# Local weighted topological pressure

Fangzhou Cai  School of Mathematics and Systems Science, Guangdong Polytechnic
Normal University, Guangzhou, 510665, PR China<EMAIL_ADDRESS>
###### Abstract.

In [D. Feng, W. Huang, Variational principle for weighted topological
pressure. J. Math. Pures Appl. (2016)], the authors studied weighted
topological pressure and established a variational principle for it. In this
paper, we introduce the notion of local weighted topological pressure and
generalize Feng and Huang’s main results to localized version.

##  1\. Introduction

###  1.1. Weighted topological entropy and pressure

We say that $(X,T)$ is a topological dynamical system (TDS for short) if $X$
is a compact metric space and $T$ is a continuous map from $X$ to $X$. We
define $M(X)$ and $M(X,T)$ as the sets of Borel probability measures and
$T$-invariant Borel probability measures on $X$. We first briefly review the
classical theory of entropy and pressure in dynamical systems. One of the most
basic theorems about entropy is variational principle [1, 2, 3]:

$h_{top}(X,T)=\sup_{\mu\in M(X,T)}h_{\mu}(T),$

where $h_{top}(X,T)$ is the topological entropy of $(X,T)$ and $h_{\mu}(T)$ is
the Kolomogorov–Sinai entropy. Motivated by statistical mechanics, Ruelle [4]
and Walters [5] introduced the topological pressure $P(T,f)$ for a real-valued
continuous function $f$ on $X$ and proved the variational principle:

$P(T,f)=\sup_{\mu\in M(X,T)}\big{(}h_{\mu}(T)+\int_{X}fd\mu\big{)}.$

Motivated by fractal geometry of self-affine carpets and sponges [6, 7, 8],
Feng and Huang [21] introduced weighted topological pressure for factor maps
between dynamical systems, and established a variational principle for it. To
be precise, let us introduce some notations first. Let $(X,T)$ and $(Y,S)$ be
two TDSs. We say $(Y,S)$ is a factor of $(X,T)$ if there exists a continuous
surjective map $\pi:X\to Y$ such that $\pi\circ T=S\circ\pi$. The map $\pi$ is
called a factor map from $X$ to $Y$. For $\mu\in M(X,T)$, we denote by
$\pi\mu\in M(Y,S)$ the push-forward of $\mu$ by $\pi.$ Denote the set of real-
valued continuous functions on $X$ by $C(X,\mathbb{R})$. Let $f\in
C(X,\mathbb{R})$ and $a_{1}>0,a_{2}\geq 0$. The main purpose of [21] is to
consider the following question:

How can one define a meaningful term $P^{(a_{1},a_{2})}(T,f)$ such that the
following variational principle holds?

$P^{(a_{1},a_{2})}(T,f)=\sup_{\mu\in
M(X,T)}\Big{(}a_{1}h_{\mu}(T)+a_{2}h_{\pi\mu}(S)+\int_{X}fd\mu\Big{)}.$

Feng and Huang’s definition for $P^{(a_{1},a_{2})}(T,f)$ was inspired from the
dimension theory of affine invariant subsets of tori, and from the “dimension”
approaches of Bowen [9] and Pesin–Pitskel [27] in defining the topological
entropy and topological pressure for arbitrary subsets. We will give the
detailed definition in the next section.

In recent years, lots of authors focus on weighted topological and measure-
theoretic entropy and pressure. In [28], Wang and Huang introduced various
weighted topological entropies from different points of view and studied their
relationships. The notion of measure-theoretic weighted entropy was also
defined and studied in [28]. In [20] the authors studied weighted topological
entropy of the set of generic points. It was proved in [20] that the weighted
topological entropy of generic points of the ergodic measure $\mu$ is equal to
the weighted measure entropy of $\mu$, which generalized the classical result
of Bowen [9]. In [29], the authors studied weighted entropy of a flow on non-
compact sets. In [30], the authors studied weighted topological entropy of
random dynamical systems. Recently, Tsukamoto [31] introduced a new approach
to weighted topological pressure and established a corresponding variational
principle for it. The new approach is a modification of the familiar
definition of topological entropy and pressure. It is very different from the
original definitions in [21]. The equivalence of the two definitions is highly
nontrivial. In [32], the authors generalized Tsukamoto’s approach to amenable
group action.

###  1.2. Local entropy and pressure

The local theory of entropy and pressure is fundamental to many areas in
dynamical systems. It has relations with entropy pairs, entropy sets, entropy
points, and entropy structure, etc. (please see [10, 11, 12, 13, 14, 15, 16,
17, 22, 23, 24, 25]). In this subsection we briefly review the local theory of
entropy and pressure. With the notion of entropy pairs [11, 13] in both
topological and measure-theoretic situations, the study of local version of
the variational principle of entropy has attracted a lot of attention.
Blanchard, Glasner and Host [12] showed the following local variational
principle: for an open cover $\mathcal{U}$ of $X$, there exists $\mu\in
M(X,T)$ such that

$\inf_{\alpha\succeq\mathcal{U}}h_{\mu}(T,\alpha)\geq h_{top}(T,\mathcal{U}),$

where the infimum is taken over all $\alpha\in\mathcal{P}_{X},$ i.e., all
finite Borel partitions of $X$, finer than $\mathcal{U}$. To make a general
investigation on the converse of the inequality, Romagnoli [22] introduced two
types of measure-theoretic entropies related to $\mathcal{U}$:

$h_{\mu}(T,\mathcal{U})=\lim_{n\to\infty}\frac{1}{n}\inf_{\alpha\succeq\mathcal{U}_{0}^{n-1}}H_{\mu}(\alpha),$
$h_{\mu}^{+}(T,\mathcal{U})=\inf_{\alpha\succeq\mathcal{U}}h_{\mu}(T,\alpha).$

He showed the following local variational principle:

$h_{top}(T,\mathcal{U})=\max_{\mu\in M(X,T)}h_{\mu}(T,\mathcal{U}),$

and the supremum can be attained by an ergodic measure. Later, Glasner and
Weiss [15] proved that if $(X,T)$ is invertible then the local variational
principle also holds for $h_{\mu}^{+}(T,\mathcal{U}):$

$h_{top}(T,\mathcal{U})=\max_{\mu\in M(X,T)}h^{+}_{\mu}(T,\mathcal{U}).$

Huang and Yi [25] generalized the above local variational principles of
entropy to the case of pressure: For any $f\in C(X,\mathbb{R})$, the local
pressure $P(T,f,\mathcal{U})$ satisfies

$P(T,f,\mathcal{U})=\max_{\mu\in
M(X,T)}\Big{(}h_{\mu}(T,\mathcal{U})+\int_{X}fd\mu\Big{)},$

and the supremum can be attained by an ergodic measure. The relative local
variational principle for entropy was proved by Huang, Ye and Zhang in [24].
In [18], Wu studied various notions of local pressure of subsets and measures,
which were defined by Carathéordory–Pesin construction.

In this paper, we study the local weighted topological pressure with respect
to a fixed open cover. We generalize the main results in [21] to localized
version. As a corallary, we get the variational principle for weighted
topological pressure, which was first proved in [21].

##  2\. Local Weighted topological pressure

In this section, we introduce the notion of local weighted topological
pressure and give some basic properties of it. First let us recall the
definition of weighted topological pressure introduced in [21].

Let $k\geq 1,$ ${\bf a}=(a_{1},\ldots,a_{k})$ with $a_{1}>0$ and $a_{i}\geq
0,i=2,\ldots,k$. Let $(X_{i},T_{i}),i=1,\ldots,k$ be TDSs with metric $d_{i}$
and $(X_{i+1},T_{i+1})$ be a factor of $(X_{i},T_{i})$ for each $i$. Let
$\tau_{i}:X_{1}\to X_{i+1}$ be the factor map and set $\tau_{0}={\rm
id}_{X_{1}}.$

######  Definition 2.1.

[21, a-weighted Bowen ball] For $x\in X_{1},n\in\mathbb{N}$ and $\epsilon>0,$
denote

$B_{n}^{\bf a}(x,\epsilon):=\\{y\in
X_{1}:d_{i}(T_{i}^{j}\tau_{i-1}x,T_{i}^{j}\tau_{i-1}y)<\epsilon,0\leq
j\leq\lceil(a_{1}+\ldots+a_{i})n\rceil-1,1\leq i\leq k\\},$

where $\lceil u\rceil$ denotes the least integer $\geq u.$

For $f\in C(X_{1},\mathbb{R})$, denote $S_{n}f=\sum_{i=0}^{n-1}f\circ
T_{1}^{i}.$ Now we give the definition of a-weighted topological pressure.

######  Definition 2.2.

[21, a-weighted topological pressure] Let $Z\subset X_{1},N\in\mathbb{N},s\geq
0,$ $\epsilon>0$ and $f\in C(X_{1},\mathbb{R})$. Define

$\Lambda^{{\bf
a},s}_{N,\epsilon}(Z,f)=\inf\sum_{j}e^{-sn_{j}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}S_{\lceil a_{1}n_{j}\rceil}f(x)},$

where the infimum is taken over all countable collections
$\\{(n_{j},A_{j})\\}_{j}$ such that $n_{j}\geq N,A_{j}$ be a Borel subset of
$B_{n_{j}}^{\bf a}(x_{j},\epsilon)$ for some $x_{j}\in X_{1}$ and
$Z\subset\bigcup_{j}A_{j}.$ Define

$\Lambda^{{\bf a},s}_{\epsilon}(Z,f)=\lim_{N\to\infty}\Lambda^{{\bf
a},s}_{N,\epsilon}(Z,f)$

and

$\begin{split}P^{{\bf a}}(T_{1},Z,\epsilon,f)&=\inf\\{s:\Lambda^{{\bf
a},s}_{\epsilon}(Z,f)=0\\}\\\ &=\sup\\{s:\Lambda^{{\bf
a},s}_{\epsilon}(Z,f)=+\infty\\}.\end{split}$

The a-weighted topological pressure for $Z$ with potential $f$ is defined as

$P^{{\bf a}}(T_{1},Z,f)=\lim_{\epsilon\to 0}P^{{\bf a}}(T_{1},Z,\epsilon,f).$

For a function $g:X_{1}\to[0,+\infty)$, define

$W^{{\bf
a},s}_{N,\epsilon}(g)=\inf\sum_{j}c_{j}e^{-sn_{j}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}S_{\lceil a_{1}n_{j}\rceil}f(x)},$

where the infimum is taken over all countable collections
$\\{(n_{j},A_{j},c_{j})\\}_{j}$ such that $n_{j}\geq N,$ $0<c_{j}<\infty$,
$A_{j}$ be a Borel subset of $B_{n_{j}}^{\bf a}(x_{j},\epsilon)$ for some
$x_{j}\in X_{1}$ and

$\sum_{j}c_{j}\chi_{A_{j}}\geq g.$

For $Z\subset X_{1},$ we set $W^{{\bf a},s}_{N,\epsilon}(Z,f)=W^{{\bf
a},s}_{N,\epsilon}(\chi_{Z}).$ Define

$W^{{\bf a},s}_{\epsilon}(Z,f)=\lim_{N\to\infty}W^{{\bf
a},s}_{N,\epsilon}(Z,f)$

and

$\begin{split}P_{W}^{{\bf a}}(T_{1},Z,\epsilon,f)&=\inf\\{s:W^{{\bf
a},s}_{\epsilon}(Z,f)=0\\}\\\ &=\sup\\{s:W^{{\bf
a},s}_{\epsilon}(Z,f)=+\infty\\}.\end{split}$

The average a-weighted topological pressure for $Z$ with potential $f$ is
defined as

$P_{W}^{{\bf a}}(T_{1},Z,f)=\lim_{\epsilon\to 0}P_{W}^{{\bf
a}}(T_{1},Z,\epsilon,f).$

If there is no confusion, we omit $T_{1},f$ and simply write $P^{{\bf
a}}(Z),P_{W}^{{\bf a}}(Z)$ and $P^{{\bf a}}(Z,\epsilon),P_{W}^{{\bf
a}}(Z,\epsilon)$ for short.

It was proved in [21] that weighted topological pressure and average weighted
topological pressure are equal:

######  Proposition 2.3.

[21, Proposition 3.5] For any $f\in C(X_{1},\mathbb{R})$ and $\epsilon>0$, we
have

$P^{{\bf a}}(T_{1},Z,6\epsilon,f)\leq P_{W}^{{\bf a}}(T_{1},Z,\epsilon,f)\leq
P^{{\bf a}}(T_{1},Z,\epsilon,f),$

hence $P^{{\bf a}}(Z,f)=P_{W}^{{\bf a}}(Z,f).$

We now define local weighted topological pressure for a fixed open cover. The
entropy case please also see [20].

######  Definition 2.4.

Let $Z\subset X_{1},N\in\mathbb{N},s\geq 0$, $f\in C(X_{1},\mathbb{R})$ and
$\mathcal{U}_{i}$ be open covers of $X_{i},i=1,\ldots,k$. Define

$\Lambda^{{\bf
a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)=\inf\sum_{j}e^{-sn_{j}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}S_{\lceil a_{1}n_{j}\rceil}f(x)},$

where the infimum is taken over all countable collections
$\\{(n_{j},A_{j})\\}_{j}$ such that $n_{j}\geq N,$ $Z\subset\bigcup_{j}A_{j}$
and $A_{j}$ be a Borel subset of some element of
$\bigvee\limits_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n_{j}\rceil-1}.$
Since $\Lambda^{{\bf a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)$ does not
decrease with $N$, define

$\Lambda^{{\bf
a},s}_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)=\lim_{N\to\infty}\Lambda^{{\bf
a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)$

and

$\begin{split}P^{{\bf
a}}(T_{1},Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)&=\inf\\{s:\Lambda^{{\bf
a},s}_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)=0\\}\\\ &=\sup\\{s:\Lambda^{{\bf
a},s}_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)=+\infty\\}.\end{split}$

For a function $g:X_{1}\to[0,+\infty)$, define

$W^{{\bf
a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(g)=\inf\sum_{j}c_{j}e^{-sn_{j}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}S_{\lceil a_{1}n_{j}\rceil}f(x)},$

where the infimum is taken over all countable collections
$\\{(n_{j},A_{j},c_{j})\\}_{j}$ such that $n_{j}\geq N,$ $0<c_{j}<\infty$,
$A_{j}$ be a Borel subset of some element of
$\bigvee\limits_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n_{j}\rceil-1}$
and $\sum_{j}c_{j}\chi_{A_{j}}\geq g.$ For $Z\subset X_{1},$ we set

$W^{{\bf a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)=W^{{\bf
a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(\chi_{Z}).$

Since $W^{{\bf a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)$ does not
decrease with $N$, define

$W^{{\bf a},s}_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)=\lim_{N\to\infty}W^{{\bf
a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)$

and

$\begin{split}P_{W}^{{\bf
a}}(T_{1},Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)&=\inf\\{s:W^{{\bf
a},s}_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)=0\\}\\\ &=\sup\\{s:W^{{\bf
a},s}_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(Z,f)=+\infty\\}.\end{split}$

If there is no confusion, we omit $T_{1},f$ and simply write $P^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k}),P_{W}^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k})$ for short. If $Z=X_{1}$, we also write
$P^{{\bf a}}(\\{\mathcal{U}_{i}\\}_{i=1}^{k})$ and $P_{W}^{{\bf
a}}(\\{\mathcal{U}_{i}\\}_{i=1}^{k})$ instead of $P^{{\bf
a}}(X_{1},\\{\mathcal{U}_{i}\\}_{i=1}^{k})$ and $P_{W}^{{\bf
a}}(X_{1},\\{\mathcal{U}_{i}\\}_{i=1}^{k}).$

For an open cover $\mathcal{U},$ denote
$diam\mathcal{U}=\max_{U\in\mathcal{U}}diamU.$ The relation between weighted
topological pressure and local weighted topological pressure is the following:

######  Theorem 2.5.

For subset $Z\subset X_{1},$ we have

$\begin{split}P^{{\bf a}}(Z)&=\sup_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}P^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k})=\lim_{\max\limits_{1\leq i\leq
k}diam\mathcal{U}_{i}\to 0}P^{{\bf a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k})\\\
&=P_{W}^{{\bf a}}(Z)=\sup_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}P_{W}^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k})=\lim_{\max\limits_{1\leq i\leq
k}diam\mathcal{U}_{i}\to 0}P_{W}^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k}),\end{split}$

where the supremum are taken over all open covers $\mathcal{U}_{i}$ of
$X_{i},i=1,\ldots,k.$

###### Proof.

By Proposition 2.3 we have $P^{{\bf a}}(Z)=P_{W}^{{\bf a}}(Z).$ We prove the
equality for $P^{{\bf a}}(Z)$, the one for $P_{W}^{{\bf a}}(Z)$ is similar.
Let $\mathcal{U}_{i}$ be open covers of $X_{i}$ and the Lebesgue number of
$\mathcal{U}_{i}$ be $\epsilon_{i},1\leq i\leq k$. Let
$\epsilon<\min_{i}\frac{\epsilon_{i}}{2}.$ Then

$P^{{\bf a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k})\leq P^{{\bf a}}(Z,\epsilon).$

Hence

$\sup_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}P^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k})\leq P^{{\bf a}}(Z).$

Let $\epsilon>0.$ If $\max\limits_{1\leq i\leq
k}diam\mathcal{U}_{i}<\epsilon$, then

$P^{{\bf a}}(Z,\epsilon)\leq P^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k})\leq\sup_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}P^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k}).$

Hence

$P^{{\bf a}}(Z)\leq\lim_{\max\limits_{1\leq i\leq k}diam\mathcal{U}_{i}\to
0}P^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k})\leq\sup_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}P^{{\bf
a}}(Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k}).$

∎

For $M\in\mathbb{N},$ $\mathcal{U}_{i}$ be open cover of $X_{i}$ and $f\in
C(X_{1},\mathbb{R})$, recall $S_{M}f=\sum_{i=0}^{M-1}f\circ T_{1}^{i}$ and
$(\mathcal{U}_{i})_{0}^{M-1}=\bigvee\limits_{l=0}^{M-1}T_{i}^{-l}\mathcal{U}_{i}.$
We have the following proposition:

######  Proposition 2.6.

Let $M\in\mathbb{N},Z\subset X_{1},f\in C(X_{1},\mathbb{R})$ and
$\mathcal{U}_{i}$ be open covers of $X_{i},i=1,\ldots,k$. We have

$P^{{\bf a}}(T_{1},Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)=\frac{1}{M}P^{{\bf
a}}(T_{1}^{M},Z,\\{(\mathcal{U}_{i})_{0}^{M-1}\\}_{i=1}^{k},S_{M}f),$
$P_{W}^{{\bf
a}}(T_{1},Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)=\frac{1}{M}P_{W}^{{\bf
a}}(T_{1}^{M},Z,\\{(\mathcal{U}_{i})_{0}^{M-1}\\}_{i=1}^{k},S_{M}f).$

###### Proof.

We prove the equality for $P^{{\bf a}}$, the one for $P_{W}^{{\bf a}}$ is
similar. Let

$h=P^{{\bf a}}(T_{1},Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f).$

For any $s<Mh,$ we have

$\Lambda^{{\bf
a},\frac{s}{M}}_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(T_{1},Z,f)=+\infty,$

hence there exists some $N$ such that

$\Lambda^{{\bf
a},\frac{s}{M}}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(T_{1},Z,f)\geq
e^{\frac{M||f||}{a_{1}}}.$

Now we prove

$\Lambda^{{\bf
a},s}_{N,\\{(\mathcal{U}_{i})_{0}^{M-1}\\}_{i=1}^{k}}(T_{1}^{M},Z,S_{M}f)\geq
1.$

Consider

$\sum_{j}e^{-sn_{j}+\frac{1}{a_{1}}\sup_{x\in A_{j}}\sum_{i=0}^{{\lceil
a_{1}n_{j}\rceil}-1}(S_{M}f)\circ T_{1}^{Mi}(x)},$

where $n_{j}\geq N,$ $Z\subset\bigcup_{j}A_{j}$ and $A_{j}$ is a Borel subset
of some element of

$\bigvee_{i=1}^{k}\bigvee_{l=0}^{\lceil(a_{1}+\ldots+a_{i})n_{j}\rceil-1}T_{1}^{-Ml}\tau_{i-1}^{-1}(\mathcal{U}_{i})_{0}^{M-1}=\bigvee_{i=1}^{k}\tau_{i-1}^{-1}(\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n_{j}\rceil
M-1}.$

Since

$\bigvee_{i=1}^{k}\tau_{i-1}^{-1}(\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n_{j}\rceil
M-1}\succeq\bigvee_{i=1}^{k}\tau_{i-1}^{-1}(\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})Mn_{j}\rceil-1}$

and $Mn_{j}>N,$ by definition we have

$\sum_{j}e^{-\frac{s}{M}Mn_{j}+\frac{1}{a_{1}}\sup_{x\in A_{j}}S_{\lceil
a_{1}Mn_{j}\rceil}f(x)}\geq e^{\frac{M||f||}{a_{1}}}.$

It is easy to check that

$\sum_{j}e^{-\frac{s}{M}Mn_{j}+\frac{1}{a_{1}}\sup_{x\in A_{j}}S_{\lceil
a_{1}Mn_{j}\rceil}f(x)}\leq\sum_{j}e^{-\frac{s}{M}Mn_{j}+\frac{M||f||}{a_{1}}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}S_{M\lceil a_{1}n_{j}\rceil}f(x)}.$

It follows that

$\Lambda^{{\bf
a},s}_{N,\\{(\mathcal{U}_{i})_{0}^{M-1}\\}_{i=1}^{k}}(T_{1}^{M},Z,S_{M}f)\geq
1.$

Let $s\to Mh$ we have

$Mh\leq P^{{\bf
a}}(T_{1}^{M},Z,\\{(\mathcal{U}_{i})_{0}^{M-1}\\}_{i=1}^{k},S_{M}f).$

Now we prove the opposite direction. Let

$h=P^{{\bf
a}}(T_{1}^{M},Z,\\{(\mathcal{U}_{i})_{0}^{M-1}\\}_{i=1}^{k},S_{M}f).$

For any $s<\frac{h}{M},$ we have

$\Lambda^{{\bf
a},Ms}_{\\{(\mathcal{U}_{i})_{0}^{M-1}\\}_{i=1}^{k}}(T_{1}^{M},Z,S_{M}f)=+\infty.$

There exists $N_{0}$ such that

$\Lambda^{{\bf
a},Ms}_{N_{0},\\{(\mathcal{U}_{i})_{0}^{M-1}\\}_{i=1}^{k}}(T_{1}^{M},Z,S_{M}f)\geq
e^{Ms(\frac{1}{a_{1}}+1)+\frac{\lceil a_{1}M+M\rceil||f||}{a_{1}}}.$

Let $N>M(N_{0}+1+\frac{1}{a_{1}})$. Now we prove that

$\Lambda^{{\bf a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(T_{1},Z,f)\geq 1.$

Consider

$\sum_{j}e^{-sn_{j}+\frac{1}{a_{1}}\sup_{x\in A_{j}}S_{\lceil
a_{1}n_{j}\rceil}f(x)},$

where $n_{j}\geq N,$ $Z\subset\bigcup_{j}A_{j}$ and $A_{j}$ is a Borel subset
of some element of
$\bigvee\limits_{i=1}^{k}\tau_{i-1}^{-1}(\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n_{j}\rceil-1}.$
Let

$\tilde{n}_{j}=\lceil\frac{n_{j}}{M}-\frac{1}{a_{1}}\rceil-1\geq N_{0}.$

Note that

$\begin{split}\bigvee_{i=1}^{k}\tau_{i-1}^{-1}(\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n_{j}\rceil-1}&\succeq\bigvee_{i=1}^{k}\tau_{i-1}^{-1}(\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})\tilde{n}_{j}\rceil
M-1}\\\
&=\bigvee_{i=1}^{k}\bigvee_{l=0}^{\lceil(a_{1}+\ldots+a_{i})\tilde{n}_{j}\rceil-1}T^{-Ml}_{1}\tau_{i-1}^{-1}(\mathcal{U}_{i})_{0}^{M-1}.\end{split}$

By definition we have

$\sum_{j}e^{-Ms\tilde{n}_{j}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}\sum_{i=0}^{{\lceil a_{1}\tilde{n}_{j}\rceil}-1}(S_{M}f)\circ
T_{1}^{Mi}(x)}\geq e^{Ms(\frac{1}{a_{1}}+1)+\frac{\lceil
a_{1}M+M\rceil||f||}{a_{1}}}.$

It is easy to check that

$\begin{split}\sum_{j}e^{-Ms\tilde{n}_{j}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}S_{M\lceil
a_{1}\tilde{n}_{j}\rceil}f(x)}&\leq\sum_{j}e^{-Ms(\frac{n_{j}}{M}-\frac{1}{a_{1}}-1)+\frac{\lceil
a_{1}M+M\rceil||f||}{a_{1}}+\frac{1}{a_{1}}\sup_{x\in A_{j}}S_{\lceil
a_{1}n_{j}\rceil}f(x)}\\\ &=\sum_{j}e^{-sn_{j}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}S_{\lceil a_{1}n_{j}\rceil}f(x)}e^{Ms(\frac{1}{a_{1}}+1)+\frac{\lceil
a_{1}M+M\rceil||f||}{a_{1}}}.\end{split}$

Hence

$\sum_{j}e^{-sn_{j}+\frac{1}{a_{1}}\sup_{x\in A_{j}}S_{\lceil
a_{1}n_{j}\rceil}f(x)}\geq 1.$

It follows that

$\Lambda^{{\bf a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(T_{1},Z,f)\geq 1.$

Let $s\to\frac{h}{M}$, we have

$\frac{h}{M}\leq P^{{\bf a}}(T_{1},Z,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f).$

∎

##  3\. lower bound

In order to prove the lower bound of the variational principle for weighted
topological pressure, in [21] the authors established weighted version of
Shannon–McMillan–Breiman theorem ([21, Proposition A.2]) and weighted version
of Brin–Katok theorem ([21, Theorem 4.1]). It was shown in [21, Proposition
4.2] that

$P^{{\bf a}}(T_{1},f)\geq\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i})+\int
fd\mu$

for any $\mu\in M(X_{1},T_{1})$ and $f\in C(X_{1},\mathbb{R})$. In this
section, we generalize the above lower bound part of the variational
principle, with the notion of local weighted topological pressure.

We first need some lemmas:

######  Proposition 3.1.

[21, Proposition A.2] Let $(X,T)$ be a TDS, $\mathcal{B}$ be the collection of
all Borel sets of $X$, $\mu\in M(X,T)$ and $k\geq 1.$ Let
$\alpha_{1},\ldots,\alpha_{k}\in\mathcal{P}_{X}$ with
$H_{\mu}(\alpha_{i})<\infty,1\leq i\leq k$, and ${\bf
a}=(a_{1},\ldots,a_{k})\in\mathbb{R}^{k}$ with $a_{1}>0$ and $a_{i}\geq
0,i=2,\ldots,k.$ Then

$\lim_{N\to+\infty}\frac{1}{N}I_{\mu}\big{(}\bigvee_{i=1}^{k}(\alpha_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}(x)=\sum_{i=1}^{k}a_{i}\mathbb{E}_{\mu}(F_{i}|\mathcal{I}_{\mu})(x)$

almost everywhere, where

$F_{i}(x)=I_{\mu}\Big{(}\bigvee_{j=i}^{k}\alpha_{j}|\bigvee_{n=1}^{\infty}T^{-n}(\bigvee_{j=i}^{k}\alpha_{j})\Big{)}(x),i=1,\ldots,k$

and $\mathcal{I}_{\mu}=\\{B\in\mathcal{B}:\mu(B\Delta T^{-1}B)=0\\}$. In
particular, if $T$ is ergodic, we have

$\lim_{N\to+\infty}\frac{1}{N}I_{\mu}\big{(}\bigvee_{i=1}^{k}(\alpha_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}(x)=\sum_{i=1}^{k}a_{i}h_{\mu}(T,\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\alpha_{j}).$

######  Lemma 3.2.

Let $(X,T)$ be a TDS, $\alpha=\\{A_{1},\ldots,A_{m}\\}\in\mathcal{P}_{X},$
$\mu\in M(X,T)$ and $\delta>0$. Then there exists an open cover
$\mathcal{U}=\\{U_{0},U_{1},\ldots,U_{m}\\}$ such that

  1. (1)

$\mu(U_{0})<\delta.$

  2. (2)

$\mu(A_{i}\Delta U_{i})<\delta,1\leq i\leq m.$

  3. (3)

$U_{1},\ldots,U_{m}$ are pairwise disjoint.

###### Proof.

We can find compact sets $K_{i}\subset A_{i}$ with
$\mu(A_{i}-K_{i})<\frac{\delta}{m},1\leq i\leq m.$ Since $K_{1},\ldots,K_{m}$
are pairwise disjoint, we can find open sets $K_{i}\subset U_{i}$ such that
$U_{1},\ldots,U_{m}$ are pairwise disjoint. It is easy to see that
$\mu(\bigcup_{i=1}^{m}U_{i})>1-\delta.$ Hence we can find an open set $U_{0}$
such that $(\bigcup_{i=1}^{m}U_{i})^{c}\subset U_{0}$ and $\mu(U_{0})<\delta.$
It is easy to see that $A_{i}\Delta
U_{i}\subset\bigcup_{i=1}^{m}(A_{i}-K_{i})$ and hence $\mu(A_{i}\Delta
U_{i})<\delta,1\leq i\leq m.$ ∎

For a finite set $M$, denote $|M|$ the cardinality of $M.$ The following
theorem is the main result of this section:

######  Theorem 3.3.

Let $\alpha_{i}\in\mathcal{P}_{X_{i}},1\leq i\leq k$ and $\mu\in
M(X_{1},T_{1})$. Then for any $\epsilon>0$, there exist open covers
$\mathcal{U}_{i}$ of $X_{i}$ such that $|\mathcal{U}_{i}|=|\alpha_{i}|+1$,
$\tau_{i-1}\mu(\mathcal{U}_{i}\Delta\alpha_{i})<\epsilon,1\leq i\leq k$ and

$\begin{split}P^{{\bf
a}}(supp\mu,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)\geq\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\alpha_{j})+\int
fd\mu-\epsilon.\end{split}$

###### Proof.

step 1: reduce to an ergodic measure.

Assume $\alpha_{i}=\\{A_{1}^{i},\ldots,A_{m_{i}}^{i}\\},1\leq i\leq k$. Set
$M=\max_{1\leq i\leq k}m_{i}.$ By [19, Lemma 4.15], there exists $\delta_{1}$
such that if
$\alpha,\beta\in\mathcal{P}_{X_{1}},|\alpha|,|\beta|\leq(M+1)^{k}$ and
$\mu(\alpha\Delta\beta)<\delta_{1},$ then

$|h_{\mu}(T_{1},\alpha)-h_{\mu}(T_{1},\beta)|<\epsilon.$

Choose $\delta>0$ such that

$\delta<\frac{1}{k}\min\\{\frac{\epsilon}{k\log(M+1)\sum_{i=1}^{k}a_{i}+||f||+\epsilon},\frac{\epsilon}{2\log(M+1)},\frac{\delta_{1}}{M+1},(\frac{\epsilon}{M+1})^{\frac{1}{2}},1\\}.$

By Lemma 3.2, for each $1\leq i\leq k$ there exists an open cover
$\mathcal{U}_{i}=\\{U_{0}^{i},U_{1}^{i},\ldots,U_{m_{i}}^{i}\\}$ such that

  * •

$\tau_{i-1}\mu(U_{0}^{i})<\delta^{2}.$

  * •

$\tau_{i-1}\mu(A_{l}^{i}\Delta U_{l}^{i})<\delta^{2},1\leq l\leq m_{i}.$

  * •

$U_{1}^{i},\ldots,U_{m_{i}}^{i}$ are pairwise disjoint.

It is easy to see that
$\tau_{i-1}\mu(\mathcal{U}_{i}\Delta\alpha_{i})<\epsilon.$ Let

$\beta_{i}=\\{(\bigcup_{l=1}^{m_{i}}U_{l}^{i})^{c},U_{1}^{i},\ldots,U_{m_{i}}^{i}\\}\in\mathcal{P}_{X_{i}}.$

Then

$\tau_{i-1}\mu(\alpha_{i}\Delta\beta_{i})<(M+1)\delta$

and hence

$\mu\Big{(}(\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\alpha_{j})\Delta(\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})\Big{)}<\delta_{1}.$

It follows that

(1)
$\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})>\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\alpha_{j})-\sum_{i=1}^{k}a_{i}\epsilon.$

Let

$\mu=\int_{M^{e}(X_{1},T_{1})}\theta dm(\theta)$

be the ergodic decomposition of $\mu$. Denote

$I_{\mu}=\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})+\int
fd\mu$

and

$I_{\theta}=\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})+\int
fd\theta$

for $\theta\in M^{e}(X_{1},T_{1})$. Then

$I_{\mu}=\int_{M^{e}(X_{1},T_{1})}I_{\theta}dm(\theta)=\int_{\\{\theta:I_{\theta}>I_{\mu}-\epsilon\\}}I_{\theta}dm(\theta)+\int_{\\{\theta:I_{\theta}\leq
I_{\mu}-\epsilon\\}}I_{\theta}dm(\theta).$

Since

$|\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j}|\leq(M+1)^{k},$

we have

$I_{\theta}\leq k\log(M+1)\sum_{i=1}^{k}a_{i}+||f||.$

Hence

$\int_{\\{\theta:I_{\theta}>I_{\mu}-\epsilon\\}}I_{\theta}dm(\theta)\leq\big{(}k\log(M+1)\sum_{i=1}^{k}a_{i}+||f||\big{)}m(\\{\theta:I_{\theta}>I_{\mu}-\epsilon\\})$

and

$\int_{\\{\theta:I_{\theta}\leq
I_{\mu}-\epsilon\\}}I_{\theta}dm(\theta)\leq(I_{\mu}-\epsilon)m(\\{\theta:I_{\theta}\leq
I_{\mu}-\epsilon\\}).$

We get

$m(\\{\theta:I_{\theta}>I_{\mu}-\epsilon\\})\geq\frac{\epsilon}{k\log(M+1)\sum_{i=1}^{k}a_{i}+||f||+\epsilon}>k\delta.$

Since for each $1\leq i\leq k,$ $\tau_{i-1}\mu(U_{0}^{i})<\delta^{2},$ it is
easy to see that

$m(\\{\theta:\theta(\tau_{i-1}^{-1}U_{0}^{i})<\delta\\})>1-\delta.$

Hence there exists $\theta\in M^{e}(X_{1},T_{1})$ such that

(2) $\theta(\tau_{i-1}^{-1}U_{0}^{i})<\delta,\ \forall\ 1\leq i\leq k$

and

(3)
$\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})+\int
fd\theta>\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})+\int
fd\mu-\epsilon.$

step 2: deal with the ergodic measure $\theta$ and finish the proof.

Here we use the method in [27] (please also see [20]). For $y\in
supp\theta,1\leq i\leq k$ and $n\in\mathbb{N},$ write $t^{i}_{n}(y)$ for the
number of integers $0\leq j\leq\lceil(a_{1}+\ldots+a_{i})n\rceil-1$ such that

$T^{j}_{i}\tau_{i-1}y\in U_{0}^{i}.$

By the Birkhoff Ergodic Theorem and (2), there exist $N_{1}\in\mathbb{N}$ and
$B_{1}\subset supp\theta$ such that $\theta(B_{1})>\frac{5}{6}$ and for all
$y\in B_{1},n\geq N_{1}$ and $1\leq i\leq k,$

$t^{i}_{n}(y)<2\delta\lceil(a_{1}+\ldots+a_{i})n\rceil.$

By Proposition 3.1, there exist $N_{2}\in\mathbb{N}$ and $B_{2}\subset
supp\theta$ such that $\theta(B_{2})>\frac{5}{6}$ and for all $y\in
B_{2},n\geq N_{2},$

$\theta\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\beta_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n\rceil-1}(y)\big{)}\leq
e^{-n\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})+n\epsilon}.$

By the Birkhoff Ergodic Theorem there exist $N_{3}\in\mathbb{N}$ and
$B_{3}\subset supp\theta$ such that $\theta(B_{3})>\frac{5}{6}$ and for all
$y\in B_{3},n\geq N_{3},$

$|\frac{1}{a_{1}n}S_{\lceil a_{1}n\rceil}f(y)-\int fd\theta|<\epsilon.$

Let $N=\max\\{N_{1},N_{2},N_{3}\\}$ and $B=B_{1}\cap B_{2}\cap B_{3}.$ We have
$\theta(B)>\frac{1}{2}.$ Let

$\lambda<\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})-\sum_{i=1}^{k}(a_{1}+\ldots+a_{i})\epsilon-2\epsilon+\int
fd\theta.$

We will prove $\Lambda^{{\bf
a},\lambda}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(supp\theta)\geq\frac{1}{2}e^{-k\epsilon}.$
Consider

$\sum_{j}e^{-\lambda n_{j}+\frac{1}{a_{1}}\sup_{x\in A_{j}}S_{\lceil
a_{1}n_{j}\rceil}f(x)},$

where $n_{j}\geq N,$ $A_{j}$ is a Borel subset of some element of
$\bigvee\limits_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n_{j}\rceil-1}$
and $supp\theta\subset\bigcup_{j}A_{j}$. For $l\geq N$, let

$\Gamma_{l}=\\{(n_{j},A_{j}):n_{j}=l,A_{j}\cap B\neq\emptyset\\}$

and

$Y_{l}=\bigcup_{(n_{j},A_{j})\in\Gamma_{l}}A_{j}.$

Denote

$L=\\{C\in\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\beta_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})l\rceil-1}:C\cap
Y_{l}\cap B\neq\emptyset\\}.$

Note that

$\sum_{C\in L}\theta(C)\geq\theta(Y_{l}\cap B)$

and $C\cap B_{2}\neq\emptyset$ for $C\in L.$ We have

$|L|\geq\theta(Y_{l}\cap
B)e^{l\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})-l\epsilon}.$

For $(n_{j},A_{j})\in\Gamma_{l},$ by the definition of $B_{1}$ we have

$\begin{split}&|\\{C\in\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\beta_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})l\rceil-1}:C\cap
A_{j}\cap B\neq\emptyset\\}|\\\
&\leq\prod_{i=1}^{k}(M+1)^{2\delta\lceil(a_{1}+\ldots+a_{i})l\rceil}\leq
e^{l\epsilon\sum_{i=1}^{k}(a_{1}+\ldots+a_{i})+k\epsilon},\end{split}$

the second inequality holds because $2\delta\log(M+1)<\epsilon.$ We have

$|\Gamma_{l}|\geq\theta(Y_{l}\cap
B)e^{l\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})-l\epsilon-l\epsilon\sum_{i=1}^{k}(a_{1}+\ldots+a_{i})-k\epsilon}.$

For $(n_{j},A_{j})\in\Gamma_{l},A_{j}\cap B_{3}\neq\emptyset,$ we have

$\begin{split}&\sum_{j}e^{-\lambda n_{j}+\frac{1}{a_{1}}\sup_{x\in
A_{j}}S_{\lceil a_{1}n_{j}\rceil}f(x)}\\\
&\geq\sum_{l=N}^{\infty}\sum_{(n_{j},A_{j})\in\Gamma_{l}}e^{-\lambda
l+\frac{1}{a_{1}}\sup_{x\in A_{j}}S_{\lceil a_{1}l\rceil}f(x)}\\\
&\geq\sum_{l=N}^{\infty}\sum_{(n_{j},A_{j})\in\Gamma_{l}}e^{-\lambda l+l\int
fd\theta-l\epsilon}\\\ &\geq\sum_{l=N}^{\infty}\theta(Y_{l}\cap B)e^{-\lambda
l+l\int
fd\theta+l\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})-2l\epsilon-l\epsilon\sum_{i=1}^{k}(a_{1}+\ldots+a_{i})-k\epsilon}\\\
&\geq\sum_{l=N}^{\infty}\theta(Y_{l}\cap
B)e^{-k\epsilon}\geq\theta(B)e^{-k\epsilon}\geq\frac{1}{2}e^{-k\epsilon}.\end{split}$

Hence $\Lambda^{{\bf
a},\lambda}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(supp\theta)\geq\frac{1}{2}e^{-k\epsilon}.$
It follows that

(4) $P^{{\bf
a}}(supp\theta,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)\geq\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\beta_{j})-\sum_{i=1}^{k}(a_{1}+\ldots+a_{i})\epsilon-2\epsilon+\int
fd\theta.$

Since $supp\theta\subset supp\mu,$ combining (1), (3) and (4) we finish the
proof. ∎

######  Remark 3.4.

In [23], the authors proved the following lemma:

######  Lemma 3.5.

[23, Lemma 9] Let $(X,T)$ be a TDS. For every $M\in\mathbb{N},\epsilon>0$,
there exists $\delta>0$ such that for every $M$-set measurable covers
$\mathcal{U}=\\{U_{1},\ldots,U_{M}\\},\mathcal{V}=\\{V_{1},\ldots,V_{M}\\}$ of
$X$ with $\mu(\mathcal{U}\Delta\mathcal{V})<\delta,$ one has

$|h^{+}_{\mu}(T,\mathcal{U})-h^{+}_{\mu}(T,\mathcal{V})|<\epsilon.$

Hence in Theorem 3.3, we can further require the open covers $\mathcal{U}_{i}$
satisfying

$|h^{+}_{\tau_{i-1}\mu}(T_{i},\mathcal{U}_{i})-h_{\tau_{i-1}\mu}(T_{i},\alpha_{i})|<\epsilon$

and

$|\sum_{i=1}^{k}a_{i}h^{+}_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})-\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\alpha_{j})|<\epsilon.$

From Theorem 3.3 we can get the lower bound part of the variational principle
for weighted topological pressure:

######  Corollary 3.6.

Let $f\in C(X_{1},\mathbb{R})$. We have

$P^{{\bf a}}(T_{1},f)\geq\sup_{\mu\in
M(X_{1},T_{1})}\big{(}\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i})+\int
fd\mu\big{)}.$

###### Proof.

Let $\alpha_{i}\in\mathcal{P}_{X_{i}},1\leq i\leq k,$ $\mu\in M(X_{1},T_{1})$
and $\epsilon>0$. By Theorem 3.3 there exist open covers $\mathcal{U}_{i}$ of
$X_{i}$ such that

$P^{{\bf
a}}(supp\mu,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)\geq\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\alpha_{j})+\int
fd\mu-\epsilon.$

By Theorem 2.5, we have

$P^{{\bf a}}(T_{1},f)\geq P^{{\bf
a}}(T_{1},\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)\geq P^{{\bf
a}}(supp\mu,\\{\mathcal{U}_{i}\\}_{i=1}^{k},f).$

Since $\alpha_{i}$ and $\epsilon$ are chosen arbitrary, we have

$P^{{\bf a}}(T_{1},f)\geq\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i})+\int
fd\mu.$

∎

##  4\. upper bound

In order to prove the upper bound of the variational principle for weighted
topological pressure, the authors in [21] applied the techniques in geometric
measure theory. They introduced the notion of average weighted topological
pressure to prove the dynamical Frostman lemma ([21, Lemma 3.3]), which played
a key role in the proof of the upper bound part of the variational principle.
In this section, we generalize the result of the upper bound part in [21] to
local case.

First we need some lemmas:

######  Lemma 4.1.

[22, Proposition 6] Let $\pi:(X,T)\to(Y,S)$ be a factor map between two TDSs
and $\mathcal{U}$ be an open cover of $Y$. Then for any $\mu\in M(X,T)$, we
have

$h_{\mu}(T,\pi^{-1}\mathcal{U})=h_{\pi\mu}(S,\mathcal{U}).$

######  Lemma 4.2.

[23, 24, 25] Let $(X,T)$ be a TDS, $\mu\in M(X,T)$ and $\mathcal{U}$ be an
open cover of $X$. If $(X,T)$ is invertible, then

$h_{\mu}^{+}(T,\mathcal{U})=h_{\mu}(T,\mathcal{U}).$

######  Lemma 4.3.

[23, Proposition 5] Let $(X,T)$ be a TDS, $\mu\in M(X,T)$ and $\mathcal{U}$ be
an open cover of $X$. Let

$\mu=\int_{M^{e}(X,T)}\theta dm(\theta)$

be the ergodic decomposition of $\mu$. Then

$h_{\mu}(T,\mathcal{U})=\int_{M^{e}(X,T)}h_{\theta}(T,\mathcal{U})dm(\theta).$

######  Lemma 4.4.

[26, Lemma 2.4] Let $(X,T)$ be a TDS, $\nu\in M(X)$ and $M\in\mathbb{N}.$
Suppose $\alpha\in\mathcal{P}_{X}$ and $|\alpha|\leq M.$ Then for any
$n,l\in\mathbb{N}$ satisfying $n\geq 2l,$ we have

$\frac{1}{n}H_{\nu}(\bigvee_{i=0}^{n-1}T^{-i}\alpha)\leq\frac{1}{l}H_{\nu_{n}}(\bigvee_{i=0}^{l-1}T^{-i}\alpha)+\frac{2l}{n}\log
M,$

where $\nu_{n}=\frac{1}{n}\sum_{i=0}^{n-1}T^{i}\nu.$

######  Lemma 4.5.

[21, Lemma 5.1] Let $(X,T)$ be a TDS and $\mu\in M(X,T)$. Suppose
$\alpha\in\mathcal{P}_{X}$ and $|\alpha|\leq M$. Write

$h(n)=H_{\frac{1}{n}\sum_{i=0}^{n-1}T^{i}\mu}(\alpha),$
$h(n,m)=H_{\frac{1}{m}\sum_{i=n}^{m+n-1}T^{i}\mu}(\alpha).$

Then

$|h(n+1)-h(n)|\leq\frac{1}{n+1}\log(3M^{2}(n+1)).$
$|h(n+m)-\frac{n}{n+m}h(n)-\frac{m}{n+m}h(n,m)|\leq\log 2.$

######  Lemma 4.6.

[21, Lemma 5.4] Let $p\in\mathbb{N}$ and
$u_{j}:\mathbb{N}\to\mathbb{R},j=1,\ldots,p$ be bounded functions with

$\lim_{n\to\infty}|u_{j}(n+1)-u_{j}(n)|=0.$

Then for any positive numbers $c_{1},\ldots,c_{p}$ and $r_{1},\ldots,r_{p}$,
we have

$\varlimsup_{n\to\infty}\sum_{j=1}^{p}\big{(}u_{j}(\lceil
c_{j}n\rceil)-u_{j}(\lceil r_{j}n\rceil)\big{)}\geq 0.$

######  Lemma 4.7.

[23, Lemma 2] Let $(X,T)$ be a TDS, $\mathcal{U}=\\{U_{1},\ldots,U_{m}\\}$ be
an open cover of $X$ and $G:\mathcal{P}_{X}\to\mathbb{R}$ be monotone in the
sense that $G(\alpha)\geq G(\beta)$ whenever $\alpha\succeq\beta.$ Then

$\inf_{\alpha\in\mathcal{P}_{X},\alpha\succeq\mathcal{U}}G(\alpha)=\inf_{\alpha:\alpha=\\{A_{1},\ldots,A_{m}\\},A_{i}\subset
U_{i},1\leq i\leq m}G(\alpha).$

Now we pay attention back to weighted topological pressure. We apply the
techniques in geometric measure theory and get the following useful lemma,
which can be seen as the local version of the dynamical Frostman lemma [21,
Lemma 3.3].

######  Lemma 4.8.

Let $s\geq 0,N\in\mathbb{N}$ and $\mathcal{U}_{i}$ be open covers of
$X_{i},i=1,\ldots,k$. Suppose that

$c=W^{{\bf a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(X_{1})>0,$

then there is a Borel probability measure $\mu$ on $X_{1}$ such that for any
$n\geq N$ and

$A\in\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n\rceil-1},$

we have

$\mu(A)\leq\frac{1}{c}e^{-ns+\frac{1}{a_{1}}\sup_{x\in A}S_{\lceil
a_{1}n\rceil}f(x)}.$

###### Proof.

Define a function $p$ on $C(X_{1})$ by

$p(g)=\frac{1}{c}W^{{\bf a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(g).$

It is easy to verify that

  1. (1)

$p(g_{1}+g_{2})\leq p(g_{1})+p(g_{2})$ for any $g_{1},g_{2}\in C(X_{1}).$

  2. (2)

$p(tg)=tp(g)$ for any $t\geq 0,g\in C(X_{1}).$

  3. (3)

$p({\bf 1})=1,0\leq p(g)\leq||g||_{\infty}$ for any $g\in C(X_{1}),$ and
$p(g)=0$ for $g\in C(X_{1})$ with $g\leq 0.$

By the Hahn-Banach theorem, we can extend the linear functional $t\to tp({\bf
1}),t\in\mathbb{R}$, from the subspace of the constant functions to a linear
functional $L:C(X_{1})\to\mathbb{R}$ satisfying $L({\bf 1})=p({\bf 1})=1$ and
$-p(-g)\leq L(g)\leq p(g)$ for any $g\in C(X_{1}).$ By the Riesz
representation theorem we can find a Borel probability measure $\mu$ on
$X_{1}$ such that $L(g)=\int gdu$ for $g\in C(X_{1}).$ By standard discussion
we can prove that for any $n\geq N$ and any
$A\in\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n\rceil-1},$
we have

$\mu(A)\leq\frac{1}{c}e^{-ns+\frac{1}{a_{1}}\sup_{x\in A}S_{\lceil
a_{1}n\rceil}f(x)}.$

∎

For open covers $\mathcal{U}_{i}$ of $X_{i}$ and $f\in C(X_{1},\mathbb{R})$,
denote

$w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})=\sup_{U\in\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i}}\\{|f(x)-f(y)|:x,y\in
U\\}.$

The following theorem is the main result of this section:

######  Theorem 4.9.

Let $\mathcal{U}_{i}$ be open covers of $X_{i},i=1,\ldots,k$ and $f\in
C(X_{1},\mathbb{R})$. Then there exists an ergodic measure $\mu\in
M(X_{1},T_{1})$ such that

$\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})\geq
P_{W}^{{\bf
a}}(T_{1},\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)-3w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int
fd\mu.$

###### Proof.

case 1: $(X_{1},T_{1})$ is invertible and zero-dimensional. We will prove
there exists an ergodic measure $\mu\in M(X_{1},T_{1})$ such that

$\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})\geq
P_{W}^{{\bf
a}}(T_{1},\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int
fd\mu.$

Write

$t_{0}(n)=0,\ t_{i}(n)=\lceil(a_{1}+\ldots+a_{i})n\rceil.$

For an open cover $\mathcal{U}=\\{U_{1},\ldots,U_{d}\\}$ of $X_{1}$, define

$\mathcal{U}^{*}=\\{\alpha\in\mathcal{P}_{X_{1}}:\alpha=\\{A_{1},\ldots,A_{d}\\},A_{m}\subset
U_{m},A_{m}\text{\ are clopen \ sets,\ }m=1,\ldots,d\\}.$

Denote $h=P_{W}^{{\bf a}}(\\{\mathcal{U}_{i}\\}_{i=1}^{k}).$ Without loss of
generality, we assume $h>0.$ Let

$M=\max_{1\leq i\leq k}|\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j}|.$

Claim: For every $\l\in\mathbb{N},$ the set

$\begin{split}&M(l)=\\{\mu\in M(X_{1},T_{1}):\forall\
\beta_{i}\in(\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})^{*},i=1,\ldots,k,\\\
&\sum_{i=1}^{k}a_{i}H_{\mu}(\bigvee_{h=0}^{l-1}T^{-h}\beta_{i})\geq
l\big{(}h-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int
fd\mu\big{)}-1-k\sum_{i=1}^{k}a_{i}\log 2\\}\end{split}$

is not empty.

proof of claim: Fix $l\in\mathbb{N}$. Choose some $s\geq 0$ such that
$h-\frac{1}{l}\leq s<h.$ By definition there exists $N\in\mathbb{N}$ such that
$W^{{\bf a},s}_{N,\\{\mathcal{U}_{i}\\}_{i=1}^{k}}(X_{1})\geq 1.$ By Lemma
4.8, there is a Borel probability measure $\nu$ on $X_{1}$ such that for any
$n\geq N$ and
$A\in\bigvee\limits_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n\rceil-1},$we
have

$\nu(A)\leq e^{-ns+\frac{1}{a_{1}}\sup_{x\in A}S_{\lceil a_{1}n\rceil}f(x)}.$

Define $\nu_{m}=\frac{1}{m}\sum_{j=0}^{m-1}T^{j}\nu$ and

$\omega_{i,n}=\frac{\sum_{j=t_{i-1}(n)}^{t_{i}(n)-1}T^{j}\nu}{t_{i}(n)-t_{i-1}(n)}$

for $1\leq i\leq k$ with $t_{i}(n)>t_{i-1}(n).$ For a Borel probability
measure $\theta$ on $X_{1}$ and $1\leq i\leq k$, define

$H_{\theta}(i)=\inf_{\beta\in(\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})^{*}}H_{\theta}(\bigvee\limits_{j=0}^{l-1}T^{-j}\beta).$

For

$n>\max\\{N,\\{\frac{2l+1}{a_{i}}:1\leq i\leq k,a_{i}\neq 0\\}\\}$

and

$\beta_{i}\in(\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})^{*},i=1,\ldots,k,$

we have

$\begin{split}\bigvee_{\begin{subarray}{c}i:1\leq i\leq k,\\\
t_{i}(n)>t_{i-1}(n)\end{subarray}}(\beta_{i})_{t_{i-1}(n)}^{t_{i}(n)-1}&\succeq\bigvee_{\begin{subarray}{c}i:1\leq
i\leq k,\\\
t_{i}(n)>t_{i-1}(n)\end{subarray}}(\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})_{t_{i-1}(n)}^{t_{i}(n)-1}\\\
&=\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})n\rceil-1}.\end{split}$

Hence for $A\in\bigvee\limits_{\begin{subarray}{c}i:1\leq i\leq k,\\\
t_{i}(n)>t_{i-1}(n)\end{subarray}}(\beta_{i})_{t_{i-1}(n)}^{t_{i}(n)-1}$, we
have

$\nu(A)\leq e^{-ns+\frac{\lceil
a_{1}n\rceil}{a_{1}}w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})+\frac{1}{a_{1}}\sup_{x\in
A}S_{\lceil a_{1}n\rceil}f(x)}.$

It follows that

$-\nu(A)\log\nu(A)\geq\big{(}ns-\frac{\lceil
a_{1}n\rceil}{a_{1}}w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\frac{1}{a_{1}}\sup_{x\in
A}S_{\lceil a_{1}n\rceil}f(x)\big{)}\nu(A)$

and

$H_{\nu}\Big{(}\bigvee_{\begin{subarray}{c}i:1\leq i\leq k,\\\
t_{i}(n)>t_{i-1}(n)\end{subarray}}\bigvee_{j=t_{i-1}(n)}^{t_{i}(n)-1}T^{-j}\beta_{i}\Big{)}\geq
ns-\frac{2\lceil
a_{1}n\rceil}{a_{1}}w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\frac{1}{a_{1}}\int
S_{\lceil a_{1}n\rceil}fdv.$

For $i$ with $1\leq i\leq k,a_{i}>0$, since $n\geq\frac{2l+1}{a_{i}}$, we have
$t_{i}(n)-t_{i-1}(n)\geq 2l$, hence by Lemma 4.4, we have

$\frac{t_{i}(n)-t_{i-1}(n)}{l}H_{\omega_{i,n}}(\bigvee_{j=0}^{l-1}T^{-j}\beta_{i})\geq
H_{\nu}\Big{(}\bigvee_{j=t_{i-1}(n)}^{t_{i}(n)-1}T^{-j}\beta_{i}\Big{)}-2l\log
M.$

It follows that

$\begin{split}&\sum_{i=1}^{k}\frac{t_{i}(n)-t_{i-1}(n)}{l}H_{\omega_{i,n}}(\bigvee_{j=0}^{l-1}T^{-j}\beta_{i})\\\
&\geq ns-\frac{2\lceil
a_{1}n\rceil}{a_{1}}w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\frac{\lceil
a_{1}n\rceil}{a_{1}}\int fd\nu_{t_{1}(n)}-2kl\log M.\end{split}$

Since

$\nu_{t_{i}(n)}=\frac{t_{i-1}(n)}{t_{i}(n)}\nu_{t_{i-1}(n)}+\frac{t_{i}(n)-t_{i-1}(n)}{t_{i}(n)}\omega_{i,n},$

by Lemma 4.5, we have

$\begin{split}t_{i}(n)H_{\nu_{t_{i}(n)}}(\bigvee_{j=0}^{l-1}T^{-j}\beta_{i})-t_{i-1}(n)H_{\nu_{t_{i-1}(n)}}(\bigvee_{j=0}^{l-1}T^{-j}\beta_{i})\\\
\geq\big{(}t_{i}(n)-t_{i-1}(n)\big{)}H_{\omega_{i,n}}\big{(}\bigvee_{j=0}^{l-1}T^{-j}\beta_{i}\big{)}-t_{i}(n)\log
2.\end{split}$

Hence

$\begin{split}&\sum_{i=1}^{k}\Big{(}t_{i}(n)H_{\nu_{t_{i}(n)}}(\bigvee_{j=0}^{l-1}T^{-j}\beta_{i})-t_{i-1}(n)H_{\nu_{t_{i-1}(n)}}(\bigvee_{j=0}^{l-1}T^{-j}\beta_{i})\Big{)}\\\
&\geq nls-\frac{2l\lceil
a_{1}n\rceil}{a_{1}}w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\frac{\lceil
a_{1}n\rceil l}{a_{1}}\int fd\nu_{t_{1}(n)}-2kl^{2}\log M-kt_{k}(n)\log
2.\end{split}$

Since $\beta_{i}$ is chosen arbitrary, we have

$\begin{split}\Theta_{n}:&=\sum_{i=1}^{k}\Big{(}t_{i}(n)H_{\nu_{t_{i}(n)}}(i)-t_{i-1}(n)H_{\nu_{t_{i-1}(n)}}(i)\Big{)}\\\
&\geq nls-\frac{2l\lceil
a_{1}n\rceil}{a_{1}}w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\frac{\lceil
a_{1}n\rceil l}{a_{1}}\int fd\nu_{t_{1}(n)}-2kl^{2}\log M-kt_{k}(n)\log
2.\end{split}$

Define

$\gamma_{n}=\sum_{i=2}^{k}t_{i}(n)\big{(}H_{\nu_{t_{i}(n)}}(i)-H_{\nu_{t_{1}(n)}}(i)\big{)}-\sum_{i=2}^{k}t_{i-1}(n)\big{(}H_{\nu_{t_{i-1}(n)}}(i)-H_{\nu_{t_{1}(n)}}(i)\big{)}.$

We have

$\Theta_{n}=\gamma_{n}+\sum_{i=1}^{k}\big{(}t_{i}(n)-t_{i-1}(n)\big{)}H_{\nu_{t_{1}(n)}}(i).$

Define

$\begin{split}w(n)=&\sum_{i=2}^{k}(a_{1}+\ldots+a_{i-1})\big{(}H_{\nu_{t_{i-1}(n)}}(i)-H_{\nu_{t_{1}(n)}}(i)\big{)}\\\
&-\sum_{i=2}^{k}(a_{1}+\ldots+a_{i})\big{(}H_{\nu_{t_{i}(n)}}(i)-H_{\nu_{t_{1}(n)}}(i)\big{)}.\end{split}$

In Lemma 4.6, we take $p=2k-2,$

$\begin{split}&u_{j}(n)=(a_{1}+\ldots+a_{j})H_{\nu_{n}}(j+1),c_{j}=a_{1}+\ldots+a_{j},1\leq
j\leq k-1;\\\
&u_{j}(n)=-(a_{1}+\ldots+a_{j-k+2})H_{\nu_{n}}(j-k+2),c_{j}=a_{1}+\ldots+a_{j-k+2},k\leq
j\leq 2k-2,\end{split}$

and take $r_{j}=a_{1},1\leq j\leq 2k-2.$ Since for any $1\leq i\leq k,$

$|\bigvee\limits_{j=0}^{l-1}T^{-j}\beta|\leq
M^{l},\forall\beta\in(\bigvee\limits_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})^{*},$

by Lemma 4.5 we have

$|H_{\nu_{n}}(i)-H_{\nu_{n+1}}(i)|\leq\frac{1}{n+1}\log(3M^{2l}(n+1)).$

Hence for any $1\leq j\leq 2k-2,$

$\lim_{n\to\infty}|u_{j}(n+1)-u_{j}(n)|=0.$

By Lemma 4.6 we have $\varlimsup\limits_{n\to\infty}w(n)\geq 0$. It follows
that

$\varlimsup_{n\to\infty}-\frac{\gamma_{n}}{n}=\varlimsup_{n\to\infty}w(n)\geq
0.$

Hence

$\varlimsup_{n\to\infty}\Big{(}\sum_{i=1}^{k}a_{i}H_{\nu_{t_{1}(n)}}(i)+l\int
fd\nu_{t_{1}(n)}\Big{)}\geq
l\big{(}s-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})\big{)}-k\sum_{i=1}^{k}a_{i}\log
2.$

We can take a subsequence $(n_{j})$ such that

$\lim_{j\to\infty}\Big{(}\sum_{i=1}^{k}a_{i}H_{\nu_{t_{1}(n_{j})}}(i)+l\int
fd\nu_{t_{1}(n_{j})}\Big{)}=\varlimsup_{n\to\infty}\Big{(}\sum_{i=1}^{k}a_{i}H_{\nu_{t_{1}(n)}}(i)+l\int
fd\nu_{t_{1}(n)}\Big{)}$

and $\nu_{t_{1}(n_{j})}$ converges. Assume $\nu_{t_{1}(n_{j})}\to\mu$. It is
easy to see that $\mu$ is $T_{1}$-invariant. For any
$\beta_{i}\in(\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})^{*},i=1,\ldots,k,$
since $\beta_{i}$ are clopen sets, we have

$\begin{split}\sum_{i=1}^{k}a_{i}H_{\mu}(\bigvee_{h=0}^{l-1}T^{-h}\beta_{i})&\geq
l\big{(}s-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i}\big{)}-\int
fd\mu)-k\sum_{i=1}^{k}a_{i}\log 2\\\ &\geq
l\big{(}h-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i}\big{)}-\int
fd\mu)-1-k\sum_{i=1}^{k}a_{i}\log 2.\end{split}$

Hence $\mu\in M(l).$ The claim is proved.

It is easy to check that $M(l)$ is closed and $M(l_{1}l_{2})\subset M(l_{1})$
for $l_{1},l_{2}\in\mathbb{N}$. Hence
$\bigcap\limits_{l\in\mathbb{N}}M(l)\neq\emptyset.$ Let
$\mu\in\bigcap\limits_{l\in\mathbb{N}}M(l).$ We have

$\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\beta_{i})\geq
h-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int fd\mu,\
\forall\beta_{i}\in(\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})^{*},i=1,\ldots,k.$

Since $X$ is zero-dimensional, there exists a fundamental base of the topology
made of clopen sets. Hence by Lemma 4.7, we have

$\sum_{i=1}^{k}a_{i}h^{+}_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})\geq
h-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int fd\mu.$

Since $(X_{1},T_{1})$ is invertible, by Lemma 4.2 we have
$h_{\mu}(T_{1},\mathcal{U})=h^{+}_{\mu}(T_{1},\mathcal{U}),$ hence (we also
can prove directly using Proposition 2.6, see [25, Proposition 4.3] )

$\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})\geq
h-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int fd\mu.$

Let

$\mu=\int_{M^{e}(X_{1},T_{1})}\theta dm(\theta)$

be the ergodic decomposition of $\mu$. By Lemma 4.3, there exists $\theta\in
M^{e}(X_{1},T_{1})$ such that

$\sum_{i=1}^{k}a_{i}h_{\theta}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})\geq
P_{W}^{{\bf
a}}(\\{\mathcal{U}_{i}\\}_{i=1}^{k})-2w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int
fd\theta.$

case 2: general case

Let $(\tilde{X}_{1},\sigma_{T_{1}})$ be the natural extension of
$(X_{1},T_{1})$. That is:

$\tilde{X}_{1}=\\{(x_{1},x_{2},\ldots)\in
X_{1}^{\mathbb{N}}:T_{1}(x_{i+1})=x_{i},i\in\mathbb{N}\\},$

$\sigma_{T_{1}}:\tilde{X}_{1}\to\tilde{X}_{1}$ is defined as

$\sigma_{T_{1}}(x_{1},x_{2},\ldots)=(T_{1}x_{1},x_{1},x_{2},\ldots).$

Let $\pi:(\tilde{X}_{1},\sigma_{T_{1}})\to(X_{1},T_{1})$ be the factor map
which project each element of $\tilde{X}_{1}$ onto is first component.
Consider the following system

$\tilde{X}_{1}\to X_{2}\to\cdots\to X_{k}.$

By definition we have

$w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})=w_{f\circ\pi}(\bigvee_{i=1}^{k}\pi^{-1}\tau_{i-1}^{-1}\mathcal{U}_{i}).$

It is easy to check that

$P_{W}^{{\bf
a}}(T_{1},\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)-w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})\leq
P_{W}^{{\bf
a}}(\sigma_{T_{1}},\\{\pi^{-1}\mathcal{U}_{1},\mathcal{U}_{2},\ldots,\mathcal{U}_{k}\\},f\circ\pi).$

By case 1, there exists $\mu\in M^{e}(\tilde{X}_{1},\sigma_{T_{1}})$ such that

$\begin{split}\sum_{i=1}^{k}a_{i}h_{\mu}(\sigma_{T_{1}},\bigvee_{j=i}^{k}\pi^{-1}\tau_{j-1}^{-1}\mathcal{U}_{j})\geq&P_{W}^{{\bf
a}}(\sigma_{T_{1}},\\{\pi^{-1}\mathcal{U}_{1},\mathcal{U}_{2},\ldots,\mathcal{U}_{k}\\},f\circ\pi)\\\
&-2w_{f\circ\pi}(\bigvee_{i=1}^{k}\pi^{-1}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int
f\circ\pi d\mu\\\ \geq&P_{W}^{{\bf
a}}(T_{1},\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)-3w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int
f\circ\pi d\mu.\end{split}$

By Lemma 4.1, we have

$\sum_{i=1}^{k}a_{i}h_{\pi\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})\geq
P_{W}^{{\bf
a}}(T_{1},\\{\mathcal{U}_{i}\\}_{i=1}^{k},f)-3w_{f}(\bigvee_{i=1}^{k}\tau_{i-1}^{-1}\mathcal{U}_{i})-\int
fd\pi\mu.$

∎

By Theorem 2.5 and Theorem 4.9 we know

$\sup_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}\sup_{\mu\in
M(X_{1},T_{1})}\Big{(}\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})+\int
fd\mu\Big{)}\geq P^{{\bf a}}(T_{1},f).$

We will show

$\sup\limits_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})=\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i}).$

Hence the left side of the inequality above is just

$\sup_{\mu\in
M(X_{1},T_{1})}\big{(}\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i})+\int
fd\mu\big{)}.$

We will give a more general result. We need the following lemma:

######  Lemma 4.10.

Let $\pi:(X,T)\to(Y,S)$ be a factor map between TDSs. Let $\mu\in M(X,T)$,
$\alpha=\\{A_{1},A_{2},\ldots,A_{k}\\}\in\mathcal{P}_{Y}$ and $\epsilon>0$.
Then there exists an open cover $\mathcal{U}$ of $Y$ with $k$ elements such
that for any $j\geq 0$ and any $\beta\in\mathcal{P}_{X}$ satisfying
$T^{-j}\pi^{-1}\mathcal{U}\preceq\beta$, we have
$H_{\mu}(T^{-j}\pi^{-1}\alpha|\beta)<\epsilon.$

###### Proof.

By [19, Lemma 4.15] there exists $\delta>0$ such that whenever
$\beta_{1},\beta_{2}\in\mathcal{P}_{X}$ with $|\beta_{1}|=|\beta_{2}|=k$ and
$\mu(\beta_{1}\Delta\beta_{2})<\delta,$ then
$H_{\mu}(\beta_{1}|\beta_{2})<\epsilon$. Take closed subsets $B_{i}\subset
A_{i}$ with

$\pi\mu(A_{i}-B_{i})<\frac{\delta}{2k^{2}},i=1,\ldots,k.$

Let $B_{0}=(\bigcup_{i=1}^{k}B_{i})^{c}$ and $U_{i}=B_{0}\cup
B_{i},i=1,\ldots,k.$ Then $\pi\mu(B_{0})<\frac{\delta}{2k}$ and
$\mathcal{U}=\\{U_{1},\ldots,U_{k}\\}$ is an open cover of $Y$. For $j\geq 0$
and $\beta\in\mathcal{P}_{X}$ satisfying
$T^{-j}\pi^{-1}\mathcal{U}\preceq\beta,$ we can find
$\beta^{\prime}=\\{C_{1},\ldots,C_{k}\\}\in\mathcal{P}_{X}$ satisfying

$C_{i}\subset T^{-j}\pi^{-1}U_{i},i=1,\ldots,k$

and $\beta\succeq\beta^{\prime}.$ Since

$T^{-j}\pi^{-1}B_{i}\subset C_{i}\subset T^{-j}\pi^{-1}U_{i},$

we have

$\mu(C_{i}\Delta
T^{-j}\pi^{-1}A_{i})\leq\mu(T^{-j}\pi^{-1}A_{i}-T^{-j}\pi^{-1}B_{i})+\mu(T^{-j}\pi^{-1}B_{0})<\frac{\delta}{k}.$

Hence

$\sum_{i=1}^{k}\mu(C_{i}\Delta T^{-j}\pi^{-1}A_{i})<\delta.$

It follows that $H_{\mu}(T^{-j}\pi^{-1}\alpha|\beta^{\prime})<\epsilon$ and
hence $H_{\mu}(T^{-j}\pi^{-1}\alpha|\beta)<\epsilon.$ ∎

We have following theorem:

######  Theorem 4.11.

For any $\mu\in M(X_{1},T_{1})$, we have

$\begin{split}\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i})&=\sup\limits_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}\varlimsup_{N\to+\infty}\frac{1}{N}H_{\mu}(\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1})\\\
&=\sup\limits_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}\varliminf_{N\to+\infty}\frac{1}{N}H_{\mu}(\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}).\end{split}$

where the supremum are taken over all open covers $\mathcal{U}_{i}$ of
$X_{i},i=1,\ldots,k.$

###### Proof.

It is easy to see that

$\begin{split}\varlimsup_{N\to+\infty}\frac{1}{N}H_{\mu}(\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1})\leq\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})\leq\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i}).\end{split}$

Now we prove the opposite direction. Let $\epsilon>0$ and
$\alpha_{i}\in\mathcal{P}_{X_{i}},i=1,\ldots,k.$ By Lemma 4.10, we can find
corresponding open covers $\mathcal{U}_{i}$ of $X_{i}$. For
$\beta\in\mathcal{P}_{X_{1}}$ with

$\beta\succeq\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1},$

we have

$\beta\succeq T_{1}^{-j}\tau_{i-1}^{-1}\mathcal{U}_{i},1\leq i\leq k,0\leq
j\leq\lceil(a_{1}+\ldots+a_{i})N\rceil-1.$

By Lemma 4.10, we have

$H_{\mu}\big{(}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}|\beta\big{)}\leq\epsilon,1\leq
i\leq k,0\leq j\leq\lceil(a_{1}+\ldots+a_{i})N\rceil-1.$

It follows that

$\begin{split}H_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\alpha_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}&\leq
H_{\mu}(\beta)+H_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\alpha_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}|\beta\big{)}\\\
&\leq
H_{\mu}(\beta)+\sum_{i=1}^{k}\sum_{j=0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}H_{\mu}\big{(}T_{1}^{-j}\tau_{i-1}^{-1}\alpha_{i}|\beta\big{)}\\\
&\leq
H_{\mu}(\beta)+\sum_{i=1}^{k}\lceil(a_{1}+\ldots+a_{i})N\rceil\epsilon.\end{split}$

Since $\beta$ is arbitrary, we have

$\begin{split}&H_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\alpha_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}\\\
&\leq
H_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}+\sum_{i=1}^{k}\lceil(a_{1}+\ldots+a_{i})N\rceil\epsilon.\end{split}$

Hence

$\begin{split}&\varliminf_{N\to+\infty}\frac{1}{N}H_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\alpha_{i}\big{)}_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1})\\\
&\leq\varliminf_{N\to+\infty}\frac{1}{N}H_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}+\sum_{i=1}^{k}(a_{1}+\ldots+a_{i})\epsilon.\end{split}$

Since $I_{\mu}\geq 0,$ by Fatou’s Lemma and Proposition 3.1 we have

$\begin{split}\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\alpha_{j})&=\int\sum_{i=1}^{k}a_{i}\mathbb{E}_{\mu}(F_{i}|\mathcal{I}_{\mu})(x)d\mu(x)\\\
&=\int\varliminf_{N\to+\infty}\frac{1}{N}I_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\alpha_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}(x)d\mu(x)\\\
&\leq\varliminf_{N\to+\infty}\int\frac{1}{N}I_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\alpha_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}(x)d\mu(x)\\\
&=\varliminf_{N\to+\infty}\frac{1}{N}H_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\alpha_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}\end{split}$

Hence

$\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i})\leq\sup\limits_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}\varliminf_{N\to+\infty}\frac{1}{N}H_{\mu}\big{(}\bigvee_{i=1}^{k}(\tau_{i-1}^{-1}\mathcal{U}_{i})_{0}^{\lceil(a_{1}+\ldots+a_{i})N\rceil-1}\big{)}.$

∎

######  Corollary 4.12.

For any $\mu\in M(X_{1},T_{1})$, we have

$\sup\limits_{\\{\mathcal{U}_{i}\\}_{i=1}^{k}}\sum_{i=1}^{k}a_{i}h_{\mu}(T_{1},\bigvee_{j=i}^{k}\tau_{j-1}^{-1}\mathcal{U}_{j})=\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i}).$

Combing Theorem 3.3 and Theorem 4.9, we have

######  Corollary 4.13.

[21, Theorem 1.4](variational principle for weighted topological pressure) For
$f\in C(X_{1},\mathbb{R}),$ we have

$P^{{\bf a}}(T_{1},f)=\sup_{\mu\in
M(X_{1},T_{1})}\Big{(}\sum_{i=1}^{k}a_{i}h_{\tau_{i-1}\mu}(T_{i})+\int
fd\mu\Big{)}.$

## References

  * [1] L. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc. 23 (1969) 679-688.
  * [2] E. Dinaburg, A correlation between topological entropy and metric entropy, Dokl. Akad. Nauk SSSR 190 (1970) 19-22.
  * [3] T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (1971) 176-180.
  * [4] D. Ruelle, Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 185 (1973) 237-251.
  * [5] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 17 (1975) 937-971.
  * [6] T. Bedford, Crinkly curves, Markov partitions and box dimension in self-similar sets, Ph.D. Thesis, University of Warwick, 1984.
  * [7] C. McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96 (1984) 1-9.
  * [8] R. Kenyon, Y. Peres, Measures of full dimension on affine-invariant sets, Ergod. Theory Dyn. Syst. 16 (1996) 307-323.
  * [9] R. Bowen, Topological entropy for noncompact sets, Trans. Am. Math. Soc. 184 (1973) 125–136.
  * [10] F. Blanchard, Fully positive topological entropy and topological mixing, Symbolic Dynamics and Its Applications (AMS Contemporary Mathematics, 135). Ed. P. Walters, American Mathematical Society, Providence, (1992) 95–105.
  * [11] F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France. 121 (1993) 565–578.
  * [12] F. Blanchard, E. Glasner, B. Host, A variation on the variational principle and applications to entropy pairs, Ergod. Theory Dyn. Syst. 17 (1997) 29–43.
  * [13] F. Blanchard, B. Host, A. Maass, S. Martinez, D. Rudolph, Entropy pairs for a measure, Ergod. Theory Dyn. Syst. 15 (1995) 621–632.
  * [14] A. Dooley, G. Zhang, Local entropy theory of a random dynamical system, Memoirs of the American Mathematical Society, 233 (2015) 1099.
  * [15] E. Glasner, B. Weiss, On the interplay between measurable and topological dynamics. Handbook of Dynamical Systems. Vol. 1B. Eds. Hasselblatt and Katok. North-Holland, Amsterdam, (2005) 597– 648\.
  * [16] E. Glasner, X. Ye, Local entropy theory, Ergod. Theory Dyn. Syst. 29 (2) (2009) 321–356.
  * [17] W. Huang, X. Ye, A local variational relation and applications, Israel J. Math. 151(1) (2006) 237–279.
  * [18] W. Wu, Local pressure of subsets and measures, J. Stat. Phys. 185 (2021), no. 2, Paper No. 9.
  * [19] P. Walters, An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.
  * [20] 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.
  * [21] D. Feng, W. Huang, Variational principle for weighted topological pressure, J. Math. Pures Appl. (9) 106 (2016) no. 3, 411–452.
  * [22] P. Romagnoli, A local variational principle for the topological entropy, Ergod. Theory Dyn. Syst. 23 (2003) 1601–1610.
  * [23] W. Huang, A. Maass, P. Romagnoli, X. Ye, Entropy pairs and a local Abramov formula for a measure theoretical entropy of open covers, Ergod. Theory Dyn. Syst. 24 (2004) 1127–1153.
  * [24] W. Huang, X. Ye, G. Zhang, A local variational principle for conditional entropy, Ergod. Theory Dyn. Syst. 26 (2006) 219–245.
  * [25] W. Huang, Y. Yi, A local variational principle of pressure and its applications to equilibrium states, Israel J. Math. 161 (2007) 29–74.
  * [26] Y. Cao, D. Feng, W. Huang, The thermodynamic formalism for sub-additive potentials, Discrete Contin. Dyn. Syst. 20 (2008) no. 3, 639–657.
  * [27] Y. Pesin, B. Pitskel, Topological pressure and the variational principle for noncompact sets, Funct. Anal. Appl. 18 (1984) 307–318.
  * [28] T. Wang, Y. Huang, Weighted topological and measure-theoretic entropy, Discrete Contin. Dyn. Syst. 39 (2019) 3941–3967.
  * [29] 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.
  * [30] K. Yang, E. Chen, Z. Lin, X. Zhou, Weighted Topological Entropy of Random Dynamical Systems, arXiv: 2207.09719.
  * [31] M. Tsukamoto, New approach to weighted topological entropy and pressure, Ergod. Theory Dyn. Syst. 43 (2023) no. 3, 1004–1034.
  * [32] J. Yang, E. Chen, R. Yang, X. Yang, Variational principle for weighted amenable topological pressure, arXiv: 2306.15509.