Bowen’s Equations for Weighted Upper Metric Mean Dimension with Potential

Abstract

In this paper, we define the weighted upper metric mean dimension with potential and derive its corresponding Bowen’s equation. Furthermore, we introduce the weighted BS metric mean dimension and the weighted packing BS metric mean dimension on the subsets, and prove Bowen’s equations for both. Additionally, we establish variational principles for these dimensions, linking them to the weighted local measure-theoretic entropy.

1. Introduction and Main Results

Motivated by the fractal geometry of self-affine carpets and sponges [1,2,3], Feng and Huang [4] introduced the concept of weighted topological pressure and established a variational principle for it. Now, let us proceed to introduce the background of our study. Let $(X_i,T_i)$ be topological dynamical systems (TDSs) for $1\le i\le k$ and $k\ge 2$. For each $1\le i\le k-1,$ we set a factor map ${\pi }_i:X_i\to X_{i+1}.$ Specifically, ${\pi }_1,\cdots ,{\pi }_{k-1}$ are continuous maps such that the following diagrams commute:

For simplicity, we define ${\pi }_0$ as the identity map on $X_1$. For $i=0,1,\cdots ,k-1,$ we define ${\tau }_i:X_1\to X_{i+1}$ with ${\tau }_i={\pi }_i\circ {\pi }_{i-1}\circ \cdots \circ {\pi }_0$. Denote by $M(X_i,T_i)$ the set of all $T_i$-invariant Borel probability measures on $X_i$ and by $E(X_i,T_i)$ the set of ergodic measures. Set $\mathbf{a}=(a_1,a_2,\cdots ,a_k)\in {\mathbb{R}}^k$, where $a_1>0$ and $a_i\ge 0$ for $i\ge 2.$ Take $\phi :X_1\to \mathbb{R}$ as a continuous function. They obtained the relationship between the weighted topological pressure and measure theoretic entropy, as follows:

$$\begin{array}{c}\hfill P^{\mathbf{a}}(T_1,\phi )=\underset{\mu \in M(X_1,T_1)}{sup}(\sum _{i=1}^k{a}_i{h}_{\mu \circ {\tau }_{i-1}^{-1}}\left(T_i\right)+\int \phi d\mu ).\end{array}$$

The connection between topological pressure and the dimension theory of dynamical systems emerges from the following result. Consider the equation

$$\begin{array}{c}\hfill P(-t\phi )=0,\end{array}$$

where $\phi$ is a function related to a given invariant set. The unique t that solves this equation is often related to the Hausdorff dimension of that set. Bowen introduced this equation in [5] during his research on quasi-circles. Since then, it has become widely known as Bowen’s equation. The Bowen’s equation not only is applied to compute or estimate the dimension of invariant sets in invertible or non-invertible dynamics [6,7] but also admits a proper generalization. In [8], Barreira and Schmeling defined a new dimension, termed the BS dimension, and proved that it is the unique root of Bowen’s equation. Inspired by the work of Jaerisch et al. [9], Xing and Chen [10] introduced induced topological pressure, which specializes the BS dimension, and established Bowen’s equation for both induced and classical topological pressures.

Metric mean dimension quantifies the complexity of infinite-entropy systems. It was first introduced by Gromov [11] and further developed by Lindenstrauss and Weiss [12]. The metric mean dimension has various applications, including embedding problems [13,14,15,16,17,18,19] and ergodic theory [17,19,20,21,22,23,24,25,26]. Yang, Chen, and Zhou [27] introduced the concepts of induced metric mean dimension and BS metric mean dimension, deriving two Bowen equations for these quantities. Wang [28] was the first to introduce weighted versions of the definitions for mean dimension and metric mean dimension.

Throughout this paper, we assume that $\mathbf{a}=(a_1,\cdots ,a_k)\in {\mathbb{R}}^k$ with $a_1>0$ and $a_i\ge 0$ for $i\ge 2$. For $k\ge 2$, let $(X_i,d_i)$ be compact metric spaces, and $(X_i,T_i)$ be topological dynamical systems. Let ${\pi }_i:X_i\to X_{i+1}$ be factor maps. Set ${\pi }_0$ to be the identity map on $X_1$. Define ${\tau }_i:X_1\to X_{i+1}$ by ${\tau }_i={\pi }_i\circ {\pi }_{i-1}\circ \cdots \circ {\pi }_0$ for $i=0,1,\cdots ,k-1$. In the first main result, we introduce the concept of weighted induced metric mean dimension with potential and derive Bowen’s equations. The first key result is established as follows:

Theorem 1.

Let $\left(X_1,T_1\right)$ be a topological dynamical system (TDS), $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$ and $\psi >0$. Assume that

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)<\infty ,\end{array}$$

then ${\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)$ is the unique root of the equation

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=0.\end{array}$$

We refer to ${\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)$ as the weighted $\psi -$induced upper metric mean dimension with potential φ, as defined in Definition 3.

Inspired by the works in [8,27,29], we define the weighted BS metric mean dimension and the weighted packing BS metric mean dimension, and then establish Bowen equations for both. The second main result is presented as follows:

Theorem 2.

Let $\left(X_1,T_1\right)$ be a topological dynamical system (TDS) and Z be a non-empty subset of $X_1$. Assume that $\phi \in C\left(X_1,\mathbb{R}\right)$ and $\phi >0$. Then,

(a) 

If ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k\right)<\infty$, then ${\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$ is the only root of equation

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(-t\phi \right)=0.\end{array}$$

(b) 

If ${\overline{\mathrm{Pmdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k\right)<\infty$, then ${\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$ is the only root of equation

$$\begin{array}{c}\hfill {\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(-t\phi \right)=0.\end{array}$$

where ${\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$, ${\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$, and ${\overline{\mathrm{Pmdim}}}_M^{\mathbf{a}}(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k)$ denote the weighted BS metric mean dimension, weighted packing BS metric mean dimension, and weighted packing upper mean dimension over Z (see Definitions 6, 7, and 5), respectively.

In [30], Feng and Huang proved that for any non-empty compact subset K, the Bowen entropy of K can be expressed as the supremum of the measure-theoretic local entropies, taken over all Borel probability measures supported on K. In [27], Yang et al. applied Feng and Huang’s approach to prove variational principles for the BS metric mean dimension and the packing BS metric mean dimension. Inspired by the ideas presented in [8,29,30], we will separately establish variational principles for the weighted BS metric mean dimension and the weighted packing BS metric mean dimension.

Theorem 3.

Let $\left(X_1,T_1\right)$ be a topological dynamical system (TDS) and K be a non-empty subset of $X_1$. Assume that $\phi \in C\left(X_1,\mathbb{R}\right)$ and $\phi >0$. Then

$$\begin{array}{c}\hfill {\overline{\mathrm{BSmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{sup\left\{{\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right):\mu \in M\left(X_1\right),\mu \left(K\right)=1\right\}}{log\frac{1}{ϵ}}},\end{array}$$

where the definition of ${\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)$ can be found in Definition 8.

In this theorem, we extend the variational principle established in [30] for Bowen entropy to the weighted BS-metric mean dimension by introducing the weight vector $\mathbf{a}=(a_1,\cdots ,a_k)$. Our framework characterizes the complexity of multi-scale dynamical systems. The proof proceeds along the following steps:

1.

Combining Definition 8, we estimate the lower bound of covering numbers.

2.

Applying the Frostman lemma, we establish the upper bound relationship between local entropy and dimension.

Theorem 4.

Let $\left(X_1,T_1\right)$ be a topological dynamical system (TDS). Assume that $\phi \in C\left(X_1,\mathbb{R}\right)$ and $\phi >0$.

(a) 

If $K\subseteq X_1$ is non-empty and compact, then

$$\begin{array}{c}\hfill {\overline{\mathrm{BSPdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{sup\left\{{\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right):\mu \in M\left(X_1\right),\mu \left(K\right)=1\right\}}{log\frac{1}{ϵ}}},\end{array}$$

where the definition of ${\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)$ can be found in Definition 8.

(b) 

If $Z\subseteq X_1$ is analytic, then

$$\begin{array}{c}\hfill {\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=sup\left\{{\overline{\mathrm{BSPdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right):K\subseteq Z\mathrm{is}\mathrm{compact}\right\}.\end{array}$$

The proof proceeds along the following steps:

1.

Construct measures on compact subsets and link covering numbers with local entropy to prove the variational principle for packing BS dimension.

2.

Inductively construct compact subsets $\left\{K_n\right\}\subseteq Z$ and use convergence to extend the result from $K_n$ to analytic Z.

2. Proof of Theorem 1

In this section, we turn our attention to the weighted upper metric mean dimension with potential, focusing on the entire space. Specifically, we introduce the concept of the weighted induced upper metric mean dimension with potential and subsequently derive Bowen’s equation for this dimension in the context of the entire phase space. For $k\ge 2$, let $(X_i,d_i)$ be compact metric spaces, and $(X_i,T_i)$ be topological dynamical systems. Let ${\pi }_i:X_i\to X_{i+1}$ be factor maps. Set ${\pi }_0$ to be the identity map on $X_1$. Define ${\tau }_i:X_1\to X_{i+1}$ by ${\tau }_i={\pi }_i\circ {\pi }_{i-1}\circ \cdots \circ {\pi }_0$ for $i=0,1,...,k-1$.

2.1. Weighted Induced Upper Metric Mean Dimension with Potential

Definition 1

(The $\mathbf{a}$-weighted metric and $\mathbf{a}$-weighted Bowen ball). For $x,y\in X_1$, $n\in \mathbb{N}$, $ϵ>0$, denote

$$\begin{array}{c}\hfill d_n^{\mathbf{a}}(x,y)=sup\{d_i(T_i^j{\tau }_{i-1}x,T_i^j{\tau }_{i-1}y):0\le j\le \lceil (a_1+\cdots +a_i)n\rceil -1,1\le i\le k\},\end{array}$$

and

$$\begin{array}{c}\hfill B_n^{\mathbf{a}}(x,ϵ)=\left\{y\in X_1:d_n^{\mathbf{a}}(x,y)<ϵ\right\}\end{array}$$

where $\lceil u\rceil$ denotes the least integer $\ge u$. We call $B_n^{\mathbf{a}}(x,ϵ)$ the n-th $\mathbf{a}$-weighted Bowen ball of radius ϵ centered at x.

Let $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$ and $\psi >0$. For all $n⩾1,x\in X,$ we set $S_{\lceil a_{1}n\rceil }\phi \left(x\right):={\sum }_{i=0}^{\lceil a_{1}n\rceil -1}\phi \left(T_1^i\left(x\right)\right)$ and $m:={\displaystyle \underset{x\in X_1}{min}}\psi >0.$

Definition 2.

Let $0<ϵ<1$ and $\phi \in C(X_1,\mathbb{R}).$ Set

$$\begin{array}{cc}\hfill & {\#}_{sep}\left(X_1,d_n^{\mathbf{a}},S_{\lceil a_{1}n\rceil }\phi ,ϵ\right)\hfill \\ \hfill & =\left\{\sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}:F_n\mathrm{is}\mathrm{an}\left(a,n,ϵ\right)-\mathrm{separated}\mathrm{set}\mathrm{of}{X}_1\right\}\hfill \end{array}$$

and

$$\begin{array}{c}\hfill P^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)=\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{log{\#}_{sep}\left(X_1,d_n^{\mathbf{a}},S_{\lceil a_{1}n\rceil }\phi ,ϵ\right)}{n}}.\end{array}$$

The weighted upper metric mean dimension with potential φ is given by

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)=\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{P^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)}{log\frac{1}{ϵ}}}.\end{array}$$

Specially, ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k\right)={\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,0\right).$

Definition 3.

Let $\left(X_1,T_1\right)$ be a topological dynamical system (TDS), $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$ and $\psi >0$. For $V>0$, set

$$\begin{array}{c}\hfill S_V:=\left\{n\in \mathbb{N}:\exists x\in X_1,\mathit{such}\mathit{that}{S}_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽V,S_{\lceil a_{1}n\rceil +1}\psi \left(x\right)>V\right\}.\end{array}$$

For all $n\in S_V,ϵ>0,$ define

$$\begin{array}{c}\hfill X_n=\left\{x\in X_1:S_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽V,S_{\lceil a_{1}n\rceil +1}\psi \left(x\right)>V\right\}\end{array}$$

and

$$\begin{array}{cc}\hfill & P_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)\hfill \\ \hfill & =sup\left\{\sum _{n\in S_V}\sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}:F_n\mathrm{is}\mathrm{an}\left(a,n,ϵ\right)-\mathrm{separated}\mathrm{set}\mathrm{of}{X}_1\right\}.\hfill \end{array}$$

We call

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)=\underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{\displaystyle \frac{1}{Vlog\frac{1}{ϵ}}}log{P}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)\end{array}$$

the weighted ψ-induced upper metric mean dimension with potential φ.

Remark 1.

(a) If $S_V\ne \varnothing ,$ then for any $n\in S_V,$ we have $\lceil a_{1}n\rceil ⩽\left[{\displaystyle \frac{V}{m}}\right]+1,$ where $\left[{\displaystyle \frac{V}{m}}\right]$ is the integer part of ${\displaystyle \frac{V}{m}}$ and $m={\displaystyle \underset{x\in X_1}{min}}\psi \left(x\right).$ In other words, $S_V$ is a finite set.

(b) 

If $\psi =1,$ then the weighted ψ-induced upper metric mean dimension with potential φ and the weighted upper metric mean dimension with potential φ are equal, that is,

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,1}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)={\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).\end{array}$$

(c) 

${\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)>-\infty .$

Analogous to the definition of classical topological pressure, we can also define the weighted $\psi$-induced upper metric mean dimension with potential $\phi$ via spanning sets.

Proposition 1.

Let $\left(X_1,T_1\right)$ be a topological dynamical system (TDS), $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$ and $\psi >0$. Set

$$\begin{array}{cc}\hfill & Q_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)\hfill \\ \hfill & =inf\left\{\sum _{n\in S_V}\sum _{x\in E_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}:E_{n}isan\left(\mathbf{a},n,ϵ\right)-spanningsetof{X}_1\right\},\hfill \end{array}$$

then

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & =\underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{\displaystyle \frac{1}{Vlog\frac{1}{ϵ}}}log{Q}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right).\hfill \end{array}$$

Proof. 

Let $0<ϵ<1,n\in S_V$. Since the largest $\left(\mathbf{a},n,ϵ\right)$-separated set with respect to $F_n$ in $X_n$ is also an $\left(\mathbf{a},n,ϵ\right)$-spanning set of $X_n$, we deduce that

$$\begin{array}{c}\hfill Q_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)⩽P_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right).\end{array}$$

Hence,

$$\begin{array}{cc}\hfill & \underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{\displaystyle \frac{1}{Vlog\frac{1}{ϵ}}}log{Q}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)\hfill \\ \hfill ⩽& \underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{\displaystyle \frac{1}{Vlog\frac{1}{ϵ}}}log{P}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)\hfill \\ \hfill =& {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).\hfill \end{array}$$

On the other hand, set $\gamma \left(ϵ\right):=sup\left\{\left|\phi \left(x\right)-\phi \left(y\right)\right|:d_1\left(x,y\right)<ϵ\right\}.$ Take $E_n$ to be an $\left(\mathbf{a},n,{\displaystyle \frac{ϵ}{2}}\right)$-spanning set of $X_n$ and $F_n$ to be an $\left(\mathbf{a},n,ϵ\right)$-separated set of $X_n$ for $n\in S_V.$ Suppose there is a mapping $\Phi :F_n\to E_n$. For any x in $F_n$, $\Phi$ maps x to $\Phi \left(x\right)$ in $E_n$ and $d_n^{\mathbf{a}}\left(x,\Phi \left(x\right)\right)<{\displaystyle \frac{ϵ}{2}}$. In this case, $\Phi :F_n\to E_n\mathrm{is}\mathrm{an}\mathrm{injective}\mathrm{mapping}.$ Therefore,

$$\begin{array}{cc}\hfill \sum _{n\in S_V}\sum _{y\in E_n}{\left({\displaystyle \frac{2}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(y\right)}& ⩾\sum _{n\in S_V}\sum _{x\in F_n}{\left({\displaystyle \frac{2}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(\Phi \left(x\right)\right)}\hfill \\ \hfill & =\sum _{n\in S_V}\sum _{x\in F_n}{\left({\displaystyle \frac{2}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(\Phi \left(x\right)\right)-S_{\lceil a_{1}n\rceil }\phi \left(x\right)+S_{\lceil a_{1}n\rceil }\phi \left(x\right)}\hfill \\ \hfill & ⩾{\left({\displaystyle \frac{2}{ϵ}}\right)}^{-a_1\gamma \left(ϵ\right)\left({\displaystyle \frac{V}{m}}+1\right)}\sum _{n\in S_V}\sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}{2}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}\hfill \\ \hfill & ⩾{\left({\displaystyle \frac{2}{ϵ}}\right)}^{-a_1\gamma \left(ϵ\right)\left({\displaystyle \frac{V}{m}}+1\right)}{2}^{-a_1\left({\displaystyle \frac{V}{m}}+1\right)\parallel \phi \parallel }\sum _{n\in S_V}\sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}.\hfill \end{array}$$

By Definition 3, we obtain

$$\begin{array}{cc}\hfill & \underset{V\to \infty }{lim\; sup}{\displaystyle \frac{1}{V}}log{Q}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{\displaystyle \frac{ϵ}{2}}\right)\hfill \\ \hfill & ⩾-{\displaystyle \frac{a_1}{m}}\gamma \left(ϵ\right)log{\displaystyle \frac{2}{ϵ}}-a_1{\displaystyle \frac{\parallel \phi \parallel }{m}}log2+\underset{V\to \infty }{lim\; sup}{\displaystyle \frac{1}{V}}log{P}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right).\hfill \end{array}$$

Taking the limit as $ϵ\to 0$ and noting that $\gamma \left(ϵ\right)\to 0,$ we obtain

$$\begin{array}{cc}\hfill \underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}& {\displaystyle \frac{1}{Vlog\frac{1}{ϵ}}}log{Q}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)⩾{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).\hfill \end{array}$$

□

2.2. Bowen’s Equation for the Weighted Upper Metric Mean Dimension with Potential

In this part, our main purpose is to prove Theorem 1. For this, we will explore the relationship between ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi )$ and ${\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ).$

Theorem 5.

Let $\left(X_1,T_1\right)$ be a topological dynamical system (TDS), $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$ and $\psi >0$. For $V>0$, define

$$\begin{array}{c}\hfill G_V:=\left\{n\in \mathbb{N}:\exists x\in X_1,\mathit{such}\mathit{that}{S}_{\lceil a_{1}n\rceil }\psi \left(x\right)>V\right\}.\end{array}$$

For all $n\in G_V$ and $ϵ>0,$ define

$$\begin{array}{c}\hfill Y_n=\left\{x\in X_1:S_{\lceil a_{1}n\rceil }\psi \left(x\right)>V\right\},\end{array}$$

$$\begin{array}{cc}\hfill & R_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)\hfill \\ \hfill & =sup\left\{\sum _{n\in G_V}\sum _{x\in F_n^{\prime }}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}:F_n^{\prime }\mathrm{is}\mathrm{an}\left(a,n,ϵ\right)-separatedsetof{Y}_n\right\}.\hfill \end{array}$$

Then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & =inf\left\{\beta \in \mathbb{R}:\underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)<\infty \right\}.\hfill \end{array}\end{array}$$

(1)

We use the convention that $inf\varnothing =\infty .$

Proof. 

For $n\in \mathbb{N},x\in X_1,$$m_{\lceil a_{1}n\rceil }\left(x\right)$ is defined by

$$\begin{array}{c}\hfill \left(m_{\lceil a_{1}n\rceil }\left(x\right)-1\right)\parallel \psi \parallel <S_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽m_{\lceil a_{1}n\rceil }\left(x\right)\parallel \psi \parallel .\end{array}$$

(2)

Then, we have

$$\begin{array}{c}\hfill -∥\psi ∥<S_{\lceil a_{1}n\rceil }\psi \left(x\right)-m_{\lceil a_{1}n\rceil }\left(x\right)∥\psi ∥⩽0,\end{array}$$

that is,

$$\begin{array}{c}\hfill \left|\beta \right|\left|S_{\lceil a_{1}n\rceil }\psi \left(x\right)-m_{\lceil a_{1}n\rceil }\left(x\right)∥\psi ∥\right|⩽\left|\beta \right|∥\psi ∥\end{array}$$

for all $\beta \in \mathbb{R}.$ Consequently, we find

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\displaystyle \frac{1}{ϵ}}\right)}^{-\beta ∥\psi ∥m_{\lceil a_{1}n\rceil }\left(x\right)}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{-\left|\beta \right|∥\psi ∥}& ⩽{\left({\displaystyle \frac{1}{ϵ}}\right)}^{-\beta S_{\lceil a_{1}n\rceil }\psi \left(x\right)}\hfill \\ \hfill & ⩽{\left({\displaystyle \frac{1}{ϵ}}\right)}^{-\beta ∥\psi ∥m_{\lceil a_{1}n\rceil }\left(x\right)}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{\left|\beta \right|∥\psi ∥}\hfill \end{array}\end{array}$$

(3)

for any $x\in X_1.$ Given $0<ϵ<1.$ We take

$$\begin{array}{cc}\hfill & R_{\psi ,V}^{\mathbf{a}\left(1\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{\left\{\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }+\left|\beta \right|\parallel \psi \parallel \right\}}_{n\in G_V},ϵ\right)\hfill \\ \hfill & =sup\{\sum _{n\in G_V}\sum _{x\in F_n^{\prime }}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }\left(x\right)-\left|\beta \right|\parallel \psi \parallel }:\hfill \\ \hfill & F_n^{\prime }\mathrm{is}\mathrm{an}\left(\mathbf{a},n,ϵ\right)-\mathrm{separated}\mathrm{set}\mathrm{of}{Y}_n\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & R_{\psi ,V}^{\mathbf{a}\left(2\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{\left\{\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }-\left|\beta \right|\parallel \psi \parallel \right\}}_{n\in G_V},ϵ\right)\hfill \\ \hfill & =sup\{\sum _{n\in G_V}\sum _{x\in F_n^{\prime }}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }\left(x\right)+\left|\beta \right|\parallel \psi \parallel }:\hfill \\ \hfill & F_n^{\prime }\mathrm{is}\mathrm{an}\left(\mathbf{a},n,ϵ\right)-\mathrm{separated}\mathrm{set}\mathrm{of}{Y}_n\}\hfill \end{array}$$

and let

$$\begin{array}{cc}\hfill & A=inf\left\{\beta \in \mathbb{R}:\underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}\left(1\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{{\rm Y}}_A,ϵ\right)<\infty \right\},\hfill \\ \hfill & B=inf\left\{\beta \in \mathbb{R}:\underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)<\infty \right\},\hfill \\ \hfill & C=inf\left\{\beta \in \mathbb{R}:\underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}\left(2\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{{\rm Y}}_C,ϵ\right)<\infty \right\},\hfill \end{array}$$

where ${{\rm Y}}_A=\left\{\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }+\left|\beta \right|\parallel \psi \parallel \right\},$ ${{\rm Y}}_C=\left\{\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }-\left|\beta \right|\parallel \psi \parallel \right\},$ $n\in G_V$.

Based on (3), we can infer that $A⩽B⩽C.$ However, to prove (1), we still need to verify that

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)⩽AandC⩽{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).\end{array}$$

We proceed with the proof in two steps.

Step 1: Let us start by proving ${\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)⩽A.$

Assume first

$$\begin{array}{c}\hfill \beta <{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).\end{array}$$

Then we can find a positive integer $\delta >0$ and a sequence $\left\{{ϵ}_k\right\}$ with $0<{ϵ}_k<1$, so that the following two conditions hold:

$$\begin{array}{c}\hfill \beta +\delta <{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right);\end{array}$$

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)=\underset{k\to \infty }{lim\; sup}\underset{V\to \infty }{lim\; sup}{\displaystyle \frac{1}{log\frac{1}{{ϵ}_k}V}}{P}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{ϵ}_k\right).\end{array}$$

In this case, there is $K_0\in \mathbb{N}$, so for all $k>K_0,$ we look at a subsequence ${\left\{V_j\right\}}_{j\in \mathbb{N}}$. As $j\to \infty$, this subsequence converges to ∞ and satisfies the following inequality:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{V_j\left(\beta +{\displaystyle \frac{\delta }{2}}\right)}<P_{\psi ,V_j}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{ϵ}_k\right).\end{array}$$

According to Definition 3, for any $j\in \mathbb{N}$ and $n\in S_{V_j},$ we have

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{V_j\left(\beta +{\displaystyle \frac{\delta }{2}}\right)}<\sum _{n\in S_{V_j}}\sum _{x\in F_n}{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)},\end{array}$$

(4)

where $F_n$ is an $\left(\mathbf{a},n,ϵ\right)$-separated set of $X_n$.

We claim that for any $V>0$. If $S_V\ne \varnothing ,$ then for any $n\in S_V,$ we have

$$\begin{array}{c}\hfill {\displaystyle \frac{V}{\parallel \psi \parallel }}-1⩽\lceil a_{1}n\rceil ⩽\left[{\displaystyle \frac{V}{m}}\right]+1,\end{array}$$

(5)

where $m=\underset{x\in X_1}{min}\psi \left(x\right)>0.$ Suppose that $n\in S_V,$ then we can find $x\in X_1$ so that $S_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽V$ and $S_{\lceil a_{1}n\rceil +1}\psi \left(x\right)>V.$ From this, it follows that $S_{\lceil a_{1}n\rceil }\psi \left(x\right)+∥\psi ∥>V.$ Hence, $V-\parallel \psi \parallel <S_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽V.$

We take any $V_{j_1}$ and note that $V_j\to \infty ,$ so we can take $V_{j_2}$ such that

$$\begin{array}{c}\hfill \left[{\displaystyle \frac{V_{j_1}}{m}}\right]+1<{\displaystyle \frac{V_{j_2}}{\parallel \psi \parallel }}-1.\end{array}$$

We claim that

$$S_{V_i}\cap S_{V_j}=\varnothing$$

with $i\ne j.$ In fact, we suppose that $S_{V_i}\cap S_{V_j}\ne \varnothing$, let $n\in S_{V_i}\cap S_{V_j}$ with $i<j$. For this n, we can pick $x_1,x_2\in X_1$, where $S_{\lceil a_{1}n\rceil }\psi \left(x_1\right)⩽V_i,S_{\lceil a_{1}n\rceil +1}\psi \left(x_1\right)>V_i$ and $S_{\lceil a_{1}n\rceil }\psi \left(x_2\right)⩽V_j,S_{\lceil a_{1}n\rceil +1}\psi \left(x_2\right)>V_j.$ In light of (5), it follows that

$$\begin{array}{c}\hfill \lceil a_{1}n\rceil ⩽\left[{\displaystyle \frac{V_i}{m}}\right]+1<{\displaystyle \frac{V_j}{\parallel \psi \parallel }}-1<\lceil a_{1}n\rceil ,\end{array}$$

which contradicts the assumption.

Since for any $j\in \mathbb{N},n\in S_{V_j},$ if $x\in F_n,$ then we have $V_j-\parallel \psi \parallel <S_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽V_j.$ According to (2), we can obtain $V_j-∥\psi ∥<m_{\lceil a_{1}n\rceil }\left(x\right)∥\psi ∥\mathrm{and}{m}_{\lceil a_{1}n\rceil }\left(x\right)∥\psi ∥<V_j+∥\psi ∥,$ that is, $\left|\parallel \psi \parallel m_{\lceil a_{1}n\rceil }\left(x\right)-V_j\right|<2\parallel \psi \parallel .$ Hence,

$$\begin{array}{c}\hfill -\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }\left(x\right)⩾-\beta V_j-2\left|\beta \right|\parallel \psi \parallel \end{array}$$

(6)

for any $\beta \in \mathbb{R}.$ Therefore, combining (4) and (6), we can obtain

$$\begin{array}{cc}\hfill & R_{\psi ,V}^{\mathbf{a}\left(1\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{\left\{\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }+\left|\beta \right|\parallel \psi \parallel \right\}}_{n\in G_V},{ϵ}_k\right)\hfill \\ \hfill & ⩾\sum _{j\in \mathbb{N},V_j-\parallel \psi \parallel >t}\sum _{n\in S_{V_j}}\sum _{x\in F_n}{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil \left(x\right)}-\left|\beta \right|\parallel \psi \parallel }\hfill \\ \hfill & ⩾{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{-3\left|\beta \right|\parallel \psi \parallel }\sum _{j\in \mathbb{N},V_j-\parallel \psi \parallel >V}\sum _{n\in S_{V_j}}\sum _{x\in F_n}{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-\beta V_j}\hfill \\ \hfill & ⩾{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{-3\left|\beta \right|\parallel \psi \parallel }\sum _{j\in \mathbb{N},V_j-\parallel \psi \parallel >V}{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{{\displaystyle \frac{\delta }{2}}{V}_j}\hfill \\ \hfill & =\infty .\hfill \end{array}$$

This yields that for any $\beta <{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right),$ we have that

$$\begin{array}{c}\hfill \underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}\left(1\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\left\{\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }+\left|\beta \right|\parallel \psi \parallel \right\},ϵ\right)=\infty \end{array}$$

is valid. Furthermore, this leads to

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)⩽A.\end{array}$$

If ${\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)=\infty ,$ suppose $P<{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).$ Then, for any $\beta <P$, we can infer that

$$\begin{array}{c}\hfill \underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}\left(1\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\left\{\beta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }+\left|\beta \right|\parallel \psi \parallel \right\},ϵ\right)=\infty .\end{array}$$

Due to the arbitrariness of P, $A=inf\varnothing =\infty$. Hence,

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)=A=\infty .\end{array}$$

Step 2: Let us now prove $C⩽{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).$ Let $\delta >0$; for any $0<ϵ<{ϵ}_0$, $0<{ϵ}_0<1$, there exists

$$\begin{array}{c}\hfill \underset{V\to \infty }{lim\; sup}log{\displaystyle \frac{1}{ϵ}}Vlog{P}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)<{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)+{\displaystyle \frac{\delta }{2}}.\end{array}$$

Consequently, there is a variable $l_0\in \mathbb{N},$ for which, when $l⩾l_0$, we have

$$\begin{array}{c}\hfill log{P}_{\psi ,lm}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,ϵ\right)<{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{lm\left({\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)+{\displaystyle \frac{2\delta }{3}}\right)},\end{array}$$

(7)

$$\begin{array}{c}\hfill {\displaystyle \frac{\delta }{3}}{l}_{0}m-△-1>0,\end{array}$$

(8)

where $△=3\parallel \psi \parallel \left(\left|{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)+\delta \right|\right),m={\displaystyle \underset{x\in X_1}{min}}\psi \left(x\right).$

For $n\in S_{lm},$ suppose that $F_n$ is an $\left(\mathbf{a},n,ϵ\right)$-separated set of $X_n$; then, for every $x\in F_n,$ it holds that $S_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽lm$ and $S_{\lceil a_{1}n\rceil +1}\psi \left(x\right)>lm,$ i.e., $lm-∥\psi ∥<S_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽lm.$ Combining (2) for $m_{\lceil a_{1}n\rceil }\left(x\right),$ we obtain

$$\begin{array}{c}\hfill \left|\parallel \psi \parallel m_{\lceil a_{1}n\rceil }\left(x\right)-lm\right|<2\parallel \psi \parallel .\end{array}$$

Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & -\left({\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)+\delta \right)\parallel \psi \parallel m_{\lceil a_{1}n\rceil }\left(x\right)\hfill \\ \hfill & ⩽-lm\left({\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)+\delta \right)+\hfill \\ & 2\parallel \psi \parallel \left({\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)+\delta \right).\hfill \end{array}\end{array}$$

(9)

When V is sufficiently large, for $n\in G_V$, assume that $F_n^{\prime }$ is an $\left(\mathbf{a},n,ϵ\right)$-separated set of $Y_n$; then for any $x\in F_n^{\prime }$, there is one and only one $l⩾l_0$ for which $\left(l-1\right)m<S_{\lceil a_{1}n\rceil }\psi \left(x\right)⩽lm.$ Hence, $S_{\lceil a_{1}n\rceil +1}\psi \left(x\right)=S_{\lceil a_{1}n\rceil }\psi \left(x\right)+\psi \left(T_1^{\lceil a_{1}n\rceil }x\right)>lm.$ It can be seen that

$$\begin{array}{cc}\hfill & R_{\psi ,V}^{\mathbf{a}\left(2\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{\left\{\eta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }-\left|\eta \right|\parallel \psi \parallel \right\}}_{n\in G_V},ϵ\right)\hfill \\ \hfill & ⩽\sum _{l⩾l_0}sup\{\sum _{n\in S_{lm}}\sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-\eta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }\left(x\right)+\left|\eta \right|\parallel \psi \parallel }:\hfill \\ \hfill & F_n\mathrm{is}\mathrm{an}\left(\mathbf{a},n,ϵ\right)-\mathrm{separated}\mathrm{set}\mathrm{of}{X}_n\}\hfill \\ \hfill & ⩽{\left({\displaystyle \frac{1}{ϵ}}\right)}^{3\left|\eta \right|\parallel \psi \parallel }\sum _{l⩾l_0}sup\{\sum _{n\in S_{lm}}\sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-lm\eta }:\hfill \\ \hfill & F_n\mathrm{is}\mathrm{an}\left(\mathbf{a},n,ϵ\right)-\mathrm{separated}\mathrm{set}\mathrm{of}{X}_n\}\hfill \\ \hfill & ⩽{\left({\displaystyle \frac{1}{ϵ}}\right)}^{△}\sum _{l⩾l_0}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{-{\displaystyle \frac{\delta }{3}}lm}={\displaystyle \frac{{\left(\frac{1}{ϵ}\right)}^{\frac{\delta }{3}{l}_{0}m-△}}{1-{\left(\frac{1}{ϵ}\right)}^{\frac{\delta }{2}m}}}\hfill \\ \hfill & <{\displaystyle \frac{ϵ}{1-{\left(\frac{1}{ϵ}\right)}^{\frac{\delta }{2}m}}}<{\displaystyle \frac{1}{1-{\left(\frac{1}{ϵ}\right)}^{\frac{\delta }{2}m}}},\hfill \end{array}$$

where $\eta ={\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)+\delta$. Thus,

$$\begin{array}{c}\hfill \underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}\left(2\right)}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\left\{\eta \parallel \psi \parallel m_{\lceil a_{1}n\rceil }\left(x\right)-\left|\eta \right|\parallel \psi \parallel \right\},ϵ\right)⩽1.\end{array}$$

Then $C⩽{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)+\delta$. When $\delta \to 0,$ it follows that

$$\begin{array}{c}\hfill C⩽{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).\end{array}$$

□

Corollary 1.

Let $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$ and $\psi >0$. Then

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & ⩾inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩽0\right\}.\hfill \end{array}$$

Proof. 

Based on the result (1) proved by Theorem 5, we let $\beta$ belong to the set

$$\begin{array}{c}\hfill \left\{\beta \in \mathbb{R}:{\displaystyle \underset{ϵ\to 0}{lim\; sup}}{\displaystyle \underset{V\to \infty }{lim\; sup}}{R}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)<\infty \right\}\end{array}$$

and define M as $M:={\displaystyle \underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}}{R}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right).$ According to the definition of $M,$ we know that there exists ${ϵ}_0\in \left(0,1\right)$. For any $ϵ\in \left(0,{ϵ}_0\right)$, we can obtain $V_0\in \mathbb{N}$. Consequently, for all $V⩾V_0$, the inequality

$$\begin{array}{c}\hfill R_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)<M+1\end{array}$$

is satisfied. Combining the definition of $P^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)$ in Definition 2, we can find a subsequence $n_j$ with the property that $n_j\to \infty$ as $j\to \infty$. As a result, the following equalities hold

$$\begin{array}{cc}\hfill P^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)& =\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{log{\#}_{sep}\left(X_1,d_n^{\mathbf{a}},S_{\lceil a_{1}n\rceil }\left(\phi -\beta \psi \right),ϵ\right)}{n}}\hfill \\ \hfill & =\underset{j\to \infty }{lim}{\displaystyle \frac{log{\#}_{sep}\left(X_1,d_{n_j}^{\mathbf{a}},S_{\lceil a_1{n}_j\rceil }\left(\phi -\beta \psi \right),ϵ\right)}{n_j}}.\hfill \end{array}$$

For any $V⩾V_0,$ we can determine a positive integer $n_j>V$ such that for every $x\in X_1$, $S_{\lceil a_{1}n\rceil }\psi \left(x\right)>V$. This indicates that $n_j\in G_V$. Assume that $F_{n_j}$ is an $\left(\mathbf{a},n_j,ϵ\right)$-separated set of $X_1$, then

$$\begin{array}{c}\hfill \sum _{x\in F_{n_j}}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_1{n}_j\rceil }\left(\phi \left(x\right)-\beta \psi \left(x\right)\right)}<M+1.\end{array}$$

This shows that $P^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)⩽0.$ Hence, we have

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩽0.\end{array}$$

Finally, we obtain that

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & ⩾inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩽0\right\}.\hfill \end{array}$$

□

The proposition that follows provides an in-depth explanation of Bowen’s equation of ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,$$\phi -\beta \psi )$ in terms of $\beta$.

Proposition 2.

Let $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$ and $\psi >0$. Given a mapping

$$\beta \in \mathbb{R}\mapsto {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right),$$

then the following statement is valid.

(a) 

For ${\beta }_0\in \mathbb{R}$, if

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_0\psi \right)=\infty ,\end{array}$$

then the map ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -·\psi \right)$ is infinite.

(b) 

For ${\beta }_0\in \mathbb{R}$, if

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_0\psi \right)<\infty ,\end{array}$$

then the map ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -·\psi \right)$ is finite, strictly decreasing, and continuous on $\mathbb{R}$. What is more, the equation

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=0\end{array}$$

has a unique (finite) root.

Proof. 

Let $0<ϵ<1$, ${\beta }_1,{\beta }_2\in \mathbb{R}$, and for any $n\in \mathbb{N},$ we have

$$\begin{array}{cc}\hfill & \sum _{x\in E}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-{\beta }_2{S}_{\lceil a_{1}n\rceil }\psi \left(x\right)-\lceil a_{1}n\rceil \left|{\beta }_1-{\beta }_2\right|\parallel \psi \parallel }\hfill \\ \hfill & ⩽\sum _{x\in E}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-{\beta }_1{S}_{\lceil a_{1}n\rceil }\psi \left(x\right)}\hfill \\ \hfill & ⩽\sum _{x\in E}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-{\beta }_2{S}_{\lceil a_{1}n\rceil }\psi \left(x\right)+\lceil a_{1}n\rceil \left|{\beta }_1-{\beta }_2\right|\parallel \psi \parallel },\hfill \end{array}$$

where E is an $\left(\mathbf{a},n,ϵ\right)$-separated set of $X_1$. Hence,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_2\psi \right)-\left|{\beta }_1-{\beta }_2\right|\parallel \psi \parallel \hfill \\ \hfill & ⩽{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_1\psi \right)\hfill \\ \hfill & ⩽{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_2\psi \right)+\left|{\beta }_1-{\beta }_2\right|\parallel \psi \parallel .\hfill \end{array}\end{array}$$

(10)

This implies that the finiteness of ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_1\psi \right)$ is equivalent to the finiteness of

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_2\psi \right).\end{array}$$

Based on the assumption of $\left(b\right)$, we move on to prove the other statements.

According to (10), we have

$$\begin{array}{cc}\hfill & \left|{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_1\psi \right)-{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_2\psi \right)\right|\hfill \\ \hfill & ⩽\left|{\beta }_1-{\beta }_2\right|\parallel \psi \parallel .\hfill \end{array}$$

Based on this inequality, we can infer that the mapping ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -·\psi \right)$ is continuous on $\mathbb{R}$.

Next, we will prove that the mapping ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -·\psi \right)$ is strictly decreasing. Take ${\beta }_1,{\beta }_2\in \mathbb{R}$ such that ${\beta }_1<{\beta }_2$ and fix $0<ϵ<1$. Suppose E is an $\left(\mathbf{a},n,ϵ\right)$-separated set of $X_1$, then we obtain

$$\begin{array}{cc}\hfill & \sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-{\beta }_2{S}_{\lceil a_{1}n\rceil }\psi \left(x\right)}\hfill \\ \hfill & ⩽\sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-{\beta }_1{S}_{\lceil a_{1}n\rceil }\psi \left(x\right)+\left({\beta }_1-{\beta }_2\right)S_{\lceil a_{1}n\rceil }\psi \left(x\right)}\hfill \\ \hfill & ⩽\sum _{x\in F_n}{\left({\displaystyle \frac{1}{ϵ}}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-{\beta }_1{S}_{\lceil a_{1}n\rceil }\psi \left(x\right)+\left({\beta }_1-{\beta }_2\right)\lceil a_{1}n\rceil m},\hfill \end{array}$$

where $m=\underset{x\in X_1}{min}\psi \left(x\right)>0.$ From these inequalities, we can conclude that

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_2\psi \right)\hfill \\ \hfill & ⩽{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_1\psi \right)-\left({\beta }_2-{\beta }_1\right)m,\hfill \end{array}$$

that is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \left({\beta }_2-{\beta }_1\right)m\hfill \\ \hfill ⩽& {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_1\psi \right)-{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_2\psi \right).\hfill \end{array}\end{array}$$

(11)

Finally, we need to prove the uniqueness of the root of the equation

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=0.\end{array}$$

When ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)=0$, only $\beta =0$ satisfies the equation

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=0.\end{array}$$

When ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\ne 0,$ suppose ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)>0$, ${\beta }_1=0,{\beta }_2=h>0$ into (11), and we obtain

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -h\psi \right)⩽{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)-hm.\end{array}$$

Therefore, the only root of the equation ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=0$ meets the condition $0<\beta ⩽{\displaystyle \frac{{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)}{m}}.$ Suppose ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)<0$; let ${\beta }_1=h<0,{\beta }_2=0$ into (11), and we obtain

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)-hm⩽{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -h\psi \right).\end{array}$$

Therefore, the only root of the equation ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=0$ meets the condition $0>\beta ⩾\frac{{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)}{m}.$ □

Proposition 3.

Let $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$, and $\psi >0$. Then

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & =inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩽0\right\}\hfill \\ \hfill & =sup\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩾0\right\}.\hfill \end{array}$$

Proof. 

If there exists ${\beta }_0\in \mathbb{R}$ such that ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -{\beta }_0\psi \right)=\infty$, then, in this case, according to Proposition 2, we know that

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=\infty \end{array}$$

for any $\beta \in \mathbb{R}$. In addition, the statement in Corollary 1 shows that

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & ⩾inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩽0\right\}.\hfill \end{array}$$

Consequently, we obtain

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & =sup\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩾0\right\}\hfill \\ \hfill & =inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩽0\right\}\hfill \\ \hfill & =inf\varnothing =\infty .\hfill \end{array}$$

If for any $\beta \in \mathbb{R}$, we have ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)\in \mathbb{R}.$ Next first prove that

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)⩽inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩽0\right\}.\end{array}$$

Suppose $\beta \in \mathbb{R}$ and ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=2a<0.$ Then there is ${ϵ}_0\in \left(0,1\right)$ with the property that, for any $ϵ\in \left(0,{ϵ}_0\right)$, we can select $N_0$ so that

$$\begin{array}{cc}\hfill sup& \left\{\sum _{x\in E}{\left(\frac{1}{ϵ}\right)}^{S_{\lceil a_{1}n\rceil }\left(\phi \left(x\right)-\beta \psi \left(x\right)\right)}:F_n\mathrm{is}\mathrm{an}\left(\mathbf{a},n,ϵ\right)-\mathrm{separated}\mathrm{set}\mathrm{of}{X}_1\right\}\hfill \\ \hfill & <{\left(\frac{1}{ϵ}\right)}^{an}.\hfill \end{array}$$

This indicates that when V is sufficiently large, we have

$$\begin{array}{cc}\hfill R_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)& ⩽\sum _{n⩾N_0}\underset{F_n}{sup}\sum _{x\in F_n}{\left(\frac{1}{ϵ}\right)}^{S_{\lceil a_{1}n\rceil }\phi \left(x\right)-\beta S_{\lceil a_{1}n\rceil }\psi \left(x\right)}\hfill \\ \hfill & ⩽\sum _{n⩾N_0}{\left(\frac{1}{ϵ}\right)}^{an}<\frac{1}{1-{ϵ}^{-a}}.\hfill \end{array}$$

By taking the limits, we can obtain

$$\underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)⩽1.$$

Then, by combining the previously proven (1), we obtain

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & =inf\left\{\beta \in \mathbb{R}:\underset{ϵ\to 0}{lim\; sup}\underset{V\to \infty }{lim\; sup}{R}_{\psi ,V}^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi ,ϵ\right)<\infty \right\}\hfill \\ \hfill & ⩽inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)<0\right\}.\hfill \end{array}$$

Combining with Proposition 2, we can prove that

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & ⩽inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)<0\right\}\hfill \\ \hfill & =inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩽0\right\}\hfill \\ \hfill & =sup\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)⩾0\right\}.\hfill \end{array}$$

On the contrary, the reverse inequality has been proven in Corollary 1. □

Proof of Theorem 1.

Based on Proposition 2, the equation

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=0\end{array}$$

has a unique root $\beta$. Then using Proposition 3, we find that $\beta$ is equal to

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right).\end{array}$$

□

3. Proof of Theorem 2

Section 3 is divided into two parts. The first part presents basic definitions and preliminary results associated with the weighted upper metric mean dimension with potential on subsets. The second part formulates Bowen’s equations for the weighted upper metric mean dimension with potential on subsets.

3.1. Several Types of Weighted Upper Metric Dimension with Potential

Definition 4.

Let $0<ϵ<1$ and $\lambda \in \mathbb{R}$. For $Z\subset X_1,\phi \in C\left(X_1,\mathbb{R}\right)$. We define

$$\begin{array}{c}\hfill {\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,N,ϵ\right)=inf\left\{\sum _{i\in I}{e}^{-n_i\lambda +\frac{1}{a_1}log\frac{1}{ϵ}{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\right\},\end{array}$$

where we take the infimum over all finite or countable covers ${\left\{B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z with $n_i⩾N.$

$$\begin{array}{c}\hfill {\overline{m}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,N,ϵ\right)=inf\left\{\sum _{i\in I}{e}^{-N\lambda +\frac{1}{a_1}log\frac{1}{ϵ}{\displaystyle \underset{y\in B_N^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_{1}N\rceil }\phi \left(y\right)}\right\},\end{array}$$

where we take the infimum over all finite or countable covers ${\left\{B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z with $n_i=N.$

Let

$$\begin{array}{cc}\hfill & {\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=\underset{N\to \infty }{lim}{\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,N,ϵ\right),\hfill \\ \hfill & {\overline{m}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=\underset{N\to \infty }{lim}{\overline{m}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,N,ϵ\right).\hfill \end{array}$$

Then we can obtain the critical values of λ from ∞ to 0 for ${\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)$ and ${\overline{m}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)$, which we respectively denote as

$$\begin{array}{cc}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right):& =inf\left\{\lambda :{\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=0\right\}\hfill \\ \hfill & =sup\left\{\lambda :{\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=\infty \right\},\hfill \\ \hfill {\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right):& =inf\left\{\lambda :{\overline{m}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=0\right\}\hfill \\ \hfill & =sup\left\{\lambda :{\overline{m}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=\infty \right\}.\hfill \end{array}$$

Put

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}\frac{{\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)}{log\frac{1}{ϵ}},\hfill \\ \hfill & {\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}\frac{{\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)}{log\frac{1}{ϵ}}.\hfill \end{array}$$

The quantities ${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$ and ${\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$ are respectively referred to as the weighted Bowen upper mean dimension with potential φ, the weighted u-upper metric mean dimension with potential φ on the set Z. Additionally, when the metrics $d_i$ for $i=1, \dots , k$ are clear, we can omit their mention from these quantities. Particularly, when $\phi =0,$ we denote

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z,{\left\{d_i\right\}}_{i=1}^k\right):={\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(0,{\left\{d_i\right\}}_{i=1}^k\right)\end{array}$$

as the weighted Bowen upper mean dimension on the set Z.

Remark 2.

Let Z be a subset of $X_1$. We define

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(Z,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & =\underset{ϵ\to 0}{lim\; sup}\underset{n\to \infty }{lim\; sup}\frac{1}{nlog\frac{1}{ϵ}}log\underset{E_n}{inf}\left\{\sum _{x\in E_n}{e}^{\frac{1}{a_1}log\frac{1}{ϵ}·S_{\lceil a_{1}n\rceil }\phi \left(x\right)}\right\},\hfill \end{array}$$

where the infimum runs over all $(\mathbf{a},n,ϵ)$-spanning sets $E_n$ of Z. Similarly, we can obtain that

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(Z,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)\hfill \\ \hfill & =\underset{ϵ\to 0}{lim\; sup}\underset{n\to \infty }{lim\; sup}\frac{1}{nlog\frac{1}{ϵ}}log\underset{F_n}{sup}\left\{\sum _{x\in F_n}{e}^{\frac{1}{a_1}log\frac{1}{ϵ}·S_{\lceil a_{1}n\rceil }\phi \left(x\right)}\right\}\hfill \\ \hfill & ={\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right),\hfill \end{array}$$

where the supremum runs over all $(\mathbf{a},n,ϵ)$-separated sets $F_n$ of Z.

Definition 5.

Let $0<ϵ<1$ and $\lambda \in \mathbb{R}$. For $Z\subset X_1,\phi \in C\left(X_1,\mathbb{R}\right)$. We define

$$\begin{array}{c}\hfill P^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,N,ϵ\right)=sup\left\{\sum _{i\in I}{e}^{-n_i\lambda +\frac{1}{a_1}log\frac{1}{ϵ}·{\displaystyle \underset{y\in {\overline{B}}_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\right\},\end{array}$$

where we take the supremum over all finite or countable pairwise disjoint closed families ${\left\{{\overline{B}}_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z with $n_i⩾N,x_i\in Z.$ The value $P^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)$ is monotonically decreasing in N, so we define

$$\begin{array}{c}\hfill P^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=\underset{N\to \infty }{lim}{P}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,N,ϵ\right).\end{array}$$

Set

$$\begin{array}{c}\hfill {\mathcal{P}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=inf\left\{\sum _{i=1}^{\infty }{P}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z_i,\phi ,\lambda ,ϵ\right):\bigcup _{i⩾1}{Z}_i\supseteq Z\right\}.\end{array}$$

Then we can obtain the critical values of λ from ∞ to 0 for ${\mathcal{P}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)$, which we denote as

$$\begin{array}{cc}\hfill {\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right):& =inf\left\{\lambda :{\mathcal{P}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=0\right\}\hfill \\ \hfill & =sup\left\{\lambda :{\mathcal{P}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,ϵ\right)=\infty \right\}.\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill {\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}\frac{{\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)}{log\frac{1}{ϵ}}.\end{array}$$

The value ${\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$ is referred to as the weighted packing upper mean dimension with potential φ on the set Z. Moreover, when $d_i$ for $i=1,...,k$ is clear, we can omit their mention from these quantities. In addition, when $\phi =0,$ we denote ${\overline{\mathrm{Pmdim}}}_M^{\mathbf{a}}\left(T_1,Z,{\left\{d_i\right\}}_{i=1}^k\right):={\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(0,{\left\{d_i\right\}}_{i=1}^k\right)$ as the weighted packing upper mean dimension on the set Z.

Proposition 4.

Let $\phi \in C\left(X_1,\mathbb{R}\right)$.

(i) 

If $Z_1\subset Z_2\subset X,$ then

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,Z_1,T_1}^{\mathbf{a}}\left(\phi \right)⩽{\overline{\mathrm{mdim}}}_{M,Z_2,T_1}^{\mathbf{a}}\left(\phi \right),\hfill \\ \hfill & {\overline{\mathrm{upmdim}}}_{M,Z_1,T_1}^{\mathbf{a}}\left(\phi \right)⩽{\overline{\mathrm{upmdim}}}_{M,Z_2,T_1}^{\mathbf{a}}\left(\phi \right),\hfill \\ \hfill & {\overline{\mathrm{Pmdim}}}_{M,Z_1,T_1}^{\mathbf{a}}\left(\phi \right)⩽{\overline{\mathrm{Pmdim}}}_{M,Z_2,T_1}^{\mathbf{a}}\left(\phi \right).\hfill \end{array}$$

(ii) 

If $Z={\bigcup }_{i⩾1}{Z}_i,$ then

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)=\underset{1⩽i⩽N}{max}{\overline{\mathrm{mdim}}}_{M,Z_i,T_1}^{\mathbf{a}}\left(\phi \right),\hfill \\ \hfill & {\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)=\underset{1⩽i⩽N}{max}{\overline{\mathrm{upmdim}}}_{M,Z_i,T_1}^{\mathbf{a}}\left(\phi \right),\hfill \\ \hfill & {\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)=\underset{1⩽i⩽N}{max}{\overline{\mathrm{Pmdim}}}_{M,Z_i,T_1}^{\mathbf{a}}\left(\phi \right).\hfill \end{array}$$

(iii) 

If any non-empty set $Z\subset X_1$, then

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)⩽{\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)⩽{\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right).\end{array}$$

Proof. 

(i) and (ii) can be directly derived from Definitions 4 and 5.

(iii) Let $ϵ\in \left(0,1\right)$, $\gamma \left(4\epsilon \right)=sup\left\{\left|\phi \left(x\right)-\phi \left(y\right)\right|:d_1\left(x,y\right)⩽4ϵ\right\},$$Z\subset X_1$, and $n\in \mathbb{N}$. Let R denote the maximal cardinality of a disjoint family ${\left\{{\overline{B}}_n^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i=1}^R$ with $x_i\in Z.$ It is obvious that ${\bigcup }_{i=1}^R{B}_n^{\mathbf{a}}\left(x_i,3ϵ\right)\supseteq Z.$

Assume that $\lambda \in \mathbb{R};$ we have

$$\begin{array}{cc}\hfill & {\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,n,3ϵ\right)\hfill \\ \hfill & ⩽\sum _{i=1}^R{e}^{-n\lambda +\frac{1}{a_1}log\frac{1}{3ϵ}·{\displaystyle \underset{y\in B_n^{\mathbf{a}}\left(x_i,3ϵ\right)}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩽\sum _{i=1}^R{e}^{-n\lambda +\frac{1}{a_1}log\frac{1}{3ϵ}·{\displaystyle \underset{y\in {\overline{B}}_n^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)+\frac{1}{a_1}\lceil a_{1}n\rceil \gamma \left(4\epsilon \right)log\frac{1}{3ϵ}}\hfill \\ \hfill & ⩽\sum _{i=1}^R{e}^{-n\lambda +\frac{1}{a_1}log\frac{1}{ϵ}·{\displaystyle \underset{y\in {\overline{B}}_n^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)-\frac{1}{a_1}\lceil a_{1}n\rceil \parallel \phi \parallel log\frac{1}{3}+\frac{1}{a_1}\lceil a_{1}n\rceil \gamma \left(4\epsilon \right)log\frac{1}{3ϵ}}\hfill \\ \hfill & ⩽P^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda -\parallel \phi \parallel log\frac{1}{3}-\gamma \left(4\epsilon \right)log\frac{1}{3ϵ},n,ϵ\right).\hfill \end{array}$$

For any ${\bigcup }_{i⩾1}{Z}_i\supseteq Z,$ we have

$$\begin{array}{cc}\hfill & {\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,3ϵ\right)\hfill \\ \hfill & ⩽\sum _{i⩾1}{\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z_i,\phi ,\lambda ,3ϵ\right)\hfill \\ \hfill & ⩽\sum _{i⩾1}{P}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z_i,\phi ,\lambda -\parallel \phi \parallel log{\displaystyle \frac{1}{3}}-\gamma \left(4\epsilon \right)log{\displaystyle \frac{1}{3ϵ}},ϵ\right).\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,3ϵ\right)\hfill \\ \hfill & ⩽{\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)+log{\displaystyle \frac{1}{3}}\parallel \phi \parallel +\gamma \left(4\epsilon \right)log{\displaystyle \frac{1}{3ϵ}}.\hfill \end{array}$$

Therefore, we can obtain ${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)⩽{\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right).$

Next, we prove that ${\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)⩽{\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right).$ We can suppose that

$$\begin{array}{c}\hfill {\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)>-\infty ,\end{array}$$

because if not, there is nothing to prove. Let $-\infty <v<{\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$, then we can pick a subsequence ${ϵ}_k$ with ${ϵ}_k\in \left(0,1\right)$ that converges to 0 when $k\to \infty$. As a result,

$$\begin{array}{c}\hfill {\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{k\to \infty }{lim}{\displaystyle \frac{{\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,{ϵ}_k\right)}{log\frac{1}{{ϵ}_k}}}>v.\end{array}$$

Hence, for any $k>K_0$ with $K_0\in \mathbb{N},$ there is ${\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,{ϵ}_k\right)>vlog{\displaystyle \frac{1}{{ϵ}_k}}.$ In this way,

$$\begin{array}{c}\hfill P^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,vlog{\displaystyle \frac{1}{{ϵ}_k}},{ϵ}_k\right)⩾{\mathcal{P}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,vlog{\displaystyle \frac{1}{{ϵ}_k}},{ϵ}_k\right)=\infty .\end{array}$$

Fix $k>K_0$, for each $N\in \mathbb{N},$ we can determine a countable pairwise disjoint closed family ${\left\{{\overline{B}}_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ with $n_i⩾N,x_i\in Z,$ so that

$$\begin{array}{c}\hfill \sum _{i\in I}{e}^{-n_{i}vlog\frac{1}{{ϵ}_k}+\frac{1}{a_1}log\frac{1}{{ϵ}_k}·{\displaystyle \underset{y\in {\overline{B}}_n^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}>1.\end{array}$$

Define $E_l=\left\{x_{n_i}:n_i=l,i\in I\right\}$ with $l⩾N.$ Then

$$\begin{array}{cc}\hfill & \sum _{l⩾N}\sum _{x\in E_l}{e}^{-lvlog{\displaystyle \frac{1}{{ϵ}_k}}+{\displaystyle \frac{1}{a_1}}log{\displaystyle \frac{1}{{ϵ}_k}}·\left(S_{\lceil a_{1}l\rceil }\phi \left(y\right)+\lceil a_{1}l\rceil \gamma \left({\epsilon }_k\right)\right)}\hfill \\ \hfill & ⩾\sum _{l⩾N}\sum _{x\in E_l}{e}^{-lvlog{\displaystyle \frac{1}{{ϵ}_k}}+{\displaystyle \frac{1}{a_1}}log{\displaystyle \frac{1}{{ϵ}_k}}·{\displaystyle \underset{y\in {\overline{B}}_l^{\mathbf{a}}\left(x,{ϵ}_k\right)}{sup}}{S}_{\lceil a_{1}l\rceil }\phi \left(y\right)}>1,\hfill \end{array}$$

where $\gamma \left(\epsilon \right):=sup\left\{\left|\phi \left(x\right)-\phi \left(y\right)\right|:d_1\left(x,y\right)⩽ϵ\right\}.$ Let $u<v$, then there is an $l_N⩾N$ such that

$$\begin{array}{c}\hfill \sum _{x\in E_{l_N}}{e}^{-l_N\left(v-\gamma \left({\epsilon }_k\right)\right)log{\displaystyle \frac{1}{{ϵ}_k}}+{\displaystyle \frac{1}{a_1}}log{\displaystyle \frac{1}{{ϵ}_k}}{S}_{\lceil a_1{l}_N\rceil }\phi \left(x\right)}>\left(1-e^{\left(u-v\right)log{\displaystyle \frac{1}{{ϵ}_k}}}\right)e^{\left(u-v\right)l_{N}log{\displaystyle \frac{1}{{ϵ}_k}}},\end{array}$$

that is,

$$\begin{array}{c}\hfill \sum _{x\in E_{l_N}}{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{S_{\lceil a_1{l}_N\rceil }\phi \left(x\right)}>\left(1-e^{\left(u-v\right)log{\displaystyle \frac{1}{{ϵ}_k}}}\right){\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{\left(u-\gamma \left({\epsilon }_k\right)\right)l_N},\end{array}$$

where $E_{l_N}$ is an $\left(\mathbf{a},l_N,{ϵ}_k\right)$-separated set of Z. Hence, we can obtain

$$\begin{array}{c}\hfill \underset{N\to \infty }{lim\; sup}{\displaystyle \frac{1}{Nlog\frac{1}{{ϵ}_k}}}log\underset{E_N}{sup}\left\{\sum _{x\in E_N}{\left({\displaystyle \frac{1}{{ϵ}_k}}\right)}^{S_{\lceil a_{1}N\rceil }\phi \left(x\right)}\right\}⩾u-\gamma \left({\epsilon }_k\right),\end{array}$$

where the supremum runs over all $\left(\mathbf{a},N,{ϵ}_k\right)$-separated sets of Z. Observe that when $k\to \infty ,$$\gamma \left({\epsilon }_k\right)\to 0$, combined with Remark 2, it can be concluded that

$$\begin{array}{c}\hfill {\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)⩾u.\end{array}$$

Finally, we let $u\to {\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$; we can obtain the desired result. □

3.2. Bowen’s Equation for Weighted Upper Metric Mean Dimension

In this subsection, we first studied some basic characteristics of functions. These functions are defined using the weighted Bowen upper metric mean dimension with potential and the weighted packing upper metric mean dimension with potential on a subset of $X_1$. Next, we defined the weighted BS metric mean dimension and the weighted packing BS metric mean dimension. Finally, we proved that they are the only roots of the corresponding Bowen’s equation.

Given a non-empty subset $Z\subset X_1$ and $\phi \in C\left(X_1,\mathbb{R}\right)$, next, we look at the following functions:

$$\begin{array}{cc}\hfill & \Phi \left(t\right)={\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t\phi ,{\left\{d_i\right\}}_{i=1}^k\right),\hfill \\ \hfill & \varphi \left(t\right)={\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t\phi ,{\left\{d_i\right\}}_{i=1}^k\right).\hfill \end{array}$$

Proposition 5.

Let $\mathbf{a}=(a_1,\cdots ,a_k)\in {\mathbb{R}}^k$ with $a_1>0$ and $a_i\ge 0$ for $i\ge 2$. Let $Z\subset X_1$ be a non-empty subset, $\phi \in C\left(X_1,\mathbb{R}\right)$, and $\phi <0$. Then for any $t\in \mathbb{R}$, we have ${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t\phi \right)>-\infty$, and ${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t\phi \right)<\infty$ if and only if ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)<\infty .$

Proof. 

Set $m={\displaystyle \underset{x\in X_1}{min}}\phi \left(x\right)$ and $0<ϵ<1.$ When $t⩾0,$ then for any N, we have

$$\begin{array}{cc}\hfill & {\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,t\phi ,tmlog{\displaystyle \frac{1}{ϵ}},N,ϵ\right)\hfill \\ \hfill & =inf\left\{\sum _{i\in I}{e}^{-n_{i}tmlog{\displaystyle \frac{1}{ϵ}}+{\displaystyle \frac{1}{a_1}}log{\displaystyle \frac{1}{ϵ}}·{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\right\},\hfill \end{array}$$

where we take the infimum over all finite or countable covers ${\left\{B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z with $n_i⩾N.$ According to Definition 3, we have ${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,ϵ\right)⩾tmlog{\displaystyle \frac{1}{ϵ}}.$ Consequently, ${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t\phi \right)⩾tm>-\infty .$ When $t<0$, take $t^{\prime }>0$. Given that $t\phi ⩾t^{\prime }\phi$ and ${\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,\phi ,\lambda ,N,ϵ\right)$ is a monotonic function with respect to $\phi ,$ we can obtain

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t\phi \right)⩾{\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t^{\prime }\phi \right)>-\infty .\end{array}$$

Next, we prove the equivalence between ${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t\phi \right)<\infty$ and ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)<\infty .$ For any $t\in \mathbb{R}$, by Definition 4, we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,0,\lambda -\left|t\right|∥\phi ∥log{\displaystyle \frac{1}{ϵ}},N,ϵ\right)=inf\left\{\sum _{i\in I}{e}^{-n_i\lambda -\left|t\right|n_i\parallel \phi \parallel log{\displaystyle \frac{1}{ϵ}}}\right\}\hfill \\ \hfill & ⩽inf\left\{\sum _{i\in I}{e}^{-n_i\lambda +t{\displaystyle \frac{1}{a_1}}log{\displaystyle \frac{1}{ϵ}}{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\right\}\hfill \\ \hfill & ⩽inf\left\{\sum _{i\in I}{e}^{-n_i\lambda +\left|t\right|n_i\parallel \phi \parallel log{\displaystyle \frac{1}{ϵ}}}\right\}={\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,0,\lambda +\left|t\right|∥\phi ∥log{\displaystyle \frac{1}{ϵ}},N,ϵ\right),\hfill \end{array}\end{array}$$

(12)

where we take the infimum over all finite or countable covers ${\left\{B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z and $n_i⩾N.$ When $\phi =0,$ we have ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)={\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(0\right).$ Then, combining this with (12), we can deduce that

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)-\left|t\right|\parallel \phi \parallel ⩽{\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t\phi \right)⩽{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)+\left|t\right|\parallel \phi \parallel .\end{array}$$

□

Proposition 6.

Let $\phi \in C\left(X_1,\mathbb{R}\right)$ with $\phi <0$. Assuming ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,X_1\right)<\infty$, then the function $\varphi \left(t\right)$ is strictly decreasing and Lipschitz, the function $\varphi \left(t\right)=0$ has a single (finite) root s, and $-{\displaystyle \frac{1}{m}}{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)⩽s⩽-{\displaystyle \frac{1}{M}}{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)$, where $m=\underset{x\in X_1}{min}\phi \left(x\right),M=\underset{x\in X_1}{max}\phi \left(x\right).$

Proof. 

Given $t_1,t_2\in \mathbb{R}$ with $t_1>t_2,$ and $0<ϵ<1$, $N\in \mathbb{N}$, choose a cover ${\left\{B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z where $n_i⩾N.$ We thus obtain

$$\begin{array}{cc}\hfill & \sum _{i\in I}{e}^{-n_i\lambda +t_1{\displaystyle \frac{1}{a_1}}log{\displaystyle \frac{1}{ϵ}}·{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩽\sum _{i\in I}{e}^{-n_i\lambda +t_2{\displaystyle \frac{1}{a_1}}log{\displaystyle \frac{1}{ϵ}}·{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)+\left(t_1-t_2\right)n_{i}Mlog{\displaystyle \frac{1}{ϵ}}}.\hfill \end{array}$$

Based on this, we can obtain

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t_1\phi \right)⩽{\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t_2\phi \right)+\left(t_1-t_2\right)M.\end{array}$$

(13)

Hence, we can conclude that $\varphi \left(t\right)$ is strictly decreasing on $\mathbb{R}$.

Likewise, we can obtain

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t_2\phi \right)+\left(t_1-t_2\right)m⩽{\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t_1\phi \right).\end{array}$$

(14)

Taking the Lipschitz constant as $-m$, we can see that

$$\begin{array}{c}\hfill \left|{\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t_1\phi \right)-{\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(t_2\phi \right)\right|⩽-m\left|t_1-t_2\right|.\end{array}$$

Let $t_1=h>0,t_2=0$ in (13), then we have

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(h\phi \right)⩽{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)-h\left(-M\right),\end{array}$$

which implies that

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\left(-{\displaystyle \frac{1}{M}}{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)\right)\phi \right)⩽0.\end{array}$$

Let $t_1=h>0,t_2=0$ in (14), then we have

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(h\phi \right)⩾{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)-h\left(-m\right).\end{array}$$

It follows that

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\left(-{\displaystyle \frac{1}{m}}{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)\right)\phi \right)⩾0.\end{array}$$

Finally, by applying the intermediate value theorem of continuous functions, it can be concluded that the function $\varphi \left(t\right)=0$ has a single (finite) root s, and

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{m}}{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right)⩽s⩽-{\displaystyle \frac{1}{M}}{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right).\end{array}$$

□

By slightly modifying Proposition 6, we can obtain the following results:

Proposition 7.

Let $\phi \in C\left(X_1,\mathbb{R}\right)$ and $\phi <0$. Assume ${\overline{\mathrm{Pmdim}}}_M^{\mathbf{a}}\left(T_1,X_1\right)<\infty$. Then the function $\Phi \left(t\right)$ is strictly decreasing, Lipschitz, and the function $\Phi \left(t\right)=0$ has a single root.

Below, we define two novel concepts: the weighted BS metric mean dimension and the weighted packing BS metric mean dimension, both of which are defined on subsets.

Definition 6.

Let $0<ϵ<1$, $N\in \mathbb{N}$ and $\lambda \in \mathbb{R}$. For $Z\subset X_1,\phi \in C\left(X_1,\mathbb{R}\right)$ and $\phi >0$. We define

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,N,ϵ\right)=inf\left\{\sum _{i\in I}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\right\},\end{array}$$

where we take the infimum over all finite or countable covers ${\left\{B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z with $n_i⩾N.$

The value $R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,N,ϵ\right)$ does not decrease with the increase in N, so we define

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)=\underset{N\to \infty }{lim}{R}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,N,ϵ\right).\end{array}$$

Then we can obtain the critical values of λ from ∞ to 0 for $R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)$, which we respectively denote as

$$\begin{array}{cc}\hfill {\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)& =inf\left\{\lambda :R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)=0\right\}\hfill \\ \hfill & =sup\left\{\lambda :R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)=\infty \right\}.\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill {\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{{\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)}{log\frac{1}{ϵ}}}.\end{array}$$

The value ${\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$ is referred to as the weighted BS upper mean dimension with respect to φ on the set Z (or simply weighted BS metric mean dimension). Moreover, when $d_i$ for $i=1,...,k$ is clear, we can omit their mention from these quantities.

Definition 7.

Let $0<ϵ<1,$ $N\in \mathbb{N}$, and $\lambda \in \mathbb{R}$. For $Z\subset X_1,\phi \in C\left(X_1,\mathbb{R}\right)$ and $\phi >0$. We define

$$\begin{array}{c}\hfill P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,N,ϵ\right)=sup\left\{\sum _{i\in I}{e}^{-\lambda \frac{1}{a_1}{\displaystyle \underset{y\in {\overline{B}}_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\right\},\end{array}$$

where we take the supremum over all finite or countable pairwise disjoint closed families ${\left\{{\overline{B}}_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z with $n_i⩾N,x_i\in Z.$ The value $P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,N,ϵ\right)$ is monotonically decreasing in N, so we define

$$\begin{array}{c}\hfill P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)=\underset{N\to \infty }{lim}{P}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,N,ϵ\right).\end{array}$$

Set

$$\begin{array}{c}\hfill {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)=inf\left\{\sum _{i=1}^{\infty }{P}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z_i,ϵ\right):\bigcup _{i⩾1}{Z}_i\supseteq Z\right\}.\end{array}$$

Then we can obtain the critical values of λ from ∞ to 0 for ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)$, which we denote as

$$\begin{array}{cc}\hfill {\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)& =inf\left\{\lambda :{\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)=0\right\}\hfill \\ \hfill & =sup\left\{\lambda :{\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right)=\infty \right\}.\hfill \end{array}$$

Let

$$\begin{array}{c}\hfill {\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{{\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)}{log\frac{1}{ϵ}}}.\end{array}$$

The value ${\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$ is referred to as the weighted packing BS upper mean dimension with respect to φ on the set Z (or simply weighted packing BS metric mean dimension). Moreover, when $d_i$ for $i=1,...,k$ is clear, we can omit their mention from these quantities.

Remark 3.

(a) For any $Z\subset X_1$, $0⩽{\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)⩽{\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right).$

(b) 

${\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(1\right)={\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,Z\right),{\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(1\right)={\overline{\mathrm{Pmdim}}}_M^{\mathbf{a}}\left(T_1,Z\right).$

Proof of Theorem 2.

Let $ϵ\in \left(0,1\right)$, for any N, by combining Definitions 4 and 6, we have

$$\begin{array}{c}\hfill {\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,-{\displaystyle \frac{\lambda \phi }{log\frac{1}{ϵ}}},0,N,ϵ\right)=R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,N,ϵ\right).\end{array}$$

If $v>{\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$, then for small-enough $ϵ>0$,

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,vlog{\displaystyle \frac{1}{ϵ}},Z,ϵ\right)<1.\end{array}$$

Therefore, we obtain that ${\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,-v\phi ,0,ϵ\right)<1$, suggesting ${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(-v\phi \right)$$⩽0.$ By proving the continuity of $\varphi$ through Proposition 6, it can be concluded that when $v\to {\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$,

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(-\left({\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)\right)\phi \right)⩽0.\end{array}$$

If $v<{\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$, we can pick a subsequence ${ϵ}_k\in \left(0,1\right)$ with ${ϵ}_k\to 0$ when $k\to \infty$, so that

$$\begin{array}{c}\hfill {\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{k\to \infty }{lim\; sup}{\displaystyle \frac{{\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,{ϵ}_k\right)}{log\frac{1}{{ϵ}_k}}}.\end{array}$$

Consequently, for large-enough k, $R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,vlog{\displaystyle \frac{1}{{ϵ}_k}},Z,{ϵ}_k\right)>1$. Hence, we can obtain ${\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,-v\phi ,0,{ϵ}_k\right)>1.$ Similarly, we infer that

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(-\left({\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)\right)\phi \right)⩾0.\end{array}$$

By Proposition 6, we can obtain that ${\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$ is the only root of the equation

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(-t\phi \right)=0.\end{array}$$

By ${\mathcal{P}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,Z,-{\displaystyle \frac{\lambda }{log\frac{1}{ϵ}}}\phi ,0,ϵ\right)={\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\lambda ,Z,ϵ\right),$ we can similarly obtain that ${\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi \right)$ is the only root of the equation ${\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(-t\phi \right)=0.$ □

Corollary 2.

Let $\phi ,\psi \in C\left(X_1,\mathbb{R}\right)$, and $\psi >0$. Then

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,0\right)={\overline{\mathrm{BSmdim}}}_{M,X_1,T_1}^{\mathbf{a}}\left(\psi \right).\end{array}$$

Proof. 

If ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k\right)=\infty ,$ by Remark 3, we can obtain

$$\begin{array}{cc}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k\right)& ={\overline{\mathrm{mdim}}}_{M,X_1,T_1}^{\mathbf{a}}\left(0,{\left\{d_i\right\}}_{i=1}^k\right)\hfill \\ \hfill & ={\overline{\mathrm{mdim}}}_{M,X_1,T_1}^{\mathbf{a}}\left(0·\left(-\psi \right),{\left\{d_i\right\}}_{i=1}^k\right)\hfill \\ \hfill & ={\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,0·\left(-\psi \right),{\left\{d_i\right\}}_{i=1}^k\right)\hfill \\ \hfill & =\infty \hfill \end{array}$$

By Proposition 2, taking $\phi =0$, we have ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,-\beta \psi ,{\left\{d_i\right\}}_{i=1}^k\right)=\infty$ for any $\beta \in \mathbb{R}$. By Proposition 3, we have

$$\begin{array}{cc}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,0\right)& =inf\left\{\beta \in \mathbb{R}:{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,-\beta \psi \right)⩽0\right\}\hfill \\ \hfill & =inf\varnothing =\infty .\hfill \end{array}$$

Let $M:=\underset{x\in X_1}{max}\psi \left(x\right)>0$, for any $ϵ\in \left(0,1\right),\lambda ⩾0,\mathrm{and}N\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\psi ,\lambda ,X_1,N,ϵ\right)& =inf\left\{\sum _{i\in I}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\right\}\hfill \\ \hfill & ⩾inf\left\{\sum _{i\in I}{e}^{-\lambda M{n}_i}\right\}\hfill \\ \hfill & ={\mathcal{M}}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,X_1,0,M\lambda ,N,ϵ\right),\hfill \end{array}$$

where we take the infimum over all finite or countable covers ${\left\{B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}}_{i\in I}$ of Z where $n_i=N.$ We thus obtain

$$\begin{array}{c}\hfill \infty ={\displaystyle \frac{{\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,X_1,{\left\{d_i\right\}}_{i=1}^k\right)}{M}}⩽{\overline{\mathrm{BSmdim}}}_{M,X_1,T_1}^{\mathbf{a}}\left(\psi ,{\left\{d_i\right\}}_{i=1}^k\right).\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,0\right)={\overline{\mathrm{BSmdim}}}_{M,X_1,T_1}^{\mathbf{a}}\left(\psi ,{\left\{d_i\right\}}_{i=1}^k\right)=\infty .\end{array}$$

For ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,X_1,{\left\{d_i\right\}}_{i=1}^k\right)<\infty$, by Remark 2, we have

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k\right)={\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(T_1,X_1,{\left\{d_i\right\}}_{i=1}^k\right)<\infty .\end{array}$$

By Theorem 1, we can obtain

$$\begin{array}{cc}\hfill & {\overline{\mathrm{mdim}}}_{M,X_1,T_1}^{\mathbf{a}}\left(-{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,0\right)·\psi ,{\left\{d_i\right\}}_{i=1}^k\right)\hfill \\ \hfill & ={\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,-{\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,0\right)·\psi ,{\left\{d_i\right\}}_{i=1}^k\right)=0.\hfill \end{array}$$

Combining with Theorem 2 and the uniqueness of the equation’s roots, we have

$$\begin{array}{c}\hfill {\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,0\right)={\overline{\mathrm{BSmdim}}}_{M,X_1,T_1}^{\mathbf{a}}\left(\psi ,{\left\{d_i\right\}}_{i=1}^k\right).\end{array}$$

□

4. Proof of Theorems 3 and 4

In this section, we mainly set up the variational principle for the weighted BS and packing BS metric mean dimension on subsets using the given weighted upper and lower local measure-theoretical BS entropies.

Definition 8.

Let $\mu \in M\left(X_1\right),\phi \in C\left(X_1,\mathbb{R}\right)$ with $\phi >0$, we set up the weighted upper and lower local measure-theoretical BS entropies of μ as

$$\begin{array}{c}\hfill {\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1\right)=\underset{ϵ\to 0}{lim}{\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right),\\ \hfill {\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1\right)=\underset{ϵ\to 0}{lim}{\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right),\end{array}$$

respectively, where

$$\begin{array}{cc}\hfill & {\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)=\int \underset{n\to \infty }{lim\; sup}-{\displaystyle \frac{a_{1}log\mu \left(B_n^{\mathbf{a}}\left(x,ϵ\right)\right)}{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}}d\mu =\int {\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,x,ϵ\right)d\mu ,\hfill \\ \hfill & {\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)=\int \underset{n\to \infty }{liminf}-{\displaystyle \frac{a_{1}log\mu \left(B_n^{\mathbf{a}}\left(x,ϵ\right)\right)}{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}}d\mu =\int {\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,x,ϵ\right)d\mu .\hfill \end{array}$$

Lemma 1

([31]). Let $r>0$ and $\mathcal{B}\left(r\right)=\{B_n^{\mathbf{a}}(x,r):x\in X,n=1,2,\dots \}$. For any family $\mathcal{F}\subset \mathcal{B}\left(r\right)$, there exists a (not necessarily countable) subfamily $\mathcal{G}\subset \mathcal{F}$ consisting of disjoint balls such that

$$\begin{array}{c}\hfill \bigcup _{B\in \mathcal{F}}B\subset \bigcup _{B_n^{\mathbf{a}}(x,r)\in \mathcal{G}}{B}_n^{\mathbf{a}}(x,3r).\end{array}$$

Definition 9.

Let $\phi \in C\left(X_1,\mathbb{R}\right)$ with $\phi >0$, and $\varphi >0$ with ϕ is a bounded function on $X_1$. Additionally, let $ϵ>0$, $N\in \mathbb{N}$, and $\lambda \in \mathbb{R}$. We define

$$\begin{array}{c}\hfill W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\psi ,\lambda ,N,ϵ\right)=inf\left\{\sum _{i=1}^{\infty }{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}\right\},\end{array}$$

(15)

where the infimum is over all finite or countable families ${\left\{(B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right),c_i)\right\}}_{i\in I}$ such that $0<c_i<\infty ,x_i\in X,n_i⩾N,$ and ${\sum }_{i\in I}{c}_i{\chi }_{B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}⩾\psi ,$ with ${\chi }_E$ denoting the characteristic function of E.

For Z to be a subset of $X_1$, set

$$\begin{array}{c}\hfill W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right):=W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{\chi }_Z,\lambda ,N,ϵ\right).\end{array}$$

Since the value $W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right)$ does not decrease with the increase in N, we define

$$\begin{array}{c}\hfill W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,ϵ\right)=\underset{N\to \infty }{lim}{W}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right).\end{array}$$

Then we can obtain the critical values of λ from ∞ to 0 for $W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,ϵ\right)$, which we denote as

$$\begin{array}{cc}\hfill {\overline{\mathrm{Wmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)& =inf\left\{\lambda :W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,ϵ\right)=0\right\}\hfill \\ \hfill & =sup\left\{\lambda :W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,ϵ\right)=\infty \right\}.\hfill \end{array}$$

Let ${\overline{\mathrm{Wmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)={\displaystyle \underset{ϵ\to 0}{lim\; sup}}{\displaystyle \frac{{\overline{\mathrm{Wmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)}{log\frac{1}{ϵ}}}.$

The value ${\overline{\mathrm{Wmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$ is referred to as the weighted BS metric mean dimension with respect to φ on the set Z. Furthermore, when d is clear, we can generally omit d from the value.

4.1. Variational Principle for the Weighted BS Metric Mean Dimension

In this subsection, we set up the variational principle for the weighted BS metric mean dimension on subsets.

Proposition 8.

Let $\phi \in C\left(X_1,\mathbb{R}\right)$ with $\phi >0$, and let $0<ϵ<1$, and $Z\subset X_1$. Then

$$\begin{array}{cc}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda +\delta ,N,6ϵ\right)& ⩽W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right)\hfill \\ \hfill & ⩽R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right),\hfill \end{array}$$

for $\lambda >0,\delta >0.$ Hence,

$$\begin{array}{c}\hfill {\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)={\overline{\mathrm{Wmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right).\end{array}$$

Proof. 

Let $Z\subset X_1,\lambda >0,$$ϵ$, and $\delta >0.$ By setting $\psi ={\chi }_Z,c_i=1$ in (15), we obtain that for any $N\in \mathbb{N},$

$$\begin{array}{c}\hfill W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right)⩽R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right).\end{array}$$

Next, we only need to prove that

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda +\delta ,N,6ϵ\right)⩽W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right).\end{array}$$

Let $N>2$, so that $n^2{e}^{-nm\delta }⩽1$ when $n⩾N,$ where $m=\underset{x\in X_1}{min}\phi \left(x\right).$ Consider a family ${\left\{\left(B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right),c_i\right)\right\}}_{i\in \mathcal{I}}$ with the properties $0<c_i<\infty ,x_i\in X_1,n_i⩾N,$ and

$$\begin{array}{c}\hfill \sum _{i\in I}{c}_i{\chi }_{B_i^{\mathbf{a}}}⩾{\chi }_Z,\end{array}$$

(16)

where $B_i^{\mathbf{a}}=B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right).$ Next, we show that

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda +\delta ,N,6ϵ\right)⩽\sum _{i\in \mathcal{I}}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)},\end{array}$$

(17)

which means $R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda +\delta ,N,6ϵ\right)⩽W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda ,N,ϵ\right).$

To simplify a later discussion, we first define the notations. For $n⩾N,k\in \mathbb{N},$ we set ${\mathcal{I}}_n=\left\{i\in \mathcal{I}:n_i=n\right\},$${\mathcal{I}}_{n,k}=\left\{i\in {\mathcal{I}}_n:i⩽k\right\}.$ For convenience, let $B_i^{\mathbf{a}}=B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right),$$5B_i^{\mathbf{a}}=B_{n_i}^{\mathbf{a}}\left(x_i,5ϵ\right)$ for $i\in \mathcal{I}.$ Suppose that $B_i\ne B_j$ whenever $i\ne j.$ Given $t>0$, we establish

$$\begin{array}{cc}\hfill & Z_{n,t}=\left\{x\in Z:\sum _{i\in {\mathcal{I}}_n}{c}_i{\chi }_{B_i^{\mathbf{a}}}\left(x\right)>t\right\},\hfill \\ \hfill & Z_{n,k,t}=\left\{x\in Z:\sum _{i\in {\mathcal{I}}_{n,k}}{c}_i{\chi }_{B_i^{\mathbf{a}}}\left(x\right)>t\right\}.\hfill \end{array}$$

We are going to prove (17) by dividing it into three steps.

Step 1: For any $n⩾N,k\in \mathbb{N}$ and $t>0,$ there is a finite set ${\mathcal{J}}_{n,k,t}\subset {\mathcal{I}}_{n,k}$ with pairwise disjoint balls $B_i^{\mathbf{a}}$ for $i\in {\mathcal{J}}_{n,k,t},$$Z_{n,k,t}\subset {\bigcup }_{i\in {\mathcal{J}}_{n,k,t}}5B_i^{\mathbf{a}}$ and

$$\begin{array}{c}\hfill \sum _{i\in {\mathcal{J}}_{n,k,t}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_n^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}⩽{\displaystyle \frac{1}{t}}\sum _{i\in {\mathcal{I}}_{n,k}}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_n^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}.\end{array}$$

As ${\mathcal{I}}_{n,k}$ is finite, we can assume that every $c_i$ is a positive rational by upward-approximating the $c_i$’s. Next, multiplying by a common denominator allows us to assume that every $c_i$ is a positive integer.

Let $m_0$ be the smallest integer such that $m_0⩾t$. Denote $\mathcal{B}=\left\{B_i^{\mathbf{a}}:i\in {\mathcal{I}}_{n,k}\right\}$, and define a mapping $v:\mathcal{B}\to \mathbb{Z}$ where $v\left(B_i^{\mathbf{a}}\right)=c_i.$ We define, by means of induction, interval-valued functions $v_0,v_1,\cdots ,v_{m_0}$ on $\mathcal{B}$, as well as subfamilies ${\mathcal{B}}_0,{\mathcal{B}}_1,\cdots ,{\mathcal{B}}_{m_0}$ of $\mathcal{B}$, commencing with $v_0=v.$ Utilizing Lemma 1, we obtain a pairwise disjoint subfamily ${\mathcal{B}}_1$ of $\mathcal{B}$ with ${\bigcup }_{B\in \mathcal{B}}{B}^{\mathbf{a}}\subset {\bigcup }_{B\in {\mathcal{B}}_1}5B^{\mathbf{a}}.$ As a consequence, it follows that $Z_{n,k,t}\subset {\bigcup }_{B\in {\mathcal{B}}_1}5B^{\mathbf{a}}.$ Then, through the repeated application of Lemma 1, we can inductively define for $j=1,\cdots ,m_0,$ disjoint subfamilies ${\mathcal{B}}_j$ of $\mathcal{B}$ with

$$\begin{array}{c}\hfill {\mathcal{B}}_j\subset \left\{B^{\mathbf{a}}\in \mathcal{B}:v_{j-1}\left(B^{\mathbf{a}}\right)⩾1\right\},Z_{n,k,t}\subset \bigcup _{B\in {\mathcal{B}}_j}5B^{\mathbf{a}},\end{array}$$

where

$$\begin{array}{c}\hfill v_j\left(B^{\mathbf{a}}\right)=\left\{\begin{array}{c}{v}_{j-1}\left(B^{\mathbf{a}}\right)-1,B^{\mathbf{a}}\in {\mathcal{B}}_j,\\ v_{j-1}\left(B^{\mathbf{a}}\right),B^{\mathbf{a}}\in \mathcal{B}\setminus {\mathcal{B}}_j.\end{array}\right.\end{array}$$

When $j<m_0$, we obtain

$$\begin{array}{c}\hfill Z_{n,k,t}\subset \left\{x:\sum _{B^{\mathbf{a}}\in \mathcal{B}:B^{\mathbf{a}}\ni x}{v}_j\left(B^{\mathbf{a}}\right)⩾m_0-j\right\}.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & \sum _{j=1}^{m_0}\sum _{B^{\mathbf{a}}\in {\mathcal{B}}_j}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & =\sum _{j=1}^{m_0}\sum _{B^{\mathbf{a}}\in {\mathcal{B}}_j}\left[v_{j-1}\left(B^{\mathbf{a}}\right)-v_j\left(B^{\mathbf{a}}\right)\right]e^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩽\sum _{B^{\mathbf{a}}\in \mathcal{B}}\sum _{j=1}^{m_0}\left[v_{j-1}\left(B^{\mathbf{a}}\right)-v_j\left(B^{\mathbf{a}}\right)\right]e^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩽\sum _{B^{\mathbf{a}}\in \mathcal{B}}v\left(B^{\mathbf{a}}\right)e^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & =\sum _{i\in {\mathcal{I}}_{n,k}}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}.\hfill \end{array}$$

Take $j_0\in \left\{1,\cdots ,m_0\right\}$ for which ${\sum }_{B^{\mathbf{a}}\in {\mathcal{B}}_{j_0}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}$ is the smallest. Then

$$\begin{array}{cc}\hfill \sum _{B^{\mathbf{a}}\in {\mathcal{B}}_{j_0}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}& ⩽{\displaystyle \frac{1}{m_0}}\sum _{i\in {\mathcal{I}}_{n,k}}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{t}}\sum _{i\in {\mathcal{I}}_{n,k}}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}.\hfill \end{array}$$

Therefore, ${\mathcal{J}}_{n,k,t}=\left\{i\in {\mathcal{I}}_{n,k}:B^{\mathbf{a}}\in {\mathcal{B}}_{j_0}\right\}$ is desired.

Step 2: For any $n⩾N,t>0,$ we obtain

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z_{n,t},\lambda +\delta ,N,6ϵ\right)⩽{\displaystyle \frac{1}{n^{2}t}}\sum _{i\in {\mathcal{I}}_n}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}.\end{array}$$

(18)

To prove the above result, let $Z_{n,t}\ne \varnothing ;$ otherwise, the proof is not necessary. Since $Z_{n,k,t}\uparrow Z_{n,t},$ when k is sufficiently large, $Z_{n,k,t}$ is non-empty. Let ${\mathcal{J}}_{n,k,t}$ denote the sets obtained from Step 1. Then, for a large-enough k, ${\mathcal{J}}_{n,k,t}$ is also non-empty. Define $E_{n,k,t}=\left\{x_i:i\in {\mathcal{J}}_{n,k,t}\right\}.$ It is well known that the family of all non-empty compact subsets of $X_1$ is compact in the Hausdorff distance (cf. Federer [32]). As a consequence, there exists a subsequence $\left(k_j\right)$ of natural numbers, and a non-empty compact set $E_{n,t}\subset X.$ As $j\to \infty ,$$E_{n,k_j,t}$ converges to $E_{n,t}$ in the Hausdorff distance. Given that, with respect to the metric $d_n^{\mathbf{a}}$, the distance between any two points within the set $E_{n,k,t}$ is at least $ϵ$, the same is true for $E_{n,t}$. Thus, $E_{n,t}$ is finite. Additionally, when j is large enough, $\#\left(E_{n,k_j,t}\right)=\#\left(E_{n,t}\right).$ Therefore,

$$\begin{array}{c}\hfill \bigcup _{x\in E_{n,t}}{B}_n^{\mathbf{a}}\left(x,5.5ϵ\right)\supseteq \bigcup _{x\in E_{n,k_j,t}}{B}_n^{\mathbf{a}}\left(x,5ϵ\right)=\bigcup _{i\in {\mathcal{J}}_{n,k_j,t}}5B_i^{\mathbf{a}}\supseteq Z_{n,k_j,t};\end{array}$$

thus, ${\bigcup }_{x\in E_{n,t}}{B}_n^{\mathbf{a}}\left(x,6ϵ\right)\supseteq Z_{n,t}.$ Additionally, when j is large enough, $\#\left(E_{n,k_j,t}\right)=\#\left(E_{n,t}\right)$, we have

$$\begin{array}{c}\hfill \sum _{x\in E_{n,t}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}⩽{\displaystyle \frac{1}{t}}\sum _{i\in {\mathcal{I}}_n}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_i^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z_{n,t},\lambda +\delta ,N,6ϵ\right)& ⩽\sum _{x\in E_{n,t}}{e}^{-\left(\lambda +\delta \right){\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B\left(x_i,6ϵ\right)}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{t{e}^{nm\delta }}}\sum _{i\in {\mathcal{I}}_n}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B\left(x_i,6ϵ\right)}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{n^{2}t}}\sum _{i\in {\mathcal{I}}_n}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B\left(x_i,6ϵ\right)}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}.\hfill \end{array}$$

Since $B\left(x_i,6ϵ\right)\supset B\left(x_i,ϵ\right),$ we deduce that

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z_{n,t},\lambda +\delta ,N,6ϵ\right)⩽{\displaystyle \frac{1}{n^{2}t}}\sum _{i\in {\mathcal{I}}_n}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_i^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}.\end{array}$$

Step 3: For every $t\in (0,1)$, we obtain

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda +\delta ,N,6ϵ\right)⩽{\displaystyle \frac{1}{t}}\sum _{i\in \mathcal{I}}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_i^{\mathbf{a}}}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}.\end{array}$$

Then (17) is valid.

To prove the above result, let $t\in (0,1).$ Note that ${\sum }_{n=N}^{\infty }{\displaystyle \frac{1}{n^2}}<1$. Hence,

$$\begin{array}{c}\hfill Z\subseteq \bigcup _{n=N}^{\infty }{Z}_{n,n^{-2}t}\end{array}$$

from (16). Then by (18) and with $R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,·,\lambda +\delta ,N,6ϵ\right)$ being an outer measure, we have

$$\begin{array}{cc}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z,\lambda +\delta ,N,6ϵ\right)& ⩽\sum _{n=N}^{\infty }{R}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,Z_{n,n^{-2}t},\lambda +\delta ,N,6ϵ\right)\hfill \\ \hfill & ⩽\sum _{n=N}^{\infty }{\displaystyle \frac{1}{t}}\sum _{i\in {\mathcal{I}}_n}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_i^{\mathbf{a}}}{sup}}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & ={\displaystyle \frac{1}{t}}\sum _{i\in \mathcal{I}}{c}_i{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{\displaystyle \underset{y\in B_i^{\mathbf{a}}}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)},\hfill \end{array}$$

which finishes the proof of the lemma. □

Lemma 2.

Let $K\subset X_1$ be a non-empty compact subset, $\lambda ⩾0,N\in \mathbb{N},ϵ>0,$$\phi \in C\left(X_1,\mathbb{R}\right)$ and $\phi >0$. Suppose that $c:=W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,K,\lambda ,N,ϵ\right)>0;$ then there exists a Borel probability measure $\mu \in M\left(X_1\right),$ such that $\mu \left(K\right)=1$, and

$$\begin{array}{c}\hfill \mu \left(B_n^{\mathbf{a}}\left(x,ϵ\right)\right)⩽{\displaystyle \frac{1}{c}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x\right)}\end{array}$$

holds for any $x\in X_1,n⩾\mathbb{N}.$

Proof. 

Certainly, $c<\infty .$ Now, on the space $C\left(X_1\right)$ of continuous real-valued functions on $X_1$, we proceed to define a function p as follows:

$$\begin{array}{c}\hfill p\left(f\right)={\displaystyle \frac{1}{c}}{W}^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{\chi }_K·f,\lambda ,N,ϵ\right),\end{array}$$

with $W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,·,\lambda ,N,ϵ\right)$ defined according to (15).

Define $\mathbf{1}\in C\left(X_1\right)$ as the constant function, and for any $x\in X_1,$$\mathbf{1}\left(x\right)\equiv 1.$ It can be readily verified that

(1)

For any $g,f\in C\left(X_1\right),p\left(f+g\right)⩽p\left(f\right)+p\left(g\right)$.

(2)

For any $t⩾0$ and $f\in C\left(X_1\right),p\left(tf\right)=tp\left(f\right).$

(3)

$p\left(\mathbf{1}\right)=1,$ for any $f\in C\left(X_1\right),$ the inequality $0⩽p\left(f\right)⩽{\parallel f\parallel }_{\infty }$ holds, and for any $g\in C\left(X_1\right)$ with $g⩽0,p\left(g\right)=0.$

By the Hahn–Banach theorem, the linear functional $t\mapsto tp\left(\mathbf{1}\right)$ for $t\in \mathbb{R}$, defined on the constant-function subspace, can be extended to $L:C\left(X_1\right)\to \mathbb{R}.$ It satisfies $L\left(\mathbf{1}\right)=p\left(\mathbf{1}\right)=1.$ For any $f\in C\left(X_1\right)$, we have $-p\left(-f\right)⩽L\left(f\right)⩽p\left(f\right).$ Additionally, if $f\in C\left(X_1\right)$ with $f⩾0$, then $p\left(-f\right)=0,$ so $L\left(f\right)⩾0.$ Given that $L\left(\mathbf{1}\right)=1$, applying the Riesz representation theorem allows us to obtain a Borel probability measure $\mu$ on $X_1.$ For all $f\in C\left(X_1\right)$, this $\mu$ satisfies $L\left(f\right)=\int fd\mu .$

Now, we show that $\mu \left(K\right)=1$. Let $E_1$ be an arbitrary compact subset of $X_1\setminus K;$ the Urysohn lemma ensures the existence of $f\in C\left(X_1\right)$ with $0⩽f⩽1,f\left(x\right)=1$ for $x\in E_1,$ and $f\left(x\right)=0$ for $x\in K$. Then $f·{\chi }_K\equiv 0$, so $p\left(f\right)=0.$ Since $\mu \left(E_1\right)=L\left(f\right)⩽p\left(f\right),$ we obtain $\mu \left(E_1\right)=0.$ As $E_1$ is any compact subset of $X_1\setminus K$; this implies $\mu \left(X_1\setminus K\right)=0,$ and thus, $\mu \left(K\right)=1.$

Finally, we show that for $x\in X_1,n⩾N,$

$$\begin{array}{c}\hfill \mu \left(B_n^{\mathbf{a}}\left(x,ϵ\right)\right)⩽{\displaystyle \frac{1}{c}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x\right)}.\end{array}$$

For any compact set $E_2\subset B_n^{\mathbf{a}}\left(x,ϵ\right)$, from the Urysohn lemma, we know that there exists $f\in C\left(X_1\right),$ so that $0⩽f⩽1,f\left(y\right)=1,$ for $y\in E_2$, and $f\left(y\right)=0,$ for $y\in X_1\setminus B_n^{\mathbf{a}}\left(x,ϵ\right).$ Then, we have $\mu \left(E_2\right)=L\left(f\right)⩽p\left(f\right).$ As $f·{\chi }_K⩽{\chi }_{B_n^{\mathbf{a}}\left(x,ϵ\right)}$ and $n⩾N,$ we obtain

$$\begin{array}{c}\hfill W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,{\chi }_K·f,\lambda ,N,ϵ\right)⩽e^{-\lambda {\displaystyle \frac{1}{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x\right)},\end{array}$$

and $p\left(f\right)⩽{\displaystyle \frac{1}{c}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x\right)}.$ Consequently,

$$\begin{array}{c}\hfill \mu \left(E_2\right)⩽e^{-\lambda {\displaystyle \frac{1}{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x\right)}.\end{array}$$

Based on this result, we obtain

$$\begin{array}{cc}\hfill \mu \left(B_n^{\mathbf{a}}\left(x,ϵ\right)\right)& ⩽sup\left\{\mu \left(E_2\right):E_2\mathrm{is}\mathrm{a}\mathrm{compact}\mathrm{subset}\mathrm{of}{B}_n^{\mathbf{a}}\left(x,ϵ\right)\right\}\hfill \\ \hfill & ⩽{\displaystyle \frac{1}{c}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x\right)}.\hfill \end{array}$$

□

Proof of Theorem 3.

We will prove that

$$\begin{array}{c}\hfill {\overline{\mathrm{BSmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{sup\left\{{\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right):\mu \in M\left(X_1\right),\mu \left(K\right)=1\right\}}{log\frac{1}{ϵ}}}.\end{array}$$

On the one hand, we seek to prove that $lhs⩾rhs.$ For $x\in X_1,n\in \mathbb{N},ϵ>0,$ by Definition 8, we have

$$\begin{array}{c}\hfill {\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)=\int {\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,x,ϵ\right)d\mu .\end{array}$$

Therefore, it suffices to prove

$$\begin{array}{c}\hfill \left(1-{\displaystyle \frac{\gamma \left(2ϵ\right)}{m}}\right)\int {\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,x,ϵ\right)d\mu ⩽R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,K,{\displaystyle \frac{ϵ}{2}}\right),\end{array}$$

for any $0<ϵ<1$.

Fix $0<ϵ<1$ and $l\in \mathbb{N}$. Denote

$$\begin{array}{c}\hfill \gamma \left(2ϵ\right)=sup\left\{\left|\phi \left(x\right)-\phi \left(y\right)\right|:d_1\left(x,y\right)<2ϵ\right\},\end{array}$$

and

$$\begin{array}{c}\hfill u_l=min\left\{l,\int {\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,x,ϵ\right)d\mu -{\displaystyle \frac{1}{l}}\right\}.\end{array}$$

Next, we can find a Borel set $A_l\subset X_1$ such that $\mu \left(A_l\right)>0$ and $N\in \mathbb{N}.$ For all $x\in A_l$ and $n⩾N,$ we have

$$\begin{array}{c}\hfill \mu \left(B_n^{\mathbf{a}}\left(x,ϵ\right)\right)⩽e^{-u_l{\displaystyle \frac{1}{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x\right)}.\end{array}$$

(19)

There is a countable or finite family $\left\{B_{n_i}^{\mathbf{a}}\left(x_i,{\displaystyle \frac{ϵ}{2}}\right)\right\}$, where $x_i\in X_1$, $n_i⩾N$, and it satisfies the condition that

$$\begin{array}{c}\hfill \bigcup _i{B}_{n_i}^{\mathbf{a}}\left(x_i,{\displaystyle \frac{ϵ}{2}}\right)\supset K\cap A_l.\end{array}$$

It is known that for any i, $B_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\cap \left(K\cap A_l\right)\ne \varnothing ,$ so there is $y_i\in B_{n_i}^{\mathbf{a}}\left(x_i,{\displaystyle \frac{ϵ}{2}}\right)\cap \left(K\cap A_l\right).$ Then, from (19), we obtain the following inequalities:

$$\begin{array}{cc}\hfill \sum _i{e}^{-u_l{\displaystyle \frac{1}{a_1}}\left(1-{\displaystyle \frac{\gamma \left(2ϵ\right)}{m}}\right){\displaystyle \underset{z\in B_{n_i}^{\mathbf{a}}\left(x_i,{\displaystyle \frac{ϵ}{2}}\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(z\right)}& ⩾\sum _i{e}^{-u_l{\displaystyle \frac{1}{a_1}}{\displaystyle \underset{z\in B_{n_i}^{\mathbf{a}}\left(x_i,{\displaystyle \frac{ϵ}{2}}\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(z\right)+u_l{\displaystyle \frac{1}{a_1}}\gamma \left(2ϵ\right)\lceil a_1{n}_i\rceil }\hfill \\ \hfill & ⩾\sum _i{e}^{-u_l{\displaystyle \frac{1}{a_1}}{\displaystyle \underset{z\in B_{n_i}^{\mathbf{a}}\left(x_i,{\displaystyle \frac{ϵ}{2}}\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(z\right)+u_l{n}_i\gamma \left(2ϵ\right)}\hfill \\ \hfill & ⩾\sum _i{e}^{-u_l{\displaystyle \frac{1}{a_1}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y_i\right)}\hfill \\ \hfill & ⩾\sum _i\mu \left(B_{n_i}^{\mathbf{a}}\left(y_i,ϵ\right)\right)\hfill \\ \hfill & ⩾\sum _i\mu \left(B_{n_i}^{\mathbf{a}}\left(x_i,{\displaystyle \frac{ϵ}{2}}\right)\right)\hfill \\ \hfill & ⩾\mu \left(K\cap A_l\right)=\mu \left(A_l\right)>0.\hfill \end{array}$$

Based on these results, we obtain

$$\begin{array}{cc}\hfill & R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\left(1-{\displaystyle \frac{\gamma \left(2ϵ\right)}{m}}\right)u_l,K,{\displaystyle \frac{ϵ}{2}}\right)\hfill \\ \hfill & ⩾R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,\left(1-{\displaystyle \frac{\gamma \left(2ϵ\right)}{m}}\right)u_l,K\cap A_l,{\displaystyle \frac{ϵ}{2}}\right)\hfill \\ \\ \hfill & ⩾\mu \left(A_l\right).\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,K,{\displaystyle \frac{ϵ}{2}}\right)⩾\left(1-{\displaystyle \frac{\gamma \left(2ϵ\right)}{m}}\right)u_l.\end{array}$$

Letting $l\to \infty$, we have

$$\begin{array}{c}\hfill \left(1-{\displaystyle \frac{\gamma \left(2ϵ\right)}{m}}\right)\int {\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,x,ϵ\right)d\mu ⩽R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,K,{\displaystyle \frac{ϵ}{2}}\right).\end{array}$$

Therefore,

$$\begin{array}{c}\hfill \left(1-{\displaystyle \frac{\gamma \left(2ϵ\right)}{m}}\right){\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)⩽R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,K,{\displaystyle \frac{ϵ}{2}}\right).\end{array}$$

Combining

$$\begin{array}{c}\hfill {\overline{\mathrm{BSmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{R^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,K,ϵ\right)}{log\frac{1}{ϵ}}},\end{array}$$

we can obtain $lhs⩾rhs.$

On the other hand, let ${\overline{\mathrm{BSmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)>0.$ By Proposition 8, we have

$$\begin{array}{c}\hfill {\overline{\mathrm{BSmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)={\overline{\mathrm{Wmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right).\end{array}$$

Suppose that $0<\lambda <{\overline{\mathrm{Wmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right).$ Then, there exists a sequence ${ϵ}_k$ with $0<{ϵ}_k<1$, and this sequence converges to 0 as $k\to \infty .$ Hence,

$$\begin{array}{c}\hfill {\overline{\mathrm{Wmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)=\underset{k\to \infty }{lim}{\displaystyle \frac{{\overline{\mathrm{Wmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,{ϵ}_k\right)}{log\frac{1}{{ϵ}_k}}}>\lambda .\end{array}$$

Thus, for a large-enough k, there is an $N_0\in \mathbb{N}.$ Then we define a positive number as

$$\begin{array}{c}\hfill c:=W^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,K,\lambda log{\displaystyle \frac{1}{{ϵ}_k}},N_0,{ϵ}_k\right)>0.\end{array}$$

From Lemma 2, a Borel probability measure $\mu \in M\left(X_1\right)$ exists. It has $\mu \left(K\right)=1$, and

$$\begin{array}{c}\hfill \mu \left(B_n^{\mathbf{a}}\left(x,{ϵ}_k\right)\right)⩽{\displaystyle \frac{1}{c}}{e}^{-\lambda {\displaystyle \frac{1}{a_1}}log{\displaystyle \frac{1}{{ϵ}_k}}{S}_{\lceil a_{1}n\rceil }\phi \left(x\right)}\end{array}$$

holds for any $x\in X_1,n⩾N_0.$ Hence,

$$\begin{array}{c}\hfill {\displaystyle \frac{sup\left\{{\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,{ϵ}_k\right):\mu \in M\left(X_1\right),\mu \left(K\right)=1\right\}}{log\frac{1}{{ϵ}_k}}}⩾{\displaystyle \frac{{\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)}{log\frac{1}{{ϵ}_k}}}⩾\lambda .\end{array}$$

Let $\lambda \to {\overline{\mathrm{Wmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right);$ combined with

$$\begin{array}{c}\hfill {\overline{\mathrm{BSmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)={\overline{\mathrm{Wmdim}}}_{M,K,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right),\end{array}$$

we can obtain $lhs⩽rhs$ for a sufficiently large k.

□

4.2. Variational Principle for the Weighted Packing BS Metric Mean Dimension

In what follows, we are going to establish the variational principle for the weighted packing BS metric mean dimension on subsets. To attain this goal, we introduce an additional concept. In a metric space, a set is an analytic set if it is the continuous image of the set $\mathcal{N}$ of infinite natural-number sequences (with product topology). It is well known that, in a Polish space, analytic subsets are closed under countable unions and intersections, and all Borel sets are analytic (cf. [32]).

Lemma 3.

Let Z be a subset of $X_1$, $s,ϵ>0.$ If $P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s,Z,ϵ\right)=\infty ,$ then for any finite interval $\left(a,b\right)\subset \mathbb{R}$ with $a⩾0$ and any $N\in \mathbb{N}$, there exists a finite disjoint family $\left\{{\overline{B}}_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}$ where $x_i\in Z,n_i⩾N$ and

$$\begin{array}{c}\hfill \sum _i{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(x_i\right)}\in \left(a,b\right).\end{array}$$

Proof. 

Select a sufficiently large $N_1⩾N$ so that $e^{-Nms}<b-a.$ Given that

$$\begin{array}{c}\hfill P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s,Z,ϵ\right)=\infty ,\end{array}$$

we can conclude that $P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s,Z,N,ϵ\right)=\infty .$ Therefore, there is a finite disjoint family $\left\{{\overline{B}}_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)\right\}$ where $x_i\in Z$ and $n_i⩾N,$ and the following inequalities hold:

$$\begin{array}{c}\hfill \sum _i{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(x_i\right)}>\sum _i{e}^{-{\displaystyle \frac{s}{a_1}}{\displaystyle \underset{y\in {\overline{B}}_{n_i}^{\mathbf{a}}\left(x_i,ϵ\right)}{sup}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(y\right)}>b.\end{array}$$

As $e^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(x_i\right)}⩽e^{-Nms}<b-a,$ we can remove elements from this collection one by one. In the end, we can make ${\sum }_i{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{n}_i\rceil }\phi \left(x_i\right)}\in \left(a,b\right).$ □

Proof of Theorem 4.

We split the proof into the following two parts:

Part 1.

$$\begin{array}{c}\hfill {\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)⩾\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{sup\left\{{\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right):\mu \in M\left(X_1\right),\mu \left(Z\right)=1\right\}}{log\frac{1}{ϵ}}},\end{array}$$

for any Borel set $Z\subset X_1.$ Given $\mu \in M\left(X_1\right)$, where $\mu \left(Z\right)=1$ and ${\overline{h}}_{\phi ,\mu }^{\mathbf{a}}(T_1,2ϵ)>0$. For $s\in \left(0,{\overline{h}}_{\phi ,\mu }^{\mathbf{a}}(T_1,2ϵ)\right)$, by Definition 8, we can select $\delta >0$ and a Borel set $A\subset Z$ satisfying $\mu \left(A\right)>0$. Then, for any $x\in A$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}-{\displaystyle \frac{a_{1}log\mu \left(B_n^{\mathbf{a}}\left(x,ϵ\right)\right)}{S_{\lceil a_{1}n\rceil }\phi \left(x\right)}}>s+\delta .\end{array}$$

Next, we prove ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right),Z,{\displaystyle \frac{ϵ}{5}}\right)=\infty$, where $m={min}_{x\in X_1}\phi \left(x\right)$ with the property that $m>0$, and $\gamma \left(ϵ\right)=sup\left\{\right|\phi \left(x\right)-\phi \left(y\right)|:d_1(x,y)⩽ϵ\}$. For this purpose, we only need to demonstrate that, for every $E\subset A$ and $\mu \left(E\right)>0$, we have

$$\begin{array}{c}\hfill P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right),E,{\displaystyle \frac{ϵ}{5}}\right)=\infty .\end{array}$$

To this end, we fix a set E like this and define

$$\begin{array}{c}\hfill E_n:=\{x\in E:\mu \left(B_n^{\mathbf{a}}(x,2ϵ)\right)<e^{-\left(\frac{s+\delta }{a_1}\right)S_{\lceil a_{1}n\rceil }\phi \left(x\right)}\}.\end{array}$$

We then obtain that, for every $N\in \mathbb{N}$, $E={\cup }_{n\ge N}{E}_n$. After that, we fix such N. Combining $\mu \left(E\right)=\mu \left({\cup }_{n\ge N}{E}_n\right)$ with ${\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{n\left(n+1\right)}}=1$, we can obtain that when $n\ge N$, $\mu \left(E_n\right)⩾{\displaystyle \frac{1}{n(n+1)}}\mu \left(E\right)$. Fix n; consider taking into account a family of closed covers for the set $E_n$, which is $\{B_n^{\mathbf{a}}(x,{\displaystyle \frac{ϵ}{5}}):x\in E_n\}$. According to Lemma 1, there is a finite pairwise disjoint subfamily ${\{B_n^{\mathbf{a}}(x,{\displaystyle \frac{ϵ}{5}}):x\in E_n\}}_{i\in I}$, where I is a finite index set, and it holds that

$$\begin{array}{c}\hfill {\cup }_{i\in I}{B}_n^{\mathbf{a}}(x_i,ϵ)\supseteq {\cup }_{x\in E_n}{B}_n^{\mathbf{a}}(x,{\displaystyle \frac{ϵ}{5}})\supseteq E_n.\end{array}$$

For any $i\in I$, we obtain

$$\begin{array}{cc}\hfill \underset{y\in {\overline{B}}_n^{\mathbf{a}}(x,{\displaystyle \frac{ϵ}{5}})}{sup}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)& ⩽S_{\lceil a_{1}n\rceil }\phi \left(x_i\right)+\lceil a_{1}n\rceil \gamma \left(ϵ\right)\hfill \\ \hfill & ⩽S_{\lceil a_{1}n\rceil }\phi \left(x_i\right)+{\displaystyle \frac{\underset{y\in {\overline{B}}_n^{\mathbf{a}}(x,\frac{ϵ}{5})}{\sup }{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}{m}}\gamma \left(ϵ\right).\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill & P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right),E,{\displaystyle \frac{ϵ}{5}}\right)\hfill \\ \hfill & ⩾P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right),E_n,{\displaystyle \frac{ϵ}{5}}\right)\hfill \\ \hfill & ⩾\sum _{iϵI}{e}^{-{\displaystyle \frac{s}{a_1}}\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right)\underset{y\in {\overline{B}}_n^{\mathbf{a}}\left(x_i,{\displaystyle \frac{ϵ}{5}}\right)}{sup}{S}_{\lceil a_{1}n\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩾\sum _{iϵI}{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x_i\right)}\hfill \\ \hfill & =\sum _{iϵI}{e}^{-{\displaystyle \frac{1}{a_1}}\left(s+\delta \right)S_{\lceil a_{1}n\rceil }\phi \left(x_i\right)}{e}^{{\displaystyle \frac{\delta }{a_1}}{S}_{\lceil a_{1}n\rceil }\phi \left(x_i\right)}\hfill \\ \hfill & ⩾e^{nm\delta }\sum _{iϵI}\mu \left({\overline{B}}_n^{\mathbf{a}}\left(x_i,ϵ\right)\right)\hfill \\ \hfill & ⩾e^{nm\delta }\mu \left(E_n\right)\hfill \\ \hfill & ⩾e^{nm\delta }{\displaystyle \frac{\mu \left(E\right)}{n\left(n+1\right)}}.\hfill \end{array}$$

As $N\to \infty$, we arrive at the result that $P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right),E,{\displaystyle \frac{ϵ}{5}}\right)=\infty .$ Therefore,

$$\begin{array}{c}\hfill {\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,{\displaystyle \frac{ϵ}{5}}\right)⩾s\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right).\end{array}$$

Furthermore, as $s\to {\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,2ϵ\right)$, for every $\mu \in M\left(X_1\right)$ satisfying $\mu \left(Z\right)=1$, we obtain

$$\begin{array}{c}\hfill {\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,2ϵ\right)\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right)⩽{\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,{\displaystyle \frac{ϵ}{5}}\right).\end{array}$$

This implies

$$\begin{array}{cc}\hfill & {\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,{\displaystyle \frac{ϵ}{5}}\right)\hfill \\ \hfill & ⩾\left(1-{\displaystyle \frac{\gamma \left(ϵ\right)}{m}}\right)sup\left\{{\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,2ϵ\right):\mu \in M\left(X_1\right),\mu \left(Z\right)=1\right\},\hfill \end{array}$$

and thus,

$$\begin{array}{c}\hfill {\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)⩾\underset{ϵ\to 0}{lim\; sup}{\displaystyle \frac{sup\left\{{\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right):\mu \in M\left(X_1\right),\mu \left(Z\right)=1\right\}}{log\frac{1}{ϵ}}}.\end{array}$$

Part 2. Let Z be an analytic subset of $X_1$, ${\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)>0$. For every $s\in \left(0,{\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)\right)$, there is a compact $K\subset Z$ and $\mu \in M\left(K\right)$ so that ${\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)\ge s$.

Given that Z is an analytic set, there is a continuous surjective mapping $\varphi :\mathcal{N}\to Z$. Denote by ${\Gamma }_{n_1,n_2,\dots ,n_p}$ the set of all $(m_1,m_2,\dots )\in \mathcal{N}$ satisfying the conditions $m_1\le n_1$, $m_2\le n_2$, …, $m_p\le n_p$. Furthermore, let $Z_{n_1,\dots ,n_p}$ be the image of ${\Gamma }_{n_1,\dots ,n_p}$ via the map $\varphi$.

Take $ϵ>0$ to be sufficiently small such that $0<s<{\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)$. Take $t\in \left(s,{\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k,ϵ\right)\right)$. Through an inductive process, we construct a sequence of finite sets ${\left(K_i\right)}_{i=1}^{\infty }$ and a sequence of finite measures ${\left({\mu }_i\right)}_{i=1}^{\infty }.$ For every $i,$ we will ensure that $K_i\subset Z$ and the measure ${\mu }_i$ has its support on the set $K_i$. Alongside the two sequences ${\left(K_i\right)}_{i=1}^{\infty }$ and ${\left({\mu }_i\right)}_{i=1}^{\infty }$, we further construct a sequence of integers $\left(n_i\right)$, a sequence of positive numbers $\left({\gamma }_i\right)$, and a sequence of integer-valued functions $(m_i:K_i\to \mathbb{N})$. Our construction method gets inspired by Joyce and Preiss’s [33] and Feng and Huang’s [30] works. The construction is split into several detailed steps.

Step 1: Construct $K_1$, ${\mu }_1$, $m_1(·)$, $n_1$, and ${\gamma }_1$.

Note that ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z,ϵ\right)=\infty$. Let

$$\begin{array}{c}\hfill H=\bigcup \{G\subseteq X_1:G\mathrm{is}\mathrm{open},P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z\cap G,ϵ\right)=0\}.\end{array}$$

Due to the separability of $X_1$, it can be concluded that $P_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z\cap H,ϵ\right)=0$. Let $Z^{\prime }=Z\setminus H=Z\cap (X_1\setminus H)$. For every open set $G\subseteq X_1$, either $Z^{\prime }\cap G=\varnothing$ (which means $G\subseteq H$) or $Z^{\prime }\cap G\ne \varnothing$ (in which case, ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z^{\prime }\cap G,ϵ\right)>0$). To show this, let us assume that, for some open set G, ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z^{\prime }\cap G,ϵ\right)=0;$ then

$$\begin{array}{cc}\hfill & {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z\cap G,ϵ\right)\hfill \\ \hfill & ⩽{\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z^{\prime }\cap G,ϵ\right)+{\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z\cap H,ϵ\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

From this, it can be inferred that $G\subset H$ and then $Z^{\prime }\cap G=\varnothing$. Since

$$\begin{array}{cc}& {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z,ϵ\right)\hfill \\ \hfill \le & {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z^{\prime },ϵ\right)+{\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z\cap H,ϵ\right)\hfill \\ \hfill =& {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z^{\prime },ϵ\right),\hfill \end{array}$$

it follows that

$$\begin{array}{c}\hfill {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z^{\prime },ϵ\right)=\infty .\end{array}$$

By Lemma 3, there exists a finite set $K_1\subset Z^{\prime }$; an integer-valued function $m_1\left(x\right)$ on $K_1$ with the collection ${\left\{{\overline{B}}_{m_1\left(x\right)}^{\mathbf{a}}(x,ϵ)\right\}}_{x\in K_1}$ is disjoint and

$$\begin{array}{c}\hfill \sum _{x\in K_1}{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_1\left(x\right)\rceil }\phi \left(x\right)}\in (1,2).\end{array}$$

Let ${\mu }_1={\sum }_{x\in K_1}{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_1\left(x\right)\rceil }\phi \left(x\right)}{\delta }_x$, with ${\delta }_x$ being the Dirac measure at x. Choose a small ${\gamma }_1>0$ so that, for every function $z,K_1\to X_1$ with $d_1(x,z\left(x\right))⩽{\gamma }_1$. Then, for all $x\in K_1$, we have

$$\begin{array}{c}\hfill \left(\overline{B}(z\left(x\right),{\gamma }_1)\cup {\overline{B}}_{m_1\left(x\right)}^{\mathbf{a}}(z\left(x\right),ϵ)\right)\cap \left(\bigcup _{y\in K_1\setminus \left\{x\right\}}(\overline{B}(z\left(y\right),{\gamma }_1)\cup {\overline{B}}_{m_1\left(y\right)}^{\mathbf{a}}(z\left(y\right),ϵ)\right)=\varnothing .\end{array}$$

(20)

In this context and subsequently, $\overline{B}\left(x,ϵ\right)$ is defined as the closed ball $\{y\in X_1:d_1(x,y)⩽ϵ\}$. Given that $K_1\subset Z^{\prime }$, for any $x\in K_1$, we have ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z\cap B\left(x,{\displaystyle \frac{{\gamma }_1}{4}}\right),ϵ\right)⩾{\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z^{\prime }\cap B\left(x,{\displaystyle \frac{{\gamma }_1}{4}}\right),ϵ\right)>0.$ Therefore, we can choose a sufficiently large $n_1\in \mathbb{N}$ such that $Z_{n_1}\supset K_1$, and for all $x\in K_1$, ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1}\cap B\left(x,{\displaystyle \frac{{\gamma }_1}{4}}\right),ϵ\right)>0$.

Step 2: Construct $K_2$, ${\mu }_2$, $m_2(·)$, $n_2$, and ${\gamma }_2$. From (20), the ball family $\overline{B}(x,{\gamma }_1){\}}_{x\in K_1}$ is pairwise disjoint. For every $x\in K_1$, since ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1}\cap B\left(x,{\displaystyle \frac{{\gamma }_1}{4}}\right),ϵ\right)>0$, similar to Step 1, we can build a finite set

$$\begin{array}{c}\hfill E_2\left(x\right)\subset Z_{n_1}\cap B\left(x,{\displaystyle \frac{{\gamma }_1}{4}}\right)\end{array}$$

and an integer-valued function

$$\begin{array}{c}\hfill m_2:E_2\left(x\right)\to \mathbb{N}\cap \left[max\{m_1\left(y\right):y\in K_1\},\infty \right)\end{array}$$

with the following:

(2-a)

For any open set G where $G\cap E_2\left(x\right)\ne \varnothing ,$${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1}\cap G,ϵ\right)>0;$

(2-b)

The elements in ${\left\{{\overline{B}}_{m_2\left(y\right)}^{\mathbf{a}}(y,ϵ)\right\}}_{y\in E_2\left(x\right)}$ are disjoint, and

$$\begin{array}{c}\hfill {\mu }_1\left(\left\{x\right\}\right)<\sum _{y\in E_2\left(x\right)}{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_2\left(y\right)\rceil }\phi \left(y\right)}<(1+2^{-2}){\mu }_1\left(\left\{x\right\}\right).\end{array}$$

To prove this, fix $x\in K_1$. Let $F=Z_{n_1}\cap B\left(x,{\displaystyle \frac{{\gamma }_1}{4}}\right)$. Define

$$\begin{array}{c}\hfill H_x:=\bigcup \{G\subseteq X_1:G\mathrm{is}\mathrm{open},{\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,F\cap G,ϵ\right)=0\}.\end{array}$$

Set $F^{\prime }=F\setminus H_x$. Similar to Step 1, we can prove

$$\begin{array}{c}\hfill {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,F^{\prime },ϵ\right)={\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,F,ϵ\right)>0.\end{array}$$

Additionally, for all open set G when $G\cap F^{\prime }\ne \varnothing$, ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,F^{\prime }\cap G,ϵ\right)>0.$ As ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,s,F^{\prime },ϵ\right)=\infty$, according to Lemma 3, we can find a finite set $E_2\left(x\right)\subseteq F^{\prime }$, along with a mapping $m_2:E_2\left(x\right)\to \mathbb{N}\cap \left[max\{m_1\left(y\right):y\in K_1\},\infty \right)$, for which condition (2-b) holds. Observe that when an open set G has $G\cap E_2\left(x\right)\ne \varnothing$, then $G\cap F^{\prime }\ne \varnothing$. Consequently,

$$\begin{array}{c}\hfill {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1}\cap G,ϵ\right)⩾{\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,F^{\prime }\cap G,ϵ\right)>0.\end{array}$$

This implies that (2-a) holds.

Given that the family ${\left\{\overline{B}(x,{\gamma }_1)\right\}}_{x\in K_1}$ is disjoint, for $x,x^{\prime }\in K_1$ with $x\ne x^{\prime },$$E_2\left(x\right)\cap E_2\left(x^{\prime }\right)=\varnothing$. Let

$$\begin{array}{c}\hfill K_2=\bigcup _{x\in K_1}{E}_2\left(x\right),{\mu }_2=\sum _{y\in K_2}{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_2\left(y\right)\rceil }\phi \left(y\right)}{\delta }_y.\end{array}$$

Based on (2-b) and (20), the elements of ${\left\{{\overline{B}}_{m_2\left(y\right)}^{\mathbf{a}}(y,ϵ)\right\}}_{y\in K_2}$ are pairwise disjoint. Therefore, we choose $0<{\gamma }_2<{\displaystyle \frac{{\gamma }_1}{4}}$ such that, for any function $z:K_2\to X_1$ satisfying the condition that $d_1(x,z\left(x\right))<{\gamma }_2$ for all $x\in K_2$, we obtain

$$\begin{array}{c}\hfill \left(\overline{B}(z\left(x\right),{\gamma }_2)\cup {\overline{B}}_{m_2\left(x\right)}^{\mathbf{a}}(z\left(x\right),ϵ)\right)\cap \left(\bigcup _{y\in K_2\setminus \left\{x\right\}}\overline{B}(z\left(y\right),{\gamma }_2)\cup {\overline{B}}_{m_2\left(y\right)}^{\mathbf{a}}(z\left(y\right),ϵ)\right)=\varnothing \end{array}$$

(21)

for all $x\in K_2$. Choose a large $n_2\in \mathbb{N}$ such that $Z_{n_1,n_2}\supset K_2$ and

$$\begin{array}{c}\hfill {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1,n_2}\cap B\left(x,{\displaystyle \frac{{\gamma }_2}{4}}\right),ϵ\right)>0\end{array}$$

for all $x\in K_2$.

Step 3. Suppose that $K_i$, ${\mu }_i$, $m_i(·)$, $n_i$, and ${\gamma }_i$ are built for $i=1,\dots ,p$. Given any function $z:K_p\to X_1$ satisfying $d_1(x,z\left(x\right))<{\gamma }_p$ for $x\in K_p$,

$$\begin{array}{c}\hfill \left(\overline{B}(z\left(x\right),{\gamma }_p)\cup {\overline{B}}_{m_p\left(x\right)}^{\mathbf{a}}(z\left(x\right),ϵ)\right)\cap \left(\bigcup _{y\in K_p\setminus \left\{x\right\}}\overline{B}(z\left(y\right),{\gamma }_p)\cup {\overline{B}}_{m_p\left(y\right)}^{\mathbf{a}}(z\left(y\right),ϵ)\right)=\varnothing \end{array}$$

(22)

for any $x\in K_p$, $Z_{n_1,\dots ,n_p}\supset K_p$ and

$$\begin{array}{c}\hfill {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1,...,n_p}\cap B\left(x,{\displaystyle \frac{{\gamma }_p}{4}}\right),ϵ\right)>0\end{array}$$

for any $x\in K_p$. Now, we construct elements for $i=p+1$ just like in Step 2.

Observe that the elements within the collection ${\left\{\overline{B}(x,{\gamma }_p)\right\}}_{x\in K_p}$ are pairwise disjoint. Since ${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1,...,n_p}\cap B\left(x,{\displaystyle \frac{{\gamma }_p}{4}}\right),ϵ\right)>0$, for any $x\in K_p$, following the procedure outlined in Step 2, we can construct a finite set

$$\begin{array}{c}\hfill E_{p+1}\left(x\right)\subseteq Z_{n_1,\dots ,n_p}\cap B(x,{\displaystyle \frac{{\gamma }_p}{4}})\end{array}$$

and an integer-valued function

$$\begin{array}{c}\hfill m_{p+1}:E_{p+1}\left(x\right)\to \mathbb{N}\cap \left[max\{m_p\left(y\right):y\in K_p\},\infty \right)\end{array}$$

with the following properties:

(3-a)

${\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1,...,n_p}\cap G,ϵ\right)>0$, for any open set G for which

$$\begin{array}{c}\hfill G\cap E_{p+1}\left(x\right)\ne \varnothing ;\end{array}$$

(3-b)

${\left\{{\overline{B}}_{m_{p+1}\left(y\right)}^{\mathbf{a}}(y,ϵ)\right\}}_{y\in E_{p+1}\left(x\right)}$ are disjoint and satisfy

$$\begin{array}{c}\hfill {\mu }_p\left(\left\{x\right\}\right)<\sum _{y\in E_{p+1}\left(x\right)}{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_{p+1}\left(y\right)\rceil }\phi \left(y\right)}<(1+2^{-p-1}){\mu }_p\left(\left\{x\right\}\right).\end{array}$$

Obviously, for different $x,x^{\prime }\in K_p,$$E_{p+1}\left(x\right)\cap E_{p+1}\left(x^{\prime }\right)=\varnothing$. We set

$$K_{p+1}=\bigcup _{x\in K_p}{E}_{p+1}\left(x\right)$$

and

$${\mu }_{p+1}=\sum _{y\in K_{p+1}}{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_{p+1}\left(y\right)\rceil }\phi \left(y\right)}{\delta }_y.$$

Using (3-b) and (22), we can see that the sets ${\left\{{\overline{B}}_{m_{p+1}\left(y\right)}^{\mathbf{a}}(y,ϵ)\right\}}_{y\in K_{p+1}}$ are disjoint. Therefore, we can choose $0<{\gamma }_{p+1}<{\displaystyle \frac{{\gamma }_p}{4}}$, and given any function $z:K_{p+1}\to X_1$ where $d_1(x,z\left(x\right))<{\gamma }_{p+1}$ for all $x\in K_{p+1}$, we have

$$\begin{array}{c}\hfill \left(\overline{B}(z\left(x\right),{\gamma }_{p+1})\cup {\overline{B}}_{m_{p+1}\left(x\right)}^{\mathbf{a}}(z\left(x\right),ϵ)\right)\cap \left({\cup }_{y\in K_{p+1}\setminus \left\{x\right\}}\overline{B}(z\left(y\right),{\gamma }_{p+1})\cup {\overline{B}}_{m_{p+1}\left(y\right)}^{\mathbf{a}}(z\left(y\right),ϵ)\right)=\varnothing .\end{array}$$

for all $x\in K_{p+1}$. Choose a large $n_{p+1}\in \mathbb{N}$ such that $Z_{n_1,\dots ,n_{p+1}}\supset K_{p+1}$ and for all $x\in K_{p+1}$

$$\begin{array}{c}\hfill {\mathcal{P}}_p^{\mathbf{a}}\left(T_1,{\left\{d_i\right\}}_{i=1}^k,\phi ,t,Z_{n_1,...,n_{p+1}}\cap B\left(x,{\displaystyle \frac{{\gamma }_{p+1}}{4}}\right),ϵ\right)>0.\end{array}$$

Just like in the previous steps, we can construct the sequences $\left(K_i\right)$, $\left({\mu }_i\right)$, $\left(m_i(·)\right)$, $\left(n_i\right)$, and $\left({\gamma }_i\right)$ by induction. Here are some of their basic properties:

(a)

The family ${\mathcal{F}}_i=\{\overline{B}(x,{\gamma }_i):x\in K_i\}$ is disjoint, for any i. Every element in ${\mathcal{F}}_{i+1}$ is part of $\left\{\overline{B}(x,{\displaystyle \frac{{\gamma }_i}{2}})\right\}$ for some $x\in K_i$.

(b)

For any $x\in K_i$ and $z\in \overline{B}(x,{\gamma }_i)$, we have ${\overline{B}}_{m_i\left(x\right)}^{\mathbf{a}}(z,ϵ)\cap {\bigcup }_{y\in K_i\setminus \left\{x\right\}}\overline{B}(y,{\gamma }_i)=\varnothing ,$ and

$$\begin{array}{cc}\hfill {\mu }_i\left(\overline{B}(x,{\gamma }_i)\right)& =e^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_i\left(x\right)\rceil }\phi \left(x\right)}⩽\sum _{y\in E_{i+1}\left(x\right)}{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_{i+1}\left(y\right)\rceil }\phi \left(y\right)}\hfill \\ \hfill & ⩽(1+2^{-i-1}){\mu }_i\left(\overline{B}(x,{\gamma }_i)\right),\hfill \end{array}$$

with $E_{i+1}\left(x\right)=B(x,{\gamma }_i)\cap K_{i+1}$.

From the second part of (b), we obtain

$${\mu }_i\left(F_i\right)\le {\mu }_{i+1}\left(F_i\right)=\sum _{F\in {\mathcal{F}}_{i+1}:F\subseteq {\mathcal{F}}_i}{\mu }_{i+1}\left(F\right)\le 1+2^{-i-1}{\mu }_i\left(F_i\right),\forall F_i\in {\mathcal{F}}_i.$$

Repeatedly using these inequalities, for any $j>i$, we have

$$\begin{array}{c}\hfill {\mu }_i\left(F_i\right)⩽{\mu }_j\left(F_i\right)⩽\prod _{n=i+1}^j\left(1+2^{-n}\right){\mu }_i\left(F_i\right)⩽C{\mu }_i\left(F_i\right),\forall F_i\in {\mathcal{F}}_i,\end{array}$$

(23)

and $C={\prod }_{n=1}^{\infty }(1+2^{-n})<\infty$.

Denote the limit point of $\left({\mu }_i\right)$ in the weak-star topology as $\tilde{\mu }$. Set $K={\bigcap }_{n=1}^{\infty }\overline{{\bigcup }_{i⩾n}{K}_i}$. The support of $\tilde{\mu }$ lies on K. Moreover, we have $K={\bigcap }_{n=1}^{\infty }\overline{{\bigcup }_{i⩾n}{K}_i}\subset {\bigcap }_{p=1}^{\infty }\overline{Z_{n_1,...,n_p}}.$ However, due to the continuity of $\varphi$, we can prove that ${\bigcap }_{p=1}^{\infty }{Z}_{n_1,...,n_p}={\bigcap }_{p=1}^{\infty }\overline{Z_{n_1,...,n_p}}$ using Cantor’s diagonal argument. Thus, K is a compact subset of Z. On the contrary, based on Equation (23), we have

$$\begin{array}{cc}\hfill e^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_i\left(x\right)\rceil }\phi \left(x\right)}& ={\mu }_i\left(\overline{B}(x,{\gamma }_i)\right)⩽\tilde{\mu }\left(B(x,{\gamma }_i)\right)\hfill \\ & ⩽C{\mu }_i\left(\overline{B}(x,{\gamma }_i)\right)⩽C{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_i\left(x\right)\rceil }\phi \left(x\right)},\forall x\in K_i.\hfill \end{array}$$

Specifically,

$$\begin{array}{c}\hfill 1\le \sum _{x\in K_1}{\mu }_1\left(B(x,{\gamma }_1)\right)\le \tilde{\mu }\left(K\right)⩽\sum _{x\in K_1}C{\mu }_1\left(B(x,{\gamma }_1)\right)\le 2C.\end{array}$$

Observe that $K\subset {\bigcup }_{x\in K_i}\overline{B}(x,{\displaystyle \frac{{\gamma }_i}{2}}).$ From the first part of (b), for any $x\in K_i$ and $z\in \overline{B}(x,{\gamma }_i)$, we have

$$\begin{array}{c}\hfill \tilde{\mu }\left({\overline{B}}_{m_i\left(x\right)}^{\mathbf{a}}(z,ϵ)\right)\le \tilde{\mu }\left(\overline{B}(x,{\displaystyle \frac{{\gamma }_i}{2}})\right)\le C{e}^{-{\displaystyle \frac{s}{a_1}}{S}_{\lceil a_1{m}_i\left(x\right)\rceil }\phi \left(x\right)}.\end{array}$$

Moreover, for any $z\in K$ and $i\in \mathbb{N}$, there exists an $x\in K_i$ so that $z\in \overline{B}(x,\frac{{\gamma }_i}{2}).$ Consequently,

$$\begin{array}{c}\hfill \tilde{\mu }\left({\overline{B}}_{m_i\left(x\right)}^{\mathbf{a}}(z,ϵ)\right)\le C{e}^{-\frac{s}{a_1}{S}_{\lceil a_1{m}_i\left(x\right)\rceil }\phi \left(x\right)}.\end{array}$$

Set $\mu =\tilde{\mu }/\tilde{\mu }\left(K\right)$. Then, it can be inferred that $\mu \in M\left(K\right)$, and there is a sequence $k_i\uparrow \infty$ for any $z\in K$ so that $\tilde{\mu }\left(B_{k_i}^{\mathbf{a}}(z,ϵ)\right)\le C{e}^{-\frac{s}{a_1}{S}_{\lceil a_1{k}_i\left(z\right)\rceil }\phi \left(z\right)}/\tilde{\mu }\left(K\right).$ It follows that ${\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1,ϵ\right)⩾s.$ □

5. Conclusions

This paper introduces weighted metric mean dimensions with potential and establishes their corresponding Bowen equations. For the weighted upper metric mean dimension with potential, we derived a crucial relationship showing that ${\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)$ is the unique root of the equation ${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi -\beta \psi \right)=0$ (Theorem 1). Regarding subsets, we defined the weighted BS metric mean dimension and the weighted packing BS metric mean dimension, and proved that they are the unique roots of the relevant Bowen equations (Theorem 2). Moreover, we established variational principles for the weighted BS metric mean dimension and the weighted packing BS metric mean dimension (Theorems 3 and 4). These variational principles provide a significant connection between the geometric and measure-theoretic aspects of dynamical systems. Our results contribute to the field of dynamical systems in several ways. First, they generalize and extend existing theories related to metric mean dimensions and Bowen’s equations. By introducing the weighted framework, we can better analyze the behavior of dynamical systems with multi-scale structures, which is relevant in many applications such as the study of self-affine carpets and sponges. Second, the variational principles established in this paper offer new perspectives for understanding the relationship between topological and measure-theoretic properties of subsets in dynamical systems. This can potentially lead to further research on the dimension theory of more complex sets and systems. Third, the methods and results presented here may inspire new approaches in related fields, such as fractal geometry and ergodic theory, where the quantification of complexity and the relationship between different types of dimensions are of great importance. For the convenience of readers, we list the abbreviations of the definitions involved in the paper in this table.

Symbols	Meanings
$d_n^{\mathbf{a}}(x,y)$	The weighted metric
$B_n^{\mathbf{a}}(x,ϵ)$	The weighted Bowen ball
$\lceil u\rceil$	The least integer $\ge u$
$\left[\frac{V}{m}\right]$	The integer part of $\frac{V}{m}$
${\overline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1\right)$	The weighted upper local measure-theoretical BS entropies of $\mu$
${\underline{h}}_{\phi ,\mu }^{\mathbf{a}}\left(T_1\right)$	The weighted lower local measure-theoretical BS entropies of $\mu$
${\overline{\mathrm{mdim}}}_M^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)$	The weighted upper metric mean dimension with potential $\phi$
${\overline{\mathrm{mdim}}}_{M,\psi }^{\mathbf{a}}\left(X_1,T_1,{\left\{d_i\right\}}_{i=1}^k,\phi \right)$	The weighted $\psi$-induced upper metric mean dimension with potential $\phi$
${\overline{\mathrm{mdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$	The weighted Bowen upper mean dimension with potential $\phi$ on the set Z
${\overline{\mathrm{upmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$	The weighted u-upper metric mean dimension with potential $\phi$ on the set Z
${\overline{\mathrm{Pmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$	The weighted packing upper mean dimension with potential $\phi$ on the set Z
${\overline{\mathrm{BSmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$	The weighted BS upper mean dimension with respect to $\phi$ on the set Z
${\overline{\mathrm{BSPdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$	The weighted packing BS upper mean dimension with respect to $\phi$ on the set Z
${\overline{\mathrm{Wmdim}}}_{M,Z,T_1}^{\mathbf{a}}\left(\phi ,{\left\{d_i\right\}}_{i=1}^k\right)$	The weighted BS metric mean dimension with respect to $\phi$ on the set Z