# The $\star$-product approach for linear ODEs: a numerical study of the
scalar case

Stefano Pozza1 Niel Van Buggenhout1
1 Charles University
Sokolovská 83 186, 75 Praha 8, Czech Republic
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Solving systems of non-autonomous ordinary differential equations (ODE) is a
crucial and often challenging problem. Recently a new approach was introduced
based on a generalization of the Volterra composition. In this work, we
explain the main ideas at the core of this approach in the simpler setting of
a scalar ODE. Understanding the scalar case is fundamental since the method
can be easily extended to the more challenging problem of systems of ODEs.
Numerical examples illustrate the method’s efficacy and its properties in the
scalar case.

###### keywords:

Ordinary Differential Equations, Volterra composition, Legendre polynomials

34A25, 65L05, 94A11

##  1 Introduction

Systems of non-autonomous linear ordinary differential equations arise in a
variety of contexts [1, 2, 3, 10, 11, 13]. Yet, their solution remains
surprisingly difficult to obtain, both formally and numerically, especially
when dealing with systems of large-to-huge size. Consider an $N\times N$
matrix $\tilde{A}(t)$ depending on the variable $t\in I\subseteq\mathbb{R}$.
The unique solution $U_{s}(t)$ of the system

$\tilde{A}(t)U_{s}(t)=\frac{d}{dt}U_{s}(t),\quad U_{s}(s)=I_{N},\quad\text{
for }t\geq s,\quad t,s\in I,$ (1)

with $I_{N}$ the $N\times N$ identity matrix, is a $N\times N$ matrix-valued
function known as the _time-ordered exponential_ of $\tilde{A}(t)$. If
$\tilde{A}(\tau_{1})\tilde{A}(\tau_{2})=\tilde{A}(\tau_{2})\tilde{A}(\tau_{1})$
for all $\tau_{1},\tau_{2}\in I$, then the time-ordered exponential can be
expressed as

$U_{s}(t)=\exp\left(\int_{s}^{t}\tilde{A}(\tau)\,\text{d}\tau\right).$

In general, however, $U_{s}$ has no known simple expression in terms of
$\tilde{A}(t)$.

In [5, 6], a new expression for the solution is given using the _path-sum
approach_ , a method able to express each element of $U_{s}(t)$ as a finite
sequence of integral equations. However, this requires solving an NP-hard
problem. In [7, 8, 9], the NP-hard problem is overcome by introducing the
_$\star$ -Lanczos method_, a constructive method able to tridiagonalize
$\tilde{A}(t)$. At the heart of both the path-sum and $\star$-Lanczos method
is a non-commutative convolution-like product, denoted by $\star$, defined
between certain distributions [12]. Thanks to this product, the solution of
(1) can be expressed through the $\star$-product inverse [7].

In this work, we aim to illustrate to the numerical mathematics community the
$\star$-product and how it can be used to solve an ODE numerically. For this
reason, we restrict the presentation to the simpler case in which the ODE (1)
is a scalar equation. While this framework may look too simple to show the
potential of the newly introduced technique, the reader should keep in mind
that the results and construction we illustrate for the scalar case can be
straightforwardly extended to the matrix case in full generality.

In Section 2, we give an introduction to the $\star$-product and the related
expression for the solution of a scalar ODE. Section 3 discretizes the
$\star$-product. As a consequence, the ODE solution can be obtained by solving
a linear system. Several properties of the linear system are numerically
investigated in Section 4. The numerical experiments in Section 5 show that
the presented strategy can compute the solution up to machine precision.
Section 6 concludes the presentation.

##  2 ODE solution by the $\star$-product approach

Given two appropriate bivariate functions
$\tilde{f}_{1}(t,s),\tilde{f}_{2}(t,s)$, the _Volterra composition_ ,
introduced by Vito Volterra (e.g., [16]), is defined as

$\big{(}\tilde{f}_{2}\star_{v}\tilde{f}_{1}\big{)}(t,s):=\int_{s}^{t}\tilde{f}_{2}(t,\tau)\tilde{f}_{1}(\tau,s)\,\text{d}\tau.$

For our purposes, it suffices to assume $\tilde{f}_{1}$ and $\tilde{f}_{2}$ to
be smooth (i.e., infinitely differentiable) on both variables over a bounded
interval $I=[0,T]$ to have a well-defined operation for every $t,s\in I$.
Therefore, from now on, a function marked with a tilde will stand for a smooth
function in both $t$ and $s$ over $I$. Since the Volterra composition is
closed for such functions, we are allowed to define the $k$th
$\star_{v}$-power of a function $\tilde{f}$, that is,
$\tilde{f}^{\star_{v}1}=\tilde{f}$, and

$\displaystyle\tilde{f}^{\star_{v}k}$
$\displaystyle:=\tilde{f}\star_{v}\tilde{f}\dots\star_{v}\tilde{f}=$
$\displaystyle=\int_{s}^{t}\tilde{f}(t,\tau_{1})\int_{s}^{\tau_{1}}\tilde{f}(\tau_{1},\tau_{2})\,\dots\int_{s}^{\tau_{k-1}}\tilde{f}(\tau_{k-2},\tau_{k-1})\tilde{f}(\tau_{k-1},s)\,\text{d}\tau_{k-1}\cdots\,\text{d}\tau_{2}\;\text{d}\tau_{1},$

for $k\geq 1$, with the convention $\tau_{0}=t$. Moreover, the operation is
also defined for univariate functions $\tilde{f}_{2}(t)$:

$\big{(}\tilde{f}_{2}(t)\star_{v}\tilde{f}_{1}(t,s)\big{)}(t,s):=\int_{s}^{t}\tilde{f}_{2}(t)\tilde{f}_{1}(\tau,s)\,\text{d}\tau=\tilde{f}_{2}(t)\int_{s}^{t}\tilde{f}_{1}(\tau,s)\,\text{d}\tau.$

It is possible to use the Volterra composition to express the solution of the
following differential equation for every initial time $s\in I$.

$\frac{d}{dt}y_{s}(t)=\tilde{f}(t)y_{s}(t),\quad y_{s}(s)=1,\quad
t\in[s,T]\subset\mathbb{R};$ (2)

see, e.g., [5]. In fact, using Picard iterations, we get

$\displaystyle\frac{d}{dt}y_{s}(t)$ $\displaystyle=\tilde{f}(t)y_{s}(t),\quad
y_{s}(s)=1$ $\displaystyle\downarrow\,\textrm{ integration }$ $\displaystyle
y_{s}(t)$
$\displaystyle=1+\int_{s}^{t}\tilde{f}(\tau)y_{s}(\tau)\textrm{d}\tau$
$\displaystyle\downarrow\,\textrm{ integration }$ $\displaystyle y_{s}(t)$
$\displaystyle=1+\int_{s}^{t}\tilde{f}(\tau)\left(1+\int_{s}^{\tau}\tilde{f}(\rho)y_{s}(\rho)\textrm{d}\rho\right)\textrm{d}\tau$
$\displaystyle=1+\int_{s}^{t}\tilde{f}(\tau)+\int_{s}^{\tau}\tilde{f}(\tau)\tilde{f}(\rho)y_{s}(\rho)\,\textrm{d}\rho\,\textrm{d}\tau$
$\displaystyle\downarrow\,\textrm{ \ldots}$ $\displaystyle y_{s}(t)$
$\displaystyle=1+\int_{s}^{t}\tilde{f}(\tau)\textrm{d}\tau+\int_{s}^{t}\tilde{f}^{\star_{v}2}(\tau)\textrm{d}\tau+\dots,$

from which we obtain the expression

$y_{s}(t)=1+\int_{s}^{t}\sum_{k=1}^{\infty}\tilde{f}^{\star_{v}k}(\tau)\;\textrm{d}\tau.$
(3)

The Volterra composition is not a product and lacks essential features, for
instance, the identity. For this reason, the Volterra composition has been
extended, obtaining the so-called $\star$-product [7] that we briefly
introduce in the following. Consider the class $\mathcal{D}(I)$ of all the
distributions $d$ that can be written as

$d(t,s)=\widetilde{d}(t,s)\Theta(t-s)+\sum_{i=0}^{N}\widetilde{d}_{i}(t,s)\delta^{(i)}(t-s),$

where $N$ is a finite integer, $\tilde{d},\tilde{d}_{i}$ are smooth bivariate
functions over $I\times I$, $\Theta(\cdot)$ stands for the Heaviside theta
function

$\Theta(t-s)=\begin{cases}1,\quad t\geq s\\\ 0,\quad t<s\end{cases},$

and $\delta^{(i)}(\cdot)$ is the $i$th derivative of the Dirac delta
distribution $\delta(\cdot)=\delta^{(0)}(\cdot)$. We can endow the class
$\mathcal{D}(I)$ with a non-commutative algebraic structure by defining the
$\star$-product as

$\big{(}f_{2}\star
f_{1}\big{)}(t,s):=\int_{I}f_{2}(t,\tau)f_{1}(\tau,s)\,\text{d}\tau,\quad
f_{1},f_{2}\in\mathcal{D}(I).$ (4)

The $\star$-product is associative over $\mathcal{D}(I)$, $\mathcal{D}(I)$ is
closed under $\star$-multiplication, and the identity element with respect to
the $\star$-product is the Dirac delta distribution, $1_{\star}:=\delta(t-s)$,
see, e.g., [7].

Consider the subclass $\mathcal{C}_{\Theta}^{\infty}(I)\subset\mathcal{D}(I)$
comprising those distributions of form

$f(t,s)=\widetilde{f}(t,s)\Theta(t-s).$

Then, the $\star$-product between
$f_{1},f_{2}\in\mathcal{C}_{\Theta}^{\infty}(I)$ reduces to a Volterra
composition

$\displaystyle\big{(}f_{2}\star f_{1}\big{)}(t,s)$
$\displaystyle=\int_{I}\widetilde{f}_{2}(t,\tau)\widetilde{f}_{1}(\tau,s)\Theta(t-\tau)\Theta(\tau-s)\,\text{d}\tau,$
$\displaystyle=\Theta(t-s)\int_{s}^{t}\widetilde{f}_{2}(t,\tau)\widetilde{f}_{1}(\tau,s)\,\text{d}\tau=\Theta(t-s)(\tilde{f}_{2}\star_{v}\tilde{f}_{1}).$

As a consequence, using (3), we can express the solution of (2) for every
$s\in I$ as

$y_{s}(t)=u(t,s)=\Theta(t-s)\star R_{\star}(f),$ (5)

where $f(t,s)=\tilde{f}(t)\Theta(t-s)$ and $R_{\star}(f)$ is the
$\star$-resolvent of $f$, i.e.,

$R_{\star}(f)=\delta(t-s)+\sum_{k=1}^{\infty}f(t,s)^{\star k},$

with $f(t,s)^{\star k}=\Theta(t-s)\tilde{f}(t)^{\star_{v}k}$. Note that the
series $\sum_{k=1}^{\infty}f(t,s)^{\star k}$ converges for every
$f\in\mathcal{C}_{\Theta}^{\infty}(I)$. The $\star$-product easily extends to
matrices composed of elements from $\mathcal{D}(I)$ by extending the scalar
multiplication appearing in the integrand in (4) to the usual matrix-matrix
multiplication; see [9] for more details.

While expression (5) is compact, the $\star$-resolvent definition hides an
infinite series of nested integrals. Therefore, at first sight, it does not
seem like a convenient expression. In the next section, we effectively solve
this problem by showing that it is possible to approximate the $\star$-product
by the usual matrix-matrix product. Consequently, for a fixed $s$, expression
(5) can be approximated relatively cheaply by solving a linear system.

##  3 Discretization of the $\star$-product

In this section, we describe an effective strategy for approximating the
$\star$-product. Consider a sequence of orthonormal functions
$\\{p_{k}\\}_{k}$ over the bounded interval $I=[0,T]$, i.e.,

$\displaystyle\int_{I}p_{k}(\tau)p_{\ell}(\tau)d\tau=\begin{cases}0,\quad\text{if
}k\neq\ell\\\ 1,\quad\text{if }k=\ell\end{cases},$

so that $\\{p_{k}\\}_{k}$ is a basis for the space of smooth functions over
$I$. Note that the functions $p_{k}$ are not in $\mathcal{D}(I)$; hence we
cannot (formally) $\star$-multiply them. Consider a distribution
$f\in\mathcal{C}_{\Theta}^{\infty}(I)$. The function
$f(t,s)=\tilde{f}(t,s)\Theta(t-s)$ is piecewise smooth, therefore, we can
choose the basis $\\{p_{k}\\}_{k}$ so that

$f(t,s)=\sum_{k=0}^{\infty}\sum_{\ell=0}^{\infty}f_{k,\ell}\,p_{k}(t)p_{\ell}(s),\quad
t\neq s,\quad t,s\in I,$ (6)

with coefficients

$f_{k,\ell}=\int_{I}\int_{I}f(\tau,\rho)p_{k}(\tau)p_{\ell}(\rho)\;\textrm{d}\rho\;\textrm{d}\tau.$

For instance, the basis $\\{p_{k}\\}_{k}$ can be set as the sequence of
shifted Legendre polynomials (e.g., [14, p. 55]). Defining the _coefficient
matrix_ $F_{M}$ and the vector $\phi_{M}(t)$ as

$F_{M}:=\begin{bmatrix}f_{0,0}&f_{0,1}&\dots&f_{0,M-1}\\\
f_{1,0}&f_{1,1}&\dots&f_{1,M-1}\\\ \vdots&\vdots&&\vdots\\\
f_{M-1,0}&f_{M-1,1}&\dots&f_{M-1,M-1}\end{bmatrix},\quad\phi_{M}(t):=\begin{bmatrix}p_{0}(s)\\\
p_{1}(s)\\\ \vdots\\\ p_{M-1}(s)\end{bmatrix},$ (7)

the truncated expansion series can be written in the matrix form:

$\displaystyle
f_{M}(t,s):=\sum_{k=0}^{M-1}\sum_{\ell=0}^{M-1}f_{k,\ell}\,p_{k}(t)p_{\ell}(s)=\phi_{M}(t)^{T}F_{M}\,\phi_{M}(s).$

Consider $f,g,h\in\mathcal{C}_{\Theta}^{\infty}(I)$ so that $h=f\star g$, and
the related coefficient matrices (7), respectively, $F_{M},G_{M},H_{M}$. By
replacing $f$ and $g$ with their expansion (6), it is not difficult to show
that the expansion coefficients for $h$ are given by

$h_{k,\ell}=\sum_{j=0}^{\infty}f_{k,j}\,g_{j,\ell}.$ (8)

As a consequence, we can approximate $H_{M}$ by the expression

$H_{M}\approx\hat{H}_{m}:=F_{M}G_{M},$ (9)

i.e., the $\star$-product can be approximated by the usual matrix-matrix
multiplication of the related coefficient matrices.

The approximation (9) is affected by a truncation error. Therefore, fixing $k$
and $\ell$, if the magnitude of the product $f_{k,j},g_{j,\ell}$ in (8) does
not decay quickly enough for $j\rightarrow\infty$, then the truncation error
$(H_{M})_{k,\ell}-(\hat{H}_{M})_{k,\ell}$ can be too large for practical
purposes. Luckily, since $f\in\mathcal{C}_{\Theta}^{\infty}$, numerical
considerations illustrate that $F_{M}$ and $G_{M}$ are numerically banded for
a certain choice of $\\{p_{k}\\}_{k}$; for instance, see Section 4 where we
choose the shifted Legendre polynomials. Therefore, $M$ does not need to be
too large to reach a small truncation error in the approximation (9),
excluding the last rows of the matrix $\hat{H}_{M}$ where the truncation error
can still be significant. Further details and explanations on this matter are
being developed and will be presented in future work. For the moment, in
Section 4, we provide numerical evidence of these claims.

To conclude the presentation, we must discuss the convergence behavior of
expansion (6). Indeed, since $f$ is discontinuos for $t=s$, the expansion may
not converge quickly (or may not converge) to $f(t,s)$ for every $t,s\in I$;
see, e.g., [14, 15] for the polynomial case. Nevertheless, fixing $s=0$, the
univariate function $f(t,0)=\tilde{f}(t,0)\Theta(t-0)=\tilde{f}(t,0)$ is
smooth over $I=[0,T]$. Therefore

$f(t,0)=\sum_{k=0}^{\infty}a_{k}p_{k}(t)=\sum_{k=0}^{\infty}p_{k}(t)\sum_{\ell=0}^{\infty}f_{k,\ell}\,p_{\ell}(0),$

with $a_{k}=\sum_{\ell=0}^{\infty}(f_{k,\ell}\,p_{\ell}(0))$. As a
consequence, we can approximate the function $f(t,0)$ by the expression

$f(t,0)\approx\phi_{M}(t)^{T}F_{M}\,\phi_{M}(0),$

and expect to reach a small enough accuracy for a (relatively) small $M$.
Section 5 illustrates with several numerical examples that it is possible to
achieve machine precision accuracy for a small value of $M$.

Consider the function $u(t,s)$ in (5). Using the previous construction, the
related coefficient matrix $U_{M}$, i.e., such that
$u(t,s)\approx\phi(t)_{M}^{T}\,U_{M}\,\phi_{M}(s)$, can be approximated by

$U_{M}\approx T_{M}(I_{M}-F_{M})^{-1},$

where $T_{M}$ is the coefficient matrix of $\Theta(t-s)$, and $F_{M}$ is the
coefficient matrix of $\tilde{f}(t)\Theta(t-s)$, with $\tilde{f}$ from (2).
Since $u(t,s)\in\mathcal{C}_{\Theta}^{\infty}$, for $s=0$ we can approximate
the solution of (2) by the formula:

$y_{0}(t)\approx\phi_{M}(t)^{T}U_{M}\,\phi_{M}(0)\approx\phi_{M}(t)^{T}T_{M}(I_{M}-F_{M})^{-1}\phi_{M}(0).$

Then, the vector $u_{M}=T_{M}x$ contains the approximated expansion
coefficients of $y_{0}(t)$, i.e.,

$y_{0}(t)\approx\phi_{M}(t)^{T}u_{M},\;\text{ for every }t\in I,$

where $x$ is the solution of the linear system

$(I_{M}-F_{M})x=\phi_{M}(0).$ (10)

##  4 Properties of the coefficient matrix

In this section, we illustrate several properties of the coefficient matrices
(7) through numerical examples. We set $I=[0,1]$, and, as the sequence of
orthonormal functions, we choose the sequence of orthonormal shifted Legendre
polynomials, i.e., the sequence of polynomials $\\{p_{k}\\}_{k}$ such that

$\displaystyle\int_{0}^{1}p_{k}(\tau)p_{\ell}(\tau)d\tau=\begin{cases}1,\quad\text{if
}k=\ell\\\ 0,\quad\text{if }k\neq\ell\end{cases},$

with $k,\ell$ the degree of the polynomial. In the following, we consider the
functions $f_{k}(t,s)=\tilde{f}_{k}(t)\Theta(t-s)$ from Table 1 and the
related $M\times M$ coefficient matrix $F^{(k)}_{M}$ defined in (7). The
numerical experiments were performed using MatLab R2022a.

Functions | $\tilde{f}_{1}=1$ | $\tilde{f}_{2}=t$ | $\tilde{f}_{3}=t^{3}$ | $\tilde{f}_{4}=\cos(t)$ | $\tilde{f}_{5}=\log(t+1)$
---|---|---|---|---|---
$M=25$
Num. band. | $1$ | $2$ | $4$ | $13$ | $20$
Spectral radius | $0.0592$ | $0.0357$ | $0.0238$ | $0.0480$ | $0.0271$
$\sigma_{\min}$ | $2.42e-3$ | $3.50e-5$ | $9.68e-9$ | $1.74e-3$ | $3.42e-5$
$\sigma_{\max}$ | $1.2732$ | $0.9447$ | $0.6864$ | $0.9694$ | $0.6938$
$M=100$
Num. band. | $1$ | $2$ | $4$ | $13$ | $20$
Spectral radius | $0.0556$ | $0.0296$ | $0.0155$ | $0.0444$ | $0.0223$
$\sigma_{\min}$ | $1.56e-4$ | $1.57e-7$ | $2.33e-13$ | $1.10e-4$ | $1.56e-7$
$\sigma_{\max}$ | $1.2732$ | $0.9447$ | $0.6864$ | $0.9694$ | $0.6938$
$M=500$
Num. band. | $1$ | $2$ | $4$ | $13$ | $20$
Spectral radius | $0.0554$ | $0.0297$ | $0.0146$ | $0.0458$ | $0.0215$
$\sigma_{\min}$ | $6.27e-6$ | $2.59e-10$ | $6.58e-19$ | $2.59e-10$ | $3.42e-5$
$\sigma_{\max}$ | $1.2732$ | $0.9447$ | $0.6864$ | $0.9694$ | $0.6938$
Table 1:  Properties of coefficient matrices $F^{(k)}_{M}$.

Table 1 reports the numerical bandwidth of each coefficient matrix
$F_{M}^{(k)}$ for $M=25,100,500$. With numerical bandwidth, we mean the
bandwidth of the matrix once all its elements with a magnitude smaller than
the machine precision have been rounded to zero. First, we observe that the
numerical bandwidth is the same for every value of $M$. Moreover, we note that
for the polynomial functions $\tilde{f}_{1},\tilde{f}_{2},\tilde{f}_{3}$, the
corresponding bandwidth is equal to the degree of the polynomial plus one.
Finally, the functions
$f_{4}(t,s)=\cos(t)\Theta(t-s),f_{5}(t,s)=\log(t+1)\Theta(t-s)$ are also
numerically banded.

$M=25$

$M=100$

$M=500$

Figure 1:  Spectrum of the coefficient matrices $F_{M}^{(1)}$ (left),
$F_{M}^{(4)}$ (right) defined in (7), for $M=25,100,500$.

Table 1 also reports the spectral radius and the minimal and maximal singular
values (respectively $\sigma_{\min}$, $\sigma_{\max}$) of each $F_{M}^{(k)}$.
While both the spectral radius and $\sigma_{\max}$ do not vary significantly
for $M=25,100,500$, $\sigma_{\min}$ becomes smaller as $M$ increases. As the
linear system (10) involves the shifted matrix $I_{M}-F_{M}^{(k)}$, it is
important to note that all the computed spectral radii are smaller than $1$.

Finally, Figure 1 presents the spectra of the matrices $F_{M}^{(1)}$ and
$F_{M}^{(4)}$ for $M=25,100,500$. For both the functions, as $M$ increases,
the spectrum tends to distribute in a circle on the right-half of the complex
plane, closer and closer to the origin. We do not report the spectrum plots of
the other matrices considered above since they display analogous behavior.

##  5 Numerical experiments

In this section, we test the numerical method explained in Section 3 on the
ODE

$\frac{d}{dt}y(t)=\tilde{f}_{k}(t)y(t),\quad y(0)=1,\quad t\in I=[0,1],$ (11)

for each function $\tilde{f}_{k}$ from Table 1. More precisely, the method
works as follows:

  1. 1.

We discretize $f_{k}(t,s)=\tilde{f}_{k}(t)\Theta(t-s)$ as described in Section
3, obtaining the matrix $F_{M}^{(k)}$. We use as an orthonormal basis the
shifted orthonormal Legendre polynomials in Section 4.

  2. 2.

Let $b$ be the numerical bandwidth of $F_{M}^{(k)}$; we define the matrix
$\hat{F}_{M}^{(k)}$ by setting the last $b$ rows of the matrix $F_{M}^{(k)}$
to zero. This has proven helpful in reducing the accumulation of truncation
errors in the last rows of the solution.

  3. 3.

We solve the (banded) linear system

$\left(I_{M}-\hat{F}_{M}^{(k)}\right)x=\phi_{M}(0),$

using the Matlab backslash \ operation.

  4. 4.

The solution of (11) is given by

$y(t)\approx\hat{y}_{M}(t):=\phi_{M}(t)^{T}u_{M},\quad\textrm{ with
}u_{M}=T_{m}\,x.$ (12)

In Table 2, we report the maximal relative error of approximation (12) over
$I$ for $M=25,100$. The relative errors were computed on an equispaced mesh of
$100$ points over $[0,1]$. As a reference value for the solution, we
considered the function $\exp(\int_{0}^{t}\tilde{f}_{k}(\tau)\,\text{d}\tau)$.
We compare our results with the maximal relative errors obtained using the
Matlab methods ode45 and ode89 with relative and absolute tolerances set equal
to $\texttt{eps}=2.2204e-16$. For $M=100$, Table 2 shows that approximation
(12) is always better than the others. On the other hand, for $M=25$, we
obtain worse results for $k=3,4,5$, showing that it is possible to calibrate
the accuracy of the solution by the matrix size.

With these experiments, we do not want to claim anything about the performance
of our method compared to well-established explicit methods such as ode45 and
ode89. The examples considered here are certainly not enough for drawing any
conclusion. The table aims to show that approximation (12) can compete in
accuracy with well-established approaches, a promising result for our future
work.

Functions | $\tilde{f}_{1}=1$ | $\tilde{f}_{2}=t$ | $\tilde{f}_{3}=t^{3}$ | $\tilde{f}_{4}=\cos(t)$ | $\tilde{f}_{5}=\log(t+1)$
---|---|---|---|---|---
$\hat{y}_{25}(t)$ | $1.20e-15$ | $1.11e-15$ | $3.36e-14$ | $1.37e-09$ | $4.04e-04$
$\hat{y}_{100}(t)$ | $1.20e-15$ | $1.11e-15$ | $8.88e-16$ | $1.22e-15$ | $9.77e-16$
ode45 | $9.76e-15$ | $4.61e-13$ | $1.53e-12$ | $1.13e-13$ | $9.72e-13$
ode89 | $1.17e-13$ | $6.69e-14$ | $3.63e-14$ | $1.13e-13$ | $9.92e-14$
Table 2:  Maximal relative error over $I=[0,1]$ of ode45, ode89 methods and of
the approximation $\hat{y}_{M}(t)$ in (12) for the solution of the ODE (11).

##  6 Conclusion and future work

In this work, we have explained how to express the solution of a scalar linear
ODE using the so-called $\star$-product. Moreover, we have shown how to derive
a numerical method from this expression and successfully tested it on several
examples. The numerical method requires solving a linear system whose
properties have also been numerically investigated. Concerning the numerical
efficiency of the introduced method, other possible approaches in the solution
of the linear system may be used – for instance, Krylov subspace methods.
Furthermore, since the solution depends continuously on the initial time
parameter $s$, we are also investigating the use of acceleration methods such
as the one in [4]. In addition, we are currently developing an efficient
method for computing the coefficient matrix $F_{m}$.

Given a smooth matrix-valued function $\tilde{A}(t)\in\mathbb{C}^{N\times N}$,
the solution of the system

$\frac{d}{dt}Y_{s}(t)=\tilde{A}(t)Y_{s}(t),\quad y(s)=I_{N},\quad t\geq
s,\quad t,s\in I,$

can also be expressed as

$Y_{s}(t)=U(t,s)=\Theta(t-s)\star R_{\star}(\tilde{A}(t)\Theta(t-s)),\quad
t,s\in I,$

following the results in [5]. Therefore, the scalar method we have described
can be generalized to the more challenging problem offered by systems of non-
autonomous linear ODEs. The results discussed in this work are thus promising
for developing new efficient methods for computing $Y_{s}(t)$.

## Acknowledgements

This work was supported by Charles University Research programs UNCE/SCI/023
and PRIMUS/21/SCI/009 and by the Magica project ANR-20-CE29-0007 funded by the
French National Research Agency.

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  * [8] Giscard, P.L. and Pozza, S.: Tridiagonalization of systems of coupled linear differential equations with variable coefficients by a Lanczos-like method.  Linear Algebra Appl. 624 (2021), 153–173.  URL https://doi.org/10.1016/j.laa.2021.04.011.
  * [9] Giscard, P.L. and Pozza, S.: A lanczos-like method for non-autonomous linear ordinary differential equations.  Boll Unione Mat Ital (2022).  URL https://doi.org/10.1007/s40574-022-00328-6.
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  * [11] Lauder, M., Knight, P., and Greenland, P.: Pulse-shape effects in intense-field laser excitation of atoms.  Opt. Acta 33 (1986), 1231–1252.
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# Cloud-Assisted Hybrid Rendering for Thin-Client Games and VR Applications

Tan Yu Wei0000-0002-7972-2828, Louiz Kim-Chan0000-0001-8981-0592, Anthony
Halim0000-0002-4340-5820, Anand Bhojan0000-0001-8105-1739
School of Computing, National University of Singapore

###### Abstract

We introduce a novel distributed rendering approach to generate high-quality
graphics in thin-client games and VR applications. Many mobile devices have
limited computational power to achieve ray tracing in real-time. Hence,
hardware-accelerated cloud servers can perform ray tracing instead and have
their output streamed to clients in remote rendering. Applying the approach of
distributed hybrid rendering, we leverage the computational capabilities of
both the thin client and powerful server by performing rasterization locally
while offloading ray tracing to the server. With advancements in 5G
technology, the server and client can communicate effectively over the network
and work together to produce a high-quality output while maintaining
interactive frame rates. Our approach can achieve better visuals as compared
to local rendering but faster performance as compared to remote rendering.

<ccs2012> <concept> <concept_id>10010147.10010371.10010372</concept_id>
<concept_desc>Computing methodologies Rendering</concept_desc>
<concept_significance>500</concept_significance> </concept> <concept>
<concept_id>10010147.10010371.10010372.10010374</concept_id>
<concept_desc>Computing methodologies Ray tracing</concept_desc>
<concept_significance>500</concept_significance> </concept> <concept>
<concept_id>10010405.10010476.10011187.10011190</concept_id>
<concept_desc>Applied computing Computer games</concept_desc>
<concept_significance>500</concept_significance> </concept> </ccs2012>

[500]Computing methodologies Rendering [500]Computing methodologies Ray
tracing [500]Applied computing Computer games

##  1 Introduction

With recent advancements in GPU hardware acceleration, ray tracing is now
feasible in real-time applications and is no longer limited to offline
rendering. Hybrid rendering techniques that combine ray tracing and
rasterization can generate higher quality visuals while maintaining
interactive frame rates on PC applications. In contrast, the graphics
capabilities of mobile and VR devices are weaker so the use of ray tracing can
lead to undesirably low frame rates. Hence, current cloud gaming providers
apply remote rendering by performing rendering on powerful cloud
infrastructure and streaming the output to the users’ access devices [HMR09].

However, thin-client devices, albeit with lower computational power, can
perform rasterization in real-time. Rather than solely relying on the server
for rendering, we leverage this limited graphics capability of the client
[CWC∗15] via distributed rendering. In particular, our approach can help to
alleviate the server’s workload so it can focus on ray tracing. The ray-traced
information can then be used to achieve better visual quality in the final
output.

Nonetheless, in order to combine the graphics capabilities of both the client
and server to meet real-time performance constraints, we require excellent
network bandwidth and latency for efficient two-way communication and data
transfer. Fortunately, this can be achieved with the newest developments in 5G
technology. With faster networks, we can improve the overall performance of
ray tracing-incorporated real-time rendering, bringing more realistic graphics
to thin-client games and VR applications.

##  2 Design

For distributed rendering, we adopt the UDP network protocol for efficient
communication between the cloud server and client device. Nonetheless, UDP
does not handle retransmission of lost packets. Retransmitted packets are also
not useful for real-time interactive applications as they will be outdated for
the current frame. Hence, in the event of a timeout, we make use of the latest
available previously received data instead.

As for hybrid rendering, we adapt the DirectX implementation of a simple
hybrid rendering pipeline [Wym18] with Lambertian shading [Kop14] and ray-
traced shadows. We first perform a G-buffer rasterization pass for deferred
shading to obtain the necessary per-pixel information required. Next, we
perform ray tracing to query the relative visibility of every light in the
scene with respect to each pixel. Lastly, we combine the G-buffer and light
visibility information in computing the final pixel colour $I$ as shown.

$I=\frac{I_{d}}{\pi}\sum_{i}k_{i}\cdot\textnormal{{saturate(}}\textnormal{\emph{$N\cdot
L_{i}$}}\textnormal{{)}}I_{i}$ (1)

In the above formula, $I_{d}/\pi$ refers to the diffuse BRDF of the pixel with
$I_{d}$ as its material diffuse colour. $N$ represents the pixel’s surface
normal while $k_{i}$, $L_{i}$ and $I_{i}$ refer to the relative visibility,
direction and intensity of light $i$ respectively.

###  2.1 Rasterization

For each frame, the client first sends the dynamic updates in scene
information including any user input and camera movement to the server for
synchronization. Rasterization is then performed to obtain per-pixel
information needed for the rest of the rendering. The server requires the
pixels’ world positions for the tracing of shadow rays, while the client
requires their world space normal vectors and material diffuse colours for
Lambertian shading.

For this particular pipeline, we perform rasterization on both the client and
server at the same time. Alternatively, to avoid repeated work, we can also
make the client or server perform rasterization and send the necessary
information required by the other end over the network. If we let the client
do the rasterization, we can leverage its limited computational capabilities
and also minimize the amount of data that needs to be transferred (i.e. world
positions only). On the other hand, rasterization on the server will most
likely be faster because of its superior graphics hardware. The server can
then proceed with ray tracing immediately rather than wait for the thin client
to complete rasterization and send its data over.

Nonetheless, we choose to perform rasterization on both ends to improve the
overall rendering performance. By performing rasterization on the server, we
minimize the time taken to obtain the visibility bitmap. Although the client
takes longer for rasterization, it is possible that the client would have
completed rasterization by the time it receives the visibility bitmap from the
server. It can then promptly proceed with Lambertian shading for the final
output.

In this case, both the client and server require raster information. However,
for other hybrid rendering pipelines, it would be better to avoid repeated
work. For instance, [Cab10] introduces a pipeline that performs direct
lighting computation in parallel with the ray tracing process. Such
independent work distribution can enable our approach to attain better
performance than pure remote rendering.

###  2.2 Ray Tracing

Shadow mapping is usually adopted to achieve real-time shadows in interactive
applications. However, ray-traced shadows produce more accurate shadow
boundaries by avoiding texture artifacts such as jaggies and shadow acne.
Hence, we perform hardware-accelerated ray tracing on the server to
incorporate high-quality ray-traced shadows in real-time.

For every pixel, we obtain its corresponding world position efficiently
through rasterization as compared to ray casting from the pixel centre. From
the world position, we then trace a shadow ray to every light in the scene. If
this light-direction ray for light $i$ is not obstructed by any object, the
light is visible from the pixel and $k_{i}=1$. The pixel is hence deemed to be
illuminated by the light. On the other hand, if the ray is obstructed by some
object, $k_{i}=0$ which indicates that the light is not visible from the
pixel. Hence, the light does not contribute to the pixel’s final colour.

This visibility boolean is stored in a frame-sized bitmap with a compact
bitmask per pixel, where every bit represents the relative visibility of a
light from the pixel. For every visible light, its direction $L_{i}$ and
intensity $I_{i}$ are then included in the computation of the final pixel
colour. The size of the per-pixel bitmask and number of overall visibility
bitmaps can be increased based on the number of lights in the scene. To
minimize the duration of visibility bitmap transfer, we apply LZ4 lossless
compression to reduce the amount of data sent over the network. This duration
can be further brought down with the help of high speed 5G networks.

##  3 Discussion

We tested our distributed hybrid rendering approach on The Modern Living Room
(CC BY) with three lights. Hardware-accelerated ray tracing was performed on a
PC with a GeForce RTX 2080 GPU. A reasonably interactive frame rate of 34 fps
was achieved with our basic prototype without substantial optimization.

With more premium GPUs out in the market and the rapid development of 5G
networks as well as edge computing, we believe that even higher frame rates
can be attained with our current implementation. Nonetheless, we are
continuing to explore optimizations in data transfer over the network as well
as between local GPU and CPUs, and aim to enhance performance further with
better data compression and transmission techniques. In the event of lost
packets, we are also working on more accurate alternatives to the raw data of
previous frames. For instance, we believe that we can use scene information
like motion vectors in conjunction with history frames to estimate the
contents of lost packets.

Eventually, we hope to bring our distributed rendering approach to more
advanced pipelines and incorporate lighting and camera effects such as
reflections, global illumination and depth of field.

## Acknowledgements

This work is supported by the Singapore Ministry of Education Academic
Research grant T1 251RES1812, “Dynamic Hybrid Real-time Rendering with
Hardware Accelerated Ray-tracing and Rasterization for Interactive
Applications”. We thank developers Nicholas Nge and Alden Tan in our lab for
their assistance.

## References

  * [Cab10] Cabeleira J. P. G.:  _Combining Rasterization and Ray Tracing Techniques to Approximate Global Illumination in Real-Time_.  Master’s thesis, Portugal, Nov. 2010.  URL: http://voltaico.net/files/article.pdf.
  * [CWC∗15] Cuervo E., Wolman A., Cox L. P., Lebeck K., Razeen A., Saroiu S., Musuvathi M.:  Kahawai: High-quality mobile gaming using gpu offload.  In _Proceedings of the 13th Annual International Conference on Mobile Systems, Applications, and Services_ (New York, NY, USA, 2015), MobiSys ’15, Association for Computing Machinery, p. 121–135.  URL: https://doi.org/10.1145/2742647.2742657, doi:10.1145/2742647.2742657.
  * [HMR09] Holthe O., Mogstad O., Ronningen L. A.:  Geelix LiveGames: Remote playing of video games.  In _2009 6th IEEE Consumer Communications and Networking Conference_ (Jan 2009), pp. 1–2.  URL: https://doi.org/10.1109/CCNC.2009.4784713, doi:10.1109/CCNC.2009.4784713.
  * [Kop14] Koppal S. J.:  _Lambertian Reflectance_.  Springer US, Boston, MA, 2014, pp. 441–443.  URL: https://doi.org/10.1007/978-0-387-31439-6_534, doi:10.1007/978-0-387-31439-6_534.
  * [Wym18] Wyman C.:  Introduction to directx raytracing.  In _ACM SIGGRAPH 2018 Courses_ (Aug. 2018), SIGGRAPH ’18.  URL: http://cwyman.org/code/dxrTutors/dxr_tutors.md.html.


#  Topological pressure of free semigroup actions for non-compact sets and
Bowen’s equation, II

Qian Xiao  School of Mathematics, South China University of Technology,
Guangzhou 510641, P.R. China<EMAIL_ADDRESS>and  Dongkui Ma
School of Mathematics, South China University of Technology, Guangzhou 510641,
P.R. China<EMAIL_ADDRESS>
###### Abstract.

Inspired to the work of Ma and Wu[11] and Climenhaga[6], we introduce the new
nation of topological pressure of a semigroup of maps by using the
Carathéodory-Pesin structure (C-P structure) with respect to arbitrary subset
in this paper. Moreover, by Bowen’s equation, we characterize the Hausdorff
dimension of an arbitrary subset, where the points of the subset have the
positive lower Lyapunov exponents and satisfy a so called tempered contraction
condition.

###### Key words and phrases:

Free semigroup actions, Topological pressure, Hausdorff dimension, Bowen’s
equation.

###### 2010 Mathematics Subject Classification:

37B40, 37C45, 37C85

* Corresponding author

##  1\.  _Introduction_

Topological pressure, as a natural extension of topological entropy, plays a
fundamentable role in dynamical systems. Ruelle [19] and Walters [25] brought
the concept of topological pressure and it was further developed by Pesin and
Pitskel [14, 15]. Pesin [14] use the “dimension” approach to the notion of
topological pressure which is based on the Carathéodory structure. which we
called it Carathéodory-Pesin structure (namely, C-P structure). It is a very
powerful tool to study dimension theory and dynamical systems.

Bowen’s equation plays a key role in dynamical systems, which is a bridge
between fractals and dynamical systems. For a long time, people have done a
lot of research on Bowen’s equation. Bowen[4] first investigated the
relationship between topological pressure and Hausdorff dimension. He showed
that, for a certain compact set $J\subset\mathbb{C}$ which is invariant set of
fractional linear transformations $f$ of the Riemann sphere, the Hausdorff
dimension $t=dim_{H}J$ is the unique root of the equation (namely, Bowen’s
equation)

(1.1) $P_{J}(-t\varphi)=0,$

where $P_{J}$ is the topological pressure of the map $f:J\rightarrow J$, and
$\varphi(z)=\log|f^{\prime}(z)|$. When $f$ is a $C^{1+\varepsilon}$ conformal
map on a Riemann manifold and $J$ is a repeller, Ruelle[19] showed that
Bowen’s equation(1.1) yields the Hausdorff dimension of $J$. Gatzouras and
Peres[18] further extended the result to the case where $f$ is $C^{1}$.
Rugh[20] redefined the conformal map and confirmed that the result remains
valid as $X$ is a metric space (not necessarily a manifold). Climenhaga [6]
characterized the relationship between the Hausdorff dimension of certain sets
$Z$ and topological pressure on $Z$, whose point has positive lower Lyapunov
exponents and satisfies tempered contraction condition. By the Bowen’s
equation, there exists extensive literature that studied the settings of Julia
sets and non-conformal repellers (see, for example,[1, 5, 7, 12, 13, 16, 17,
22, 23, 24]).

The above related results are focused on a single maps. Later, Bufetov [3]and
Biś[2] respectively proposed two kinds of topological entropies to a semigroup
action generated by finite continuous maps. Moreover, Ma and Wu[11] adopt C-P
structure to derive the idea of topological entropy and topological pressure
on any subsets, which generalized the definition of Biś[2]. Later, Lin et
al[10] provided the notion of topological pressure for a free semigroup action
on a compact metric space, which is extended the result of Bufetov [3]. Then
Ju et al. [9] extended this concept to arbitrary subsets by using C-P
structure and gave some properties and applications. On the basis of the
notion of Ju, Ma and Wang[9], Xiao and Ma [26] introduced the concept of
topological pressure of free semigroup actions on arbitrary subsets.
Furthermore, inspired by the results of Climenhaga[6], Xiao and Ma [26] gave
the relationship between the topological pressure on $Z\subset X$ and the
Hausdorff dimension of a certain subset $Z$, whose points have positive lower
Lyapunov exponents and satisfy a tempered contraction condition.

Motivated by Ma and Wu [11], we give the new notion of topological pressure of
a semigroup of continuous maps with finite functions on arbitrary subset of
$X$ by using C-P structure. Meanwhile, we also obtain some properties of it.
Furthermore, in the meaning of the new topological pressure defined in this
paper, we give the relationship between the Hausdorff dimension of a certain
subset $Z$ and the topological pressure on $Z\subset X$ by Bowen equation.

The paper is organized as follows. In section 2, we review some preliminaries.
In section 3, we give the new definitions of the topological pressure and
lower and upper capacity topological pressures of a semigroup by using the C-P
structure. Moreover, the equivalent definition is given in an alternative way
and several of their properties are provided. In section 4, we show the
Theorem 4.1.

##  2\. _Preliminaries_

Let $(X,d)$ be a compact metric space. Consider a semigroup $G$ of continuous
transformations of $X$ into itself. The semigroup $G$ is assumed to be
finitely generated, namely, there exists a finite set
$G_{1}=\\{id_{X},f_{0},f_{1},\cdots,f_{k-1}\\}$ such that
$G=\bigcup_{n\in\mathbb{N}}G_{n}$, where $G_{n}=\\{g_{1}\circ\cdots\circ
g_{n}:g_{1},\cdots,g_{n}\in G_{1}\\}$. Obviously, $G_{m}\subset G_{n}$ for all
$m\leq n$.

Let $F_{k}^{+}$ be the set of all finite words of alphabet
$\\{0,1,2,\ldots,k-1\\}$. For any $w\in F_{k}^{+}$, $|w|$ stands for the
length of $w$, that is, the number of symbols in $w$. Denote
$F_{k}^{+}(n)=\\{w\in F_{k}^{+}:|w|=n,n\in\mathbb{N}\\}$. Obviously,
$F_{k}^{+}$ with respect to the law of composition is a free semigroup with
$k$ generators. We write ${w}^{\prime}\leq{w}$ if there exists a word
$w^{\prime\prime}\in F_{k}^{+}$ such that $w=w^{\prime\prime}w^{\prime}$. For
$w=i_{1}\ldots i_{n}\in F_{k}^{+}$, denote $\overline{w}=i_{n}\ldots i_{1}$.
Given a real number $\delta>0$, $w\in F_{k}^{+}$ and a point $x\in X$, define
the $(w,\delta)$-Bowen ball at $x$ by

$B_{w}(x,\delta)=\\{y\in
X:d(f_{w^{\prime}}(x),f_{w^{\prime}}(y))<\delta,~{}{\rm
for}~{}w^{\prime}\leq\overline{w}\\}.$

Let $(X,d)$ be a metric space and $\mathcal{P}(Z,r)$ denote the collection of
countable open covers $\\{U_{i}\\}_{i=1}^{\infty}$ of $Z$ for which
$diamU_{i}<r$ for all $i$. Given a subset $Z\subset X,t\geq 0$, define

$\mathcal{H}(Z,t,r)=\inf_{\mathcal{P}(Z,r)}\left\\{\sum\limits_{U_{i}\in\mathcal{P}(Z,r)}({\rm
diam}(U_{i}))^{t}\right\\}.$

As $r$ decreases, $\mathcal{H}(Z,t,r)$ increases. Therefore, there exists the
limit

$\mathcal{H}(Z,t)=\lim\limits_{r\rightarrow 0}\mathcal{H}(Z,t,r).$

We call $\mathcal{H}(Z,t)$ the _t-dimensional Hausdorff measure_ of $Z$.
Moreover, there is a critical value of $t$ at which $\mathcal{H}(Z,t)$ jumps
from $\infty$ to $0$, which is called the _Hausdorff dimension_ of $Z$. For
details, see [8]. Formally,

$dim_{H}(Z)=\inf\\{t>0:\mathcal{H}(Z,t)=0\\}=\sup\\{t>0:\mathcal{H}(Z,t)=\infty\\}.$

One may equivalently define Hausdorff dimension of using covers by open balls
rather than arbitrary open sets. Let $\mathcal{P}^{b}(Z,r)$ denote the
collection of countable open balls covers
$\\{B(x_{i},r_{i})\\}_{i=1}^{\infty}$ of $Z$ with $x_{i}\in Z$ and $r_{i}<r$
for all $i$, and then define

$\mathcal{H}^{b}(Z,t,r)=\inf_{\mathcal{P}^{b}(Z,r)}\left\\{\sum\limits_{B(x_{i},r_{i})\in\mathcal{P}^{b}(Z,r)}({\rm
diam}B(x_{i},r_{i}))^{t}\right\\}.$

Finally, using the same procedure as above we can also define
$\mathcal{H}^{b}(Z,t)$ and $dim_{H}^{b}Z$. Furthermore, we have known
$dim_{H}^{b}Z=dim_{H}Z$ from [6, 8].

##  3\. _Topological pressure and lower and upper capacity topological
pressures of a free semigroup action and their properties_

###  3.1. Topological pressure and lower and upper capacity topological
pressures

Let $X$ be a compact metric space with metric $d$. Given
$\varphi_{0},\cdots,\varphi_{k-1}\in C(X,\mathbb{R})$, denote
$\Phi=\\{\varphi_{0},\cdots,\varphi_{k-1}\\}$. Let $w=i_{1}i_{2}\ldots
i_{n}\in F_{k}^{+}$ and $f_{w}=f_{i_{1}}\circ f_{i_{2}}\circ\ldots\circ
f_{i_{n}}$, where $i_{j}=2,\ldots,k$ for all $j=1,\ldots,n$ . Obviously,
$f_{ww^{\prime}}=f_{w}f_{w^{\prime}}$. For $w=i_{1}i_{2}\cdots i_{n}\in
F_{k}^{+}$, denote

$S_{w}\Phi(x):=\varphi_{i_{1}}(x)+\varphi_{i_{2}}(f_{i_{1}}(x))+\cdots+\varphi_{i_{n}}(f_{i_{n-1}i_{n-2}\cdots
i_{1}}(x)).$

Define a new metric $d_{n}$ on $X$ by

$d_{n}(x,y)=\max\left\\{d(g(x),g(y):g\in G_{n}\right\\}.$

Clearly, if $n_{1}\leq n_{2}$, then $d_{n_{1}}(x_{1},x_{2})\leq
d_{n_{2}}(x_{1},x_{2})$ for all $x_{1},x_{2}\in X$.

Considering a real number $\delta>0$ and a point $x\in X$, we can define the
$(n,\delta)$-Bowen ball at $x$ by

$B_{n}(x,\delta)=\\{y\in X:d_{n}(x,y)<\delta\\}.$

Given $Z\subset X,\alpha\in\mathbb{R}$ and $N>0$, define

$\displaystyle
M(Z,G_{1},\Phi,\alpha,\delta,N):=\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\left(-\alpha\cdot
n+\sup_{y\in
B_{n}(x,\delta)}\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(y)\right)\right\\},$

where the infimum runs over all finite or countable subcollections
$\mathcal{G}=\\{B_{n}(x,\delta)\\}$ such that $x\in X,~{}n\geq N$ and
$\bigcup\limits_{B_{n}(x,\delta)\in\mathcal{G}}B_{n}(x,\delta)\supset Z.$

We can easily verify that the function $M(Z,G_{1},\Phi,\alpha,\delta,N)$ is
non-decreasing as $N$ increases. Therefore, there exists the limit

$m(Z,G_{1},\Phi,\alpha,\delta)=\lim\limits_{N\to\infty}M(Z,G_{1},\Phi,\alpha,\delta,N).$

Moreover, we can also define

$\displaystyle R(Z,G_{1},\Phi,\alpha,\delta,N)$
$\displaystyle=\inf\limits_{\mathcal{G}_{N}}\left\\{\sum\limits_{B_{N}(x,\delta)\in\mathcal{G}_{N}}\exp\left(-\alpha\cdot
N+\sup_{y\in
B_{N}(x,\delta)}\frac{1}{k^{N}}\sum_{|w|=N}S_{w}\Phi(y)\right)\right\\},$

where the infimum is taken over all finite or countable subcollections
$\mathcal{G}_{N}$ covering $Z$( i.e.,
$\bigcup\limits_{B_{N}(x,\delta)\in\mathcal{G}_{N}}B_{N}(x,\delta)\supset Z$).

Let

$\underline{r}(Z,G_{1},\Phi,\alpha,\delta)=\liminf_{N\to\infty}R(Z,G_{1},\Phi,\alpha,\delta,N),$
$\overline{r}(Z,G_{1},\Phi,\alpha,\delta)=\limsup_{N\to\infty}R(Z,G_{1},\Phi,\alpha,\delta,N).$

By the construction of C-P structure (for details see [14]), there exist
unique critical values respectively such that the quantities
$m(Z,G_{1},\Phi,\alpha,\delta)$, $\underline{r}(Z,G_{1},\Phi,\alpha,\delta)$,
$\overline{r}(Z,G_{1},\Phi,\alpha,\delta)$ jump from $\infty$ to $0$. We
denote them as
$P_{Z}(G_{1},\Phi,\delta),\underline{CP}_{Z}(G_{1},\Phi,\delta),$ and
$\overline{CP}_{Z}(G_{1},\Phi,\delta)$ respectively. Accordingly, we have that

$P_{Z}(G_{1},\Phi,\delta)=\inf\\{\alpha:m(Z,G_{1},\Phi,\alpha,\delta)=0\\}=\sup\\{\alpha:m(Z,G_{1},\Phi,\alpha,\delta)=\infty\\},$
$\underline{CP}_{Z}(G_{1},\Phi,\delta)=\inf\\{\alpha:\underline{r}(Z,G_{1},\Phi,\alpha,\delta)=0\\}=\sup\\{\alpha:\underline{r}(Z,G_{1},\Phi,\alpha,\delta)=\infty\\},$
$\overline{CP}_{Z}(G_{1},\Phi,\delta)=\inf\\{\alpha:\overline{r}(Z,G_{1},\Phi,\alpha,\delta)=0\\}=\sup\\{\alpha:\overline{r}(Z,G_{1},\Phi,\alpha,\delta)=\infty\\}.$

Similar to the argument of Pesin[14] and Ma and Wu [11], the limits
$\lim\limits_{\delta\rightarrow 0}P_{Z}(G_{1},\Phi,\delta)$,
$\lim\limits_{\delta\rightarrow
0}\underline{CP}_{Z}(G_{1}$,$\Phi,\delta),\lim\limits_{\delta\rightarrow
0}\overline{CP}_{Z}(G_{1},\Phi,\delta)$ exist respectively.

######  Definition 3.1.

For any set $Z\subset X$, we call the following quantities:

$P_{Z}(G_{1},\Phi):=\lim\limits_{\delta\rightarrow
0}P_{Z}(G_{1},\Phi,\delta),$
$\underline{CP}_{Z}(G_{1},\Phi):=\lim\limits_{\delta\rightarrow
0}\underline{CP}_{Z}(G_{1},\Phi,\delta),$
$\overline{CP}_{Z}(G_{1},\Phi):=\lim\limits_{\delta\rightarrow
0}\overline{CP}_{Z}(G_{1},\Phi,\delta),$

the topological pressure and lower and upper capacity topological pressures on
the set $Z$ respectively.

######  Remark 3.1.

(1) It is easy to see that
$P_{Z}(G_{1},\Phi)\leq\underline{CP}_{Z}(G_{1},\Phi)\leq\overline{CP}_{Z}(G_{1},\Phi)$.

(2) If $\varphi_{0}=\cdots=\varphi_{k-1}=\varphi$, it is easy to verify
$P_{Z}(G_{1},\Phi)\leq P_{Z}(G_{1},\varphi)$, where $P_{Z}(G_{1},\varphi)$ is
defined by Ma and Wu [11].

(3) When $\varphi_{0}=\cdots=\varphi_{k-1}=0$, it is the topological entropy
$h_{Z}(G_{1})$ defined by Ma and Wu [11].

###  3.2. Equivalent definition by using the center of Bowen’s ball

Given $Z\subset X,\alpha\in\mathbb{R}$ and $N>0$, we can define new functions
as follows

$M^{\prime}(Z,G_{1},\Phi,\alpha,\delta,N)=\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\left(-\alpha\cdot
n+\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)\right)\right\\}$

where the infimum runs over all finite or countable subcollections
$\mathcal{G}=\\{B_{n}(x,\delta):x\in Z,n\geq N\\}$ such that
$\bigcup\limits_{B_{n}(x,\delta)\in\mathcal{G}}B_{n}(x,\delta)\supset Z,$ and

$R^{\prime}(Z,G_{1},\Phi,\alpha,\delta,N)=\inf\limits_{\mathcal{G}_{N}}\left\\{\sum\limits_{B_{N}(x,\delta)\in\mathcal{G}_{N}}\exp\left(-\alpha\cdot
N+\frac{1}{k^{N}}\sum_{|w|=N}S_{w}\Phi(x)\right)\right\\},$

where the infimum runs over all finite or countable subcollections
$\mathcal{G}_{N}=\\{B_{N}(x,\delta):x\in Z\\}$ such that
$\bigcup\limits_{B_{n}(x,\delta)\in\mathcal{G}}B_{n}(x,\delta)\supset Z.$

By the same procedure as above, we can get
$m^{\prime}(Z,G_{1},\Phi,\alpha,\delta),\
\underline{r^{\prime}}(Z,G_{1},\Phi,\alpha,\delta)$ and
$\overline{r^{\prime}}(Z,G_{1},\Phi,\alpha,\delta)$ respectively. Finally, for
any $Z\subset X$, we denote the respective critical values by

$P^{\prime}_{Z}(G_{1},\Phi,\delta),\
\underline{CP^{\prime}}_{Z}(G_{1},\Phi,\delta),\
\overline{CP^{\prime}}_{Z}(G_{1},\Phi,\delta).$

######  Theorem 3.1.

For any set $Z\subset X$, the following hold:

$P_{Z}(G_{1},\Phi)=\lim\limits_{\delta\rightarrow
0}P^{\prime}_{Z}(G_{1},\Phi,\delta),$
$\underline{CP}_{Z}(G_{1},\Phi)=\lim\limits_{\delta\rightarrow
0}\underline{CP^{\prime}}_{Z}(G_{1},\Phi,\delta),$
$\overline{CP}_{Z}(G_{1},\Phi)=\lim\limits_{\delta\rightarrow
0}\overline{CP^{\prime}}_{Z}(G_{1},\Phi,\delta).$

###### Proof.

Our approach is inspired by the work of Pesin in [14]. Firstly, given
$\delta>0$, let

$\varepsilon(\delta):=\sup\\{|\varphi_{i}(x)-\varphi_{i}(y)|:d(x,y)<\delta,i=0,1,\cdots,k-1\\}.$

Owing to the fact that $\varphi_{i}\in C(X,\mathbb{R})$ and $X$ is compact,
$\varphi_{i}$ is uniformly continuous. Hence $\varepsilon(\delta)$ is finite
and $\lim\limits_{\delta\rightarrow 0}\varepsilon(\delta)=0$. Moreover, for
any $y\in B_{n}(x,\delta),~{}|w|=n$, we have

$\left|S_{w}\Phi(x)-S_{w}\Phi(y)\right|\leq n\varepsilon(\delta).$

Consider the collection $\mathcal{G}=\\{B_{n}(x,\delta):x\in Z,n\geq N\\}$
such that
$\bigcup\limits_{B_{n}(x,\delta)\in\mathcal{G}}B_{n}(x,\delta)\supset Z$, then

$\displaystyle M(Z,G_{1},\Phi,\alpha,\delta,N)$
$\displaystyle\leq\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\left(-\alpha
n+\sup_{y\in B_{n}(x,\delta)}\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(y)\right)$
$\displaystyle\leq\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\Big{(}-\alpha
n+\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)+n\varepsilon(\delta)\Big{)}$
$\displaystyle=\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\Big{(}-n(\alpha-\varepsilon(\delta))+\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)\Big{)}.$

Thus we can get

$M(Z,G_{1},\Phi,\alpha,\delta,N)\leq
M^{\prime}(Z,G_{1},\Phi,\alpha-\varepsilon(\delta),\delta,N).$

Taking the limit $N\rightarrow\infty$ yields

$m(Z,G_{1},\Phi,\alpha,\delta)\leq
m^{\prime}(Z,G_{1},\Phi,\alpha-\varepsilon(\delta),\delta).$

Therefore,

$P_{Z}(G_{1},\Phi,\delta)\leq
P^{\prime}_{Z}(G_{1},\Phi,\delta)+\varepsilon(\delta),$

and as $\delta\rightarrow 0$, that is, $\varepsilon(\delta)\rightarrow 0$, we
obtain

$P_{Z}(G_{1},\Phi)\leq\liminf_{\delta\rightarrow
0}P^{\prime}_{Z}(G_{1},\Phi,\delta).$

Next, choosing $\mathcal{G}=\\{B_{n}(x,\delta/2):x\in X,n\geq N\\}$ such that
$\bigcup\limits_{B_{n}(x,\delta/2)\in\mathcal{G}}B_{n}(x,\delta/2)\supset Z$,
we may assume without loss of generality that for every
$B_{n}(x,\delta/2)\in\mathcal{G}$, we have $B_{n}(x,\delta/2)\cap
Z\neq\emptyset.$ Thus, for each such $B_{n}(x,\delta/2)$, we choose $y\in
B_{n}(x,\delta/2)\cap Z$; we see that $B_{n}(x,\delta/2)\subset
B_{n}(y,\delta).$ Set $\mathcal{G}^{\prime}=\\{B_{n}(y,\delta):y\in
B_{n}(x,\delta/2)\cap Z\\}$, then
$\bigcup\limits_{B_{n}(y,\delta)\in\mathcal{G}^{\prime}}B_{n}(y,\delta)\supset
Z$ and we can get

$\displaystyle M(Z,G_{1},\Phi,$ $\displaystyle\alpha,\delta/2,N)$
$\displaystyle=\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta/2)\in\mathcal{G}}\exp\left(-\alpha\cdot
n+\sup_{z\in
B_{n}(x,\delta/2)}\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(z)\right)\right\\}$
$\displaystyle\geq\inf\limits_{\mathcal{G}^{\prime}}\left\\{\sum\limits_{B_{n}(y,\delta)\in\mathcal{G}^{\prime}}\exp\left(-\alpha\cdot
n+\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(y)-n\varepsilon(\delta)\right)\right\\}$
$\displaystyle\geq
M^{\prime}(Z,G_{1},\Phi,\alpha+\varepsilon(\delta),\delta,N).$

Hence

$m(Z,G_{1},\Phi,\alpha,\delta/2)\geq
m^{\prime}(Z,G_{1},\Phi,\alpha+\varepsilon(\delta),\delta).$

As a result,

$P_{Z}(G_{1},\Phi,\delta/2)\geq
P^{\prime}_{Z}(G_{1},\Phi,\delta)-\varepsilon(\delta),$

and taking the limit $\delta\rightarrow 0$ gives

$P_{Z}(G,\Phi)\geq\limsup_{\delta\rightarrow 0}P^{\prime}_{Z}(G,\Phi,\delta),$

which finishes the proof of the first. The existence of the two other limits
can be proved in a similar way. ∎

###  3.3. Properties of topological pressure and lower and upper capacity
topological pressures

The following basic properties of topological pressure and lower and upper
capacity topological pressures of a semigroup can be verified directly from
the basic properties of the Carathéodory-Pesin dimension [14] and definitions.

######  Proposition 3.1.

(1) $P_{\emptyset}(G_{1},\Phi)\leq 0.$

(2) $P_{Z_{1}}(G_{1},\Phi)\leq P_{Z_{2}}(G_{1},\Phi)$, if $Z_{1}\subset
Z_{2}$.

(3) $P_{Z}(G_{1},\Phi)=\sup\limits_{i\geq 1}P_{Z_{i}}(G_{1},\Phi)$, where
$Z=\cup_{i\geq 1}Z_{i}$ and $Z_{i}\subset X,i=1,2,\cdots$.

######  Proposition 3.2.

(1) $\underline{CP}_{\emptyset}(G_{1},\Phi)\leq
0,~{}\overline{CP}_{\emptyset}(G_{1},\Phi)\leq 0.$

(2) $\underline{CP}_{Z_{1}}(G_{1},\Phi)\leq\underline{CP}_{Z_{2}}(G_{1},\Phi)$
and $\overline{CP}_{Z_{1}}(G_{1},\Phi)\leq\overline{CP}_{Z_{2}}(G_{1},\Phi)$,
if $Z_{1}\subset Z_{2}$.

(3) $\underline{CP}_{Z}(G_{1},\Phi)\geq\sup\limits_{i\geq
1}\underline{CP}_{Z_{i}}(G_{1},\Phi)$ and
$\overline{CP}_{Z}(G_{1},\Phi)\geq\sup\limits_{i\geq
1}\overline{CP}_{Z_{i}}(G_{1},\Phi)$, where $Z=\cup_{i\geq 1}Z_{i}$ and
$Z_{i}\subset X,i=1,2,\cdots$.

(4) If $g:X\rightarrow X$ is a homeomorphism which commutes with $G_{1}$
(i.e., $f_{i}\circ g=g\circ f_{i}$, for all $f_{i}\in G_{1}$), then

$P_{Z}(G_{1},\Phi)=P_{g(Z)}(G_{1},\Phi\circ g^{-1}),$
$\underline{CP}_{Z}(G_{1},\Phi)=\underline{CP}_{g(Z)}(G_{1},\Phi\circ
g^{-1}),$ $\overline{CP}_{Z}(G_{1},\Phi)=\overline{CP}_{g(Z)}(G_{1},\Phi\circ
g^{-1}),$

where $\Phi\circ g^{-1}=\\{\varphi_{0}\circ g^{-1},\cdots,\varphi_{k-1}\circ
g^{-1}\\}$.

We also point out the _continuity property of topological pressure and lower
and upper capacity topological pressures._

######  Theorem 3.2.

If $\Phi=\\{\varphi_{0},\cdots,\varphi_{k-1}\\}$ and
$\Psi=\\{\psi_{0},\cdots,\psi_{k-1}\\}$, we have

$|P_{Z}(G_{1},\Phi)-P_{Z}(G_{1},\Psi)|\leq\max_{0\leq i\leq
k-1}\|\varphi_{i}-\psi_{i}\|,$
$|\underline{CP}_{Z}(G_{1},\Phi)-\underline{CP}_{Z}(G_{1},\Psi)|\leq\max_{0\leq
i\leq k-1}\|\varphi_{i}-\psi_{i}\|,$
$|\overline{CP}_{Z}(G_{1},\Phi)-\overline{CP}_{Z}(G_{1},\Psi)|\leq\max_{0\leq
i\leq k-1}\|\varphi_{i}-\psi_{i}\|,$

where $\|\cdot\|$ denotes the supremum norm in the space of continuous
functions on X.

###### Proof.

Note that for any $w\in F_{k}^{+},~{}x\in X$,

$S_{w}\Phi(x)-S_{w}\Psi(x)\leq|w|\max_{0\leq i\leq
k-1}\|\varphi_{i}-\psi_{i}\|,$

and

$\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)-\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Psi(x)\leq
n\max_{0\leq i\leq k-1}\|\varphi_{i}-\psi_{i}\|.$

Therefore,

$\displaystyle M^{\prime}(Z,G_{1},\Phi,\alpha,\delta,N)\leq
M^{\prime}(Z,G_{1},\Psi,\alpha-\max_{0\leq i\leq
k-1}\|\varphi_{i}-\psi_{i}\|,\delta,N).$

Thus,

$\displaystyle M^{\prime}(Z,G_{1},\Phi,\alpha,\delta,N)\leq
M^{\prime}(Z,G_{1},\Psi,\alpha-\max_{0\leq i\leq
k-1}\|\varphi_{i}-\psi_{i}\|,\delta,N).$

Taking limit $N\rightarrow\infty$ yields

$\displaystyle m^{\prime}(Z,G_{1},\Phi,\alpha,\delta)\leq
m^{\prime}(Z,G_{1},\Psi,\alpha-\max_{0\leq i\leq
k-1}\|\varphi_{i}-\psi_{i}\|,\delta).$

Therefore,

$P^{\prime}_{Z}(G_{1},\Phi,\delta)\leq
P^{\prime}_{Z}(G_{1},\Psi,\delta)+\max_{0\leq i\leq
k-1}\|\varphi_{i}-\psi_{i}\|.$

Similarly, we can also get

$P^{\prime}_{Z}(G_{1},\Phi,\delta)\geq
P^{\prime}_{Z}(G_{1},\Psi,\delta)-\max_{0\leq i\leq
k-1}\|\varphi_{i}-\psi_{i}\|.$

Let $\delta\rightarrow 0$, and we obtain

$P_{Z}(G_{1},\Psi)-\max_{0\leq i\leq k-1}\|\varphi_{i}-\psi_{i}\|\leq
P_{Z}(G_{1},\Phi)\leq P_{Z}(G_{1},\Psi)+\max_{0\leq i\leq
k-1}\|\varphi_{i}-\psi_{i}\|,$

which establishes the first inequality. ∎

##  4\. _Bowen’s equation_

In this section, using the topological pressure defined in Section 3.1, we
give a qualitative characterization of the Hausdoff dimension of the set $Z$,
whose points have the positive lower Lyapunov exponents and satisfy a tempered
contraction condition.

######  Definition 4.1.

Let $(X,d)$ be a compact metric space and $f:X\longrightarrow X$ a continuous
map. A continuous map $f:X\longrightarrow X$ is called _conformal_ with factor
$a(x)$ if for every $x\in X$, we have

$a(x)=\lim_{y\rightarrow x}\frac{d(f(x),f(y))}{d(x,y)},$

where $a:X\longrightarrow[0,\infty)$ is continuous.

Let $(X,d)$ be a compact metric space and $G$ be a semigroup of continuous
transformations of $X$ generated by $G_{1}$,where
$G_{1}=\\{id_{X},f_{0},\cdots,f_{k-1}\\}$ and $f_{0},\ldots,f_{k-1}$ are
conformal with factor $a_{i}(x),~{}i=0,\cdots,k-1$. Set $\Phi=\\{\log
a_{0},\cdots,\log a_{k-1}\\}$.

For $w=i_{1}i_{2}\cdots i_{n}\in F_{k}^{+}$, denote

$S_{w}\Phi(x):=\log a_{i_{1}}(x)+\log a_{i_{2}}(f_{i_{1}}(x))+\cdots+\log
a_{i_{n}}(f_{i_{n-1}i_{n-2}\cdots i_{1}}(x))$

and

$\displaystyle\lambda_{w}(x)$ $\displaystyle=\frac{1}{|w|}S_{w}\Phi(x),$

where

$f_{i_{n-1}i_{n-2}\cdots i_{1}}:=f_{i_{n-1}}\circ f_{i_{n-2}}\circ\cdots\circ
f_{i_{1}}.$

Let

$\displaystyle\lambda_{n}(x):$
$\displaystyle=\frac{1}{n}\left(\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)\right)$
$\displaystyle=\frac{1}{n}\Bigg{(}\frac{\sum_{i_{1}\in
F_{k}^{+}(1)}\varphi_{i_{1}}(x)}{k}+\frac{\sum_{i_{1}i_{2}\in
F_{k}^{+}(2)}\varphi_{i_{2}}(f_{i_{1}}(x))}{k^{2}}+\cdots+$
$\displaystyle\frac{\sum_{i_{1}i_{2}\cdots i_{n-1}\in
F_{k}^{+}(n-1)}\varphi_{i_{n-1}}(f_{i_{n-2}\cdots
i_{1}}(x))}{k^{n-1}}+\frac{\sum_{i_{1}i_{2}\cdots i_{n}\in
F_{k}^{+}(n)}\varphi_{i_{n}}(f_{i_{n-1}\cdots i_{1}}(x))}{k^{n}}\Bigg{)}.$

It is easy to obtain

$\lambda_{n}(x)=\frac{1}{k^{n}}\sum_{|w|=n}\lambda_{w}(x).$

Set

$\underline{\lambda}(x)=\liminf_{n\rightarrow\infty}\lambda_{n}(x),$

and

$\overline{\lambda}(x)=\limsup_{n\rightarrow\infty}\lambda_{n}(x),$

then we call $\underline{\lambda}(x)$ and $\overline{\lambda}(x)$ the _lower
and upper Lyapunov exponents_ of the semigroup $G$ at $x$. If the two are
equal, their common value is called the _Lyapunov exponent_ at $x$:

$\lambda(x)=\lim_{n\rightarrow\infty}\lambda_{n}(x).$

Given $E\subset\mathbb{R}$, let $\mathcal{A}(E)$ be the set of points along
whose orbits all the asymptotic exponential expansion rates of the $G$ lie in
$E$:

$\mathcal{A}(E)=\\{x\in
X:[\underline{\lambda}(x),\overline{\lambda}(x)]\subseteq E\\}.$

In particular, $\mathcal{A}((0,\infty))$ is the set of all points for which
$\underline{\lambda}(x)>0$ and
$\mathcal{A}(\alpha)=\mathcal{A}(\\{\alpha\\})$.

In [6], Climenhaga introduced the so-called tempered contraction condition. On
the base of [6], we propose the following so-called _tempered contraction
condition_ :

(4.1) $\inf_{\begin{subarray}{c}n\in\mathbb{N}\\\ w^{\prime}\in
F_{k}^{+}(m)\\\ 0<m\leq
n\end{subarray}}\bigg{\\{}\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)-S_{w^{\prime}}\Phi(x)+n\varepsilon\bigg{\\}}>-\infty,~{}\text{for
any}~{}\varepsilon>0.$

Denote $\mathcal{B}$ as the set of all points in $X$ which satisfy (4.1) .

######  Example 4.1.

Suppose $X:=[0,1]$ and $f_{0}=2x~{}(mod~{}1),f_{1}=2x+1/2~{}(mod~{}1)$, then
$G_{1}=\\{id_{X},f_{0},f_{1}\\},\Phi=\\{\log a_{0},\log a_{1}\\}=\\{\log
2,\log 2\\}$. Thus, $\mathcal{B}=X$.

######  Theorem 4.2.

Let $(X,d)$ be a compact metric space and $G$ be a semigroup of continuous
transformations of $X$ generated by $G_{1}$,where
$G_{1}=\\{id_{X},f_{0},\cdots,f_{k-1}\\}$ and $f_{0},\ldots,f_{k-1}$ are
conformal with factor $a_{i}(x),~{}i=0,\cdots,k-1$. Suppose that $f_{i}$ has
no critical points and singularities, that is, $0<a_{i}(x)<\infty$ for all
$x\in X$ and $i\in\\{0,\cdots,k-1\\}$. Consider
$Z\subset\mathcal{A}((0,\infty))\bigcap\mathcal{B}$ and $\Phi=\\{\log
a_{0},\cdots,\log a_{k-1}\\}$. Then the Hausdorff dimension of $Z$ is given by

$\displaystyle dim_{H}Z=t^{*}$ $\displaystyle=\sup\\{t\geq
0:P_{Z}(G_{1},-t\Phi)>0\\}$ $\displaystyle=\inf\\{t\geq
0:P_{Z}(G_{1},-t\Phi)\leq 0\\}.$

Furthermore, if $Z\subset\mathcal{A}((\alpha,\infty))\bigcap\mathcal{B}$ for
some $\alpha>0$, then $t^{*}$ is the unique root of Bowen’s equation

$P_{Z}(G_{1},-t\Phi)=0.$

Finally, if $Z\subset\mathcal{A}(\alpha)$ for some $\alpha>0$, then
$P_{Z}(G_{1},-t\Phi)=h_{Z}(G_{1})-t\alpha$, and hence

$dim_{H}Z=\frac{1}{\alpha}h_{Z}(G_{1}).$

Here $h_{Z}(G_{1})$ is the topological entropy on $Z$ defined by Ma and Wu in
[11] and $P_{Z}(G_{1},-t\Phi)$ denotes the topological pressure on $Z$ (see
Section 3.1).

Before proving the Theorem 4.1, we first give the relevant properties and
auxiliary lemmas.

######  Proposition 4.1.

Let $f_{i}:X\longrightarrow X$ be as in Theorem 4.2. Fix
$0<\alpha\leq\beta<\infty$ and $Z\subset\mathcal{A}([\alpha,\beta])$, then

(1) for any $t\in\mathbb{R}$ and $h>0$, we have

(4.2) $P_{Z}(G_{1},-t\Phi)-\beta h\leq P_{Z}(G_{1},-(t+h)\Phi)\leq
P_{Z}(G_{1},-t\Phi)-\alpha h.$

(2) The equation $P_{Z}(G_{1},-t\Phi)=0$ has unique root $t^{*}$ and

$\frac{h_{Z}(G_{1})}{\beta}\leq t^{*}\leq\frac{h_{Z}(G_{1})}{\alpha}.$

(3) If $\alpha=\beta$, then $t^{*}=\frac{h_{Z}(G_{1})}{\alpha}$.
Here $h_{Z}(G_{1})$ is the topological entropy defined by Ma and Wu [11],
$\Phi=\\{\log a_{0},\cdots,\log a_{k-1}\\}$ and $t\Phi=\\{t\cdot\log
a_{0},\cdots,t\cdot\log a_{k-1}\\}$.

###### Proof.

(1) For arbitrary $\varepsilon>0$ and $m\geq 1$, consider

$Z_{m}=\Big{\\{}x\in
Z:\lambda_{n}(x)\in(\alpha-\varepsilon,\beta+\varepsilon),~{}\text{for
any}~{}n\geq m\Big{\\}},$

and it is easy to verify $Z=\cup_{m=1}^{\infty}Z_{m}$. Now fix
$t\in\mathbb{R},~{}h>0,~{}m>0$ and $N\geq m$, then for any $\delta>0$,
$p\in\mathbb{R}$, we can get

$\displaystyle M^{\prime}(Z_{m},$ $\displaystyle G_{1},-(t+h)\Phi,p,\delta,N)$
$\displaystyle=\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\bigg{(}-p\cdot
n-(t+h)\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)\bigg{)}\right\\}$
$\displaystyle\leq\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\bigg{(}-p\cdot
n-t\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)-h\cdot
n(\alpha-\varepsilon)\bigg{)}\right\\}$
$\displaystyle=\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\bigg{(}-(p+h(\alpha-\varepsilon)\cdot
n-t\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)\bigg{)}\right\\}$
$\displaystyle=M^{\prime}(Z_{m},G_{1},-t\Phi,p+h(\alpha-\varepsilon),\delta,N),$

where $\mathcal{G}$ covers $Z_{m}$ and for any
$B_{n}(x,\delta)\in\mathcal{G}$, $n\geq N$. Then

$m^{\prime}(Z_{m},G_{1},-(t+h)\Phi,p,\delta)\leq
m^{\prime}(Z_{m},G_{1},-t\Phi,p+h(\alpha-\varepsilon),\delta).$

It follows that

$P^{\prime}_{Z_{m}}(G_{1},-(t+h)\Phi,\delta)\leq
P^{\prime}_{Z_{m}}(G_{1},-t\Phi,\delta)-h(\alpha-\varepsilon).$

Letting $\delta\rightarrow 0$, we obtain

$P_{Z_{m}}(G_{1},-(t+h)\Phi)\leq
P_{Z_{m}}(G_{1},-t\Phi)-h(\alpha-\varepsilon).$

Taking the supremum of $m\geq 1$ and by the Proposition 3.1, we can get

$P_{Z}(G_{1},-(t+h)\Phi)\leq P_{Z}(G_{1},-t\Phi)-h(\alpha-\varepsilon).$

Since $\varepsilon>0$ is arbitrary, the right of inequality (4.2) is
established.

Using the similar calculation in this manner, we also get

$\displaystyle M^{\prime}(Z_{m},G_{1},-(t+h)\Phi,p,\delta,N)\geq
M^{\prime}(Z_{m},G_{1},-t\Phi,p+h(\beta+\varepsilon),\delta,N).$

It follows that

$\displaystyle m^{\prime}(Z_{m},G_{1},-(t+h)\Phi,p,\delta)\geq
m^{\prime}(Z_{m},G_{1},-t\Phi,p+h(\beta+\varepsilon),\delta).$

Hence,

$P^{\prime}_{Z_{m}}(G_{1},-(t+h)\Phi,\delta)\geq
P^{\prime}_{Z_{m}}(G_{1},-t\Phi,\delta)-h(\beta+\varepsilon).$

Let $\delta\rightarrow 0$ and we can get

$P_{Z_{m}}(G_{1},-(t+h)\Phi)\geq
P_{Z_{m}}(G_{1},-t\Phi)-h(\beta+\varepsilon).$

Take the supremum over all $m\geq 1$, and by the Proposition 3.1 we can get

$P_{Z}(G_{1},-(t+h)\Phi)\geq P_{Z}(G_{1},-t\Phi)-h(\beta+\varepsilon).$

Since $\varepsilon>0$ is arbitrary, the left of inequality (4.2) can be given.
This complete the proof of the inequality (4.2).

(2) Note that the map $t\mapsto P_{Z}(G_{1},-t\Phi)$ is continuous and
strictly decreasing by (1). First let $t=0$ and $h=\frac{h_{Z}(G_{1})}{\beta}$
on the left of inequality (4.2), and then we have

$P_{Z}(G_{1},-\frac{h_{Z}(G_{1})}{\beta}\Phi)\geq
P_{Z}(G_{1},0)-h_{Z}(G_{1})=0.$

Second let $t=0$ and $h=\frac{h_{Z}(G_{1})}{\alpha}$ on the right of
inequality (4.2), we can get

$P_{Z}(G_{1},-\frac{h_{Z}(G_{1})}{\alpha}\Phi)\leq
P_{Z}(G_{1},0)-h_{Z}(G_{1})=0.$

Consequently, we can obtain the desired result by Intermediate Value Theorem.

(3) It follows from (2) immediately. ∎

######  Lemma 4.3.

Let $f_{i}:X\longrightarrow X$ be as in the Theorem 4.2. Then given any
$x\in\mathcal{B}$ and $\varepsilon>0$, there exist
$\delta_{0}=\delta_{0}(\varepsilon)>0$, $\eta=\eta(x,\varepsilon)>0$ such that
for each $n\in\mathbb{N}$ and $0<\delta<\delta_{0}$,

(4.3) $B(x,\eta\delta e^{-n(\lambda_{n}(x)+\varepsilon)})\subset
B_{n}(x,\delta)\subset B(x,\delta e^{-n(\lambda_{n}(x)-\varepsilon)}).$

###### Proof.

Owing to the fact that $f_{i}$ is conformal with factor $a_{i}(x)>0$ for each
$i\in\\{0,1,2\cdots,k-1\\}$, we have

$\lim_{y\rightarrow x}\frac{d(f_{i}(x),f_{i}(y))}{d(x,y)}=a_{i}(x).$

Since $a_{i}(x)>0$ everywhere, we can take logarithms of the above equation
and get

$\lim_{y\rightarrow x}\bigg{(}\log d(f_{i}(x),f_{i}(y))-\log
d(x,y)\bigg{)}=\log a_{i}(x).$

Then we can extend it to a continuous function $\zeta_{i}:X\times
X\longrightarrow\mathbb{R}$

$\zeta_{i}(x,y)=\begin{cases}\log d(f_{i}(x),f_{i}(y)-\log d(x,y)\ \ x\neq
y,\\\ \log a_{i}(x)\ \ x=y.\end{cases}$

Similar to the proof of Lemma 6.1 in [26], for any $\varepsilon>0$ we can also
get

(4.4) $d(f_{i}(x),f_{i}(y))e^{-(\log
a_{i}(x)+\varepsilon)}<d(x,y)<d(f_{i}(x),f_{i}(y))e^{-(\log
a_{i}(x)-\varepsilon)},$

whenever the middle quantity is less than $\delta$.

Now we prove the second half of (4.3). Let $\Phi=\\{\log a_{0},\cdots,\log
a_{k-1}\\}$ and for any $y\in B_{n}(x,\delta)$, we have $d(g(x),g(y))<\delta$
for all $g\in G_{n}$. Then for any $w=i_{1}i_{2}\cdots i_{n}\in F^{+}_{k}(n)$,
repeated application of the second inequality in (4.4) yields

$\displaystyle d(x,y)$ $\displaystyle<d(f_{i_{1}}(x),f_{i_{1}}(y))e^{-(\log
a_{i_{1}}(x)-\varepsilon)}$
$\displaystyle<d(f_{i_{2}i_{1}}(x),f_{i_{2}i_{1}}(y))e^{-(\log
a_{i_{2}}(f_{i_{1}}(x))-\varepsilon)}e^{-(\log a_{i_{1}}(x)-\varepsilon)}$
$\displaystyle<\cdots$
$\displaystyle<d(f_{\overline{w}}(x),f_{\overline{w}}(y))e^{-(S_{w}\Phi(x)-|w|\varepsilon)}$
$\displaystyle<\delta e^{-|w|(\lambda_{w}(x)-\varepsilon)}.$

Since $w\in F^{+}_{k}(n)$ is arbitrary, we can obtain

$d(x,y)^{k^{n}}<\prod_{|w|=n}\delta e^{-n(\lambda_{w}(x)-\varepsilon)},$

and then we have

$d(x,y)<\left(\prod_{|w|=n}\delta
e^{-n(\lambda_{w}(x)-\varepsilon)}\right)^{\frac{1}{k^{n}}}=\delta
e^{-n(\lambda_{n}(x)-\varepsilon)}.$

Thus

$B_{n}(x,\delta)\subset B(x,\delta e^{-n(\lambda_{n}(x)-\varepsilon)}).$

Next we show the first inclusion in (4.3). Note that for any fixed
$w=i_{1}i_{2}\cdots i_{n}\in F^{+}_{k}(n)$, if $d(x,y)<\delta$, then by the
first inequality in (4.4) we get

$d(f_{i_{1}}(x),f_{i_{1}}(y))<d(x,y)e^{\log a_{i_{1}}(x)+\varepsilon}.$

Then if $d(x,y)<\delta e^{-(\log a_{i_{1}}(x)+\varepsilon)}$, we have
$d(f_{i_{1}}(x),f_{i_{1}}(y))<\delta$ and so

$\displaystyle d(f_{i_{2}i_{1}}(x),f_{i_{2}i_{1}}(y))$
$\displaystyle<d(f_{i_{1}}(x),f_{i_{1}}(y))e^{\log
a_{i_{2}}(f_{i_{1}}x)+\varepsilon}$ $\displaystyle<d(x,y)e^{\log
a_{i_{2}}(f_{i_{1}}x)+\varepsilon}e^{\log a_{i_{1}}(x)+\varepsilon}.$

Using this method repeatedly, we can obtain that if

$d(x,y)<\delta e^{-|w^{\prime}|(\lambda_{w^{\prime}}(x)+\varepsilon)}$

for each $\overline{w^{\prime}}\leq\overline{w}$, we have
$d(f_{\overline{w^{\prime}}}(x),f_{\overline{w^{\prime}}}(y))<\delta$, namely,
$y\in B_{w}(x,\delta)$. Therefore

$B(x,\delta\min_{\overline{w^{\prime}}\leq\overline{w}}e^{-|w^{\prime}|(\lambda_{w^{\prime}}(x)+\varepsilon)})\subset
B_{w}(x,\delta).$

Because of $B_{n}(x,\delta)=\cap_{|w|=n}B_{w}(x,\delta)$, it is easy to get

(4.5)
$B(x,\delta\min_{\overline{w^{\prime}}\leq\overline{w},|w|=n}e^{-|w^{\prime}|(\lambda_{w^{\prime}}(x)+\varepsilon)})\subset
B_{n}(x,\delta).$

Now we find what $\eta$ should be for any
$\overline{w^{\prime}}\leq\overline{w}$ and $|w|=n$, and we observe that

$\displaystyle\frac{e^{-n(\lambda_{n}(x)+2\varepsilon)}}{e^{-|w^{\prime}|(\lambda_{w^{\prime}}(x)+\varepsilon)}}$
$\displaystyle=\frac{e^{-\frac{1}{k^{n}}\Sigma_{|w|=n}S_{w}\Phi(x)-2n\varepsilon}}{e^{-S_{w^{\prime}}\Phi(x)-|w^{\prime}|\varepsilon}}$
$\displaystyle=e^{-\big{(}\frac{1}{k^{n}}\Sigma_{|w|=n}S_{w}\Phi(x)-S_{w^{\prime}}\Phi(x)+2n\varepsilon-|w^{\prime}|\varepsilon\big{)}}$
$\displaystyle\leq
e^{-\big{(}\frac{1}{k^{n}}\Sigma_{|w|=n}S_{w}\Phi(x)-S_{w^{\prime}}\Phi(x)+n\varepsilon\big{)}}.$

Because $x$ satisfies the tempered contraction condition, there exists
$\eta=\eta(x,\varepsilon)>0$ such that for arbitrary $w\in
F_{k}^{+}(n),~{}\overline{w^{\prime}}\leq\overline{w}$,

$\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)-S_{w^{\prime}}\Phi(x)+n\varepsilon>\log\eta,$

and hence

$e^{-\big{(}\frac{1}{k^{n}}\Sigma_{|w|=n}S_{w}\Phi(x)-S_{w^{\prime}}\Phi(x)+n\varepsilon\big{)}}<\frac{1}{\eta}.$

Then for any $|w|=n,\ \overline{w^{\prime}}\leq\overline{w}$, we have

$\eta
e^{-n(\lambda_{n}(x)+2\varepsilon)}<e^{-|w^{\prime}|(\lambda_{w^{\prime}}(x)+\varepsilon)},$

and combining with (4.5)

$B(x,\delta\eta e^{-n(\lambda_{n}(x)+2\varepsilon)})\subset B_{n}(x,\delta).$

Taking $\delta_{0}=\delta_{0}(\varepsilon/2)$ gives

$B(x,\eta\delta e^{-n(\lambda_{n}(x)+\varepsilon)})\subset B_{n}(x,\delta).$

∎

######  Lemma 4.4.

Let $f_{i}:X\longrightarrow X$ satisfy the conditions of Theorem 4.2 and fix
$Z\subset\mathcal{A}((\alpha,\infty))\cap\mathcal{B}$, where
$0<\alpha<\infty$. Let $t^{*}$ be the unique real number with
$P_{Z}(G_{1},-t^{*}\Phi)=0$, whose existence and uniqueness are guaranteed by
Proposition 4.1, where $\Phi=\\{\log a_{0},\cdots,~{}\log a_{k-1}\\}$. Then
$dim_{H}Z=t^{*}$.

###### Proof.

Firstly we show $dim_{H}Z\leq t^{*}$. Given $m\geq 1$, and let

$Z_{m}=\\{x\in Z:\lambda_{n}(x)>\alpha~{}~{}\text{for all}~{}n\geq m\\}$

and then it is obvious to examine that $Z=\bigcup_{m=1}^{\infty}Z_{m}$. For
any $t>t^{*}$, we can get $P_{Z}(G_{1},-t\Phi)<0$. Thus there exists
$\varepsilon>0$ with $P_{Z}(G_{1},-t\Phi)<-t\varepsilon$ and by Lemma 4.3
there exists $\delta_{0}=\delta_{0}(\varepsilon)$ such that for all $x\in
Z_{m},~{}0<\delta<\delta_{0}$, and $n\geq m$, we have

(4.6) $diamB_{n}(x,\delta)\leq 2\delta e^{-n(\lambda_{n}(x)-\varepsilon)}\leq
2\delta e^{-n(\alpha-\varepsilon)}.$

For given $N\geq m$ and $0<\delta<\delta_{0}$, we have

$\displaystyle M^{\prime}(Z_{m},G_{1},-t\Phi,-t\varepsilon,\delta,N)$
$\displaystyle=\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\bigg{(}-(-t\varepsilon)\cdot
n-t\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)\bigg{)}\right\\}$
$\displaystyle=\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\bigg{(}-t\cdot
n(\lambda_{n}(x)-\varepsilon)\bigg{)}\right\\}$
$\displaystyle\geq\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}(\frac{1}{2\delta}diamB_{n}(x,\delta))^{t}\right\\}$
$\displaystyle\geq\inf\limits_{\mathcal{P}(Z_{m},2\delta
e^{-N(\alpha-\varepsilon)})}\left\\{\sum_{U_{i}\in\mathcal{P}(Z_{m},2\delta
e^{-N(\alpha-\varepsilon)})}(\frac{1}{2\delta}diamU_{i})^{t}\right\\}$
$\displaystyle=(2\delta)^{-t}\mathcal{H}(Z_{m},t,2\delta
e^{-N(\alpha-\varepsilon)}),$

where $\mathcal{G}$ covers $Z_{m}$ satisfying for any
$B_{n}(x,\delta)\in\mathcal{G}$, $n\geq N$ and $\mathcal{P}(Z_{m},2\delta
e^{-N(\alpha-\varepsilon)})$ denotes the collection of open covers
$\\{U_{i}\\}$ of $Z_{m}$ for which $diamU_{i}<2\delta
e^{-N(\alpha-\varepsilon)}$ for all $i$.
Take the limit as $N\rightarrow\infty$ and we get

(4.7)
$m^{\prime}(Z_{m},G_{1},-t\Phi,-t\varepsilon,\delta)\geq(2\delta)^{-t}\mathcal{H}(Z_{m},t).$

Moreover, we have

$-t\varepsilon>P_{Z}(G_{1},-t\Phi)\geq
P_{Z_{m}}(G_{1},-t\Phi)=\lim_{\delta\rightarrow
0}P^{\prime}_{Z_{m}}(G_{1},-t\Phi,\delta),$

and for sufficiently small $\delta>0$, we have
$-t\varepsilon>P^{\prime}_{Z_{m}}(G_{1},-t\Phi,\delta)$. Hence
$\mathcal{H}(Z_{m},t)=0$ by (4.7), which means $dim_{H}(Z_{m})\leq t$. Then
taking the union over all $m$ gives $dim_{H}(Z)\leq t$ for all $t>t^{*}$.
Hence $dim_{H}(Z)\leq t^{*}$.

Next we show the opposite inequality, $dim_{H}(Z)\geq t^{*}$. Consider
$t<t^{*}$, and then we manage to prove $dim_{H}(Z)\geq t$. Assume $t>0$,
otherwise there is nothing left to prove. By the Proposition 4.1, we know
$t^{*}$ is the unique real number such that $P_{Z}(G_{1},-t^{*}\Phi)=0$. Since
the pressure function $P_{Z}(G_{1},-t\Phi)$ with respect to $t$ is decreasing,
we have $P_{Z}(G_{1},-t\Phi)>0$. Hence we can choose $\varepsilon>0$
satisfying

$P_{Z}(G_{1},-t\Phi)>t\varepsilon>0.$

Let $\delta_{0}=\delta_{0}(\varepsilon)$ be given in Lemma 4.3. For given
$n\in\mathbb{N},~{}m\geq 1$, consider the set

$Z_{m}=\\{x\in
Z:(\ref{7.3})~{}holds~{}with~{}\eta=e^{-m}~{},~{}0<\delta<\delta_{0}\\}.$

It is easy to observe that $Z=\bigcup\limits_{m=1}^{\infty}Z_{m}$ and hence
$P_{Z}(G_{1},-t\Phi)=\sup\limits_{m\geq 1}P_{Z_{m}}(G_{1},-t\Phi)$. Then there
exists $m\in\mathbb{N}$ such that $P_{Z_{m}}(G_{1},-t\Phi)>t\varepsilon$, and
we can choose sufficiently small $\delta$ with $0<\delta<\delta_{0}$ such that

(4.8) $P^{\prime}_{Z_{m}}(G_{1},-t\Phi,\delta)>t\varepsilon.$

Let $\beta=\max_{i}\\{\sup_{x\in X}\log a_{i}(x)\\}<\infty$. For any
$n\in\mathbb{N},~{}x\in X$, denote $s_{n}(x)=e^{-m}\delta
e^{-n(\lambda_{n}(x)+\varepsilon)}$, and observe that

$\frac{s_{n}(x)}{s_{n+1}(x)}=\frac{e^{-n(\lambda_{n}(x)+\varepsilon)}}{e^{-(n+1)(\lambda_{n+1}(x)+\varepsilon)}}=e^{(n+1)\lambda_{n+1}(x)-n\lambda_{n}(x)+\varepsilon}\leq
e^{\beta+\varepsilon}.$

Furthermore, for given $x\in Z_{m}$ and $r>0$ enough small, there exists
$n=n(x,r)\in\mathbb{N}$ such that

(4.9) $s_{n}(x)e^{-(\beta+\varepsilon)}\leq s_{n+1}(x)\leq r\leq
s_{n}(x)=e^{-m}\delta e^{-n(\lambda_{n}(x)+\varepsilon)}.$

For this value $n$, by Lemma 4.3 we have

$B(x,r)\subset B_{n}(x,\delta).$

Then given any $\\{B(x_{i},r_{i})\\}$ such that $Z_{m}\subset\bigcup
B(x_{i},r_{i})$, we can get $Z_{m}\subset\bigcup B_{n_{i}}(x_{i},\delta)$,
where $~{}n_{i}=n_{i}(x_{i},r_{i})$ satisfies (4.9).

Moreover, for every $n\in\mathbb{N}$ and $x\in X$, we have
$\lambda_{n}(x)\leq\beta$ and thus $s_{n}(x)\geq\delta
e^{-\big{(}m+n(\beta+\varepsilon)\big{)}}$. From (4.9), it yields that for
$n=n(x,r)$ we have

$\delta e^{-\big{(}m+(n+1)(\beta+\varepsilon)\big{)}}\leq r,$

and then

$n\geq\frac{-\log r+\log\delta-m}{\beta+\varepsilon}-1.$

Denote $N:=N(r,\delta)=\frac{-\log r+\log\delta-m}{\beta+\varepsilon}-1$ and
note that for every fixed $\delta>0$, we have $\lim_{r\rightarrow
0}N(r,\delta)=\infty$.

Therefore, for all $r>0,~{}0<\delta<\delta_{0}$, by applying (4.9) we can get

$\displaystyle\mathcal{H}^{b}(Z_{m},t,r)$
$\displaystyle=\inf_{\mathcal{P}^{b}(Z_{m},r)}\left\\{\sum\limits_{B(x_{i},r_{i})\in\mathcal{P}^{b}(Z_{m},r)}(2r_{i})^{t}\right\\}$
$\displaystyle\geq\inf\limits_{\mathcal{G}^{\prime}}\left\\{\sum\limits_{B_{n_{i}}(x_{i},\delta)\in\mathcal{G}^{\prime}}\bigg{(}2e^{-(\beta+\varepsilon)}s_{n_{i}}({x_{i}})\bigg{)}^{t}\right\\}$
$\displaystyle=(2\delta)^{t}e^{-t(m+\beta+\varepsilon)}\inf\limits_{\mathcal{G}^{\prime}}\left\\{\sum\limits_{B_{n_{i}}(x_{i},\delta)\in\mathcal{G}^{\prime}}e^{-tn_{i}(\lambda_{n_{i}}(x_{i})+\varepsilon)}\right\\}$
$\displaystyle=(2\delta)^{t}e^{-t(m+\beta+\varepsilon)}\inf\limits_{\mathcal{G}^{\prime}}\left\\{\sum\limits_{B_{n_{i}}(x_{i},\delta)\in\mathcal{G}^{\prime}}\exp\bigg{(}-t\varepsilon
n_{i}-t\frac{1}{k^{n_{i}}}\sum_{|w|=n_{i}}S_{w}\Phi(x_{i})\bigg{)}\right\\}$
$\displaystyle\geq(2\delta)^{t}e^{-t(m+\beta+\varepsilon)}\inf\limits_{\mathcal{G}}\left\\{\sum\limits_{B_{n}(x,\delta)\in\mathcal{G}}\exp\bigg{(}-t\varepsilon
n-t\frac{1}{k^{n}}\sum_{|w|=n}S_{w}\Phi(x)\bigg{)}\right\\}$
$\displaystyle=(2\delta)^{t}e^{-t(m+\beta+\varepsilon)}M^{\prime}(Z_{m},G_{1},-t\Phi,t\varepsilon,\delta,N),$

where $\mathcal{P}^{b}(Z_{m},r)$ denotes the collection of countable open
balls covers $\\{B(x_{i},r_{i})\\}$ of $Z_{m}$ for which $r_{i}<r$ for all
$i$, $\mathcal{G}^{\prime}$ denotes the collection of countable Bowen balls
covers $\\{B_{n_{i}}(x_{i},\delta)\\}_{i=1}^{\infty}$ of $Z_{m}$ with for any
$B_{n_{i}}(x_{i},\delta)\in\mathcal{G}^{\prime},n_{i}\geq N$,
$B(x_{i},r_{i})\subset B_{n_{i}}(x_{i},\delta)$ and $\mathcal{G}$ denotes the
collection of countable Bowen balls covers $\\{B_{n}(x,\delta)\\}$ of $Z_{m}$
with for any $B_{n}(x,\delta)\in\mathcal{G}^{\prime},n\geq N$.
Consequently, we can get

$\mathcal{H}^{b}(Z_{m},t,r)\geq(2\delta)^{t}e^{-t(m+\beta+\varepsilon)}M^{\prime}(Z_{m},G_{1},-t\Phi,t\varepsilon,\delta,N).$

Taking $r\rightarrow 0$, it is easy to see that the quantity on the right goes
to $\infty$ by (4.8), and thus we have $\mathcal{H}^{b}(Z_{m},t)=\infty$.
Therefore,

$dim_{H}Z\geq dim_{H}Z_{m}\geq t,$

and since $t<t^{*}$ is arbitrary, this establishes the Lemma. ∎

###### Proof of Theorem 4.1.

Choose a decreasing positive numbers sequence $\\{\alpha_{m}\\}$ with
converging to $0$, and set $Z_{m}=\mathcal{A}((\alpha_{m},\infty))\bigcap Z$,
then Lemma 4.4 applies to $Z_{m}$ and we can get
$Z=\bigcup_{m=1}^{\infty}Z_{m}$. Let $t_{m}$ be the unique real number with

$P_{Z_{m}}(G_{1},-t_{m}\Phi)=0$

for every $m$ and whose existence and uniqueness are guaranteed by Proposition
4.1. Hence, by Lemma 4.4 we obtain

$dim_{H}Z_{m}=t_{m}.$

Denote $t^{*}=\sup_{m}t_{m}$, then $dim_{H}Z=t^{*}$. Then it is left to prove

(4.10) $t^{*}=\sup\\{t\geq 0:P_{Z}(G_{1},-t\Phi)>0\\}.$

Considering $t\geq 0$, we have

$P_{Z}(G_{1},-t\Phi)=\sup_{m}P_{Z_{m}}(G_{1},-t\Phi).$

On the one hand, for any $t<t^{*}$, there exists $t_{m}$ with $t<t_{m}$ and
hence $P_{Z_{m}}(G_{1},-t\Phi)>0$. Then $t\in\\{t\geq
0:P_{Z}(G_{1},-t\Phi)>0\\}$ and thus $t\leq\sup\\{t\geq
0:P_{Z}(G_{1},-t\Phi)>0\\}$. Therefore, $t^{*}\leq\sup\\{t\geq
0:P_{Z}(G_{1},-t\Phi)>0\\}$.

On the other hand, for arbitrary $t<\sup\\{t\geq 0:P_{Z}(G_{1},-t\Phi)>0\\}$,
there exists $t_{j}>t$ with $P_{Z}(G_{1},-t_{j}\Phi)>0$. Then there exists
$Z_{m}$ such that $P_{Z_{m}}(G_{1},-t_{j}\Phi)>0$ and thus $t_{j}<t_{m}$. It
follows that $t<t_{j}<t_{m}<t^{*}$. So $\sup\\{t\geq
0:P_{Z}(G_{1},-t\Phi)>0\\}\leq t^{*}$. This establishes (4.10).

Finally, it can be yielded from (4.10) and continuity of $t\mapsto
P_{Z}(G_{1},-t\Phi)$ that $P_{Z}(G_{1},-t^{*}\Phi)=0$. If
$Z\subset\mathcal{A}((\alpha,\infty))$ for some $\alpha>0$, then Proposition
4.1 guarantees that $t^{*}$ is in fact the unique root of Bowen’s equation. ∎

Acknowledgement. The work was supported by National Natural Science Foundation
of China (grant no.11771149, 11671149) and Guangdong Natural Science
Foundation 2018B0303110005.

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