# Assouad Dimensions and Lower Dimensions of Some Moran Sets

Authors: JiaQing Xiao
Journal: Journal of Mathematics
 (2020)

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## 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 ratioc∗>0.


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## Body

## 1. Introduction

Let us begin with the definition of the Assouad dimension and the lower dimension. Forr>0,E⊆Rd, and NrE 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 setF⊆Rd is defined by dimAF=infα≥0. There exists a constant c>0 such that, for any 0<r<R, and x∈F, NrBx,R∩F≤cR/rα.
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 bydimLF=infα≥0. There exists a constant c>0 such that, for any 0<r<R≤F, and x∈F, NrBx,R∩F≥cR/rα.
The lower dimension is not monotonic, and the modified lower dimension is defined by(1)dimMLF=supdimLE:E⊆F.
The 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 c∗>0 and the paper [7] conjecture that the conclusion remains true if the condition c∗>0 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 c∗>0, 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 nkk≥1 be a sequence of positive integers. Define D0=∅, and for any k≥1, set Dm,k=im,im+1,…,ik;1≤ij≤nj,m≤j≤k, Dk=D1,k, and D=∪k≥0Dk. If σ=σ1,σ2,…,σk∈Dk,τ=τ1,…,τm−k∈Dk+1,m, let σ∗τ=σ1,…,σk,τ1,…,τm−k. And if σ∈Dk, remark σl=σ1,…,σl for 1≤l≤k.Definition 3.
Suppose thatJ⊂Rd is a compact set with intJ¯=J. Let ϕk be a sequence of positive real vectors with ϕk=ck,1,ck,2,…,ck,nk and ∑j=1nkck,j≤1,k∈N. We say the collection F=Jσ;σ∈D of closed subsets of J fulfills the Moran structure if it satisfies the following Moran structure conditions (MSC):(1)
Forσ∈D,Jσ is geometrically similar to J, i.e., there exists a similarity Sσ:Rd⟶Rd such that Jσ=SσJ. For convenience, we write J∅=J(2)
For allk≥0 and σ∈Dk,Jσ∗1,Jσ∗2,…,Jσ∗nk+1 are subsets of Jσ and satisfy that intJσ∗i∩intjσ∗j=∅i≠j(3)
For anyk≥1 and σ∈Dk−1,1≤j≤nk,(2)Jσ∗jJσ=ck,j,where A denotes the diameter of A.
Suppose thatF is a collection of closed subsets of J fulfilling the Moran structure, set(3)Ek=∪σ∈DkJσ,E=∩k≥0Ek.
It is ready to see thatE is a nonempty compact set. The set E=EF is called the Moran set associated with the collection F.
LetFk=Jσ;σ∈Dk and F=∪k≥0Fk. The elements of Fk are called kth-level basic sets of E and the elements of F are called the basic sets of E. Suppose that the set J and the sequences nk and ϕk are given. We denote by M=MJ,nk,ϕk the class of the Moran sets satisfying the MSC. We call MJ,nk,ϕk the Moran class associated with the triplet J,nk,ϕk.Remark 1.
From the above definition, we see that if the Moran setsE1,E2∈MJ,nk,ϕk and E1≠E2, then the relative positions of kth-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 M=MJ,nk,ϕk be a Moran class. Let c∗:infci,jandcσ=c1,σ1,…,ck,σkforσ=σ1,…,σk∈Dk. Let(4)s∗=liminfk⟶∞sk,s∗=limsupk⟶∞sk,where sk satisfies the following equation:(5)∏i=1k∑j=1nici,jsk=∑σ∈Dkcσsk=1.
Set(6)h¯=liminfk⟶∞∑σ∈DkJσs∗,h¯=limsupk⟶∞∑σ∈DkJσs∗.
We can now present the main result of Hua et al. [8].Theorem 1.
(see [8]). Suppose that M=MJ,nk,ϕk is a Moran class satisfying c∗>0. Then, for any E∈M,(7)dimHE=s∗,dimPE=dim¯BE=s∗.Li [7] computed the Assouad dimension of a fairly general (and important) class of Moran fractals.Theorem 2.
(see [7]). Suppose that M=MJ,nk,ϕk is a Moran class satisfying c∗>0. Then, for any E∈M,(8)dimPE=dim¯BE=dimAE=s∗.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 thatM=MJ,nk,ϕk is a Moran class satisfying c∗>0 and 0<h¯≤h¯<∞.Then, for any E∈M,(9)dimLE=dimMLE=dimHE=s∗.Lemma 1.
There exists a probability measurev supported by the Moran E such that(10)vJσ0=Jσ0s∗∑σ∈DkJσs∗,for any k≥1 and σ0∈Dk.Proof.
Take a sequence of probability measuresvmm≥1 supported by E such that(11)vmJσ0=Jσ0s∗∑σ∈DmJσs∗,for any σ0∈Dm.
More precisely, we can constructvm as follows.
First, we distribute the unit mass among themth-level basic elements according to (11). Inductively, suppose that we have already distributed the mass of proportion vmJσ to a kth-level basic set Jσσ∈Dk,k≥m; then, we distribute the mass concentrated on Jσ evenly to each of its k+1th-level basic subsets, i.e.,(12)vmJσ∗j=ck+1,js∗∑l=1nk+1ck+1,ls∗vmJσ,for 1≤j≤nk+1.
Repeating the above procedure, we get the desired measure.
Now, fix somem≥1; for any k<m and σ0∈Dk, we obtain(13)vmJσ0=∑τ∈Dk+1,mvmJσ0∗τ.
Combining it with (11), we have(14)∑σ∈DmJσs∗vmJσ0=∑τ∈Dk+1,mJσ0∗τs∗.
For anyσ1∈Dk, by the definitions of E,(15)Jσ1∗τJσ1=Jσ0∗τJσ0,and thus by (14),(16)Jσ1s∗∑σ∈DmJσs∗vmJσ0=Jσ0s∗∑τ∈Dk+1,mJσ1∗τs∗.
This gives(17)∑σ1∈DkJσ1s∗∑σ∈DmJσs∗vmJσ0=Jσ0s∗∑σ1∈Dk,τ∈Dk+1,mJσ1∗τs∗.
Observing that(18)∑σ∈DmJσs∗=∑σ1∈Dk,τ∈Dk+1,mJσ1∗τs∗,one obtains(19)vmJσ0=Jσ0s∗∑σ∈DkJσs∗.
To summarize, we obtain a sequence of probability measuresvmm≥1 supported by E and satisfy (10) for any k≤m and σ0∈Dk.
Now, Hellys theorem [9] enables us to extract a subsequence vmnn≥1 converging weakly to a limit measure v.
To verify thatv fulfills the desired requirements, we fix some k≥0 and σ0∈Dk. Then, by the properties of the weak convergence,(20)lim¯n⟶∞vmnJσ0≤vJσ0.
Combining with (19), this implies(21)vJσ0≥Jσ0s∗∑σ∈DkJσs∗.
On the other hand, take anε>0 small enough so that the ε-neighborhood Jε of Jσ0 is separated from the other mth-level basic set; then, vmnJε=vmnJσ0. By the properties of weak convergence, the following holds:(22)lim¯n⟶∞vmnJε≥vJε≥vJσ0.
Combining with (19) yields(23)vJσ0≤Jσ0s∗∑σ∈DkJσs∗.
We have for anyk≥1 and σ0∈Dk,(24)vJσ0=Jσ0s∗∑σ∈DkJσs∗.
Finally, for anyx which is not in E, since E is a closed set, there exists an open set U containing x and separated from E, and thus, vU≤lim¯n⟶∞vmnU=0, which asserts that v is supported by E.Lemma 2.
(see [1, 2]). If F⊂Rd is closed, then(25)dimLE≤dimMLE≤dimHE≤dimPE≤dim¯BE≤dimLE.Forσ∈D, we denote by σ- the word obtained by deleting the last letter of σ. For γ>0, we define Γγ by Γγ=σ∈Dcσ<γ≤cσ−.Lemma 3.
(see [7], Lemma 3.1). If c∗>0, there exists a constant l0 such that #τ∈ΓγBx,γ∩Jτ≠∅≤l0 for all x∈E and γ>0.Remark 2.
Some subtly different definitions of the low dimension are given as follows:dimLF=infα≥0. There exist two constant c>0 and b>0 such that, for any 0<r<R<b≤F, and x∈F, NrBx,R∩F≥cR/rα.
It is easy to check this definition and Definition2 coincides.Proof of Theorem 3.
Fixx∈E and 0<r<R; there exists σ0∈ΓR/J such that x∈Jσ0and Jσ0⊂Bx,R. By Lemma 1, there exists a probability measure v supported by E, such that(26)vJσ0=Jσ0s∗∑σ∈Dσ0Jσs∗,vJσ0=∑τ∈Γr/J,Jτ⊂Jσ0vJτ=∑τ∈Γr/J,Jτ⊂Jσ0Jτs∗∑σ∈DτJσs∗,where σ0 denotes the rank of Jσ0, i.e., Jσ0 is the σ0th-level basic set. This implies that(27)Jσ0s∗∑σ∈Dσ0Jσs∗≤#τ∈ΓrJ,Jτ⊂Jσ0rs∗∑σ∈DτJσs∗.
By Remark2, let R be small enough such that ∑σ∈Dσ0Jσs∗<2h¯ and ∑σ∈DτJσs∗>h¯/2. Therefore,(28)#τ∈ΓrJ,Jτ⊂Jσ0≥c∗s∗∑σ∈DτJσs∗∑σ∈Dσ0Jσs∗Rrs∗>c∗s∗h¯4h¯Rrs∗.
Using Lemma3 and (28), we attain(29)NrBx,R∩E>c∗s∗l0h¯4h¯Rrs∗,which implies dimLE>s∗ 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 thatJ is the interval [0, 1] and ck,j=ck for any k>0, 1≤j≤nk, in Definition 3. For all k>0, σ∈Dk, distJσ∗j,Jσ∗j+1=distJσ∗j+1,Jσ∗j+21≤j≤nk+1−2, and the left endpoint of Jσ∗1 is the left endpoint of Jσ and the right endpoint of Jσ∗nk+1 is the right endpoint of Jσ. The set E∈MJ,nk,ck is called the homogeneous Cantor set. Write E=CJ,nk,ck.Theorem 4.
(see [10]). Suppose E=CJ,nk,ck. Then,(30)dimHE=liminfk⟶∞logn1n2,…,nk−logc1c2,…,ck,dimPE=dim¯BE=limsupk⟶∞logn1n2,…,nk+1−logc1c2,…,ck+lognk+1.Theorem 5.
SupposeE=CJ,nk,ck. If limk⟶∞nk=∞, then dimAE=1.Proof.
Takex as the left endpoint of some basic elements of order k and R=c1c2,…,ck; then, c1c2,…,ck is the length of basic elements of order k. It is obvious that x∈E. Notice that(31)c1c2,…,ck+1+c1c2,…,ck1−nk+1ck+1nk+1−1=c1c2,…,ck1−ck+1nk+1−1.
Here,c1c2,…,ck+1 is the length of the basic elements of order k + 1 of the Moran set, and c1c2,…,ck1−nk+1ck+1/nk+1−1 is the length of the interval among the basic elements of order k + 1 of the Moran set E. Take r=c1c2,…,ck1−ck+1/nk+1−1; then,(32)NrBx,R∩E≥nk+1=1−ck+1nk+1nk+1−1Rr≥12Rr.
Note thatR/r=nk+1−1/1−ck+1⟶∞ when k⟶∞; therefore, dimAE=1.Theorem 6.
SupposeE=CJ,nk,ck. If limk⟶∞nkck=0, then dimLE=0.Proof.
Takex as the left endpoint of some basic elements of order k + 1 and(33)R=c1c2,…,ck+1+c1c2,…,ck1−nk+1ck+1nk+1−1=c1c2,…,ck1−ck+1nk+1−1.
Here,c1c2,…,ck+1 is the length of the basic elements of order k + 1 of the Moran set and c1c2,…,ck1−nk+1ck+1/nk+1−1 is the length of the interval among the basic elements of order k + 1 of the Moran set E. Take r=c1c2,…,ck+1. It is obvious that(34)NrBx,R∩E≤3=3Rr0.
Note thatR/r=1−ck+1/nk+1−1ck+1⟶∞ when k⟶∞; therefore, dimLE=0.Example 1.
Takenk=3kandck=4−k, and then dimLE=0, dimHE=dimPE=dim¯BE=log3/log4, and dimAE=1.Remark 3.
By Theorem5 and Theorem 6, we give a negative answer to the conjecture in the paper [8, see Remark 1].



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