# Relative weighted topological pressure

Zhengyu Yin  Department of Mathematics, Nanjing University, Nanjing 210093,
People’s Republic of China

###### Abstract.

In this article, we introduce the notion of relative weighted topological
pressures, and we prove two variational principles. We generalize the Relative
Variational Principle of Ledrappier and Walter to a weighted version, also,
prove a variational principle which is a relative version of the result proved
by Tsukamoto (New approach to weighted topological entropy and pressure,
Ergodic Theory Dynam. systems. 43 (2023) 1004-1034) while involving the
question mentioned by Feng and Huang (Variational principle for weighted
topological pressure, J. Math. Pures Appl. 106 (2016), 411-452) that is: Given
$0\leq\omega\leq 1$, let $\pi:(X,T)\to(Y,S)$ and $\varphi:(Y,S)\to(Z,R)$ be
factor maps, there exists a relativized variational principle for weighted
pressures which has the form

$P^{\omega}_{Z}(\pi,T,f)=\sup_{\mu\in\mathcal{M}(X,T)}\left(\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\right),$

where $P^{\omega}_{Z}(\pi,T,f)$ is the relative weighted topological pressure
and $h_{\\{\cdot\\}}(\cdot|R)$ are the conditional metric entropy w.r.t
$(Z,R)$ in the paper.

###### Key words and phrases:

dynamical system, relative pressure, weighted pressure, variational
principles, zero-dimensional principal extension

###### 1991 Mathematics Subject Classification:

28A80, 28D20, 37A35, 37B40, 37C45, 37D35

##  1\. Introduction

Let $(X,T)$ be a topological dynamical system (TDS for short), namely, $X$ is
a compact metric space and $T:X\to X$ a surjective continuous from $X$ to
itself. Let $\mathcal{M}(X)$ $(\mathcal{M}(X,T))$ be the sets of Borel
probability measures ($T$-invariant Borel probability measures) on $X$. Recall
that there is a fundamental relation between topological entropy and measure-
theoretical entropy, namely, the variational principle (see [4, 9, 10]):
$h_{top}(X,T)=\sup\\{h_{\mu}(X,T):\mu\in\mathcal{M}(X,T)\\}$, where
$h_{top}(X,T)$ is the topological entropy defined in [1] by Adler etc., and
$h_{\mu}(T)$ is the Kolmogorov-Sinai entropy. Motivated by statistical
mechanics, given $f\in C(X,\mathbb{R})$ a real-valued continuous function on
$X$, topological pressure was first introduced by Ruelle [16], and further
studied by Walters [19], who also developed a variational principle, that is,
$P(T,f)=\sup\left\\{h_{\mu}(X,T)+\int_{X}fd\mu:\mu\in\mathcal{M}(X,T)\right\\}$.

Now, let $\pi:(X,T)\to(Y,S)$ be a factor map between TDSs. Inspired by the
fractal geometry and the dimension theory of affine invariant subsets of tori.
In 2016, Feng and Huang [8] consider and prove the following interesting
question for factor maps: Is there a meaningful term
$\mathcal{P}^{(a_{1},a_{2})}$ such that it is the supremum of
$\\{a_{1}h_{\mu}(T)+a_{2}h_{\pi\mu}(S)+\int_{X}fd\mu|\mu\in\mathcal{M}(X,T)\\}.$
To be precise, let $\pi_{i}:(X_{i},T_{i})\to(X_{i+1},T_{i+1})$,
$i=1,\cdots,n-1$ be factor maps and $a_{i}\geq 0$ with $a_{1}>0$, for every
$f\in C(X_{1},\mathbb{R})$ real-valued continuous function on $X_{1}$, they
define a meaningful term $\mathcal{P}^{\bm{a}}(T_{1},f)$, where
$\bm{a}=\\{a_{1},\cdots,a_{n}\\}$ and also proved a variational principle:

$\displaystyle\mathcal{P}^{\bm{a}}(T_{1},f)=\sup_{\mu\in\mathcal{M}(X_{1},T_{1})}\left\\{\sum_{i=1}^{n}a_{i}h_{\mu_{i}}(X_{i},T_{i})+\int_{X_{1}}fd\mu\right\\}.$
(1.1)

More recently, Tsukamoto [18] introduced a new approach to weighted
topological pressure for factor map $\pi:(X,T)\to(Y,S)$ and proved a
corresponding variational principle for it. The definitions between [8] and
[18] are very different and the equivalence of the two definitions is highly
nontrivial (see [18]). After that, given factor maps
$\pi_{i}:(X_{i},T_{i})\to(X_{i+1},T_{i+1})$ by using a similar method with
[18], Alibabaei [2] defined the weighted topological pressure for $n$ factor
maps and established a variational principle for it. Moreover, many authors
have been investigating the relevant properties of weighted pressures, and the
interested reader may refer to e.g. [21, 23, 22, 17].

This paper is motivated by the question left in [8, secion 7.3], that is, is
there a relative version of the weighted variational principle (1.1)? Inspired
the work of [18], given $0\leq\omega\leq 1$, let $\pi:(X,T)\to(Y,S)$ and
$\phi:(Y,S)\to(Z,R)$ be factor maps with $\psi=\varphi\circ\pi,$ we shall
define the $\omega$-relative weighted topological pressure/entropy
$P^{\omega}_{Z}(\pi,T,f)$ with respect to the common factor $(Z,R)$ and
respond this question, namely, we shall prove

$P^{\omega}_{Z}(\pi,T,f)=\sup_{\mu\in\mathcal{M}(X,T)}\left(\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\right),$

where $h_{\\{\cdot\\}}(\cdot|R)$ is the conditional entropy with respect to
$(Z,R)$. In addition, by reviewing the classical relative variational
principle proved by Leddapier and Walters in [13], we also extend their result
to the weighted version.

The paper is organized as follows. In Section 2, we define the relative
weighted topological pressure and prove some basic properties about it, and
the main results are also mentioned in this section. In Section 3, we recall
the definition and some fundamental results for conditional metric entropy. In
Section 4, we prove the variational principles for relative weighted pressures
and the notion of the relative zero-dimensional principle extension is also
discussed, which plays an important role in the proof of the main theorem.

##  2\. Relative weighted topological pressure

In this section, following the ideas of Misiurewicz in [14] and Tsukamoto in
[18], we introduce new invariants for factors between TDSs.

###  2.1. Relative weighted topological pressures

Let $(X,T)$ be a TDS, denote $\mathcal{B}(X)$ the collection of all Borel
subsets of $X$. For a TDS $(X,T)$, we call a collection $\mathcal{U}$ of
subsets a cover of $X$, if $\underset{U\in\mathcal{U}}{\bigcup}U=X$. We denote
by $\mathcal{C}^{o}_{X}$ (respectively $\mathcal{C}_{X}$, $\mathcal{P}_{X}$)
the class of finite open covers (respectively finite covers, finite partition)
of $X$. Given $\mathcal{U},\mathcal{V}\in\mathcal{C}_{X}$, we write
$\mathcal{V}\preceq\mathcal{U}$ if for any $U\in\mathcal{U}$, there is some
$V\in\mathcal{V}$ such that $U\subset V$, we also set
$\mathcal{V}\vee\mathcal{U}=\\{V\cap U:V\in\mathcal{V},U\in\mathcal{U}\\}$.
Given $m<n\in\mathbb{N}$ and $\mathcal{U}\in\mathcal{C}_{X}$, we put
$\mathcal{U}^{n}_{m}=\bigvee_{i=m}^{n}T^{-i}\mathcal{U}$. Let
$\alpha\in\mathcal{P}_{X}$ and $x\in X$, set $\alpha(x)$ the element of
$\alpha$ which $x$ belongs to.

Let $(X,T)$ and $(Y,S)$ be two TDSs, we say $Y$ is a factor of $X$, if there
exists a continuous map $\pi:X\to Y$ such that $\pi\circ T=S\circ\pi$.

Let $(X,T)$ and $(Y,S)$ be TDSs with metrics $d$ and $d^{\prime}$ respectively
and $f\in C(X,\mathbb{R})$. Put $\mathcal{F}(\mathbb{N})$ the class of all
finite subset of $\mathbb{N}$. For any finite subset $F\subset\mathbb{N}$, we
define compatible metrics with respect to $F$ on $X$ and $Y$ respectively by

$d_{F}(x_{1},x_{2})=\max_{k\in
F}d(T^{k}x_{1},T^{k}x_{2}),~{}~{}d^{\prime}_{F}(y_{1},y_{2})=\max_{k\in
F}d^{\prime}(S^{k}y_{1},S^{k}y_{2}).$

For a natural number $n\in\mathbb{N}$, we denote $d_{n}$ and $d^{\prime}_{n}$
by taking $F=\\{0,\cdots,n-1\\}.$ We also define a continuous function
$\mathbb{S}_{F}f:X\to\mathbb{R}$ by

$\mathbb{S}_{F}f=\sum_{g\in F}f(T^{g}x).$

We simply write $\mathbb{S}_{n}f$ for $F=\\{0,\cdots,n-1\\}.$

Now, let $\pi:(X,T)\to(Y,S)$ and $\varphi:(Y,S)\to(Z,R)$ be two factor maps
between TDSs and $\psi=\varphi\circ\pi:(X,T)\to(Z,R)$. Given $\varepsilon>0$,
$\Omega\subset X$, $F\in\mathcal{F}(\mathbb{N})$ and $f\in C(X,\mathbb{R})$,
we define

$\displaystyle P(\Omega,f,F,\varepsilon)$ $\displaystyle=$
$\displaystyle\inf\left\\{\sum_{i=1}^{k}e^{\sup_{U_{i}}S_{F}f}:\begin{array}[]{l}\text{there
exist open subsets}~{}U_{1},\cdots,U_{k}\text{ of }X\text{with}\\\
\Omega\subset U_{1}\cup\cdots\cup U_{k}\text{ and
}\mathrm{diam}(U_{i},d_{F})<\varepsilon\text{ for}\\\ \text{all}~{}1\leq i\leq
k\end{array}\right\\},$

and simply write $P(\Omega,f,n,\varepsilon)$ for $F=\\{0,\cdots,n-1\\}.$

By taking the upper limit of the average, we define

$P(\Omega,f,\varepsilon)=\limsup_{n\to\infty}\frac{1}{n}P(\Omega,f,n,\varepsilon)\text{
and }$ $P(\Omega,f)=\lim_{\varepsilon\to 0}P(\Omega,f,\varepsilon).$

Let $0\leq\omega\leq 1$ be a real number, $K\subset Z$ and
$F\in\mathcal{F}(\mathbb{N})$ we set

$\displaystyle P_{Z}^{\omega}(\pi,K,f,F,\varepsilon)$ (2.1) $\displaystyle=$
$\displaystyle\inf\left\\{\sum_{i=1}^{k}(P(\pi^{-1}(V_{i}),f,F,\varepsilon))^{\omega}:\begin{array}[]{l}\text{there
exist open subsets}~{}V_{1},\cdots,V_{k}\text{ of }Y\text{with}\\\
\varphi^{-1}(K)\subset V_{1}\cup\cdots\cup V_{k}\text{ and
}\mathrm{diam}(V_{i},d^{\prime}_{F})<\varepsilon\text{ for}\\\
\text{all}~{}1\leq i\leq k\end{array}\right\\}.$ (2.5)

If $F=\\{0,\cdots,n-1\\}$, we write $P_{Z}^{\omega}(\pi,K,f,n,\varepsilon)$.

Also,

$P_{Z}^{\omega}(\pi,K,f,\varepsilon)=\limsup_{n\to\infty}\frac{1}{n}\log
P_{Z}^{\omega}(\pi,K,f,n,\varepsilon),\text{ and }$
$P_{Z}^{\omega}(\pi,K,f)=\lim_{\varepsilon\to
0}P_{Z}^{\omega}(\pi,K,f,\varepsilon).$

If $K=\\{z\\}$ is a point of $Z$, we denote

$P_{Z}^{\omega}(\pi,z,f,n,\varepsilon)=P_{Z}^{\omega}(\pi,\\{z\\},f,n,\varepsilon)\text{
and }P_{Z}^{\omega}(\pi,z,f)=P_{Z}^{\omega}(\pi,\\{z\\},f).$

Let $\pi:(X,T)\to(Y,S)$ be a factor map between TDSs, in [13] the authors
proved a Leddrapier-Walter’s type relative variational principle of $\pi$,
that is,

######  Theorem 2.1.

([13, Theorem 2.1]) For any $f\in C(X,\mathbb{R})$ and
$\nu\in\mathcal{M}(Y,S)$. Then

$\sup\\{h_{\mu}(T|S)+\mu(f)|\mu\in\mathcal{M}(X,T)\text{ and
}\nu=\pi\mu\\}=\int_{Y}P(\pi^{-1}(y),f)d\nu(y).$

Now, let $\pi:(X,T)\to(Y,S)$ and $\varphi:(Y,S)\to(Z,R)$ be two factor maps
between TDSs and $\psi=\varphi\circ\pi:(X,T)\to(Z,R)$, with the definition of
(2.1), we prove a Leddrapier-Walter’s type $\omega$-relative variational
principal.

######  Theorem 2.2.

For any $f\in C(X,\mathbb{R})$ and $\kappa\in\mathcal{M}(Z,R)$. Then

$\displaystyle\sup\left\\{\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu|\mu\in\mathcal{M}(X,T)\text{
and }\kappa=\psi\mu\right\\}$
$\displaystyle=\int_{Z}P_{Z}^{\omega}(\pi,z,f)d\kappa(z).$

######  Remark 2.3.

Let $m\in\mathbb{N}$, if necessary, we put $T^{m}$ into quantities, like
$P_{Z}^{\omega}(\pi,T^{m},K,f)$, to emphasize the factor relationship of $m$
iterations between TDSs, that is, for $\pi:(X,T^{m})\to(Y,S^{m})$ and
$\varphi:(Y,S^{m})\to(Z,R^{m}).$

For completeness, we shall verify the existence of limit of the quantity we
shall define.

Recall that we say a discrete infinite countable semigroup $\mathbb{N}$
satisfies F$\phi$lner condition, if for every finite subset
$K\subset\mathbb{N}$ and $\varepsilon>0$, there is a non-empty finite subset
$F\subset\mathbb{N}$ such that

$|(k+F)\backslash F|<\varepsilon|F|\text{ for all }k\in K.$

Equivalently, $\mathbb{N}$ satisfies F$\phi$lner condition if for any
$s\in\mathbb{N}$, there is a sequence $\\{F_{i}\\}_{i\in\mathbb{N}}$ of non-
empty finite subsets of $\mathbb{N}$ such that

$\lim_{i\to\infty}\frac{|(s+F_{i})\backslash F_{i}|}{|F_{i}|}=0,\text{ for all
}s\in\mathbb{N}.$

Moreover, we say a countable semigroup $\mathbb{N}$ amenable if it satisfies
F$\phi$lner condition. In [3], the authors proved a key Theorem to verify the
convergence of a sequence of quantities under the action of a cancellative
amenable semigroup $\mathbb{N}$-action. That is,

######  Theorem 2.4.

([3, Corollary 1.2]) Let $\mathbb{N}$ be a cancellative amenable semigroup and
let $\mathcal{F}(\mathbb{N})$ denote the set of all finite subsets of
$\mathbb{N}$. $h:\mathcal{F}(\mathbb{N})\to\mathbb{R}$ is a function satisfies
the following conditions:

  1. 1.

$h$ is subadditive, i.e.,

$h(E\cup F)\leq h(E)+h(F)\text{ for all }E,F\in\mathcal{F}(\mathbb{N});$

  2. 2.

$h$ is right-subinvariant, i.e.,

$h(E+s)\leq h(E)\text{ for all }s\in S\text{ and
}E\in\mathcal{F}(\mathbb{N}).$

Then there exists $\lambda\geq 0$, depending only on $h$, such that the
sequence $\\{h(F_{i})\backslash|F_{i}|\\}_{i\in\mathbb{N}}$ converges to
$\lambda$ for every F$\phi$lner sequence $\\{F_{i}\\}_{i\in\mathbb{N}}$.

######  Lemma 2.5.

For any $K\subset Z$ the quantity $\log
P_{Z}^{\omega}(\pi,K,f,\cdot,\varepsilon):\mathcal{F}(\mathbb{N})\to\mathbb{R}$
is subadditive, that is, for any finite subsets $E,F\subset\mathbb{Z}$,

$\displaystyle\log P_{Z}^{\omega}(\pi,K,f,E\cup F,\varepsilon)\leq\log
P_{Z}^{\omega}(\pi,K,f,E,\varepsilon)+\log
P_{Z}^{\omega}(\pi,K,f,F,\varepsilon).$ (2.6)

###### Proof..

Indeed, let $\mathcal{V}=\\{V_{1},\cdots,V_{n}\\}$ be a class of open subsets
of $Y$ with $\mathrm{diam}(V_{i},d^{\prime}_{E})\leq\varepsilon$, where
$i=1,\cdots,n$ and $\varphi^{-1}(K)\subset V_{1}\cup\cdots\cup V_{n}$, and
$\mathcal{U}=\\{U_{1},\cdots,U_{m}\\}$ is another class of open subsets of $Y$
with $\mathrm{diam}(U_{j},d^{\prime}_{F})\leq\varepsilon$, where
$j=1,\cdots,m$ and $\varphi^{-1}(K)\subset U_{1}\cup\cdots\cup U_{m}$. Then

$\displaystyle\left(\sum_{i=1}^{n}(P^{\omega}_{Z}(\pi^{-1}(V_{i}),f,E,\varepsilon))^{\omega}\right)\cdot\left(\sum_{j=1}^{m}(P^{\omega}_{Z}(\pi^{-1}(U_{j}),f,F,\varepsilon))^{\omega}\right)$
$\displaystyle\geq\sum_{i=1}^{n}\sum_{j=1}^{m}\left(P_{Z}^{\omega}(\pi^{-1}(V_{i}),f,E,\varepsilon)\cdot
P_{Z}^{\omega}(\pi^{-1}(U_{j}),f,F,\varepsilon)\right)^{\omega}$
$\displaystyle\geq\sum_{i,j}P^{\omega}_{Z}(\pi^{-1}(V_{i}\cap U_{j}),f,E\cup
F,\varepsilon)$

since $\mathcal{V}\vee\mathcal{U}$ is a class of open subsets with
$\mathrm{diam}(U\cap V,d^{\prime}_{E\cup F})\leq\varepsilon$

$\displaystyle\geq P_{Z}^{\omega}(\pi,K,f,E\cup F,\varepsilon).$

$\mathcal{V}$ and $\mathcal{U}$ can be taken arbitrarily, so we have

$P_{Z}^{\omega}(\pi,K,f,E\cup F,\varepsilon)\leq
P_{Z}^{\omega}(\pi,K,f,E,\varepsilon)\cdot
P_{Z}^{\omega}(\pi,K,f,F,\varepsilon),$

and (2.6) holds. ∎

######  Lemma 2.6.

For any $\varepsilon>0$, the function $\sup_{z\in Z}\log
P^{\omega}_{Z}(\pi,z,f,\cdot,\varepsilon):\mathcal{F}(\mathbb{N})\to\mathbb{R}$
is right-subinvariant.

###### Proof..

Let $E\in\mathcal{F}(\mathbb{N})$ and $z\in Z$, suppose
$\mathcal{V}=\\{V_{1},\cdots,V_{p}\\}$ is an open cover of $\varphi^{-1}(z)$
with $\mathrm{diam}(V_{j},d_{E}^{\prime})\leq\varepsilon$, $j=1,\cdots,p$ and
$\mathcal{U}=\\{U_{1},\cdots,U_{q}\\}$ a class of open subsets in $X$ with
$\mathrm{diam}(U_{i},d_{E}^{\prime})\leq\varepsilon$, $i=1,\cdots,q$ such that

$\sum_{j=1}^{p}\left(\sum_{U_{ij}}e^{\sup_{U_{ij}}\mathbb{S}_{E}f}\right)^{\omega}\leq
P^{\omega}_{Z}(\pi,z,f,E,\varepsilon)+\delta,$

where $U_{ij}\in\mathcal{U}$ with $V_{j}\subset\bigcup_{U_{ij}}U_{ij}$ and
$\delta$ is a sufficiently small positive number.

Assume $m\in\mathbb{N}$, then $S^{-m}\mathcal{V}$ is a family of open subsets
of $Y$ with $\mathrm{diam}(S^{-m}V_{j},d^{\prime}_{E+m})$ and
$\varphi^{-1}(R^{-m}(z))$ and $T^{-m}\mathcal{U}$ a class of open subsets of
$X$ such that

$\displaystyle\sum_{S^{-m}V_{j}}\left(\sum_{T^{-m}U_{ij}}e^{\sup_{T^{-m}U_{ij}}\mathbb{S}_{E+m}f}\right)^{\omega}=\sum_{j=1}^{m}\left(\sum_{U_{ij}}e^{\sup_{U_{ij}}\mathbb{S}_{E}f}\right)^{\omega}.$

Since $\mathcal{V}$ and $\mathcal{U}$ are arbitrarily chosen, we obtain

$\log P^{\omega}_{Z}(\pi,R^{-m}(z),f,E+m,\varepsilon)\leq\log
P^{\omega}_{Z}(\pi,z,f,E,\varepsilon).$

As $R:Z\to Z$ is a surjective map, by taking supremum all over $z\in Z$ for
both sides of the above inequality, we have

$\sup_{z\in Z}\log P^{\omega}_{Z}(\pi,z,f,E+m,\varepsilon)\leq\sup_{z\in
Z}\log P^{\omega}_{Z}(\pi,z,f,E,\varepsilon).$

∎

Now, combine Lemma 2.5 and Lemma 2.6, by using Theorem 2.4, we introduce

######  Definition 2.7.

Let $\pi:(X,T)\to(Y,S)$ and $\varphi:(Y,S)\to(Z,R)$ be two factor maps and
$0\leq\omega\leq 1$. Given $f\in C(Y,\mathbb{R})$, the $\omega$-relative
weighted topological pressure with respect to $(Z,R)$ is defined by

$P^{\omega}_{Z}(\pi,T,f)=\lim_{\varepsilon\to
0}\left(\lim_{n\to\infty}\frac{\sup_{z\in Z}\log
P^{\omega}_{Z}(\pi,z,f,n,\varepsilon)}{n}\right).$

When $f\equiv 0$, we define the $\omega$-relative weighted topological entropy
with respect to $(Z,R)$ by

$h^{\omega}_{Z}(\pi,T)=\lim_{\varepsilon\to
0}\left(\lim_{n\to\infty}\frac{\sup_{z\in Z}\log
P^{\omega}_{Z}(\pi,z,0,n,\varepsilon)}{n}\right).$

######  Remark 2.8.

Let $\pi:(X,T)\to(Y,S)$ and $\varphi:(Y,S)\to(Z,R)$ be two factor maps between
TDSs and $\psi=\varphi\circ\pi:(X,T)\to(Z,R)$, we have

  1. 1.

When $Z=\\{*\\}$ a singleton, above $P_{Z}^{\omega}(\pi,z,f,N,\varepsilon)$
induces the $\omega$-weighted topological pressure which is defined in [18] by

$P^{\omega}_{*}(\pi,T,f)=\lim_{\varepsilon\to
0}\left(\lim_{n\to\infty}\displaystyle\frac{\log
P^{\omega}(\pi,*,f,n,\varepsilon)}{n}\right).$

  2. 2.

If $\omega=0$ and $f\equiv 0$, $P_{Z}^{\omega}(\pi,z,f,N,\varepsilon)$ induces
the topological conditional entropy of $\varphi$ by

$h_{top}(Y,S|Z,R)=\lim_{\varepsilon\to
0}\left(\lim_{n\to\infty}\frac{\sup_{z\in Z}\log
P_{Z}^{0}(\pi,z,f,n,\varepsilon)}{n}\right).$

One can see [6, Chapter 6] for more details.

Moreover, we state our main result for $\omega$-relative weighted topological
pressure.

######  Theorem 2.9.

(Variational principle for $\omega$-relative weighted topological pressure)
For any $0\leq\omega\leq 1$

$P^{\omega}_{Z}(\pi,T,f)=\sup_{\mu\in\mathcal{M}(X,T)}\left(\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\right).$

For convenience, we put

$P^{\omega}_{Z,var}(\pi,T,f)=\sup_{\mu\in\mathcal{M}(X,T)}\left(\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\right).$

In particular, when $f\equiv 0$, we can directly get a corollary for the
variational principle of $\omega$-relative weighted topological entropy.

######  Corollary 2.10.

(Variational principle for $\omega$-relative weighted topological entropy) For
any $0\leq\omega\leq 1,$

$h^{\omega}_{Z}(\pi,T)=\sup_{\mu\in\mathcal{M}(X,T)}\left(\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)\right).$

###  2.2. Basic Properties

In Theorem 2.2, we need to check that $P_{Z}^{\omega}(\pi,z,f)$ is a Borel
measurable function. Recall that a real-valued function $f$ defined on a
compact metric space $Z$ is called upper semi-continuous  (written as u.s.c)
if the set $\\{z\in Z:f(z)<r\\}$ is open. Besides, every u.s.c function $f$ of
$Z$ is Borel measurable.

To prove the upper semi-continuous property, we need a fundamental result of
general topology.

######  Lemma 2.11.

([12, Theorem 3.10]) Let $\pi:X\to Y$ be a surjective continuous map between
compact metric spaces, then $\pi$ is a closed map if and only if given an open
subset $U$ of $X$, the union of all members of fibers of $\pi$ which are
subsets of $U$ is open.

######  Proposition 2.12.

(upper semi-continuous) Let $f\in C(X,\mathbb{R})$,
$P_{Z}^{\omega}(\pi,z,f,n,\varepsilon)$ is the quantity defined abpve, then

  1. 1.

the function $z\mapsto P_{Z}^{\omega}(\pi,z,f,n,\varepsilon)$ is u.s.c for any
$n\in\mathbb{N}$ and $z\in Z$, and hence the function $z\mapsto
P_{Z}^{\omega}(\pi,z,f)$ of $Z$ to $\mathbb{R}$ is Borel measurable.

  2. 2.

For each $\varepsilon>0$ there is a constant $C(\varepsilon)>0$ such that

$\displaystyle\frac{1}{n}\log
P_{Z}^{\omega}(\pi,z,f,n,\varepsilon)<C(\varepsilon)\text{ for all }n\geq
1\text{ and }z\in Z.$

###### Proof..

1\. Suppose $P_{Z}^{\omega}(\pi,z,f,n,\varepsilon)<C(z,\varepsilon)$ for some
positive number $C(z,\varepsilon)>0$, then there is an open cover
$\mathcal{V}=\\{V_{1},\cdots,V_{p}\\}$ of $\varphi^{-1}(z)$ with
$\mathrm{diam}(V_{i},d^{\prime}_{n})<\varepsilon$, $i=1,\cdots,p.$ It is clear
that $\pi:(Y,S)\to(Z,R)$ is a closed map, so by Lemma 2.11, there is an open
subset $W$ of $V_{1}\cup\cdots\cup V_{p}$ consisting of fibers of $\pi$ and
$\varphi(W)$ is also an open subset of $Z$111Every closed continuous map is a
quotient map, see [12] Chapter 3.. Hence, for any $z\in W$
$P_{Z}^{\omega}(\pi,z,f,n,\varepsilon)<C(z,\varepsilon)$, which means
$z\mapsto P_{Z}^{\omega}(\pi,z,f,n,\varepsilon)$ is upper semi-continuous and
the function $z\mapsto P_{Z}^{\omega}(\pi,z,f)$ from $Z$ to $\mathbb{R}$ is
Borel measurable.

2\. Let $N(\varepsilon,Y)$ be the smallest number of open sets of diameter
$\varepsilon$ required to cover $Y$ and $N(\varepsilon,X)$ the smallest number
of open sets of diameter $\varepsilon$ required to cover $X$. Then

$\displaystyle
P^{\omega}_{Z}(\pi,z,f,n,\varepsilon)\leq\sum_{j=1}^{N(\varepsilon,Y)^{n}}\left(\sum_{i=1}^{N(\varepsilon,X)^{n}}e^{n||f||}\right)^{\omega}\text{
for all }z\in Z.$

Hence, by letting $C(\varepsilon)=\log N(\varepsilon,Y)+\omega\log
N(\varepsilon,X)+||f||$ we obtain

$\displaystyle\frac{1}{n}\log
P_{Z}^{\omega}(\pi,z,f,n,\varepsilon)<C(\varepsilon)\text{ for all }n\geq
1\text{ and }z\in Z.$

∎

######  Proposition 2.13.

Let $\pi:(X_{1},T_{1})\to(X_{2},T_{2})$ and
$\varphi:(X_{2},T_{2})\to(X_{3},T_{3})$ with $\psi=\varphi\circ\pi$ and
$k\in\mathbb{N}$. For any $z\in X_{3},$ we have

$P_{Z}^{\omega}(\pi,T_{1}^{k},z,\mathbb{S}^{T_{1}}_{k}f)=kP_{Z}^{\omega}(\pi,T_{1},z,f),$

and

$P_{Z}^{\omega}(\pi,T_{1}^{k},\mathbb{S}^{T_{1}}_{k}f)=kP_{Z}^{\omega}(\pi,T_{1},f).$

###### Proof..

Let $d^{1},d^{2}$ be compatible metrics on $X_{1},X_{2}$, respectively. For
any $\varepsilon>0$, there is a $0<\delta<\varepsilon$ such that
$d^{i,T_{i}}_{k}(w^{i}_{1},w^{i}_{2})<\varepsilon$ whenever
$d^{i,T_{i}}(w^{i}_{1},w^{i}_{2})<\delta$ $(i=1,2)$, where
$(w^{i}_{1},w^{i}_{2})\in X_{i}\times X_{i}$. Then for any $n\in\mathbb{N}$,
$d_{kn}^{i,T_{i}}(w^{i}_{1},w^{i}_{2})<\varepsilon$ whenever
$d_{n}^{i,T^{k}_{i}}(w^{i}_{1},w^{i}_{2})<\delta$. Therefore, if
$\mathcal{V}_{2}$ is an open cover of $\varphi^{-1}(z)$ with
$d_{n}^{2,T_{2}^{k}},(\mathcal{V}_{2})<\delta,$ then
$d^{2,T_{2}}_{kn}(\mathcal{V}_{2})<\varepsilon$. Hence,

$P^{\omega}(\pi,T_{1},z,f,kn,\varepsilon)\leq
P^{\omega}(\pi,T_{1}^{k},z,\mathbb{S}_{k}^{T_{1}}f,n,\delta).$

It is obvious that

$P^{\omega}(\pi,T_{1},z,\mathbb{S}_{k}^{T_{1}}f,n,\varepsilon)\leq
P^{\omega}(\pi,T_{1},z,f,kn,\varepsilon).$

So,

$P_{Z}^{\omega}(\pi,T_{1}^{m},z,\mathbb{S}^{T_{1}}_{k}f)=kP_{Z}^{\omega}(\pi,T_{1},z,f),$

and

$P_{Z}^{\omega}(\pi,T_{1}^{k},\mathbb{S}^{T_{1}}_{k}f)=kP_{Z}^{\omega}(\pi,T_{1},f).$

∎

######  Proposition 2.14.

Let $(X_{i},T_{i})$ $(i=1,2,3)$ and $(Z,R)$ be TDSs admitting the following
commutative diagram:

(2.11)

Then for any $z\in Z$ and $f\in C(X_{2},\mathbb{R})$,

$P^{\omega}_{Z}(\pi_{1},T_{2},z,f)\leq
P^{\omega}_{Z}(\pi_{1}\circ\pi_{2},T_{3},z,f\circ\pi_{2}),$

and

$P^{\omega}_{Z}(\pi_{1},T_{2},f)\leq
P^{\omega}_{Z}(\pi_{1}\circ\pi_{2},T_{3},f\circ\pi_{2}).$

###### Proof..

Let $d^{i}$ be compatible metrics on $X_{i}$, where $i=1,2,3$. To prove the
inequality, let $\varepsilon>0$, there is a $0<\delta<\varepsilon$ such that
$d^{3}(w_{1}^{3},w_{2}^{3})<\delta$ implies that
$d^{2}(\pi_{2}(w_{1}^{3}),\pi_{2}(w_{2}^{3}))<\varepsilon,$ where
$(w^{3}_{1},w^{3}_{2})\in X_{3}\times X_{3}.$ Then for any $n>0$,

$d_{n}^{3}(w_{1}^{3},w_{2}^{3})<\delta\Rightarrow
d_{n}^{2}(\pi_{2}(w_{1}^{3}),\pi_{2}(w_{2}^{3}))<\varepsilon.$

Hence, for any $\Omega\subset X_{3}$, we have

$P(\pi_{2}(\Omega),T_{2},f,n,\varepsilon)\leq
P(\Omega,T_{3},f\circ\pi_{2},n,\delta).$

Let $\mathcal{V}_{1}=\\{V_{1},\cdots,V_{p}\\}$ be a family of open subsets
with $d^{1}_{n}(V_{i})<\delta$ covering $\varphi^{-1}(z)$. Then for any
$i=1,\cdots,p$

$P(\pi_{1}^{-1}(V_{i}),T_{2},f,n,\varepsilon)\leq
P(\pi_{2}^{-1}(\pi_{1}^{-1}(V_{i})),T_{3},f\circ\pi_{2},n,\delta).$

Therefore,

$P_{Z}^{\omega}(\pi_{1},T_{2},f,n,\varepsilon)\leq
P_{Z}^{\omega}(\pi_{1}\circ\pi_{2},T_{2},f\circ\pi_{2},n,\delta).$

So,

$P^{\omega}_{Z}(\pi_{1},T_{2},z,f)\leq
P^{\omega}_{Z}(\pi_{1}\circ\pi_{2},T_{3},z,f\circ\pi_{2}),$

and

$P^{\omega}_{Z}(\pi_{1},T_{2},f)\leq
P^{\omega}_{Z}(\pi_{1}\circ\pi_{2},T_{3},f\circ\pi_{2}).$

∎

The following property is a relative version of the Lemma 2.4 in [18].

######  Proposition 2.15.

Assume that the following solid line commutative diagram exists among the
dynamical systems $(X,T)$, $(Y,S)$, $(Y^{\prime},S^{\prime})$ and $(Z,R)$:

(2.18)

Then there is a dynamical system $(X^{\prime},T^{\prime})$ satisfying the
commutative diagram as above such that given $f\in C(X,\mathbb{R})$ for any
$z\in Z$,

$P^{\omega}_{Z}(\pi,T,z,f)\leq P^{\omega}_{Z}(\Pi,T^{\prime},z,f\circ\eta)$

and

$P^{\omega}_{Z}(\pi,T,f)\leq P^{\omega}_{Z}(\Pi,T^{\prime},f\circ\eta).$

###### Proof..

We define $X^{\prime}$ as

$X^{\prime}=X\times_{Y}Y^{\prime}=\\{(x,y)\in X\times
Y^{\prime}|\pi(x)=\xi(y)\\},$

and $(X\times_{Y}Y^{\prime},T\times S^{\prime})$ is a topological dynamical
system with the action $(T\times S^{\prime})(x,y)=(Tx,S^{\prime}y)$, The
mappings $\pi$ and $\Pi$ are projections from $X^{\prime}$ to $X$ and
$Y^{\prime}$, respectively.

Let $d$, $d^{\prime}$ and $\widetilde{d}$ be metrics on $X$, $Y$ and
$Y^{\prime}$, respectively. We define a metric $\rho$ on $X^{\prime}$ by

$\rho((x_{1},y_{1}),(x_{2},y_{2}))=\max(d(x_{1},x_{2}),\widetilde{d}(y_{1},y_{2})).$

For any $A\subset Y^{\prime}$, we have

$\pi^{-1}(\xi(A))=\eta(\Pi^{-1}(A)).$

Since $\rho((x_{1},y_{1}),(x_{2},y_{2}))<\varepsilon$ implies
$d(x_{1},x_{2})<\varepsilon$, we have

$P(\eta(\Omega),T,f,n,\varepsilon)\leq P(\Omega,T\times
S^{\prime},f\circ\eta,n,\varepsilon),$

for any $\Omega\subset X^{\prime}$. Therefore, for any $A\subset Y^{\prime}$,

$\displaystyle P(\pi^{-1}\xi(A),T,f,n,\varepsilon)$
$\displaystyle=P(\eta(\Pi^{-1}(A)),T,f,n,\varepsilon)$ $\displaystyle\leq
P(\Pi^{-1}(A),T,f\circ\eta,n,\varepsilon).$

For any $\varepsilon>0$, there is $0<\delta<\varepsilon$ such that
$\widetilde{d}(y_{1},y_{2})<\delta$ implies
$d^{\prime}(\xi(y_{1}),\xi(y_{2}))<\varepsilon$.

Now, for a fixed $z\in Z$, we claim that

$P_{Z}^{\omega}(\pi,T,z,f,n,\varepsilon)\leq P_{Z}^{\omega}(\Pi,T\times
S^{\prime},z,f\circ\eta,n,\delta).$

Take a positive number $C$ with

$P_{Z}^{\omega}(\Pi,T\times S^{\prime},z,f\circ\eta,n,\delta)<C.$

Then there is an open cover $\phi^{-1}(z)\subset V_{1}\cup\cdots\cup V_{p}$
such that $V_{i}\subset Y^{\prime}$ and
$\widetilde{d}_{N}^{S^{\prime}}(V_{i})<\delta$ for all $1\leq i\leq p$, and

$\sum_{i=1}^{n}(P(\Pi^{-1}(V_{i}),T\times
S^{\prime},f\circ\eta,\delta))^{\omega}<C.$

We can find compact subsets $A_{i}\subset V_{i}$ satisfying
$\phi^{-1}(z)\subset A_{1}\cup\cdots\cup A_{p}$, and

$\displaystyle\sum_{i=1}^{n}(P(\pi^{-1}\xi(A_{i})),T,f,n,\varepsilon)^{\omega}$
$\displaystyle\leq\sum_{i=1}^{n}(P(\Pi^{-1}(A_{i})),T\times
S^{\prime},f\circ\eta,n,\varepsilon)^{\omega}$
$\displaystyle\leq\sum_{i=1}^{n}(P(\Pi^{-1}(A_{i})),T\times
S^{\prime},f\circ\eta,n,\delta)^{\omega}$
$\displaystyle\leq\sum_{i=1}^{n}(P(\Pi^{-1}(V_{i})),T\times
S^{\prime},f\circ\eta,n,\varepsilon)^{\omega}<C.$

Since each $\xi(A_{i})$ is a compact subset of $Y$ with
$d_{N}^{{}^{\prime}S}(\xi(A_{i}))<\varepsilon$, there are open subsets
$\xi(A_{i})\subset W_{i}$ covering $\psi^{-1}(z)$ with
$d_{N}^{{}^{\prime}S}(W_{i})<\varepsilon$ for $1\leq i\leq p$ and

$\sum_{i=1}^{n}(P(\pi^{-1}(W_{i})),T,f,n,\varepsilon)^{\omega}<C.$

Since $C$ is taken arbitrarily, we have

$P_{Z}^{\omega}(\pi,T,z,f,n,\varepsilon)\leq P_{Z}^{\omega}(\Pi,T\times
S^{\prime},z,f\circ\eta,n,\delta).$

Thus,

$P^{\omega}_{Z}(\pi,T,z,f)\leq P^{\omega}_{Z}(\Pi,T^{\prime},z,f\circ\eta)$

and

$P^{\omega}_{Z}(\pi,T,f)\leq P^{\omega}_{Z}(\Pi,T^{\prime},f\circ\eta).$

∎

##  3\. Conditional measure-theoretic entropy

Recall that $\mathcal{M}(X)$ denote the family of all Borel probability
measure on $X$ and $\mathcal{M}(X,T)$ $(E(X,T))\subset\mathcal{M}(X)$ the
class of $T$-invariant (ergodic) probability measures on $X$, let
$\mathcal{A}\in\mathcal{P}_{X}$ and $\mu\in\mathcal{M}(X)$, recall the
information entropy of $\alpha$ with respect to $\mu$ is defined by

$H_{\mu}(\mathcal{A})=\sum_{A\in\alpha}-\mu(A)\log\mu(A).$

In this article, we assume $\mu(\varnothing)\log\mu(\varnothing)=0$ for empty
set.

Let $\mu$ be a probability measure on $X$ (not necessarily $T$-invariant) and
a measurable partition $\mathcal{A}\in\mathcal{P}_{X}$. For each
$A\in\mathcal{A}$ with $\mu(A)>0$, it induce a probability measure

$\mu_{A}(B):=\frac{\mu(A\cap B)}{\mu(A)}.$

Let $\mathcal{B}$ be another measurable partition of $X$, the conditional
entropy of $\mathcal{B}$ with respect to $\mathcal{A}$ is defined by

$H_{\mu}(\mathcal{B}|\mathcal{A})=\sum_{A\in\mathcal{A}}\mu(A)H_{\mu_{A}}(\mathcal{B})=\int_{X}H_{\mu_{A(x)}}(\mathcal{B})d\mu(x).$

The following properties are well-known (see e.g. [15],[11]).

######  Proposition 3.1.

Let $(X,\mathcal{B}_{X},\mu)$ be a measure space, $\mathcal{A},\mathcal{B}$
and $\mathcal{C}$ are measurable partitions with
$H_{\mu}(\mathcal{B}|\mathcal{A}),H_{\mu}(\mathcal{C}|\mathcal{A})<\infty$.

  1. 1.

If $\mathcal{B}\preceq\mathcal{C}$, then $H_{\mu}(\mathcal{B}|\mathcal{A})\leq
H_{\mu}(\mathcal{C}|\mathcal{A})$;

  2. 2.

$H_{\mu}(\mathcal{B}\vee\mathcal{C}|\mathcal{A})=H_{\mu}(\mathcal{B}|\mathcal{A})+H_{\mu}(\mathcal{C|\mathcal{B}}\vee\mathcal{A})$;

  3. 3.

$H_{\mu}(\mathcal{C}|\mathcal{A})\leq
H_{\mu}(\mathcal{C}|\mathcal{B})+H_{\mu}(\mathcal{B}|\mathcal{A})$;

  4. 4.

if $\mathcal{A}\preceq\mathcal{C}$, then $H_{\mu}(\mathcal{B}|\mathcal{C})\leq
H_{\mu}(\mathcal{B}|\mathcal{A}).$

Now, let $(X,T)$ and $(Y,S)$ be two TDSs and $\pi:X\to Y$ a factor map between
them, we proceed by defining the conditional metric entropy of
$\mu\in\mathcal{M}(X,T)$ relative $(Y,S)$.

Put $\epsilon_{Y}$ the partition of $Y$ into points. Let
$\mu\in\mathcal{M}(X,T)$, then in the Lebesgue space $(X,\mu)$, the measurable
partition $\pi^{-1}(\epsilon_{Y})$ generates the $\sigma$-algebra
$\pi^{-1}(\mathcal{B}(Y))$ which is also $T$-invariant. We simply denote by
$\mu_{x}=\mu_{x}^{\pi^{-1}\epsilon_{Y}}$ the conditional measures uniquely
determined by the measurable partition $\pi^{-1}\epsilon_{Y}$ for $\mu-$a.e.
$x\in X$, we write $\mu_{y}:=\mu_{x}$ if $\pi(x)=y$, let
$\mu=\int_{Y}\mu_{y}d\nu(y)$ be the disintegration of $\mu$ over $\nu$ where
$\nu=\pi\mu$ and $\mu_{y}$ is supporting on $\pi^{-1}(y)$. Given
$\alpha\in\mathcal{P}_{X}$, the conditional entropy of $\alpha$ relative to
$Y$ is defined by

$H_{\mu}(\alpha|Y)=\int_{Y}H_{\mu_{y}}(\alpha)d\nu(y).$

If $\mu\in\mathcal{M}(X,T)$, the conditional entropy of $\alpha$ relative to
$(Y,S)$ is defined by

$h_{\mu}(T,\alpha|Y)=\lim_{n\to\infty}\frac{1}{n}H_{\mu}(\alpha^{n-1}_{0}|Y),$

and

$h_{\mu}(T|S)=\sup\\{h_{\mu}(T,\alpha|Y)|\alpha\in\mathcal{P}_{X}\\}.$

To end this section, we include some basic properties. The following results
are well-known.

######  Theorem 3.2.

(Rohlin-Abramov theorem, see [5, Proposition 3.7] and [13, Lemma 3.1]) Let
$\pi:(X,\mathcal{B}_{X},\mu,T)\to(Y,\mathcal{B}_{Y},\nu,S)$ and
$\phi:(Y,\mathcal{B}_{Y},\nu,S)\to(Z,\mathcal{B}_{Z},\kappa,R)$ be two
measure-preserving maps, then

$h_{\mu}(T|R)=h_{\mu}(T|S)+h_{\nu}(S|R).$

######  Lemma 3.3.

([13, Lemma 3.2]) Let $\pi:(X,T)\to(Y,S)$ be a factor map betwwen TDSs,

  1. 1.

Let $\mathcal{A}$ be a partition of $X$ and let $\mu_{1},\mu_{2}$ be Borel
probability measures on $\mathcal{B}_{X}$. If $0\leq\omega\leq 1$ then

$\omega H_{\mu_{1}}(\mathcal{A}|Y)+(1-\omega)H_{\mu_{2}}(\mathcal{A}|Y)\leq
H_{\omega\mu_{1}+(1-\omega)\mu_{2}}(\mathcal{A}|Y).$

  2. 2.

The function $\mu\mapsto h_{\mu}(T|S)$ is affine from $\mathcal{M}(X,T)$ to
$\mathbb{R}.$

######  Lemma 3.4.

([20, Lemma 9.9]) Let $c_{i}\in\mathbb{R}$, $p_{i}\geq 0$, $i=1,\cdots,m,$ and
$\sum_{i=1}^{m}p_{i}=1$, then we have

$\sum_{i=1}^{m}p_{i}(c_{i}-\log p_{i})\leq\log\sum_{i=1}^{m}e^{c_{i}},$

and the equality holds iff $p_{i}=e^{c_{i}}/\sum_{i=1}^{m}e^{c_{i}}$ for all
$i=1,\cdots,m.$

##  4\. Variational principle for $\omega$-relative weighted pressure

The purpose of this section is to prove the Theorem 2.9 and Theorem 2.2. The
zero-dimensional principal extension also plays an important role in the
following discussion.

###  4.1. Proof of one side of Theorem 2.2 and Theorem 2.9

######  Proposition 4.1.

Let $\pi:(X,T)\to(Y,S)$ and $\varphi:(Y,S)\to(Z,R)$ be factor maps with
$\psi=\varphi\circ\pi$, for any $0\leq\omega\leq 1$, $f\in C(X,\mathbb{R})$
and $\mu\in\mathcal{M}(X,T),$

  1. 1.

We obtain

$\omega h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\leq
P^{\omega}_{Z}(\pi,T,f),$

that is,

$P^{\omega}_{Z,var}(\pi,T,f)\leq P^{\omega}_{Z}(\pi,T,f).$

  2. 2.

Moreover, if $\kappa=\psi\mu\in\mathcal{M}(Z,R)$, then

$\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\leq\int_{Z}P_{Z}^{\omega}(\pi,z,f)d\kappa(z)$

###### Proof..

1\. We use a similar approach as in [18] by applying amplification trick. That
is, we shall prove that there is a constant $C>0$ such that for any
$k\in\mathbb{N}$,

$\omega
h_{\mu}(T^{k}|R^{k})+(1-\omega)h_{\pi\mu}(S^{k}|R^{k})+\omega\int_{X}\mathbb{S}^{T}_{k}fd\mu\leq
P^{\omega}_{Z}(\pi,T^{k},\mathbb{S}_{k}^{T}f)+C.$

Then by Proposition 2.13, we have

$\omega h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\leq
P^{\omega}_{Z}(\pi,T,f)+\frac{C}{k}.$

The conclusion holds as $k\to\infty$.

Let $\mathcal{A}$ be a measurable partition of $Y$ and $\mathcal{B}$ a
partition of $X$, we shall verify that there is a $C>0$ such that

$\omega
h_{\mu}(T,\mathcal{B}|Z)+(1-\omega)h_{\pi\mu}(S,\mathcal{A}|Z)+\omega\int_{X}fd\mu\leq
P^{\omega}_{Z}(\pi,T,f)+C.$

For each $n\in\mathbb{N}$, we denote
$\mathcal{A}^{n}=\bigvee_{i=0}^{n-1}T^{-i}\mathcal{A}=\\{A_{1}^{n},\cdots,A_{\alpha}^{n}\\}$.
Given $z\in Z$, put $\mathcal{A}^{n,z}$ the partition of $\varphi^{-1}(z)$ and
the element in which has the form $A_{i}^{n,z}:=A_{i}^{n}\cap\varphi^{-1}(z)$.
Take a compact subset $C_{i}^{n,z}\subset A_{i}^{n,z}$ for each $1\leq
i\leq\alpha$ satisfying

$\sum_{i=1}^{\alpha}\nu_{z}(A_{i}^{n,z}\backslash
C_{i}^{n,z})<\frac{1}{\log\alpha},$

where $\nu_{z}=\pi\mu|_{\varphi^{-1}(z)}$. We put
$C^{n,z}_{0}=Y\backslash(C^{n,z}_{1}\cup\cdots\cup C^{n,z}_{\alpha})$ and
$\mathcal{C}^{n,z}=\\{C^{n,z}_{0},C^{n,z}_{1},\cdots,C^{n,z}_{\alpha}\\}.$

######  Claim 4.2.

For each $n\in\mathbb{N}$, the class of partitions $\\{\mathcal{C}^{n,z}|z\in
Z\\}$ satisfying

$\displaystyle
H_{\nu}(S,\mathcal{A}^{n}|Z)\leq\int_{Z}H_{\nu_{z}}(\mathcal{C}^{n,z})d\kappa_{z}+1,$
(4.1)

where $\kappa=\psi\mu$. Hence,

$h_{\nu}(S,\mathcal{A}|Z)=\lim_{n\to\infty}\frac{1}{n}H_{\nu}(S,\mathcal{A}^{n}|Z)\leq\liminf_{n\to\infty}\frac{1}{n}\int_{Z}H_{\nu_{z}}(\mathcal{C}^{n,z})d\kappa_{z}.$

###### Proof..

By Proposition 3.1 we notice that

$H_{\nu_{y}}(\mathcal{A}^{n})\leq
H_{\nu_{y}}(\mathcal{C}^{n,z})+H_{\nu_{y}}(\mathcal{A}^{n}|\mathcal{C}^{n,z}),$

and

$H_{\nu_{y}}(\mathcal{A}^{n}|\mathcal{C}^{n,z})=\nu_{y}(C_{0})\sum_{i=1}^{\alpha}\left(-\frac{\nu_{y}(A^{n,z}_{i}\cap
C_{0}^{n,z})}{\nu_{y}(C_{0}^{n,z})}\log\frac{\nu_{y}(A^{n,z}_{i}\cap
C_{0}^{n,z})}{\nu_{y}(C_{0}^{n,z})}\right)\leq\nu_{y}(C_{0})\log\alpha.$

Therefore, (4.1) is obtained. ∎

Consider $\mathcal{B}^{n,z}=\mathcal{B}^{n}\vee\pi^{-1}(\mathcal{C}^{n,z}),$
which is a partition of $\psi^{-1}(z)$ and has the form

$\mathcal{B}^{n,z}\vee\pi^{-1}(\mathcal{C}^{n,z})=\\{B^{n,z}_{ij}|0\leq
i\leq\alpha,1\leq j\leq\beta_{i}\\},$

and

$\pi^{-1}(C_{i}^{n,z})=\bigcup_{j=1}^{\beta_{i}}B^{n,z}_{ij}\text{ }(0\leq
i\leq\alpha).$

For each $B^{n,z}_{ij}$ $(0\leq i\leq\alpha,1\leq j\leq\beta_{i})$, we take a
compact subset $D^{n,z}_{ij}\subset B^{n,z}_{ij}$ such that

$\sum_{i=0}^{\beta_{i}}\log\beta_{i}\left(\sum_{j=1}^{\beta_{i}}\mu_{z}(B^{n,z}_{ij}\backslash
D^{n,z}_{ij})\right)<1.$

Put

$D_{i0}^{n,z}=\pi^{-1}(C_{i}^{n,z})\backslash\bigcup_{j=1}^{\beta_{i}}B_{ij}^{n,z}\text{
}(0\leq i\leq\alpha).$

Define

$\mathcal{D}^{n,z}=\\{D_{ij}^{n,z}|0\leq i\leq\alpha,0\leq j\leq\beta_{i}\\}.$

######  Claim 4.3.

For each $n\in\mathbb{N}$, $\mathcal{D}^{n,z}$ is a partition of
$\psi^{-1}(z)$ and the class $\\{\mathcal{D}^{n,z}|z\in Z\\}$ satisfying

$H_{\mu}(T,\mathcal{B}^{n}|Z)\leq\int_{Z}H_{\mu_{z}}(\mathcal{D}^{n,z})d\kappa_{z}+1.$

Hence,

$h_{\mu}(T,\mathcal{B}|Z)=\lim_{n\to\infty}\frac{1}{n}H_{\mu}(T,\mathcal{B}^{n}|Z)\leq\liminf_{n\to\infty}\frac{1}{n}\int_{Z}H_{\mu_{z}}(\mathcal{D}^{n,z})d\kappa_{z}.$

###### Proof..

$\pi^{-1}(\mathcal{C}^{n,z})\preceq\mathcal{D}^{n,z}$ is obvious. By the
construction,

$\displaystyle H_{\mu}(T,\mathcal{B}^{n}|Z)$
$\displaystyle=\int_{Z}H_{\mu_{z}}(\mathcal{B}^{n,z})d\kappa_{z}$
$\displaystyle\leq\int_{Z}H_{\mu_{z}}(\mathcal{B}^{n,z}\vee\pi^{-1}(\mathcal{C}^{n,z}))d\kappa_{z}$
$\displaystyle\leq\int_{Z}H_{\mu_{z}}(\mathcal{D}^{n,z})+H_{\mu_{z}}(\mathcal{B}^{n,z}\vee\pi^{-1}(\mathcal{C}^{n,z})|\mathcal{D}^{n,z})d\kappa_{z}.$

Since $D_{ij}^{n,z}\subset B_{ij}^{n,z}$ for $0\leq i\leq\alpha$ and $1\leq
j\leq\beta_{i}$, hence,

$\displaystyle
H_{\mu_{z}}(\mathcal{B}^{n,z}\vee\pi^{-1}(\mathcal{C}^{n,z})|\mathcal{D}^{n,z})$
$\displaystyle=\sum_{i=0}^{\alpha}\mu(D^{n,z}_{i0})\sum_{j=1}^{\beta_{i}}\left(-\frac{\mu_{z}(D^{n,z}_{i0}\cap
B^{n,z}_{ij})}{\mu_{z}(D^{n,z}_{i0})}\log\frac{\mu_{z}(D^{n,z}_{i0}\cap
B^{n,z}_{ij})}{\mu_{z}(D^{n,z}_{i0})}\right)$
$\displaystyle\leq\sum_{i=0}^{\alpha}\mu_{z}(D_{ij}^{n,z})\log\beta_{i}$
$\displaystyle<1.$

Thus, the conclusion holds. ∎

Foe each $n\in\mathbb{N}$ and $z\in Z$, let $\mathcal{C}^{n,z}$ and
$\mathcal{D}^{n,z}$ be the partitions defined as above. From the definition of
conditional entropy, by Claim 4.2 and 4.3 we shall prove

$\displaystyle\omega\liminf_{n\to\infty}\frac{1}{n}\int_{Z}H_{\mu_{z}}(\mathcal{D}^{n,z})d\kappa_{z}+(1-\omega)\liminf_{n\to\infty}\frac{1}{n}\int_{Z}H_{\nu_{z}}(\mathcal{C}^{n,z})d\kappa_{z}+\omega\int_{X}fd\mu$
$\displaystyle\leq P^{\omega}_{Z}(\pi,T,f)+C,$

for some $C>0$. So by Claim 4.2 and 4.3, we have

$\omega
h_{\mu}(T,\mathcal{B}|Z)+(1-\omega)h_{\nu}(S,\mathcal{A}|Z)+\omega\int_{X}fd\mu\leq
P^{\omega}_{Z}(\pi,T,f)+C.$

For any $f\in C(X,\mathbb{R})$ and $n\in\mathbb{N}$, by measure decomposition,

$\int_{X}fd\mu=\int_{Z}\int_{X}\mathbb{S}_{n}fd\mu_{z}d\kappa(z),$

and,

$\displaystyle\omega\liminf_{n\to\infty}\frac{1}{n}\int_{Z}H_{\mu_{z}}(\mathcal{D}^{n,z})d\kappa_{z}+(1-\omega)\liminf_{n\to\infty}\frac{1}{n}\int_{Z}H_{\nu_{z}}(\mathcal{C}^{n,z})d\kappa_{z}+\omega\int_{X}fd\mu$
$\displaystyle\leq\liminf_{n\to\infty}\frac{1}{n}\left(\int_{Z}\left(\omega
H_{\mu_{z}}(\mathcal{D}^{n,z})+(1-\omega)H_{\nu_{z}}(\mathcal{C}^{n,z})+\omega\int_{X}\mathbb{S}_{n}fd\mu_{z}\right)d\kappa(z)\right).$

Notice that for each $z\in Z$, $\nu_{z}=\pi\mu_{z}$, and from the construction
of $\mathcal{D}^{n,z}$ and $\mathcal{C}^{n,z}$,
$\pi^{-1}(\mathcal{C}^{n,z})\preceq\mathcal{D}^{n,z}$. Hence

$\displaystyle\omega
H_{\mu_{z}}(\mathcal{D}^{n,z})+(1-\omega)H_{\nu_{z}}(\mathcal{C}^{n,z})$
$\displaystyle=\omega
H_{\mu_{z}}(\mathcal{D}^{n,z})+(1-\omega)H_{\mu_{z}}(\pi^{-1}(\mathcal{C}^{n,z}))$
$\displaystyle=H_{\nu_{z}}(\mathcal{C}^{n,z})+\omega
H_{\mu_{z}}(\mathcal{D}^{n,z}|\pi^{-1}(\mathcal{C}^{n,z})).$

Therefore,

$\displaystyle\omega\liminf_{n\to\infty}\frac{1}{n}\int_{Z}H_{\mu_{z}}(\mathcal{D}^{n,z})d\kappa_{z}+(1-\omega)\liminf_{n\to\infty}\frac{1}{n}\int_{Z}H_{\nu_{z}}(\mathcal{C}^{n,z})d\kappa_{z}+\omega\int_{X}fd\mu$
$\displaystyle\leq\liminf_{n\to\infty}\frac{1}{n}\left(\int_{Z}\left(\underbrace{H_{\nu_{z}}(\mathcal{C}^{n,z})+\omega
H_{\mu_{z}}(\mathcal{D}^{n,z}|\pi^{-1}(\mathcal{C}^{n,z}))+\omega\int_{X}\mathbb{S}_{n}fd\mu_{z}}_{\rm{(I)}}\right)d\kappa(z)\right).$

For $C\in\mathcal{C}^{n,z}$, we define

$\mathcal{D}^{n,z}_{C}=\\{D\in\mathcal{D}^{n,z}|D\cap\pi^{-1}(C)\neq\varnothing\\}=\\{D\in\mathcal{D}^{n,z}|D\subset\pi^{-1}(C)\\}.$

Also,

$\pi^{-1}(C)=\bigsqcup_{D\in\mathcal{D}^{n,z}_{C}}D.$

For each $C\in\mathcal{C}^{n,z}$ with $\nu_{z}(C)>0$, and
$D\in\mathcal{D}_{C}^{n,z}$, we write

$\mu_{z}(D|C)=\frac{\mu_{z}(D)}{\mu_{z}(\pi^{-1}(C))}.$

It is clear that

$\sum_{D\in\mathcal{D}_{C}^{n,z}}\mu_{z}(D|C)=1.$

######  Claim 4.4.

For each $z\in Z$ and $n\in\mathbb{N}$, we obtain a upper bound for
$\rm{(1)}$, that is,

$H_{\nu_{z}}(\mathcal{C}^{n,z})+\omega
H_{\mu_{z}}(\mathcal{D}^{n,z}|\pi^{-1}(\mathcal{C}^{n,z}))+\omega\int_{X}\mathbb{S}_{n}fd\mu_{z}\leq\log\sum_{C\in\mathcal{C}^{n,z}}\left(\sum_{D\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D}\mathbb{S}_{n}f}\right)^{\omega}.$

###### Proof..

First,

$\displaystyle\int_{X}\mathbb{S}_{n}fd\mu_{z}$
$\displaystyle=\sum_{D\in\mathcal{D}^{n,z}}\int_{D}\mathbb{S}_{n}fd\mu_{z}\leq\sum_{D\in\mathcal{D}^{n,z}}\mu_{z}(D)\sup_{D}\mathbb{S}_{n}f$
$\displaystyle=\sum_{C\in\mathcal{C}^{n,z}}\nu_{z}(C)\left(\sum_{C\in\mathcal{D}_{C}^{n,z}}\mu_{z}(D|C)\sup_{D}\mathbb{S}_{n}f\right).$

where $\nu_{z}(C)>0$. Hence,

$\displaystyle
H_{\mu_{z}}(\mathcal{D}^{n,z}|\pi^{-1}(\mathcal{C}^{n,z}))+\int_{X}\mathbb{S}_{n}fd\mu_{z}$
$\displaystyle\leq\sum_{C\in\mathcal{C}^{n,z}}\nu_{z}(C)\left(\sum_{D\in\mathcal{D}_{C}^{n,z}}\left(-\mu_{z}(D|C)\log\mu_{z}(D|C)+\mu_{z}(D|C)\sup_{D}\mathbb{S}_{n}f\right)\right).$

By Lemma 3.4, we have $\rm{(II)}$,

$\sum_{D\in\mathcal{D}_{C}^{n,z}}\left(-\mu_{z}(D|C)\log\mu_{z}(D|C)+\mu_{z}(D|C)\sup_{D}\mathbb{S}_{n}f\right)\leq\log\sum_{D\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D}}\mathbb{S}_{n}f.$

By using Lemma 3.4 again. Therefore,

$\displaystyle H_{\nu_{z}}(\mathcal{C}^{n,z})+\omega
H_{\mu_{z}}(\mathcal{D}^{n,z}|\pi^{-1}(\mathcal{C}^{n,z}))+\omega\int_{X}\mathbb{S}_{n}fd\mu_{z}$
$\displaystyle\leq\sum_{C\in\mathcal{C}^{n,z}}\left(-\nu_{z}(C)\log\nu_{z}(C)+\nu_{z}(C)\log\left(\sum_{D\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D}}\mathbb{S}_{n}f\right)^{\omega}\right)$
$\displaystyle\leq\log\sum_{C\in\mathcal{C}^{n,z}}\left(\sum_{D\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D}}\mathbb{S}_{n}f\right)^{\omega}.$

∎

Let $d$ and $d^{\prime}$ be metrics on $X$ and $Y$, respectively. For any
$n\in\mathbb{N}$ and $z\in Z$, since $C^{n,z}_{i}$ $(1\leq i\leq\alpha)$ are
mutually disjoint compact subset of $\varphi^{-1}(z)$ and $D^{n,z}_{ij}$
$(0\leq i\leq\alpha,1\leq j\leq\beta_{i})$ are mutually disjoint compact
subsets of $X$. Hence, there is a $\varepsilon(n,z)>0$ such that:

  1. 1.

for any $y\in C_{i}^{n,z}$ and $y^{\prime}\in C_{i^{\prime}}^{n,z}$ with
$i\neq i^{\prime}$,

$\varepsilon(n,z)<d^{\prime}(y,y);$

  2. 2.

for any $x\in D_{ij}^{n,z}$ and $x^{\prime}\in D_{ij^{\prime}}^{n,z}$ with
$j\neq j^{\prime}$,

$\varepsilon(n,z)<d(x,x^{\prime}).$

######  Claim 4.5.

For any $z\in Z$ and $n\in\mathbb{N}$:

  1. 1.

If a subset $V\subset Y$ with $d^{\prime}_{n}(V)<\varepsilon(n,z)$, then the
member of $C^{n,z}\in\mathcal{C}^{n,z}$ having non-empty intersection with $V$
at most $2^{n}$, namely,

$|\\{C\in\mathcal{C}^{n,z}|C\cap V\neq\varnothing\\}|\leq 2^{n}.$

  2. 2.

If a subset $U\subset X$ with $d_{n}(U)<\varepsilon(n,z)$, then for each
$C\in\mathcal{C}^{n,z}$, the number of $D^{n,z}\in\mathcal{D}_{C}^{n,z}$
having non-empty intersection with $U$ is at most $2^{n}$:

$|\\{D\in\mathcal{D}_{C}^{n,z}|D\cap U\neq\varnothing\\}|\leq 2^{n}.$

###### Proof..

$(1)$ For each $0\leq k<n$, the set $S^{k}V$ may have non-empty intersection
with $C_{0}^{n,z}$ and at most one set
$\\{C_{1}^{n,z},\cdots,C_{\alpha}^{n,z}\\}$. Hence, the statement holds.

$(2)$ For each $C_{i}\in\mathcal{C}^{n,z}$, $\mathcal{D}_{C}^{n,z}$ is a
partition with $D_{ij}^{n,z}$ compact subsets of $X$, where
$j=1\cdots,\beta_{i}$. For each $0\leq k<n,$ the set $T^{k}U$ may have non-
empty intersection with $D_{ij}^{n,z}$ at most one of
$\\{D_{i1}^{n,z},\cdots,D_{i\beta_{i}}^{n,z}\\}$. The above statement follows
from this. ∎

Let $n\in\mathbb{N}$ be a natural number, suppose there is an open cover
$\varphi^{-1}(z)\subset V^{n,z}_{1}\cup\cdots\cup V^{n,z}_{k}$ with
$d(V_{i},d^{\prime}_{n})<\varepsilon$ for all $1\leq i\leq k$. Moreover,
suppose that for each $1\leq i\leq k,$ we are given an open cover
$\pi^{-1}(V^{n,z}_{i})=U^{n,z}_{i1}\cup\cdots\cup U^{n,z}_{im_{i}}$ with
$d(U^{n,z}_{ij},d_{n})<\varepsilon$ for all $1\leq j\leq m_{i}$. For each
$z\in Z$, we are going to prove that

$\displaystyle\log\sum_{C\in\mathcal{C}^{n,z}}\left(\sum_{D\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D}\mathbb{S}_{n}f}\right)^{\omega}\leq
2n\log
2+\log\sum_{i=1}^{k}\left(\sum_{j=1}^{m_{i}}e^{\sup_{U^{n,z}_{ij}}\mathbb{S}_{n}f}\right)^{\omega}.$
(4.2)

Suppose (4.2) is proved. Then from Claim 4.4, we obtain,

$\displaystyle H_{\nu_{z}}(\mathcal{C}^{n,z})+\omega
H_{\mu_{z}}(\mathcal{D}^{n,z}|\pi^{-1}(\mathcal{C}^{n,z}))+\omega\int_{X}\mathbb{S}_{n}fd\mu_{z}\leq
2n\log
2+\log\sum_{i=1}^{k}\left(\sum_{j=1}^{m_{i}}e^{\sup_{U_{ij}}\mathbb{S}_{n}f}\right)^{\omega}.$
(4.3)

Then by Claim 4.2 and 4.3, we have

$\displaystyle\omega
H_{\mu}(S,\mathcal{B}^{n}|Z)+(1-\omega)H_{\nu}(T,\mathcal{A}^{n}|Z)+\omega\int_{X}fd\mu$
$\displaystyle\leq\int_{Z}H_{\nu_{z}}(\mathcal{C}^{n,z})+\omega
H_{\mu_{z}}(\mathcal{D}^{n,z}|\pi^{-1}(\mathcal{C}^{n,z}))d\kappa_{z}+\omega\int_{X}\mathbb{S}_{n}fd\mu$
$\displaystyle\leq 2n\log 2+\sup_{z\in Z}\log
P_{Z}^{\omega}(\pi,z,f,n,\varepsilon),$

the last inequality is given by taking over $\\{V^{n,z}_{i}\\}$ and
$\\{U^{n,z}_{ij}\\}$ satisfying the above assumption.

Let $n\to\infty$ and divide by $n$, we have

$\displaystyle\omega
h_{\mu}(T,\mathcal{B}|Z)+(1-\omega)h_{\nu}(\mathcal{A}|Z)+\omega\int_{X}fd\mu$
$\displaystyle=\omega\lim_{n\to\infty}\frac{1}{n}\left(\omega
H_{\mu}(S,\mathcal{B}^{n}|Z)+(1-\omega)H_{\nu}(T,\mathcal{A}^{n}|Z)+\omega\int_{X}fd\mu\right)$
$\displaystyle\leq 2\log 2+\lim_{n\to\infty}\left(\frac{\sup_{z\in Z}\log
P^{\omega}_{Z}(\pi,z,f,n,\varepsilon)}{n}\right).$

Taking $\varepsilon\to 0$, we obtain:

$\omega
h_{\mu}(T,\mathcal{B}|Z)+(1-\omega)h_{\nu}(\mathcal{A}|Z)+\omega\int_{X}fd\mu\leq
2\log 2+P^{\omega}_{Z}(\pi,T,f).$

So the rest is to prove (4.2).

Given $n\in\mathbb{N}$ and $z\in Z$. For each $D^{n,z}\in\mathcal{D}^{n,z}$,
we have

$e^{\sup_{D^{n,z}}\mathbb{S}_{n}f}\leq\sum_{U_{ij^{n,z}}\cap
D^{n,z}\neq\varnothing}e^{\sup_{U_{ij}^{n,z}}\mathbb{S}_{n}f}.$

Here the sum is taken over all index $(i,j)$ such that $U_{ij}^{n,z}$ has non-
empty intersection with $D.$

Let $C\in\mathcal{C}^{n,z}.$ We define $\mathcal{V}_{C}$ as the set of $1\leq
i\leq k$ such that $V^{n,z}_{i}\cap C\neq\varnothing$. By Claim 4.5 $(2),$ we
get

$\sum_{D^{n,z}\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D^{n,z}}\mathbb{S}_{n}f}\leq
2^{n}\sum_{i\in\mathcal{V}_{C}}\sum_{j=1}^{m_{j}}e^{\sup_{U_{ij}^{n,z}}\mathbb{S}_{n}f}.$

Therefore,

$\displaystyle\left(\sum_{D^{n,z}\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D^{n,z}}\mathbb{S}_{n}f}\right)^{\omega}$
$\displaystyle\leq
2^{n\omega}\left(\sum_{i\in\mathcal{V}_{C}}\sum_{j=1}^{m_{j}}e^{\sup_{U_{ij}^{n,z}}\mathbb{S}_{n}f}\right)^{\omega}$
$\displaystyle\leq
2^{n\omega}\sum_{i\in\mathcal{V}_{C}}\left(\sum_{j=1}^{m_{j}}e^{\sup_{U_{ij}^{n,z}}\mathbb{S}_{n}f}\right)^{\omega}.$

######  Remark 4.6.

The last inequality holds since for $0\leq\omega\leq 1$ and non-negative
numbers $x,y,$

$(x+y)^{\omega}\leq x^{\omega}+y^{\omega}.$

Hence,

$\sum_{C\in\mathcal{C}^{n,z}}\left(\sum_{D^{n,z}\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D^{n,z}}\mathbb{S}_{n}f}\right)^{\omega}\leq
2^{n\omega}\sum_{C\in\mathcal{C}^{n,z}}\left(\sum_{i\in\mathcal{V}_{C}}\left(\sum_{j=1}^{m_{i}}e^{\sup_{U^{n,z}_{ij}}\mathbb{S}_{n}f}\right)^{\omega}\right).$

By Claim 4.5 $(1)$, for each $1\leq i\leq k$, the number
$C\in\mathcal{C}^{n,z}$ satisfying $i\in\mathcal{V}_{C}$ is at most $2^{n}.$
So

$2^{n\omega}\sum_{C\in\mathcal{C}^{n,z}}\left(\sum_{i\in\mathcal{V}_{C}}\left(\sum_{j=1}^{m_{j}}e^{\sup_{U^{n,z}_{ij}}\mathbb{S}_{n}f}\right)^{\omega}\right)\leq
2^{n\omega}\cdot
2^{n}\sum_{i=1}^{k}\left(\sum_{j=1}^{m_{i}}e^{\sup_{U^{n,z}_{ij}}\mathbb{S}_{n}f}\right)^{\omega}.$

Therefore,

$\sum_{C\in\mathcal{C}^{n,z}}\left(\sum_{D^{n,z}\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D^{n,z}}\mathbb{S}_{n}f}\right)^{\omega}\leq
2^{n\omega}\cdot
2^{n}\sum_{i=1}^{k}\left(\sum_{j=1}^{m_{i}}e^{\sup_{U^{n,z}_{ij}}\mathbb{S}_{n}f}\right)^{\omega}.$

Taking the logarithm,

$\displaystyle\log\sum_{C\in\mathcal{C}^{n,z}}\left(\sum_{D^{n,z}\in\mathcal{D}_{C}^{n,z}}e^{\sup_{D^{n,z}}\mathbb{S}_{n}f}\right)^{\omega}$
$\displaystyle\leq(n+n\omega)\log
2+\log\sum_{i=1}^{k}\left(\sum_{j=1}^{m_{i}}e^{\sup_{U^{n,z}_{ij}}\mathbb{S}_{n}f}\right)^{\omega}$
$\displaystyle\leq 2n\log
2+\sum_{i=1}^{k}\left(\sum_{j=1}^{m_{i}}e^{\sup_{U^{n,z}_{ij}}\mathbb{S}_{n}f}\right)^{\omega},$

which proves (4.2).

2\. Let $\mathcal{A}$ and $\mathcal{B}$ be the partitions of $Y$ and $X$
mentioned in 1. and repeat the process in 1. Then use (4.3) with Claim 4.2,
Claim 4.3 and Claim 4.5 we have

$\displaystyle\omega
h_{\mu}(T,\mathcal{B}|Z)+(1-\omega)h_{\nu}(S,\mathcal{B}|Z)+\omega\int fd\mu$
$\displaystyle\leq\limsup_{n\to\infty}\frac{1}{n}\int_{Z}2n\log 2+\log
P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)d\kappa(z)+1$
$\displaystyle\leq\int_{Z}\limsup_{n\to\infty}\frac{1}{n}\left(2n\log 2+\log
P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)\right)d\kappa(z)+1,$

and by Fatou’s Lemma and Proposition 2.12

$\displaystyle\leq 2\log 2+\int_{Z}P^{\omega}_{Z}(\pi,z,f)d\kappa(z)+1.$

Since $\mathcal{A}$ and $\mathcal{B}$ are arbitrary,

$\displaystyle\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\leq\int_{Z}P^{\omega}_{Z}(\pi,z,f)d\kappa(z)+1+2\log
2.$

Finally, with Proposition 2.13, that is, the amplification trick, we get the
conclusion we want, ∎

###  4.2. Relative zero-dimensional principal extension

Let $\pi:(X,T)\to(Y,S)$ be a factor map between TDSs and $d$ a metric on $X$,
the topological conditional entropy of $\pi$ is defined by (also, see Remark
2.8)

$h_{top}(X,T|Y,S)=\lim_{\varepsilon\to
0}\left(\lim_{n\to\infty}\frac{\sup_{y\in Y}\log
N(\pi^{-1}(y),n,\varepsilon)}{n}\right),$

where $N(\pi^{-1}(y),n,\varepsilon)$ denotes the least number of open cover
$\mathcal{U}$ of $\pi^{-1}(y)$ with $\mathrm{diam}(U,d_{n})<\varepsilon$ for
all $U\in\mathcal{U}$, that is,
$N(\pi^{-1}(y),n,\varepsilon)=\min\\{|\mathcal{U}|\pi^{-1}(y)\subset\cup_{U\in\mathcal{U}}U,\text{
where }\mathrm{diam}(U,d_{n})<\varepsilon\\}$.

Recall that the factor map $\pi:(X,T)\to(Y,S)$ is said to be principal if
$h_{top}(X,T|Y,S)=0$. We need the following results about principal
extensions.

######  Theorem 4.7.

([6, Corollary 6.8.9], [18, Remark 5.2]) Let $\pi:(X,T)\to(Y,S)$ be a factor
map with $h_{top}(Y,S)<\infty$, then $\pi$ is principal if and only if for any
$\mu\in\mathcal{M}(X,T)$

$h_{\mu}(T)=h_{\pi\mu}(S).$

The following property is proved in [18], we restate it without detailed proof
here for a relative version.

######  Lemma 4.8.

([18, Lemma 5.3 for $Z=\\{*\\}$]) Let $(X,T)$, $(Y,S)$ and $(Z,R)$ be TDSs
satisfying the following commutative diagram

(4.10)

where

$X\times_{Y}Y^{\prime}=\\{(x,y)\in X\times Y^{\prime}|\pi(x)=\xi(y)\\}.$

If $\xi$ is a principal extension between $(Y^{\prime},S^{\prime})$ and
$(Y,S)$, then $\eta$ is also a principal extension for $(X,T)$ and
$(X\times_{Y}Y^{\prime},T\times S^{\prime})$.

Recall that a compact metric space $X$ is said to be zero-dimensional if it
has a base consisting of clopen sets. For a topological dynamical system
$(X,T)$, the following significant result can be found in [6, Theorem 7.6.1]
and also [7, Theorem 3.1].

######  Theorem 4.9.

Let $(X,T)$ be a TDS, there is an extension map
$\pi:(X^{\prime},T^{\prime})\to(X,T)$ such that

  1. 1.

$\pi:(X^{\prime},T^{\prime})\to(X,T)$ is principal;

  2. 2.

$X^{\prime}$ is a zero-dimensional compact metrizable space.

We now state a key result of a relative version of the zero-dimensional
extension as follows.

######  Proposition 4.10.

([18, Corollary 5.5 for $Z=\\{*\\}$]) Let $\pi(X,T)\to(Y,S)$ and
$\varphi:(Y,S)\to(Z,R)$ be factor maps with $\psi=\varphi\circ\pi$. Then there
is a commutative diagram as follows,

(4.19)

satisfying the following conditions.

  1. 1.

The factor maps $\eta$, $\rho$ (hence $\eta\circ\rho$) and $\xi$ are principal
extensions. Besides, $X^{\prime}$ and $Y^{\prime}$ are zero-dimensional;

  2. 2.

For any $0\leq\omega\leq 1$ and $z\in Z$ we have

$P^{\omega}_{Z}(\pi,T,z,f)\leq
P^{\omega}_{Z}(\Pi^{\prime},T^{\prime},z,f\circ\eta\circ\rho)$

hence,

$P^{\omega}_{Z}(\pi,T,f)\leq
P^{\omega}_{Z}(\Pi^{\prime},T^{\prime},f\circ\eta\circ\rho).$

And,

$P^{\omega}_{Z,var}(\Pi^{\prime},T^{\prime},f\circ\eta\circ\rho)\leq
P^{\omega}_{Z,var}(\pi,T,f).$

###### Proof..

By Theorem 4.9 there is a zero-dimensional principal extension
$\xi:(Y^{\prime},S^{\prime})\to(Y,S)$. Consider the joining of $(X,T)$ and
$(Y^{\prime},S^{\prime})$ relative to $(Y,S)$ and the projections
$\eta:X\times_{Y}Y^{\prime}\to X$ and $\Pi:X\times_{Y}Y^{\prime}\to
Y^{\prime}$. Then by Proposition 2.15, for any $0\leq\omega\leq 1$,

$P^{\omega}_{Z}(\pi,T,f)\leq P^{\omega}_{Z}(\Pi,T\times
S^{\prime},f\circ\eta).$

For any $\mu\in\mathcal{M}(X\times_{Y}Y^{\prime},T\times S^{\prime})$, by
Lemma 4.8 $\eta$ is principal, also by Rohlin-Abramov Theorem 3.2, hence,

$h_{\mu}(T\times S^{\prime}|R)=h_{\eta\mu}(T|R)\text{ and
}h_{\Pi\mu}(S^{\prime}|R)=h_{\xi\Pi\mu}(S|R)=h_{\pi\eta\mu}(S|R).$

Therefore, we have

$\displaystyle P^{\omega}_{Z,var}$ $\displaystyle(\Pi,T\times
S^{\prime},f\circ\eta)$
$\displaystyle=\sup_{\mu\in\mathcal{M}(X\times_{Y}Y^{\prime},T\times
S^{\prime})}\left\\{\omega h_{\mu}(T\times
S^{\prime}|R)+(1-\omega)h_{\Pi\mu}(S^{\prime}|R)+\omega\int_{X\times_{Y}Y^{\prime}}f\circ\eta
d\mu\right\\}$
$\displaystyle=\sup_{\mu\in\mathcal{M}(X\times_{Y}Y^{\prime},T\times
S^{\prime})}\left\\{\omega
h_{\eta\mu}(T|R)+(1-\omega)h_{\pi\eta\mu}(S|R)+\omega\int_{X}fd(\eta\mu)\right\\}$
$\displaystyle\leq P^{\omega}_{Z,var}(\pi,T,f).$

By applying Theorem 4.9 again, there is a zero-dimensional principal extension
$\rho:(X^{\prime},T^{\prime})\to(X\times_{Y}Y^{\prime},T\times S^{\prime})$ as
above.

By Proposition 2.14, we obtain

$P_{Z}^{\omega}(\Pi,T\times S^{\prime},f\circ\eta)\leq
P_{Z}^{\omega}(\Pi^{\prime},T^{\prime},f\circ\eta\circ\rho).$

Since $\rho$ is principal, we have

$P^{\omega}_{Z,var}(\Pi^{\prime},T^{\prime},f\circ\eta\circ\rho)\leq
P^{\omega}_{Z,var}(\Pi,T\times S^{\prime},f\circ\eta).$

We also conclude that

$P^{\omega}_{Z}(\pi,T,f)\leq
P_{Z}^{\omega}(\Pi^{\prime},T^{\prime},f\circ\eta\circ\rho),\text{
}P^{\omega}_{Z,var}(\Pi^{\prime},T^{\prime},f\circ\eta\circ\rho)\leq
P^{\omega}_{Z,var}(\pi,T,f).$

∎

###  4.3. Proof of the other side of Theorem 2.2 and Theorem 2.9

######  Proposition 4.11.

Let $\pi:(X,T)\to(Y,S)$ be a factor map between zero-dimensional TDSs, suppose
there is a factor map $\varphi:(Y,S)\to(Z,R)$. Then for any $0\leq\omega\leq
1$ and a continuous function $f:X\to\mathbb{R}$,

$P^{\omega}_{Z}(\pi,T,f)\leq P^{\omega}_{Z,var}(\pi,T,f).$

###### Proof..

Let $\mathcal{A}$ be a clopen cover of $Y$ with
$\mathrm{diam}(A,d^{\prime})<\varepsilon$, for all $A\in\mathcal{A}$, for each
$z\in Z$, we set

$\mathcal{A}^{z}=\\{A\cap\varphi^{-1}(z)|A\in\mathcal{A}\\}.$

Due to $\dim(X)=0,$ for each $1\leq i\leq\alpha$, we can also take a clopen
partition

$\pi^{-1}(A_{i})=\bigcup_{j=1}^{\beta_{ij}}B_{ij}\text{ with
}\mathrm{diam}(B_{ij},d)<\varepsilon\text{ for all }1\leq j\leq\beta_{i}.$

Set

$\mathcal{B}^{z}=\\{B\cap\pi^{-1}(A^{z})|A^{z}\in\mathcal{A}^{z}\\}.$

Each element of $\mathcal{A}^{z}$ and $\mathcal{B}^{z}$ and are clopen for the
fibers $\varphi^{-1}(z)$ and $\psi^{-1}(z)$ under the subspace topology,
respectively. For each $z\in Z$, we have
$\pi^{-1}(\mathcal{A}^{z})\preceq\mathcal{B}^{z}.$ For each $n\in\mathbb{N},$
we set

$\mathcal{A}^{n,z}=\\{A\cap\varphi^{-1}(z)|A\in\mathcal{A}^{n}\\},$

and

$\mathcal{B}^{n,z}=\\{B\cap\pi^{-1}(A^{n,z})|A^{n,z}\in\mathcal{A}^{n,z}\\}.$

For each $A^{n,z}\in\mathcal{A}^{n,z}$, put

$\displaystyle\mathcal{B}_{A^{n,z}}^{n,z}=\\{B^{n,z}\in\mathcal{B}^{n,z}|B^{n,z}\cap\pi^{-1}(A^{n,z})\neq\varnothing\\}=\\{B^{n,z\in\mathcal{B}^{n,z}}|B^{n,z}\subset\pi^{-1}(A^{n,z})\\}.$
(4.20)

We have

$\pi^{-1}(A^{n,z})=\bigcup_{B^{n,z}\in\mathcal{B}_{A^{n,z}}^{n,z}}B^{n,z}.$

Set

$W_{A^{n,z}}=\sum_{B^{n,z}\in\mathcal{B}_{A^{n,z}}^{n,z}}e^{\sup_{B^{n,z}}\mathbb{S}_{n}f},$

and define

$W_{n,z}=\sum_{A^{n,z}\in\mathcal{A}^{n,z}}\left(W_{A^{n,z}}\right)^{\omega}.$

######  Claim 4.12.

For each $n\in\mathbb{N}$ and $z\in Z$, $W_{A^{n,z}}$ and $W_{n,z}$ are
defined as above, then

$P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)\leq W_{n,z}.$

###### Proof..

Let $0<\delta<\varepsilon$ be a positive number and
$A^{n,z}\in\mathcal{A}^{n,z}$, for each $B^{n,z}\in\mathcal{B}^{n,z}$, we can
find an open subset $\widetilde{B}^{n,z}\supseteq B^{n,z}$ such that

$\mathrm{diam}(\widetilde{B}^{n,z},d_{n})<\varepsilon\text{ with
}\pi(\widetilde{B}^{n,z})\subset A^{n}\text{ and
}e^{\sup_{\widetilde{B}^{n,z}}\mathbb{S}_{n}f}\leq
e^{\sup_{B^{n,z}}\mathbb{S}_{n}f}+\delta.$

By Lemma 2.11, these open subsets can be chosen such that
$\widetilde{A}^{n,z}:=\pi(\bigcup\widetilde{B}^{n,z})$ is an open set in $Y$
which covers $\varphi^{-1}(z)$. Similarly, we can define the same class
$\widetilde{\mathcal{B}}^{n,z}$ ,$\widetilde{\mathcal{A}}^{n,z}$ and
$\widetilde{\mathcal{B}}^{n,z}_{\widetilde{A}^{n,z}}$ as $\mathcal{B}^{n,z},$
$\mathcal{A}^{n,z}$ and $\mathcal{B}_{A^{n,z}}^{n,z}$, respectively, write

$\widetilde{W}_{\widetilde{A}^{n,z}}\sum_{\widetilde{B}^{n,z}\in\widetilde{\mathcal{B}}_{\widetilde{A}^{n,z}}^{n,z}}e^{\sup_{\widetilde{B}^{n,z}}\mathbb{S}_{n}f},$

and
$\widetilde{W}_{n,z}=\sum_{\widetilde{A}^{n,z}\in\widetilde{\mathcal{A}}^{n,z}}(W_{\widetilde{A}^{n,z}})^{\omega}.$
It is obvious that
$P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)\leq\widetilde{W}^{n,z}.$ Now, letting
$\delta$ be arbitrarily small, we obtain $W_{n,z}\geq\widetilde{W}_{n,z}.$ So
the conclusion is completed. ∎

Let $\varepsilon_{0}$ be small enough, for each $n\in\mathbb{N}$ there is a
point $z^{\prime}\in Z$ such that

$\displaystyle\sup_{z\in Z}\log
P_{Z}^{\omega}(\pi,T,z,f,n,\varepsilon)\leq\log
P_{Z}^{\omega}(\pi,T,z^{\prime},f,n,\varepsilon)+\varepsilon_{0}.$ (4.21)

Hence, by Claim 4.12, it is enough to prove that there is a
$\mu\in\mathcal{M}(X,T)$ satisfying

$\displaystyle\omega
h_{\mu}(T,\mathcal{B}|Z)+(1-\omega)h_{\pi\mu}(S,\mathcal{A}|Z)+\omega\int_{X}fd\mu$
$\displaystyle\geq\lim_{n\to\infty}\frac{\log W_{n,z^{\prime}}}{n}$
$\displaystyle\geq\lim_{n\to\infty}\frac{\sup_{z\in
Z}\log\left(P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)-\varepsilon_{0}\right)}{n}.$

Let $n\in\mathbb{N}$ be a natural number and $z\in Z$ a point satisfying
condition (4.21). We assume that the elements of $\mathcal{B}^{n,z}$ and
$\mathcal{A}^{n,z}$ are all non-empty. For each $B^{n,z}\in\mathcal{B}^{n,z}$,
we denote by $\mathcal{A}^{n,z}(B^{n,z})$ the unique element in
$\mathcal{A}^{n,z}$ containing $\pi(B^{n,z}).$

Since each $B^{n,z}\in\mathcal{B}^{n,z}$ is compact, we take a point
$x_{B^{n,z}}\in B^{n,z}$ satisfying
$\mathbb{S}_{n}f(x_{B^{n,z}})=\sup_{B^{n,z}}\mathbb{S}_{n}f.$ Define a
probability measure on $X$ by

$\displaystyle\sigma_{n}$
$\displaystyle=\frac{1}{W_{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}}(W_{A^{n,z}})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\cdot\delta_{x_{B^{n,z}}}$
$\displaystyle=\frac{1}{W_{n,z}}\sum_{A^{n,z}\in\mathcal{A}^{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}_{A^{n,z}}}(W_{A^{n,z}})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\cdot\delta_{x_{B^{n,z}}},$

where $\delta_{x_{B^{n,z}}}$ is the probability measure mass on the point
$x_{B^{n,z}}.$ We set

$\displaystyle\mu_{n}=\frac{1}{n}\sum_{s=0}^{n-1}T^{s}\sigma_{n}.$

Under the topology of weak*, we can take a subsequence $\\{\mu_{n_{k},z}\\}$
converging to an invariant measure $\mu\in\mathcal{M}(X,T),$ we shall prove
that

$\omega
h_{\mu}(T,\mathcal{B}|Z)+(1-\omega)h_{\pi\mu}(S,\mathcal{A}|Z)+\omega\int_{X}fd\mu\geq\lim_{n\to\infty}\frac{\sup_{z\in
Z}\log\left(P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)-\varepsilon_{0}\right)}{n}.$

######  Claim 4.13.

Let $z\in Z$ be a point satisfying condition (4.21). For any natural number
$n\in\mathbb{N}$, we have

$\displaystyle\omega
H_{\sigma_{n}}(\mathcal{B}^{n}|Z)+(1-\omega)H_{\pi\sigma_{n}}(\mathcal{A}^{n}|Z)+\omega\int_{X}\mathbb{S}_{n}fd\sigma_{n}$
$\displaystyle=\log W_{n,z}$ $\displaystyle\geq\sup_{z\in
Z}\log(P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)-\varepsilon_{0}).$

###### Proof..

From the construction of the probability measure $\sigma_{n},$ for each
$B^{n}\in\mathcal{B}^{n},$ we have

$\sigma_{n}(B)=\sigma_{n}(B^{n,z})=\frac{(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}}{W_{n,z}}e^{\mathbb{S}_{n}f(x_{B^{n,z}})},$

where $B^{n,z}$ is a subset of $B^{n}$ concentrating on $\psi^{-1}(z)$, that
is, $B^{n,z}\subset\psi^{-1}(z).$ Then

$\displaystyle\int_{Z}H_{(\sigma_{n})_{z}}(\mathcal{B}^{n})d\kappa_{n}(z)$
$\displaystyle=H_{\sigma_{n}}(\mathcal{B}^{n}|Z)=H_{\sigma_{n}}(\mathcal{B}^{n,z}|Z)$
$\displaystyle=-\sum_{B^{n,z}\in\mathcal{B}^{n,z}}\frac{(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}}{W_{n,z}}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\log\left(\frac{(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}}{W_{n,z}}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\right)$
$\displaystyle=\frac{\log
W_{n,z}}{W_{n,z}}\underbrace{\sum_{B^{n,z}\in\mathcal{B}^{n,z}}(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}}_{(I)}$
$\displaystyle-\frac{\omega-1}{W_{n,z}}\underbrace{\sum_{B^{n,z}\in\mathcal{B}^{n,z}}(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\log
W_{\mathcal{A}^{n,z}(B^{n,z})}}_{(II)}$
$\displaystyle-\underbrace{\sum_{B^{n,z}\in\mathcal{B}^{n,z}}\frac{(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}}{W_{n,z}}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\mathbb{S}_{n}f(x_{B^{n,z}})}_{(III)},$

where $\kappa_{n}=\psi\sigma_{n}$. The term $(I)$ can be calculated by

$(I)=\sum_{A^{n,z}\in\mathcal{A}^{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}_{A^{n,z}}}(W_{A^{n,z}})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}=W_{n,z}.$

Term $(II)$ is obtained by

$(II)=\sum_{A^{n,z}\in\mathcal{A}^{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}_{A^{n,z}}}(W_{A^{n,z}})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\log
W_{A^{n,z}}=\sum_{A^{n,z}\in\mathcal{A}^{n,z}}(W_{A^{n,z}})^{\omega}\log
W_{A^{n,z}}.$

For term $(III)$, we have

$\int_{X}\mathbb{S}_{n}fd\sigma_{n}=\frac{1}{W_{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}}(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\mathbb{S}_{n}f(x_{B^{n,z}})=(III).$

Therefore,

$\displaystyle
H_{\sigma_{n}}(\mathcal{B}^{n}|Z)+\int_{X}\mathbb{S}_{n}fd\sigma_{n}=\log
W_{n,z}-\frac{\omega-1}{W_{n,z}}\sum_{A^{n,z}\in\mathcal{A}^{n,z}}(W_{A^{n,z}})^{\omega}\log
W_{A^{n,z}}.$ (4.22)

Moreover, we have

$\pi\sigma=\frac{1}{W_{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}}(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\cdot\delta_{\pi(x_{B^{n,z}})}.$

For each non-empty $A\in\mathcal{A}^{n},$

$\displaystyle\pi\sigma_{n}(A)$
$\displaystyle=\frac{1}{W_{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}_{A^{n,z}}}(W_{\mathcal{A}^{n,z}(B^{n,z})})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}$
$\displaystyle=\frac{1}{W_{n,z}}(W_{A^{n,z}})^{\omega},$

where $\mathcal{A}^{n,z}(B^{n,z})=A^{n,z}$ for
$B^{n,z}\in\mathcal{B}^{n,z}_{A^{n,z}}$. Then,

$H_{\pi\sigma}(\mathcal{A}^{n}|Z)=\log
W_{n,z}-\omega\sum_{A^{n,z}\in\mathcal{A}^{n,z}}\frac{(W_{A^{n,z}})^{\omega}}{W_{n,z}}\log
W_{A^{n,z}}.$

Combining this with (4.22), we obtain

$\omega
H_{\sigma_{n}}(\mathcal{B}^{n}|Z)+(1-\omega)H_{\pi\sigma_{n}}(\mathcal{A}^{n}|Z)+\omega\int_{X}\mathbb{S}_{n}fd\sigma_{n}=\log
W_{n,z}.$

∎

The proof of the following claim is standard (See the proof of the variational
principle in [20]), but for the sake of completeness, we will write it out.

######  Claim 4.14.

Let $m$ and $n$ be positive integers. We have

$\frac{1}{m}H_{\mu_{n}}(\mathcal{B}^{m}|Z)\geq\frac{1}{n}H_{\sigma_{n}}(\mathcal{B}^{n}|Z)-\frac{2m\log|\mathcal{B}|}{n},$
$\frac{1}{m}H_{\pi\mu_{n}}(\mathcal{A}^{m}|Z)\geq\frac{1}{n}H_{\pi\sigma_{n}}(\mathcal{A}^{n}|Z)-\frac{2m\log|\mathcal{A}|}{n},$

where $|\cdot|$ is the cardinal operator.

###### Proof..

It is clear from the construction that $\sigma_{n}$ is concentrated on
$\psi^{-1}(z)$ and $\pi\sigma_{n}$ is concentrated on $\varphi^{-1}(z)$, we
have

$H_{\sigma_{n}}(\mathcal{B}^{n}|Z)=H_{\sigma_{n}}(\mathcal{B}^{n}),$

and

$H_{\pi\sigma_{n}}(\mathcal{A}^{n}|Z)=H_{\pi\sigma_{n}}(\mathcal{A}^{n}).$

Here, we provide the proof for $\mathcal{B}^{m}$, the case of
$\mathcal{A}^{m}$ is similar.

We assume that $1<m<n$, and for $0\leq l<m,$ let $a(l)$ denote the integer
part of $(n-l)m^{-1}$, so that $n=l+a(l)m+r$ with $0\leq r<q$. Then

$\mathcal{B}^{n}=\bigvee_{i=0}^{n-1}T^{-i}\mathcal{B}=\left(\bigvee_{j=0}^{a(l)-1}T^{-(l+jm)}\bigvee_{i=0}^{m-1}T^{-i}\mathcal{B}\right)\vee\bigvee_{t\in
S_{l}}T^{-t}\mathcal{B},$

where $S_{l}$ is a subset of $\\{0,1,\cdots,n-1\\}$ with cardinality at most
$2m$. By Proposition 3.1, it follows that

$\displaystyle
H_{\sigma_{n}}\left(\bigvee_{i=0}^{n-1}T^{-i}\mathcal{B}|Z\right)$
$\displaystyle\leq\sum_{j=0}^{a(l)-1}H_{\sigma_{n}}\left(T^{-(l+jm)}\bigvee_{i=0}^{m-1}T^{-i}\mathcal{B}|Z\right)+2m\log|\mathcal{B}|$
$\displaystyle\leq\sum_{j=0}^{a(l)-1}H_{T^{(l+jm)}\sigma_{n}}\left(\bigvee_{i=0}^{m-1}T^{-i}\mathcal{B}|Z\right)+2m\log|\mathcal{B}|.$

Sum this inequality over $l\in\\{0,1,\cdots,m-1\\},$ we have that

$\displaystyle
mH_{\sigma_{n}}\left(\bigvee_{i=0}^{n-1}T^{-i}\mathcal{B}|Z\right)$
$\displaystyle\leq\sum_{t=0}^{n-1}H_{T^{t}\sigma_{n}}\left(\bigvee_{i=0}^{m-1}T^{-i}\mathcal{B}|Z\right)+2m^{2}\log|\mathcal{B}|$
$\displaystyle\leq
nH_{\mu_{n}}\left(\bigvee_{i=0}^{m-1}T^{-i}\mathcal{B}|Z\right)+2m^{2}\log|\mathcal{B}|,$

where the second inequality depends on the general property of the conditional
entropy of partitions
$H_{\sum_{i}p_{i}\mu_{i}}(\mathcal{B}|\mathcal{R})\geq\sum_{i}p_{i}H_{\mu_{i}}(\mathcal{B}|\mathcal{R})$
which holds for any finite partition $\mathcal{B}$, $\sigma$-algebra
$\mathcal{R}$, Borel probability measures $\mu_{i}$, and positive numbers
$p_{i}$ with $p_{1}+\cdots+p_{n}=1$ (See Lemma 3.3). Dividing by $nm$ in the
above inequality, we attain

$\frac{1}{n}H_{\sigma_{n}}\left(\bigvee_{i=0}^{n-1}T^{-i}\mathcal{B}|Z\right)\leq\frac{1}{m}H_{\mu_{n}}\left(\bigvee_{i=0}^{m-1}T^{-i}\mathcal{B}|Z\right)+\frac{2m}{n}\log|\mathcal{B}|,$

which completes the proof of the claim. ∎

From the construction of $\mu_{n}$, we have

$\int_{X}fd\mu_{n}=\frac{1}{n}\int_{X}\sum_{i=0}^{n-1}f\circ
T^{i}d\sigma_{n}=\frac{1}{n}\int_{X}\mathbb{S}_{n}fd\sigma_{n}.$

Then Claim 4.14 implies that

$\displaystyle\frac{\omega}{m}$ $\displaystyle
H_{\mu_{n}}(\mathcal{B}^{m}|Z)+\frac{1-\omega}{m}H_{\pi\mu_{n}}(\mathcal{A}^{m}|Z)+\omega\int_{X}fd\mu_{n}$
$\displaystyle\geq\frac{\omega}{n}H_{\sigma_{n}}(\mathcal{B}^{n}|Z)+\frac{1-\omega}{n}H_{\sigma_{n}}(\mathcal{A}^{n}|Z)+\frac{\omega}{n}\int_{X}fd\sigma_{n}-\frac{2m(\log|\mathcal{A}|\cdot|\mathcal{B}|)}{n}$
$\displaystyle=\frac{\log
W_{n,z}}{n}-\frac{2m(\log|\mathcal{A}|\cdot|\mathcal{B}|)}{n}$
$\displaystyle\geq\frac{\sup_{z\in
Z}\log(P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)-\varepsilon_{0})}{n}-\frac{2m(\log|\mathcal{A}|\cdot|\mathcal{B}|)}{n},$

where the last inequality is obtained from Claim 4.13.

For each $n\in\mathbb{N}$ the boundary of $\mathcal{A}^{n}$ and
$\mathcal{B}^{n}$ have measure zero, so by taking $\mu_{n_{k}}\to\mu$ as
$k\to\infty$, we have

$\displaystyle\frac{\omega}{m}H_{\mu}(\mathcal{B}^{m}|Z)+\frac{1-\omega}{m}H_{\pi\mu}(\mathcal{A}^{m}|Z)+\omega\int_{X}fd\mu$
$\displaystyle\geq\lim_{n\to\infty}\frac{\sup_{z\in
Z}\log(P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)-\varepsilon_{0})}{n}.$

Finally, letting $m\to\infty$, we get

$\omega
h_{\mu}(T,\mathcal{B}|Z)+(1-\omega)h_{\pi\mu}(S,\mathcal{A}|Z)+\omega\int_{X}fd\mu\geq\lim_{n\to\infty}\frac{\sup_{z\in
Z}\log P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)}{n}$

∎

######  Theorem 4.15.

Let $\pi:(X,T)\to(Y,S)$ be a factor map between TDSs, $(Z,R)$ is a factor of
$(Y,S)$ indeuced by $\varphi$ and $f\in C(X,\mathbb{R})$. Then for any
$0\leq\omega\leq 1$, we have

$P^{\omega}_{Z}(\pi,T,f)=P^{\omega}_{Z,var}(\pi,T,f).$

###### Proof..

On the one hand, by Proposition 4.1, we have obtained that

$P^{\omega}_{Z,var}(\pi,T,f)\leq P^{\omega}_{Z}(\pi,T,f).$

On the other hand, there is a factor map
$\Pi^{\prime}:(X^{\prime},T^{\prime})\to(Y^{\prime},S^{\prime})$ between zero-
dimensional systems with $f^{\prime}\in C(X^{\prime},\mathbb{R})$ (See the
definition of $f^{\prime}$ in Proposition 4.10) satisfying the conditions of
Proposition 4.10 such that

$P^{\omega}_{Z}(\pi,T,f)\leq
P^{\omega}_{Z}(\Pi^{\prime},T^{\prime},f^{\prime}),$

and

$P^{\omega}_{Z,var}(\Pi^{\prime},T^{\prime},f^{\prime})\leq
P^{\omega}_{Z,var}(\pi,T,f).$

Moreover, by Proposition 4.11, we have

$P^{\omega}_{Z}(\Pi^{\prime},T^{\prime},f^{\prime})\leq
P^{\omega}_{Z,var}(\Pi^{\prime},T^{\prime},f^{\prime}).$

Therefore, we can conclude that

$P^{\omega}_{Z}(\pi,T,f)=P^{\omega}_{Z,var}(\pi,T,f).$

∎

######  Proposition 4.16.

Let $\kappa\in\mathcal{M}(Z,R)$ and let $z$ be a generic point for $\kappa$
namely,

$\displaystyle\frac{1}{n}\sum_{i=0}^{n-1}\delta_{R^{i}z}\to\kappa\text{ as
}n\to\infty.$

Let $\varepsilon>0$. There exists $\mu\in\mathcal{M}(X,T)$ with $\nu=\pi\mu$
and $\kappa=\psi\mu$ such that

$\displaystyle P^{\omega}_{Z}(\pi,T,z,f,\varepsilon)\leq\omega
h_{\mu}(T|R)+(1-\omega)h_{\nu}(S|R)+\omega\int_{X}fd\mu.$

###### Proof..

Because of Proposition 4.10, we just have to prove this conclusion for both
$X$ and $Y$ are 0-dimensional.

Let $\mathcal{A}$ and $\mathcal{B}$ be clopen partitions in $Y$ and $X$ with
$\mathrm{diam}(A,d^{\prime})<\varepsilon$ for all $A\in\mathcal{A}$ and
$\mathrm{diam}(B,d)<\varepsilon$ for all $B\in\mathcal{B}$, respectively. For
each $n\in\mathbb{N}$, the symbols $\mathcal{A}^{n,z}$, $\mathcal{B}^{n,z}$
and $W_{n,z}$ are defined as in Proposition 4.11 such that

$P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon)\leq W_{n,z},$

see Claim 4.12.

$\sigma_{n}$ are defined as in Proposition 4.11, that is,

$\displaystyle\sigma_{n}$
$\displaystyle=\frac{1}{W_{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}}(W_{A^{n,z}})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\cdot\delta_{x_{B^{n,z}}}$
$\displaystyle=\frac{1}{W_{n,z}}\sum_{A^{n,z}\in\mathcal{A}^{n,z}}\sum_{B^{n,z}\in\mathcal{B}^{n,z}_{A^{n,z}}}(W_{A^{n,z}})^{\omega-1}e^{\mathbb{S}_{n}f(x_{B^{n,z}})}\cdot\delta_{x_{B^{n,z}}},$

and

$\displaystyle\mu_{n}=\frac{1}{n}\sum_{s=0}^{n-1}T^{s}\sigma_{n}.$

Then $\psi\sigma_{n}=\delta_{z}$ and

$\psi\mu_{n}=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{R^{i}z}.$

Choose a subsequence $\\{n_{j}\\}$ so that

$\frac{1}{n_{j}}P^{\omega}_{Z}(\pi,T,z,f,n_{j},\varepsilon)\to
P^{\omega}_{Z}(\pi,T,z,f,\varepsilon)$

and $\mu_{n_{j}}$ converges. Let $\mu_{n_{j}}\to\mu$. Then $\nu=\pi\mu$ ,
$\kappa=\psi\mu$ and $\mu\in\mathcal{M}(X,T)$. Following a similar discussion
in Claim 4.13, we conclude that

$\omega
H_{\sigma_{n}}(\mathcal{B}^{n}|Z)+(1-\omega)H_{\pi\sigma_{n}}(\mathcal{A}^{n}|Z)+\omega\int_{X}\mathbb{S}_{n}fd\sigma_{n}=\log
W_{n,z}\geq\log P^{\omega}_{Z}(\pi,T,z,f,n,\varepsilon).$

Moreover, with a similar discussion in Claim 4.14, we have

$\displaystyle P^{\omega}_{Z}(\pi,T,z,f,\varepsilon)\leq\omega
h_{\mu}(T|R)+(1-\omega)h_{\nu}(S|R)+\omega\int_{X}fd\mu.$

∎

######  Theorem 4.17.

Let $\pi:(X,T)\to(Y,S)$ be a factor mao between TDSs, $(Y,S)$ is a extension
of $(Z,R)$ induced by $\varphi$ and $f\in C(X,\mathbb{R})$. Then for any
$0\leq\omega\leq 1$, we have

$\displaystyle\sup\left\\{\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu|\mu\in\mathcal{M}(X,T)\text{
and }\kappa=\psi\mu\right\\}$
$\displaystyle=\int_{Z}P_{Z}^{\omega}(\pi,z,f)d\kappa(z).$

###### Proof..

On the one hand.

1\. Suppose $\kappa$ is ergodic, that is, $\kappa\in E(Z,R)$. Let $z$ be a
generic point for $\kappa$ and $\varepsilon>0$. By Proposition 4.16,

$P^{\omega}_{Z}(\pi,T,z,f,\varepsilon)\leq\sup_{\pi\mu=\kappa}\left\\{\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\right\\}=a.$

Since $\kappa$-a.e. $z\in Z$ are generic, we have

$\int_{Z}P^{\omega}_{Z}(\pi,T,z,f,\varepsilon)\leq a.$

But $P^{\omega}_{Z}(\pi,T,z,f,\varepsilon)\nearrow P^{\omega}_{Z}(\pi,T,z,f)$
as $\varepsilon\to 0$ and the function $P^{\omega}_{Z}(\pi,T,z,f,\varepsilon)$
are clearly bounded from below by $-||f||$. So
$\int_{Z}P^{\omega}_{Z}(\pi,T,z,f)d\kappa(z)\leq a$ if $\kappa$ is ergodic.

2\. If $\kappa$ is not ergodic, let
$\kappa=\int_{E(Z,R)}\kappa_{\alpha}d\rho(\alpha)$ be its ergodic
decomposition.

Let $\delta>0$, define

$\displaystyle K_{\delta}=\\{(\tau,\mu)$ $\displaystyle\in
E(Z,R)\times\mathcal{M}(X,T)|\psi\mu=\tau,$ $\displaystyle\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\geq\int_{Z}P^{\omega}_{Z}(\pi,T,z,f)d\kappa(z)-\delta\\}.$

Now $K_{\delta}$ is a measurable subset of $E(Z,R)\times\mathcal{M}(X,T)$ and
we have shown above that $K_{\delta}$ projects onto $E(Z,R)$. Hence there is a
section $K_{\delta}$, that is, a measurable map
$\phi_{\delta}:E(Z,R)\to\mathcal{M}(X,T)$ such that

$\rho\left(\\{\tau|(\tau,\phi_{\delta}(\tau))\in K_{\delta}\\}\right)=1.$

Define $\mu_{\delta}$ by
$\mu_{\delta}=\int\phi_{\delta}(\kappa_{\alpha})d\rho(\alpha).$

Then $\mu_{\delta}\in\mathcal{M}(X,T),$ $\nu_{\delta}=\pi\mu_{\delta}$ and
$\kappa=\psi\mu_{\delta}$ satisfying

$\displaystyle\omega
h_{\mu_{\delta}}(T|R)+(1-\omega)h_{\nu_{\delta}}(S|R)+\omega\int_{X}fd\mu$
$\displaystyle=\int\omega
h_{\phi_{\delta}(\kappa_{\alpha})}(T|R)+(1-\omega)h_{\pi\phi_{\delta}(\kappa_{\alpha})}(S|R)d\rho(\alpha)+\omega\int\left(\int
fd\phi_{\delta}(\kappa_{\alpha})\right))d\rho(\alpha)$
$\displaystyle\geq\int\left(\int
P^{\omega}_{Z}(\pi,T,z,f)d\kappa(z)-\delta\right)d\rho(\alpha)$
$\displaystyle=\int P^{\omega}_{Z}(\pi,T,z,f)d\kappa(z)-\delta.$

Therefore,

$\int P^{\omega}_{Z}(\pi,T,z,f)d\kappa\leq\sup_{\pi\mu=\kappa}\left\\{\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\right\\}.$

On the other hand, From Proposition 4.1 we have known that

$\int P^{\omega}_{Z}(\pi,T,z,f)d\kappa\geq\sup_{\pi\mu=\kappa}\left\\{\omega
h_{\mu}(T|R)+(1-\omega)h_{\pi\mu}(S|R)+\omega\int_{X}fd\mu\right\\}.$

Then the conclusion follows. ∎

## References

  * [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309–319.
  * [2] N. Alibabaei. Weighted topological pressure revisited. arXiv:2307.16772. 2023.
  * [3] T. Ceccherini-Silberstein, M. Coornaert, F. Krieger: An analogue of Fekete’s lemma for subadditive functions on cancellative amenable semigroups, J. Anal. Math. 124 (2014) 59–81.
  * [4] E. Dinaburg, A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR 190 (1970) 19-22.
  * [5] A.H. Dooley, G.H. Zhang, Local Entropy of a Random Dynamical System, Mem. Amer. Math. Soc., vol. 233, American Mathematical Society, Providence, RI, 2015, pp. 1–120.
  * [6] T. Downarowicz. Entropy in Dynamical Systems. Cambridge University Press, Cambridge, 2011.
  * [7] T. Downarowicz and D. Huczek. Zero-dimensional principal extensions. Acta Appl. Math. 126 (2013), 117–129.
  * [8] D. Feng and W. Huang, Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016), 411-452.
  * [9] L. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969) 679-688.
  * [10] T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (1971) 176-180.
  * [11] H. Hu, Y. Hua, W. Wu, Unstable entropies and variational principle for partially hyperbolic diffeomorphisms, Adv. Math. 321 (2017) 31–68.
  * [12] J. L. Kelley, General Topology, GTM 27, Springer, Berlin, 1955.
  * [13] F. Ledrappier and P. Walters, A relativized variational principle for continuous transformations, J. London Math. Soc. (2), 16 (1977), 568576.
  * [14] M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.
  * [15] V.A. Rohlin, On the Fundamental Ideas of Measure Theory, Amer. Math. Soc. Transl., vol. 71, 1952, 55 pp.
  * [16] D. Ruelle, Statistical mechanics on a compact set with Zv action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 185 (1973) 237-251.
  * [17] J. Shen, L. Xu, X. Zhou, Weighted entropy of a flow on non-compact sets, J. Dynam. Differential Equations. 32 (2020) no. 1, 181–203.
  * [18] M. Tsukamoto, New approach to weighted topological entropy and pressure, Ergodic Theory Dynam. systems. 43 (2023) 1004-1034.
  * [19] P. Walters, A variational principle for the pressure of continuous transformations. Am. J. Math. 97, (1975) 937-971.
  * [20] P. Walters. An Introduction to Ergodic Theory. Springer-Verlag, New York, 1982.
  * [21] T. Wang, Y. Huang, Weighted topological and measure-theoretic entropy, Discrete Contin. Dyn. Syst. 39 (2019) 3941–3967.
  * [22] J. Yang, E. Chen, R. Yang, X. Yang, Variational principle for weighted amenable topological pressure, arXiv: 2306.15509.
  * [23] C. Zhao, E. Chen, X. Zhou, Z. Yin, Weighted topological entropy of the set of generic points in topological dynamical systems, J. Dynam. Differential Equations. 30 (2018) no. 3, 937–955.

E-mail<EMAIL_ADDRESS>