Quasisymmetric Minimality on Packing Dimension for Homogeneous Perfect Sets

Abstract

The quasisymmetric minimality for fractal sets is a hot research topic for scholars focused on the fractal geometry and quasisymmetric mappings. In this paper, we study the quasisymmetric minimality on packing dimension for homogeneous perfect sets. By using some mathematical tools such as the mass distribution principle, we find that a special class of homogeneous perfect sets with packing dimension 1 is quasisymmetrically packing minimal. Our result generalizes the results in the references.

1. Introduction

Let $f:{\mathbb{R}}^n\to {\mathbb{R}}^n(n\ge 1)$ be a homeomorphism; if there exists a homeomorphism $\eta :[0,\infty )\to {\mathbb{R}}^n$, such that any triple points $a,b,x\in {\mathbb{R}}^n$ satisfy

$${\displaystyle \frac{\left|f(x)-f(a)\right|}{\left|f(x)-f(b)\right|}}\le \eta ({\displaystyle \frac{\left|x-a\right|}{\left|x-b\right|}}),$$

(1)

where the distance between the two point $a,b\in {\mathbb{R}}^n$ is denoted by $\left|a-b\right|$, then f is called a $n-$dimensional quasisymmetric mapping. The quasisymmetric mappings contain the Lipschitz mappings, but the fractal dimensions of the fractal sets may not be invariant under the quasisymmetric mappings, where the Lipschitz mappings preserve the fractal dimensions. How the quasisymmetric mappings change the fractal dimensions has always been a hot topic for researchers focused on the fractal geometry and quasisymmetric mappings; quasisymmetrically minimal sets are very important research objects in this subject. Suppose $E\subset {\mathbb{R}}^n$, then E is called a quasisymmetrically Hausdorff minimal set if for any $n-$dimensional quasisymmetric mapping f, we have ${dim}_{H}f(E)\ge {dim}_{H}E$, where the Hausdorff dimension of the set $E\subset {\mathbb{R}}^n$ is denoted by ${dim}_{H}E$ (the definition can be seen in [1,2]). Similarly, we can define a quasisymmetrically packing minimal set based on the packing dimension (the packing dimension of the set $E\subset {\mathbb{R}}^n$ is denoted by ${dim}_{P}E$; the definition can be seen in [1,2]).

In recent years, scholars have conducted a great deal of research on quasisymmetrically minimal sets. There are some typical results for quasisymmetric Hausdorff minimality in Euclidean space: Tyson and Gehring obtained some results on quasisymmetric Hausdorff minimality in ${\mathbb{R}}^n(n\ge 2)$—see [3,4,5]—Bishop and Kovalev obtained some results on quasisymmetric Hausdorff minimality in $\mathbb{R}$—see [6,7]—Hakobyan, Dai, Wang, et al. obtained some results on quasisymmetric Hausdorff minimality in the Moran sets on the real line—see [8,9,10,11]. Compared with quasisymmetric Hausdorff minimality, there are few results on quasisymmetric packing minimality. Kovalev showed in [7] that if $E\subset \mathbb{R}$ is a quasisymmetrically packing minimal set, then ${dim}_{P}E=0$ or ${dim}_{P}E=1$. It is obvious that any set with packing dimension 0 is a quasisymmetrically packing minimal set; therefore, we focus on the quasisymmetric packing minimality on the sets in $\mathbb{R}$ with packing dimension 1.

• Li, Wu, and Xi [12] discovered that two large classes of Moran sets in $\mathbb{R}$ (under the condition $\underset{k\ge 1}{sup}{n}_k<\infty$) with packing dimension 1 are quasisymmetrically packing minimal;
• Wang and Wen proved in [10] that all uniform Cantor sets (there is no requirement for the value of $\underset{k\ge 1}{sup}{n}_k$) with packing dimension 1 are quasisymmetrically packing minimal;
• Li, Qiao and Lou [13] and Li, Fu and Yang [14] proved that some large classes of special homogeneous Moran sets in $\mathbb{R}$ (under the condition $\underset{k\ge 1}{sup}{n}_k<\infty$) with packing dimension 1 are quasisymmetrically packing minimal, which generalized a result in [12] in some sense.

A large class of the homogeneous perfect sets with packing dimension 1, which contains all uniform Cantor sets with packing dimension 1, is proved to be quasisymmetrically packing minimal in this paper, which generalizes a result presented in [10].

This paper is organized as follows. In Section 2, we recall the definition of homogeneous perfect sets and provide some lemmas. In Section 3, we state our main result. The proof of our main result is given in Section 4 and Section 5.

2. Preliminaries

We study the quasisymmetric minimality on packing dimension for homogeneous perfect sets in this paper; the homogeneous perfect sets is a class of important fractal sets, which has been widely studied by many scholars focused on the fractal geometry. We first recall its definition in Section 2.1.

2.1. Homogeneous Perfect Sets

Wen and Wu gave the definition of the homogeneous perfect sets in [15], now we recall it.

Let the sequences ${\left\{c_k\right\}}_{k\ge 1}\subset {\mathbb{R}}^{+}$ and ${\left\{n_k\right\}}_{k\ge 1}\subset {\mathbb{Z}}^{+}$ with $n_k\ge 2$ and $n_k{c}_k<1$ for any $k\ge 1$. For any $k\ge 1$, let $D_k=\left\{i_1{i}_2\cdots i_k:1\le i_j\le n_j,1\le j\le k\right\}$, $D_0=\varnothing$ and $D={\cup }_{k\ge 0}{D}_k$. If $\sigma ={\sigma }_1{\sigma }_2\cdots {\sigma }_k\in D_k$, $\tau ={\tau }_1{\tau }_2\cdots {\tau }_m$, where $1\le {\tau }_i\le n_{k+i}$ for any $1\le i\le m$, let $\sigma \ast \tau ={\sigma }_1{\sigma }_2\cdots {\sigma }_k{\tau }_1{\tau }_2\cdots {\tau }_m\in D_{k+m}$.

Definition 1

(Homogeneous perfect sets [15]). For a closed interval $I_0\subset \mathbb{R}$ with $I_0\ne \varnothing$, which we call the initial interval, we say the collection of closed subintervals $\mathcal{I}=\left\{I_{\sigma }:\sigma \in D\right\}$ of $I_0$ has homogeneous perfect structure if it satisfies the following:

(1)

$I_{\varnothing }=I_0$;

(2)

for any $k\ge 1$ and $\sigma \in D_{k-1}$, $I_{\sigma \ast 1},\cdots ,I_{\sigma \ast n_k}$ are closed subintervals of $I_{\sigma }$ with $min(I_{\sigma \ast (l+1)})\ge max(I_{\sigma \ast l})$ for any $1\le l\le n_k-1$, where for any set $A\subset \mathbb{R}$, $min(A)=sup\left\{x:x\le aforanya\in A\right\}$, $max(A)=inf\left\{y:y\ge aforanya\in A\right\}$;

(3)

for any $k\ge 1$ and $\sigma \in D_{k-1}$, $1\le j\le n_k$,

$${\displaystyle \frac{\left|I_{\sigma \ast j}\right|}{\left|I_{\sigma }\right|}}=c_k,$$

(2)

where the diameter of the set A is denoted by $\left|A\right|$;

(4)

there exists a sequence ${\left\{{\xi }_{k,l}\right\}}_{k\ge 1,0\le l\le n_k}\subset {\mathbb{R}}^{+}\cup \left\{0\right\}$ satisfying for any $k\ge 1$ and $\sigma \in D_{k-1}$,

$$min(I_{\sigma \ast 1})-min(I_{\sigma })={\xi }_{k,0};$$

(3)

$$min(I_{\sigma \ast (l+1)})-max(I_{\sigma \ast l})={\xi }_{k,l}(\forall 1\le l\le n_k-1);$$

(4)

$$max(I_{\sigma })-max(I_{\sigma \ast n_k})={\xi }_{k,n_k}.$$

(5)

If $\mathcal{I}$ has homogeneous perfect structure, let $E_k={\cup }_{\sigma \in D_k}{I}_{\sigma }$ for any $k\ge 0$, then $E={\cap }_{k\ge 0}{E}_k=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ is called a homogeneous perfect set. For any $k\ge 0$, let ${\mathcal{E}}_k=\left\{I_{\sigma }:\sigma \in D_k\right\}$, then any $I_{\sigma }$ in ${\mathcal{E}}_k$ is called a k-order basic interval of E.

Remark 1.

If $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ is a homogeneous perfect set, then E is a homogeneous Moran set with $E\in \mathcal{M}(I_0,\left\{n_k\right\},\left\{c_k\right\})$, where the definition of the homogeneous Moran sets can be seen in [16].

Remark 2.

If $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ with ${\xi }_{k,0}={\xi }_{k,n_k}=0$, ${\xi }_{k,i}={\xi }_{k,j}$ for any $k\ge 1$ and $1\le i<j\le n_k$, then E is a uniform Cantor set, where the definition of the uniform Cantor set can be seen in [10]. If $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ with $n_k=2$, $c_k={\displaystyle \frac{1}{3}}$, ${\xi }_{k,0}={\xi }_{k,2}=0$ and ${\xi }_{k,1}={\displaystyle \frac{1}{3}}$ for any $k\ge 1$, then E is the Cantor ternary set.

2.2. Some Lemmas

We need the following lemmas to finish our proof.

The next lemma provides the packing dimension formula for some homogeneous perfect sets.

Lemma 1

([17]). Suppose that $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$, and there exists a real number $\chi \ge 1$ such that ${max}_{1\le l\le n_k-1}{\xi }_{k,l}\le \chi {min}_{1\le l\le n_k-1}{\xi }_{k,l}$ for any $k\ge 1$, then

$${dim}_{P}E=\underset{k\to \infty }{\overline{lim}}{\displaystyle \frac{log{n}_1\cdots n_k{n}_{k+1}}{-log\frac{c_1{c}_2\cdots c_k-{\xi }_{k+1,0}-{\xi }_{k+1,n_{k+1}}}{n_{k+1}}}}.$$

(6)

If E is the Cantor ternary set, then $n_k=2$, $c_k={\displaystyle \frac{1}{3}}$, ${\xi }_{k,0}={\xi }_{k,2}=0$, and ${\xi }_{k,1}={\displaystyle \frac{1}{3}}$ for any $k\ge 1$, we can easily obtain that ${dim}_{P}E={\displaystyle \frac{log2}{log3}}$ by Lemma 1; if E is a uniform Cantor set, then ${\xi }_{k,0}={\xi }_{k,n_k}=0$, ${\xi }_{k,i}={\xi }_{k,j}$ for any $k\ge 1$ and $1\le i<j\le n_k$, we obtain that ${dim}_{P}E={\overline{lim}}_{k\to \infty }{\displaystyle \frac{log{n}_1{n}_2\cdots n_{k+1}}{-log{c}_1{c}_2\cdots c_k+log{n}_{k+1}}}$ by Lemma 1, which is equal to the results in the reference [16].

The quasisymmetric packing minimality of the fractal sets is related to the packing dimension of the sets. In Lemma 6 of this paper, for studying the quasisymmetric packing minimality, we use Lemma 1 to obtain some properties of the homogeneous perfect sets in Theorem 1 of this paper.

We need the mass distribution principle to estimate packing dimension of the image sets of the quasisymmertic mappings.

Lemma 2

(Mass distribution principle [2]). Suppose that $s\ge 0$, let μ be a Borel probability measure on a Borel set $E\subseteq \mathbb{R}$. If there is a positive constant C, such that

$$\underset{r\to 0}{\underline{lim}}{\displaystyle \frac{\mu (B(x,r))}{r^s}}\le C$$

(7)

for any $x\in E$, then ${dim}_{P}E\ge s.$

We can use Lemma 2 to estimate the lower bound of the packing dimension for some typical fractal sets, see in the reference [18]. Some scholars used Lemma 2 to estimate the lower bound of the packing dimension for studying the quasisymmetric packing minimality for some fractal sets, such as the references [7,10,12,13,14]. We also use Lemma 2 to estimate the lower bound of the packing dimension of the quasisymmertic image sets in this paper.

The following lemma shows some relationships between the lengths for the image sets of the quasisymmertic mappings and the lengths for the original sets.

Lemma 3

([19]). Let $f:\mathbb{R}\to \mathbb{R}$ be a 1-dimensional quasisymmetric mapping, then there exist positive real numbers $\beta >0$, $K_{\rho }>0$ and $0<p\le 1\le q$ such that

$$\beta {({\displaystyle \frac{\left|I^{\prime }\right|}{\left|I\right|}})}^q\le {\displaystyle \frac{\left|f(I^{\prime })\right|}{\left|f(I)\right|}}\le 4{({\displaystyle \frac{\left|I^{\prime }\right|}{\left|I\right|}})}^p,$$

(8)

where $I^{\prime }$ and I are any intervals satisfying $I^{\prime }\subseteq I$, and

$${\displaystyle \frac{\left|f(\rho I)\right|}{\left|f(I)\right|}}\le K_{\rho },$$

(9)

where for any $\rho >0$, $\rho I$ denotes the interval with the same center of the interval I, and $\left|\rho I\right|=\rho \left|I\right|$.

For studying the quasisymmetric minimality (include the Hausdorff minimality and the packing minimality) of some fractal sets, scholars used Lemma 3 to estimate the lengths for the image sets, such as the references [8,9,11,12,14]; we also carry this out in this paper.

3. Main Result

We obtain that a special class of homogeneous perfect sets with packing dimension 1 is quasisymmetrically packing minimal, the specific content is as follows.

Theorem 1.

Suppose $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$, and there exists a real number $\chi \ge 1$, such that ${max}_{1\le l\le n_k-1}{\xi }_{k,l}\le \chi {min}_{1\le l\le n_k-1}{\xi }_{k,l}$ for any $k\ge 1$. If ${dim}_{P}E=1$, then for any 1-dimensional quasisymmetric mapping f, we have ${dim}_{P}f(E)=1$.

Remark 3.

In Theorem 1.2 of [10], Wang and Wen proved that for any uniform Cantor sets E, if ${dim}_{P}E=1$, then we have ${dim}_{P}f(E)=1$ for any 1-dimensional quasisymmetric mapping f. It is obvious that the homogeneous perfect sets satisfying the condition of Theorem 1 in this paper contain the uniform Cantor sets. Thus Theorem 1 in this paper generalizes Theorem 1.2 in [10].

4. The Reconstruction of Homogeneous Perfect Sets

In order to prove the Theorem 1, we reconstruct the homogeneous perfect set $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ and represent it as an equivalent form which is easier to discuss in our proof.

For any $k\ge 0$, $\sigma \in D_k$, let $I_{\sigma }^{\ast }$ be a closed subinterval of $I_{\sigma }$ satisfying the following conditions:

(A)

$min(I_{\sigma }^{\ast })-min(I_{\sigma })={\xi }_{k+1,0}$, $max(I_{\sigma })-max(I_{\sigma }^{\ast })={\xi }_{k+1,n_{k+1}}$;

(B)

$\left|I_{\sigma }^{\ast }\right|={\sum }_{l=1}^{n_{k+1}-1}{\xi }_{k+1,l}+c_1{c}_2\cdots c_{k+1}{n}_{k+1}$,

which implies that for any $k\ge 0$, $\sigma \in D_k$, the most left point of $I_{\sigma }^{\ast }$ coincides with the most left point of $I_{\sigma \ast 1}$ and the most right point of $I_{\sigma }^{\ast }$ is coincides with the most right point of $I_{\sigma \ast n_{k+1}}$.

Let $I_0^{\ast }=I_{\varnothing }^{\ast }$ and denote ${\delta }_k=\left|I_{\sigma }^{\ast }\right|$, ${\delta }_0=\left|I_0^{\ast }\right|$ for any $k\ge 1$ and $\sigma \in D_k$. Suppose that $E_k^{\ast }={\cup }_{\sigma \in D_k}{I}_{\sigma }^{\ast }$ for any $k\ge 0$ and $\sigma \in D_k$, notice that $I_{\sigma }^{\ast }$ deletes the redundant gaps of $I_{\sigma }$; then, it obvious that $E={\cap }_{k\ge 0}{E}_k={\cap }_{k\ge 0}{E}_k^{\ast }$.

In fact, $E=E(I_0^{\ast },\left\{n_k^{\ast }\right\},\left\{c_k^{\ast }\right\},\left\{{\xi }_{k,l}^{\ast }\right\})$ is a homogeneous perfect set with the following parameters for any $k\ge 1$, where ${\mathcal{I}}^{\ast }=\{I_{\omega }^{\ast }:\omega \in D\}$ has a homogeneous perfect structure:

(1)

$I_0^{\ast }=I_0-[min(I_0),min(I_0)+{\xi }_{1,0})-(max(I_0)-{\xi }_{1,n_1},max(I_0)]$;

(2)

$c_k^{\ast }={\displaystyle \frac{{\delta }_k}{{\delta }_{k-1}}}$, $n_k^{\ast }=n_k$;

(3)

${\xi }_{k,l}^{\ast }={\xi }_{k,l}+{\xi }_{k+1,0}+{\xi }_{k+1,n_{k+1}}(\forall 1\le l\le n_k-1)$, ${\xi }_{k,0}^{\ast }={\xi }_{k+1,0}$, ${\xi }_{k,n_k}^{\ast }={\xi }_{k+1,n_{k+1}}$.

For any $k\ge 1$, denote

$$N_k^{\ast }=n_1^{\ast }{n}_2^{\ast }\cdots n_k^{\ast },{\delta }_k^{\ast }={\delta }_0{c}_1^{\ast }{c}_2^{\ast }\cdots c_k^{\ast }.$$

(10)

$${\underline{\alpha }}_k^{\ast }=\underset{1\le l\le n_k-1}{min}{\xi }_{k,l}^{\ast },{\overline{\alpha }}_k^{\ast }=\underset{1\le l\le n_k-1}{max}{\xi }_{k,l}^{\ast },$$

(11)

then $N_k^{\ast }$ is the number of the k-order basic interval of E in $E_k^{\ast }$, ${\delta }_k$ is the length of each k-order basic interval of E in $E_k^{\ast }$.

It is easy to obtain that

$${\xi }_{k,0}^{\ast }+{\xi }_{k,n_k}^{\ast }\le {\underline{\alpha }}_k^{\ast },$$

(12)

and if E satisfies the condition of Theorem 1, then

$${\overline{\alpha }}_k^{\ast }\le \chi {\underline{\alpha }}_k^{\ast },$$

(13)

where $\chi$ is the constant in Theorem 1.

The folllowing lemma gives a new form of the homogeneous perfect sets in Theorem 1.

Lemma 4.

Suppose $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ satisfies the condition of Theorem 1, then there exists ${\left\{H_m\right\}}_{m\ge 0}$, which is a sequence of closed sets with length decreasing, such that $E={\cap }_{k\ge 0}{E}_k={\cap }_{k\ge 0}{E}_k^{\ast }={\cap }_{m\ge 0}{H}_m$. Furthermore, ${\left\{H_m\right\}}_{m\ge 0}$ satisfies the following:

(1)

For any $m\ge 0$, $H_m$ is a union of a finite number of closed intervals whose interiors are disjoint, which are called the branches of $H_m$. Denote ${\mathcal{H}}_m=\{A:A\mathrm{is}\mathrm{a}\mathrm{branch}\mathrm{of}{H}_m\}$;

(2)

${\left\{E_k^{\ast }\right\}}_{k\ge 0}\subset {\left\{H_m\right\}}_{m\ge 0}$ and $H_{m_k}=E_k^{\ast }$ for any $k\ge 0$;

(3)

There exists $M\in {\mathbb{Z}}^{+}$ with $M>2\chi$ such that each branch of $H_{m-1}$ contains at most $M^2$ branches of $H_m$ for any $m\ge 1$, where χ is the constant in Theorem 1;

(4)

For any $m\ge 0$, ${max}_{I\in {\mathcal{H}}_m}\left|I\right|\le 2\chi {min}_{I\in {\mathcal{H}}_m}\left|I\right|$.

Proof. 

Let $M=\left[2\chi \right]+1$. For any $k\ge 1$, let $i_k$ be the positive integer satisfying the following conditions: If $n_k^{\ast }\in [2,M)$, then $i_k=1$; If $n_k^{\ast }\in [M,+\infty )$, then $i_k$ is the positive integer satisfying $n_k^{\ast }\in [M^{i_k},M^{i_k+1})$. Define $m_0=0,m_k={\sum }_{l=1}^k{i}_l,$ then $m_k=m_{k-1}+i_k.$

For any $k\ge 0$, define $H_{m_k}=E_k^{\ast }$ and ${\mathcal{H}}_{m_k}=\{I_{\omega }^{\ast }:\omega \in D_k\}$, which means all branches of $H_{m_k}$ are all k-order basic intervals in $E_k^{\ast }$. Next, we construct $H_m$ for any $k\ge 1$ and $m_{k-1}<m<m_k$,.

(1):

If $n_k^{\ast }\in [M,M^2)$, then $i_k=1$ and $m_k=m_{k-1}+1$; we have nothing to do.

(2):

If $n_k^{\ast }\in [M^2,+\infty )$, then $i_k\ge 2$, and there are $a_0,a_1,···,a_{i_k-1}$, such that $a_j\in \left\{0,1,\cdots ,M-1\right\}$ for any $j\in \left\{0,1,\cdots ,i_k-1\right\}$ and

$$n_k^{\ast }=a_0+a_{1}M+a_2{M}^2+\cdots +a_{i_k-1}{M}^{i_k-1}+M^{i_k}.$$

(14)

For any $k\ge 1$ and $\sigma \in D_{k-1}$, since $H_{m_{k-1}}=E_{k-1}^{\ast }$, $H_{m_{k-1}}$ has $N_{k-1}^{\ast }$ branches and for any $I_{\sigma }^{\ast }\in {\mathcal{H}}_{m_{k-1}}$, the number of the k-order basic intervals in $E_k^{\ast }$ contained in $I_{\sigma }^{\ast }$ is $n_k^{\ast }$, denote $I_{\sigma \ast 1}^{\ast },\cdots ,I_{\sigma \ast n_k^{\ast }}^{\ast }$.

Now we begin to construct $H_{m_{k-1}+i}$ for any $1\le i\le i_k-1$.

Let $[T_1,T_2,\cdots ,T_t]$ be the smallest closed interval containing the t closed intervals $T_1,T_2,\cdots ,T_t$. For example, if $A=[0,1],B=[2,4],C=\left[7.8\right]$, we have $[A,B,C]=[0,8]$.

(a)

For any $I_{\sigma }^{\ast }\in {\mathcal{H}}_{m_{k-1}}$, let $l_1=a_1+a_{2}M+\cdots +a_{i_k-1}{M}^{i_k-2}+M^{i_k-1}$, then $n_k^{\ast }=M{l}_1+a_0=a_0(l_1+1)+(M-a_0)l_1$. Define

$$I_1^{\sigma ,1}=[I_{\sigma \ast 1}^{\ast },\cdots ,I_{\sigma \ast (l_1+1)}^{\ast }],$$

$$I_2^{\sigma ,1}=[I_{\sigma \ast (l_1+2)}^{\ast }\cdots I_{\sigma \ast (2l_1+2)}^{\ast }],$$

⋯

$$I_{a_0}^{\sigma ,1}=[I_{\sigma \ast ((a_0-1)l_1+a_0)}^{\ast }\cdots I_{\sigma \ast (a_0{l}_1+a_0)}^{\ast }],$$

$$I_{a_0+1}^{\sigma ,1}=[I_{\sigma \ast (a_0{l}_1+a_0+1)}^{\ast },\cdots ,I_{\sigma \ast (a_0{l}_1+a_0+l_1)}^{\ast }],$$

⋯

$$I_M^{\sigma ,1}=[I_{\sigma \ast (n_k^{\ast }+1-l_1)}^{\ast },\cdots ,I_{\sigma \ast n_k^{\ast }}^{\ast }].$$

Then for any $1\le i\le a_0$, $I_i^{\sigma ,1}$ contains $l_1+1$ k-order basic intervals of E, for any $a_0+1\le i\le M$, $I_i^{\sigma ,1}$ contains $l_1$ k-order basic intervals of E. Let $H_{m_{k-1}+1}={\bigcup }_{\sigma \in D_{k-1}}{\bigcup }_{i=1}^M{I}_i^{\sigma ,1}$, and let the M closed intervals $I_1^{\sigma ,1},\cdots ,I_M^{\sigma ,1}$ be the M branches of $H_{m_{k-1}+1}$ in $I_{\sigma }^{\ast }$, then each branch of $H_{m_{k-1}}$ contains M branches of $H_{m_{k-1}+1}$ and it is easy to obtain that ${max}_{I\in {\mathcal{H}}_{m_{k-1}+1}}\left|I\right|\le 2\chi {min}_{I\in {\mathcal{H}}_{m_{k-1}+1}}\left|I\right|$.

(b)

If $i_k=2$, then $m_k=m_{k-1}+2$, and $H_{m_{k-1}+1}$ is defined as above, $H_{m_{k-1}}=E_{k-1}^{\ast }$, $H_{m_k}=E_k^{\ast }$. This completes the construction of $H_{m_{k-1}+i}$ for any $1\le i\le i_k-1$.

(c)

If $i_k\ge 3$, then we continue to construct $H_{m_{k-1}+2}$. Let $l_2=a_2+a_{3}M+\cdots +a_{i_k-2}{M}^{i_k-3}+M^{i_k-2}$, then $l_1=M{l}_2+a_1$, $n_k^{\ast }=M^2{l}_2+a_{1}M+a_0=a_0(M{l}_2+a_1+1)+(M-a_0)(M{l}_2+a_1)$.

For any $I_i^{\sigma ,1}\in {\mathcal{H}}_{m_{k-1}+1}(\sigma \in D_{k-1},1\le i\le M)$, we divide our construction into the following two cases:

(c1): If $1\le i\le a_0$, then the number of the k-order basic intervals contained in each $I_i^{\sigma ,1}$ is $l_1+1=M{l}_2+a_1+1=(l_2+1)(a_1+1)+l_2(M-a_1-1)$. Define

$$I_{i\ast 1}^{\sigma ,1}=[I_{\sigma \ast ((i-1)l_1+i)}^{\ast },\cdots ,I_{\sigma \ast ((i-1)l_1+i+l_2)}^{\ast }],$$

$$I_{i\ast 2}^{\sigma ,1}=[I_{\sigma \ast ((i-1)l_1+i+l_2+1)}^{\ast },\cdots ,I_{\sigma \ast ((i-1)l_1+i+2l_2+1)}^{\ast }],$$

⋯

$$I_{i\ast (a_1+1)}^{\sigma ,1}=[I_{\sigma \ast ((i-1)l_1+i+a_1{l}_2+a_1)}^{\ast },\cdots ,I_{\sigma \ast ((i-1)l_1+i+(a_1+1)l_2+a_1)}^{\ast }],$$

$$I_{i\ast (a_1+2)}^{\sigma ,1}=[I_{\sigma \ast ((i-1)l_1+i+(a_1+1)(l_2+1))}^{\ast },\cdots ,I_{\sigma \ast ((i-1)l_1+i+(a_1+1)(l_2+1)+l_2-1)}^{\ast }],$$

⋯

$$I_{i\ast M}^{\sigma ,1}=[I_{\sigma \ast (i{l}_1+i+1-l_2)}^{\ast },\cdots ,I_{\sigma \ast (i{l}_1+i)}^{\ast }].$$

Then for any $1\le j\le a_1+1$, $I_{i\ast j}^{\sigma ,1}$ contains $l_2+1$ k-order basic intervals of E, for any $a_1+2\le j\le M$, $I_{i\ast j}^{\sigma ,1}$ contains $l_2$ k-order basic intervals of E.

(c2): If $a_0+1\le i\le M$, then the number of the k-order basic intervals contained in each $I_i^{\sigma ,1}$ is $l_1=M{l}_2+a_1=a_1(l_2+1)+(M-a_1)l_2$. Define

$$I_{i\ast 1}^{\sigma ,1}=[I_{\sigma \ast ((i-1)l_1+a_0+1)}^{\ast },\cdots ,I_{\sigma \ast ((i-1)l_1+a_0+1+l_2)}^{\ast }],$$

$$I_{i\ast 2}^{\sigma ,1}=[I_{\sigma \ast ((i-1)l_1+a_0+l_2+2)}^{\ast },\cdots ,I_{\sigma \ast ((i-1)l_1+a_0+2l_2+2)}^{\ast }],$$

⋯

$$I_{i\ast a_1}^{\sigma ,1}=[I_{\sigma \ast ((i-1)l_1+a_0+(a_1-1)l_2+a_1)}^{\ast },\cdots ,I_{\sigma \ast ((i-1)l_1+a_0+a_1{l}_2+a_1)}^{\ast }],$$

$$I_{i\ast (a_1+1)}^{\sigma ,1}=[I_{\sigma \ast ((i-1)l_1+a_0+a_1{l}_2+a_1+1)}^{\ast },\cdots ,I_{\sigma \ast ((i-1)l_1+a_0+a_1{l}_2+a_1+l_2)}^{\ast }],$$

⋯

$$I_{i\ast M}^{\sigma ,1}=[I_{\sigma \ast (i{l}_1+a_0+1-l_2)}^{\ast },\cdots ,I_{\sigma \ast (i{l}_1+a_0)}^{\ast }].$$

Then for any $1\le j\le a_1$, $I_{i\ast j}^{\sigma ,1}$ contains $l_2+1$ k-order basic intervals of E, for any $a_1+1\le j\le M$, $I_{i\ast j}^{\sigma ,1}$ contains $l_2$ k-order basic intervals of E.

Let $H_{m_{k-1}+2}={\bigcup }_{\sigma \in D_{k-1}}{\bigcup }_{i=1}^M{\bigcup }_{j=1}^M{I}_{i\ast j}^{\sigma ,1}$, and let the M closed intervals $I_{i\ast 1}^{\sigma ,1},I_{i\ast 2}^{\sigma ,1},\cdots ,I_{i\ast M}^{\sigma ,1}$ be the M branches of $H_{m_{k-1}+2}$ in $I_i^{\sigma ,1}$, then each branch of $H_{m_{k-1}+1}$ contains M branches of $H_{m_{k-1}+2}$ and it is easy to obtain that ${max}_{I\in {\mathcal{H}}_{m_{k-1}+2}}\left|I\right|\le 2\chi {min}_{I\in {\mathcal{H}}_{m_{k-1}+2}}\left|I\right|$.

(d)

If $i_k=3$, then $m_k=m_{k-1}+3$, and $H_{m_{k-1}+1}$, $H_{m_{k-1}+2}$ are defined as above, $H_{m_{k-1}}=E_{k-1}^{\ast }$, $H_{m_k}=E_k^{\ast }$. This completes the construction of $H_{m_{k-1}+i}$ for any $1\le i\le i_k-1$.

(e)

If $i_k\ge 4$, then $m_k=m_{k-1}+i_k$. If $H_{m_{k-1}+i-1}(3\le i\le i_k-1)$ has been constructed, we repeat the method of the construction of $H_{m_{k-1}+i-1}$ from $H_{m_{k-1}+i-2}$ to define $H_{m_{k-1}+i}$ from $H_{m_{k-1}+i-1}$. Then $H_{m_{k-1}+1},H_{m_{k-1}+2},···H_{m_{k-1}+i_k-1}$ are defined, and we can obtain that for any $1\le i\le i_k-1$, each branch of $H_{m_{k-1}+i-1}$ contains M branches of $H_{m_{k-1}+i}$ and ${max}_{I\in {\mathcal{H}}_{m_{k-1}+i}}\left|I\right|\le 2\chi {min}_{I\in {\mathcal{H}}_{m_{k-1}+i}}\left|I\right|$. This completes the construction of $H_{m_{k-1}+i}$ for any $1\le i\le i_k-1$.

(f)

For any $i_k\ge 2$, suppose $H_{m_{k-1}+i}$ has been constructed for any $1\le i\le i_k-1$. Notice that each branch of $H_{m_{k-1}+i-1}(2\le i\le i_k-1)$ contains M branches of $H_{m_{k-1}+i}$; thus, each branch of $H_{m_{k-1}}$ contains $M^{i_k-1}$ branches of $H_{m_{k-1}+i_k-1}$. Notice that $m_k=m_{k-1}+i_k$ and $H_{m_k}=E_k^{\ast }$ for any $k\ge 0$ implies that each branch of $H_{m_{k-1}}$ contains $n_k^{\ast }$ branches of $H_{m_k}$, then each branch of $H_{m_{k-1}+i_k-1}$ contains at most $M^2$ branches of $H_{m_k}$(otherwise, if there exists a branch of $H_{m_{k-1}+i_k-1}$ containing $M^{\ast }>M^2$ branches of $H_{m_k}$, then any branch of $H_{m_{k-1}+i_k-1}$ contains $M^{\ast }$ or $M^{\ast }+1$ or $M^{\ast }-1$ branches of $H_{m_k}$, which implies that $n_k^{\ast }>M^2\times M^{i_k-1}=M^{i_k+1}$; it is contrary to $M_{i_k}\le n_k^{\ast }<M^{i_k+1}$).

(g)

Since $H_{m_k}=E_k^{\ast }$ for any $k\ge 0$, we have ${max}_{I\in {\mathcal{H}}_{m_k}}\left|I\right|={min}_{I\in {\mathcal{H}}_{m_k}}\left|I\right|$ for any $k\ge 0$.

We finish the construction of ${\left\{H_m\right\}}_{m\ge 0}$ which satisfies (1)–(4) of Lemma 4. □

Remark 4.

Without loss of generality, we assume that $I_0^{\ast }=[0,1]$; then, $H_{m_0}=E_0^{\ast }=[0,1]$ and ${\delta }_0=1$.

Lemma 5.

Let $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ satisfies the condition of Theorem 1, ${\left\{H_m\right\}}_{m\ge 0}$ be the length decreasing sequence in Lemma 4 and the total length of all branches of each $H_m$ is denoted by $l(H_m)$, then for any $k\ge 1$ and $m_{k-1}<m<m_k$,

$$l(H_{m_k})=N_k^{\ast }{\delta }_k^{\ast },(1-{\displaystyle \frac{2\chi }{M}})N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }\le l(H_m)\le N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }.$$

(15)

Proof. 

Since $H_{m_k}=E_k^{\ast }$ for any $k\ge 1$ and ${\left\{H_m\right\}}_{m\ge 0}$ is a length decreasing sequence, it is obvious that $l(H_{m_k})=l(E_k^{\ast })=N_k^{\ast }{\delta }_k^{\ast }$ and $l(H_m)\le l(H_{m_{k-1}})=l(E_{k-1}^{\ast })=N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }$ for any $k\ge 1$ and $m_{k-1}<m<m_k$. Then we only need to verify that $(1-{\displaystyle \frac{2\chi }{M}})N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }\le l(H_m)$ for any $k\ge 1$ and $m_{k-1}<m<m_k$.

We can see a fact from the construction of ${\left\{H_m\right\}}_{m\ge 0}$: In order to get $H_{m_k-1}$, we remove a half open and half closed interval of length ${\xi }_{k,0}^{\ast }$ and a half open and half closed interval of length ${\xi }_{k,n_k}^{\ast }$ from each branch of $H_{m_{k-1}}$, and remove $\left[{\sum }_{j=0}^{i_k-2}{M}^j(M-1)\right]N_{k-1}^{\ast }=(M^{i_k-1}-1)N_{k-1}^{\ast }$ open intervals which the lengths of them are at most ${\overline{\alpha }}_k^{\ast }$ from $E_{k-1}^{\ast }=H_{m_{k-1}}$. Then by (12), we have

$$\begin{array}{cc}\hfill l(H_{m_k-1})& \ge N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }-N_{k-1}^{\ast }({\xi }_{k,0}^{\ast }+{\xi }_{k,n_k}^{\ast })-(M^{i_k-1}-1)N_{k-1}^{\ast }{\overline{\alpha }}_k^{\ast }\hfill \\ \hfill & \ge N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }-N_{k-1}^{\ast }{\underline{\alpha }}_k^{\ast }-(M^{i_k-1}-1)N_{k-1}^{\ast }{\overline{\alpha }}_k^{\ast }\hfill \\ \hfill & \ge N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }-N_{k-1}^{\ast }{\overline{\alpha }}_k^{\ast }-(M^{i_k-1}-1)N_{k-1}^{\ast }{\overline{\alpha }}_k^{\ast }\hfill \\ \hfill & \ge N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }-M^{i_k-1}{N}_{k-1}^{\ast }{\overline{\alpha }}_k^{\ast }.\hfill \end{array}$$

(16)

Notice that $n_k^{\ast }\ge 2$ and

$$M^{i_k}\le n_k^{\ast }<M^{i_k+1},$$

(17)

then by (13), we have

$$\begin{array}{cc}\hfill l(H_{m_k-1})& \ge N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }-{\displaystyle \frac{n_k^{\ast }}{M}}{N}_{k-1}^{\ast }{\overline{\alpha }}_k^{\ast }\hfill \\ \hfill & \ge N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }-{\displaystyle \frac{2(n_k^{\ast }-1)}{M}}{N}_{k-1}^{\ast }{\overline{\alpha }}_k^{\ast }\hfill \\ \hfill & \ge N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }-{\displaystyle \frac{2}{M}}{N}_{k-1}^{\ast }\chi (n_k^{\ast }-1){\underline{\alpha }}_k^{\ast }\hfill \\ \hfill & \ge N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }-{\displaystyle \frac{2\chi }{M}}{N}_{k-1}^{\ast }{\delta }_{k-1}^{\ast }\hfill \\ \hfill & \ge (1-{\displaystyle \frac{2\chi }{M}})N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }.\hfill \end{array}$$

(18)

Notice that ${\left\{H_m\right\}}_{m\ge 0}$ is a length decreasing sequence, then

$$l(H_m)\ge l(H_{m_k-1})\ge (1-{\displaystyle \frac{2\chi }{M}})N_{k-1}^{\ast }{\delta }_{k-1}^{\ast }$$

(19)

for any $k\ge 1$ and $m_{k-1}<m<m_k$. □

5. The Quasisymmetric Packing Minimality on Homogeneous Perfect Sets

In this section, using Lemma 2, we first define a measure on the quasisymmertic image sets; then, we give some notations and use Lemma 1 to obtain some properties of the homogeneous perfect sets in Theorem 1 for the following proof. Then, through Lemma 3 and some geometric structures of the sets, we estimate the relationships between the measures and the diameters of the some basic intervals of the image sets. Finally, we use the relationships to estimate the lower bound of the packing dimension of the quasisymmertic image sets by Lemma 2, and finish the proof of Theorem 1.

5.1. The Measure ${\mu }_d$

Let $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$, which satisfies the condition of Theorem 1, and f be a 1-dimensional quasisymmetric mapping, ${\left\{H_m\right\}}_{m\ge 0}$ be the length decreasing sequence in Lemma 4. In order to complete the proof of Theorem 1 by Lemma 2, we need to define a positive and finite Borel measure on $f(E)$.

For any $m\ge 0$, let $J_m$ be the image set of a branch of $H_m$ under f; it is obvious that image sets of all branches of $H_m$ under f constitute $f(H_m)$. We call $J_m$ a branch of $f(H_m)$. Let $J_{m,1}\cdots ,J_{m,N(J_m)}$ be all branches of $f(H_{m+1})$ contained in $J_m$, where $N(J_m)$ is the number of the branches of $f(H_{m+1})$ contained in $J_m$; then $N(J_m)\le M^2$. For any $d\in (0,1)$, $m\ge 0$ and $1\le i\le N(J_m)$, by the measure extension theorem, there is a probability Borel measure ${\mu }_d$ on $f(E)$ satisfying

$${\mu }_d(J_{m,i})={\displaystyle \frac{{\left|J_{m,i}\right|}^d}{{\sum }_{j=1}^{N(J_m)}{\left|J_{m,j}\right|}^d}}{\mu }_d(J_m).$$

(20)

For any $m\ge 1$, let k satisfy $m_{k-1}<m\le m_k$, and denote

$$\mathrm{\Lambda }(m)=\underset{I\in {\mathcal{H}}_{m-1},J\in {\mathcal{H}}_m,J\subset I}{max}{\displaystyle \frac{\left|J\right|}{\left|I\right|}},\lambda (m)=\underset{I\in {\mathcal{H}}_{m-1},J\in {\mathcal{H}}_m,J\subset I}{min}{\displaystyle \frac{\left|J\right|}{\left|I\right|}};$$

(21)

$$\mathrm{\Gamma }(m)={\displaystyle \frac{{\overline{\alpha }}_k^{\ast }}{{min}_{I\in {\mathcal{H}}_{m-1}}\left|I\right|}},\gamma (m)={\displaystyle \frac{{\underline{\alpha }}_k^{\ast }}{{max}_{I\in {\mathcal{H}}_{m-1}}\left|I\right|}}.$$

(22)

Through Lemma 4, we have

$$\lambda (m)\le \mathrm{\Lambda }(m)\le 4{\chi }^2\lambda (m),\gamma (m)\le \mathrm{\Gamma }(m)\le 2{\chi }^2\gamma (m),$$

(23)

and

$$l(H_m)\le \prod _{i=1}^m{M}^2\mathrm{\Lambda }(i),l(H_m)\le \prod _{i=1}^m(1-\gamma (i)).$$

(24)

We have the following results.

Lemma 6.

If ${dim}_{P}E=1$, then there exists a subsequence ${\left\{a_k\right\}}_{k\ge 0}$ of ${\left\{m_k-1\right\}}_{k\ge 0}$, such that

(1)

$$\underset{k\to \infty }{lim}{(l(H_{a_k}))}^{{\displaystyle \frac{1}{a_k}}}=1;$$

(25)

(2)

Let $S_{\epsilon }(m)=\left\{j:1\le j\le m,\gamma (j)\le \epsilon \right\}$ for any $\epsilon \in (0,1)$ and $m\ge 1$, then

$$\underset{k\to \infty }{lim}{\displaystyle \frac{\#S_{\epsilon }(a_k)}{a_k}}=1,$$

(26)

where the cardinality is denoted by #;

(3)

For any $p\in (0,1]$, we have

$$\underset{k\to \infty }{lim}{(\prod _{j\in S_{\epsilon }(a_k)}(1-{(\gamma (j))}^p))}^{{\displaystyle \frac{1}{a_k}}}=1$$

(27)

for sufficiently small $\epsilon \in (0,1)$.

Proof. 

(1) By ${dim}_{P}E=1$ and Lemma 1, we have

$$\underset{k\to \infty }{\overline{lim}}{\displaystyle \frac{{log}_M{n}_1\cdots n_k{n}_{k+1}}{-{log}_M\frac{c_1{c}_2\cdots c_k-{\xi }_{k+1,0}-{\xi }_{k+1,n_{k+1}}}{n_{k+1}}}}=1.$$

(28)

Notice that

$$N_{k+1}^{\ast }=n_1\cdots n_k{n}_{k+1},{\delta }_k^{\ast }=c_1\cdots c_k-{\xi }_{k+1,0}-{\xi }_{k+1,n_{k+1}},n_{k+1}^{\ast }=n_{k+1},$$

(29)

then

$$\underset{k\to \infty }{\overline{lim}}{\displaystyle \frac{{log}_M{N}_k^{\ast }}{{log}_M{N}_k^{\ast }-{log}_M{N}_{k-1}^{\ast }{\delta }_{k-1}^{\ast }}}=\underset{k\to \infty }{\overline{lim}}{\displaystyle \frac{{log}_M{N}_{k+1}^{\ast }}{-{log}_M\frac{{\delta }_k^{\ast }}{n_{k+1}^{\ast }}}}=1,$$

(30)

which implies that

$$\underset{k\to \infty }{\overline{lim}}{(N_{k-1}^{\ast }{\delta }_{k-1}^{\ast })}^{{\displaystyle \frac{1}{{log}_M{N}_k^{\ast }}}}=1.$$

(31)

Since $M^{i_k}\le n_k^{\ast }<M^{i_k+1}$, $m_k=i_1+\cdots +i_k$, we have $N_k^{\ast }=n_1^{\ast }\cdots n_k^{\ast }<M^{m_k+k}\le M^{2m_k}$, then ${log}_M{N}_k^{\ast }\le 2m_k$. Notice that $m_k\ge 2$; then we have $m_k-1\ge {\displaystyle \frac{m_k}{2}}\ge {\displaystyle \frac{{log}_M{N}_k^{\ast }}{4}}$. Through Lemma 5, we have

$$\begin{array}{cc}\hfill \underset{k\to \infty }{\overline{lim}}{(l(H_{m_k-1}))}^{{\displaystyle \frac{1}{m_k-1}}}& \ge \underset{k\to \infty }{\overline{lim}}{[(1-{\displaystyle \frac{2\chi }{M}})(N_{k-1}^{\ast }{\delta }_{k-1}^{\ast })]}^{{\displaystyle \frac{1}{m_k-1}}}\hfill \\ \hfill & \ge \underset{k\to \infty }{\overline{lim}}{[(1-{\displaystyle \frac{2\chi }{M}})(N_{k-1}^{\ast }{\delta }_{k-1}^{\ast })]}^{{\displaystyle \frac{4}{{log}_M{N}_k^{\ast }}}}\hfill \\ \hfill & =1.\hfill \end{array}$$

(32)

Notice that

$$\underset{k\to \infty }{\overline{lim}}{(l(H_{m_k-1}))}^{{\displaystyle \frac{1}{m_k-1}}}\le \underset{k\to \infty }{\overline{lim}}{(l(H_{m_k-1}))}^0=1,$$

(33)

then

$$\underset{k\to \infty }{\overline{lim}}{(l(H_{m_k-1}))}^{{\displaystyle \frac{1}{m_k-1}}}=1.$$

(34)

Therefore, there exists a subsequence ${\left\{a_k\right\}}_{k\ge 0}$ of ${\left\{m_k-1\right\}}_{k\ge 0}$, such that

$$\underset{k\to \infty }{lim}{(l(H_{a_k}))}^{{\displaystyle \frac{1}{a_k}}}=1.$$

(35)

(2) Since $l(H_m)\le {\prod }_{j=1}^m(1-\gamma (j))$ for any $m\ge 1$, we have $l(H_{a_k})\le {\prod }_{j=1}^{a_k}(1-\gamma (j))$, thus

$${(l(H_{a_k}))}^{{\displaystyle \frac{1}{a_k}}}\le {(\prod _{j=1}^{a_k}(1-\gamma (j)))}^{{\displaystyle \frac{1}{a_k}}}\le 1-{\displaystyle \frac{1}{a_k}}\sum _{j=1}^{a_k}\gamma (j).$$

(36)

Let $k\to \infty$, combining the result of (1), we obtain that $1\le 1-{lim}_{k\to \infty }{\displaystyle \frac{1}{a_k}}{\sum }_{j=1}^{a_k}\gamma (j)$, which implies that

$$\underset{k\to \infty }{lim}{\displaystyle \frac{1}{a_k}}\sum _{j=1}^{a_k}\gamma (j)=0.$$

(37)

Notice that

$${\displaystyle \frac{a_k-\#S_{\epsilon }(a_k)}{a_k}}={\displaystyle \frac{\#\{i:1\le i\le a_k,\gamma (i)>\epsilon \}}{a_k}}\le {\displaystyle \frac{1}{a_k\epsilon }}\sum _{i=1}^{a_k}\gamma (i),$$

(38)

by the equality (37), we get

$$\underset{k\to \infty }{lim}{\displaystyle \frac{\#S_{\epsilon }(a_k)}{a_k}}=1.$$

(39)

(3) For any $p\in (0,1]$, by the Jensen’s inequality, we have

$${\displaystyle \frac{1}{a_k}}\sum _{j=1}^{a_k}{(\gamma (j))}^p\le {({\displaystyle \frac{1}{a_k}}\sum _{j=1}^{a_k}\gamma (j))}^p,$$

(40)

combining the equality (37), we have

$$\underset{k\to \infty }{lim}{\displaystyle \frac{1}{a_k}}\sum _{j=1}^{a_k}{(\gamma (j))}^p=0.$$

(41)

Since $log(1-x)\ge -2x$ for any $x\in (0,{\displaystyle \frac{1}{2}})$, we get

$$\begin{array}{cc}\hfill log{(\prod _{j\in S_{\epsilon }(a_k)}(1-{(\gamma (j))}^p))}^{{\displaystyle \frac{1}{a_k}}}& ={\displaystyle \frac{1}{a_k}}\sum _{j\in S_{\epsilon }(a_k)}log(1-{(\gamma (j))}^p)\hfill \\ \hfill & \ge {\displaystyle \frac{-2}{a_k}}\sum _{j\in S_{\epsilon }(a_k)}{(\gamma (j))}^p\hfill \\ \hfill & \ge {\displaystyle \frac{-2}{a_k}}\sum _{j=1}^{a_k}{(\gamma (j))}^p\hfill \end{array}$$

(42)

for sufficiently small $\epsilon \in (0,1)$. Together with (41), we obtain that

$$\underset{k\to \infty }{lim}{(\prod _{j\in S_{\epsilon }(a_k)}(1-{(\gamma (j))}^p))}^{{\displaystyle \frac{1}{a_k}}}=1.$$

(43)

□

5.2. The Estimate of ${\mu }_d$

Let $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ satisfy the condition of Theorem 1 with ${dim}_{P}E=1$; let f be a 1-dimensional quasisymmetric mapping, ${\left\{H_m\right\}}_{m\ge 0}$ be the length decreasing sequence in Lemma 4, and ${\left\{a_k\right\}}_{k\ge 1}$ be the sequence in Lemma 6. In using Lemma 2 to prove Theorem 1, we first estimate ${\mu }_d(J)$ for any branch J of $f(H_{a_k})$ for any $k\ge 1$.

Proposition 1.

For any $k\ge 1$ and any branch of $f(H_{a_k})$, denoted by J, there is a positive constant $C($independent of $J)$ satisfying ${\mu }_d(J)\le C{\left|J\right|}^d$ for any $d\in (0,1)$.

Proof. 

For any $d\in (0,1)$ and $k\ge 1$, let $J=J_{a_k}$ be a branch of $f(H_{a_k})$. For any $0\le j\le a_k-1$, let $J_j$ be a branch of $f(H_j)$ satisfying

$$J=J_{a_k}\subset J_{a_k-1}\subset \cdots \subset J_1\subset J_0=f(I_0^{\ast }).$$

(44)

With loss of generality, suppose $J_0=I_0^{\ast }=[0,1]$. By the definition of ${\mu }_d$, it is obvious that

$${\mu }_d(J_{a_k})={\displaystyle \frac{{\left|J_{a_k}\right|}^d}{{\sum }_{i=1}^{N(J_{a_k}-1)}\left|J_{a_k-1,i}\right|}}·{\displaystyle \frac{{\left|J_{a_k-1}\right|}^d}{{\sum }_{i=1}^{N(J_{a_k}-2)}\left|J_{a_k-2,i}\right|}}\cdots {\displaystyle \frac{{\left|J_1\right|}^d}{{\sum }_{i=1}^{N(J_0)}\left|J_{0,i}\right|}}{\left|J_0\right|}^d,$$

(45)

thus

$${\displaystyle \frac{{\mu }_d(J_{a_k})}{{\left|J_{a_k}\right|}^d}}=\prod _{j=0}^{a_k-1}{\displaystyle \frac{{\left|J_j\right|}^d}{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}}.$$

(46)

We start to estimate ${\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{{\left|J_j\right|}^d}}$ for any $0\le j\le a_k-1$. For any $k\ge 0$, $\sigma \in D_k$, let $L_{\sigma \ast 0}=[min(I_{\sigma }^{\ast }),min(I_{\sigma \ast 1}^{\ast }))$, $L_{\sigma \ast n_k^{\ast }}=(max(I_{\sigma \ast n_k^{\ast }}^{\ast }),max(I_{\sigma }^{\ast })]$. For any $0\le j\le a_k-1$, let $J_{j,1},J_{j,2},\cdots ,J_{j,N{(}_{J_j})}\subset J_j\subset f(H_j)$ be branches of $f(H_{j+1})$ located from left to right in $J_j$ and $G_{j,1},\cdots ,G_{j,N(J_j)-1}$ be gaps between $J_{j,1},J_{j,2},\cdots ,J_{j,N{(}_{J_j})}$. Let

$$I_j=f^{-1}(J_j),I_{j,l}=f^{-1}(J_{j,l})(\forall 1\le l\le N(J_j));$$

(47)

$$L_{j,l}=f^{-1}(G_{j,l})(\forall 1\le l\le N(J_j)-1).$$

(48)

It is obvious that

(1)

$I_j=f^{-1}(J_j)$ is a branch of $H_j$ and any $I_{j,l}\subset I_j$ is a branch of $H_{j+1}$;

(2)

$L_{j,1},\cdots ,L_{j,N(J_j)-1}$ are gaps between $I_{j,1},I_{j,2},\cdots ,I_{j,N(J_j)}$.

By Lemma 3, we have

$${\displaystyle \frac{\left|J_{j,l}\right|}{\left|J_j\right|}}\ge \beta {({\displaystyle \frac{\left|I_{j,l}\right|}{\left|I_j\right|}})}^q\ge \beta {(\lambda (j+1))}^q\ge {(4{\chi }^2)}^{-q}\beta {(\mathrm{\Lambda }(j+1))}^q,$$

(49)

$${\displaystyle \frac{\left|G_{j,l}\right|}{\left|J_j\right|}}\le 4{({\displaystyle \frac{\left|L_{j,l}\right|}{\left|I_j\right|}})}^p\le 4{(\mathrm{\Gamma }(j+1))}^p\le 4(2{\chi }^2\gamma (j+1)){)}^p\le 8{\chi }^2{(\gamma (j+1))}^p.$$

(50)

It follows from (49) that

$${\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{{\left|J_j\right|}^d}}\ge {(4{\chi }^2)}^{-dq}{\beta }^d{(\mathrm{\Lambda }(j+1))}^{dq}.$$

(51)

Next, we make another estimation of ${\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{{\left|J_j\right|}^d}}$ for any $(j+1)\in S_{\epsilon }(a_k)$ with sufficiently small $\epsilon \in (0,1)$.

Let $k\ge 1$ satisfy $m_{k-1}\le j<m_k$ and $I_j\subset I_{\sigma }^{\ast }(\sigma \in D_{k-1})$; together $N(J_j)\le M^2$ with (50), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}\left|J_{j,i}\right|}{\left|J_j\right|}}& \ge {\displaystyle \frac{\left|J_j\right|-{\sum }_{l=1}^{N(J_j)-1}\left|G_{j,l}\right|-\left|f(L_{\sigma \ast 0})\right|-\left|f(L_{\sigma \ast n_k^{\ast }})\right|}{\left|J_j\right|}}\hfill \\ \hfill & \ge 1-8M^2{\chi }^2{(\gamma (j+1))}^p-4({({\displaystyle \frac{{\xi }_{k,0}^{\ast }}{\left|I_j\right|}})}^p+{({\displaystyle \frac{{\xi }_{k,n_k}^{\ast }}{\left|I_j\right|}})}^p).\hfill \end{array}$$

(52)

Notice that ${\xi }_{k,0}^{\ast }+{\xi }_{k,n_k}^{\ast }\le {\underline{\alpha }}_k^{\ast }\le {\overline{\alpha }}_k^{\ast },$ then

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}\left|J_{j,i}\right|}{\left|J_j\right|}}& \ge 1-8M^2{\chi }^2{(\gamma (j+1))}^p-8{({\displaystyle \frac{\left|{\overline{\alpha }}_k^{\ast }\right|}{\left|I_j\right|}})}^p\hfill \\ \hfill & \ge 1-8M^2{\chi }^2{(\gamma (j+1))}^p-8{(\mathrm{\Gamma }(j+1))}^p\hfill \\ \hfill & \ge 1-8M^2{\chi }^2{(\gamma (j+1))}^p-8{(2{\chi }^2\gamma (j+1))}^p\hfill \\ \hfill & \ge 1-8{\chi }^2(M^2+2){(\gamma (j+1))}^p,\hfill \end{array}$$

(53)

which implies that

$${\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{{\left|J_j\right|}^d}}\ge {\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{{({\sum }_{i=1}^{N(J_j)}\left|J_{j,i}\right|)}^d}}\times {(1-8{\chi }^2(M^2+2){(\gamma (j+1))}^p)}^d.$$

(54)

For any $1\le l\le N(J_j)$, if $(j+1)\in S_{\epsilon }(a_k)$ and $\epsilon \in (0,{\displaystyle \frac{1}{2M^2{\chi }^2}})$, by the construction and properties of ${\left\{H_m\right\}}_{m\ge 0}$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\left|I_{j,l}\right|}{\left|I_j\right|}}& \ge {\displaystyle \frac{\left|I_j\right|-{\sum }_{l=1}^{N(J_j)-1}\left|L_{j,l}\right|-{\xi }_{k,0}^{\ast }-{\xi }_{k,n_k}^{\ast }}{2M^2\chi \left|I_j\right|}}\hfill \\ \hfill & \ge {\displaystyle \frac{\left|I_j\right|-(M^2-1){\overline{\alpha }}_k^{\ast }-{\overline{\alpha }}_k^{\ast }}{2\chi M^2\left|I_j\right|}}\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2\chi M^2}}(1-M^2\mathrm{\Gamma }(j+1))\hfill \\ \hfill & \ge {\displaystyle \frac{1-2{\chi }^2{M}^2\epsilon }{2\chi M^2}}.\hfill \end{array}$$

(55)

By Lemma 3, we obtain

$$1>{\displaystyle \frac{\left|J_{j,l}\right|}{\left|J_j\right|}}\ge \beta {({\displaystyle \frac{1-2{\chi }^2{M}^2\epsilon }{2\chi M^2}})}^q,$$

(56)

which implies that

$$1\ge {\displaystyle \frac{\left|J_{j,l}\right|}{{max}_{1\le i\le N(J_j)}\left|J_{j,i}\right|}}\ge \beta {({\displaystyle \frac{1-2{\chi }^2{M}^2\epsilon }{2\chi M^2}})}^q.$$

(57)

Notice that for any $d\in (0,1)$ and $x_1,\cdots ,x_k\in (0,1]$,

$${\displaystyle \frac{1+x_1^d+\cdots +x_k^d}{{(1+x_1+\cdots +x_k)}^d}}\ge {(1+max\left\{x_1,\cdots ,x_k\right\})}^{1-d},$$

(58)

thus

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{{({\sum }_{i=1}^{N(J_j)}\left|J_{j,i}\right|)}^d}}& =({\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d)/{({max}_{1\le i\le N(J_j)}\left|J_{j,i}\right|)}^d}{{({\sum }_{i=1}^{N(J_j)}\left|J_{j,i}\right|)}^d/{({max}_{1\le i\le N(J_j)}\left|J_{j,i}\right|)}^d}}\hfill \\ \hfill & \ge {(1+\beta {({\displaystyle \frac{1-2{\chi }^2{M}^2\epsilon }{2\chi M^2}})}^q)}^{1-d}\hfill \\ \hfill & \triangleq \kappa >1.\hfill \end{array}$$

(59)

Combining (54) with (59), we obtain

$${\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{{\left|J_j\right|}^d}}\ge \kappa \times {(1-8{\chi }^2(M^2+2){(\gamma (j+1))}^p)}^d$$

(60)

for any $\epsilon \in (0,{\displaystyle \frac{1}{2M^2{\chi }^2}})$ and $(j+1)\in S_{\epsilon }(a_k)$.

Let $M_1$ be an integer satisfying $M_1>8{\chi }^2(M^2+2)$. Notice that $1-M_{1}x>{(1-x)}^{M_1}$ if $x>0$ is sufficiently small; then, if $\epsilon >0$ is sufficiently small, for $(j+1)\in S_{\epsilon }(a_k)$, we have

$${\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{{\left|J_j\right|}^d}}\ge \kappa \times {(1-{(\gamma (j+1))}^p)}^{M_{1}d}.$$

(61)

For sufficiently small $\epsilon >0$, combining (51) with (61), notice that (24) holds; we have

$$\begin{array}{cc}\hfill \prod _{j=0}^{a_k-1}{\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{|J_{j,i}{|}^d}}& \ge \prod _{0\le j\le a_k-1,(j+1)\notin S_{\epsilon }(a_k)}{(4{\chi }^2)}^{-dq}{\beta }^d{(\mathrm{\Lambda }(j+1))}^{dq}\hfill \\ & \times \prod _{0\le j\le a_k-1,(j+1)\in S_{\epsilon }(a_k)}\kappa {(1-{(\gamma (j+1))}^p)}^{M_{1}d}\hfill \\ \hfill & =\prod _{0\le j\le a_k-1,(j+1)\notin S_{\epsilon }(a_k)}{(4{\chi }^2)}^{-dq}{\beta }^d\hfill \\ & \times \prod _{0\le j\le a_k-1,(j+1)\notin S_{\epsilon }(a_k)}{(\mathrm{\Lambda }(j+1))}^{dq}\hfill \\ \hfill & \times \prod _{0\le j\le a_k-1,(j+1)\in S_{\epsilon }(a_k)}\kappa {(1-{(\gamma (j+1))}^p)}^{M_{1}d}\hfill \\ \hfill & \ge {({(4M^2{\chi }^2)}^{-dq}{\beta }^d)}^{a_k-\#S_{\epsilon }(a_k)}\times {(l(H_{a_k}))}^{dq}\hfill \\ & \times \prod _{0\le j\le a_k-1,(j+1)\in S_{\epsilon }(a_k)}\kappa {(1-{(\gamma (j+1))}^p)}^{M_{1}d}.\hfill \end{array}$$

(62)

By (2) of Lemma 6, we have

$$\underset{k\to \infty }{lim}{({(4M^2{\chi }^2)}^{-dq}{\beta }^d)}^{{\displaystyle \frac{a_k-\#S_{\epsilon }(a_k)}{a_k}}}=1,$$

(63)

by (1) of Lemma 6, we have

$$\underset{k\to \infty }{lim}{(l(H_{a_k}))}^{{\displaystyle \frac{dq}{a_k}}}=1.$$

(64)

Notice that

$$\begin{array}{cc}\hfill & \prod _{0\le j\le a_k-1,(j+1)\in S_{\epsilon }(a_k)}\kappa {(1-{(\gamma (j+1))}^p)}^{M_{1}d}\hfill \\ & ={\kappa }^{\#S_{\epsilon }(a_k)}{(\prod _{0\le j\le a_k-1,(j+1)\in S_{\epsilon }(a_k)}(1-{(\gamma (j+1))}^p))}^{M_{1}d},\hfill \end{array}$$

(65)

combining (2) and (3) of Lemma 6, we have

$$\begin{array}{cc}\hfill & \underset{k\to \infty }{lim}{(\prod _{0\le j\le a_k-1,(j+1)\in S_{\epsilon }(a_k)}\kappa {(1-{(\gamma (j+1))}^p)}^{M_{1}d})}^{{\displaystyle \frac{1}{a_k}}}\hfill \\ & =\underset{k\to \infty }{lim}{\kappa }^{{\displaystyle \frac{\#S_{\epsilon }(a_k)}{a_k}}}{(\prod _{0\le j\le a_k-1,(j+1)\in S_{\epsilon }(a_k)}(1-{(\gamma (j+1))}^p))}^{{\displaystyle \frac{M_{1}d}{a_k}}}\hfill \\ & =\kappa .\hfill \end{array}$$

(66)

Together with (63), (64) and (66), we obtain

$$\underset{k\to \infty }{\underline{lim}}{(\prod _{j=0}^{a_k-1}{\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{|J_{j,i}{|}^d}})}^{{\displaystyle \frac{1}{a_k}}}\ge \kappa >1,$$

(67)

which implies that

$$\underset{k\to \infty }{\underline{lim}}\prod _{j=0}^{a_k-1}{\displaystyle \frac{{\sum }_{i=1}^{N(J_j)}{\left|J_{j,i}\right|}^d}{|J_{j,i}{|}^d}}=\infty .$$

(68)

Combining (68) with the equality (46), we obtain a constant $C>0$, such that

$${\mu }_d(J)\le C{\left|J\right|}^d.$$

(69)

□

5.3. The Proof of Theorem 1

The Proof of Theorem 1.

Now we start to finish the proof of Theorem 1. Let ${\left\{a_k\right\}}_{k\ge 1}$ be the sequence in Lemma 6. For any $x\in f(E)$, since $h_x(r)=\left|f^{-1}(B(x,r))\right|$ is a continuous mapping and $\underset{r\to 0}{lim}{h}_x(r)=0$, there exists a sequence ${\left\{r_k\right\}}_{k\ge 1}$, satisfying

$$\underset{I\in {\mathcal{H}}_{a_k}}{min}\left|I\right|\le h_x(r_k)<\underset{I\in {\mathcal{H}}_{a_k-1}}{min}\left|I\right|,$$

(70)

then $f^{-1}(B(x,r_k))$ meets at most two branches of $H_{a_k-1}$; thus, it meets at most $2M^2$ branches of $H_{a_k}$, and $B(x,r_k)$ meets at most $2M^2$ branches of $f(H_{a_k})$.

Let $R_1,R_2,\cdots ,R_l(1\le l\le 2M^2)$ be the branches of $f(H_{a_k})$ meeting $B(x,r_k)$, then

$$B(x,r_k)\cap f(E)\subset R_1\cup R_2\cup \cdots \cup R_l.$$

(71)

By Proposition 1, we get

$${\mu }_d(B(x,r_k))={\mu }_d(B(x,r_k)\cap f(E))\le \sum _{j=1}^l{\mu }_d(R_j)\le C\sum _{j=1}^l{\left|R_j\right|}^d.$$

(72)

Notice that

$$\underset{I\in {\mathcal{H}}_{a_k}}{min}\left|I\right|\le \left|f^{-1}(B(x,r_k))\right|,\underset{I\in {\mathcal{H}}_{a_k}}{max}\left|I\right|\le 2\chi \underset{I\in H_{a_k}}{min}\left|I\right|,$$

(73)

then for any $1\le j\le l$

$$\left|f^{-1}(R_j)\right|\le \underset{I\in {\mathcal{H}}_{a_k}}{max}\left|I\right|\le 2\chi \underset{I\in {\mathcal{H}}_{a_k}}{min}\left|I\right|\le 2\chi \left|f^{-1}(B(x,r_k))\right|.$$

(74)

Notice that $B(x,r_k)\cap R_j\ne \varnothing$ for any $1\le j\le l$, then

$$f^{-1}(R_j)\subset 6\chi f^{-1}(B(x,r_k)),$$

(75)

where for any $\rho >0$, $\rho I$ denotes the interval with the same center of the interval I, and $\left|\rho I\right|=\rho \left|I\right|$. By Lemma 3, there is a constant $K_{6\chi }$, such that

$$|R_j|=\left|f(f^{-1}(R_j))\right|\le \left|f(6\chi f^{-1}(B(x,r_k)))\right|\le K_{6\chi }\left|B(x,r_k)\right|=2K_{6\chi }{r}_k.$$

(76)

Combining (72) and (76), we have

$$\begin{array}{cc}\hfill {\mu }_d(B(x,r_k))& \le C\sum _{j=1}^l{\left|R_j\right|}^d\hfill \\ \hfill & \le C·2M^2{(2K_{6\chi }{r}_k)}^d\hfill \\ \hfill & =4K_{6\chi }^d{M}^{2}C{(r_k)}^d\hfill \\ \hfill & \triangleq C_1{(r_k)}^d.\hfill \end{array}$$

(77)

Since $\underset{k\to \infty }{lim}{r}_k=0$, then for any $x\in f(E)$

$$\underset{r\to 0}{\underline{lim}}{\displaystyle \frac{{\mu }_d(B(x,r))}{r^d}}\le C_1.$$

(78)

By Lemma 2 and (78), we obtain that ${dim}_{P}f(E)\ge d$. Notice that $d\in (0,1)$ is arbitrary; then we have

$${dim}_{P}f(E)\ge 1,$$

(79)

which implies that ${dim}_{P}f(E)=1$.

We finish the proof of Theorem 1. □

Remark 5.

Compared with the uniform Cantor sets in [10], the homogeneous perfect sets do not require the condition “${\xi }_{k,0}={\xi }_{k,n_k}=0$, ${\xi }_{k,i}={\xi }_{k,j}$ for any $k\ge 1$ and $1\le i<j\le n_k$ (which means for any $k\ge 1$ and any $(k-1)-$order basic interval $I_{\sigma }$, the furthest left point of $I_{\sigma }$ coincides with the furthest left point of $I_{\sigma \ast 1}$, the furthest right point of $I_{\sigma }$ coincides with the furthest right point of $I_{\sigma \ast n_k}$, and the lengths of the gaps contained in the same basic interval are equal)”. To weaken the influence of the inequal lengths for the gaps and the random structures for the basic intervals of the homogeneous perfect sets, we first reconstruct the homogeneous perfect sets, then carry out the research by referring to the methods of the research of the uniform Cantor sets.

6. Conclusions and Prospects

In this paper, we prove that the homogeneous perfect set $E=E(I_0,\left\{n_k\right\},\left\{c_k\right\},\left\{{\xi }_{k,l}\right\})$ with packing dimension 1, which exists as a real number $\chi \ge 1$ with ${max}_{1\le l\le n_k-1}{\xi }_{k,l}\le \chi {min}_{1\le l\le n_k-1}{\xi }_{k,l}$ for any $k\ge 1$, is a quasisymmetrically packing minimal set.

Since the homogeneous perfect sets in Theorem 1 are a class of the sets which have some requirements for the gaps between the basic intervals, our work has some limitations. We hope that we can apply our result to more fractals sets which do not have strict requirements for the gaps, such as the homogeneous Moran sets, the general one-dimensional Moran sets (the definitions can be seen in [20,21,22]), or we can obtain other results about the quasisymmetric packing minimality of homogeneous perfect sets under some conditions which are different from the condition “${max}_{1\le l\le n_k-1}{\xi }_{k,l}\le \chi {min}_{1\le l\le n_k-1}{\xi }_{k,l}$ for any $k\ge 1$”.