On the Fractal Measures and Dimensions of Image Measures on a Class of Moran Sets

Abstract

In this work, we focus on the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure that determines the modified lower box dimension Moran fractal sets. The equivalence of these measures for a class of Moran is shown by having a strong separation condition. We give a sufficient condition for the equality of the Hewitt–Stromberg dimension, Hausdorff dimension, and packing dimensions. As an application, we obtain some relevant conclusions about the Hewitt–Stromberg measures and dimensions of the image measure of a $\tau$-invariant ergodic Borel probability measures. Moreover, we give some statistical interpretation to dimensions and corresponding geometrical measures.

1. Introduction

The measure theory has an important role in formulating calculus on fractal sets [1,2,3,4,5,6,7]. The most important (and well-known) measures in fractal geometry are the Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure. These measures have been investigated by several authors, highlighting their importance in the study of local properties of fractals and products of fractals [8,9,10,11,12,13,14,15,16,17,18,19]. Such measures appear explicitly, for example, in Pesin’s monograph and implicitly in Mattila’s text [20,21]. Finer measures have been used in the built analysis on a wider class of fractal sets [1]. Fractal Cantor-like sets and their measures were investigated in [7]. The quasi-Lipschitz equivalence of the Moran fractals was studied utilizing the Hausdorff dimension [22]. One of the purposes of this paper is to define and study a class of the Hewitt-Stromberg measures on Moran fractal sets. While Hausdorff and packing measures are defined using coverings and packing by families of sets with diameters that are less than a given positive number $\delta$, say, the Hewitt-Stromberg measures are defined using ’packing of balls’ with a fixed diameter $\delta$.

A Mathematical Theory of Communication, Claude E. Shannon’s 1948 article [23], was the first to establish the idea of entropy. Entropy is “a measure of the uncertainty associated with a random variable”, according to Wikipedia. In this context, “message” refers to a particular realization of the random variable, and “term” usually refers to the Shannon entropy, which measures the anticipated value of the information contained in a message, typically in unit-like bits. The Shannon entropy, on the other hand, measures the average amount of information that is lost when one does not know the value of the random variable. The introduction of Shannon entropy can be regarded as one of the most significant achievements during the previous fifty years in the literature on probabilistic uncertainty. Entropy created the framework for the thorough knowledge of communication theory. Several academic fields, such as statistical thermodynamics, spectral analysis, urban and regional planning, image reconstruction, business, queuing theory, economics, finance, operations research, biology, and manufacturing, among others, have used the concept of entropy. These applications will be discussed in the following section. This section reviews entropy as well as the related ideas of maximum entropy and directed divergence.

The Kolmogorov entropy is a crucial metric for describing how chaotic a system is. With relation to the phase point’s location on the attractor, it provides the average rate of information loss. The Rényi entropy [24] can be used to calculate the Kolmogorov entropy. The Shannon entropy in information theory is a particular instance of Rényi entropy. The thermodynamic entropy is the result of the Shannon entropy and Boltzmann constant. The fractal dimension is what distinguishes fractal formations. The family of fractal dimensions is limitless. In an e-dimensional space, a generalized fractal dimension can be defined. There is a direct relationship between the Rényi entropy and the generalized fractal dimension. The main purpose of this paper is to study some formulas for some fractal dimensions and measures of the image of measures that are computed using entropy on some Moran sets.

In the present paper, we investigate a class of Moran sets in general metric spaces and we discuss some proprieties and the equivalence of the fractal measures on these sets. As an application of the main result, we obtain similar formulas of measures and dimensions of the image of $\tau$-invariant measures in symbolic space using entropy as in the classical case of self-similar sets. We give some interesting examples; in particular, we discuss a group of Moran sets and provide some statistical explanations for the dimensions and associated geometrical measures.

The outline of the paper is as follows: In the next section, we define modified Hausdorff and packing measures. In Section 3, Moran fractal sets are defined and strong separation conditions are given. Moreover, the equivalence of the Hausdorff measure, the packing measure, and the Hewitt-Stromberg measures are proved. Section 4 gives some results about the Hewitt–Stromberg measures and dimensions of the images of $\tau$-invariant ones in symbolic space using entropy. Section 5 presents a conclusion.

2. The Fractal Measures

We first recall the definition of the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure. Throughout this paper, $B(x,r)$ stands for the closed ball

$$B(x,r)=\left\{y\in {\mathbb{R}}^n|\mathrm{d}(x,y)\le r\right\}.$$

For $t\ge 0$, $E\subseteq {\mathbb{R}}^n$ and $\delta >0$, we define the packing pre-measure,

$${\displaystyle {\overline{\mathcal{P}}}^t\left(E\right)=\underset{\delta >0}{inf}sup\left\{\sum _i{\left(2r_i\right)}^t|{\left(B(x_i,r_i)\right)}_i\mathrm{is}\mathrm{a}\mathrm{centered}\delta -\mathrm{packing}\mathrm{of}E\right\}.}$$

In a similar way, we define the Hausdorff pre-measure,

$${\displaystyle {\overline{\mathcal{H}}}^t\left(E\right)=\underset{\delta >0}{sup}inf\left\{\sum _i{\left(2r_i\right)}^t|{\left(B(x_i,r_i)\right)}_i\mathrm{is}\mathrm{a}\mathrm{centered}\delta -\mathrm{covering}\mathrm{of}E\right\}.}$$

Function ${\overline{\mathcal{H}}}^t$ is $\sigma$-sub-additive but not increasing, and function ${\overline{\mathcal{P}}}^t$ is increasing but not $\sigma$-sub-additive. That is why we introduce the modifications of the Hausdorff and packing measures ${\mathcal{H}}^t$ and ${\mathcal{P}}^t$:

$${\displaystyle {\mathcal{H}}^t\left(E\right)=\underset{F\subseteq E}{sup}{\overline{\mathcal{H}}}^t\left(F\right)\mathrm{and}{\mathcal{P}}^t\left(E\right)=\underset{E\subseteq {\bigcup }_i{E}_i}{inf}\left\{\sum _i{\overline{\mathcal{P}}}^t\left(E_i\right)|E_i\subseteq {\mathbb{R}}^n,\forall i\right\}.}$$

The Hewitt–Stromberg pre-measure of E is defined by ${\displaystyle D^t\left(E\right)=\underset{r\to 0}{lim\; inf}{N}_r\left(E\right){\left(2r\right)}^t,}$ where $N_r\left(E\right)$ is the minimum number of closed balls with diameter r, needed to cover E (the largest number of disjoint balls of radius r with centers in E). The Hewitt–Stromberg measure of E is defined by

$$d^t\left(E\right)=inf\left\{\sum _{i=1}^{+\infty }{D}^t\left(E_i\right)|E\subset \bigcup _i{E}_i\mathrm{and}\mathrm{the}{E}_i^{\prime }\mathrm{s}\mathrm{are}\mathrm{bounded}\right\}.$$

The functions ${\mathcal{H}}^t$, ${\mathcal{P}}^t$, and $d^t$ are metric outer measures and, thus, measures on the Borel family of subsets of ${\mathbb{R}}^n$ (see [15,16,25,26,27,28,29]). An important feature of the Hausdorff measure, packing measure, and Hewitt–Stromberg measure is that

$${\mathcal{H}}^t\left(E\right)\le d^t\left(E\right)\le {\mathcal{P}}^t\left(E\right)\mathrm{for}\mathrm{any}\mathrm{bounded}\mathrm{set}E\subseteq {\mathbb{R}}^n.$$

The Hausdorff dimension, the packing dimension, and the Hewitt–Stromberg dimension (modified lower box dimension) can be defined by

$$\begin{array}{ccc}{dim}_H\left(E\right)\hfill & =\hfill & inf\{t\ge 0\left|{\mathcal{H}}^t\left(E\right)=0\right\}=sup\{t\ge 0\left|{\mathcal{H}}^t\left(E\right)=+\infty \right\},\hfill \\ {dim}_P\left(E\right)\hfill & =\hfill & inf\{t\ge 0\left|{\mathcal{P}}^t\left(E\right)=0\right\}=sup\{t\ge 0\left|{\mathcal{P}}^t\left(E\right)=+\infty \right\},\hfill \\ {\underline{dim}}_{MB}\left(E\right)\hfill & =\hfill & inf\{t\ge 0\left|d^t\left(E\right)=0\right\}=sup\{t\ge 0\left|d^t\left(E\right)=+\infty \right\}.\hfill \end{array}$$

It follows immediately from the definitions that

$${dim}_H\left(E\right)\le {\underline{dim}}_{MB}\left(E\right)\le {\overline{dim}}_{MB}\left(E\right)={dim}_P\left(E\right),$$

where ${\overline{dim}}_{MB}$ is the modified upper box dimension (see for example [1,15,29,30]).

3. The Equivalence of Fractal Measures on Moran Fractal Sets

We will start by defining the Moran sets. Let ${\left\{n_k\right\}}_k$ and ${\left\{{\mathrm{\Phi }}_k\right\}}_{k\ge 1}$ be, respectively, two sequences of positive integers and positive vectors, such that

$${\displaystyle {\mathrm{\Phi }}_k=(c_{k_1},c_{k_2},\dots ,c_{k_{n_k}}),\sum _{j=1}^{n_k}{c}_{kj}\le 1,k\in \mathbb{N}.}$$

For any $m,k\in \mathbb{N}$, such that $m\le k$, let

$$D_{m,k}=\{(i_m,i_{m+1},\dots ,i_k)\left|1\le i_j\le n_j,m\le j\le k\right\}$$

and

$$D_k=D_{1,k}=\{(i_1,i_2,\dots ,i_k)\left|1\le i_j\le n_j,1\le j\le k\right\}.$$

We also set $D_0=\varnothing$ and ${\displaystyle D={\cup }_{k\ge 0}{D}_k,}$ Considering $\sigma =(i_1,i_2,\dots ,i_k)\in D_k$, $\tau =(j_1,j_2,\dots ,j_m)\in D_{k+1,m}$, we set

$$\sigma \ast \tau =(i_1,i_2,\dots ,i_k,j_1,j_2,\dots ,j_m).$$

Definition 1 

([31,32]). Let X be a complete metric space and $I\subset X$ a compact set with no empty interior (for convenience, we assume that the diameter of I is 1). The collection $\mathcal{F}=\{I_{\sigma }|\sigma \in D\}$ of subsets of I is called having Moran structure if

1.

For any $(i_1,i_2,\dots ,i_k)\in D_k$, $I_{i_1{i}_2\dots i_k}$ is similar to I. That is, there exists a similar transformation

$$\begin{array}{cccc}{S}_{i_1{i}_2\dots i_k}:\hfill & X\hfill & \to \hfill & X\hfill \\ & I\hfill & \mapsto \hfill & I_{i_1{i}_2\dots i_k},\hfill \end{array}$$

where we assume that $I_{\varnothing }=I$.

2.

For all $k\ge 1$, $(i_1,i_2,\dots ,i_{k-1})\in D_{k-1}$, $I_{i_1{i}_2\dots i_k}\left(i_k\in \{1,2,\dots ,n_k\}\right)$ are subsets of $I_{i_1{i}_2\dots i_{k-1}}$ and

$$I_{i_1{i}_2\dots i_{k-1},i_k}^{\circ }\cap I_{i_1{i}_2\dots i_{k-1},i_k^{\prime }}^{\circ }=\varnothing ,1\le i_k<i_k^{\prime }\le n_k,$$

where $I^{\circ }$ denotes the interior of I.

3.

For all $k\ge 1$ and $1\le j\le n_k$, taking $(i_1,i_2,\dots ,i_{k-1},j)\in D_k$, we have

$${\displaystyle 0<c_{kj}=c_{i_1{i}_2\dots i_{k-1}j}=\frac{|I_{i_1{i}_2\dots i_{k-1}j}|}{|I_{i_1{i}_2\dots i_{k-1}}|}<1,k\ge 2,}$$

where $|I|$ denotes the diameter of I.

Suppose that $\mathcal{F}$ is a collection of subsets of I having Moran structure. We call ${\displaystyle E=\bigcap _{k\ge 1}\bigcup _{\sigma \in D_k}{I}_{\sigma },}$ a Moran set determined by $\mathcal{F}$, and call ${\mathcal{F}}_k=\left\{I_{\sigma },\sigma \in D_k\right\}$ the k-order fundamental sets of E. I is called the original set of E. We assume ${\displaystyle \underset{k\to +\infty }{lim}\underset{\sigma \in D_k}{max}|I_{\sigma }|=0.}$ Then, for all $i\in D$, the set ${\displaystyle \left(\bigcap _{n\ge 1}{I}_{i_1{i}_2\dots i_n}\right)}$ is a single point. We shall denote it by $\phi \left(i\right)$. For all $w=(i_1,i_2,\dots i_k,\dots )\in D$, we use the abbreviation ${w|}_k$ for the first k elements of the sequence,

$$I_k\left(w\right)=I_{{w|}_k}=I_{i_1{i}_2\dots i_k},\mathrm{and}{c}_n\left(w\right)=c_{i_1{i}_2\dots i_n}.$$

(1)

Here, we consider a class of Moran sets E which satisfy a special property called the strong separation condition (SSC), i.e., Take any $I_{\sigma }\in \mathcal{F}$. Let $I_{\sigma *1},I_{\sigma *2},\dots ,I_{\sigma *n_{k+1}}$ be the $(k+1)$-order fundamental subsets. We say that $I_{\sigma }$ satisfies the (SSC) if $\mathrm{dist}(I_{\sigma *i},I_{\sigma *j})\ge {\delta }_k|I_{\sigma }|,\mathrm{for}\mathrm{all}i\ne j,$ where ${\left({\delta }_k\right)}_k$ is a sequence of positive real numbers, such that ${\displaystyle 0<\delta =\underset{k}{inf}{\delta }_k<1.}$

We focus on the equivalence of the centered Hausdorff measure, the packing measure, and the Hewitt–Stromberg measure for the Moran sets meeting the strong separation requirement. We outline the resources and preliminary findings that will be used in the validation of our major findings.

Lemma 1. 

Let μ be a finite Borel measure on ${\mathbb{R}}^n$ and E be a bounded Borel set. Then there exist two positive constants $A_1,A_2$, such that

$$\begin{array}{c}\hfill {\displaystyle A_1{d}^t\left(E\right)\underset{x\in E}{inf}\left\{\underset{r\to 0}{lim\; inf}\frac{\mu \left(B(x,r)\right)}{r^t}\right\}\le \mu \left(E\right)\le A_2{d}^t\left(E\right)\underset{x\in E}{sup}\left\{\underset{r\to 0}{lim\; sup}\frac{\mu \left(B(x,r)\right)}{r^t}\right\}.}\end{array}$$

Proof. 

Follows directly from Theorem 2.1 in [32].    □

Lemma 2 

([31,32]). Suppose that $E\subset I$ is a Moran set satisfying (SSC) and let μ be a finite Borel measure, such that $supp\mu \subset E$. Then, there exist positive constants $c_i,1\le i\le 4$ relating to $\delta ,t$, with which the following inequalities hold for any $\phi \left(i\right)\in E$

$$\begin{array}{c}\hfill c_1\underset{n\to +\infty }{lim\; inf}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^t}\le \underset{r\to 0}{lim\; inf}\frac{\mu \left(B(\phi \left(i\right),r)\right)}{r^t}\le c_2\underset{n\to +\infty }{lim\; inf}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^t}\end{array}$$

and

$$\begin{array}{c}\hfill c_3\underset{n\to +\infty }{lim\; sup}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^t}\le \underset{r\to 0}{lim\; sup}\frac{\mu \left(B(\phi \left(i\right),r)\right)}{r^t}\le c_4\underset{n\to +\infty }{lim\; sup}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^t}.\end{array}$$

Let $\mu$ be a finite Borel measure on ${\mathbb{R}}^n$, we define the dimension of a measure $\mu$ by

$${\underline{dim}}_{MB}\left(\mu \right)=sup\{t\ge 0\left|d^t\left(\mu \right)=+\infty \right\}=inf\{t\ge 0\left|d^t\left(\mu \right)=0\right\},$$

where

$${\displaystyle d^t\left(\mu \right)=\underset{E\subset {\mathbb{R}}^n}{inf}\{d^t\left(E\right)\left|\mu ({\mathbb{R}}^n\setminus E)=0\right\}.}$$

Definition 2. 

We say that two Borel measures μ and ν are equivalent and we write $\mu \sim \nu$ if for any Borel set A, we have $\mu \left(A\right)=0\iff \nu \left(A\right)=0.$

Theorem 1. 

Suppose that $E\subset I$ is a Moran set satisfying (SSC). Let μ be a finite Borel measure such that $supp\mu \subset E$.

1.

Suppose that there exists γ, such that

$$\begin{array}{c}{\displaystyle \underset{n\to +\infty }{lim}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^t}=\left\{\begin{array}{ccc}0\hfill & if\hfill & t<\gamma ,\hfill \\ +\infty \hfill & if\hfill & t>\gamma ,\hfill \end{array}\right.\mathit{for}\mathit{any}i\in D.}\hfill \end{array}$$

Then, ${\underline{dim}}_{MB}\left(E\right)=\gamma ={\underline{dim}}_{MB}\left(\mu \right)$.

2.

Suppose that μ satisfies a stronger condition at γ, i.e.,

$$\begin{array}{c}\hfill {\displaystyle 0<\underset{n\to +\infty }{lim}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^{\gamma }}<+\infty ,\mathit{for}\mathit{any}i\in D.}\end{array}$$

(2)

Then, $\mu \sim d^{\gamma }\mathrm{on}E$.

Proof. 

1.

From Lemmas 1 and 2, we can see that if $t>\gamma$, then

$$\begin{array}{ccc}\hfill d^t\left(E\right)& \le & {\displaystyle A_1^{-1}\mu \left(E\right)\underset{x\in E}{sup}\left\{{\displaystyle \underset{r\to 0}{lim\; sup}\frac{r^t}{\mu \left(B(x,r)\right)}}\right\}\le C\mu \left(E\right)\underset{i\in D}{sup}\left\{{\displaystyle \underset{n\to +\infty }{lim\; sup}\frac{{|I_n\left(i\right)|}^t}{\mu \left(I_n\left(i\right)\right)}}\right\}}\hfill \\ & =& {\displaystyle C\mu \left(E\right)\underset{i\in D}{sup}{\left\{{\displaystyle \underset{n\to +\infty }{lim\; inf}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^t}}\right\}}^{-1}=0,}\hfill \end{array}$$

where $C=A_1^{-1}{c}_2^{-1}$. It follows that ${\underline{dim}}_{MB}\left(E\right)\le \gamma$.

On the other hand, we can also see that if $t<\gamma$, then for all set $F\subset E$ such that $\mu \left(F\right)>0$, we have

$$\begin{array}{ccc}\hfill d^t\left(F\right)& \ge & {\displaystyle A_2^{-1}\mu \left(F\right)\underset{x\in F}{inf}\left\{{\displaystyle \underset{r\to 0}{lim\; inf}\frac{r^t}{\mu \left(B(x,r)\right)}}\right\}\ge C_1\mu \left(F\right)\underset{i\in {\phi }^{-1}\left(F\right)}{inf}\left\{{\displaystyle \underset{n\to +\infty }{lim\; inf}\frac{{|I_n\left(i\right)|}^t}{\mu \left(I_n\left(i\right)\right)}}\right\}}\hfill \\ & =& {\displaystyle C_1\mu \left(F\right)\underset{i\in {\phi }^{-1}\left(F\right)}{inf}{\left\{{\displaystyle \underset{n\to +\infty }{lim\; sup}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^t}}\right\}}^{-1}=+\infty ,}\hfill \end{array}$$

where $C_1=A_2^{-1}{c}_4^{-1}$. This leads to ${\underline{dim}}_{MB}\left(\mu \right)\ge \gamma$. Thus, ${\underline{dim}}_{MB}\left(\mu \right)\ge {\underline{dim}}_{MB}\left(E\right).$ From ${\underline{dim}}_{MB}\left(\mu \right)\le {\underline{dim}}_{MB}\left(E\right)$, we have ${\underline{dim}}_{MB}\left(\mu \right)=\gamma ={\underline{dim}}_{MB}\left(E\right).$

2.

Suppose that $\mu$ satisfies (2) and set

$${\mathrm{\Omega }}_n=\left\{{\displaystyle i\in D|\underset{r\to 0}{lim\; sup}\frac{r^{\gamma }}{\mu \left(B(\phi \left(i\right),r)\right)}<n}\right\}.$$

Then ${\displaystyle D\subset \bigcup _{n\ge 1}{\mathrm{\Omega }}_n}$. Suppose that $\mu \left(B\right)=0,$ for any $B\subset E$. Then, there exists a sequence of open sets ${\left\{G_k\right\}}_k$, such that $B\subset G_k$ and ${\displaystyle \mu \left(G_k\right)\le \frac{1}{k}}$, for all $k.$ Putting ${\mu }_k\left(A\right)=\mu (A\cap G_k)$ and taking into account that

$${\displaystyle \underset{r\to 0}{lim\; sup}\frac{r^{\gamma }}{{\mu }_k\left(B(x,r)\right)}=\underset{r\to 0}{lim\; sup}\frac{r^{\gamma }}{\mu \left(B(x,r)\right)},\mathrm{for}\mathrm{all}x\in B.}$$

From Lemma 1, we can see that, for any $k,n\in \mathbb{N}$ and $x\in B$, we have

$$\begin{array}{ccc}\hfill d^t\left(B\cap \phi \left({\mathrm{\Omega }}_n\right)\right)& \le & {\displaystyle A_1^{-1}{\mu }_k\left(B\right)\underset{x\in B\cap \phi \left({\mathrm{\Omega }}_n\right)}{sup}\left\{{\displaystyle \underset{r\to 0}{lim\; sup}\frac{r^{\gamma }}{{\mu }_k\left(B(x,r)\right)}}\right\}}\hfill \\ & \le & {\displaystyle A_1^{-1}\mu \left(G_k\right)\underset{x\in B\cap \phi \left({\mathrm{\Omega }}_n\right)}{sup}\left\{{\displaystyle \underset{r\to 0}{lim\; sup}\frac{r^{\gamma }}{\mu \left(B(x,r)\right)}}\right\}\le A_1^{-1}\frac{n}{k}.}\hfill \end{array}$$

Letting $k\to +\infty$, we obtain $d^t\left(B\cap \phi \left({\mathrm{\Omega }}_n\right)\right)=0,$ for any $n\in \mathbb{N}.$ This leads to

$${\displaystyle d^t\left(B\right)\le \sum _{n\ge 1}{d}^t\left(B\cap \phi \left({\mathrm{\Omega }}_n\right)\right)=0.}$$

On the other hand, if we assume that $d^t\left(B\right)=0$, for all $B\subset E$, then setting

$${\mathrm{\Omega }}_n^{\prime }=\left\{{\displaystyle i\in D|\underset{r\to 0}{lim\; inf}\frac{r^{\gamma }}{\mu \left(B\left(\phi \right(i),r)\right)}>\frac{1}{n}}\right\},$$

we easily obtain, from Lemma 1, where

$$\begin{array}{ccc}\hfill 0=d^t\left(B\cap \phi \left({\mathrm{\Omega }}_n^{\prime }\right)\right)& \ge & {\displaystyle A_2^{-1}\mu \left(B\cap \phi \left({\mathrm{\Omega }}_n^{\prime }\right)\right)\underset{x\in B\cap \phi \left({\mathrm{\Omega }}_n^{\prime }\right)}{inf}\left\{{\displaystyle \underset{r\to 0}{lim\; inf}\frac{r^{\gamma }}{\mu \left(B(x,r)\right)}}\right\}}\hfill \\ & \ge & {\displaystyle \frac{A_2^{-1}}{n}\mu \left(B\cap \phi \left({\mathrm{\Omega }}_n^{\prime }\right)\right).}\hfill \end{array}$$

This implies that $\mu \left(B\cap \phi \left({\mathrm{\Omega }}_n^{\prime }\right)\right)=0$, for any $n\in \mathbb{N}$. Now, thanks to the fact that ${\displaystyle \mu \left(\phi \left({\cup }_{n\ge 1}{\mathrm{\Omega }}_n^{\prime }\right)\right)=1,}$ we deduce that $\mu \left(B\right)=0$, which ends the proof.

   □

Example 1. 

We will consider in this example a special case in which the conditions of all the numbers $c_{i_1,i_2,\dots ,i_n}$ depend only on the length and the last variable, i.e.,

$$c_{i_1,i_2,\dots ,i_n}=c_{n,i_n}.$$

We define the pressure function π and the Gibbs measure ${\mu }_t$ defined respectively by

$$\pi \left(t\right)=\underset{n\to +\infty }{lim}{\pi }_n\left(t\right)\mathit{where}{\pi }_n\left(t\right)=\frac{1}{n}log\left(\sum _{(i_1,i_2,\dots ,i_n)\in D_n}\prod _{j=1}^n{c}_{j,i_j}^t\right).$$

(3)

and

$${\mu }_t(i_1{i}_2\dots i_n)=\prod _{j=1}^n\frac{c_{j,i_j}^t}{Z_j^t},$$

(4)

where $Z_n^t={\sum }_j{c}_{n,j}^t$ for any $(i_1,i_2,\dots ,i_n)\in D_n$, $n\in \mathbb{N}$. It is not difficult to see that π is strictly decreasing and continuous. Moreover, we have

$${\mu }_t(i_1{i}_2\dots i_n)={|I_n\left(w\right)|}^t{\left(\prod _{j=1}^n{Z}_j^t\right)}^{-1}.$$

Since, for all n, we have

$$\sum _{(i_1,i_2,\dots i_n)\in D_n}\prod _{j=1}^n{c}_{j,i_j}^t=\prod _{j=1}^n{Z}_j^t$$

then

$$\underset{n\to +\infty }{lim}\frac{{\mu }_t(i_1{i}_2\dots i_n)}{|I_n{\left(w\right)|}^{\gamma }}=\underset{n\to +\infty }{lim}|I_n{\left(w\right)|}^{t-\gamma }\left(e^{-log{\prod }_{j=1}^n{Z}_j^t}\right)=\underset{n\to +\infty }{lim}{|I_n\left(w\right)|}^{t-\gamma }\left(e^{-n{\pi }_n\left(t\right)}\right).$$

Now, suppose that $t:=\alpha$ is the unique number, such that

$$\sum _{(i_1,i_2,\dots ,i_n)\in D_n}{|I_{i_1{i}_2\dots i_n}|}^{\alpha }=1.$$

It is clear that $\pi \left(\alpha \right)=0$, and then

$$\underset{n\to +\infty }{lim}\frac{{\mu }_{\alpha }(i_1{i}_2\dots i_n)}{|I_n{\left(w\right)|}^{\gamma }}=+\infty \mathit{if}\gamma >\alpha$$

and

$$\underset{n\to +\infty }{lim}\frac{{\mu }_{\alpha }(i_1{i}_2\dots i_n)}{|I_n{\left(w\right)|}^{\gamma }}=0\mathit{if}\gamma <\alpha .$$

It follows from Theorem 1 that $\alpha ={\underline{dim}}_{MB}\left(E\right)={\underline{dim}}_{MB}\left({\mu }_{\alpha }\right)$.

Theorem 2. 

Suppose that $E\subset I$ is a Moran set satisfying (SSC). Let μ be a finite Borel measure with $supp\mu \subset E$ and $0<\alpha <n$ such that

$${\displaystyle 0<\underset{n\to +\infty }{lim}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^{\alpha }}<+\infty \mathit{for}\mathit{any}i\in D.}$$

Then

$${dim}_H\left(E\right)={\underline{dim}}_{MB}\left(E\right)={\overline{dim}}_{MB}\left(E\right)={dim}_P\left(E\right)=\alpha .$$

In addition,

$${\mathcal{H}}^{\alpha }\sim d^{\alpha }\sim {\mathcal{P}}^{\alpha }\mathit{on}E.$$

Proof. 

Follows from Theorem 1 and ([32], Theorems 4.2 and 4.3). □

Example 2. 

Let $I=[0,1]$, $n_k=2$ and $c_{kj}=\frac{1}{3}$ far all $k\ge 1$ and $1\le j\le n_k$. In this case, the Moran set E is the classical ternary Cantor set. Let $\alpha =\frac{log2}{log3}$ and μ be a probability measure on I such that

$$\mu \left(I_n\left(i\right)\right)=\left\{\begin{array}{cc}|I_n{\left(i\right)|}^{\alpha }\hfill & \mathit{if}i\in D,\hfill \\ \\ 0\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

It is clear that $supp\mu \subset E$ and

$${\displaystyle \underset{n\to +\infty }{lim}\frac{\mu \left(I_n\left(i\right)\right)}{{|I_n\left(i\right)|}^{\alpha }}=1.}$$

Therefore,

$${dim}_H\left(E\right)={\underline{dim}}_{MB}\left(E\right)={\overline{dim}}_{MB}\left(E\right)={dim}_P\left(E\right)=\frac{log2}{log3}\mathit{and}{\mathcal{H}}^{\alpha }\sim d^{\alpha }\sim {\mathcal{P}}^{\alpha }\mathit{on}E.$$

Example 3. 

Let $A=\{a,b\}$ be a two-letter alphabet, and $A^{*}$ the free monoid generated by A. Let F be the homomorphism on $A^{*},$ defined by $F\left(a\right)=ab$ and $F\left(b\right)=a$. It is easy to see that $F^n\left(a\right)=F^{n-1}\left(a\right)F^{n-2}\left(a\right)$. We denote by $\left|F^n\left(a\right)\right|$ the length of the word $F^n\left(a\right)$, thus

$$F^n\left(a\right)=s_1{s}_2\cdots s_{\left|F^n\left(a\right)\right|},s_i\in A.$$

Therefore, as $n\to +\infty$, we have the infinite sequence

$$\omega =\underset{n\to +\infty }{lim}{F}^n\left(a\right)=s_1{s}_2{s}_3\cdots s_n\cdots \in {\{a,b\}}^{\mathbb{N}},$$

which is called the Fibonacci sequence. For any $n\ge 1,$ write ${\omega }_n={\left.\omega \right|}_n=s_1{s}_2\cdots s_n$. We denote by ${\left|{\omega }_n\right|}_a$ the number of the occurrence of the letter $a\mathrm{in}{\omega }_n,\mathrm{and}{\left|{\omega }_n\right|}_b$ the number of occurrences of $b.$ Then, ${\left|{\omega }_n\right|}_a+{\left|{\omega }_n\right|}_b=n$. It follows that ${\displaystyle \underset{n\to +\infty }{lim}\frac{{\left|{\omega }_n\right|}_a}{n}=\eta ,}$ where ${\eta }^2+\eta =1$.

Let $0<r_a<\frac{1}{2},0<r_b<\frac{1}{3},r_a,r_b\in \mathbb{R}$. In the Moran construction above, let

$$n_k=\left\{\begin{array}{cc}2,\hfill & \mathit{if}{s}_k=a\hfill \\ 3,\hfill & \mathit{if}{s}_k=b,\hfill \end{array}\right.$$

$$c_{k_j}=c_k=\left\{\begin{array}{cc}{r}_a,\hfill & \mathit{if}{s}_k=a\hfill \\ r_b,\hfill & \mathit{if}{s}_k=b\hfill \end{array},1\le j\le n_k.\right.$$

Then we construct the homogeneous Moran set related to the Fibonacci sequence and denote it by $E:=E\left(\omega \right)=\left(I,\left\{n_k\right\},\left\{c_k\right\}\right)$. Through the construction of $E,$ we have

$$\left|I_{\sigma }\right|=r_a^{{\left|{\omega }_k\right|}_a}{r}_b^{{\left|{\omega }_k\right|}_b},\forall \sigma \in D_k.$$

There exists a probability measure ${\nu }_{\alpha }$ supported by E such that for any $k\ge 1$ and ${\sigma }_0\in D_k$,

$${\nu }_{\alpha }\left(I_{{\sigma }_0}\right)=\frac{{\left|I_{{\sigma }_0}\right|}^{\alpha }}{{\displaystyle \sum _{\sigma \in D_k}{\left|I_{\sigma }\right|}^{\alpha }}},\mathit{for}\mathit{all}\alpha \in \mathbb{R}.$$

Let $\alpha =\frac{-log2-\eta log3}{log{r}_a+\eta log{r}_b},$ where ${\eta }^2+\eta =1$. It is clear that there exists a positive constant c, such that

$$\underset{n\to +\infty }{lim}\frac{{\nu }_{\alpha }\left(I_n\left(w\right)\right)}{|I_n{\left(w\right)|}^{\gamma }}=c\underset{n\to +\infty }{lim}{|I_n\left(w\right)|}^{\alpha -\gamma }\mathit{for}\mathit{all}\gamma \in \mathbb{R}.$$

This implies that

$$\underset{n\to +\infty }{lim}\frac{{\nu }_{\alpha }\left(I_n\left(w\right)\right)}{|I_n{\left(w\right)|}^{\gamma }}=+\infty \mathit{if}\gamma >\alpha$$

and

$$\underset{n\to +\infty }{lim}\frac{{\nu }_{\alpha }\left(I_n\left(w\right)\right)}{|I_n{\left(w\right)|}^{\gamma }}=0\mathit{if}\gamma <\alpha .$$

Finally, Theorem 1 gives that ${\underline{dim}}_{MB}\left(E\right)={\underline{dim}}_{MB}\left({\nu }_{\alpha }\right)=\frac{-log2-\eta log3}{log{r}_a+\eta log{r}_b},$ where ${\eta }^2+\eta =1$.

Remark 1. 

As we analyze the Hewitt–Stromberg measures and dimensions, our findings are closely comparable to those in the cited work [32], for which M. Dai also demonstrates similar findings for the Hausdorff and packing measures. Even though the results are parallel, we want to point out that the goal of our paper is to draw some useful inferences about the Hewitt–Stromberg measures and dimensions of the image measure of τ-invariant ergodic Borel probability measures, and provide a statistical explanation for the dimensions and associated geometrical measures in the next section.

4. The Dimension of Image Measure

In this section, we use entropy to derive analogous formulas for the Hewitt–Stromberg measures and the image dimensions of $\tau$-invariant measures in a symbolic space. We focus on a particular class of Moran sets and provide some statistical explanations for the dimensions and associated geometrical measures.

Let E be a Moran set satisfying the strong separation condition and $\mu$ be a probability measure defined on the $\sigma$-algebra $\mathcal{F}$ of subsets of D. Wa said that $\tau$ preserves the measure $\mu$ (or $\mu$ is $\tau$-invariant), if

$$\mu \left({\tau }^{-1}\left(B\right)\right)=\mu \left(B\right),B\in \mathcal{F}.$$

An invariant probability measure is said to be ergodic if for every set $B\in \mathcal{F}$, ${\tau }^{-1}\left(B\right)=B$ has either zero or full measure.

Theorem 3 

([33]). Let μ be the τ-invariant ergodic probability measure on D and $f\in L^1(D,\mathbb{R})$. Then

$$\underset{n\to +\infty }{lim}\frac{1}{n}\sum _{k=0}^{n-1}f\circ {\tau }^k={\int }_{D}fd\mu ,\mu a.e.$$

We define the entropy $h_{\mu }\left(\tau \right)$ of $\tau$ with respect to the measure $\mu$ by

$$h_{\mu }\left(\tau \right)=\underset{n\to +\infty }{lim}\frac{1}{n}\sum _{w\in D_n}\mu (i_1{i}_2,\dots i_n)log\mu (i_1{i}_2\dots i_n).$$

In this section, we will study the image measure of $\mu$ by $\phi$ denoted by ${\phi }^{*}\mu$ and defined by

$${\phi }^{*}\mu \left(B\right)=\mu \circ {\phi }^{-1}\left(B\right),\mathrm{for}\mathrm{all}B\in \mathcal{B}\left(\mathcal{X}\right)$$

where $\mathcal{B}\left(\mathcal{X}\right)$ is the Borel algebra on $\mathcal{X}$ and $\phi :D⟶E$ is a measurable map. We will assume through this section that

$$0<\underset{n\to +\infty }{lim\; inf}\prod _{j=1}^n\frac{c_j\left(w\right)}{c_j\left(\tau \left(w\right)\right)}\le \underset{n\to +\infty }{lim\; sup}\prod _{j=1}^n\frac{c_j\left(w\right)}{c_j\left(\tau \left(w\right)\right)}<+\infty$$

(5)

and

$$\underset{n\to +\infty }{lim\; inf}-\frac{1}{n}log|I_n\left(w\right)|=\underset{n\to +\infty }{lim\; sup}-\frac{1}{n}log|I_n\left(w\right)|$$

(6)

for all $w=(i_1,i_2,\dots i_k,\dots )\in D$. Our main result in this section is the following.

Theorem 4. 

Let E be a Moran set satisfying the strong separation condition and μ be a τ-invariant ergodic Borel probability measures on D.

1.

One has

$${\underline{dim}}_{MB}{\phi }^{*}\mu =h_{\mu }\left(\tau \right){\left(\int \underset{n\to +\infty }{lim}\frac{-1}{n}\sum _{j=1}^{n}ln{c}_j\left(i\right)d\mu \right)}^{-1}.$$

2.

If we denote by $\alpha ={\underline{dim}}_{MB}\left({\phi }^{*}\mu \right)$ then

$$d^{\alpha }\left({\phi }^{*}\mu \right)=0\iff \underset{n\to +\infty }{lim}\frac{{\phi }^{*}\mu \left(I_n\left(w\right)\right)}{{\prod }_{j=1}^n{c}_j{\left(w\right)}^{\alpha }}=+\infty \mu -a.e.w.$$

$$0<d^{\alpha }\left({\phi }^{*}\mu \right)<+\infty \iff 0<\underset{n\to +\infty }{lim}\frac{{\phi }^{*}\mu \left(I_n\left(w\right)\right)}{{\prod }_{j=1}^n{c}_j{\left(w\right)}^{\alpha }}<+\infty \mu -a.e.w.$$

$$d^{\alpha }\left({\phi }^{*}\mu \right)=+\infty \iff \underset{n\to +\infty }{lim}\frac{{\phi }^{*}\mu \left(I_n\left(w\right)\right)}{{\prod }_{j=1}^n{c}_j{\left(w\right)}^{\alpha }}=0\mu -a.e.w.$$

Proof. 

1.

We recall that $|I_n\left(w\right)|={\prod }_{j=1}^n{c}_j\left(w\right),$ for all $w=(i_1,i_2,\dots i_k,\dots )\in D$ and then using (6), we set

$${\mathrm{\Gamma }}^{+}\left(w\right)=\underset{n\to +\infty }{lim}-\frac{1}{n}\sum _{j=1}^{n}log{c}_j\left(w\right).$$

We consider, for $\alpha \in \mathbb{R}$, the set

$$D_{\mu }\left(\alpha \right)=\left\{w\in D,{\mathrm{\Gamma }}^{+}\left(w\right)=\alpha \mathrm{and}\underset{n\to +\infty }{lim}\frac{1}{n}log\mu \left(w_{|n}\right)=-h_{\mu }\left(\tau \right)\right\}.$$

From (5) we have ${\mathrm{\Gamma }}^{+}\left(w\right)={\mathrm{\Gamma }}^{+}\left(\tau \left(w\right)\right)>0$ and since $\mu$ is ergodic, it follows from Theorem 3 that there exists ${\alpha }_{\mu }$ such that

$${\mathrm{\Gamma }}^{+}\left(w\right)={\alpha }_{\mu }=\int {\mathrm{\Gamma }}^{+}\left(w\right)d\mu ,\mu -\mathrm{a}.\mathrm{e}.\mathrm{on}D,$$

(7)

and

$$\underset{n\to +\infty }{lim}\frac{1}{n}log\mu \left(w_{|n}\right)=-h_{\mu }\left(\tau \right)\mu a.e.,$$

by Shannon–McMillan–Breiman Theorem. Therefore, $\mu \left(D_{\mu }\left({\alpha }_{\mu }\right)\right)=1$, which implies by using (7), for $ϵ>0$ that

$$\left\{\begin{array}{cc}{\left[\prod _{j=1}^n{c}_j\left(w\right)\right]}^{h_{\mu }\left(\tau \right)/{\alpha }_{\mu }}\le e^{-n{h}_{\mu }\left(\tau \right)+nϵh_{\mu }\left(\tau \right)/{\alpha }_{\mu }},\hfill & \\ & \\ \mu \left(w_{|n}\right)\ge e^{-n{h}_{\mu }\left(\tau \right)+nϵ},\hfill \end{array}\right.$$

for almost every $w\in D_{\mu }\left({\alpha }_{\mu }\right)$ and n big enough. Now, we put $B\in \mathcal{B}\left(\mathcal{X}\right)$, such that ${\phi }^{-1}\left(B\right)=D_{\mu }\left({\alpha }_{\mu }\right)$. For all $w\in {\phi }^{-1}\left(B\right)$ with $B\subset E$, we have

$$\frac{{\phi }^{*}\mu \left(I_n\left(w\right)\right)}{{\prod }_{j=1}^n{c}_j{\left(w\right)}^t}\ge e^{nϵ(1-t)}\ge 1,$$

for $t=\frac{h_{\mu }\left(\tau \right)}{{\alpha }_{\mu }}$. By using Lemma 1, where there exists a positive constant $A_1$, such that

$$\begin{array}{c}\hfill {\displaystyle A_1{d}^t\left(B\right)\underset{w\in {\phi }^{-1}\left(B\right)}{inf}\left(\underset{n\to +\infty }{lim\; inf}\frac{{\phi }^{*}\mu \left(I_n\left(w\right)\right)}{{\prod }_{j=1}^n{c}_j{\left(w\right)}^t}\right)\le {\phi }^{*}\mu \left(B\right)}\end{array}$$

which implies that

$${\underline{dim}}_{MB}\left(B\right)\le \frac{h_{\mu }\left(\tau \right)}{{\alpha }_{\mu }}\mathrm{and}\mathrm{then}{\underline{dim}}_{MB}\left({\phi }^{*}\mu \right)\le \frac{h_{\mu }\left(\tau \right)}{{\alpha }_{\mu }}.$$

(8)

On the other hand, let B, such that ${\phi }^{*}\mu \left(B\right)=1$ and take $C={\phi }^{-1}\left(B\right)\cap D_{\mu }\left({\alpha }_{\mu }\right)$. Since $\mu \left(C\right)=1$, then, using again Lemma 1, we can deduce that

$${\underline{dim}}_{MB}\left(B\right)\ge {\underline{dim}}_{MB}\left(C\right)\ge \frac{h_{\mu }\left(\tau \right)}{{\alpha }_{\mu }}$$

and then we have equality using (8).

2.

For all $w\in D$, we set

$${\displaystyle B\left(w\right)=\underset{n\to +\infty }{lim}\frac{{\phi }^{*}\mu \left(I_n\left(w\right)\right)}{{\prod }_{j=1}^n{c}_j{\left(w\right)}^{\alpha }}.}$$

If we assume that $d^{\alpha }\left({\phi }^{*}\mu \right)=0$ then, for each $n\in \mathbb{N}$, there exists $B_n\subset E$, such that

$${\phi }^{*}\mu (E\setminus B_n)=0\mathrm{and}{d}^{\alpha }\left(B_n\right)<\frac{1}{n}.$$

Now consider the following sets

$$B^0=\left\{w\in D,B\left(w\right)=0\right\},$$

$$B^{+}=\left\{w\in D,1/k<B\left(w\right)<k\right\}$$

and

$$B^{\infty }=\left\{w\in D,B\left(w\right)=+\infty \right\}.$$

Since $\mu$ is ergodic, these sets are either null or full with respect to the measure $\mu$. Now, Suppose that $\mu \left(B\right)=1$, where $B\in \{B^0,B^{+},B^{\infty }\}$, and put $Z_{n,k}=B_n\cap \phi \left(B_k\right)$ where

$$\left\{\begin{array}{cc}{B}_k=\left\{w\in D,\frac{1}{k}\le B\left(w\right)<+\infty \right\}\hfill & \mathrm{if}B=B^{+},\hfill \\ \\ B_k=\left\{w\in D,B\left(w\right)=0\right\}\hfill & \mathrm{if}B=B^0,\hfill \\ \\ B_k=\left\{w\in D,B\left(w\right)=+\infty \right\}\hfill & \mathrm{if}B=B^{\infty }.\hfill \end{array}\right.$$

It is clear that $B_k⟶B$ as $k\to +\infty$, which implies that ${\displaystyle \underset{k\to +\infty }{lim}{\phi }^{*}\mu \left(Z_{n,k}\right)=1}$, for any $n\in \mathbb{N}$. Moreover,

$$d^{\alpha }\left(Z_{n,k}\right)\le d^{\alpha }\left(B_n\right)\le 1/n,\forall k\in \mathbb{N}.$$

which gives, from Lemma 1, where there exists a positive constant $A_2$, such that

$${\phi }^{*}\mu \left(Z_{n,k}\right)\le A_2{d}^{\alpha }\left(Z_{n,k}\right)\underset{w\in {\phi }^{-1}\left(Z_{n,k}\right)}{inf}B\left(w\right).$$

Choose k large enough to satisfy ${\phi }^{*}\mu \left(Z_{n,k}\right)>1/2$. Then

$$1/2\le A_2{d}^{\alpha }\left(Z_{n,k}\right)\underset{w\in {\phi }^{-1}\left(Z_{n,k}\right)}{inf}B\left(w\right)\le \frac{A_2}{n}\underset{w\in {\phi }^{-1}\left(Z_{n,k}\right)}{inf}B\left(w\right),$$

which creates a contradiction if $B=B^0$. Now, if $B=B^{+}$, then the above inequalities imply that

$$\frac{n}{2A_2}\le k,$$

for any n and any sufficiently large k, which is not possible. Finally, we conclude that $B=B^{\infty }$.

The two other assertions may be proved in the same way. In what follows, we will only prove the second one and keep the other to the readers. We suppose conversely that $B=B^{\infty }$$\mu$ a.e. w. It follows from Lemma 1 that $d^{\alpha }\left(B\right)=0$ whenever ${\phi }^{*}\mu \left(B\right)=1$ and then

$$d^{\alpha }\left({\phi }^{*}\mu \right))=0.$$

   □

Example 4. 

Consider again the special case studied in Example 1 in which the conditions of all the numbers $c_{i_1,i_2,\dots ,i_n}$ depend only on the length and the last variable, i.e.,

$$c_{i_1,i_2,\dots ,i_n}=c_{n,i_n}.$$

Let α be the unique number, such that

$$\sum _{(i_1,i_2,\dots ,i_n)\in D_n}{|I_{i_1{i}_2\dots i_n}|}^{\alpha }=1,$$

and ${\mu }_{\alpha }$ the associated Gibbs measure (see (4)). Then

$${\phi }^{*}{\mu }_{\alpha }\left(I_n\left(w\right)\right)={\mu }_{\alpha }(i_1{i}_2\dots i_n)={|I_n\left(w\right)|}^{\alpha }{\left(\prod _{j=1}^n{Z}_j^{\alpha }\right)}^{-1}.$$

Recall the definition of the pressure function π (see (3)), then

$$\underset{n\to +\infty }{lim}\frac{{\phi }^{*}{\mu }_{\alpha }\left(I_n\left(w\right)\right)}{|I_n{\left(w\right)|}^{\gamma }}=\underset{n\to +\infty }{lim}|I_n{\left(w\right)|}^{\alpha -\gamma }\left(e^{-log{\prod }_{j=1}^n{Z}_j^{\alpha }}\right)=\underset{n\to +\infty }{lim}{|I_n\left(w\right)|}^{\alpha -\gamma }\left(e^{-n{\pi }_n\left(\alpha \right)}\right).$$

which implies that

$$\underset{n\to +\infty }{lim}\frac{{\phi }^{*}\mu \left(I_n\left(w\right)\right)}{|I_n{\left(w\right)|}^{\gamma }}=+\infty \mathit{if}\gamma >\alpha$$

and

$$\underset{n\to +\infty }{lim}\frac{{\phi }^{*}\mu \left(I_n\left(w\right)\right)}{|I_n{\left(w\right)|}^{\gamma }}=0\mathit{if}\gamma <\alpha .$$

It follows from Theorem 1 that

$$\alpha ={\underline{dim}}_{MB}\left(E\right)={\underline{dim}}_{MB}\left({\phi }^{*}{\mu }_{\alpha }\right).$$

Moreover, if we assume that

$$0<\underset{n\to +\infty }{lim}\sum _{(i_1,i_2,\dots ,i_n)\in D_n}\prod _{j=1}^n{c}_{j,i_j}^{\alpha }<+\infty$$

then by using Theorems 1 and 4 we have

$$d^{\alpha }\sim {\phi }^{*}{\mu }_{\alpha }\mathit{on}E.$$

5. Conclusions

In this study, we focus on the characteristics of a class of Moran sets for the centered Hausdorff measure, the Packing measure, and the Hewitt–Stromberg measure. We primarily explore the equivalence of these measures for a class of Moran sets meeting the strong separation criterion. We specifically address a class of Moran sets in general metric spaces with respect to the equality of the three aforementioned dimensions and their accompanying measures. Using entropy, we derive measurements and dimensions of the image of $\tau$-invariant measures in symbolic spaces that are analogous to those obtained in the traditional situation of self-similar sets. Moreover, we provide several intriguing examples, focusing in particular on a class of Moran sets and providing some statistical explanations for dimensions and corresponding geometrical measures.