, s
i
∈
A. For k
≥
1, write ω
k
=
ω
∣
k
=
s
1
s
2
⋯
s
k; then |
ω
k
|
=
k. We denote by |
ω
k
|
a
i the number of occurrences of the letter a
i in ω
k. If for any a
i
∈
A, lim
⁡
k
→
∞
⁡
(
|
ω
k
|
a
i
/
k
)
=
η
i
>
0, then we say that the sequence ω has the frequency vector η
=
(
η
1
,
η
2
,
…
,
η
m
). It is easy to see that ∑
i
=
1
m
|
ω
k
|
a
i
=
k and ∑
j
=
1
m
η
j
=
1. For η
=
(
η
1
,
η
2
,
…
,
η
m
), let
(7)
A
η
ℕ
=
{
ω
=
{
s
k
}
k
≥
1
;
s
k
∈
A
,
lim
⁡
k
→
∞
|
ω
k
|
a
i
k
=
η
i
,
1
≤
i
≤
m
}
.
For 1
≤
i
≤
m, let m
i
∈
ℕ and let c
i be a positive real number with m
i
c
i
≤
1. For ω
∈
A
ℕ, in the homogeneous Moran construction above, for any k
≥
1 if s
k
=
a
i take n
k
=
m
i, r
k
=
c
i. Then we construct the homogeneous Moran set relating to ω
∈
A
η
ℕ and denote it by E
(
ω
)
=
{
J
,
{
n
k
}
,
{
r
k
}
}.Remark 4.
In this paper, we assume thatJ
σ
∈
ℱ
k
(
k
≥
1
), let J
σ
*
1
,
J
σ
*
2
,
…
,
J
σ
*
n
k
+
1 be the n
k
+
1 basic intervals of order k
+
1 contained in J
σ arranged from left to right. For all 1
≤
j
≤
n
k
+
1
-
1, let d
(
J
σ
*
j
,
J
σ
*
(
j
+
1
)
)
≥
Δ
k
|
J
σ
|, where {
Δ
k
} is a sequence of positive real number. Let Δ
=
inf
⁡
k
≥
1
⁡
Δ
k. In this paper we suppose Δ
>
0.


## 1.2. Moran Measure

LetP
a
i
=
(
p
i
1
p
i
2
⋯
p
i
m
i
)
(
1
≤
i
≤
m
) be probability vectors; that is, p
i
j
>
0 and ∑
j
=
1
m
i
p
i
j
=
1
(
1
≤
i
≤
m
). For any k
≥
1, σ
∈
D
k, from Section 1.1, we know σ
=
σ
1
σ
2
⋯
σ
k
∈
D
k where σ
k
∈
{
1,2
,
…
,
m
i
}, if s
k
=
a
i. For σ
=
σ
1
σ
2
⋯
σ
k, define σ
(
a
i
) as follows: let ω
k
=
s
1
⋯
s
k, e
1
<
e
2
<
⋯
e
|
ω
k
|
a
i be the occurrences of the letter a
i in ω
k; then σ
(
a
i
)
=
σ
e
1
σ
e
2
⋯
σ
e
|
ω
k
|
a
i. For convenience we will write σ
(
a
i
)
=
σ
i
1
σ
i
2
⋯
σ
i
|
ω
k
|
a
i, where σ
i
j
∈
{
1,2
,
…
,
m
i
}
(
1
≤
i
≤
m
). In fact, σ
(
a
1
)
*
σ
(
a
2
)
*
⋯
*
σ
(
a
m
) is a rearrangement of σ
=
σ
1
⋯
σ
k. We make the convention that σ
(
a
i
)
=
ϕ if |
ω
k
|
a
i
=
0.Now define(8)
p
σ
(
a
i
)
=
p
i
σ
i
1
⋯
p
i
σ
i
|
ω
k
|
a
i
,
1
≤
i
≤
m
.
It is obvious that ∑
σ
∈
D
k
∏
i
=
1
m
p
σ
(
a
i
)
=
1 for any k
≥
1. We make the convention that p
σ
(
a
i
)
=
1 if σ
(
a
i
)
=
ϕ.Letμ be a mass distribution on E
(
ω
), such that for any J
σ
∈
ℱ
k, σ
∈
D
k,
(9)
μ
(
J
σ
)
=
p
σ
(
a
1
)
p
σ
(
a
2
)
⋯
p
σ
(
a
m
)
,
and μ
(
∑
σ
∈
D
k
J
σ
)
=
1. Since μ is related to ω, we denote it by μ
(
ω
). Here μ
(
ω
) is a homogeneous Moran measure on E
(
ω
), and it is an extension of the self-similar measure by Hutchinson [2].


## 1.3. Main Results

From now on, we assume thatE
(
ω
) is a homogeneous Moran fractal defined in Section 1.1, and μ
(
ω
) is a probability measure introduced in Section 1.2. The notationa D, D
k, J
σ, μ
(
J
σ
), P
σ
(
a
i
)
(
1
≤
i
≤
m
) are as above; in the following, if σ
∈
D
k, τ
∈
D
k
+
1
,
k
+
n, let r
σ
=
|
J
σ
|, p
σ
=
μ
(
J
σ
) and r
τ
=
|
J
σ
*
τ
|
/
|
J
σ
|, p
τ
=
μ
(
J
σ
*
τ
)
/
μ
(
J
σ
). Now we define an auxiliary function β
(
q
) as follows. For each q
∈
ℝ and k
≥
1, there is a unique number β
k
(
q
) such that
(10)
∑
σ
∈
D
k
‍
p
σ
q
r
σ
β
k
(
q
)
=
1
.
By simple calculation, we get
(11)
lim
⁡
k
→
∞
β
k
(
q
)
=
β
(
q
)
=
∑
i
=
1
m
η
i
log
⁡
(
∑
j
=
1
m
i
p
i
j
q
)
∑
i
=
1
m
η
i
log
⁡
c
i
.Proposition 5 (see [3, Proposition 2.3]).
For allq
∈
ℝ, β
(
q
) defined by (10) satisfies the following:(i)
β
(
0
)
=
dim
⁡
E
;
(ii)
β
(
q
)
is strictly decreasing, and
lim
⁡
q
→
∓
∞
⁡
β
(
q
)
=
±
∞
;
(iii)
β
(
q
)
is convex in
q
, and
β
(
q
)
is strictly convex if and only if
log
⁡
p
i
j
/
log
⁡
c
i
is not the same for all
1
≤
j
≤
m
i
,
i
=
1,2
,
…
,
m
.Letμ be a Borel probability measure on ℝ
d; let
(12)
Θ
(
q
;
r
)
=
sup
⁡
∑
i
‍
μ
(
B
(
x
i
,
r
)
)
q
,
r
>
0
,
q
∈
ℝ
,
where the supremum is taken over all families of disjoint closed balls {
B
(
x
i
,
r
)
}
i with x
i
∈
supp
⁡
μ. If lim
⁡
r
→
0
(
log
⁡
Θ
(
q
;
r
)
/
log
⁡
r
) exists, we call that the L
q
-spectrum τ
(
q
) of μ exists; that is,
(13)
τ
(
q
)
=
lim
⁡
r
→
0
log
⁡
Θ
(
q
;
r
)
-
log
⁡
r
.
Peres and Solomyak [4] give alternative definition of L
q
-spectrum. Let μ be a Borel probability measure on ℝ
d. Let D
n be the partition of ℝ
d into grid boxes ∏
i
=
1
d
‍
[
k
i
2
-
n
,
(
k
i
+
1
)
2
-
n
) with k
i
∈
ℤ. For q
>
0, denote τ
n
(
q
)
(
μ
)
=
∑
Q
∈
D
n
‍
(
μ
Q
)
q. If lim
⁡
n
→
0
(
log
⁡
τ
n
(
q
)
(
μ
)
/
-
n
log
⁡
2
) exists, we call that the L
q
-spectrum τ
0
(
q
) of μ exists; that is,
(14)
τ
0
(
q
)
=
lim
⁡
n
→
0
log
⁡
τ
n
(
q
)
(
μ
)
-
n
log
⁡
2
.
Peres and Solomyak [4] prove that τ
(
q
)
=
τ
0
(
q
).Proposition 6 (see [5]).
Letμ be a Moran measure supported on the homogeneous Moran fractals E; then
(15)
τ
(
q
)
=
τ
0
(
q
)
=
β
(
q
)
.The Legendre transform of β is the function β
*
:
(
α
min
⁡
,
α
max
⁡
)
→
ℝ defined by
(16)
β
*
(
α
)
=
inf
⁡
q
∈
ℝ
{
β
(
q
)
+
α
q
}
,
where α
min
⁡
=
inf
⁡
q
∈
ℝ
β
′
(
q
), α
max
⁡
=
sup
⁡
q
∈
ℝ
β
′
(
q
).LetX be a complete separable metric space and μ a finite Borel measure on X. In the multifractal analysis one is interested in the size of the following level sets:
(17)
X
α
=
{
x
∈
X
:
lim
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
α
}
,
X
α
=
Tl
=
x
∈
X
:
lim
⁡
r
→
0
-
∞
≤
α
≤
+
∞
.
The space X has the following natural decomposition:
(18)
X
=
⋃
-
∞
≤
α
≤
+
∞
‍
X
α
∪
X
o
,
where
(19)
X
o
=
{
x
∈
X
:
lim
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
does
not
exist
}
.
The set X
o is called the set of divergence points and the point x for which the limit lim
⁡
r
→
0
⁡
(
log
⁡
μ
(
B
(
x
,
r
)
)
/
log
⁡
r
) 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 K. 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 ℝ
d, 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 K 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.
Letμ be a Moran measure supported on the homogeneous Moran fractals E
(
Ω
) associated with the sequences Ω of which the frequency of the letter exists as above. Set
(20)
E
q
1
,
q
2
=
{
limsup
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
2
)
x
∈
E
(
Ω
)
aaa
∣
β
′
(
q
1
)
=
liminf
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
a
a
a
a
a
a
a
a
a
a
<
limsup
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
2
)
}
.
Then
(21)
dim
⁡
E
q
1
,
q
2
=
β
*
(
β
′
(
q
1
)
)
.By Theorem7 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 thatα
∈
ℝ is such that α
=
-
β
′
(
q
) for some q
∈
ℝ. Then for any δ
>
0, ρ
>
0, (I)
there existd
∈
(
0
,
ρ
), ℓ
≥
d
-
β
*
(
α
)
+
δ
(
q
+
1
), an integral number N, and u
1
,
…
,
u
ℓ
∈
D
N satisfying the following properties:
(a)
d
1
+
δ
≤
r
u
i
≤
d
1
-
δ for all 1
≤
i
≤
ℓ,
(b)
d
α
+
δ
≤
p
u
i
≤
d
α
-
δ for all 1
≤
i
≤
ℓ;
(II)
there exist an integral numbern
0 such that for any integer k
>
n
0, there exist d
∈
(
0
,
ρ
), ℓ
≥
d
-
β
*
(
α
)
+
δ
(
q
+
1
), an integral number N, and u
1
,
…
,
u
ℓ
∈
D
k
+
1
,
N
+
1 satisfying the following properties:
(a)
d
1
+
δ
≤
r
u
i
≤
d
1
-
δ for all 1
≤
i
≤
ℓ;
(b)
d
α
+
δ
≤
p
u
i
≤
d
α
-
δ for all 1
≤
i
≤
ℓ.Proof.
For the givenδ
>
0, we choose a small 0
<
ϵ
<
1 such that
(22)
(
α
-
δ
2
)
ϵ
≤
β
(
q
)
-
β
(
q
+
ϵ
)
≤
(
α
+
δ
2
)
ϵ
,
(23)
(
α
-
δ
2
)
ϵ
≤
β
(
q
-
ϵ
)
-
β
(
q
)
≤
(
α
+
δ
2
)
ϵ
.
Using (10), we can pick 0
<
γ
<
min
⁡
{
ϵ
δ
/
6,1
} and an integral number n
0 such that for any integer n
>
n
0,
(24)
r
1
r
2
⋯
r
n
<
min
⁡
{
ρ
,
3
-
1
/
γ
}
,
(25)
r
σ
-
β
(
q
)
+
γ
≤
∑
σ
∈
D
n
‍
p
σ
q
≤
r
σ
-
β
(
q
)
-
γ
,
(26)
r
σ
-
β
(
q
+
ϵ
)
+
γ
≤
∑
σ
∈
D
n
‍
p
σ
q
+
ϵ
≤
r
σ
-
β
(
q
+
ϵ
)
-
γ
.
Set
(27)
𝔹
1
=
{
σ
∈
D
n
:
p
σ
≥
r
σ
α
-
δ
}
,
𝔹
2
=
{
σ
∈
D
n
:
p
σ
≥
r
σ
α
-
δ
}
,
𝔹
3
=
{
σ
∈
D
n
:
r
σ
α
+
δ
<
p
σ
<
r
σ
α
-
δ
}
.
Then
(28)
∑
σ
∈
𝔹
1
‍
p
σ
q
=
∑
σ
∈
𝔹
1
‍
p
σ
q
+
ϵ
p
σ
-
ϵ
≤
r
σ
-
β
(
q
+
ϵ
)
-
γ
-
ϵ
(
α
-
δ
)
≤
r
σ
-
β
(
q
)
+
ϵ
δ
/
2
-
γ
.
Similarly, we have
(29)
∑
σ
∈
𝔹
2
‍
p
σ
q
=
∑
σ
∈
𝔹
2
‍
p
σ
q
-
ϵ
p
σ
ϵ
≤
r
σ
-
β
(
q
-
ϵ
)
-
γ
+
ϵ
(
α
+
δ
)
≤
r
σ
-
β
(
q
)
+
ϵ
δ
/
2
-
γ
.
These two inequalities together with (22)–(26) imply
(30)
∑
σ
∈
𝔹
3
‍
p
σ
q
=
(
∑
σ
∈
D
n
‍
-
∑
σ
∈
𝔹
1
‍
-
∑
σ
∈
𝔹
2
‍
)
p
σ
q
≥
r
σ
-
β
(
q
)
+
γ
-
2
r
σ
-
β
(
q
)
+
ϵ
δ
/
2
-
γ
=
r
σ
-
β
(
q
)
+
2
γ
(
r
σ
-
γ
-
2
r
σ
(
ϵ
δ
/
2
)
-
3
γ
)
≥
r
σ
-
β
(
q
)
+
2
γ
(
r
σ
-
γ
-
2
)
(
by
0
<
γ
<
min
⁡
{
ϵ
δ
6
,
1
}
)
≥
r
σ
-
β
(
q
)
+
2
γ
(
by
(
24
)
)
.
Note that for each σ
∈
𝔹
3, p
σ
≤
max
⁡
{
r
σ
(
α
±
δ
)
q
}
=
r
σ
α
q
-
|
δ
|
q
=
(
r
1
r
2
⋯
r
n
)
α
q
-
|
δ
|
q. Hence,
(31)
∑
σ
∈
𝔹
3
‍
p
σ
q
≤
(
#
𝔹
3
)
(
r
1
r
2
⋯
r
n
)
α
q
-
|
δ
|
q
which combining with (30) yields
(32)
#
𝔹
3
≥
(
r
1
r
2
⋯
r
n
)
-
β
(
q
)
-
α
q
+
δ
|
q
|
+
2
γ
=
(
r
1
r
2
⋯
r
n
)
-
β
*
(
α
)
+
δ
|
q
|
+
2
γ
≥
(
r
1
r
2
⋯
r
n
)
-
β
*
(
α
)
+
δ
(
|
q
|
+
1
)
.
We suppose N is an integral number satisfying N
>
n
0 and d
=
r
1
r
2
⋯
r
N. This completes the proof of (I).
Next we prove (II). Note that∑
σ
∈
D
k
‍
p
σ
q
r
σ
β
k
(
q
)
=
1; we get
(33)
β
k
(
q
)
=
log
⁡
∑
σ
∈
D
k
p
σ
q
-
log
⁡
r
σ
=
∑
i
=
1
m
η
i
log
⁡
∑
j
=
1
m
i
p
i
j
q
+
∑
i
=
1
m
(
(
|
ω
k
|
α
i
/
k
)
-
η
i
)
log
⁡
∑
j
=
1
m
i
p
i
j
q
-
(
∑
i
=
1
m
η
i
log
⁡
c
i
+
∑
i
=
1
m
(
(
|
ω
k
|
α
i
/
k
)
-
η
i
)
log
⁡
c
i
)
,
log
⁡
∑
σ
∈
D
k
+
1
,
k
+
n
p
σ
q
-
log
⁡
r
σ
=
(
∑
i
=
1
m
‍
η
i
log
⁡
∑
j
=
1
m
1
‍
p
i
j
q
aaa
+
∑
i
=
1
m
(
(
(
k
+
n
)
(
(
|
ω
k
+
n
|
α
i
/
(
k
+
n
)
)
-
η
i
)
‍
a
a
a
a
a
a
a
a
a
a
-
k
(
(
|
ω
k
|
α
i
/
k
)
-
η
i
)
)
×
(
n
)
-
1
)
a
a
a
×
log
⁡
∑
j
=
1
m
i
‍
p
i
j
q
)
a
a
×
(
-
(
∑
i
=
1
m
‍
η
i
log
⁡
c
i
a
a
a
a
a
a
a
a
+
∑
i
=
1
m
(
(
(
k
+
n
)
(
(
|
ω
k
+
n
|
α
i
/
(
k
+
n
)
)
-
η
i
)
‍
a
a
a
a
a
a
a
a
a
a
a
a
a
a
-
k
(
(
|
ω
k
|
α
i
/
k
)
-
η
i
)
)
×
(
n
)
-
1
)
a
a
a
a
a
a
a
a
×
log
⁡
c
i
∑
i
=
1
m
‍
)
∑
i
=
1
m
‍
η
i
log
⁡
c
i
)
-
1
.
Note that lim
⁡
n
→
∞
β
n
(
q
)
=
β
(
q
) and lim
⁡
k
→
∞
⁡
(
|
ω
k
|
a
i
/
k
)
=
η
i for 1
≤
i
≤
m and using (33), we can choose an integral N
1 such that for any integral k
,
n
>
N
1, the following properties are satisfied; that is,
(34)
β
(
q
)
-
γ
≤
∑
i
=
1
m
η
i
log
⁡
∑
j
=
1
m
i
p
i
j
q
-
∑
i
=
1
m
η
i
log
⁡
c
i
≤
β
(
q
)
+
γ
,
∑
i
=
1
m
η
i
log
⁡
∑
j
=
1
m
i
p
i
j
q
-
∑
i
=
1
m
η
i
log
⁡
c
i
-
γ
≤
log
⁡
∑
σ
∈
D
k
+
1
,
k
+
n
p
σ
q
-
log
⁡
r
σ
≤
∑
i
=
1
m
η
i
log
⁡
∑
j
=
1
m
i
p
i
j
q
-
∑
i
=
1
m
η
i
log
⁡
c
i
+
γ
.
Combining (34) we get
(35)
β
(
q
)
-
2
γ
≤
log
⁡
∑
σ
∈
D
k
+
1
,
k
+
n
p
σ
q
-
log
⁡
r
σ
≤
β
(
q
)
+
2
γ
.
Therefore
(36)
r
σ
-
(
β
(
q
)
-
2
γ
)
≤
∑
σ
∈
D
k
+
1
,
k
+
n
‍
p
σ
q
<
r
σ
-
(
β
(
q
)
+
2
γ
)
.
Using the same method, we can choose an integral N
2 such that for any integral k
,
n
>
N
2,
(37)
r
σ
-
(
β
(
q
+
ϵ
)
-
2
γ
)
≤
∑
σ
∈
D
k
+
1
,
k
+
n
‍
p
σ
q
+
ϵ
≤
r
σ
-
(
β
(
q
+
ϵ
)
+
2
γ
)
.
Set n
0
=
max
⁡
{
N
1
,
N
2
}, and (36) and (37) satisfy simultaneously. We suppose N is an integral number satisfying N
>
n
0 and
(38)
(
∏
i
=
1
m
‍
(
c
i
η
i
)
)
N
<
ρ
,
∏
i
=
1
m
‍
(
c
i
η
i
)
δ
≤
∏
i
=
1
m
‍
c
i
(
(
k
+
N
)
(
(
|
ω
k
+
N
|
α
i
/
(
k
+
N
)
)
-
η
i
)
-
k
(
(
|
ω
k
|
α
i
/
k
)
-
η
i
)
)
/
N
≤
∏
i
=
1
m
‍
(
c
i
η
i
)
-
δ
.
Set d
=
(
∏
i
=
1
m
‍
c
i
η
i
)
N. Replace (25) and (26) with (36) and (37); we can prove (II) by the same method with (I).Lemma 9.
Letμ be a Moran measure supported on the homogeneous Moran fractals E
(
Ω
) associated with the sequences Ω of which the frequency of the letter exists as above. Set
(39)
E
q
1
,
q
2
=
{
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
2
)
x
∈
E
(
Ω
)
aaa
∣
β
′
(
q
1
)
=
liminf
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
a
a
a
<
limsup
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
2
)
}
.
Then
(40)
dim
⁡
E
q
1
,
q
2
≤
β
*
(
β
′
(
q
1
)
)
.Proof.
Letf
=
dim
⁡
E
q
1
,
q
2, then for any 0
<
ε
<
f, H
f
-
ε
(
E
q
1
,
q
2
)
=
∞; Using (39), there exist E
q
1
,
q
2
∘
⊂
E
q
1
,
q
2 and a number sequence {
r
i
}
↑
0 such that
(41)
H
f
-
ε
(
E
q
1
,
q
2
∘
)
>
1
,
(42)
μ
(
B
(
x
,
r
i
)
)
≥
3
1
/
q
1
r
i
β
′
(
q
1
)
+
ε
,
for any x
∈
E
q
1
,
q
2
∘. We can choose 0
<
δ
≤
(
1
/
2
)
r
0 such that H
δ
f
-
ε
(
E
q
1
,
q
2
∘
)
≥
1. For any r
i
≤
δ and 2
-
(
n
i
+
1
)
≤
r
i
<
2
-
n
i, we consider grid boxes [
k
2
-
n
i
,
(
k
+
1
)
2
-
n
i
); there exist 3 adjacent grid boxes A
,
A
1
,
A
2 such that B
(
x
,
r
i
)
⊂
A
∪
A
1
∪
A
2, and there x
∈
A and A
1
,
A
2 are neighbours with A. Therefore there exists A
0
∈
{
A
,
A
1
,
A
2
} such that μ
(
A
0
)
≥
r
i
β
′
(
q
1
)
+
ε
≥
2
-
n
i
(
β
′
(
q
1
)
+
ε
) (by β
′
(
q
1
)
<
0). On the other hand, notice that #
{
C
∈
D
n
i
∣
C
∩
E
q
1
,
q
2
∘
≠
⌀
}
≥
2
-
n
i
(
ε
-
f
)
H
δ
f
-
ε
(
E
q
1
,
q
2
∘
)
≥
2
-
n
i
(
ε
-
f
) (by  (41)). Thus
(43)
#
{
C
∈
D
k
∣
μ
(
A
0
)
≥
2
-
n
i
(
β
′
(
q
1
)
+
ε
)
}
≥
1
3
2
-
n
i
(
ε
-
f
)
.
Thus
(44)
τ
n
i
(
q
1
)
(
μ
)
=
∑
Q
∈
D
n
i
‍
(
μ
Q
)
q
1
≥
2
-
n
i
q
1
(
β
′
(
q
1
)
+
ε
)
2
-
n
i
(
ε
-
f
)
.
Therefore
(45)
lim
⁡
i
→
∞
log
⁡
τ
n
i
(
q
1
)
(
μ
)
-
log
⁡
2
n
i
≥
-
q
1
(
β
′
(
q
1
)
+
ε
)
+
f
-
ε
.
Using (14) and (15), notice that lim
⁡
n
→
∞
(
log
⁡
τ
n
(
q
1
)
(
μ
)
/
-
log
⁡
2
n
)
=
β
(
q
1
); we have
(46)
f
≤
β
(
q
1
)
+
q
1
(
β
′
(
q
1
)
+
ε
)
+
ε
.
Thus
(47)
dim
⁡
E
q
1
,
q
2
≤
β
*
(
β
′
(
q
1
)
)
.


## 3. Proof of Theorem7

Lemma 10.
SupposeΔ
>
0 (Δ is from Remark 4), for all r
>
0, and x
∈
J
σ
∈
F, and choose k
,
l
∈
ℕ such that
(48)
|
J
σ
∣
k
+
1
(
x
)
|
≤
r
<
|
J
σ
∣
k
(
x
)
|
,
Δ
|
J
σ
∣
l
+
1
(
x
)
|
≤
r
<
Δ
|
J
σ
∣
l
(
x
)
|
,
where J
σ
∣
l
+
1
(
x
) denote the basic elements of order l
+
1 that contains the point x. Then
(49)
(
i
)
J
σ
|
k
+
1
(
x
)
⊆
B
(
x
,
r
)
,
E
∩
B
(
x
,
r
)
⊆
J
σ
|
l
+
1
(
x
)
,
(
i
i
)
∃
Δ
0
>
0
s
u
c
h
t
h
a
t
k
-
l
≤
Δ
0
.Proof.
It is obvious thatJ
σ
|
k
+
1
(
x
)
⊆
B
(
x
,
r
). Now let y
∈
E
∩
B
(
x
,
r
), but y
∈
-
J
σ
|
l
+
1
(
x
); then there exists j
<
l
+
1 such that τ
|
j
=
σ
|
j, and τ
|
j
+
1
≠
σ
|
j
+
1, and x
∈
J
σ
∣
j
+
1
(
x
), y
∈
J
τ
∣
j
+
1
(
y
). Therefore,
(50)
|
y
-
x
|
≥
d
(
J
τ
∣
j
+
1
(
x
)
,
J
σ
∣
j
+
1
(
y
)
)
|
y
-
x
|
≥
Δ
j
|
J
σ
∣
j
(
x
)
|
≥
Δ
|
J
σ
∣
l
(
x
)
|
>
r
,
which is a contradiction since y
∈
E
∩
B
(
x
,
r
); therefore E
∩
B
(
x
,
r
)
⊆
J
σ
|
l
+
1
(
x
).
SinceΔ
<
1 and k
≥
l. (48) imply that
(51)
1
≤
|
J
σ
∣
k
(
x
)
|
Δ
|
J
σ
∣
l
+
1
(
x
)
|
≤
(
max
⁡
{
c
i
}
)
k
-
l
-
1
Δ
,
which yields
(52)
k
-
l
≤
1
+
log
⁡
Δ
logmax
{
c
i
}
≜
Δ
0
.Proof of Theorem7.
For anyq
2
<
q
1, define a number sequence {
α
i
}
i
=
1
∞ in the following manner:
(53)
q
i
=
{
q
1
,
i
is
odd
number
;
q
2
,
i
is
even
number
,
(54)
α
i
=
{
β
′
(
q
1
)
,
i
is
odd
number
;
β
′
(
q
2
)
,
i
is
even
number
.
We choose a positive sequence {
δ
i
}
i
=
1
∞
↓
0. For {
α
i
} and {
δ
i
}, there exist an integral number sequence {
n
0
(
i
)
}
i
=
1
∞ (n
0
(
i
) corresponds to n
0 in Lemma 8) and a real number {
d
i
}
i
=
1
∞
(
1
>
d
1
>
d
2
>
⋯
) such that the follow properties are satisfied.
Forα
1
=
β
′
(
q
1
) and δ
1, using Lemma 8 we can pick u
11
(
1
)
,
u
12
(
1
)
,
…
,
u
1
ℓ
(
1
)
(
1
)
∈
D
N
(
1
)
,
u
21
(
1
)
,
u
22
(
1
)
,
…
,
u
2
,
ℓ
(
1
)
(
1
)
∈
D
N
(
1
)
+
1,2
N
(
1
)
,
…
,
u
M
1
1
(
1
)
,
u
M
1
2
(
1
)
…
,
u
M
1
ℓ
(
1
)
(
1
)
∈
D
(
M
1
-
1
)
N
(
1
)
+
1
,
M
1
N
(
1
) satisfying (
d
1
)
α
1
+
δ
1
≤
p
u
j
l
(
1
)
≤
(
d
1
)
α
1
-
δ
1, (
d
1
)
1
+
δ
1
≤
r
u
j
l
(
1
)
≤
(
d
1
)
1
-
δ
1 for all 1
≤
j
≤
M
1
;
1
≤
l
≤
ℓ
(
1
) and ℓ
(
1
)
≥
(
d
1
)
-
β
*
(
α
1
)
+
δ
1
(
q
+
1
).
For1
≤
j
≤
M
1, remark B
j
(
1
)
=
{
u
j
1
(
1
)
,
u
j
2
(
1
)
,
…
,
u
j
ℓ
(
1
)
(
1
)
}.
By the same step, forα
i and δ
i, we also construct B
j
(
i
)
=
{
u
j
1
(
i
)
,
u
j
2
(
i
)
…
,
u
j
ℓ
(
i
)
(
i
)
}
(
1
≤
j
≤
M
i
) such that (a)
ℓ
(
i
)
≥
(
d
i
)
-
β
*
(
α
i
)
+
δ
i
(
q
+
1
);
(b)
u
j
1
(
i
)
,
u
j
2
(
i
)
,
…
,
u
j
,
ℓ
(
i
)
(
i
)
∈
D
∑
k
=
1
i
-
1
‍
M
i
N
(
k
)
+
(
j
-
1
)
N
(
i
)
+
1
,
∑
k
=
1
i
-
1
‍
M
i
N
(
k
)
+
j
N
(
i
);
(c)
d
i
1
+
δ
i
≤
r
u
j
l
(
i
)
≤
d
i
1
-
δ
i for 1
≤
j
≤
M
i, 1
≤
l
≤
ℓ
(
i
);
(d)
d
i
α
i
+
δ
i
≤
p
u
j
l
(
i
)
≤
d
i
α
i
-
δ
i for 1
≤
j
≤
M
i, 1
≤
l
≤
ℓ
(
i
).
Also we let{
M
i
}
i
=
1
∞ be a sequence of integers large enough such that
(e)
M
i
N
(
i
)
>
n
0
(
i
);
(f)
d
i
M
i
<
(
d
i
+
1
)
2
i for each i
∈
ℕ;
(g)
lim
⁡
k
→
∞
(
∑
i
=
1
k
‍
M
i
log
⁡
d
i
/
M
k
+
1
log
⁡
d
k
+
1
)
=
0.
Now we define a sequence of subsets of symbol setΣ
* in the following manner:
(55)
B
1
(
1
)
,
…
,
B
M
1
(
1
)
,
B
1
(
2
)
,
…
,
B
M
2
(
2
)
,
…
,
B
1
(
i
)
,
…
,
B
M
i
(
i
)
,
…
and relabel them as {
B
n
*
}
n
=
1
∞. Let
(56)
F
q
1
,
q
2
=
⋂
n
=
1
∞
‍
⋃
v
1
∈
B
1
*
,
…
,
v
n
∈
B
n
*
‍
J
v
1
v
2
⋯
v
n
.
It is easy to check that F
q
1
,
q
2 is the homogeneousMoran set which is a subset of the homogeneousMoran set
E
(
Ω
).
Next we show that(57)
liminf
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
1
)
,
limsup
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
2
)
,
∀
x
∈
F
q
1
,
q
2
,
dim
⁡
H
F
q
1
,
q
2
≥
β
*
(
β
′
(
q
1
)
)
.
Letx
∈
F
q
1
,
q
2, then there exist v
i
∈
B
i
*
(
i
=
1,2
,
…
) such that
(58)
{
x
}
=
lim
⁡
n
→
∞
J
v
1
⋯
v
n
.
Let n
k
=
M
1
+
M
2
+
⋯
+
M
2
k
-
1; there exists r
n
k such that
(59)
|
J
v
1
⋯
v
n
k
+
1
|
≤
r
n
k
<
|
J
v
1
⋯
v
n
k
|
.
Using Lemma 10, we attain
(60)
log
⁡
p
v
1
⋯
v
n
k
+
Δ
0
log
⁡
min
⁡
1
≤
i
≤
m
,
1
≤
j
≤
m
i
⁡
{
p
i
,
j
}
log
⁡
r
v
1
⋯
v
n
k
+
1
≤
log
⁡
μ
(
B
(
x
,
r
n
k
)
)
log
⁡
r
n
k
≤
log
⁡
p
v
1
⋯
v
n
k
+
1
log
⁡
r
v
1
⋯
v
n
k
.
Thus to calculate lim
⁡
k
→
∞
(
log
⁡
μ
(
B
(
x
,
r
n
k
)
)
/
log
⁡
r
n
k
), we need to estimate p
v
1
⋯
v
M and r
v
1
⋯
v
M for M
=
n
k
,
n
k
+
1. By (c) and (d), we obtain
(61)
∏
i
=
1
2
k
-
1
‍
d
i
M
i
(
α
i
+
δ
i
)
d
2
k
M
2
k
(
α
2
k
+
δ
2
k
)
≤
p
v
1
⋯
v
n
k
+
1
≤
p
v
1
⋯
v
n
k
≤
∏
i
=
1
2
k
-
1
‍
d
i
M
i
(
α
i
-
δ
i
)
d
2
k
(
M
2
k
-
1
)
(
α
2
k
-
δ
2
k
)
,
(62)
∏
i
=
1
2
k
-
1
‍
d
i
M
i
(
1
+
δ
i
)
d
2
k
M
2
k
(
1
+
δ
2
k
)
≤
r
v
1
⋯
v
n
k
+
1
≤
r
v
1
⋯
v
n
k
≤
∏
i
=
1
2
k
-
1
‍
d
i
M
i
(
1
-
δ
i
)
d
2
k
(
M
2
k
-
1
)
(
1
-
δ
2
k
)
.
By (54), (60)–(62), and (g), we obtain
(63)
lim
⁡
k
→
∞
log
⁡
μ
(
B
(
x
,
r
n
k
)
)
log
⁡
r
n
k
=
β
′
(
q
2
)
.
Let n
′
k
=
M
1
+
M
2
+
⋯
+
M
2
k
+
1
-
1; there exists r
n
k
′ such that
(64)
|
J
v
1
⋯
v
n
k
′
+
1
|
≤
r
n
k
′
<
|
J
v
1
⋯
v
n
k
′
|
.
By the same method, we obtain
(65)
lim
⁡
k
→
∞
log
⁡
μ
(
B
(
x
,
r
n
k
′
)
)
log
⁡
r
n
k
′
=
β
′
(
q
1
)
.
Next we showβ
′
(
q
1
)
≤
liminf
⁡
r
→
0
(
log
⁡
μ
(
B
(
x
,
r
)
)
/
log
⁡
⁡
(
r
)
)
<
limsup
⁡
r
→
0
(
log
⁡
μ
(
B
(
x
,
r
)
)
/
log
⁡
(
r
)
)
≤
β
′
(
q
2
). For r
>
0 small enough, there is a unique large integer n such that
(66)
|
J
v
1
⋯
v
n
+
1
|
≤
r
<
|
J
v
1
⋯
v
n
|
.
Using the same method as above, we attain
(67)
log
⁡
p
v
1
⋯
v
n
+
Δ
0
log
⁡
min
⁡
1
≤
i
≤
m
,
1
≤
j
≤
m
i
⁡
{
p
i
,
j
}
log
⁡
r
v
1
⋯
v
n
+
1
≤
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
≤
log
⁡
p
v
1
⋯
v
n
+
1
log
⁡
r
v
1
⋯
v
n
.
Now we estimate p
v
1
⋯
v
M and r
v
1
⋯
v
M for M
=
n, n
+
1. For large n, write n
=
∑
i
=
1
k
‍
M
i
+
p with 1
≤
p
≤
M
k
+
1. In the case that 1
≤
p
≤
M
k
+
1
-
1, by (c) and (d), we obtain
(68)
∏
i
=
1
k
‍
d
i
M
i
(
α
i
+
3
δ
i
)
d
k
+
1
(
p
+
1
)
(
α
k
+
1
+
δ
k
+
1
)
≤
p
v
1
⋯
v
n
+
1
≤
p
v
1
⋯
v
n
≤
∏
i
=
1
k
‍
d
i
M
i
(
α
i
-
δ
i
)
d
k
+
1
p
(
α
k
+
1
-
δ
k
+
1
)
,
(69)
∏
i
=
1
k
‍
d
i
M
i
(
1
+
δ
i
)
d
k
+
1
(
p
+
1
)
(
1
+
δ
k
+
1
)
≤
r
v
1
⋯
v
n
+
1
≤
r
v
1
⋯
v
n
≤
∏
i
=
1
k
‍
d
i
M
i
(
1
-
δ
i
)
d
k
+
1
p
(
1
-
δ
k
+
1
)
.
In the other case p
=
M
k
+
1, we have the similar inequalities where the lower bounds for p
v
1
⋯
v
n and r
v
1
⋯
v
n in (68) and (69) are replaced, respectively, by
(70)
(
∏
i
=
1
k
+
1
‍
d
i
M
i
(
α
i
+
δ
i
)
)
d
k
+
2
(
α
k
+
2
+
δ
k
+
2
)
≤
p
v
1
⋯
v
n
+
1
<
p
v
1
⋯
v
n
≤
∏
i
=
1
k
+
1
‍
d
i
M
i
(
α
i
+
δ
i
)
,
(71)
(
∏
i
=
1
k
+
1
‍
d
i
M
i
(
1
+
δ
i
)
)
d
k
+
2
(
1
+
δ
k
+
2
)
≤
r
v
1
⋯
v
n
+
1
<
r
v
1
⋯
v
n
≤
∏
i
=
1
k
+
1
‍
d
i
M
i
(
1
+
δ
i
)
.
By (54), using the inequalities (67)–(71), (f), and (g), we obtain
(72)
β
′
(
q
1
)
≤
liminf
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
<
limsup
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
≤
β
′
(
q
2
)
.
Combining (63), (65), and (72), we obtain
(73)
liminf
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
1
)
,
limsup
⁡
r
→
0
log
⁡
μ
(
B
(
x
,
r
)
)
log
⁡
r
=
β
′
(
q
2
)
,
∀
x
∈
F
q
1
,
q
2
.
To provedim
⁡
H
F
q
1
,
q
2
≥
β
*
(
q
1
), recall that F
q
1
,
q
2 is the homogeneousMoran set which is a subset of the homogeneousMoran set
E
(
Ω
). For large n, write n
=
∑
i
=
1
k
‍
M
i
+
p with 1
≤
p
≤
M
k
+
1. By (a) and (c), we have
(74)
∏
s
=
1
n
‍
#
B
s
*
≥
∏
i
=
1
k
‍
d
i
M
i
(
-
β
*
(
α
i
)
+
δ
i
(
q
i
+
1
)
)
d
k
+
1
p
(
-
β
*
(
α
k
+
1
)
+
δ
k
+
1
(
q
k
+
1
+
1
)
)
and for any v
1
∈
B
1
*
,
…
,
v
n
∈
B
n
*
(75)
r
v
n
≥
d
k
+
1
1
+
δ
k
+
1
,
r
v
1
⋯
v
n
≤
∑
i
=
1
k
‍
d
i
M
i
(
1
-
δ
i
)
d
k
+
1
p
(
1
-
δ
k
+
1
)
.
Using (75) and (f), we have
(76)
lim
⁡
n
→
∞
log
⁡
r
v
n
log
⁡
r
v
1
⋯
v
n
=
0
.
This implies the condition (5) in Proposition 3; we have dim
⁡
H
F
q
1
,
q
2
=
liminf
⁡
n
→
∞
s
n, where s
n satisfies the equation
(77)
∑
v
1
∈
B
1
*
⋯
v
n
∈
B
n
*
‍
(
r
v
1
⋯
v
n
)
s
n
=
1
.
It follows that
(78)
dim
⁡
H
F
q
1
,
q
2
≥
liminf
⁡
n
→
∞
log
⁡
(
∏
s
=
1
n
#
B
s
*
)
-
log
⁡
r
v
1
⋯
v
n
.
This, together with (74)-(75), yields
(79)
dim
⁡
H
F
q
1
,
q
2
≥
liminf
⁡
k
→
∞
(
(
∑
i
=
1
k
‍
M
i
(
β
*
(
α
i
)
-
δ
i
(
q
i
+
1
)
)
log
⁡
d
i
a
a
a
a
a
a
a
a
a
a
+
p
(
β
*
(
α
k
+
1
)
-
δ
k
+
1
(
q
k
+
1
+
1
)
)
log
⁡
d
k
+
1
∑
i
=
1
k
‍
)
a
a
a
a
a
a
a
a
×
(
∑
i
=
1
k
‍
M
i
(
1
-
δ
i
)
log
⁡
d
i
+
p
(
1
-
δ
k
+
1
)
log
⁡
d
k
+
1
)
-
1
)
≥
liminf
⁡
k
→
∞
(
(
∑
i
=
1
k
‍
M
i
(
β
*
(
β
′
(
q
1
)
)
-
δ
i
(
q
i
+
1
)
)
log
⁡
d
i
a
a
a
a
a
a
a
a
a
a
+
p
(
β
*
(
β
′
(
q
1
)
)
-
δ
k
+
1
(
q
k
+
1
+
1
)
)
log
⁡
d
k
+
1
∑
i
=
1
k
‍
)
a
a
a
a
a
a
a
a
a
×
(
∑
i
=
1
k
‍
M
i
(
1
-
δ
i
)
log
⁡
d
i
+
p
(
1
-
δ
k
+
1
)
log
⁡
d
k
+
1
)
-
1
)
≥
β
*
(
β
′
(
q
1
)
)
(
by
(
54
)
and
β
*
(
β
′
(
q
1
)
)
<
β
*
(
β
′
(
q
2
)
)
since
β
*
is
concave
(
see
[
3
]
)
β
*
(
β
′
(
q
1
)
)
<
β
*
(
β
′
(
q
2
)
)
)
.
Using Lemma9, we finish proof of Theorem 7.



---

*Source: 161756-2014-04-07.xml*