Weighted topological pressure revisited

Abstract

Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9) 106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys. 43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

1. Introduction

1.1. Dynamical systems and entropy

Topological pressure and its variational principle have been significant in several fields, including the dimension theory of dynamical systems. Recently, Feng and Huang devised an innovative invariant called weighted topological pressure for factor maps between dynamical systems and proved its variational principle [FH16]. Their work inspired Tsukamoto to suggest a new definition of this invariant [Tsu22]. He also established a variational principle, revealing the non-trivial coincidence of the two definitions. Tsukamoto focused on the simplest case with two dynamical systems.

In this paper, we extend Tsukamoto’s definition to the case of an arbitrary number of dynamical systems and prove its variational principle. With our result, we can plainly calculate the Hausdorff dimension of self-affine sponges, a topic studied by Kenyon and Peres [KP96]. Furthermore, we will show in §6 that we can determine the Hausdorff dimension of certain sofic sets embedded in higher-dimensional Euclidean space.

We review the basic notions of dynamical systems in this subsection. Refer to the book of Walters [Wal82] for the details.

A pair $(X, T)$ is called a dynamical system if X is a compact metrizable space and $T: X \rightarrow X$ is a continuous map. A map $\pi : X \rightarrow Y$ between dynamical systems $(X, T)$ and $(Y, S)$ is said to be a factor map if $\pi $ is a continuous surjection and $\pi \circ T = S \circ \pi $ . We sometimes write as $\pi : (X, T) \rightarrow (Y, S)$ to clarify the dynamical systems in question.

For a dynamical system $(X, T)$ , denote its topological entropy by $h_{\mathrm {top}}(T)$ . Let $P(f)$ be the topological pressure for a continuous function $f: X \rightarrow \mathbb {R}$ (see §2 for the definition of these quantities). Let $\mathscr {M}^T(X)$ be the set of T-invariant probability measures on X and $h_\mu (T)$ the measure-theoretic entropy for $\mu \in \mathscr {M}^T(X)$ (see §3.2). The variational principle then states that [Din70, Gm71, Gw69, Ru73, Wal75]

$$ \begin{align*} P(f) = \sup_{\mu \in \mathscr{M}^T(X)} \bigg( h_\mu(T) + \int_X f\,d\mu \bigg). \end{align*} $$

1.2. Background

We first look at self-affine sponges to understand the background of weighted topological entropy introduced by Feng and Huang. Let $m_1, m_2, \ldots , m_r$ be natural numbers with $m_1 \leq m_2 \leq \cdots \leq m_r$ . Consider an endomorphism T on $\mathbb {T}^r = \mathbb {R}^r/\mathbb {Z}^r$ represented by the diagonal matrix $A = \mathrm {diag}(m_1, m_2, \ldots , m_r)$ . For $D \subset \prod _{i=1}^r \{0, 1, \ldots , m_i-1\}$ , define

$$ \begin{align*} K(T, D) = \bigg\{\! \sum_{n=0}^{\infty} A^{-n}e_n \in \mathbb{T}^r \bigg| e_n \in D \bigg\}. \end{align*} $$

This set is compact and T-invariant, that is, $TK(T, D) = K(T, D)$ .

These sets for $r = 2$ are known as Bedford–McMullen carpets or self-affine carpets. Figure 1 exhibits a famous example, the case of $D = \{(0,0), (1,1), (0,2)\} \subset \{0, 1\} \times \{0, 1, 2\}$ . The analysis of these sets is complicated compared with ‘self-similar’ sets. Bedford [Bed84] and McMullen [McM84] independently studied these sets and showed that, in general, their Hausdorff dimension is strictly smaller than their Minkowski dimension (also known as box-counting dimension). Figure 1 has Hausdorff dimension $\log _2{(1+2^{\log _3{2}})} = 1.349 \cdots $ and Minkowski dimension $1 + \log _3{\tfrac {3}{2}} = 1.369 \cdots $ .

Figure 1 First four generations of a Bedford–McMullen carpet.

The sets $K(T, D)$ for $r \geq 3$ are called self-affine sponges. Kenyon and Peres [KP96] calculated their Hausdorff dimension for the general case (see Theorem 1.5 in this section). In addition, they showed the following variational principle for the Hausdorff dimension of $K(T, D)$ :

(1.1) $$ \begin{align} \mathrm{dim}_H K(T, D) = \sup_{\mu \in \mathscr{M}^T(\mathbb{T}^r)}{ \bigg\{ \frac{1}{\log{m_r}}h_{\mu}(T) + \sum_{i=2}^r \bigg( \frac{1}{\log{m_{r-i+1}}} - \frac{1}{\log{m_{r-i+2}}} \bigg) h_{\mu_i}(T_i) \bigg\}}. \end{align} $$

Here, the endomorphism $T_i$ on $\mathbb {T}^{r-i+1}$ is defined from $A_i = \mathrm {diag}(m_1, m_2, \ldots , m_{r-i+1})$ , and $\mu _i$ is defined as the push-forward measure of $\mu $ on $\mathbb {T}^{r-i+1}$ by the projection onto the first $r-i+1$ coordinates. Feng and Huang’s definition of weighted topological entropy of $K(T, D)$ equals $\mathrm {dim}_H K(T, D)$ with a proper setting.

1.3. Original definition of the weighted topological pressure

Motivated by the geometry of self-affine sponges described in the previous subsection, Feng and Huang introduced a generalized notion of pressure. Consider dynamical systems $(X_i, T_i)$ ( $i=1, 2, \ldots , r$ ) and factor maps $\pi _i: X_i \rightarrow X_{i+1}\ (i=1, 2, \ldots , r-1)$ :

We refer to this as a sequence of dynamical systems. Let $\boldsymbol {w} = (w_1, w_2, \ldots , w_r)$ be a vector with $w_1> 0$ and $w_i \geq 0$ for $i \geq 2$ . Feng and Huang [FH16] ingeniously defined the $\boldsymbol {w}$ -weighted topological pressure $P^{\boldsymbol {w}}_{\mathrm {FH}}(f)$ for a continuous function $f:X_1 \rightarrow \mathbb {R}$ and established the variational principle [FH16, Theorem 1.4]:

(1.2) $$ \begin{align} P^{\boldsymbol{w}}_{\mathrm{FH}}(f) = \sup_{\mu \in \mathscr{M}^{T_1}(X_1)} \bigg( \sum_{i=1}^r w_i h_{{\pi^{(i-1)}}_*\mu} (T_i) + w_1 \int_{X_1} f\,d\mu \bigg). \end{align} $$

Here, $\pi ^{(i)}$ is defined by

$$ \begin{gather*} \pi^{(0)} = \mathrm{id}_{X_1}: X_1 \to X_1, \\ \pi^{(i)} = \pi_i \circ \pi_{i-1} \circ \cdots \circ \pi_1: X_1 \to X_{i+1}, \end{gather*} $$

and ${\pi ^{(i-1)}}_* \mu $ is the push-forward measure of $\mu $ by $\pi ^{(i-1)}$ on $X_i$ . The $\boldsymbol {w}$ -weighted topological entropy $h^{\boldsymbol {w}}_{\mathrm {top}}(T_1)$ is the value of $P^{\boldsymbol {w}}_{\mathrm {FH}}(f)$ when $f \equiv 0$ . In this case, equation (1.2) becomes

(1.3) $$ \begin{align} h^{\boldsymbol{w}}_{\mathrm{top}}(T_1) = \sup_{\mu \in \mathscr{M}^{T_1}(X_1)} \bigg( \sum_{i=1}^r w_i h_{{\pi^{(i-1)}}_*\mu} (T_i) \bigg). \end{align} $$

We will explain here Feng and Huang’s method of defining $h^{\boldsymbol {w}}_{\mathrm {top}}(T_1)$ . For the definition of $P^{\boldsymbol {w}}_{\mathrm {FH}}(f)$ , see their original paper [FH16].

Let n be a natural number and $\varepsilon $ a positive number. Let $d^{(i)}$ be a metric on $X_i$ . For $x \in X_1$ , define the nth $\boldsymbol {w}$ -weighted Bowen ball of radius $\varepsilon $ centered at x by

$$ \begin{align*} B^{\boldsymbol{w}}_n(x, \varepsilon) = \bigg\{ y \in X_1 \bigg|\! \begin{array}{l} d^{(i)} \! ( T^j_i(\pi^{(i-1)}(x)), T^j_i(\pi^{(i-1)}(y))) < \varepsilon \text{ for every} \\ 0 \leq j \leq \left\lceil {(w_1 + \cdots + w_i)n} \right\rceil \text{ and } 1 \leq i \leq k. \end{array} \!\!\bigg\}. \end{align*} $$

Consider $\Gamma = \{ B^{\boldsymbol {w}}_{n_j}(x_j, \varepsilon ) \}_j$ , an at-most countable cover of $X_1$ by weighted Bowen balls. Let $n(\Gamma ) = \min _j n_j$ . For $s \geq 0$ and $N \in \mathbb {N}$ , let

$$ \begin{align*} \Lambda^{\boldsymbol{w}, s}_{N, \varepsilon} = \inf \bigg\{\! \sum_j e^{-sn_j} \bigg| \Gamma = \{ B^{\boldsymbol{w}}_{n_j}(x_j, \varepsilon) \}_j \text{ covers } X_1 \text{ and } n(\Gamma) \geq N \bigg\}. \end{align*} $$

This quantity is non-decreasing as $N \to \infty $ . The following limit hence exists:

$$ \begin{align*} \Lambda^{\boldsymbol{w}, s}_{\varepsilon} = \lim_{N \to \infty} \Lambda^{\boldsymbol{w}, s}_{N, \varepsilon}. \end{align*} $$

There is a value of s where $\Lambda ^{\boldsymbol {w}, s}_{\varepsilon }$ jumps from $\infty $ to $0$ , which we will denote by $h^{\boldsymbol {w}}_{\mathrm {top}}(T_1, \varepsilon )$ :

$$ \begin{align*} \Lambda^{\boldsymbol{w}, s}_{\varepsilon} = \begin{cases}\infty & (s < h^{\boldsymbol{w}}_{\mathrm{top}}(T_1, \varepsilon)), \\ 0 & (s> h^{\boldsymbol{w}}_{\mathrm{top}}(T_1, \varepsilon)). \end{cases} \end{align*} $$

The value $h^{\boldsymbol {w}}_{\mathrm {top}}(T_1, \varepsilon )$ is non-decreasing as $\varepsilon \to 0$ . Therefore, we can define the $\boldsymbol {w}$ -weighted topological entropy $h^{\boldsymbol {w}}_{\mathrm {top}}(T_1)$ by

$$ \begin{align*} h^{\boldsymbol{w}}_{\mathrm{top}}(T_1) = \lim_{\varepsilon \to 0} h^{\boldsymbol{w}}_{\mathrm{top}}(T_1, \varepsilon). \end{align*} $$

An important point about this definition is that in some dynamical systems, such as self-affine sponges, the quantity $h^{\boldsymbol {w}}_{\mathrm {top}}(T_1)$ is directly related to the Hausdorff dimension of $X_1$ .

Example 1.1. Consider the self-affine sponges introduced in §1.2. Define $p_i: \mathbb {T}^{r-i+1} \rightarrow \mathbb {T}^{r-i}$ by

$$ \begin{align*} p_i(x_1, x_2, \ldots, x_{r-i}, x_{r-i+1}) = (x_1, x_2, \ldots, x_{r-i}). \end{align*} $$

Let $X_1 = K(T, D)$ , $X_i = p_{i-1} \circ p_i \circ \cdots \circ p_1(X_1)$ , and $T_i: X_i \rightarrow X_i$ be the endomorphism defined by $A_i = \mathrm {diag}(m_1, m_2, \ldots , m_{r-i+1})$ . Define the factor maps $\pi _i: X_i \rightarrow X_{i+1}$ as the restrictions of $p_i$ . Let

(1.4) $$ \begin{align} \boldsymbol{w} = \bigg( \frac{\log{m_1}}{\log{m_r}}, \quad \frac{\log{m_1}}{\log{m_{r-1}}} - \frac{\log{m_1}}{\log{m_r}}, \ldots , \quad \frac{\log{m_1}}{\log{m_2}} - \frac{\log{m_1}}{\log{m_3}}, \quad 1 - \frac{\log{m_1}}{\log{m_2}} \bigg). \end{align} $$

Then each nth $\boldsymbol {w}$ -weighted Bowen ball is approximately a square of side length $\varepsilon m_1^{-n}$ . Therefore,

(1.5) $$ \begin{align} \mathrm{dim}_H K(T, D) = \frac{h^{\boldsymbol{w}}_{\mathrm{top}}(T_1)}{\log{m_1}}. \end{align} $$

1.4. Tsukamoto’s approach and its extension

Following the work of Feng and Huang [FH16] described in §1.3, Tsukamoto [Tsu22] published an intriguing approach to these invariants. There, he gave a new definition of the weighted topological pressure for a factor map between two dynamical systems:

He then proved the variational principle using his definition, showing the surprising coincidence of the two definitions. His definition of weighted topological entropy allowed for relatively easy calculations for sets like self-affine carpets.

We will extend Tsukamoto’s idea, redefine the weighted topological pressure for a sequence of dynamical systems of arbitrary length, and establish the variational principle. Here we will explain our definition in the case $f \equiv 0$ . See §2 for the general setting. We will not explain Tsukamoto’s definition itself since it is obtained by letting $r=2$ in the following argument.

Consider a sequence of dynamical systems:

Take a metric $d^{(i)}$ on $X_i$ . Let ${\boldsymbol {a}}=(a_1, a_2, \ldots , a_{r-1})$ with $0 \leq a_i \leq 1$ for each i. Let N be a natural number and $\varepsilon $ a positive number. We define a new metric $d^{(i)}_N$ on $X_i$ by

$$ \begin{align*} d^{(i)}_N(x_1, x_2) = \max_{0\leq n < N} d^{(i)}({T_i}^n x_1, {T_i}^n x_2). \end{align*} $$

We inductively define a quantity $\#^{\boldsymbol {a}}_i(\Omega , N, \varepsilon )$ for $\Omega \subset X_i$ . For $\Omega \subset X_1$ , set

$$ \begin{align*} \#^{\boldsymbol{a}}_1(\Omega, N, \varepsilon) &= \min \bigg\{ n \in \mathbb{N} \bigg|\! \begin{array}{l} \text{There exists an open cover } \{U_j\}_{j=1}^n \text{ of } \Omega \\ \text{with } \mathrm{diam}(U_j, d_N^{(1)}) < \varepsilon \text{ for all } 1 \leq j \leq n \end{array}\!\!\bigg\}. \end{align*} $$

(The quantity $\#^{\boldsymbol {a}}_1(\Omega , N, \varepsilon )$ is independent of the parameter $\boldsymbol {a}$ . However, we use this notation for the convenience of what follows.) Let $\Omega \subset X_{i+1}$ . Suppose $\#^{\boldsymbol {a}}_i$ is already defined. We set

$$ \begin{align*} & \#^{\boldsymbol{a}}_{i+1}(\Omega, N, \varepsilon)\\&\quad = \min \bigg\{ \sum_{j=1}^n ( \#^{\boldsymbol{a}}_i(\pi_i^{-1}(U_j), N, \varepsilon) )^{a_i} \bigg|\! \begin{array}{l} n \in \mathbb{N}, \{U_j\}_{j=1}^n \text{ is an open cover of } \Omega \\ \text{with } \mathrm{diam}(U_j, d_N^{(i+1)}) < \varepsilon \text{ for all } 1 \leq j \leq n \end{array} \!\!\bigg\}. \end{align*} $$

We define the topological entropy of ${\boldsymbol {a}}$ -exponent $h^{\boldsymbol {a}}(\boldsymbol {T})$ , where $\boldsymbol {T} = (T_i)_i$ , by

$$ \begin{align*} h^{\boldsymbol{a}}(\boldsymbol{T}) = \lim_{\varepsilon \to 0} \bigg( \lim_{N \to \infty} \frac{\log{\#^{\boldsymbol{a}}_r(X_r, N, \varepsilon)}}{N} \bigg). \end{align*} $$

This limit exists since $\log {\#^{\boldsymbol {a}}_r(X_r, N, \varepsilon )}$ is sub-additive in N and non-decreasing as $\varepsilon $ tends to $0$ .

From ${\boldsymbol {a}}=(a_1, a_2, \ldots , a_{r-1})$ , we define a probability vector (that is, all entries are non-negative, and their sum is 1) $\boldsymbol {w_a} = (w_1, \ldots , w_r)$ by

$$ \begin{align*} \left\{\! \begin{array}{l} w_1 = a_1 a_2 a_3 \cdots a_{r-1},\\ w_2 = (1-a_1) a_2 a_3 \cdots a_{r-1}, \\ w_3 = (1-a_2) a_3 \cdots a_{r-1}, \\ \vdots \\ w_{r-1} = (1-a_{r-2}) a_{r-1}, \\ w_r = 1- a_{r-1}. \end{array} \right.\!\! \end{align*} $$

The following theorem is a direct consequence of our main result in Theorem 2.1.

Theorem 1.2. For ${\boldsymbol {a}}=(a_1, a_2, \ldots , a_{r-1})$ with $0 \leq a_i \leq 1$ for each i,

(1.6) $$ \begin{align} h^{\boldsymbol{a}}(\boldsymbol{T}) = \sup_{\mu \in \mathscr{M}^{T_1}(X_1)} \bigg( \sum_{i=1}^r w_i h_{{\pi^{(i-1)}}_*\mu} (T_i) \bigg). \end{align} $$

The strategy of the proof is adopted from Tsukamoto’s paper. However, there are some additional difficulties. Let $h^{\boldsymbol {a}}_{\mathrm {var}}(\boldsymbol {T})$ be the right-hand side of equation (1.6). We use the ‘zero-dimensional trick’ for proving $h^{\boldsymbol {a}}(\boldsymbol {T}) \leq h^{\boldsymbol {a}}_{\mathrm {var}}(\boldsymbol {T})$ , meaning we reduce the proof to the case where all dynamical systems are zero-dimensional. Merely taking a zero-dimensional extension for each $X_i$ does not work. Therefore, we realize this by taking step by step extensions of the whole sequence of dynamical systems (see §3.3). Then we show $h^{\boldsymbol {a}}(\boldsymbol {T}) \leq h^{\boldsymbol {a}}_{\mathrm {var}}(\boldsymbol {T})$ by using an appropriate measure, the definition of which is quite sophisticated (see $\sigma _N$ in the proof of Theorem 4.1). In proving $h^{\boldsymbol {a}}(\boldsymbol {T}) \geq h^{\boldsymbol {a}}_{\mathrm {var}}(\boldsymbol {T})$ , the zero-dimensional trick can not be used. The proof, therefore, requires a detailed estimation of these quantities for arbitrary covers, which is more complicated than the original argument in [Tsu22].

Theorem 1.2 and Feng and Huang’s version of variational principle in equation (1.3) yield the following corollary.

Corollary 1.3. For ${\boldsymbol {a}}=(a_1, a_2, \ldots , a_{r-1})$ with $0 < a_i \leq 1$ for each i,

$$ \begin{align*} h^{\boldsymbol{a}}(\boldsymbol{T}) = h^{\boldsymbol{w_a}}_{\mathrm{top}}(T_1). \end{align*} $$

This corollary is rather profound, connecting the two seemingly different quantities. We can calculate the Hausdorff dimension of self-affine sponges using this result as in the following example. Additionally, we will show in §6 that we can now determine the Hausdorff dimension of certain sofic sets in higher-dimensional Euclidean space.

Example 1.4. Let us take another look at self-affine sponges. Kenyon and Peres [KP96, Theorem 1.2] calculated their Hausdorff dimension as follows. Recall the notation in §1.2 and that $m_1 \leq m_2 \leq \cdots \leq m_r$ .

Theorem 1.5. Define a sequence of real numbers $(Z_j)_j$ as follows. Let $Z_r$ be the indicator of D, namely, $Z_r(i_1, \ldots , i_r) = 1$ if $(i_1, \ldots , i_r) \in D$ and $0$ otherwise. Define $Z_{r-1}$ by

$$ \begin{align*} Z_{r-1}(i_1, \ldots, i_{r-1}) = \sum_{i_r = 0}^{m_r-1} Z_r(i_1, \ldots, i_{r-1}, i_r). \end{align*} $$

More generally, if $Z_{j+1}$ is already defined, let

$$ \begin{align*} Z_j(i_1, \ldots, i_j) = \sum_{i_{j+1} = 0}^{m_{j+1}-1} Z_{j+1}(i_1, \ldots, i_j, i_{j+1})^{\log{m_{j+1}}/\log{m_{j+2}}}. \end{align*} $$

Then

$$ \begin{align*} \mathrm{dim}_H K(T, D) = \frac{\log{Z_0}}{\log{m_1}}. \end{align*} $$

We can prove this result in a fairly elementary way by Corollary 1.3 without requiring measure theory on the surface. Set $a_i = \log _{m_{r-i+1}} m_{r-i}$ for each $1 \leq i \leq r-1$ , then $\boldsymbol {w_a}$ equals $\boldsymbol {w}$ in equation (1.4). Combining equation (1.5) and Corollary 1.3, we have

$$ \begin{align*} \mathrm{dim}_H K(T, D) = \frac{h^{\boldsymbol{w_a}}_{\mathrm{top}}(T_1)}{\log{m_1}} = \frac{h^{\boldsymbol{a}}(\boldsymbol{T})}{\log{m_1}}. \end{align*} $$

Hence, we need to show the following claim.

Claim 1.6. We have

$$ \begin{align*} h^{\boldsymbol{a}}(\boldsymbol{T}) = \log{Z_0}. \end{align*} $$

Proof. Observe first that taking the infimum over closed covers instead of open ones in the definition of $h^{\boldsymbol {a}}(\boldsymbol {T})$ does not change its value. Define a metric $d^{(i)}$ on each $X_i$ by

$$ \begin{align*} d^{(i)} (x, y) = \min_{n \in {\mathbb{Z}}^{r-i+1}} \left\lvert {x-y-n} \right\rvert. \end{align*} $$

Let

$$ \begin{align*} D_j = \{ (e_1, \ldots, e_j) | \text{ there are } e_{j+1}, \ldots, e_r \text{ with } (e_1, \ldots, e_r) \in D \}. \end{align*} $$

Define $p_i: D_{r-i+1} \rightarrow D_{r-i}$ by $p_i(e_1, \ldots , e_{r-i+1}) = (e_1, \ldots , e_{r-i})$ . Fix $0 < \varepsilon < {1}/{m_r}$ and take a natural number n with $m_1^{-n} < \varepsilon $ . Fix a natural number N and let $\psi _i: D_{r-i+1}^{N+n} \rightarrow D_{r-i}^{N+n}$ be the product map of $p_i$ , that is, $\psi _i(v_1, \ldots , v_{N+n}) = (p_i(v_1), \ldots , p_i(v_{N+n}))$ .

For $x \in D_{r-i+1}^{N+n}$ , define (recall that $A_i = \mathrm {diag}(m_1, m_2, \ldots , m_{r-i+1})$ )

$$ \begin{align*} U^{(i)}_x = \bigg\{ \sum_{k=0}^{\infty} A_i^{-k} e_k \in X_i \bigg| e_k \in D_{r-i+1} \text{ for each } k \text{ and } (e_1, \ldots, e_{N+n}) = x \bigg\}. \end{align*} $$

Then $\{U^{(i)}_x\}_{x \in D^{N+n}_{r-i+1}}$ is a closed cover of $X_i$ with $\mathrm {diam}(U^{(i)}_x, d^{(i)}_N) < \varepsilon $ . For $x, y \in D^{N+n}_{r-i+1}$ , we write $x \backsim y$ if and only if $U^{(i)}_x \cap U^{(i)}_y \ne \varnothing $ . We have for any i and $x \in D_{r-i}^{N+n}$ ,

$$ \begin{align*} \pi_i^{-1}(U^{(i+1)}_x) \subset \bigcup_{\substack{x' \in D_{r-i}^{N+n} \\ x' \backsim x}} \bigcup_{y \in {\psi_i}^{-1}(x')} U^{(i)}_y. \end{align*} $$

Notice that for each $x \in D_{r-i}^{N+n}$ , the number of $x' \in D_{r-i}^{N+n}$ with $x' \backsim x$ is not more than $3^r$ . Therefore, for every $v = (v_1^{(1)}, \ldots , v_{N+n}^{(1)}) \in D_{r-1}^{N+n}$ , there are $(v_1^{(k)}, \ldots , v_{N+n}^{(k)}) \in D_{r-1}^{N+n}$ , $k = 2, 3, \ldots , L$ , and $L \leq 3^r$ , with

$$ \begin{align*} \#^{\boldsymbol{a}}_1(\pi_1^{-1}(U^{(2)}_v), N, \varepsilon) \leq \sum_{k=1}^L Z_{r-1}(v_1^{(k)})\cdots Z_{r-1}(v_{N+n}^{(k)}). \end{align*} $$

We inductively continue while considering that the multiplicity is at most $3^r$ and obtain

$$ \begin{align*} & \#^{\boldsymbol{a}}_r(X_r, N, \varepsilon) \\&\leq 3^{r(r-1)} \sum_{x_1 \in D^{N+n}_1}\bigg( \sum_{x_2 \in {\psi_{r-1}}^{-1}(x_1)} \bigg( \cdots \bigg( \sum_{x_{r-2} \in {\psi_3}^{-1}(x_{r-3})} \\&\quad\bigg( \sum_{\substack{(v_1, \ldots, v_{N+n}) \in {\psi_2}^{-1}(x_{r-2}) \\ v_j \in D_{r-1} \text{ for each } j }} ( Z_{r-1}(v_1)\cdots Z_{r-1}(v_{N+n}))^{a_1} \bigg)^{a_2} \bigg)^{a_3} \cdots \bigg)^{a_{r-2}}\bigg)^{a_{r-1}}\\&= 3^{r(r-1)} \bigg\{\! \sum_{x_1 \in D_1}\!\! \bigg(\! \sum_{x_2 \in p_{r-1}^{-1}(x_1)}\!\! \bigg(\! \cdots \!\bigg(\! \sum_{x_{r-1} \in p_2^{-1}(x_{r-2})}\!\! {Z_{r-1}(x_1, \ldots, x_{r-1})}^{a_1} \!\bigg)^{a_2}\!\cdots\! \bigg)^{a_{r-2}}\bigg)^{a_{r-1}} \bigg\}^{N+n} \\&= 3^{r(r-1)} {Z_0}^{N+n}. \end{align*} $$

Therefore,

$$ \begin{align*} h^{\boldsymbol{a}}(\boldsymbol{T}) = \lim_{\varepsilon \to 0} \bigg( \lim_{N \to \infty} \frac{\log{\#^{\boldsymbol{a}}_r(X_r, N, \varepsilon)}}{N} \bigg) \leq \log{Z_0}. \end{align*} $$

Next, we prove $h^{\boldsymbol {a}}(\boldsymbol {T}) \geq \log {Z_0}$ . We fix $0 < \varepsilon < {1}/{m_r}$ and use $\varepsilon $ -separated sets. Take and fix $\boldsymbol {s} = (t_1, \ldots , t_r) \in D$ , and set $\boldsymbol {s}_i = (t_1, \ldots , t_{r-i+1})$ . Fix a natural number N and let $\psi _i: D_{r-i+1}^N \rightarrow D_{r-i}^N$ be the product map of $p_i$ as in the previous definition. Define

$$ \begin{align*} Q_i = \bigg\{ \sum_{k=1}^N {A_i}^{-k} e_k + \sum_{k=N+1}^{\infty} {A_i}^{-k} \boldsymbol{s}_i \in X_i \bigg| e_1, \ldots, e_N \in D_{r-i+1} \bigg\}. \end{align*} $$

Then $Q_i$ is an $\varepsilon $ -separated set with respect to the metric $d^{(i)}_N$ on $X_i$ . Consider an arbitrary open cover $\mathscr {F}^{(i)}$ of $X_i$ for each i with the following properties (this $(\mathscr {F}^{(i)})_i$ is defined as a chain of open (N, $\varepsilon $ )-covers of $(X_i)_i$ in Definition 3.1).

1. (1) For every i and $V \in \mathscr {F}^{(i)}$ , we have $\mathrm {diam}(V, d^{(i)}_N) < \varepsilon $ .

2. (2) For each $1 \leq i \leq r-1$ and $U \in \mathscr {F}^{(i+1)}$ , there is $\mathscr {F}^{(i)}(U) \subset \mathscr {F}^{(i)}$ such that

   $$ \begin{align*} \pi_i^{-1}(U) \subset \bigcup \mathscr{F}^{(i)}(U) \end{align*} $$
   and
   $$ \begin{align*} \mathscr{F}^{(i)} = \bigcup_{U \in \mathscr{F}^{(i+1)}} \mathscr{F}^{(i)}(U). \end{align*} $$

We have $\#(V \cap Q_i ) \leq 1$ for each $V \in \mathscr {F}^{(i)}$ by (1). Let $(e^{(2)}_1, e^{(2)}_2, \ldots , e^{(2)}_N) \in D_{r-1}^N$ and suppose $U \in \mathscr {F}^{(2)}$ satisfies

$$ \begin{align*} \sum_{k=1}^N {A_2}^{-k} e^{(2)}_k + \sum_{k=N+1}^{\infty} {A_2}^{-k} \boldsymbol{s}_2 \in U \cap Q_2. \end{align*} $$

Then $\pi _1^{-1}(U)$ contains at least $Z_{r-1}(e^{(2)}_1)\cdots Z_{r-1}(e^{(2)}_N)$ points of $Q_1$ . Hence,

$$ \begin{align*} \#^{\boldsymbol{a}}_1(\pi_1^{-1}(U), N, \varepsilon)\geq Z_{r-1}(e^{(2)}_1)\cdots Z_{r-1}(e^{(2)}_N). \end{align*} $$

We continue this reasoning inductively and get

$$ \begin{align*} & \#^{\boldsymbol{a}}_r(X_r, N, \varepsilon) \\&\geq \sum_{e^{(r)} \in D^N_1}\bigg( \sum_{e^{(r-1)} \in {\psi_{r-1}}^{-1}(e^{(r)})} \bigg( \cdots \bigg( \sum_{e^{(3)} \in {\psi_3}^{-1}(e^{(4)})} \\&\quad\bigg( \sum_{\substack{(e^{(2)}_1, \ldots, e^{(2)}_N) \in {\psi_2}^{-1}(e^{(3)}) \\ e^{(2)}_j \in D_{r-1} \text{ for each } j }} ( Z_{r-1}(e^{(2)}_1)\cdots Z_{r-1}(e^{(2)}_N))^{a_1} \bigg)^{a_2} \bigg)^{a_3} \cdots \bigg)^{a_{r-2}}\bigg)^{a_{r-1}}\\&= \bigg\{\! \sum_{x_1 \in D_1} \!\bigg(\! \sum_{x_2 \in p_{r-1}^{-1}(x_1)} \!\bigg(\! \cdots \bigg( \sum_{x_{r-1} \in p_2^{-1}(x_{r-2})} {Z_{r-1}(x_1, \ldots, x_{r-1})}^{a_1} \!\bigg)^{a_2} \cdots \!\bigg)^{a_{r-2}}\bigg)^{a_{r-1}} \bigg\}^N \\&= {Z_0}^N. \end{align*} $$

This implies

$$ \begin{align*} h^{\boldsymbol{a}}(\boldsymbol{T}) \geq \log{Z_0}. \end{align*} $$

We conclude that

$$ \begin{align*} h^{\boldsymbol{a}}(\boldsymbol{T}) = \log{Z_0}.\\[-34pt] \end{align*} $$

We would like to mention the work of Barral and Feng [BF12, Fe11], and of Yayama [Ya11]. These papers studied the related invariants when $(X_i, T_i) (i =1, \ldots , r)$ are subshifts over finite alphabets. In this subshift case, our definition of $h^{\boldsymbol {a}}(\boldsymbol {T})$ (and its pressure version in §2) is essentially the same as that given in [BF12, Theorem 3.1]. Hence, we can say that our definition generalizes the approach in [BF12, Theorem 3.1] from subshifts to general dynamical systems.

2. Weighted topological pressure

Here, we introduce the generalized, new definition of weighted topological pressure. Let $(X_i, T_i)$ ( $i\hspace{-1pt}=\hspace{-1pt}1, 2, \ldots , r$ ) be dynamical systems and $\pi _i: X_i\hspace{-1pt} \rightarrow\hspace{-1pt} X_{i+1}\, (i\hspace{-1pt}=\hspace{-1pt}1, 2, \ldots , r\hspace{-1pt}-\hspace{-1pt}1)$ factor maps. For a continuous function $f: X_1 \to \mathbb {R}$ and a natural number N, set

$$ \begin{align*} S_N f (x) = f(x) + f(T_1 x) + f(T_1^2 x) + \cdots + f(T_1^{N-1}x). \end{align*} $$

Let $d^{(i)}$ be a metric on $X_i$ . Recall that we defined a new metric $d^{(i)}_N$ on $X_i$ by

$$ \begin{align*} d^{(i)}_N(x_1, x_2) = \max_{0\leq n < N} d^{(i)}({T_i}^n x_1, {T_i}^n x_2). \end{align*} $$

We may write these as $S_N^{T_1}f$ or $d^{T_i}_N$ to clarify the maps $T_1$ and $T_i$ in the definitions above.

Let ${\boldsymbol {a}}=(a_1, a_2, \ldots , a_{r-1})$ with $0 \leq a_i \leq 1$ for each i and $\varepsilon $ a positive number. We inductively define a quantity $P^{\boldsymbol {a}}_i(\Omega , f, N, \varepsilon )$ for $\Omega \subset X_i$ . For $\Omega \subset X_1$ , set

$$ \begin{align*} & P^{\boldsymbol{a}}_1(\Omega, f, N, \varepsilon)\\ &\quad = \inf \bigg\{\sum_{j=1}^n \exp ( \sup_{U_j} S_N f) \bigg|\! \begin{array}{l} n \in \mathbb{N}, \{U_j\}_{j=1}^n \text{ is an open cover of } \Omega \\ \text{with } \mathrm{diam}(U_j, d_N^{(1)}) < \varepsilon \text{ for all } 1 \leq j \leq n \end{array} \!\!\bigg\}. \end{align*} $$

Let $\Omega \subset X_{i+1}$ . If $P^{\boldsymbol {a}}_i$ is already defined, let

$$ \begin{align*} & P^{\boldsymbol{a}}_{i+1}(\Omega, f, N, \varepsilon) \\&\quad= \inf \bigg\{ \sum_{j=1}^n ( P^{\boldsymbol{a}}_i(\pi_i^{-1}(U_j), f, N, \varepsilon))^{a_i} \bigg|\! \begin{array}{l} n \in \mathbb{N}, \{U_j\}_{j=1}^n \text{ is an open cover of } \Omega \\ \text{with } \mathrm{diam}(U_j, d_N^{T_{i+1}}) < \varepsilon \text{ for all } 1 \leq j \leq n \end{array} \!\!\bigg\}. \end{align*} $$

We define the topological pressure of ${\boldsymbol {a}}$ -exponent $P^{\boldsymbol {a}}(f)$ by

$$ \begin{align*} P^{\boldsymbol{a}}(f) = \lim_{\varepsilon \to 0} \bigg( \lim_{N \to \infty} \frac{\log{P^{\boldsymbol{a}}_r(X_r, f, N, \varepsilon)}}{N} \bigg). \end{align*} $$

This limit exists since $\log {P^{\boldsymbol {a}}_r(X_r, f, N, \varepsilon )}$ is sub-additive in N and non-decreasing as $\varepsilon $ tends to $0$ . When $r=1$ , this coincides with the standard definition of the topological pressure $P(f)$ on $(X_1, T_1)$ . The topological entropy $h_{\mathrm {top}}(T_1)$ is the value of $P(f)$ when $f \equiv 0$ . When we want to clarify the maps $T_i$ and $\pi _i$ used in the definition of $P^{\boldsymbol {a}}(f)$ , we will denote it by $P^{\boldsymbol {a}}(f, \boldsymbol {T})$ or $P^{\boldsymbol {a}}(f, \boldsymbol {T}, \boldsymbol {\pi })$ with $\boldsymbol {T}=(T_i)_{i=1}^r$ and $\boldsymbol {\pi } = (\pi _i)_{i=1}^r$ .

Recall that we defined a probability vector $\boldsymbol {w_a} = (w_1, \ldots , w_r)$ from ${\boldsymbol {a}}=(a_1, a_2, \ldots , a_{r-1})$ by

(2.1) $$ \begin{align} \left\{\! \begin{array}{l} w_1 = a_1 a_2 a_3 \cdots a_{r-1},\\ w_2 = (1-a_1) a_2 a_3 \cdots a_{r-1}, \\ w_3 = (1-a_2) a_3 \cdots a_{r-1}, \\ \vdots \\ w_{r-1} = (1-a_{r-2}) a_{r-1}, \\ w_r = 1- a_{r-1}. \end{array} \right.\!\! \end{align} $$

Let

$$ \begin{align*} \pi^{(0)} &= \mathrm{id}_{X_1}: X_1 \to X_1, \\ \pi^{(i)} &= \pi_i \circ \pi_{i-1} \circ \cdots \circ \pi_1: X_1 \to X_{i+1}. \end{align*} $$

We can now state the main result of this paper.

Theorem 2.1. Let $(X_i, T_i)$ ( $i=1, 2, \ldots , r$ ) be dynamical systems and $\pi _i: X_i \rightarrow X_{i+1}\ (i=1, 2, \ldots , r-1)$ factor maps. For any continuous function $f: X_1 \to \mathbb {R}$ ,

(2.2) $$ \begin{align} P^{\boldsymbol{a}}(f) = \sup_{\mu \in \mathscr{M}^{T_1}(X_1)} \bigg( \sum_{i=1}^r w_i h_{{\pi^{(i-1)}}_*\mu} (T_i) + w_1 \int_{X_1} f\,d\mu \bigg). \end{align} $$

We define $P^{\boldsymbol {a}}_{\mathrm {var}}(f)$ to be the right-hand side of this equation, where ‘var’ is the abbreviation of ‘variational’. Then we need to prove

$$ \begin{align*} P^{\boldsymbol{a}}(f) = P^{\boldsymbol{a}}_{\mathrm{var}}(f). \end{align*} $$

3. Preparation

In this section, we prepare several tools which will be used in the proof of Theorem 2.1.

3.1. Basic properties and tools

Let $(X_i, T_i)$ ( $i=1, 2, \ldots , r$ ) be dynamical systems, $\pi _i: X_i \rightarrow X_{i+1}\ (i=1, 2, \ldots , r-1)$ factor maps, $\boldsymbol {a} = (a_1, \ldots , a_{r-1}) \in [0, 1]^{r-1}$ , and $f: X_1 \to \mathbb {R}$ a continuous function.

We will use the following notions in §§3.3 and 5.

Definition 3.1. Consider a cover $\mathscr {F}^{(i)}$ of $X_i$ for each i. For a natural number N and a positive number $\varepsilon $ , the family $(\mathscr {F}^{(i)})_i$ is said to be a chain of ( $\boldsymbol {N}$ , $\boldsymbol {\varepsilon }$ )-covers of $(X_i)_i$ if the following conditions are true.

1. (1) For every i and $V \in \mathscr {F}^{(i)}$ , we have $\mathrm {diam}(V,d^{(i)}_N) < \varepsilon $ .

2. (2) For each $1 \leq i \leq r-1$ and $U \in \mathscr {F}^{(i+1)}$ , there is $\mathscr {F}^{(i)}(U) \subset \mathscr {F}^{(i)}$ such that

   $$ \begin{align*} \pi_i^{-1}(U) \subset \bigcup \mathscr{F}^{(i)}(U) \end{align*} $$
   and
   $$ \begin{align*} \mathscr{F}^{(i)} = \bigcup_{U \in \mathscr{F}^{(i+1)}} \mathscr{F}^{(i)}(U). \end{align*} $$

Moreover, if all the elements of each $\mathscr {F}^{(i)}$ are open/closed/compact, we call $(\mathscr {F}^{(i)})_i$ a chain of open/closed/compact ( $\boldsymbol {N}$ , $\boldsymbol {\varepsilon }$ )-covers of $(X_i)_i$ .

Remark 3.2. Note that we can rewrite $P^{\boldsymbol {a}}_r(X_r, f, N, \varepsilon )$ using chains of open covers as follows. For a chain of (N, $\varepsilon $ )-covers $(\mathscr {F}^{(i)})_i$ of $(X_i)_i$ , let

$$ \begin{align*} & \mathscr{P}^{\boldsymbol{a}}\bigg( f, N, \varepsilon, (\mathscr{F}^{(i)})_i \bigg)\\&\quad = \sum_{U^{(r)} \in \mathscr{F}^{(r)}} \bigg( \sum_{U^{(r-1)} \in \mathscr{F}^{(r-1)}(U^{(r)})} \bigg( \cdots \bigg( \sum_{U^{(1)} \in \mathscr{F}^{(1)}(U^{(2)})} e^{\sup_{U^{(1)}}S_Nf} \bigg)^{a_1} \cdots \bigg)^{a_{r-2}}\bigg)^{a_{r-1}}.\end{align*} $$

Then

$$ \begin{align*} & P^{\boldsymbol{a}}_r(X_r, f, N, \varepsilon) \\ &\quad = \inf{ \{ \mathscr{P}^{\boldsymbol{a}}( f, N, \varepsilon, (\mathscr{F}^{(i)})_i ) | (\mathscr{F}^{(i)})_i \text{ is a chain of open } (N, \varepsilon)\text{-covers of } (X_i)_i \} }. \end{align*} $$

Just like the classic notion of pressure, we have the following property.

Lemma 3.3. For any natural number m,

$$ \begin{align*} P^{\boldsymbol{a}}(S_m^{T_1}f, \boldsymbol{T}^m) = mP^{\boldsymbol{a}}(f, \boldsymbol{T}), \end{align*} $$

where $\boldsymbol {T}^m = ({T_i}^m)_{i=1}^r$ .

Proof. Fix $\varepsilon> 0$ . It is obvious from the definition of $P^{\boldsymbol {a}}_1$ that for any $\Omega _1 \subset X_1$ and a natural number N,

$$ \begin{align*} P^{\boldsymbol{a}}_1(\Omega_1, S_m^{T_1}f, \boldsymbol{T}^m, N, \varepsilon) \leq P^{\boldsymbol{a}}_1(\Omega_1, f, \boldsymbol{T}, mN, \varepsilon). \end{align*} $$

Let $\Omega _{i+1} \subset X_{i+1}$ . By induction on i, we have

$$ \begin{align*} P^{\boldsymbol{a}}_i(\Omega_{i+1}, S_m^{T_1}f, \boldsymbol{T}^m, N, \varepsilon) \leq P^{\boldsymbol{a}}_i(\Omega_{i+1}, f, \boldsymbol{T}, mN, \varepsilon). \end{align*} $$

Thus,

(3.1) $$ \begin{align} P^{\boldsymbol{a}}_r(S_m^{T_1}f, \boldsymbol{T}^m, N, \varepsilon) \leq P^{\boldsymbol{a}}_r(f, \boldsymbol{T}, mN, \varepsilon). \end{align} $$

There exists $0 < \delta < \varepsilon $ such that for any $1 \leq i \leq r$ ,

$$ \begin{align*} d^{(i)}(x, y) < \delta \implies d_m^{T_i}(x, y) < \varepsilon\quad (\text{for } x, y \in X_i). \end{align*} $$

Then

(3.2) $$ \begin{align} d_N^{T_i^m}(x, y) < \delta \implies d_{mN}^{T_i}(x, y) < \varepsilon\quad (\text{for } x, y \in X_i \text{ and } 1 \leq i \leq r). \end{align} $$

Let $i=1$ in equation (3.2), then we have for any $\Omega _1 \subset X_1$ ,

$$ \begin{align*} P^{\boldsymbol{a}}_1(\Omega_1, f, \boldsymbol{T}, mN, \varepsilon) \leq P^{\boldsymbol{a}}_1(\Omega_1, S_m^{T_1}f, \boldsymbol{T}^m, N, \delta). \end{align*} $$

Take $\Omega _{i+1} \subset X_{i+1}$ . Again by induction on i and by equation (3.2), we have

$$ \begin{align*} P^{\boldsymbol{a}}_i(\Omega_{i+1}, f, \boldsymbol{T}, mN, \varepsilon) \leq P^{\boldsymbol{a}}_i(\Omega_{i+1}, S_m^{T_1}f, \boldsymbol{T}^m, N, \delta). \end{align*} $$

Hence,

$$ \begin{align*} P^{\boldsymbol{a}}_r(f, \boldsymbol{T}, mN, \varepsilon) \leq P^{\boldsymbol{a}}_r(S_m^{T_1}f, \boldsymbol{T}^m, N, \delta).\end{align*} $$

Combining with equation (3.1), we have

$$ \begin{align*} P^{\boldsymbol{a}}_r(S_m^{T_1}f, \boldsymbol{T}^m, N, \varepsilon) \leq P^{\boldsymbol{a}}_r(f, \boldsymbol{T}, mN, \varepsilon) \leq P^{\boldsymbol{a}}_r(S_m^{T_1}f, \boldsymbol{T}^m, N, \delta). \end{align*} $$

Therefore,

$$ \begin{align*} P^{\boldsymbol{a}}(S^{T_1}_m f, \boldsymbol{T}^m) = mP^{\boldsymbol{a}}(f, \boldsymbol{T}).\\[-36pt] \end{align*} $$

We will later use the following standard lemma of calculus.

Lemma 3.4

1. (1) For $0 \leq a \leq 1$ and non-negative numbers $x, y$ ,

   $$ \begin{align*} (x+y)^a \leq x^a+y^a. \end{align*} $$

2. (2) Suppose that non-negative real numbers $p_1, p_2, \ldots , p_n$ satisfy $\sum _{i=\mathrm {1}}^n p_i = \mathrm {1}$ . Then for any real numbers $x_1, x_2, \ldots , x_n$ , we have

   $$ \begin{align*} \sum_{i=1}^n ( -p_i \log{p_i} +x_i p_i) \leq \log{\sum_{i=1}^n e^{x_i}}. \end{align*} $$
   In particular, letting $x_1=x_2=\cdots =x_n=0$ gives
   $$ \begin{align*} \sum_{i=1}^n(-p_i \log{p_i}) \leq \log{n}. \end{align*} $$
   Here, $0 \cdot \log {0}$ is defined as $0$ .

The proof for item (1) is elementary. See [Wal82, §9.3, Lemma 9.9] for item (2).

3.2. Measure theoretic entropy

In this subsection, we will introduce the classical measure-theoretic entropy (also known as Kolmogorov–Sinai entropy) and state some of the basic lemmas we need to prove Theorem 2.1. The main reference is the book of Walters [Wal82].

Let $(X, T)$ be a dynamical system and $\mu \in \mathscr {M}^T(X)$ . A set $\mathscr {A} = \{A_1, \ldots , A_n\}$ is called a finite partition of X with measurable elements if $X = A_1 \cup \cdots \cup A_n$ , each $A_i$ is a measurable set, and $A_i \cap A_j = \varnothing $ for $i \ne j$ . In this paper, a partition is always finite and consists of measurable elements.

Let $\mathscr {A}$ and $\mathscr {A}'$ be partitions of X. We define a new partition $\mathscr {A} \vee \mathscr {A}'$ by

$$ \begin{align*} \mathscr{A} \vee \mathscr{A}' = \{ A \cap A' | A \in \mathscr{A} \text{ and } A' \in \mathscr{A}' \}. \end{align*} $$

For a natural number N, we define a refined partition $\mathscr {A}_N$ of $\mathscr {A}$ by

$$ \begin{align*} \mathscr{A}_N = \mathscr{A} \vee T^{-1}\mathscr{A} \vee T^{-2}\mathscr{A} \vee \cdots \vee T^{-(N-1)}\mathscr{A}, \end{align*} $$

where $T^{-i}\mathscr {A} = \{ T^{-i}(A) | A \in \mathscr {A}\}$ is a partition for $i \in \mathbb {N}$ .

For a partition $\mathscr {A}$ of X, let

$$ \begin{align*} H_\mu(\mathscr{A}) = - \sum_{A \in \mathscr{A}} \mu(A) \log{(\mu(A))}. \end{align*} $$

We set

$$ \begin{align*} h_\mu(T, \mathscr{A}) = \lim_{N \to \infty} \frac{H_\mu(\mathscr{A}_N)}{N}. \end{align*} $$

This limit exists since $H_\mu (\mathscr {A}_N)$ is sub-additive in N. The measure theoretic entropy $h_\mu (T)$ is defined by

$$ \begin{align*} h_\mu(T) = \sup \{ h_\mu(T, \mathscr{A}) | \mathscr{A} \text{ is a partition of } X \}. \end{align*} $$

Let $\mathscr {A}$ and $\mathscr {A}'$ be partitions. Their conditional entropy is defined by

$$ \begin{align*} H_\mu(\mathscr{A} | \mathscr{A}') = - \sum_{\substack{A' \in \mathscr{A}' \\ \mu(A') \ne 0}} \mu(A') \sum_{A \in \mathscr{A}} \frac{\mu(A \cap A')}{\mu(A')} \log{\bigg( \frac{\mu(A \cap A')}{\mu(A')} \bigg)}. \end{align*} $$

Lemma 3.5

1. (1) $H_\mu (\mathscr {A})$ is sub-additive in $\mathscr {A}$ : that is, for partitions $\mathscr {A}$ and $\mathscr {A}'$ ,

   $$ \begin{align*} H_\mu(\mathscr{A} \vee \mathscr{A}') \leq H_\mu(\mathscr{A}) + H_\mu(\mathscr{A}'). \end{align*} $$

2. (2) $H_\mu (\mathscr {A})$ is concave in $\mu $ : that is, for $\mu , \nu \in \mathscr {M}^T(X)$ and $0 \leq t \leq 1$ ,

   $$ \begin{align*} H_{(1-t)\mu+t\nu}(\mathscr{A}) \geq (1-t)H_\mu(\mathscr{A}) + tH_\nu(\mathscr{A}). \end{align*} $$

3. (3) For partitions $\mathscr {A}$ and $\mathscr {A}'$ ,

   $$ \begin{align*} h_\mu(T, \mathscr{A}) \leq h_\mu(T, \mathscr{A}') + H_\mu(\mathscr{A}' | \mathscr{A}). \end{align*} $$

For the proof, confer with [Wal82, Theorem 4.3(viii), §4.5] for item (1), [Wal82, Remark, §8.1] for item (2), and [Wal82, Theorem 4.12, §4.5] for item (3).

3.3. Zero-dimensional principal extension

Here we will see how we can reduce the proof of $P^{\boldsymbol {a}}(f) \leq P^{\boldsymbol {a}}_{\mathrm {var}}(f)$ to the case where all dynamical systems are zero- dimensional.

First, we review the definitions and properties of (zero-dimensional) principal extension. The introduction here closely follows Tsukamoto’s paper [Tsu22] and the book of Downarowicz [Dow11]. Suppose $\pi : (Y, S) \rightarrow (X, T)$ is a factor map between dynamical systems. Let d be a metric on Y. We define the conditional topological entropy of $\pi $ by

$$ \begin{align*} h_{\mathrm{top}}(Y, S | X, T) = \lim_{\varepsilon \to 0} \bigg( \lim_{N \to \infty} \frac{\sup_{x \in X} \log{\#(\pi^{-1}(x), N, \varepsilon)}}{N} \bigg). \end{align*} $$

Here,

$$ \begin{align*} \#(\pi^{-1}(x), N, \varepsilon) &= \min \bigg\{ n \in \mathbb{N} \bigg|\! \begin{array}{l} \text{There exists an open cover } \{U_j\}_{j=1}^n \text{ of } \pi^{-1}(x) \\ \text{with } \mathrm{diam}(U_j, d_N) < \varepsilon \text{ for all } 1 \leq j \leq n \end{array} \!\!\bigg\}. \end{align*} $$

A factor map $\pi : (Y, S) \rightarrow (X, T)$ between dynamical systems is said to be a principal factor map if

$$ \begin{align*} h_{\mathrm{top}}(Y, S | X, T) = 0. \end{align*} $$

Also, $(Y, S)$ is called a principal extension of $(X, T)$ .

The following theorem is from [Dow11, Corollary 6.8.9].

Theorem 3.6. Suppose $\pi : (Y, S) \rightarrow (X, T)$ is a principal factor map. Then $\pi $ preserves measure-theoretic entropy, namely,

$$ \begin{align*} h_\mu(S) = h_{\pi_*\mu}(T) \end{align*} $$

for any S-invariant probability measure $\mu $ on Y.

More precisely, it is proved in [Dow11, Corollary 6.8.9] that $\pi $ is a principal factor map if and only if it preserves measure-theoretic entropy, provided that $h_{\mathrm {top}}(X, T) < \infty $ .

Suppose $\pi : (X_1, T_1) \rightarrow (X_2, T_2)$ and $\phi : (Y, S) \rightarrow (X_2, T_2)$ are factor maps between dynamical systems. We define a fiber product $(X_1 \times _{X_2}^{} Y, T_1 \times S)$ of $(X_1, T_1)$ and $(Y, S)$ over $(X_2, T_2)$ by

$$ \begin{align*} X_1 \times_{X_2}^{} Y = \{ (x, y) \in X_1 \times Y | \pi(x) = \phi(y) \}, \end{align*} $$

$$ \begin{align*} T_1 \times S: X_1 \times_{X_2}^{} Y \ni (x, y) \longmapsto (T_1(x), S(y)) \in X_1 \times_{X_2}^{} Y. \end{align*} $$

We have the following commutative diagram:

(3.3)

Here, $\pi '$ and $\psi $ are restrictions of the projections onto Y and $X_1$ , respectively:

$$ \begin{align*} \pi': X_1 \times_{X_2}^{} Y \ni (x, y) \longmapsto y \in Y, \end{align*} $$

$$ \begin{align*} \psi: X_1 \times_{X_2}^{} Y \ni (x, y) \longmapsto x \in X_1. \end{align*} $$

Since $\pi $ and $\phi $ are surjective, both $\pi '$ and $\psi $ are factor maps. The following lemma is proved in [Tsu22, Lemma 5.3].

Lemma 3.7. If $\phi $ is a principal extension in the diagram in equation (3.3), then $\psi $ is also a principal extension.

A dynamical system $(Y, S)$ is said to be zero-dimensional if there is a clopen basis of the topology of Y, where clopen means any element in the basis is both closed and open. A basic example of a zero-dimensional dynamical system is the Cantor set $\{ 0, 1 \}^{\mathbb {N}}$ with the shift map.

A principal extension $(Y, S)$ of $(X, T)$ is called a zero-dimensional principal extension if $(Y, S)$ is zero-dimensional. The following important theorem can be found in [Dow11, Theorem 7.6.1].

Theorem 3.8. For any dynamical system, there is a zero-dimensional principal extension.

Let $(Y_i, R_i)$ ( $i=1, 2, \ldots , m$ ) be dynamical systems, $\pi _i: Y_i \rightarrow Y_{i+1}\, (i=1, 2, \ldots , m-1)$ factor maps, and $\boldsymbol {a} = (a_1, \ldots , a_{m-1}) \in [0, 1]^{m-1}$ . Fix $2 \leq k \leq m-1$ and take a zero-dimensional principal extension $\phi _k: (Z_k, S_k) \rightarrow (Y_k, R_k)$ . For each $1 \leq i \leq k-1$ , let $(Y_i \times _{Y_k} Z_k, R_i \times S_k)$ be the fiber product and $\phi _i: Y_i \times _{Y_k} Z_k \rightarrow Y_i$ be the restriction of the projection as in the earlier definition. We have

By Lemma 3.7, $\phi _i$ is a principal factor map. We define $\Pi _i: Y_i \times _{Y_k} Z_k \rightarrow Y_{i+1} \times _{Y_k} Z_k$ by $\Pi _i(x, y) = ( \pi _i(x), y )$ for each $1 \leq i \leq k-2$ , and $\Pi _{k-1}: Y_{k-1} \times _{Y_k} Z_k \rightarrow Z_k$ as the projection. Then we have the following commutative diagram:

(3.4)

Let

$$ \begin{align*} \begin{aligned} (Z_i, S_i) &= (Y_i \times^{}_{Y_k} Z_k, R_i \times S_k)\quad\text{for } 1 \leq i \leq k-1,\quad (Z_i, S_i)\! =\! (Y_i, R_i) \quad\text{for } k\!+\!1 \leq i \leq m, \\\Pi_k &= \pi_k \circ \phi_k: Z_k \rightarrow Y_{k+1}, \Pi_i = \pi_i: Z_i \rightarrow Z_{i+1}\quad \text{for } k+1 \leq i \leq m-1, \\ \phi_i &= \mathrm{id}_{Z_i}: Z_i \rightarrow Z_i \quad\text{for } k+1 \leq i \leq m. \end{aligned} \end{align*} $$

Lemma 3.9. In the settings above,

$$ \begin{align*} P^{\boldsymbol{a}}_{\mathrm{var}}(f, \boldsymbol{R}, \boldsymbol{\pi}) \geq P^{\boldsymbol{a}}_{\mathrm{var}}(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}) \end{align*} $$

and

$$ \begin{align*} P^{\boldsymbol{a}}(f, \boldsymbol{R}, \boldsymbol{\pi}) \leq P^{\boldsymbol{a}}(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}). \end{align*} $$

Here, $\boldsymbol {R} = (R_i)_i$ , $\boldsymbol {\pi } = (\pi _i)_i$ , $\boldsymbol {S} = (S_i)_i$ and $\boldsymbol {\Pi } = (\Pi _i)_i$ .

Proof. We remark that the following proof does not require $Z_k$ to be zero-dimensional. Let

$$ \begin{align*} \pi^{(0)} &= \mathrm{id}_{Y_1}: Y_1 \to Y_1, \\ \pi^{(i)} &= \pi_i \circ \pi_{i-1} \circ \cdots \circ \pi_1: Y_1 \to Y_{i+1}, \end{align*} $$

and

$$ \begin{align*} \Pi^{(0)} &= \mathrm{id}_{Z_1}: Z_1 \to Z_1, \\ \Pi^{(i)} &= \Pi_i \circ \Pi_{i-1} \circ \cdots \circ \Pi_1: Z_1 \to Z_{i+1}. \end{align*} $$

Let $\nu \in \mathscr {M}^{S_1}(Y_1)$ and $1 \leq i \leq m$ . Since all the horizontal maps in equation (3.4) are principal factor maps, we have

$$ \begin{align*} h_{{\Pi^{(i-1)}}_*\nu}(S_i) = h_{(\phi_i)_*{\Pi^{(i-1)}}_*\nu}(R_i) = h_{{\pi^{(i-1)}}_*(\phi_1)_*\nu}(R_i). \end{align*} $$

It follows that

$$ \begin{align*} P^{\boldsymbol{a}}_{\mathrm{var}}(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}) &= \sup_{\nu \in \mathscr{M}^{S_1}(Z_1)} \bigg( \sum_{i=1}^{m} w_i h_{{\Pi^{(i-1)}}_*\nu} (S_i) + w_1 \int_{Z_1} f \circ \phi_1\,d\nu \bigg) \\ &= \sup_{\nu \in \mathscr{M}^{S_1}(Z_1)} \bigg( \sum_{i=1}^{m} w_i h_{{\pi^{(i-1)}}_*(\phi_1)_*\nu}(R_i) + w_1 \int_{Y_1} f\,d( (\phi_1)_*\nu ) \bigg) \\ &\leq \sup_{\mu \in \mathscr{M}^{T_1}(Y_1)} \bigg( \sum_{i=1}^{m} w_i h_{{\pi^{(i-1)}}_*\mu} (R_i) + w_1 \int_{Y_1} f\,d\mu \bigg) \\ &= P^{\boldsymbol{a}}_{\mathrm{var}}(f, \boldsymbol{R}, \boldsymbol{\pi}). \end{align*} $$

(The reversed inequality is generally true by the surjectivity of factor maps, yielding equality. However, we do not use this fact.)

Let $d^i$ be a metric on $Y_i$ for each i and $\widetilde {d^k}$ a metric on $Z_k$ . We define a metric $\widetilde {d^i}$ on $(Z_i, S_i)$ for $1 \leq i \leq k-1$ by

$$ \begin{align*} &\widetilde{d^i}( (x_1, y_1), (x_2, y_2) )\\ &\quad= \max\{ d^i(x_1, x_2), \widetilde{d^k}(y_1, y_2) \}\quad ((x_1, y_1), (x_2, y_2) \in Z_i = Y_i \times_{Y_k} Z_k). \end{align*} $$

Set $\widetilde {d^i} = d^i$ for $k+1 \leq i \leq m$ . Take an arbitrary positive number $\varepsilon $ . There exists $0 < \delta < \varepsilon $ such that for every $1 \leq i \leq m$ ,

(3.5) $$ \begin{align} \widetilde{d^i}(x, y) < \delta \implies d^i( \phi_i(x), \phi_i(y) ) < \varepsilon \quad (x, y \in Z_i). \end{align} $$

Let N be a natural number. We claim that

$$ \begin{align*} P^{\boldsymbol{a}}_r(f, \boldsymbol{R}, \boldsymbol{\pi}, N, \varepsilon) \leq P^{\boldsymbol{a}}_r(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}, N, \delta). \end{align*} $$

Take $M> 0$ with

$$ \begin{align*} P^{\boldsymbol{a}}_r(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}, N, \delta) < M.\end{align*} $$

Then there exists a chain of open (N, $\delta $ )-covers $(\mathscr {F}^{(i)})_i$ of $(Z_i)_i$ (see Definition 3.1 and Remark 3.2) with

$$ \begin{align*} \mathscr{P}^{\boldsymbol{a}}(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}, N, \delta, (\mathscr{F}^{(i)})_i ) < M. \end{align*} $$

We can find a compact set $C_U \subset U$ for each $U \in \mathscr {F}^{(m)}$ such that $\bigcup _{U \in \mathscr {F}^{(m)}} C_U = Z_m$ . Let $\mathscr {K}^{(m)} := \{ C_U | U \in \mathscr {F}^{(m)} \}$ . Since $\Pi _{m-1}^{-1}(C_U) \subset \Pi _{m-1}^{-1}(U)$ is compact for each $U \in \mathscr {F}^{(m)}$ , we can find a compact set $E_V \subset V$ for each $V \in \mathscr {F}^{(m-1)}(U)$ such that $\Pi _{m-1}^{-1}(C_U) \subset \bigcup _{V \in \mathscr {F}^{(k)}(U)} E_V$ . Let $\mathscr {K}^{(m-1)}(C_U) := \{ E_V | V \in \mathscr {F}^{(m-1)}(U) \}$ and $\mathscr {K}^{(m-1)} := \bigcup _{C \in \mathscr {K}^{(m)}} \mathscr {K}^{(m-1)}(C)$ . We continue likewise and obtain a chain of compact (N, $\delta $ )-covers $(\mathscr {K}^{(i)})_i$ of $(Z_i)_i$ with

$$ \begin{align*} \mathscr{P}^{\boldsymbol{a}}(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}, N, \delta, (\mathscr{K}^{(i)})_i) \leq \mathscr{P}^{\boldsymbol{a}}(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}, N, \delta, (\mathscr{F}^{(i)})_i ) < M. \end{align*} $$

Let $\phi _i(\mathscr {K}^{(i)}) = \{ \phi _i(C) | C \in \mathscr {K}^{(i)} \}$ for each i. Note that for any $\Omega \subset Z_i$ ,

$$ \begin{align*} \pi_{i-1}^{-1} ( \phi_{i} ( \Omega ) ) = \phi_{i-1} ( \Pi_{i-1}^{-1} ( \Omega ) ). \end{align*} $$

This and equation (3.5) assure that $(\phi _i(\mathscr {K}^{(i)}))_i$ is a chain of compact (N, $\varepsilon $ )-covers of $(Y_i)_i$ . We have

$$ \begin{align*} \mathscr{P}^{\boldsymbol{a}}( f, \boldsymbol{R}, \boldsymbol{\pi}, N, \varepsilon, (\phi_i(\mathscr{K}^{(i)}))_i) &=\mathscr{P}^{\boldsymbol{a}} ( f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}, N, \delta, (\mathscr{K}^{(i)})_i) < M. \end{align*} $$

Since f is continuous and each $\phi _i(\mathscr {K}^{(i)})$ is a closed cover, we can slightly enlarge each set in $\phi _i(\mathscr {K}^{(i)})$ and create a chain of open (N, $\varepsilon $ )-covers $(\mathscr {O}^{(i)})_i$ of $(Y_i)_i$ satisfying

$$ \begin{align*} \mathscr{P}^{\boldsymbol{a}}( f, \boldsymbol{R}, \boldsymbol{\pi}, N, \varepsilon, (\mathscr{O}^{(i)})_i ) < M. \end{align*} $$

Therefore,

$$ \begin{align*} P^{\boldsymbol{a}}_r(f, \boldsymbol{R}, \boldsymbol{\pi}, N, \varepsilon) \leq \mathscr{P}^{\boldsymbol{a}}( f, \boldsymbol{R}, \boldsymbol{\pi}, N, \varepsilon, (\mathscr{O}^{(i)})_i) < M. \end{align*} $$

Since $M> P^{\boldsymbol {a}}_r(f \circ \phi _1, \boldsymbol {S}, \boldsymbol {\Pi }, N, \delta )$ was chosen arbitrarily, we have

$$ \begin{align*} P^{\boldsymbol{a}}_r(f, \boldsymbol{R}, \boldsymbol{\pi}, N, \varepsilon) \leq P^{\boldsymbol{a}}_r(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}, N, \delta). \end{align*} $$

This implies

$$ \begin{align*} P^{\boldsymbol{a}}(f, \boldsymbol{R}, \boldsymbol{\pi}) \leq P^{\boldsymbol{a}}(f \circ \phi_1, \boldsymbol{S}, \boldsymbol{\Pi}).\\[-35pt] \end{align*} $$

The following proposition reduces the proof of $P^{\boldsymbol {a}}(f) \leq P^{\boldsymbol {a}}_{\mathrm {var}}(f)$ in the next section to the case where all dynamical systems are zero-dimensional.

Proposition 3.10. For all dynamical systems $(X_i, T_i)$ ( $i=1, 2, \ldots , r$ ) and factor maps $\pi _i: X_i \rightarrow X_{i+1}\ (i=1, 2, \ldots , r-1)$ , there are zero-dimensional dynamical systems $(Z_i, S_i)\ (i=1, 2, \ldots , r)$ and factor maps $\Pi _i: Z_i \rightarrow Z_{i+1}\ (i=1, 2, \ldots , r-1)$ with the following property; for every continuous function $f: X_1 \rightarrow \mathbb {R}$ , there exists a continuous function $g: Z_1 \rightarrow \mathbb {R}$ with

$$ \begin{align*} P^{\boldsymbol{a}}_{\mathrm{var}}(f, \boldsymbol{T}, \boldsymbol{\pi}) \geq P^{\boldsymbol{a}}_{\mathrm{var}}(g, \boldsymbol{S}, \boldsymbol{\Pi}) \end{align*} $$

and

$$ \begin{align*} P^{\boldsymbol{a}}(f, \boldsymbol{T}, \boldsymbol{\pi}) \leq P^{\boldsymbol{a}}(g, \boldsymbol{S}, \boldsymbol{\Pi}). \end{align*} $$

Proof. We will first construct zero-dimensional dynamical systems $(Z_i, S_i)$ ( $i=1, 2, \ldots , r$ ) and factor maps $\Pi _i: Z_i \rightarrow Z_{i+1}\ (i=1, 2, \ldots , r-1)$ alongside the following commutative diagram of dynamical systems and factor maps:

(3.6)

where all the horizontal maps are principal factor maps.

By Theorem 3.8, there is a zero-dimensional principal extension $\psi _r: (Z_r, S_r) \rightarrow (X_r, T_r)$ . The set $\{*\}$ is the trivial dynamical system, and the maps $X_r \rightarrow \{*\}$ and $Z_r \rightarrow \{*\}$ send every element to $*$ . For each $1 \leq i \leq r-1$ , the map $X_i \times _{X_r}^{} Z_r \rightarrow X_i$ in the following diagram is a principal factor map by Lemma 3.7:

For $1 \leq i \leq r-2$ , define $\pi _i^{(2)}: X_i \times _{X_r}^{} Z_r \rightarrow X_{i+1} \times _{X_r}^{} Z_r$ by

$$ \begin{align*} \pi_i^{(2)}(x, z) = (\pi_i(x), y). \end{align*} $$

Then every horizontal map in the right two rows of diagram (3.6) is a principal factor map. Next, take a zero-dimensional principal extension $\psi _{r-1}: (Z_{r-1}, S_{r-1}) \rightarrow (X_{r-1} \times _{X_r}^{} Z_r, T_{r-1} \times S_r)$ and let $\Pi _{r-1} = \pi _{r-1}^{(2)} \circ \psi _{r-1}$ . The rest of diagram (3.6) is constructed similarly, and by Lemma 3.7, each horizontal map is a principal factor map.

Let $f: X_1 \rightarrow \mathbb {R}$ be a continuous map. Applying Lemma 3.9 to the right two rows of diagram (3.6), we get

$$ \begin{align*} P^{\boldsymbol{a}}_{\mathrm{var}}(f, \boldsymbol{T}, \boldsymbol{\pi}) \geq P^{\boldsymbol{a}}_{\mathrm{var}}(f \circ \phi_1, \boldsymbol{S^{(2)}}, \boldsymbol{\Pi^{(2)}}) \end{align*} $$

and

$$ \begin{align*} P^{\boldsymbol{a}}(f, \boldsymbol{T}, \boldsymbol{\pi}) \leq P^{\boldsymbol{a}}(f \circ \phi_1, \boldsymbol{S^{(2)}}, \boldsymbol{\Pi^{(2)}}) \end{align*} $$

for $\boldsymbol {\Pi ^{(2)}} = (\pi ^{(2)}_i)_i$ and $\boldsymbol {S^{(2)}} = (T_i \times S_r)_i$ . Again by Lemma 3.9,

$$ \begin{align*} P^{\boldsymbol{a}}_{\mathrm{var}}(f \circ \phi_1, \boldsymbol{S^{(2)}}, \boldsymbol{\Pi^{(2)}}) \geq P^{\boldsymbol{a}}_{\mathrm{var}}(f \circ \phi_1 \circ \phi_2, \boldsymbol{S^{(3)}}, \boldsymbol{\Pi^{(3)}}) \end{align*} $$

and

$$ \begin{align*} P^{\boldsymbol{a}}(f \circ \phi_1, \boldsymbol{S^{(2)}}, \boldsymbol{\Pi^{(2)}}) \leq P^{\boldsymbol{a}}(f \circ \phi_1 \circ \phi_2, \boldsymbol{S^{(3)}}, \boldsymbol{\Pi^{(3)}}) \end{align*} $$

where $\boldsymbol {\Pi ^{(3)}} = ( (\pi ^{(3)}_i)_{i=1}^{r-2}, \Pi _{r-1})$ , and $\boldsymbol {S^{(3)}}$ is the collection of maps associated with $Z_r$ and the third row from the right of diagram (3.6). We continue inductively and obtain the desired inequalities, where g is taken as $f \circ \phi _1 \circ \phi _2 \circ \cdots \circ \phi _r$ .

4. Proof of $P^{\boldsymbol {a}}(f) \leq P^{\boldsymbol {a}}_{\mathrm {var}}(f)$

Let $\boldsymbol {a} = (a_1, \ldots , a_{r-1}) \in [0, 1]^{r-1}$ . Recall that we defined $(w_1, \ldots , w_r)$ by

$$ \begin{align*} \left\{\! \begin{array}{l} w_1 = a_1 a_2 a_3 \cdots a_{r-1},\\w_2 = (1-a_1) a_2 a_3 \cdots a_{r-1}, \\w_3 = (1-a_2) a_3 \cdots a_{r-1}, \\\kern55pt \vdots \\ w_{r-1} = (1-a_{r-2}) a_{r-1}, \\ w_r = 1- a_{r-1} \end{array} \right. \end{align*} $$

and $P^{\boldsymbol {a}}_{\mathrm {var}}(f)$ by

$$ \begin{align*} P^{\boldsymbol{a}}_{\mathrm{var}}(f) = \sup_{\mu \in \mathscr{M}^{T_1}(X_1)} \bigg( \sum_{i=1}^r w_i h_{{\pi^{(i-1)}}_*\mu} (T_i) + w_1 \int_{X_1} f\,d\mu \bigg), \end{align*} $$

where

$$ \begin{align*} \pi^{(0)} &= \mathrm{id}_{X_1}: X_1 \to X_1, \\ \pi^{(i)} &= \pi_i \circ \pi_{i-1} \circ \cdots \circ \pi_1: X_1 \to X_{i+1}. \end{align*} $$

The following theorem suffices by Proposition 3.10 in proving $P^{\boldsymbol {a}}(f) \leq P^{\boldsymbol {a}}_{\mathrm {var}}(f)$ for arbitrary dynamical systems.

Theorem 4.1. Suppose $(X_i, T_i)$ ( $i=1, 2, \ldots , r$ ) are zero-dimensional dynamical systems and $\pi _i: X_i \rightarrow X_{i+1}\ (i=1, 2, \ldots , r-1)$ are factor maps. Then we have

$$ \begin{align*} P^{\boldsymbol{a}}(f) \leq P^{\boldsymbol{a}}_{\mathrm{var}}(f) \end{align*} $$

for any continuous function $f: X_1 \rightarrow \mathbb {R}$ .

Proof. Let $d^{(i)}$ be a metric on $X_i$ for each $i=1, 2, \ldots , r$ . Take a positive number $\varepsilon $ and a natural number N. First, we will backward inductively define a finite clopen partition $\mathscr {A}^{(i)}$ of $X_i$ for each i. Since $X_r$ is zero-dimensional, we can take a sufficiently fine finite clopen partition $\mathscr {A}^{(r)}$ of $X_r $ . That is, each $A \in \mathscr {A}^{(r)}$ is both open and closed, and $\mathrm {diam}(A, d^{(r)}_N) < \varepsilon $ . Suppose $\mathscr {A}^{(i+1)}$ is defined. For each $A \in \mathscr {A}^{(i+1)}$ , take a clopen partition $\mathscr {B}(A)$ of $\pi _{i}^{-1} (A) \subset X_{i}$ such that any $B \in \mathscr {B}(A)$ satisfies $\mathrm {diam}(B, d^{(i)}_N) < \varepsilon $ . We let $\mathscr {A}^{(i)} = \bigcup _{A \in \mathscr {A}^{(i+1)}} \mathscr {B}(A)$ . Then $\mathscr {A}^{(i)}$ is a finite clopen partition of $X_i$ . We define

$$ \begin{align*} \mathscr{A}^{(i)}_N = \mathscr{A}^{(i)} \vee T_i^{-1}\mathscr{A}^{(i)} \vee T_i^{-2}\mathscr{A}^{(i)} \vee \cdots \vee T_i^{-(N-1)}\mathscr{A}^{(i)}. \end{align*} $$

We employ the following notation. For $i<j$ and $A \in \mathscr {A}^{(j)}_N$ , let $\mathscr {A}^{(i)}_N(A)$ be the set of ‘children’ of A:

$$ \begin{align*} \mathscr{A}^{(i)}_N(A) = \{B \in \mathscr{A}^{(i)}_N | \pi_{j-1} \circ \pi_{j-2} \circ \cdots \circ \pi_i(B) \subset A \}. \end{align*} $$

Also, for $B \in \mathscr {A}^{(i)}_N$ and $i<j$ , we denote by $\widetilde {\pi }_j B$ the unique ‘parent’ of B in $\mathscr {A}^{(j)}_N$ :

$$ \begin{align*} \widetilde{\pi}_j B = A \in \mathscr{A}^{(j)}_N\quad\text{such that } \pi_{j-1} \circ \pi_{j-2} \circ \cdots \circ \pi_{i}(B) \subset A. \end{align*} $$

We will evaluate $P^{\boldsymbol {a}}(f, N, \varepsilon )$ from above using $\{ \mathscr {A}^{(i)} \}$ . Let $A \in \mathscr {A}^{(2)}_N$ , and start by setting

$$ \begin{align*} Z^{(1)}_N (A) = \sum_{B \in \mathscr{A}^{(1)}_N(A)} e^{\sup_B S_N f}. \end{align*} $$

Let $A \in \mathscr {A}^{(i+1)}_N$ . If $Z^{(i-1)}_N$ is already defined, set

$$ \begin{align*} Z^{(i)}_N (A) = \sum_{B \in \mathscr{A}^{(i)}_N(A)} ( Z^{(i-1)}_N (B))^{a_{i-1}}. \end{align*} $$

We then define $Z_N$ by

$$ \begin{align*} Z_N = \sum_{A \in \mathscr{A}^{(r)}_N} ( Z^{(r-1)}_N (A) )^{a_{r-1}}. \end{align*} $$

It is straightforward from the construction that

$$ \begin{align*} P^{\boldsymbol{a}}_r(X_r, f, N, \varepsilon) \leq Z_N. \end{align*} $$

Therefore, we only need to prove that there is a $T_1$ -invariant probability measure $\mu $ on $X_1$ such that

$$ \begin{align*} \sum_{i=1}^r w_i h_{{\pi^{(i-1)}}_* \mu} (T_i, \mathscr{A}^{(i)}) + w_1 \int_{X_1} f\,d\mu \geq \lim_{N \to \infty} \frac{\log Z_N}{N}. \end{align*} $$

Since each $A \in \mathscr {A}^{(1)}_N$ is closed, we can choose a point $x_A \in A$ so that

$$ \begin{align*} S_N f(x_A) = \sup_A S_N f. \end{align*} $$

We define a probability measure $\sigma ^{}_N$ on $X_1$ by

$$ \begin{align*} \sigma_N^{} &= \frac{1}{Z_N} \sum_{A \in \mathscr{A}^{(1)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_{r}A)}^{a_{r-1}-1}{Z_N^{(r-2)}(\widetilde{\pi}_{r-1}A)}^{a_{r-2}-1} \\ &\quad \times \cdots \times {Z_N^{(2)}(\widetilde{\pi}_3 A)}^{a_{2}-1}{Z_N^{(1)}(\widetilde{\pi}_2 A)}^{a_{1}-1}e^{S_N f(x_A)} \delta_{x_A}, \end{align*} $$

where $\delta _{x^{}_A}$ is the Dirac measure at $x_A$ . This is indeed a probability measure on $X_1$ since

$$ \begin{align*} \sigma_N^{}(X_1) &= \frac{1}{Z_N} \sum_{A \in \mathscr{A}^{(1)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_{r}A)}^{a_{r-1}-1}{Z_N^{(r-2)}(\widetilde{\pi}_{r-1}A)}^{a_{r-2}-1} \\ &\qquad \times \cdots \times {Z_N^{(2)}(\widetilde{\pi}_3 A)}^{a_{2}-1}{Z_N^{(1)}(\widetilde{\pi}_2 A)}^{a_{1}-1}e^{S_N f(x_A)} \\ &\quad=\frac{1}{Z_N} \sum_{A_r \in \mathscr{A}^{(r)}_N} {Z_N^{(r-1)}(A_r)}^{a_{r-1}-1} \sum_{A_{r-1} \in \mathscr{A}^{(r-1)}_N(A_r)} {Z_N^{(r-2)}(A_{r-1})}^{a_{r-2}-1} \\ &\qquad \cdots \!\sum_{A_3 \in \mathscr{A}^{(3)}_N(A_4)} Z_N^{(2)}(A_3)^{a_2-1}\! \sum_{A_2 \in \mathscr{A}^{(2)}_N(A_3)} {Z_N^{(1)}(A_2)}^{a_{1}-1} \underbrace{\sum_{A_1 \in \mathscr{A}^{(1)}_N(A_2)} e^{S_N f(x_{A_1})}}_{= Z_N^{(1)}(A_2)}\\ &\quad = \frac{1}{Z_N} \sum_{A_r \in \mathscr{A}^{(r)}_N} {Z_N^{(r-1)}(A_r)}^{a_{r-1}-1} \sum_{A_{r-1} \in \mathscr{A}^{(r-1)}_N(A_r)} {Z_N^{(r-2)}(A_{r-1})}^{a_{r-2}-1} \\ &\qquad \cdots \sum_{A_3 \in \mathscr{A}^{(3)}_N(A_4)} Z_N^{(2)}(A_3)^{a_2-1} \underbrace{\sum_{A_2 \in \mathscr{A}^{(2)}_N(A_3)} {Z_N^{(1)}(A_2)}^{a_{1}}}_{= Z_N^{(2)}(A_3)}\\ &\quad = \cdots = \frac{1}{Z_N} \sum_{A_r \in \mathscr{A}^{(r)}_N} {Z_N^{(j-1)}(A_r)}^{a_{r-1}} = 1. \end{align*} $$

Although $\sigma _N^{}$ is not generally $T_1$ -invariant, the following well-known trick allows us to create a $T_1$ -invariant measure $\mu $ . We begin by setting

$$ \begin{align*} \mu_N^{} = \frac{1}{N} \sum_{k=0}^{N-1} {{T_1}^k}_* \sigma_N^{}. \end{align*} $$

Since $X_1$ is compact, we can take a sub-sequence of $(\mu _N^{})_N$ so that it weakly converges to a probability measure $\mu $ on $X_1$ . Then $\mu $ is $T_1$ -invariant by the definition of $\mu _N$ . We will show that this $\mu $ satisfies

$$ \begin{align*} \sum_{i=1}^r w_i h_{{\pi^{(i-1)}}_* \mu} (T_i, \mathscr{A}^{(i)}) + w_1 \int_{X_1} f\,d\mu \geq \lim_{N \to \infty} \frac{\log Z_N}{N}. \end{align*} $$

We first prove

$$ \begin{align*} \sum_{i=1}^r w_i H_{{\pi^{(i-1)}}_* \sigma_N} (\mathscr{A}^{(i)}_N) + w_1 \int_{X_1} \! S_N f\,d\mu = \log Z_N. \end{align*} $$

To simplify the notation, let

$$ \begin{align*} \sigma_N^{(i)} &= {\pi^{(i-1)}}_* \sigma_N^{} \\ &= \frac{1}{Z_N} \sum_{B \in \mathscr{A}^{(1)}_N}{Z_N^{(r-1)}(\widetilde{\pi}_r B)}^{a_{r-1}-1} \cdots {Z_N^{(1)}(\widetilde{\pi}_2 B)}^{a_{1}-1}e^{S_N f(x_B)} \delta_{\pi^{(i)} (x_B)} \end{align*} $$

and

$$ \begin{align*} W_N^{(j)} = \sum_{A \in \mathscr{A}^{(j+1)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(j+1)}(\widetilde{\pi}_{j+2} A)}^{a_{j+1}-1}{Z_N^{(j)}(A)}^{a_{j}} \log{( Z_N^{(j)}(A) )}. \end{align*} $$

Claim 4.2. We have the following equations:

$$ \begin{align*} H_{\sigma_N^{}}(\mathscr{A}^{(1)}_N) &= \log{Z_N} - \int_{X_1} \! S_N f\,d\sigma_N^{} -\sum_{j=1}^{r-1} \frac{a_{j}-1}{Z_n} W_N^{(j)}, \\ H_{\sigma_N^{(i)}}(\mathscr{A}^{(i)}_N) &= \log{Z_N} - \frac{a_{i-1}}{Z_n} W_N^{(i-1)} - \sum_{j=i}^{r-1} \frac{a_{j}-1}{Z_n} W_N^{(j)}\quad (\textit{for } 2 \leq i \leq r ). \end{align*} $$

Here, $\sum _{j=r}^{r-1}\ ((a_{j}-1)/Z_n) W_N^{(j)}$ is defined to be $0$ .

Proof. Let $A \in \mathscr {A}^{(1)}_N$ . We have

$$ \begin{align*} \sigma_N^{}(A) = \frac{1}{Z_N} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(1)}(\widetilde{\pi}_2 A)}^{a_{1}-1}e^{S_N f(x_A)}. \end{align*} $$

Then

$$ \begin{align*} & H_{\sigma_N^{}}(\mathscr{A}^{(1)}_N) = - \sum_{A \in \mathscr{A}^{(1)}_N} \sigma_N^{}(A) \log{(\sigma_N^{}(A))} \\ &\quad = \log{Z_N} - \underbrace{\sum_{A \in \mathscr{A}^{(1)}_N} \sigma_N^{}(A) S_N f(x_A)}_{(\mathrm{I})} \\ &\qquad - \!\sum_{j=1}^{r-1}\!\frac{a_j-1}{Z_N} \underbrace{\!\sum_{A \in \mathscr{A}^{(1)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1}\!\cdots\! {Z_N^{(1)}(\widetilde{\pi}_2 A)}^{a_{1}-1} e^{S_N f(x_A)}\!\log{(Z_N^{(j)}(\widetilde{\pi}_{j+1} A) )}}_{(\mathrm{I}\mathrm{I})}. \end{align*} $$

For term $(\mathrm {I})$ , we have

$$ \begin{align*} \int_{X_1} \! S_N f\,d\sigma_N^{} &= \frac{1}{Z_N} \sum_{A \in \mathscr{A}^{(1)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_{r}A)}^{a_{r-1}-1}\\ &\quad \cdots {Z_N^{(2)}(\widetilde{\pi}_3 A)}^{a_{2}-1}{Z_N^{(1)}(\widetilde{\pi}_2 A)}^{a_{1}-1}e^{S_N f(x_A)} S_N f(x_A) \\ &= (\mathrm{I}). \end{align*} $$

We will show that $(\mathrm {I}\mathrm {I}) = W_N^{(j)}$ . Let $A' \in \mathscr {A}^{(j+1)}_N$ . Then any $A \in \mathscr {A}^{(1)}_N(A')$ satisfies $\widetilde {\pi }_{j+1} A = A'$ . Hence,

$$ \begin{align*} (\mathrm{I}\mathrm{I}) &=\!\!\sum_{A' \in \mathscr{A}^{(j+1)}_N}\! \sum_{A \in \mathscr{A}^{(1)}_N(A')} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(1)}(\widetilde{\pi}_2 A)}^{a_{1}-1}e^{S_N f(x_A)} \log{(Z_N^{(j)}(\widetilde{\pi}_{j+1} A))} \\ &=\sum_{A' \in \mathscr{A}^{(j+1)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_r A')}^{a_{r-1}-1} \cdots {Z_N^{(j+1)}(\widetilde{\pi}_{j+2} A')}^{a_{j+1}-1} {Z_N^{(j)}(A')}^{a_{j}-1} \log{(Z_N^{(j)} (A'))} \\ &\quad \times \underbrace{\sum_{A \in \mathscr{A}^{(1)}_N(A')} {Z_N^{(j-1)}(\widetilde{\pi}_j A)}^{a_{j-1}-1} \cdots {Z_N^{(1)}(\widetilde{\pi}_2 A)}^{a_{1}-1}e^{S_N f(x_A)}.}_{(\mathrm{I}\mathrm{I})'} \end{align*} $$

The term $(\mathrm {I}\mathrm {I})'$ can be calculated similarly to how we showed $\sigma _N^{}(X_1)=1$ . Namely,

$$ \begin{align*} (\mathrm{I}\mathrm{I})' &= \sum_{A_j \in \mathscr{A}^{(j)}_N(A')} {Z_N^{(j-1)}(A_j)}^{a_{j-1}-1} \hspace{-10pt} \sum_{A_{j-1} \in \mathscr{A}^{(j-1)}_N(A_j)} {Z_N^{(j-2)}(A_{j-1})}^{a_{j-2}-1} \\ &\quad\cdots \sum_{A_3 \in \mathscr{A}^{(3)}_N(A_4)} Z_N^{(2)}(A_3)^{a_2-1} \sum_{A_2 \in \mathscr{A}^{(2)}_N(A_3)} {Z_N^{(1)}(A_2)}^{a_{1}-1} \underbrace{\sum_{A_1 \in \mathscr{A}^{(1)}_N(A_2)} e^{S_N f(x_{A_1})}}_{= Z_N^{(1)}(A_2)}\\ &= \cdots = \sum_{A_j \in \mathscr{A}^{(j)}_N(A')} {Z_N^{(j-1)}(A_j)}^{a_{j-1}} = Z^{(j)}_N(A'). \end{align*} $$

Thus, we get

$$ \begin{align*} (\mathrm{I}\hspace{-0.5pt}\mathrm{I}) &= \sum_{A \in \mathscr{A}^{(j+1)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(j+1)}(\widetilde{\pi}_{j+2} A)}^{a_{j+1}-1} \cdot {Z_N^{(j)}(A)}^{a_{j}} \log{( Z_N^{(j)}(A) )} \\ &= W_N^{(j)}. \end{align*} $$

This completes the proof of the first assertion.

Next, let $2 \leq i \leq r$ . For any $A \in \mathscr {A}^{(i)}_N$ ,

$$ \begin{align*} \sigma_N^{(i)}(A)&= \frac{1}{Z_n} \sum_{\substack{B \in \mathscr{A}^{(1)}_N, \\ \pi^{(i)} (x_B) \in A}} {Z_N^{(r-1)}(\widetilde{\pi}_r B)}^{a_{r-1}-1} \cdots {Z_N^{(1)}(\widetilde{\pi}_2 B)}^{a_{1}-1}e^{S_N f(x_B)}\\ &=\frac{1}{Z_n} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(i-1)}(\widetilde{\pi}_i A)}^{a_{i-1}-1} \\ &\quad \times \sum_{B \in \mathscr{A}^{(1)}_N(A)} {Z_N^{(i-2)}(\widetilde{\pi}_{i-1} B)}^{a_{i-2}-1} \cdots {Z_N^{(1)}(\widetilde{\pi}_2 B)}^{a_{1}-1}e^{S_N f(x_B)}. \end{align*} $$

As in the evaluation of term $(\mathrm {I}\hspace {-0.5pt}\mathrm {I})'$ , we have

$$ \begin{align*} \sum_{B \in \mathscr{A}^{(1)}_N(A)} {Z_N^{(i-2)}(\widetilde{\pi}_{i-1} B)}^{a_{i-2}-1} \cdots {Z_N^{(1)}(\widetilde{\pi}_2 B)}^{a_{1}-1}e^{S_N f(x_B)} = {Z^{(i-1)}_N(A)}^{a_{i-1}}.\end{align*} $$

Hence,

$$ \begin{align*} \sigma_N^{(i)}(A) &= \frac{1}{Z_n} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(i)}(\widetilde{\pi}_{i+1} A)}^{a_{i}-1} {Z_N^{(i-1)}(A)}^{a_{i-1}}. \end{align*} $$

Therefore,

$$ \begin{align*} &H_{\sigma_N^{(i)}}(\mathscr{A}^{(i)}_N) = - \sum_{A \in \mathscr{A}^{(i)}_N} \sigma_N^{(i)}(A) \log{\sigma_N^{(i)}(A)} \\ &=\log{Z_N} - \frac{1}{Z_n} \sum_{A \in \mathscr{A}^{(i)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(i)}(\widetilde{\pi}_{i+1} A)}^{a_{i}-1} {Z_N^{(i-1)}(A)}^{a_{i-1}} \\ &\quad \times \log{( {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(i)}(\widetilde{\pi}_{i+1} A)}^{a_{i}-1} {Z_N^{(i-1)}(A)}^{a_{i-1}})}\\ &=\log{Z_N}\!-\! \frac{a_{i-1}}{Z_n}\!\!\sum_{A \in \mathscr{A}^{(i)}_N}\! {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1}\! \cdots\! {Z_N^{(i)}(\widetilde{\pi}_{i+1} A)}^{a_{i}-1} {Z_N^{(i-1)}(A)}^{a_{i-1}} \log{( Z_N^{(i-1)}(A))} \\ &\quad\! -\!\! \sum_{j=i}^{r-1}\! \frac{a_{j}-1}{Z_n} \underbrace{\!\sum_{A \in \mathscr{A}^{(i)}_N}\!{Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1}\! \cdots\! {Z_N^{(i)}(\widetilde{\pi}_{i+1} A)}^{a_{i}-1} {Z_N^{(i-1)}(A)}^{a_{i-1}}\! \log\! {( Z_N^{(j)}(\widetilde{\pi}_{j+1} A) )}}_{(\mathrm{I}\hspace{-0.5pt}\mathrm{I}\hspace{-0.5pt}\mathrm{I})}. \end{align*} $$

Note that we can calculate term $(\mathrm {I}\hspace {-0.5pt}\mathrm {I}\hspace {-0.5pt}\mathrm {I})$ as

$$ \begin{align*} & \!\sum_{A \in \mathscr{A}^{(i)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_r A)}^{a_{r-1}-1} \cdots {Z_N^{(i)}(\widetilde{\pi}_{i+1} A)}^{a_{i}-1} {Z_N^{(i-1)}(A)}^{a_{i-1}} \log{( Z_N^{(j)}(\widetilde{\pi}_{j+1} A))} \\&\!=\!\!\sum_{A_{j+1} \!\in\! \mathscr{A}^{(j+1)}_N}\!\! {Z_N^{(r-1)}(\widetilde{\pi}_r A_{j+1})}^{a_{r-1}-1} \!\cdots {Z_N^{(j+1)}(\widetilde{\pi}_{j+2} A_{j+1})}^{a_j-1} {Z_N^{(j)}(A_{j+1})}^{a_{j-1}-1}\!\log\!{( Z_N^{(j)}(A_{j+1}))} \\&\!\quad\times\!\! \sum_{A_j \in \mathscr{A}^{(j)}_N(A_{j+1})}\! {Z^{(j-1)}_N(A_j)}^{a_{j-2}-1}\! \cdots \!\! \sum_{A_{i+1}\! \in\! \mathscr{A}^{(i\!+\!1)}_N(A_{i+2})}\! {Z^{(i)}_N(A_{i\!+\!1})}^{a_{i+1}\!-\!1}\!\! \underbrace{\!\sum_{A_i \in \mathscr{A}^{(i)}_N(A_{i+1})} \!\!{Z^{(i-1)}_N(A_i)}^{a_{i-1}}}_{= Z^{(i)}_N(A_{i+1})}\\&\!= \cdots = \sum_{A_{j+1} \in \mathscr{A}^{(j+1)}_N} {Z_N^{(r-1)}(\widetilde{\pi}_r A_{j+1})}^{a_{r-1}-1} \\&\!\quad \times \cdots \times {Z_N^{(j+1)}(\widetilde{\pi}_{j+2} A_{j+1})}^{a_j-1} {Z_N^{(j)}(A_{j+1})}^{a_{j-1}} \log{( Z_N^{(j)}(A_{j+1}))}. \end{align*} $$

We conclude that

$$ \begin{align*} H_{\sigma_N^{(i)}}(\mathscr{A}^{(i)}_N) = \log{Z_N} - \frac{a_{i-1}}{Z_n} W_N^{(i-1)} - \sum_{j=i}^{r-1} \frac{a_{j}-1}{Z_n} W_N^{(j)}. \end{align*} $$

This completes the proof of the claim.

By this claim,

$$ \begin{align*} \sum_{i=1}^r w_i H_{\sigma_N^{(i)}}\! (\mathscr{A}^{(i)}_N) \!+\! w_1 \int_{X_1} \! S_N f\,d\mu &= \log{Z_N}\! -\! \sum_{i=2}^r \frac{w_i a_{i-1}}{Z_n} W_N^{(i-1)} \!-\! \sum_{i=1}^{r-1} \sum_{j=i}^{r-1} \frac{w_i(a_{j}\!-\!1)}{Z_n} W_N^{(j)}.\end{align*} $$

However, we have

$$ \begin{align*} \sum_{i=2}^r w_i a_{i-1} W_N^{(i-1)} + \sum_{i=1}^{r-1} \sum_{j=i}^{r-1} w_i(a_{j}-1) W_N^{(j)} = 0. \end{align*} $$

Indeed, the coefficient of $W_N^{(k)}$ ( $1 \leq k \leq r-1$ ) is

$$ \begin{align*} w_{k+1}a_k + (a_k - 1) \sum_{i=1}^k w_i &= w_{k+1}a_k + (a_k - 1) a_k a_{k+1} \cdots a_{r-1} \\ &= a_k \{ w_{k+1} - (1-a_k) a_{k+1} a_{k+2} \cdots a_{r-1} \} = 0. \end{align*} $$

Thus, we have

(4.1) $$ \begin{align} \sum_{i=1}^r w_i H_{\sigma_N^{(i)}} (\mathscr{A}^{(i)}_N) + w_1 \int_{X_1} \! S_N f\,d\mu = \log{Z_N}. \end{align} $$

Let $\mu ^{(i)} = {\pi ^{(i-1)}}_* \mu $ and $\mu ^{(i)}_N = {\pi ^{(i-1)}}_* \mu _N^{}$ .

Lemma 4.3. Let N and M be natural numbers. For any $1 \leq i \leq r$ ,

$$ \begin{align*} \frac{1}{M} H_{\mu^{(i)}_N} (\mathscr{A}^{(i)}_M) \geq \frac{1}{N} H_{\sigma_N^{(i)}}(\mathscr{A}^{(i)}_N) - \frac{2M \log{ \lvert {\mathscr{A}^{(i)}} \rvert }}{N}. \end{align*} $$

Here, $ \lvert {\mathscr {A}^{(i)}} \rvert $ is the number of elements in $\mathscr {A}^{(i)}$ .

Suppose this is true, and let N and M be natural numbers. Together with equation (4.1), we obtain the following evaluation:

$$ \begin{align*} \sum_{i=1}^r \frac{w_i}{M} H_{\mu^{(i)}_N}(\mathscr{A}^{(i)}_M) + w_1 \int_{X_1} \! f\,d\mu_N^{} & \geq \sum_{i=1}^r \frac{w_i}{N} H_{\sigma_N^{(i)}}(\mathscr{A}^{(i)}_N) \\ &\quad -\sum_{i=1}^r \frac{2M \log{ \lvert {\mathscr{A}^{(i)}} \rvert }}{N} + \frac{w_1}{N} \int_{X_1} \! S_N f\,d\sigma_N^{} \\ &= \frac{\log{Z_N}}{N} - \sum_{i=1}^r \frac{2M \log{ \lvert {\mathscr{A}^{(i)}} \rvert }}{N}. \end{align*} $$

Let $N = N_k \to \infty $ along the sub-sequence $(N_k)$ for which $\mu _{N_k}^{} \rightharpoonup \mu $ . This yields

$$ \begin{align*} \sum_{i=1}^r \frac{w_i}{M} H_{\mu^{(i)}}(\mathscr{A}^{(i)}_M) + w_1 \int_{X_1} \! f\,d\mu \geq \lim_{N \to \infty} \frac{\log{Z_N}}{N}. \end{align*} $$

We let $M \to \infty $ and get

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu^{(i)}}(T_i, \mathscr{A}^{(i)}) + w_1 \int_{X_1} \! f\,d\mu \geq \lim_{N \to \infty} \frac{\log{Z_N}}{N}. \end{align*} $$

Hence,

$$ \begin{align*} P^{\boldsymbol{a}}_{\mathrm{var}}(f) \geq P^{\boldsymbol{a}}(f). \end{align*} $$

We are left to prove Lemma 4.3.

Proof of Lemma 4.3

This statement appears in the proof of variational principle in [Wal82, Theorem 8.6], and Tsukamoto also proves it in [Tsu22, Claim 6.3]. The following proof is taken from the latter. We will explain for $i=1$ ; the same argument works for all i.

Let $\mathscr {A} = \mathscr {A}^{(1)}$ . Recall that $\mu _N^{} = ({1}/{N}) \sum _{k=0}^{N-1} {{T_1}^k}_* \sigma _N^{}$ . Since the entropy function is concave (Lemma 3.5), we have

$$ \begin{align*} H_{\mu_N^{}}(\mathscr{A}_M) \geq \frac{1}{N} \sum_{k=0}^{N-1} H_{{{T_1}^k}_* \sigma_N^{}}(\mathscr{A}_M) = \frac{1}{N} \sum_{k=0}^{N-1} H_{\sigma_N^{}}(T_1^{-k}\mathscr{A}_M). \end{align*} $$

Let $N = qM + r$ with $0 \leq r < M$ , then

(4.2) $$ \begin{align} \sum_{k=0}^{N-1} H_{\sigma_N^{}}(T_1^{-k}\mathscr{A}_M) &= \sum_{s=0}^q \sum_{t=0}^{M-1} H_{\sigma_N^{}}(T_1^{-sM-t}\mathscr{A}_M) - \sum_{k=N}^{qM+M-1} H_{\sigma_N^{}}(T_1^{-k}\mathscr{A}_M) \nonumber \\ &\geq \sum_{t=0}^{M-1} \sum_{s=0}^q H_{\sigma_N^{}}(T_1^{-sM-t}\mathscr{A}_M) - M \log{ \left\lvert {\mathscr{A}_M} \right\rvert } \nonumber \\ &\geq \sum_{t=0}^{M-1} \sum_{s=0}^q H_{\sigma_N^{}}(T_1^{-sM-t}\mathscr{A}_M) - M^2 \log{ \left\lvert {\mathscr{A}} \right\rvert }. \end{align} $$

We will evaluate $\sum _{s=0}^q H_{\sigma _N^{}}(T_1^{-sM-t}\mathscr {A}_M)$ from below for each $0 \leq t \leq M-1$ . First, observe that

$$ \begin{align*} T_1^{-sM-t}\mathscr{A}_M = \bigvee_{j=0}^{M-1} T_1^{-sM-t-j}\mathscr{A}. \end{align*} $$

We have

$$ \begin{align*} \{ sM + t + j | 0 \leq s \leq q, 0 \leq j \leq M - 1 \} = \{ t, t + 1, \ldots, t + qM + M - 1 \} \end{align*} $$

without multiplicity. Therefore,

$$ \begin{align*} H_{\sigma_N^{}}(\mathscr{A}_N) &\leq H_{\sigma_N^{}}\bigg( \bigvee_{k=0}^{t+(q+1)M-1} T_1^{-k} \mathscr{A} \bigg) \quad \text{by } N <t+(q+1)M \\ &\leq \sum_{s=0}^q H_{\sigma_N^{}}(T_1^{-sM-t}\mathscr{A}_M) + \sum_{k=0}^{t-1}H_{\sigma_N^{}}(T_1^{-k}\mathscr{A}) \quad \text{by Lemma 3.5}. \end{align*} $$

This implies

$$ \begin{align*} \sum_{s=0}^q H_{\sigma_N^{}}(T_1^{-sM-t}\mathscr{A}_M) &\geq H_{\sigma_N^{}}(\mathscr{A}_N) - \sum_{k=0}^{t-1}H_{\sigma_N^{}}(T_1^{-k}\mathscr{A}) \\ &\geq H_{\sigma_N^{}}(\mathscr{A}_N) - M \log{ \left\lvert {\mathscr{A}} \right\rvert } \quad \text{by } t < M. \end{align*} $$

Now, we sum over t and obtain

$$ \begin{align*} \sum_{t=1}^{M-1} \sum_{s=0}^q H_{\sigma_N^{}}(T_1^{-sM-t}\mathscr{A}_M) &\geq M H_{\sigma_N^{}}(\mathscr{A}_N) - M^2 \log{ \left\lvert {\mathscr{A}} \right\rvert }.\end{align*} $$

Combining with equation (4.2), this implies

$$ \begin{align*} \sum_{k=0}^{N-1} H_{\sigma_N^{}}(T_1^{-k}\mathscr{A}_M) \geq M H_{\sigma_N^{}}(\mathscr{A}_N) - 2M^2 \log{ \left\lvert {\mathscr{A}} \right\rvert }. \end{align*} $$

It follows that

$$ \begin{align*} \frac{1}{M} H_{\mu_N^{}}(\mathscr{A}_N) \geq \frac{1}{MN} \sum_{k=0}^{N-1} H_{\sigma_N^{}}(T_1^{-k}\mathscr{A}_M) \geq \frac{1}{N} H_{\sigma_N^{}}(\mathscr{A}_N) - \frac{2M \log{ \left\lvert {\mathscr{A}} \right\rvert }}{N}.\\[-39pt] \end{align*} $$

This completes the proof of Theorem 4.1.

5. Proof of $P^{\boldsymbol {a}}_{\mathrm {var}}(f) \leq P^{\boldsymbol {a}}(f)$

It seems difficult to implement the zero-dimensional trick to prove $P^{\boldsymbol {a}}_{\mathrm {var}}(f) \leq P^{\boldsymbol {a}}(f)$ . Hence, the proof is more complicated.

Theorem 5.1. Suppose that $(X_i, T_i)$ ( $i=1, 2, \ldots , r$ ) are dynamical systems and $\pi _i: X_i \rightarrow X_{i+1}\ (i=1, 2, \ldots , r-1)$ are factor maps. Then we have

$$ \begin{align*} P^{\boldsymbol{a}}_{\mathrm{var}}(f) \leq P^{\boldsymbol{a}}(f) \end{align*} $$

for any continuous function $f: X_1 \rightarrow \mathbb {R}$ .

Proof. Take and fix $\mu \in \mathscr {M}^{T_1}(X_1)$ . Let $\mu _i= {\pi ^{(i-1)}}_* \mu $ . We need to prove

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu_i}(T_i) + w_1 \int_{X_1} f\,d\mu \leq P^{\boldsymbol{a}}(f, \boldsymbol{T}). \end{align*} $$

However, the following argument assures that giving an evaluation up to a constant is sufficient: suppose there is a positive number C which does not depend on f nor $(T_i)_i$ satisfying

(5.1) $$ \begin{align} \sum_{i=1}^r w_i h_{\mu_i}(T_i) + w_1 \int_{X_1} f\,d\mu \leq P^{\boldsymbol{a}}(f, \boldsymbol{T}) + C. \end{align} $$

Applying this to $S_mf$ and $\boldsymbol {T}^m = ({T_i}^m)_i$ for $m \in \mathbb {N}$ yields

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu_i}({T_i}^m) + w_1 \int_{X_1} \! S_mf\,d\mu \leq P^{\boldsymbol{a}}(S_mf, \boldsymbol{T}^m) + C. \end{align*} $$

We employ Lemma 3.3 and get

$$ \begin{align*} m \sum_{i=1}^r w_i h_{\mu_i}(T_i) + m w_1 \int_{X_1} \! f\,d\mu \leq mP^{\boldsymbol{a}}(f, \boldsymbol{T}) + C. \end{align*} $$

Divide by m and let $m \to \infty $ . We obtain the desired inequality

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu_i}(T_i) + w_1 \int_{X_1} \! f\,d\mu \leq P^{\boldsymbol{a}}(f, \boldsymbol{T}). \end{align*} $$

Therefore, we only need to prove inequality (5.1).

Let $\mathscr {A}^{(i)} = \{ A^{(i)}_1, A^{(i)}_2, \ldots , A^{(i)}_{m_i} \}$ be an arbitrary partition of $X_i$ for each i. We will prove

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu_i}(T_i, \mathscr{A}^{(i)}) + w_1 \int_{X_1} \! f\,d\mu \leq P^{\boldsymbol{a}}(f, \boldsymbol{T}) + C. \end{align*} $$

We start by approximating elements of $\mathscr {A}^{(i)}$ with compact sets using backward induction. For $1\leq i \leq r$ , let

$$ \begin{align*} \Lambda_i^{0} &= \{0, 1, \ldots, m_r\} \times \{0, 1, \ldots, m_{r-1}\} \times \cdots \times \{0, 1, \ldots, m_{i+1}\} \times \{0, 1, \ldots, m_i\},\\\Lambda_i &= \{0, 1, \ldots, m_r\} \times \{0, 1, \ldots, m_{r-1}\} \times \cdots \times \{0, 1, \ldots, m_{i+1}\} \times \{1, 2, \ldots, m_i\}.\end{align*} $$

We will denote an element $(j_r, j_{r-1}, \ldots , j_i)$ in $\Lambda _i^{0}$ or $\Lambda _i$ by $j_r j_{r-1} \cdots j_i$ . For each $A^{(r)}_j \in \mathscr {A}^{(r)}$ , take a compact set $C^{(r)}_j \subset A^{(r)}_j$ such that

$$ \begin{align*} \log{m_r} \cdot \sum_{j=1}^{m_r} \mu_r\big(A^{(r)}_j \setminus C^{(r)}_j\big) < 1. \end{align*} $$

Define $C^{(r)}_0$ as the remainder of $X_r$ , which may not be compact:

$$ \begin{align*} C^{(r)}_0 = \bigcup_{j=1}^{m_r} A^{(r)}_j \setminus C^{(r)}_j = X_r \setminus \bigcup_{j=1}^{m_r} C^{(r)}_j. \end{align*} $$

Then $\mathscr {C}^{(r)} := \{ C^{(r)}_0, C^{(r)}_1, \ldots , C^{(r)}_{m_r} \}$ is a measurable partition of $X_r$ .

Next, consider the partition $\pi _{r-1}^{-1}(\mathscr {C}^{(r)}) \vee \mathscr {A}^{(r-1)}$ of $X_{r-1}$ . For $j_r j_{r-1} \in \Lambda _{r-1}$ , let

$$ \begin{align*} B^{(r-1)}_{j_r j_{r-1}} = \pi_{r-1}^{-1}(C^{(r)}_{j_r}) \cap A^{(r-1)}_{j_{r-1}}. \end{align*} $$

Then

$$ \begin{align*} \pi_{r-1}^{-1}(\mathscr{C}^{(r)}) \vee \mathscr{A}^{(r-1)} = \{ B^{(r-1)}_{j_r j_{r-1}} | j_r j_{r-1} \in \Lambda_{r-1} \}, \end{align*} $$

and for each $j_r \in \Lambda _r^0$ ,

$$ \begin{align*} \bigcup_{j_{r-1}=1}^{m_{r-1}} B^{(r-1)}_{j_r j_{r-1}} = \pi_{r-1}^{-1}(C^{(r-1)}_{j_r}). \end{align*} $$

For each $j_rj_{r-1} \in \Lambda _{r-1}$ , take a compact set $C^{(r-1)}_{j_r j_{r-1}} \subset B^{(r-1)}_{j_r j_{r-1}}$ (which could be empty) such that

$$ \begin{align*} \log{ \left\lvert {\Lambda_{r-1}} \right\rvert } \cdot \sum_{j_r = 0}^{m_r} \sum_{j_{r-1}=1}^{m_{r-1}} \mu_{r-1}\big(B^{(r-1)}_{j_r j_{r-1}} \setminus C^{(r-1)}_{j_r j_{r-1}}\big) < 1. \end{align*} $$

Define $C^{(r-1)}_{j_r 0}$ as the remainder of $\pi _{r-1}^{-1}(C^{(r)}_{j_r})$ :

$$ \begin{align*} C^{(r-1)}_{j_r 0} = \pi_{r-1}^{-1}(C^{(r)}_{j_r}) \setminus \bigcup_{j_{r-1}=1}^{m_{r-1}} C^{(r-1)}_{j_r j_{r-1}}. \end{align*} $$

Then $\mathscr {C}^{(r-1)} = \{ C^{(r-1)}_{j_r j_{r-1}} | j_r j_{r-1} \in \Lambda _{r-1}^{0} \} $ is a measurable partition of $X_{r-1}$ .

Continue in this manner, and suppose we have obtained the partition $\mathscr {C}^{(k)} = \{ C^{(k)}_J | J \in \Lambda _k^{0} \}$ of $X_k$ for $k = i+1, i+2, \ldots , r$ . We will define $\mathscr {C}^{(i)}$ . Each element in $\pi _i^{-1}(\mathscr {C}^{(i+1)}) \vee \mathscr {A}^{(i)}$ can be expressed using $J' \in \Lambda _{i+1}^0$ and $j_i \in \{1,2, \ldots , m_i\}$ by

$$ \begin{align*} B_{J'j_i}^{(i)} = \pi_i^{-1}(C_{J'}^{(i+1)}) \cap A_{j_i}^{(i)}. \end{align*} $$

Choose a compact set $C_J^{(i)} \subset B_J^{(i)}$ for each $J \in \Lambda _{i}$ so that

$$ \begin{align*} \log{ \left\lvert {\Lambda_i} \right\rvert } \cdot \sum_{J' \in \Lambda_{i+1}^0} \sum_{j_i=1}^{m_i} \mu_i \big( B^{(i)}_{J'j_i} \setminus C^{(i)}_{J'j_i} \big) < 1. \end{align*} $$

Finally, for $J' \in \Lambda _{j+1}^{0}$ , let

$$ \begin{align*} C_{J'0}^{(i)} = \pi_i^{-1}(C_{J'}^{(i+1)}) \setminus \bigcup_{j_i=1}^{m_i} C_{J'j_i}^{(i)}. \end{align*} $$

Set $\mathscr {C}^{(i)} = \{ C^{(i)}_J | J \in \Lambda _i^{0} \}$ . This is a partition of $X_i$ .

Lemma 5.2. For $\mathscr {C}^{(i)}$ constructed above, we have

$$ \begin{align*} h_{\mu_i}(T_i, \mathscr{A}^{(i)}) \leq h_{\mu_i}(T_i, \mathscr{C}^{(i)}) + 1. \end{align*} $$

Proof. By Lemma 3.5,

$$ \begin{align*} h_{\mu_i}(T_i, \mathscr{A}^{(i)}) &\leq h_{\mu_i} \! (T_i, \mathscr{A}^{(i)} \vee \pi_i^{-1}(\mathscr{C}^{(i+1)})) \\ &\leq h_{\mu_i}(T_i, \mathscr{C}^{(i)}) + H_{\mu_i}( \mathscr{A}^{(i)} \vee \pi_i^{-1}(\mathscr{C}^{(i+1)}) | \mathscr{C}^{(i)}). \end{align*} $$

Since $C^{(i)}_J \subset B^{(i)}_J$ for $J \in \Lambda _i$ ,

$$ \begin{align*} & H_{\mu_i}( \mathscr{A}^{(i)} \vee \pi_i^{-1}(\mathscr{C}^{(i+1)}) | \mathscr{C}^{(i)} ) \\ &\quad = - \sum_{\substack{J \in \Lambda_i^0 \\ \mu_i(C_J^{(i)}) \ne 0}} \mu_i(C_J^{(i)}) \sum_{K \in \Lambda_i} \frac{\mu_i(B_{K}^{(i)} \cap C_J^{(i)})}{\mu_i(C_J^{(i)})} \log{\bigg( \frac{\mu_i(B_{K}^{(i)} \cap C_J^{(i)})}{\mu_i(C_J^{(i)})} \bigg)} \\ &\quad= - \sum_{\substack{J' \in \Lambda_{i+1}^0 \\ \mu_i(C_{J'0}^{(i)}) \ne 0}} \mu_i(C_{J'0}^{(i)}) \sum_{j_i=1}^{m_i} \frac{\mu_i(B_{J' j_i}^{(i)} \cap C_{J'0}^{(i)})}{\mu_i(C_{J'0}^{(i)})} \log{\bigg( \frac{\mu_i(B_{J' j_i}^{(i)} \cap C_{J'0}^{(i)})}{\mu_i(C_{J'0}^{(i)})} \bigg)}. \end{align*} $$

By Lemma 3.4, we have

$$ \begin{align*} - \sum_{j_i=1}^{m_i} \frac{\mu_i(B_{J' j_i}^{(i)} \cap C_{J'0}^{(i)})}{\mu_i(C_{J'0}^{(i)})} \log{\bigg( \frac{\mu_i(B_{J' j_i}^{(i)} \cap C_{J'0}^{(i)})}{\mu_i(C_{J'0}^{(i)})} \bigg)} \leq \log{ \left\lvert {\Lambda_i} \right\rvert }. \end{align*} $$

Therefore,

$$ \begin{align*}\hspace{-25pt} H_{\mu_i}( \mathscr{A}^{(i)} \vee \pi_i^{-1}(\mathscr{C}^{(i+1)}) | \mathscr{C}^{(i)} ) &\leq \log{ \left\lvert {\Lambda_i} \right\rvert } \sum_{J' \in \Lambda_{i+1}^0} \mu_i\bigg(\pi_i^{-1}(C_{J'}^{(i+1)}) \setminus \bigcup_{j_i=1}^{m_i} C_{J'j_i}^{(i)}\bigg) < 1.\\[-49pt] \end{align*} $$

Recall the definition of $\boldsymbol {w}$ in equation (2.1). We have

$$ \begin{align*} & \sum_{i=1}^r w_ih_{\mu_i}(T_i, \mathscr{C}^{(i)}) + w_1 \int_{X_1} f\,d\mu \\ &\quad = \lim_{N \to \infty} \frac{1}{N} \bigg\{ H_{\mu_r}(\mathscr{C}^{(r)}_N) + a_1a_2 \cdots a_{r-1} N \int_{X_1} f\,d\mu \\ &\qquad + \sum_{i=1}^{r-1} a_i a_{i+1} \cdots a_{r-1}\bigg(H_{\mu_i}(\mathscr{C}^{(i)}_N) - H_{\mu_{i+1}}(\mathscr{C}^{(i+1)}_N)\bigg) \bigg\}\\ &\quad= \lim_{N \to \infty} \frac{1}{N} \bigg\{ H_{\mu_r}(\mathscr{C}^{(r)}_N) + a_1a_2 \cdots a_{r-1} \int_{X_1} S_Nf\,d\mu \\ &\qquad + \sum_{i=1}^{r-1} a_i a_{i+1} \cdots a_{r-1} H_{\mu_i}(\mathscr{C}^{(i)}_N | \pi_i^{-1}(\mathscr{C}^{(i+1)}_N) ) \bigg\}. \end{align*} $$

Here, we used the relation

$$ \begin{align*} H_{\mu_i}(\mathscr{C}^{(i)}_N) - H_{\mu_{i+1}}(\mathscr{C}^{(i+1)}_N) &= H_{\mu_i}(\mathscr{C}^{(i)}_N) - H_{\mu_i}(\pi_i^{-1}(\mathscr{C}^{(i+1)}_N)) \\ &=H_{\mu_i}(\mathscr{C}^{(i)}_N | \pi_i^{-1}(\mathscr{C}^{(i+1)}_N)). \end{align*} $$

We fix N and evaluate from above the following terms using backward induction:

(5.2) $$ \begin{align} H_{\mu_r}(\mathscr{C}^{(r)}_N) + a_1a_2 \cdots a_{r-1} \int_{X_1} S_Nf\,d\mu + \sum_{i=1}^{r-1} a_i a_{i+1} \cdots a_{r-1} H_{\mu_i}(\mathscr{C}^{(i)}_N | \pi_i^{-1}(\mathscr{C}^{(i+1)}_N)). \end{align} $$

First, consider the term

$$ \begin{align*}a_1a_2 \cdots a_{r-1} \bigg( H_{\mu}(\mathscr{C}^{(1)}_N | \pi_1^{-1}(\mathscr{C}^{(2)}_N)) + \int_{X_1} S_Nf\,d\mu \bigg). \end{align*} $$

For $C \in \mathscr {C}^{(i+1)}_N$ , let $\mathscr {C}^{(i)}_N(C) = \{ D \in \mathscr {C}^{(i)}_N | \pi _i(D) \subset C \}$ , then by Lemma 3.4,

$$ \begin{align*} & H_{\mu}(\mathscr{C}^{(1)}_N | \pi_1^{-1}(\mathscr{C}^{(2)}_N) ) + \int_{X_1} S_Nf\,d\mu & \\ & \quad\leq \sum_{\substack{C \in \mathscr{C}^{(2)}_N \\ \mu_2(C) \ne 0}} \mu_2(C) \bigg\{ \sum_{D \in \mathscr{C}^{(1)}_N(C)} \bigg( -\frac{\mu(D)}{\mu_2(C)}\log{\frac{\mu(D)}{\mu_2(C)}} + \frac{\mu(D)}{\mu_2(C)} \sup_{D} S_Nf \bigg) \bigg\} & \\ &\quad \leq \sum_{C \in \mathscr{C}^{(2)}_N} \mu_2(C) \log{ \sum_{D \in \mathscr{C}^{(1)}_N(C)} e^{\sup_DS_Nf}}. & \end{align*} $$

Applying this inequality to equation (5.2), the following term appears:

(5.3) $$ \begin{align} a_2a_3 \cdots a_{r-1} \bigg( H_{\mu_2}(\mathscr{C}^{(2)}_N | \pi_2^{-1}(\mathscr{C}^{(3)}_N) ) + a_1 \!\sum_{C \in \mathscr{C}^{(2)}_N} \mu_2(C) \log{ \!\sum_{D \in \mathscr{C}^{(1)}_N(C)} e^{\sup_DS_Nf}} \bigg). \end{align} $$

This can be evaluated similarly using Lemma 3.4 as

$$ \begin{align*} & H_{\mu_2}(\mathscr{C}^{(2)}_N | \pi_2^{-1}(\mathscr{C}^{(3)}_N) ) + a_1 \sum_{C \in \mathscr{C}^{(2)}_N} \mu_2(C) \log{ \sum_{D \in \mathscr{C}^{(1)}_N(C)} e^{\sup_DS_Nf}} \\ &\quad =\!\!\sum_{\substack{C \in \mathscr{C}^{(3)}_N \\ \mu_3(C) \ne 0}} \!\mu_3(C) \bigg\{ \sum_{D \in \mathscr{C}^{(2)}_N(C)}\! \bigg( -\frac{\mu_2(D)}{\mu_3(C)}\log{\frac{\mu_2(D)}{\mu_3(C)}}\! + \! \frac{\mu_2(D)}{\mu_3(C)} \!\log\!{ \bigg( \sum_{E \in \mathscr{C}^{(1)}_N(D)}\! e^{\sup_ES_Nf} \bigg)^{a_1} } \bigg) \bigg\} \\ &\quad \leq \sum_{C \in \mathscr{C}^{(3)}_N} \mu_3(C) \log{ \sum_{D \in \mathscr{C}^{(2)}_N(C)}\bigg( \sum_{E \in \mathscr{C}^{(1)}_N(D)} e^{\sup_ES_Nf} \bigg)^{a_1}}. \end{align*} $$

Continue likewise and obtain the following upper bound for equation (5.2):

(5.4) $$ \begin{align} \log{ \sum_{C^{(r)} \in \mathscr{C}^{(r)}_N} \bigg( \sum_{C^{(r-1)} \in \mathscr{C}^{(r-1)}_N(C^{(r)})} \bigg( \cdots \bigg( \sum_{C^{(1)} \in \mathscr{C}^{(1)}_N(C^{(2)})} e^{\sup_{C^{(1)}}S_Nf} \bigg)^{a_1} \cdots \bigg)^{a_{r-2}}\bigg)^{a_{r-1}}}. \end{align} $$

For $1\leq i \leq r$ , let $\mathscr {C}^{(i)}_c = \{ C \in \mathscr {C}^{(i)} | C \text { is compact} \}$ . There is a positive number $\varepsilon _i$ such that $d^{(i)}(y_1, y_2)> \varepsilon _i$ for any $C_1, C_2 \in \mathscr {C}^{(i)}_c$ and $y_1 \in C_1, y_2 \in C_2$ . Fix a positive number $\varepsilon $ with

(5.5) $$ \begin{align} \varepsilon < \min_{1\leq i \leq r} \varepsilon_i. \end{align} $$

Let $\mathscr {F}^{(i)}$ be a chain of open (N, $\varepsilon $ )-covers of $X_i$ (see Definition 3.1). Consider

(5.6) $$ \begin{align} & \log{ \mathscr{P}^{\boldsymbol{a}}( f, N, \varepsilon, (\mathscr{F}^{(i)})_i ) }\notag \\ &\ = \log{ \sum_{U^{(r)} \in \mathscr{F}^{(r)}} \bigg( \sum_{U^{(r-1)} \in \mathscr{F}^{(r-1)}(U^{(r)})} \bigg( \cdots \bigg( \sum_{U^{(1)} \in \mathscr{F}^{(1)}(U^{(2)})} e^{\sup_{U^{(1)}}S_Nf} \bigg)^{a_1} \cdots \bigg)^{a_{r-2}}\bigg)^{a_{r-1}}}. \end{align} $$

We will evaluate equation (5.4) from above by equation (5.6) up to a constant. We need the next lemma.

Lemma 5.3

1. (1) For any $V \subset X_r$ with $\mathrm {diam}(V,d^{(r)}_N) < \varepsilon $ ,

   $$ \begin{align*} | \{ D \in \mathscr{C}^{(r)}_N | D \cap V \ne \varnothing \} | \leq 2^N. \end{align*} $$

2. (2) Let $1\leq i \leq r-1$ and $C \in \mathscr {C}^{(i+1)}_N$ . For any $V \subset X_i$ with $\mathrm {diam}(V,d^{(i)}_N) < \varepsilon $ ,

   $$ \begin{align*} | \{ D \in \mathscr{C}^{(i)}_N(C) | D \cap V \ne \varnothing \} | \leq 2^N. \end{align*} $$

Proof. (1) $D \in \mathscr {C}^{(r)}_N$ can be expressed using $C^{(r)}_{k_s} \in \mathscr {C}^{(r)}$ ( $s =0, 1, \ldots , N-1$ ) as

$$ \begin{align*} D = C^{(r)}_{k_0^{}} \cap T_r^{-1} C^{(r)}_{k_1^{}} \cap T_r^{-2} C^{(r)}_{k_2^{}} \cap \cdots \cap T_r^{-N+1} C^{(r)}_{k_{N-1}^{}}. \end{align*} $$

If $D \cap V \ne \varnothing $ , we have $T_r^{-s}(C^{(r)}_{k_s}) \cap V \ne \varnothing $ for every $0 \leq s \leq N-1$ . Then for each s,

$$ \begin{align*} \varnothing \ne T_r^s\bigg( T_r^{-s}(C^{(r)}_{k_s}) \cap V \bigg) \subset C^{(r)}_{k_s} \cap T_r^s(V). \end{align*} $$

By equation (5.5), each $k_s$ is either $0$ or one of the elements in $\{ 1, 2, \ldots , m_r \}$ . Therefore, there are at most $2^N$ such sets.

(2) The proof works in the same way as in item (1). C can be written using $J_k \in \Lambda _{i+1}^0$ ( $k=0, 1, \ldots , N-1$ ) as

$$ \begin{align*} C = C^{(i+1)}_{J_0} \cap T_{i+1}^{-1} C^{(i+1)}_{J_1} \cap T_{i+1}^{-2} C^{(i+1)}_{J_2} \cap \cdots \cap T_{i+1}^{-N+1} C^{(i+1)}_{J_{N-1}}. \end{align*} $$

Then any $D \in \mathscr {C}^{(i)}_N(C)$ is of the form

$$ \begin{align*} D = C^{(i)}_{J_0k_0} \cap T_i^{-1}C^{(i)}_{J_1k_1} \cap T_i^{-2}C^{(i)}_{J_2k_2} \cap \cdots \cap T_i^{-N+1}C^{(i)}_{J_{N-1}k_{N-1}} \end{align*} $$

with $0 \leq k_l \leq m_i$ ( $l = 1, 2, \ldots , N-1$ ). If $D \cap V \ne \varnothing $ , then each $k_l$ is either $0$ or one of the elements in $\{ 1, 2, \ldots , m_i \}$ . Therefore, there are at most $2^N$ such sets.

For any $C^{(1)} \in \mathscr {C}^{(1)}_N$ , there is $V \in \mathscr {F}^{(1)}$ with $V \cap C^{(1)} \ne \varnothing $ and

$$ \begin{align*} e^{\sup_{C^{(1)}}S_ Nf} \leq e^{\sup_V S_N f}. \end{align*} $$

Let $C^{(2)} \in \mathscr {C}^{(2)}_N$ , then by Lemma 5.3,

$$ \begin{align*} \sum_{C^{(1)} \in \mathscr{C}^{(1)}_N(C^{(2)})} e^{\sup_{C^{(1)}}S_ Nf} \leq \sum_{\substack{U \in \mathscr{F}^{(2)} \\ U \cap C^{(2)} \ne \varnothing}} 2^N \sum_{V \in \mathscr{F}^{(1)}(U)} e^{\sup_V S_N f}. \end{align*} $$

By Lemma 3.4,

$$ \begin{align*} \bigg( \sum_{C^{(1)} \in \mathscr{C}^{(1)}_N(C^{(2)})} e^{\sup_{C^{(1)}}S_ Nf} \bigg)^{a_1} \leq 2^{a_1N} \sum_{\substack{U \in \mathscr{F}^{(2)} \\ U \cap C^{(2)} \ne \varnothing}} \bigg( \sum_{V \in \mathscr{F}^{(1)}(U)} e^{\sup_V S_N f} \bigg)^{a_1}. \end{align*} $$

For $C^{(3)} \in \mathscr {C}^{(3)}_N$ , we apply Lemmas 5.3 and 3.4 similarly and obtain

$$ \begin{align*} &\bigg( \sum_{C^{(2)} \in \mathscr{C}^{(2)}_N(C^{(3)})} \bigg( \sum_{C^{(1)} \in \mathscr{C}^{(1)}_N(C^{(2)})} e^{\sup_{C^{(1)}}S_ Nf} \bigg)^{a_1} \bigg)^{a_2} \\ &\quad \leq 2^{a_1a_2N} 2^{a_2N} \sum_{\substack{O \in \mathscr{F}^{(3)} \\ O \cap C^{(3)} \ne \varnothing}} \bigg( \sum_{U \in \mathscr{F}^{(2)}(O)} \bigg( \sum_{V \in \mathscr{F}^{(1)}(U)} e^{\sup_V S_N f} \bigg)^{a_1} \bigg)^{a_2}. \end{align*} $$

We continue this reasoning and get

$$ \begin{align*} &\!\kern0.1pt\!\sum_{C^{(r)} \in \mathscr{C}^{(r)}_N} \bigg( \sum_{C^{(r-1)} \in \mathscr{C}^{(r-1)}_N(C^{(r)})} \bigg( \cdots \bigg( \sum_{C^{(1)} \in \mathscr{C}^{(1)}_N(C^{(2)})} e^{\sup_{C^{(1)}}S_Nf} \bigg)^{a_1} \cdots \bigg)^{a_{r-2}}\bigg)^{a_{r-1}} \\& \quad \leq 2^{\alpha N}\! \sum_{U^{(r)} \in \mathscr{F}^{(r)}} \!\bigg( \sum_{U^{(r-1)} \in \mathscr{F}^{(r-1)}(U^{(r)})} \bigg( \cdots \bigg( \sum_{U^{(1)} \in \mathscr{F}^{(1)}(U^{(2)})} e^{\sup_{U^{(1)}}S_Nf} \bigg)^{a_1} \cdots \bigg)^{a_{r-2}}\bigg)^{a_{r-1}}. \end{align*} $$

Here, $\alpha $ stands for $\sum _{i=1}^{r-1} a_i a_{i+1}\cdots a_{r-1}$ . We take the logarithm of both sides; the left-hand side equals equation (5.4), which is an upper bound for equation (5.2). Furthermore, consider the infimum over the chain of open (N, $\varepsilon $ )-covers $(\mathscr {F}^{(i)})_i$ on the right-hand side. By Remark 3.2, this yields

$$ \begin{align*} &H_{\mu_r}(\mathscr{C}^{(r)}_N) + a_1a_2 \cdots a_{r-1} \int_{X_1} S_Nf\,d\mu + \sum_{i=1}^{r-1} a_i a_{i+1} \cdots a_{r-1} H_{\mu_i}(\mathscr{C}^{(i)}_N | \pi_i^{-1}(\mathscr{C}^{(i+1)}_N) ) \\ &\quad \leq \log{P^{\boldsymbol{a}}_r(X_r, f, N, \varepsilon)} + \alpha N \log{2}. \\ \end{align*} $$

Divide by N, then let $N \to \infty $ and $\varepsilon \to 0$ . We obtain

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu_i}(T_i, \mathscr{C}^{(i)}) + w_1 \int_{X_1} f\,d\mu \leq P^{\boldsymbol{a}}(f, \boldsymbol{T}) + \alpha \log{2}. \end{align*} $$

Lemma 5.2 yields

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu_i}(T_i, \mathscr{A}^{(i)}) + w_1 \int_{X_1} f\,d\mu \leq P^{\boldsymbol{a}}(f, \boldsymbol{T}) + \alpha \log{2} + r. \end{align*} $$

We take the supremum over the partitions $(\mathscr {A}^{(i)})_i$ :

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu_i}(T_i) + w_1 \int_{X_1} f\,d\mu \leq P^{\boldsymbol{a}}(f, \boldsymbol{T}) + \alpha \log{2} + r. \end{align*} $$

By the argument at the beginning of this proof, we conclude that

$$ \begin{align*} \sum_{i=1}^r w_i h_{\mu_i}(T_i) + w_1 \int_{X_1} f\,d\mu \leq P^{\boldsymbol{a}}(f, \boldsymbol{T}).\\[-42pt] \end{align*} $$

6. Example: sofic sets

Kenyon and Peres [KP96-2] calculated the Hausdorff dimension of sofic sets in $\mathbb {T}^2$ . In this section, we will see that we can calculate the Hausdorff dimension of certain sofic sets in $\mathbb {T}^d$ with arbitrary d. We give an example for the case $d=3$ .

6.1. Definition of sofic sets

This subsection is referred to [KP96-2]. Weiss [We82] defined sofic systems as subshifts which are factors of shifts of finite type. Boyle, Kitchens, and Marcus proved in [BKM85] that this is equivalent to the following definition.

Definition 6.1. [KP96-2, Proposition 3.6]

Consider a finite directed graph $G = \langle V, E \rangle $ in which loops and multiple edges are allowed. Suppose its edges are colored in l colors in a ‘right-resolving’ fashion: every two edges emanating from the same vertex have different colors. Then the set of color sequences that arise from infinite paths in G is called the sofic system.

Let $m_1 \leq m_2 \leq \cdots \leq m_r$ be natural numbers, T an endomorphism on $\mathbb {T}^r = \mathbb {R}^r/\mathbb {Z}^r$ represented by the diagonal matrix $A = \mathrm {diag}(m_1, m_2, \ldots , m_r)$ , and $D = \prod _{i=1}^r \{0, 1, \ldots , m_i-1\}$ . Define a map $R_r: D^{\mathbb {N}} \rightarrow \mathbb {T}^r$ by

$$ \begin{align*} R_r((e^{(n)})_{n=1}^{\infty}) = \bigg( \sum_{k=0}^{\infty} \frac{e^{(k)}_1}{{m_1}^k}, \ldots, \sum_{k=0}^{\infty} \frac{e^{(k)}_r}{{m_r}^k} \bigg), \end{align*} $$

where $e^{(k)} = (e^{(k)}_1, \ldots , e^{(k)}_r) \in D$ for each k. Suppose the edges in some finite directed graph are labeled by the elements in D in the right-resolving fashion, and let $S \subset D^{\mathbb {N}}$ be the resulting sofic system. The image of S under $R_r$ is called a sofic set.

6.2. An example of a sofic set

Here we will look at an example of a sofic set and calculate its Hausdorff dimension via its weighted topological entropy. Let $D = \{0, 1\} \times \{0, 1, 2\} \times \{0, 1, 2, 3\}$ , and consider the directed graph $G = \langle V, E \rangle $ with $V = \{1, 2, 3\}$ and D-labeled edges E in Figure 2.

Figure 2 Directed graph G.

Let $Y_1 \subset D^{\mathbb {N}}$ be the resulting sofic system. Let $C = \{0, 1\} \times \{0, 1, 2\}$ and $B = \{0, 1\}$ . Define $p_1: D \rightarrow C$ and $p_2: C \rightarrow B$ by

$$ \begin{align*} p_1(i, j, k) = (i, j), \quad p_2(i, j) = i. \end{align*} $$

Let $p_1^{\mathbb {N}}: D^{\mathbb {N}} \rightarrow C^{\mathbb {N}}$ and $p_2^{\mathbb {N}}: C^{\mathbb {N}} \rightarrow B^{\mathbb {N}}$ be the product map of $p_1$ and $p_2$ , respectively. Set $Y_2 = p_1^{\mathbb {N}}(Y_1)$ and $Y_3 = p_2^{\mathbb {N}}(Y_2)$ . Note that $Y_2 = \{ (0, 0), (1,0), (0, 1) \}^{\mathbb {N}}$ and $Y_3 = \{0, 1\}^{\mathbb {N}}$ , meaning they are full shifts.

The sets $X_i = R_i(Y_i) (i = 1, 2, 3)$ are sofic sets. Define $\pi _1: X_1 \rightarrow X_2$ and $\pi _2: X_2 \rightarrow X_3$ by

$$ \begin{align*} \pi_1(x, y, z) = (x, y), \quad \pi_2(x, y) = x. \end{align*} $$

Furthermore, let $T_1$ , $T_2$ , and $T_3$ be the endomorphism on $X_1$ , $X_2$ , and $X_3$ represented by the matrices $\mathrm {diag}(2, 3, 4)$ , $\mathrm {diag}(2, 3)$ , and $\mathrm {diag}(2)$ , respectively. Then $(X_i, T_i)_i$ and $(\pi _i)_i$ form a sequence of dynamical systems.

For a natural number N, denote by $Y_i|_N$ the restriction of $Y_i$ to its first N coordinates, and let $p_{i, N}: Y_i|_N \rightarrow Y_{i+1}|_N$ be the projections for $i = 1, 2$ . As in Example 1.4, we have for any exponent $\boldsymbol {a} = (a_1, a_2) \in [0, 1]^2$ ,

$$ \begin{align*} h^{\boldsymbol{a}}(\boldsymbol{T}) = \lim_{N \to \infty} \frac{1}{N} \log{ \sum_{u \in \{0, 1\}^N} {\bigg( \sum_{v \in {p_{2, N}}^{-1}(u)} {| {p_{1, N}}^{-1}(v) |}^{a_1} \bigg)}^{a_2}}. \end{align*} $$

Now, let us evaluate $| {p_{1, N}}^{-1}(v) |$ using matrix products. This idea of using matrix products is due to Kenyon and Peres [KP96-2]. Fix $(a, b) \in {\{0, 1\}}^2$ and let

$$ \begin{align*} a_{ij} = | \{ e \in E | e \text{ is from } j \text{ to } i \text{ and the first two coordinates of its label are } (a, b) \} |. \end{align*} $$

Define a $3 \times 3$ matrix by $A_{(a, b)} = (a_{ij})_{ij}$ . Then we have

$$ \begin{align*} A_{(0, 0)} = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{pmatrix},\quad A_{(0, 1)} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 2 \\ \end{pmatrix},\quad A_{(1, 0)} = \begin{pmatrix} 1 & 2 & 2 \\ 0 & 1 & 2 \\ 2 & 2 & 1 \\ \end{pmatrix},\quad A_{(1, 1)} = O. \end{align*} $$

Note that ${A_{(0, 0)}}^2 = A_{(0, 1)}$ and ${A_{(0, 0)}}^3 = A_{(1, 0)}$ . For $v = (v_1, \ldots , v_N) \in Y_2|_N$ , we have

$$ \begin{align*} | {p_{1, N}}^{-1}(v) | \asymp \| A_{v_1} A_{v_2} \cdots A_{v_N} \|. \end{align*} $$

Here, $A \asymp B $ means there is a constant $c> 0$ independent of N with $c^{-1}B \leq A \leq cB$ . For $\alpha = ({1 + \sqrt {5}})/{2}$ , we have $\alpha ^2 = \alpha + 1$ and

$$ \begin{align*} A_{(0, 0)} \begin{pmatrix} \alpha \\ 1 \\ \alpha \\ \end{pmatrix} = \begin{pmatrix} 1 + \alpha \\ \alpha \\ 1 + \alpha \\ \end{pmatrix} = \alpha \begin{pmatrix} \alpha \\ 1 \\ \alpha \\ \end{pmatrix}\!, \quad A_{(0, 1)} \begin{pmatrix} \alpha \\ 1 \\ \alpha \\ \end{pmatrix} = \alpha^2 \begin{pmatrix} \alpha \\ 1 \\ \alpha \\ \end{pmatrix}\!, \quad A_{(1, 0)} \begin{pmatrix} \alpha \\ 1 \\ \alpha \\ \end{pmatrix} = \alpha^3 \begin{pmatrix} \alpha \\ 1 \\ \alpha \\ \end{pmatrix}\!. \end{align*} $$

Therefore,

where $\unicode{x3bb} _{(0,0)} = \alpha $ , $\unicode{x3bb} _{(0,1)} = \alpha ^2$ , $\unicode{x3bb} _{(1,0)} = \alpha ^3$ .

Take $u \in Y_3 = {\{0, 1\}}^{\mathbb {N}}$ and suppose there are n numbers of zeros in u. Also, if there are k numbers of $(0, 0)$ terms in $v = (v_1, \ldots , v_N) \in {p_{2, N}}^{-1}(u)$ , there are $n - k$ numbers of $(0, 1)$ terms and $N - n$ numbers of $(1, 0)$ terms in v. Then,

Therefore (recall that $Y_2 = \{(0,0), (1,0), (0,1)\}^{\mathbb {N}}$ ),

This implies

$$ \begin{align*} \sum_{u \in \{0, 1\}^N} {\bigg( \sum_{v \in {p_{2, N}}^{-1}(u)} {| {p_{1, N}}^{-1}(v) |}^{a_1} \bigg)}^{a_2} &\asymp \sum_{n=0}^N {N \choose n}{( \alpha^{a_1} + \alpha^{2a_1})}^{a_2n} \alpha^{3a_1a_2(N-n)} \\ &= \{ {( \alpha^{a_1} + \alpha^{2a_1} )}^{a_2} + \alpha^{3a_1a_2}\}^N. \end{align*} $$

We conclude that

$$ \begin{align*} h^{\boldsymbol{a}}(\boldsymbol{T}) &= \lim_{N \to \infty} \frac{1}{N} \log{\{ {( \alpha^{a_1} + \alpha^{2a_1} )}^{a_2} + \alpha^{3a_1a_2} \}^N} \\ &= \log{ \bigg\{ {\bigg( {\bigg( \frac{1 + \sqrt{5}}{2} \bigg)}^{a_1} + {\bigg( \frac{3 + \sqrt{5}}{2} \bigg)}^{a_1} \bigg)}^{a_2} + {(2+\sqrt{5})}^{a_1a_2} \bigg\}}. \end{align*} $$

As in Example 1.4, the Hausdorff dimension of $X_1$ is obtained by letting $a_1 = \log _4{3}$ and $a_2 = \log _3{2}$ :

$$ \begin{align*} \mathrm{dim}_H(X_1) &= \frac{h^{\boldsymbol{a}}(\boldsymbol{T})}{\log{2}} = \log_2{ \bigg\{ {\bigg( {\bigg( \frac{1 + \sqrt{5}}{2} \bigg)}^{\log_4{3}} + {\bigg( \frac{3 + \sqrt{5}}{2} \bigg)}^{\log_4{3}} \bigg)}^{\log_3{2}} + \sqrt{{(2+\sqrt{5})}} \bigg\}} \\ &= 2.1061\cdots. \end{align*} $$