Topological pressures of a factor map for iterated function systems

1.   Introduction

Let $(X, T)$ be a topological dynamical system (TDS for short), where $X$ is a compact metric space with metric $d$ and $T$ is a continuous map from $X$ to itself. As we know, the topological entropy and its generalization called the topological pressure play important roles in the field of dynamical systems.

In 1971, Bowen ^[3] considered a factor map $\phi: (X, T) \rightarrow (Y, S)$ between two TDSs, and proved that

where $h_{top}(T, K)$ is the topological entropy of a compact subset $K\subset X$ ^[3]. Topological pressure was firstly introduced by Ruelle ^[21]. Later on, other definitions of topological pressure, via open covers and spanning sets, were proposed by Walters ^[23], and they were further explored by Pesin and Pitskel ^[20]. Pesin ^[19] utilized Carathéodory structures to give a dimensional definition for topological pressure. Recently, there have been generalizations of topological pressure for other systems, e.g., ^[11] for non-autonomous discrete dynamical systems and ^[16] for free semigroup actions.

Recently, Fang et al. ^[7] and Oprocha ^[18] further considered the topological entropy of subsets, and they extended the inequality (1.1) to the topological entropy for non-compact subsets of a factor map. Later on, a variety of versions of inequality (1.1) were established for different systems. Li et al. ^[13] generalized the results in ^[7,18] to topological pressure, and established formulas for the topological pressure of non-compact subsets of a factor map. Some applications of the inequality can be seen in ^[8,13]. Recently, Zhao et al. ^[28] proved an inequality of packing pressure of a factor map. Zhao et al. ^[29] gave an inequality of topological pressure under free semigroup action of a factor map. Liu et al. ^[15] established an inequality of Pesin-Pitskel topological pressure of a factor map for nonautonomous dynamical systems. For some more details about related concepts of entropy and pressure, see ^[4,17,19].

As it is well known, several versions of the topological entropy under free semigroup actions have been proposed by Biś ^[25], Bufetov ^[5], and Wang et al. ^[25]. Subsequently, analogous to the classical topological pressure, many scholars proposed different versions of topological pressure under free semigroup actions, which are natural generalizations of topological entropies under free semigroup actions. Motivated by Bufetov's entropy ^[5], Lin et al. ^[14] extended the notion of Bufetov's entropy to the concept of topological pressure of free semigroup actions (here we remark it as pressure from ^[14]), and then a partial variational principle for free semigroup actions was established. Furthermore, the opposite inequality was proved by Carvalho et al. ^[6], and the complete variational principle for free semigroup was obtained. Recently, Wang et al. ^[24] generalized the topological entropy in ^[25] to an another version of topological pressure of free semigroup actions (we remark it as pressure from ^[24]), and they also established a partial variational principle for this version of pressure of free semigroup actions. At the same time, analogous to the topological pressure given by the Carathéodory structure, which was given in Pesin's work^[19]. Ma et al. ^[16] used the Carathéodory structure to propose the notions of topological pressure and topological entropy under free semigroup actions. Ju et al. ^[12] introduced the notions of the topological entropy and lower and upper capacity topological entropies of free semigroup actions on non-compact subsets, which generalized the concept of the topological entropy of free semigroup actions defined by Bufetov ^[5]. One can see some recent relevant results under free semigroup actions, such as for topological entropy ^[10,16], for topologica pressure ^[26,27,30] and for inverse pressure ^[2].

In this paper, we mainly investigate the pressure from ^[24] of free semigroup actions with $m$ generators. Let $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$ be a family of continuous self-maps on a compact metric space $(X, d)$. The iterated function system $(X, \mathscr{F})$ (IFS for short) is the action of the semigroup generated by $\left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$ on a compact metric space $(X, d)$. Following the works of ^[3,24,29], notice that the authors ^[29] have investigated the pressure from ^[14] of a factor map for IFSs, while in the present paper, we shall continue this work with respect to pressure from ^[24]. More precisely, let $(X, \mathscr{F})$ and $(Y, \mathscr{G})$ be two iterated function systems, where $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$, $\mathscr{G} = \left\{g_{0}, g_{1}, \ldots, g_{m-1}\right\}$. If there is a surjective and continuous map $\phi:X\to Y$ such that $\phi \circ f_{i} = g_{i} \circ \phi$ for any $0\leq i \leq m-1$, then we say that $(Y, \mathscr{G})$ is a factor system of $(X, \mathscr{F})$ and $\phi$ is a factor map from $(X, \mathscr{F})$ to $(Y, \mathscr{G})$. Moreover, when $\phi$ is a homeomorphism, we say that $(X, \mathscr{F})$ is conjugate to $(Y, \mathscr{G})$. Given a factor map $\phi: X \rightarrow Y$, we establish an inequality for pressure from ^[24] of a factor map, which generalizes results in ^[3] to pressure from ^[24] for IFSs, and this inequality is different from the result in ^[29].

Besides, we further investigate the power rule of pressure from ^[24]. For an IFS $(X, \mathscr{F})$, denote $\mathscr{F}^{k} = \left\{h_{1} \circ h_{2} \circ \cdots \circ h_{k}: h_{1}, h_{2}, \ldots, h_{k} \in\right. \mathscr{F}\}$. Then we explore the features associated with pressure from ^[24] of $(X, \mathscr{F}^{k})$. We also derive a power rule formula for pressure from ^[24], which extends the result in ^[22] to pressure from ^[24] for IFSs.

This manuscript is structured as follows. In Section 2, we recall several concepts of topological pressure of IFSs. In Section 3, we shall investigate the topological pressure of a factor map. In Section 4, we explore the power rule of a topological pressure.

2.   Preliminaries

Let $C(X, \mathbb{R})$ denote the collection of all real-valued continuous functions of $X$ equipped with the supremum norm. The sets of natural, nonnegative integers and real numbers are represented by $\mathbb{N}$, $\mathbb{Z}^{+}$, and $\mathbb{R}$, respectively. We adopt the notation $\#(\cdot)$ to represent the cardinality of a finite set.

In this section, we recall several versions of the notions of topological pressures for free semigroup actions given in ^[14,24]. Denote by $F_{m}^{+}$ the set of all finite words of symbols $0, 1, \ldots, m-1.$ For every $\nu \in F_{m}^{+}, |\nu|$ denotes the length of $\nu$. If $u, \nu \in F_{m}^{+}$, let $\nu u$ be the word concatenation of $\nu$ and $u$. It is obvious that $F_{m}^{+}$ associated with this concatenation is a free semigroup with $m$ generators. We remark that $u \prec\nu$ if there is a word $\rho$ with $\nu = u \rho$, furthermore, we remark that $u \preccurlyeq\nu$ if $u \prec\nu$ or $u = \nu.$ Let $f_{u} = f_{u_{k-1}} \circ f_{u_{k-2}} \circ \cdots \circ f_{u_{0}}$ and $f_{u}^{-1} = f_{u_{0}}^{-1} \circ f_{u_{1}}^{-1} \circ \cdots \circ f_{u_{k-1}}^{-1}$ where $u = u_{0} u_{1} \cdots u_{k-1} \in\{0, 1, \ldots, m-1\}^{k}$ and $f_{u_{i}}^{-1}$ is the preimage operator of $f_{u_{i}}$ ($i = 0, \cdots, k-1$).

Let $(X, \mathscr{F})$ be an IFS, $u \in F_{m}^{+}$. The max metric on $X$ is given by $d_{u}(a, b) = \max\limits_{\nu\prec u} d\left(f_{\nu}(a), f_{\nu}(b)\right).$ For any $\epsilon > 0$, we say that a subset $E \subset X$ is an $(\mathscr{F}, u, \epsilon)$-spanning set for $X$, if for any $a \in X$, there exists $b \in E$ with $d_{u}(a, b) < \epsilon$. For any $\epsilon > 0$, we say a subset $F \subset X$ is an $(\mathscr{F}, u, \epsilon)$-separated set of $X$, if for any $a, b \in F, a \neq b$, one has $d_{u}(a, b) \geq \epsilon.$

Given $n \in \mathbb{N}, u = u_{0} u_{1} \cdots u_{n-1} \in F_{m}^{+}, |u| = n$, for $i\in \{0, \cdots, n-1\}$, let $f_{u\mid_{i}} = f_{u_{i-1}} \circ f_{u_{i-2}} \circ \cdots \circ f_{u_{0}}$, where $f_{u\mid_{0}} = id$. For any $\Psi \in C(X; \mathbb{R})$, any $u \in F_{m}^{+}$, $|u| = n, x \in X$, denote

Now, we recall the notions of the topological pressure given in ^[14,24].

2.1.   Topological pressure of IFSs

Inspired by Bufetov's entropy ^[5], Lin et al. ^[14] introduced the topological pressure of IFSs via spanning sets and separated sets. They obtained the equivalence of the definitions of topological pressure of IFSs, by using spanning sets or separated sets.

Definition 2.1. ^[14] Given $\Psi \in C(X; \mathbb{R})$, define

The pressure from ^[14] of IFSs is given by

Definition 2.2. ^[14] Given $\Psi \in C(X; \mathbb{R}),$ define

Proposition 2.3. ^[14] Let $(X, \mathscr{F})$ be an IFS, one has

Remark 2.4. When $\Psi = 0$, we remark that $h_{B}(\mathscr{F}) = P(0, \mathscr{F})$, where $h_{B}(\mathscr{F})$ is Bufetov's entropy ^[5]. Clearly, the notion of pressure from ^[14] of IFSs is still valid for any $\mathcal{F}$-subinvariant compact subset $K \subset X$.

Later, Wang et al. ^[24] introduced another version of the topological pressure of IFSs, which is different from the notion of topological pressure given in ^[14].

Definition 2.5 ^[24] For any $\Psi \in C(X; \mathbb{R})$, let

The pressure from ^[24] of IFSs is given by

Similarly, Wang et al. ^[24] also obtained the equivalence for pressure from ^[24] of IFSs between the notions via spanning sets and separated sets.

Definition 2.6. ^[24] Given $\Psi \in C(X; \mathbb{R}),$ define

Proposition 27.. ^[14] Let $(X, \mathscr{F})$ be an IFS, we have

Remark 2.8. (1) It is clear that, the notion of pressure from ^[24] of IFSs still holds for any $\mathcal{F}$-subinvariant compact subset $K \subset X$ (i.e. $f(K) \subset K$ for all $f\in \mathcal{F}$).

(2) It is not hard to check that $P^{W}(\Psi, \mathscr{F})\leq P(\Psi, \mathscr{F})$ ^[24].

(3) When $\Psi = 0$, we remark that $h^{W}(\mathscr{F}) = P^{W}(0, \mathscr{F})$, where $h^{W}(\mathscr{F})$ is the topological entropy given in ^[25], and $h^{W}(\mathscr{F})$ can also be presented by open covers ^[25]. More precisely, let $\mathcal{C}_{X}$ and $\mathcal{C}_{X}^{o}$ be the set of finite covers and finite open covers, respectively. For $n\in\mathbb{N}$ and $\mathscr{U}_1, \mathscr{U}_2, \ldots, \mathscr{U}_n\in \mathcal{C}_{X}$, we denote

For any non-empty subset $E \subset X$ and $\mathscr{U}\in \mathcal{C}_{X}$, let $\mathscr{N}(\mathscr{U}\mid E)$ be the minimum among the cardinalities of the subsets of $\mathscr{U}$ that covers $E$, and we simply write $\mathscr{N}(\mathscr{U}\mid X)$ as $\mathscr{N}(\mathscr{U})$, define by

the topological entropy of $\mathscr{F}$ on the cover $\mathscr{U}$. The topological entropy of $\mathscr{F}$ is given by

Following the idea of classical topological entropy ^[23], the authors ^[25] stated that $h_{\text {top}}(\mathscr{F}) = h^{W}(\mathscr{F})$, while, we point out that "$\limsup$" can also be replaced by "$\lim$" in (2.1), and we put the statement in Appendix A.3.

3.   Topological pressure of a factor map

In this section, we study the topological pressure of a factor map, and several inequalities are established.

Theorem 3.1. Let $\phi: (X, \mathscr{F})\to (Y, \mathscr{G})$ be a factor map between two IFSs, and $\Psi \in C(Y; \mathbb{R})$. Then

Proof. As $\phi: X \rightarrow Y$ is continuous, then for $\epsilon > 0$, there are $\tau > 0$ and $0 < \tau \leq \epsilon$ such that $\rho(\phi(x), \phi(y)) > \epsilon$ implies that $d(x, y) > \tau.$ Given $u \in F_{m}^{+}$, we select an $(\mathscr{G}, u, \epsilon)$-separated set $Z_{u, n}(Y) \subset Y$. As $\phi$ is a surjective map, we can take $Z_{u, n}(X)$, and $Z_{u, n}(X)$ includes only one point from any $\phi^{-1}(z), z \in Z_{u, n}(Y)$, and does not contain any other points. As $Z_{u, n}(Y)$ is a $(\mathscr{G}, u, \epsilon)$-separated set, then we get that for $y_{1}\neq y_{2} \in Z_{u, n}(Y)$, $\rho_{u}\left(y_{1}, y_{2}\right) > \epsilon$. By definition of $Z_{u, n}(X)$, there exist $x_{1}, x_{2} \in Z_{u, n}(X)$ such that $\phi\left(x_{1}\right) = y_{1}$, $\phi\left(x_{2}\right) = y_{2}$. This indicates that

Thus, $Z_{u, n}(X)$ is a $(u, \tau, \mathscr{F})$-separated set of $X$. Hence,

Hence, we have

It follows that

and then $P^{{W}}(\Psi, \mathscr{G}, \epsilon) \leq P^{{W}}(\Psi \circ \phi, \mathscr{F}, \tau).$ This indicates that $P^{{W}}(\Psi, \mathscr{G}) \leq P^{{W}}(\Psi \circ \phi, \mathscr{F}).$ □

As an immediate conclusion, we have.

Corollary 3.2. Let $\phi: (X, \mathscr{F})\to (Y, \mathscr{G})$ be a topological conjugacy between two IFSs, and $\Psi \in C(Y; \mathbb{R})$. Then

Following the ideas given by Ghys, Langevin, and Walczak ^[9], in 2004 Biś ^[1] gave a concept of topological entropy of free semigroup actions. Let $(X, {d})$ be a compact metric space, $\mathscr{F} = \left\{{f}_{0}, \ldots, {f}_{{m}-1}\right\}$, ${f}_{0}, \ldots, {f}_{{m}-1}$ be continuous maps from $X$ to itself, and $\mathscr{F}^{n} = \left\{g_{1}{\circ} g_{2}{\circ} \ldots{\circ} {g}_{{n}}: {g}_{1}, \ldots, g_{n} \in \mathscr{F}\right\}$. Let $K\subset X$, we say that $Z\subset X$ is an $({n}, \epsilon)$-spanning set of $K$, if for each $a\in K$, there is $b\in Z$ with

Denote

Definition 3.3. ^[1] The Biś's topological entropy is given by

Notice that, Biś's topological entropy is not lower than others ^[5,25], i.e., $h^{W}(\mathscr{F})\leq h_{B}(\mathscr{F})\leq h_{\text{Biś}}(X, \mathscr{F}\cup \{id\})$.

Analogous to the topological entropy of a subset given by Bowen ^[4], the notion of Biś's topological entropy can be extended to a subset as follows. Let $K\subset X$; the Biś's topological entropy of $K$ is given by

By using Biś's topological entropy, we establish an inequality about pressure from ^[24] of a factor map for IFSs, which is similar to Bowen's inequality ^[3].

Theorem 3.4. Let $\phi: (X, \mathscr{F})\to (Y, \mathscr{G})$ be a factor map between two IFSs, and $\Psi \in C(Y; \mathbb{R})$. Then

Especially, if $P^{W}(\Psi, \mathscr{G}) < +\infty$, then

Proof. For any $\tau > 0$, denote

Assume that $\beta = \sup\nolimits_{y \in Y} h_{\text{Biś}}\left(\phi^{-1}(y), \mathscr{F}\cup \{id\}\right) < \infty$, otherwise, there is nothing to prove. For $\epsilon > 0, n \in \mathbb{N}$, and each $y\in Y$, let $F_{y} \subset \phi^{-1}(y)$ be an $(n, \epsilon)$-spanning set of $\phi^{-1}(y)$. By the definition of $h_{\text{Biś}}\left(\phi^{-1}(y), \mathscr{F}\cup \{id\}\right)$, one has $h_{\text{Biś}}\left(\phi^{-1}(y), \mathscr{F}\cup \{id\}\right) \geq h_{\text{Biś}}\left(\phi^{-1}(y), \mathscr{F}\cup \{id\}, \epsilon\right)$ where $h_{\text{Biś}}\left(\phi^{-1}(y), \mathscr{F}\cup \{id\}, \epsilon\right) = \limsup \limits_{n \rightarrow \infty} \frac{1}{n} \log s_n\left(\epsilon, \phi^{-1}(y)\right).$

For above $\epsilon > 0$. Take any $\zeta > 0$. Then for every $y \in Y$, there is $m(y)$ such that

where $M\left(m(y), \epsilon, \mathscr{F}, \phi^{-1}(y)\right) = \#({F_{y}})$, and $F_{y}$ is the minimal cardinality $(m(y), \epsilon)$-spanning set of $\phi^{-1}(y)$.

Next, for $u \in F_{m}^{+}$, we define

where $|u| = n$ and $d_{u}(c, z) = \max\limits_{\nu \prec u} d\left(f_{\nu}(c), f_{\nu}(z)\right)$. Since $F_{y}$ is an $(m(y), \epsilon)$-spanning set of $\phi^{-1}(y)$, this indicates that $F_{y}$ is also an $(\mathscr{F}, u, \epsilon)$-spanning set for $\phi^{-1}(y)$ for every $u\in F_{m}^{+}$ with $|u| = m(y)$. Here we remark that $F_{y}^{u}: = F_{y}$. Hence, $U_{y} = \bigcup_{z \in F_{y}^{u}} D_{m(y)}(\mathscr{F}, u, z, 2 \epsilon)\supset \phi^{-1}(y).$

For every $y \in Y,$ we have $\left(X \backslash U_{y}\right) \cap \bigcap_{r > 0} \phi^{-1}\left(\overline{B_{\rho}(y, r)}\right) = \emptyset,$ where $B_{\rho}(y, r) = \{z \in Y: \rho(z, y) < r\}$. By the finite intersection property, there is $W_{y} = B_{\rho}(y, r)$ such that $\phi^{-1}\left(W_{y}\right)\subset U_{y}$. Let $\left\{W_{y_{1}}, W_{y_{2}}, \ldots, W_{y_{p}}\right\}$ covers $Y$ and $\delta$ be its Lebesgue number under metric $\rho$.

Based on the notion of $P^{W}(\Psi, \mathscr{G})$, then there is an $(\mathscr{G}, u, \delta)$-spanning set $E_{u, n}$ of $Y$ such that

Suppose above $\delta$ is small enough such that for every $i\in [1, p]$, $u = |m\left(y_{i}\right)|$, we have

For every $y \in E_{u, n}, 0\leq j\leq n-1$, take $c_{j}(y) \in\left\{y_{1}, \ldots, y_{p}\right\}$ such that

where $g_{u\mid_{j}}(y) = g_{u_{j}} \circ \cdots \circ g_{u_{0}}, u = u_{0} \cdots u_{j} \cdots u_{n-1}$.

Recursively, define $t_{0}(y) = 0$ and $t_{s+1}(y) = t_{s}(y)+m\left(c_{t_{s}(y)}(y)\right)$ until one gets $t_{k(y)+1}(y) \geq n$, and take $k(y) = k$. For $y \in E_{u, n}, x_{0} \in F_{c_{t_{0}(y)}(y)}^{u}, \ldots, x_{k} \in F_{c_{t_{k}(y)}{(y)}}^{u}$, denote

Claim. (Ⅰ) Denote $\mathscr{V} = \left\{V\left(y; x_{0}, \ldots, x_{s}\right): y \in E_{u, n}, x_{s} \in F_{c_{t_{s}(y)}(y)}^{u}, s\in [0, k(y)]\right\}$, then $\mathscr{V}$ covers $X$. (Ⅱ) For every $u \in F_{m}^{+}$, with $|u| = n$, then any $(\mathscr{F}, u, 4 \epsilon)$-separated set intersects each element of $\mathscr{V}$ in at most one point.

Proof of Claim. (Ⅰ). Given $x \in X,$ as $E_{u, n}$ is an $(\mathscr{G}, u, \delta)$-spanning set of $Y$, then there is a $y \in E$ with $\rho_{u}(y, \phi(x)) = \max\limits_{\nu\prec u} \rho\left(g_{u\mid_{j}}(y), g_{u\mid_{j}}(\phi(x))\right) < \delta$ for each $j\in [0, n-1]$. For every $s\in [0, k(y)],$ we have $\phi \circ f_{u\mid_{t_{s}(y)}}(x) = g_{u\mid_{t_{s}(y)}}(\phi(x)) \in W_{c_{s}(y)}.$ Hence, there exists $x_{s} \in F_{c_{t_{s}(y)}(y)}^{u}$ such that $d\left(f_{u\mid_{t+t_{s}(y)}}(x), f_{u\mid_{t}} \left(x_{s}\right)\right) < 2 \epsilon$ for every $t\in [0, m(c_{t_{s}(y)}(y)-1]$ and $s\in [0, k(y)]$. This indicates that $x \in V\left(y; x_{0}, \ldots, x_{k}\right)$. We finish the proof of Claim (I).

(Ⅱ). For every $z, c \in V\left(y; x_{0}, \ldots, x_{k}\right)$, every $t\in [0, m(c_{t_{s}(y)}(y)-1]$ and $s\in [0, k(y)]$, one has

We finish the proof of Claim (II).

For any $(\mathscr{F}, u, 4 \epsilon)$-separated set $H_{u, n}$ of $X$, we are to estimate the upper bound of $\sum_{x \in H_{u, n}} e^{S_{u, n} \Psi \circ \phi(x)}.$ For every $y \in E_{u, n}$, denote $\mathscr{V}_{y}: = \left\{V\left(y; x_{0}, \ldots, x_{k}\right): x_{i} \in F_{c_{t_{i}(y)}(y)}^{u}, \forall i\in[0, k]\right\}.$ Set $\mathscr{V} = \cup_{y \in E_{u, n}} \mathscr{V}_{y}$, where $\#\left(\mathscr{V}_{y}\right) = \prod_{i = 0}^{k = k(y)} M\left(m\left(c_{t_{i}(y)}(y)\right), \epsilon, \mathscr{F}, \phi^{-1}(y)\right).$ By (3.1), we have

where $A = \max\{m(y_1), \cdots, m(y_p)\}.$

For every $x \in H_{u, n}$, there exist $y \in E_{u, n}$, $x_{0} \in F_{c_{t_{1}(y)}(y)}^{u}, \ldots, x_{k} \in F_{c_{t_{k}(y)}(y)}^{u}$ with $x \in V\left(y; x_{0}, \ldots, x_{k}\right).$ Thus

Notice that for any $r\in[0, m\left(c_{t_{i}(y)}(y)\right)-1]$ and $i\in[0, k]$, one has

This indicates that

Combining the above arguments, for $i\in[0, k(y)]$, we have

Hence, we deduce that

and then

It follows that

Combining the above arguments, we obtain

Hence, we deduce that

Thus,

This indicates that

Taking $n \to \infty$, $\epsilon \to 0,$

By the arbitrariness of $\zeta$, we derive that

□

Particularly, taking $\Psi = 0$, we get:

Corollary 3.5. Let $\phi: (X, \mathscr{F})\to (Y, \mathscr{G})$ be a factor map between two IFSs. Then

Especially, if $h^{W}(\mathscr{G}) < +\infty$, then

Moreover, we consider the question of whether the following inequalities hold?

Question 3.6.

or

4.   Power rule of topological pressure

In this section, we explore power rules of pressure from ^[24]. Let $(X, \mathscr{F})$ be an IFS, where $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$. We continue to study the pressure from ^[24] of $(X, \mathscr{F}^{k})$, where

Theorem 4.1. Let $(X, \mathscr{F})$ be an IFS with $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$. Let $\Psi \in C(Y; \mathbb{R})$ and $\Psi\geq 0$. Then

Proof. Take $u = u_{0} u_{1} \cdots u_{k-1} u_{k} \cdots u_{nk-1} \in F_{m}^{+}$, where $|u| = nk$, and $f_{u} = f_{u_{nk-1}} \circ \cdots \circ f_{u_{0}}$. Let

for $i = 0, 1, \ldots, n-1$ and $\tau = \tau_{0} \tau_{1} \cdots \tau_{n-1}$. It is clear that $h_{\tau_{i}} \in \mathscr{F}^{k}$. Next, we are to show

For any $\epsilon > 0$, assume $E$ is an $(\mathscr{F}, u, \epsilon)$-spanning set of $X$. Then for every $a \in X$, there is {$b \in E$} such that $d_{u}(a, b) = \max\limits_{\nu\prec u} d\left(f_{\nu}(a), f_{\nu}(b)\right) < \epsilon.$ This implies that

Hence, we deduce that $E$ is also an $(\mathscr{F}^{k}, \tau, \epsilon)$-spanning set of $X$. Moreover, one has

and notice that $\Psi \geq 0$, then we have

This indicates that

Hence,

This yields that

□ Particularly, taking $\Psi = 0$, we have.

Corollary 4.2. Let $(X, \mathscr{F})$ be an IFS with $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots\right., \left.f_{m-1}\right\}$. Then

A.    

Lemma A.1. ^[23] Let $\left\{a_{n}\right\}_{n \geq 1}$ be a sequence of real numbers such that $a_{n+p} \leq a_{n}+ a_{p}$ for all $n, p$. Then $\lim\limits_{n \rightarrow \infty} a_{n} / n$ exists and equals $\inf _{n} a_{n} / n$. (The limit could be $-\infty$, but if the $(a_{n})$ is bounded from below, then the limit will be non-negative.)

Lemma A.2. ^[23] For $\mathscr{U}, \mathscr{V} \in {C}_{X}$, we have $\mathscr{N}(\mathscr{U}\vee \mathscr{V})\leq \mathscr{N}(\mathscr{U})\cdot\mathscr{N}(\mathscr{V})$. Moreover, for any continuous map $f:X\rightarrow X$, we have $\mathscr{N}(f^{-1}\mathscr{U})\leq \mathscr{N}(\mathscr{U}).$

Theorem A.3. Let $(X, \mathscr{F})$ be an IFS with $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$. If $\mathscr{U} \in {C}_{X}$, then

exists and is equal to $\inf\limits_{n} \frac{1}{n}\left[\frac{1}{m^{n}} \sum\limits_{|w| = n} \log \mathscr{N}\left(\bigvee_{\nu \preccurlyeq w} f_{\nu}^{-1} \mathscr{U}\right)\right]$.

Proof.

By Lemma A.1, we just need to show that

Take any $\mu = i_{0} \cdots i_{n_{1}-1} i_{n_{1}} \cdots i_{n_{1}+n_{2}-1}, \mu^{(1)} = i_{0} \cdots i_{n_{1}-1}, \mu^{(2)} = i_{n_{1}} \cdots i_{n_{1}+n_{2}-1}$, i.e.,

Since $f_{\mu} = f_{i_{n_{1}+n_{2}-1}} \circ \cdots f_{i_{n_{1}}}\circ f_{i_{n_{1}}-1}\cdots \circ f_{i_{n_{0}}}, f_{\mu}^{-1} = f_{i_{0}}^{-1} \cdots\circ f_{i_{n_{1}}-1}^{-1}\circ f_{i_{n_{1}}}^{-1}\cdots \circ f_{i_{n_{1}+n_{2}-1}}^{-1}$, and then we have

Hence, combining the above arguments, we have

Hence,

We finished the proof. □

By Theorem A.3, we have the following results, and their proofs are standard; one can refer to [23, Chapter 7] for some details.

Theorem A.4. Let $(X, \mathscr{F})$ be an IFS with $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$, and $\left\{\mathscr{U}_{n}\right\}_{n = 0}^{\infty}\subset C_{X}^{o}$ with $\lim \limits_{n \rightarrow \infty} diam\left(\mathscr{U}_{n}\right) = 0$, where $diam\left(\mathscr{U}_{n}\right) = \sup\{diam(U), U\in \mathscr{U}_{n}\}$. Then $\lim \limits_{n \rightarrow \infty} h^{W}(\mathscr{F}, \mathscr{U}_n) = h^{W}(\mathscr{F}).$

Corollary A.5. Let $(X, \mathscr{F})$ be an IFS with $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$. Then

Theorem A.6. Let $(X, \mathscr{F})$ be an IFS with $\mathscr{F} = \left\{f_{0}, f_{1}, \ldots, f_{m-1}\right\}$. Then

Remark A.7. Notice that, in ^[25], the definition of $h^{W}(\mathscr{F})$ is given as follows.

B.   Conclusions

In this research, we discuss the topological pressure for iterated function systems on a compact metric space. Firstly, a formula of topological pressure of a factor map is established, which generalizes the result in [Trans. Amer. Math. Soc., 1971,153: 401–414]. Finally, we also study the power rule of a topological pressure for iterated function systems. These results enrich the theory of topological pressure for iterated function systems.

Author contributions

Zhongxuan Yang: Conceptualization, Writing-original draft, Methodology, Writing-review & editing; Xiaojun Huang: Conceptualization, Methodology, Funding acquisition; Jiajun Zhang: Conceptualization, Writing-original draft, Methodology, Writing-review & editing. All authors of this article have been contributed equally. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We would like to thank the anonymous referees for their careful reading and valuable suggestions that have helped us to improve the initial manuscript.

The research was partially supported by NNSF of China (No. 12461012) and NSF of Chongqing (No. CSTB2024NSCQ-MSX1246).

Conflict of interest

The authors declare no conflicts of interest regarding the publication of this paper.