A homeomorphism mapping $f\colon X\to Y$, where $X$ and $Y$ are two metric spaces, is said to be quasisymmetric if there is a homeomorphism $\eta\colon[0, \infty)\to[0, \infty)$ such that

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

for all triples $a, b, x$ of distinct points in $X$. Here we follow the notation in Heinonen [1] by using $|x-y|$ to denote the distance between the two points $x$ and $y$ in every metric space. In particular, we also say that $f$ is an $n$-dimensional quasisymmetric mapping when $X=Y=\mathbb{R}^n$.

Deflnition 1 We call a set $E\subset \mathbb{R}^{n}$ quasisymmetrically packing-minimal, if $\dim_{P}f(E)\geq \dim_{P}E$ for any $n$-dimensional quasisymmetric mapping $f$.

In this paper, we will show that a large class of Moran sets in $\mathbb{ R}^{1}$ of packing dimension 1 have quasisymmetric packing-minimality.

Similarly, we call a set $E\subset \mathbb{R}^{n}$ is quasisymmetrically Hausdorff-minimal, if $\dim_{H}f(E)\geq \dim_{H}E$ for any $n$-dimensional quasisymmetric mapping $f$. Recall some results on the Hausdorff dimensions of quasisymmetric images. First, $n$-dimensional quasisymmetric mappings are locally Hölder continuous [2], so if $\dim_{H}E=0$, then $\dim_{H}f(E)=0$ and $E$ is quasisymmetrically Hausdorff-minimal. In Euclidean space $\mathbb{R}^{n}$ with $n\geq 2, $ Gehring [3, 4] obtained that for any subset $E\subset \mathbb{R}^{n}$ of Hausdorff dimension $n, $ its quasisymmetric image also has Hausdorff dimension $n$, so $E$ is quasisymmetrically Hausdorff-minimal. If $0<\dim_{H}E<1$, there are $1$-dimensional quasisymmetric mappings $ f_{\varepsilon }$ and $F_{\varepsilon }$ such that $\dim_{H}f_{\varepsilon }(E)<\varepsilon $ (see [5]) and $\dim_{H}F_{\varepsilon }(E)>1-\varepsilon $ (see [6]), that is any $E \subset \mathbb{R}^{1}$ satisfies $0<\dim_{H}E<1$ is not quasisymmetrically Hausdorff-minimal.

For $\mathbb{R}^{1}$, Tukia [7] found an interesting fact, quite different from Gehring's result for $\mathbb{R}^{n}$ with $n\geq 2$, that there exists $E\subset \mathbb{R}^{1}$ such that $\dim_{H}E=1$ and $\dim_{H}f(E)<1$ for some $1$-dimensional quasisymmetric mapping $f$, so $E$ is not quasisymmetrically Hausdorff-minimal.

There is a question: what kinds of sets in $\mathbb{R}^{1}$ are quasisymmetrically Hausdorff-minimal?

For $\mathbb{R}^{1}$, many works were devoted to the quasisymmetrically Hausdorff-minimal set, i.e., the subset $E\subset \mathbb{R}^{1}$ satisfying $\dim_{H}f(E)\geq \dim_{H}E$ for any $1$-dimensional quasisymmetric mapping $f$.

Kovalev [5] showed that any quasisymmetrically Hausdorff-minimal set in $\mathbb{R}^{1}$ with $\dim_{H}E> 0$ has full Hausdorff dimension $1.$ Hakobyan [8] proved that middle interval Cantor sets of Hausdorff dimension $1$ are all quasisymmetrically Hausdorff-minimal. Hu and Wen [9] obtained that some uniform Cantor sets of Hausdorff dimension $1\ $are quasisymmetrically Hausdorff-minimal. Dai, Wen, Xi and Xiong [10] found a large class of Moran sets of Hausdorff dimension $1$ which are quasisymmetrically Hausdorff-minimal.

Compared with quasisymmetric Hausdorff-minimality, there are few results on quasisymmetric packing-minimality.

Kovalev [5] showed that any quasisymmetrically packing-minimal set in $\mathbb{R}^{1}$ with $\dim_{p}E> 0$ has packing dimension $1.$ Li, Wu and Xi [11] find two classes of Moran sets of packing dimension $1$ which are quasisymmetrically packing-minimal. Wang and Wen [12] obtained that the uniform Cantor sets of packing dimension $1$ are quasisymmetrically packing-minimal.

In this paper, we will show that a result of [11] is not accidents. In fact, a larger class of Moran sets on the line with packing dimension $1$ is quasisymmetrically packing-minimal (Theorem 1).

This paper is organized as follows. In Section 2, we state our main results and give the introduction to the Moran sets. Some preliminaries are given in Section 3, including quasisymmetric mappings, Moran sets and certain probability measure supported on the quasisymmetric image. The key of this paper is to get the estimate in Lemma 1 for the above measure. Section 4 is the proof of Theorem 1.

Before the statement of theorems, we introduce the notion of Moran sets in $\mathbb{R}^{1}$. Let $\{n_{k}\}_{k\geq 1}\subset \mathbb{N}$ and $ \{c_{k, j}\}_{1\leq j\leq n_{k}}\subset \mathbb{R}^{+}$ be sequences satisfying $n_{k}\geq 2$ and $\sum\limits_{j=1}^{n_{k}}c_{k, j}<1$ for any $k\geq 1$, set $\Omega _{k}=\bigl\{\sigma =\sigma _{1}\cdots \sigma _{k}\colon $ $ \sigma _{j}\in \lbrack 1, n_{j}]\cap \mathbb{N}$ for all $1\leq j\leq k$$ \bigr\}$ and $\Omega _{0}=\{\emptyset \}$ with empty word $\emptyset $. Write $\Omega =\bigcup\limits_{k\geq 0}\Omega _{k}$ and $(\sigma _{1}\cdots \sigma _{k})\ast \sigma _{k+1}=\sigma _{1}\cdots \sigma _{k}\sigma _{k+1}$. Let $ I\subset \mathbb{R}^{1}$ be a closed interval. Denote by $|A|$ the diameter of $A\subset \mathbb{R}^{n}$. We say that $\mathcal{F}=\{I_{\sigma }\colon $ $\sigma \in \Omega \}$, which is a collection of closed intervals, has Moran structure $(I, \{n_{k}\}, \{c_{k, j}\})$, if $I_{\emptyset }=I$ and for any $ \sigma \in \Omega _{k-1}$, $I_{\sigma \ast 1}$, $\cdots $, $I_{\sigma \ast n_{k}}$, whose interiors are pairwise disjoint, are subintervals of $ I_{\sigma }$ such that

$ \begin{equation} |I_{\sigma \ast j}|/|I_{\sigma }|=c_{k, j}\text{ for all }j\text{.} \end{equation} $

(2.1)

Then a Moran set determined by $\mathcal{F}$ is defined by

$ \begin{equation} E(\mathcal{F})=\bigcap\limits_{k\geq 1}\bigcup\limits_{\sigma \in \Omega _{k}}I_{\sigma } \text{, } \end{equation} $

(2.2)

where any $I_{\sigma }$ in $\mathcal{F}$ is called a basic interval of rank $k$ if $\sigma \in \Omega _{k}$. Denote by $\mathcal{M} (I, \{n_{k}\}, \{c_{k, j}\})$ the class of all Moran sets associated with $I$, $ \{n_{k}\}$ and $\{c_{k, j}\}.$

For the class $\mathcal{M}(I, \{n_{k}\}, \{c_{k, j}\}), $ we write

$ \begin{equation*} D_{k}=\max\limits_{1\leq j\leq n_{k}}c_{k, j}\text{, }\quad c_{\ast }=\inf\limits_{k, j}c_{k, j} \end{equation*} $

and

$ \begin{equation*} s_{\ast }=\liminf\limits_{k\rightarrow \infty }s_{k}\quad \text{and}\quad s^{\ast }=\limsup\limits_{k\rightarrow \infty }s_{k}\text{, } \end{equation*} $

where $s_{k}$ is defined by the equation

$ \begin{equation*} \prod\limits_{i=1}^{k}\sum\limits_{j=1}^{n_{i}}(c_{i, j})^{s_{k}}=1\text{.} \end{equation*} $

If $\sigma \in \Omega_{k-1}, k \ge 1$, let $I_{\sigma }^{L}$ (or $I_{\sigma }^{R}$) be the most left (or the most right) one of $I_{\sigma \ast 1}$, $\cdots $, $I_{\sigma \ast n_{k}}$.Write $r_{\ast}=\inf\limits_{\sigma \in \Omega}\min \bigl\{ \frac{\mid I_{\sigma }^{L} \mid}{\mid I_{\sigma }\mid}, \frac{\mid I_{\sigma }^{R} \mid}{\mid I_{\sigma }\mid} \bigr\} $.

Some probability of quasisymmetric mappings and Moran sets can be seen in [13] and [14].

The main result of paper are stated as follows

Theorem 1 Suppose $E\in \mathcal{M}(I, \{n_{k}\}, \{c_{k, j}\})$, $r_{\ast}>0$ and $ \sup_{k}n_{k}<\infty $, and there exist a costant $l> 1$ such that $ \sum\limits_{i=1}^{n_{k}}({\mid I_{{\sigma }\ast{i}} \mid})\ge l ({{\mid I_{\sigma }^{L} \mid+\mid I_{\sigma }^{R} \mid} })$ for any $\sigma \in \Omega_{k-1}$ and $k\ge 2$. If $\dim_{P}E=1$, then $\dim_{P}f(E)=1$ for any $1$ -dimensional quasisymmetric mappings $f$.

Remark 1 Without loss of generality, suppose $\frac{\mid I_{\sigma }^{L} \mid}{\mid I_{\sigma }\mid}=c_{k, 1}$ and $\frac{\mid I_{\sigma }^{R} \mid}{\mid I_{\sigma }\mid}=c_{k, n_{k}}$ for $\sigma \in \Omega_{k}$, $k\ge 1$, the conditions of Theorem 1 implies $c_{k, 1}$ and $c_{k, n_{k}}$ is neither too "large" nor too "small", and for $2 \le i \le n_{k}-1, c_{k, i}$ may be very "small", even $c_{\ast}=0$, but $ \sum\limits_{i=2}^{n_{k}-1}c_{k, i} $ is not too "small".

Remark 2 Notice that the condition "and there exist a costant $l> 1$ such that $ \sum\limits_{i=1}^{n_{k}}({\mid I_{{\sigma }\ast{i}} \mid})\ge l ({{\mid I_{\sigma }^{L} \mid+\mid I_{\sigma }^{R} \mid} })$ for any $\sigma \in \Omega_{k-1}$ and $k\ge 2$" implies $n_{k} \geq 3$ for all $k \geq 1$. Notice that If $E\in \mathcal{M}(I, \{n_{k}\}, \{c_{k, j}\})$, then $ E\in \mathcal{M}(I, \{N_{k}\}, \{C_{k, q}\})$, where $N_{k}=n_{2k-1}\cdot n_{2k}\geq 3$ and $ C_{k, (i-1)n_{2k-1}+j}=c_{2k-1, i}\cdot c_{2k, j}$ for${}$ $1\leq i\leq n_{2k-1}, 1\leq j\leq n_{2k}$, so without loss of generality, we always assume that $n_{k}\geq 3$ for all $k \geq 1$ in this paper.

Example 1 Let $E\in \mathcal{M}(I, \{n_{k}\}, \{c_{k, j}\})$ with $c_{\ast }>0$. If $n_{k} \geq 3$ for all $k \geq 1$, then $r_{\ast}\ge c_{\ast} >0$, $ \sup_{k}n_{k}<\infty $ and $\sum\limits_{i=1}^{n_{k}}({\mid I_{{\sigma }\ast{i}} \mid})\ge (1+c_{\ast})({{\mid I_{\sigma }^{L} \mid+\mid I_{\sigma }^{R} \mid}})$ for any $\sigma \in \Omega_{k-1}$ and $k\ge 2$; if $\inf_{k} n_{k}=2$, then $ E\in \mathcal{M}(I, \{N_{k}\}, \{C_{k, q}\})$, where $N_{k}$ and $ C_{k, q}$ are defined the same as the above remark ($N_{k} \geq 3$), it is easy to obtain that $E\in \mathcal{M}(I, \{N_{k}\}, \{C_{k, q}\})$ satisfies the conditions of Theorem 1. Then by Theorem 1, if $\dim_{P}E=1$, we have $\dim_{P}f(E)=1$ for any $1$-dimensional quasisymmetric mapping $f$.

Therefore Theorem 1 extends the results of Theorem 2 in [11].

Example 2 Let $E$ be an uniform Cantor set (see [12]) with $c_{\ast }>0$. If $n_{k} \geq 3$, then $r_{\ast}=c_{\ast} >0$, $ \sup_{k}n_{k}<\infty $ and $\sum\limits_{i=1}^{n_{k}}({\mid I_{{\sigma }\ast{i}} \mid})\ge (1+c_{\ast})({{\mid I_{\sigma }^{L} \mid+\mid I_{\sigma }^{R} \mid}})$ for any $\sigma \in \Omega_{k-1}$ and $k\ge 2$; if $\inf_{k} n_{k}=2$, then $ E\in \mathcal{M}(I, \{N_{k}\}, \{C_{k, q}\})$, where $N_{k}$ and $ C_{k, q}$ are defined the same as the above remark($N_{k} \geq 3$), it is easy to obtain that $E\in \mathcal{M}(I, \{N_{k}\}, \{C_{k, q}\})$ satisfies the conditions of Theorem 1. Then by Theorem 1, if $\dim_{P}E=1$, we have $\dim_{P}f(E)=1$ for any $1$-dimensional quasisymmetric mapping $f$.

Therefore Theorem 1 extends the results of Theorem 1.2 in [12] when $c_{\ast} >0$.

Before the proofs of the two theorems, we give some preliminaries

The following fact on packing dimension can be found in Proposition 2.3 of [15].

Lemma 1 Let $E\subset \mathbb{R}^{n}$ be a Borel set, and $\mu $ a probability measure supported on $E$. If there exists $E^{\prime }\subset E$ with $\mu (E^{\prime })>0$ and a constant $c>0$ such that

$ \begin{equation*} \liminf\limits_{r\rightarrow 0}\frac{\mu (B(x, r))}{r^{s}}\leq c\text{ for all }x\in E^{\prime }\text{, } \end{equation*} $

then $\dim_{P}E\geq s$.

We need some properties on quasisymmetry. For closed interval $I$, set $\rho I$ be a closed interval with a length of $\rho |I|$ and with the same center with $I$.

From [16], it is easy to check the following lemma.

Lemma 2 Suppose $f:\mathbb{R}^{1}\rightarrow \mathbb{R}^{1}$ is quasisymmetric, there exist constants $\lambda $, $K_{\rho }>0$, $q\geq 1$ and $ p\in (0, 1]$ such that

$ \begin{equation} \frac{\mid f(\rho I)\mid }{\mid f(I)\mid }\leq K_{\rho } \end{equation} $

(3.1)

and

$ \begin{equation} \lambda (\frac{\mid I\mid }{\mid I^{\prime }\mid })^{q}\leq \frac{\mid f(I)\mid }{\mid f(I^{\prime })\mid }\leq 4(\frac{\mid I\mid }{\mid I^{\prime }\mid })^{p}, \label{3} \end{equation} $

(3.2)

whenever closed interval $I$, $I^{\prime }$ satisfying $I\subset I^{\prime }$

The following lemma comes from [17].

Lemma 3 Suppose $E$ is a Moran set satisfying the following conditions

$(1)$ $\sup\limits_{k}n_{k}<\infty $;

$(2)$ $0<\inf\limits_{k}D_{k}\le \sup\limits_{k}D_{k}<1$.

Then we have $\dim_{P}E=s^{\ast }.$

It is easy to verify that if for a Moran set $E\in \mathcal{M}(I, \{n_{k}\}, \{c_{k, j}\})$, the conditions of Theorem 1 hold, then $E$ satisfies $\sup\limits_{k}n_{k}<\infty $ and $0<\inf\limits_{k}D_{k}\le \sup\limits_{k}D_{k}<1$, by Lemma 3, $\dim_{P}E=s^{\ast }$.

The length of $\sigma \in \Omega _{k}$ will be denoted by $|\sigma |(=k)$.

Fix a $1$-dimensional quasisymmetric mapping $f:\mathbb{R}^{1}\rightarrow \mathbb{R}^{1}$. Given a Moran set $E$ and its basic interval $I_{\sigma }$ of $E$ with rank $k$, we also call $f(I_{\sigma })$ a basic interval of $ f(E) $ with rank $k$ for convenience. Let $J_{\sigma }=f(I_{\sigma }).$

Fix $d\in (0, 1)$. We will define a probability measure $\mu _{d}$ on $f(E)$ as follows.

Without loss of generality, we set $I=[0,1]$ the initial interval of $E$.

Let $\mu _{d}(f([0,1]))=1$, for every $k\geq 1$, and for every basic interval $J_{\sigma }$ of rank $k-1$, we define

$ \begin{equation*} \mu _{d}(J_{\sigma \ast j^{\prime }})=\frac{|J_{\sigma \ast j^{\prime }}|^{d} }{\sum\limits_{j=1}^{n_{k}}\left\vert J_{\sigma \ast j}\right\vert ^{d}}\mu _{d}(J_{\sigma }) \end{equation*} $

for $1\leq j^{\prime }\leq n_{k}$.

The next proposition can be found in [11].

Proposition 1 Suppose $E$ is the Moran set satisfying $\sup\limits_{k}n_{k}<\infty $ and

$ \begin{equation} \lim\limits_{k\rightarrow \infty }\frac{\text{card}\{1\leq i\leq k\colon D_{i}\leq \alpha \}}{k}=1 \label{pro1} \end{equation} $

(3.3)

for some constant $\alpha \in (0, 1)$. If s$^{\ast }=1$, then there exists a subsequence $\{k_{t}\}_{t}$ and a constant $c>0$ such that

$ \begin{equation} \mu _{d}(J_{\sigma })\leq c\left\vert J_{\sigma }\right\vert ^{d} \end{equation} $

(3.4)

for any basic interval $J_{\sigma }$ of $f(E)$ with $|\sigma |\in \{k_{t}\}_{t}$.

By Proposition 1, we have the corollary below.

Corollary 1 Suppose $E$ is the Moran satisfies the conditions of Theorem 1. If $\dim_{P}E=1, $ then there exists a subsequence $\{k_{t}\}_{t}$ and a constant $c>0 $ such that

$ \begin{equation*} \mu _{d}(J_{\sigma })\leq c\left\vert J_{\sigma }\right\vert ^{d} \end{equation*} $

for any basic interval $J_{\sigma }$ of $f(E)$ with $|\sigma |\in \{k_{t}\}_{t}$.

Proof Since $n_{k}\geq 2$, $r_{\ast }>0$, we have $D_{k}\leq 1-r_{\ast }<1$. Take $\alpha =1-r_{\ast }$, we have

$ \begin{equation*} \lim\limits_{k\rightarrow \infty }\frac{\text{card}\{1\leq i\leq k\colon D_{k}\leq 1-r_{\ast }\}}{k}=1, \end{equation*} $

notice that $\dim_{p}E=s^{\ast }$ by Lemma 3 and Proposition 1, the corollary follows.

Let $\{k_{t}\}_{t}$ be the subsequence in Proposition 1. Let

$ \begin{equation*} B_{t}=\biggl\{x\in f(E):f^{-1}(x)\in I_{\sigma }^{L}\cup I_{\sigma }^{R} \text{ for some }|\sigma |=k_{t}-1\biggr\} \end{equation*} $

and $B=\bigcup\limits_{s=1}^{\infty }\bigcap\limits_{t\geq s}B_{t}\text{.}$

Lemma 4 Suppose that $n_{k}\geq 3$ and $c_{\ast }>0$. Then there exists a constant $\epsilon >0$ such that

$ \begin{equation*} \frac{\mu _{d}(J_{\sigma}^ {L})+\mu _{d}(J_{\sigma}^ {R})}{\mu _{d}(J_{\sigma })}\leq 1-\epsilon \end{equation*} $

for all $\sigma $ with $|\sigma |=k-1$ and $1\leq j_{1}, j_{2}\leq n_{k}$.

Proof With out loss of generality, we let $J_{\sigma}^ {L}=J_{\sigma \ast 1}, J_{\sigma}^ {R}=J_{\sigma \ast n_{k}}$, since $n_{k}\geq 3$. Take $i_{0}(1\leq i_{0}\leq n_{k})$ as follows

Case 1 If $\max\limits_{1\leq j\leq n_{k}}|J _{\sigma \ast i}|\neq \max \bigl\{|J _{\sigma \ast 1}|, |J _{\sigma \ast n_{k}}|\bigr\}$, pick $i_{0}$ such that $\max\limits_{1\leq i\leq n_{k}}|J _{\sigma \ast i}|=|J _{\sigma \ast i_{0}}|$, then $2\leq i_{0}\leq n_{k}-1$, we have

$ \begin{eqnarray*} &&\frac{\mu _{d}(J_{\sigma}^ {L})+\mu _{d}(J_{\sigma}^ {R})}{\mu _{d}(J_{\sigma })} =\frac{\mu _{d}(J_{\sigma \ast 1})+\mu _{d}(J_{\sigma \ast n_{k}})}{\mu _{d}(J_{\sigma })} =\frac{\left\vert J_{\sigma \ast 1}\right\vert ^{d}+\left\vert J_{\sigma \ast n_{k}}\right\vert ^{d}}{\sum\limits_{i=1}^{n_{k}}\left\vert J_{\sigma \ast i}\right\vert ^{d}}\\ &\leq& 1-\frac{\left\vert J_{\sigma \ast i_{0}}\right\vert ^{d}}{ \sum\limits_{i=1}^{n_{k}}\left\vert J_{\sigma \ast i}\right\vert ^{d}} \leq 1-\frac{\left\vert J_{\sigma \ast i_{0}}\right\vert ^{d}}{ (\sup\limits_{k}n_{k})(\max\limits_{1\leq j\leq n_{k}}\left\vert J_{\sigma \ast i}\right\vert )^{d}} = 1-\frac{1}{ \sup\limits_{k}n_{k}} \text{. } \end{eqnarray*} $

Pick $\epsilon=\frac{1}{\sup\limits_{k}n_{k}}$, we have

$ \frac{\mu _{d}(J_{\sigma}^ {L})+\mu _{d}(J_{\sigma}^ {R})}{\mu _{d}(J_{\sigma })}\leq 1-\epsilon. $

Case 2 If $\max\limits_{1\leq j\leq n_{k}}|J _{\sigma \ast i}|=\max \bigl\{|J _{\sigma \ast 1}|, |J _{\sigma \ast n_{k}}|\bigr\}$, pick $i_{0}$ such that $|I _{\sigma \ast i_{0}}|=\max\limits_{2\leq i\leq n_{k}-1}|I _{\sigma \ast i}|$, we have

$ \begin{eqnarray*} &&\frac{\mu _{d}(J_{\sigma}^ {L})+\mu _{d}(J_{\sigma}^ {R})}{\mu _{d}(J_{\sigma })} =\frac{\mu _{d}(J_{\sigma \ast 1})+\mu _{d}(J_{\sigma \ast n_{k}})}{\mu _{d}(J_{\sigma })} =\frac{\left\vert J_{\sigma \ast 1}\right\vert ^{d}+\left\vert J_{\sigma \ast n_{k}}\right\vert ^{d}}{\sum\limits_{i=1}^{n_{k}}\left\vert J_{\sigma \ast i}\right\vert ^{d}}\\ &\leq& 1-\frac{\left\vert J_{\sigma \ast i_{0}}\right\vert ^{d}}{ \sum\limits_{i=1}^{n_{k}}\left\vert J_{\sigma \ast i}\right\vert ^{d}} \leq 1-\frac{\left\vert J_{\sigma \ast i_{0}}\right\vert ^{d}}{ (\sup\limits_{k}n_{k})(\max\limits_{1\leq j\leq n_{k}}\left\vert J_{\sigma \ast i}\right\vert )^{d}}. \end{eqnarray*} $

Assume $\left\vert J_{\sigma \ast 1}\right\vert =\max\limits_{i}\left\vert J_{\sigma \ast i}\right\vert $. Since

$ \begin{eqnarray*} &&\sup\limits_{k}n_{k}|I _{\sigma \ast i_{0}}|\geq\sum\limits_{2\le i\le n_{k}-1}|I_{\sigma \ast i}| =\sum\limits_{1\le i\le n_{k}}|I_{\sigma \ast i}|-|I_{\sigma \ast 1}|-|I_{\sigma \ast n_{k}}|\\ &\geq&(l-1)(|I_{\sigma \ast 1}|+|I_{\sigma \ast n_{k}}|) \geq(l-1)(1+\frac{1}{r_{\ast}})|I_{\sigma \ast 1}|\end{eqnarray*} $

and $\frac{|I_{\sigma \ast 1}|}{|I_{\sigma}|}\ge r_{\ast}, $ where $l$ and $r_{\ast}$ are obtained in Theorem 1, then there exists constants $\delta_{1}>0$ and $\delta_{2}>0$, such that $\frac{|I _{\sigma \ast i_{0}}|}{|I _{\sigma}|}\ge \delta_{1}$ and $\frac{|I _{\sigma \ast 1}|}{|I _{\sigma\ast i_{0}}|}\le \delta_{2}$.

By Lemma 2, we have

$ \begin{eqnarray*} \lambda (\frac{\left\vert I_{\sigma \ast i_{0}}\right\vert }{\left\vert I_{\sigma }\right\vert })^{q} &\leq &\frac{\left\vert J_{\sigma \ast i_{0}}\right\vert }{\left\vert J_{\sigma }\right\vert }\leq 4(\frac{ \left\vert I_{\sigma \ast i_{0}}\right\vert }{\left\vert I_{\sigma }\right\vert })^{p}\text{, } \\ \lambda (\frac{\left\vert I_{\sigma \ast 1}\right\vert }{\left\vert I_{\sigma }\right\vert })^{q} &\leq &\frac{\left\vert J_{\sigma \ast 1}\right\vert }{\left\vert J_{\sigma }\right\vert }\leq 4(\frac{ \left\vert I_{\sigma \ast 1}\right\vert }{\left\vert I_{\sigma }\right\vert })^{p}\leq 4(\frac{\delta_{2} \left\vert I_{\sigma \ast i_{0}}\right\vert }{\left\vert I_{\sigma }\right\vert })^{p}\text{, } \end{eqnarray*} $

which imply

$ \begin{equation*} \frac{\left\vert J_{\sigma \ast i_{0}}\right\vert }{\left\vert J_{\sigma \ast 1}\right\vert }\geq \frac{\lambda }{4\delta_{2} ^{p}}(\frac{\left\vert I_{\sigma \ast i_{0}}\right\vert }{\left\vert I_{\sigma }\right\vert } )^{q-p}\geq \frac{\lambda }{4\delta_{2} ^{p}}(\delta_{1})^{q-p}\text{.} \end{equation*} $

Therefore

$ \begin{eqnarray*} \frac{\mu _{d}(J_{\sigma}^ {L})+\mu _{d}(J_{\sigma}^ {R})}{\mu _{d}(J_{\sigma })}\leq1-\frac{\left\vert J_{\sigma \ast i_{0}}\right\vert ^{d}}{ (\sup\limits_{k}n_{k})(\left\vert J_{\sigma \ast 1}\right\vert )^{d}}\leq 1-\epsilon \text{, } \end{eqnarray*} $

where

$ \epsilon =\frac{\lambda ^{d}}{4^{d}\delta_{2}^{pd}}(\delta_{1})^{(q-p)d}/\sup\limits_{k}n_{k}. $

Proposition 2 $\mu _{d}(B)=0$ for $d\in (0, 1)$.

Proof It suffices to prove $ \mu _{d}(\bigcap\limits_{t\geq s}B_{t})=0\text{ for any }s\text{.}$

Let

$ \begin{equation*} E_{k}=\bigcup\limits_{\substack{ |\sigma |=k, \\ (\bigcap\limits_{t\geq s}B_{t})\bigcap J_{\sigma }\neq \varnothing }}J_{\sigma }\text{.} \end{equation*} $

Notice that $\bigcap\limits_{t\geq s}B_{t}=\bigcap\limits_{k=1}^{\infty }E_{k}$ and $ E_{k+1}\subset E_{k}\subset E_{k-1}\subset \cdots $, where

$ \begin{equation} \frac{\mu _{d}(E_{k+1})}{\mu _{d}(E_{k})}\leq 1\text{.} \label{one} \end{equation} $

(4.1)

For any $\sigma $, let $J_{\sigma }^{L}=f(I_{\sigma }^{L})$, $J_{\sigma }^{R}=f(I_{\sigma }^{R})$. Therefore, for $t\geq s$,

$ \begin{equation*} E_{k_{t}}=\bigcup\limits_{\substack{ |\sigma |=k_{t}-1, \\ (\bigcap\limits _{t\geq s}B_{t})\bigcap\limits J_{\sigma }\neq \varnothing }}\left( J_{\sigma }^{L}\bigcup\limits J_{\sigma }^{R}\right) \text{.} \end{equation*} $

By Lemma 4, we have

$ \begin{equation} \frac{\mu _{d}(E_{k_{t}})}{\mu _{d}(E_{k_{t}-1})}\leq \sup\limits_{|\sigma |=K_{t}-1}\frac{\mu _{d}(J_{\sigma }^{L})+\mu _{d}(J_{\sigma }^{R})}{\mu _{d}(J_{\sigma })}\leq 1-\epsilon \text{, } \label{two} \end{equation} $

(4.2)

it follows from (4.1) and (4.2) that

$ \begin{eqnarray*} \frac{\mu _{d}(E_{k_{t}})}{\mu _{d}(E_{k_{t-1}})} &=&\frac{\mu _{d}(E_{k_{t}})}{\mu _{d}(E_{k_{t}-1})}\cdot \frac{\mu _{d}(E_{k_{t}-1})}{ \mu _{d}(E_{k_{t}-2})}\cdots \frac{\mu _{d}(E_{k_{t-1}+1})}{\mu _{d}(E_{k_{t-1}})} \leq 1-\epsilon \text{, } \end{eqnarray*} $

which implies

$ \begin{equation*} \mu _{d}(\bigcap\limits _{t\geq s}B_{t})=\lim\limits_{t\rightarrow \infty }\mu _{d}(E_{k_{t}})\leq \lim\limits_{t\rightarrow \infty }(1-\epsilon )^{t-s}\mu _{d}(E_{k_{s}})=0\text{.} \end{equation*} $

Next we finish the proof of Theorem 1.

From the proposition above, we have $\mu _{d}(f(E)\backslash B)=1>0$. Fix $x\in (f(E)\backslash B)$, then we can pick $t_{n}\uparrow \infty $ satisfies $f^{-1}(x)\in I_{\sigma }\backslash (I_{\sigma }^{L}\bigcup\limits I_{\sigma }^{R})$ with some $|\sigma |=k_{t_{n}}-1$.

Notice that $g_{x}(\alpha )=\left\vert f^{-1}(B(x, \alpha ))\right\vert $ is continuous. We can pick $r_{n}$ such that

$ \begin{equation} \left\vert f^{-1}(B(x, r_{n}))\right\vert =\min \bigl\{|I_{\sigma}^ {L}|, |I_{\sigma}^ {R}|\bigr\}\text{, } \label{7} \end{equation} $

(4.3)

notice that $r_{n}\rightarrow 0$ when $n\rightarrow \infty $.

Since $f^{-1}(x)\in I_{\sigma }\backslash (I_{\sigma }^{L}\bigcup\limits I_{\sigma }^{R})$ and $\left\vert f^{-1}(B(x, r_{n}))\right\vert =\min \bigl\{|I_{\sigma}^ {L}|, |I_{\sigma}^ {R}|\bigr\}$, we have

$ \begin{equation*} f^{-1}(B(x, r_{n}))\subset I_{\sigma }\text{, } \end{equation*} $

which implies $B(x, r_{n})\subset J_{\sigma }$.

Let $J_{\sigma \ast j_{1}}$, $J_{\sigma \ast j_{2}}$, $\cdots $, $J_{\sigma \ast j_{l^{\prime}}}$ $(1\leq l^{\prime}\leq n_{k_{t_{n}}})$ be the basic intervals of rank $ k_{t_{n}}$ meeting $B(x, r_{n})$. Then $(B(x, r_{n})\bigcap\limits f(E))\subset \bigcup\limits _{i=1}^{l^{\prime}}J_{\sigma \ast j_{i}}$.

Using the conclusion of Corollary 2, we get

$ \begin{eqnarray} &&\mu _{d}(B(x, r_{n}))=\mu _{d}(B(x, r_{n})\bigcap\limits f(E)) \notag \\ &\leq &\mu _{d}(\bigcup\limits _{i=1}^{l^{\prime}}J_{\sigma \ast j_{i}})\leq \sum\limits_{i=1}^{l^{\prime}}\mu _{d}(J_{\sigma \ast j_{i}}) \label{888} \leq c\sum\limits_{i=1}^{l^{\prime}}\left\vert J_{\sigma \ast j_{i}}\right\vert ^{d}\text{. } \notag \end{eqnarray} $

Since $r_{\ast}>0$, for any $i$, there exists a constant $\delta \geq 1$,

$ \begin{equation*} \left\vert I_{\sigma \ast j_{i}}\right\vert \leq \delta \min \bigl\{|I_{\sigma}^ {L}|, |I_{\sigma}^ {R}|\bigr\} =\delta \left\vert f^{-1}(B(x, r_{n}))\right\vert \text{, } \end{equation*} $

hence $\left\vert I_{\sigma \ast j_{i}}\right\vert \subset (3\delta )f^{-1}(B(x, r_{n}))$, where $\delta \geq 1$. By Lemma 2, we have

$ \begin{equation*} \left\vert J_{\sigma \ast j_{i}}\right\vert =\left\vert f(I_{\sigma \ast j_{i}})\right\vert \leq \left\vert f((3\delta )f^{-1}(B(x, r_{n})))\right\vert \leq K_{3\delta }\left\vert (B(x, r_{n}))\right\vert =(2K_{3\delta })r_{n}\text{, } \end{equation*} $

where $K_{3\delta }>0$ is a constant. This together with (4.4) gives

$ \begin{equation*} \mu _{d}(B(x, r_{n})\leq \lbrack 2^{d}(\sup\limits_{i}n_{i})c(K_{3\delta })^{d}]\cdot r_{n}^{d}\text{.} \end{equation*} $

Let $n\rightarrow \infty $, then for any $x\in f(E)\backslash B$, there exists a constant $C^{\prime }>0$, such that

$ \begin{equation*} \liminf\limits_{r\rightarrow 0}\frac{\mu _{d}(B(x, r))}{r^{d}}\leq C^{\prime }\text{, } \end{equation*} $

it follows from $\mu _{d}(f(E)\backslash B)>0$ and Lemma 1 that $\dim_{p}f(E)\geq d$. Let $d\rightarrow 1$, we have

$ \begin{equation*} \dim_{p}f(E)=1\text{.} \end{equation*} $