# Quantization dimensions for the bi-Lipschitz recurrent Iterated function
systems

Amit Priyadarshi  Department of Mathematics, Indian Institute of Technology
Delhi, New Delhi, India 110016<EMAIL_ADDRESS>,  Mrinal K.
Roychowdhury  School of Mathematical and Statistical Sciences, University of
Texas Rio Grande Valley, 1201 West University Drive, Edinburg, TX 78539-2999,
USA<EMAIL_ADDRESS>and  Manuj Verma  Department of
Mathematics, Indian Institute of Technology Delhi, New Delhi, India 110016
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we estimate the quantization dimensions of the Borel
probability measures supported on the limit sets of the bi-Lipschitz recurrent
iterated function systems under the strong open set condition in terms of the
spectral radius.

###### Key words and phrases:

Quantization dimension, Hausdorff dimension, Box dimension, Recurrent iterated
function systems, Probability measures

###### 1991 Mathematics Subject Classification:

28A80, 60E05, 94A34

##  1\. INTRODUCTION

The term ‘quantization’ was first used in signal processing and data
compression theory, and quantization is a process of approximating a
probability measure by a discrete probability measure supported on a finite
set. In quantization theory, there are two main types of problems. The first
is to determine the quantization dimension (see [7]) of a probability measure,
and second is to find out the optimal sets (see [7]) of the order $n$ for a
probability measure. There is some literature available on determining the
$n$-optimal set; see [5, 27] for more details. In this paper, we focus on the
estimation of the quantization dimension. A few decades ago, Zador (see [31])
introduced the term ‘quantization dimension’ and gave a formula for this. He
also compared this dimension with the Hausdorff dimension of the probability
measure and the box-counting dimension of the support of the probability
measure. Thus, the quantization dimension is introduced as a new type of
fractal dimension spectrum. There are some techniques to determine the fractal
dimension of sets and the graphs of continuous functions such as operator
theoretic method [2, 18, 19], function spaces technique [3, 12], Fourier
transform method [6], mass distribution technique [1, 6, 10], oscillation
technique [6, 17], potential theoretic method [6] etc. The fractal dimension
for the fractional integrals of some continuous functions can be found in [4,
15]. The basic results on the quantization dimension and its relationship to
other fractal dimensions are also provided in Graf-Luschgy’s book (see [7]).

Given a Borel probability measure $\mu$ on $\mathbb{R}^{k}$, a number
$r\in(0,\infty)$ and $n\in\mathbb{N}$, the $n$th quantization error of order
$r$ for $\mu$ is defined by

$V_{n,r}(\mu):=\inf\Big{\\{}\int\rho(x,A)^{r}d\mu(x):A\subset\mathbb{R}^{k},\,\text{card}(A)\leq
n\Big{\\}},$

where $\rho(x,A)$ represents the Euclidean distance of the point $x$ from the
set $A$. If $\int\|x\|^{r}d\mu(x)<\infty$ and the support of $\mu$ contains
infinitely many elements, then there exists a set $A$ with $\text{card}(A)=n$
such that $V_{n,r}(\mu)=\int\rho(x,A)^{r}d\mu(x)$ (see [7]). The set $A$ for
which the above infimum occurs is called an $n$-optimal set of order $r.$ Let
$e_{n,r}(\mu)=V_{n,r}^{\frac{1}{r}}(\mu)$. We define the quantization
dimension of order $r$ of $\mu$ by

$D_{r}=D_{r}(\mu):=\lim_{n\to\infty}\frac{\log
n}{-\log\big{(}e_{n,r}(\mu)\big{)}},$

provided the limit exists. If the limit does not exist, then we define the
lower and the upper quantization dimensions by taking the limit inferior and
the limit superior of the sequence and are denoted by $\underline{D}_{r}$ and
$\overline{D}_{r}$, respectively.

Recall that, the system $\mathcal{J}=\\{\mathbb{R}^{k};f_{1},f_{2},\cdots
f_{N}\\}$ is called an iterated function system (IFS), if each $f_{i}$ is a
contraction map on $\mathbb{R}^{k}$ such that $\|f_{i}(x)-f_{i}(y)\|\leq
s_{i}\|x-y\|$, where $0<s_{i}<1.$ By the Banach contraction theorem, there
exists a unique nonempty compact set $E\subset\mathbb{R}^{k}$ such that
$E=\cup_{i=1}^{N}f_{i}(E)$. Furthermore, if $(p_{1},p_{2},\cdots,p_{N})$ is a
probability vector corresponding to the IFS $\mathcal{J}$, then there exists a
unique Borel probability measure $\mu$ supported on the set $E$ such that
$\mu=\sum_{i=1}^{N}p_{i}\mu\circ f_{i}^{-1}.$ We call the set $E$ as the
attractor and $\mu$ is the invariant measure of the IFS $\mathcal{J}.$ We say
that the IFS $\mathcal{J}$ satisfies the strong separation condition (SSC) if
$f_{i}(E)\cap f_{j}(E)=\emptyset$ for all $i\neq j$, and the open set
condition (OSC) if there exists a nonempty bounded open set
$U\subset\mathbb{R}^{k}$ such that $f_{i}(U)\subseteq U$, $f_{i}(U)\cap
f_{j}(U)=\emptyset$ for all $1\leq i\neq j\leq N,$ and the strong open set
condition (SOSC) if the IFS $\mathcal{J}$ satisfies the open set condition
with $U\cap E\neq\emptyset$ (see [6, 28]). In [11], Hochman introduced a
weaker separation condition namely ‘exponential separation condition’ and
proved the dimension drop conjecture under this separation condition.

Let $\\{\mathbb{R}^{k};f_{1},f_{2},\cdots,f_{N}\\}$ be a similarity IFS with
probability vector $(p_{1},p_{2},\cdots,p_{N})$ such that
$\|f_{i}(x)-f_{i}(y)\|=s_{i}\|x-y\|$, where $0<s_{i}<1$ for each
$i\in\\{1,2,\cdots,N\\}$. Let $E$ be the attractor and $\mu$ be the invariant
measure corresponding to this similarity IFS. Under the OSC, Graf and Luschgy
(see [8]) proved that the quantization dimension $D_{r}$ of this invariant
measure $\mu$ is uniquely determined by the following formula

(1) $\displaystyle\sum_{i=1}^{N}(p_{i}s_{i}^{r})^{\frac{D_{r}}{r+D_{r}}}=1.$

After that, Lindsay and Mauldin (see [16]) generalized the Graf-Luschgy’s
result (see [8]) to $F$-conformal measure associated with conformal IFS
consisting of finite conformal mappings. From the paper ‘Self-conformal
multifractal measures’ (see[20]), we understand that the multifractal spectrum
is the Legendre transform of the function beta, and in Lindsay-Mauldin’s paper
[16] this function beta is called the temperature function. Also, the
quantization dimension function is connected with temperature function as can
be seen from Lindsay-Mauldin’s paper [16], Graf-Luschgy’s paper [8] and
Roychowdhury’s paper [21, 22, 23, 24]. For more details on the multifractal
analysis of measures and temperature function see [6, 14, 20, 29]. Recently,
Kesseböhmer et al. [13] generalized the result of Lindsay-Mauldin [16] and
proved that the quantization dimension $D_{r}$ of any self-conformal measure
exists, and it is determined by the intersection point of the $L^{q}$-spectrum
and the line $y=rq.$ Graf and Luschgy (see [9]) proved that the quantization
dimension of self-similar measure with respect to the geometric mean error is
equal to the Hausdorff dimension of the measure. After that, Zhu (see [33])
extended the result of Graf and Luschgy (see [9]) to the self-conformal
measure. In [32], he also determined a sufficient condition under which the
upper quantization dimension is monotone.

In [23], Roychowdhury extended the Graf-Luschgy’s result [8] to the self-
similar recurrent IFS. Later, he (see [25]) estimated the bounds of the
quantization dimension of the invariant probability measure supported on the
attractor of the bi-Lipschitz IFS in terms of temperature function under the
SOSC and some restrictions on the bi-Lipschitz constants. This result also
generalizes the result of Graf and Luschgy (see [8]) in more general setting.
In [30], Verma et al. determined the bounds of the quantization dimension of
the invariant measure supported on the attractor of the bi-Lipschitz IFS under
the SOSC and also determined the quantization dimension of measure supported
on the graph of fractal transformation. In this paper, we are working on the
bi-Lipschitz recurrent IFS, consisting of a general class of bi-Lipschitz
mappings. Recurrent IFS is a generalization of the IFS. A detailed account of
the recurrent IFS and the fractal dimension of its limit set can be found in
[1, 2, 26]. In this paper, we determine the bounds of the quantization
dimension of the Borel probability measure supported on the limit set of the
bi-Lipschitz recurrent IFS under the SOSC in terms of spectral radius. Thus,
our results generalizes the result of Graf and Luschgy [7], Verma et al. [30]
and Roychowdhury [23, 25] in more general setting.

The paper is organized as follows. In the upcoming section, we provide some
definitions and basic results. In the last section, we prove our main result.

##  2\. preliminaries

In this section, we give some basic results and definitions.

######  Definition 2.1.

Let $A$ be a subset of $\mathbb{R}^{k}$. The Voronoi region of $a\in A$ is
defined by

$W(a|A)=\\{x\in X:\rho(x,a)=\rho(x,A)\\}$

and the set $\\{W(a|A):a\in A\\}$ is called the Voronoi diagram of $A$.

######  Lemma 2.2.

(see [7]) Let $\mu$ be a Borel probability measure on $\mathbb{R}^{k}$ with
compact support $A_{*}$ and $r\in(0,\infty)$. Let $A_{n}$ be an $n$-optimal
set for measure $\mu$ of order $r$. Define

$\|A_{n}\|_{\infty}=\max\limits_{a\in A_{n}}~{}~{}\max\limits_{x\in
W(a|A_{n})\cap A_{*}}\rho(x,a).$

Then

$\Big{(}\frac{\|A_{n}\|_{\infty}}{2}\Big{)}^{r}\min\limits_{x\in
A_{*}}\mu\Big{(}B\Big{(}x,\frac{\|A_{n}\|_{\infty}}{2}\Big{)}\Big{)}\leq
V_{n,r}(\mu).$

Note that, if $\mu$ be a Borel probability measure on $\mathbb{R}^{k}$ with
compact support, then for every $n\in\mathbb{N}$, there exists a finite set
$A_{n}\subset\mathbb{R}^{k}$ such that

$V_{n,r}(\mu)=\int\rho(x,A_{n})^{r}d\mu(x).$

This $A_{n}$ is an $n$-optimal set for measure $\mu$ of order $r$.

######  Lemma 2.3.

(see [7]) Let $\mu$ be a Borel probability measure on $\mathbb{R}^{k}$ with
compact support. Then, for any $r\in(0,\infty)$

$V_{n,r}(\mu)\to 0,~{}~{}\text{as}~{}~{}n\to\infty.$

######  Proposition 2.4.

(see [7])

  1. (1)

If $0\leq t_{1}<\overline{D}_{r}<t_{2},$ then

$\limsup_{n\to\infty}ne_{n,r}^{t_{1}}=\infty~{}~{}~{}\text{and}~{}~{}~{}\lim_{n\to\infty}ne_{n,r}^{t_{2}}=0.$

  2. (2)

If $0\leq t_{1}<\underline{D}_{r}<t_{2},$ then

$\liminf_{n\to\infty}ne_{n,r}^{t_{2}}=0~{}~{}~{}\text{and}~{}~{}~{}\lim_{n\to\infty}ne_{n,r}^{t_{1}}=\infty.$

Let $X\subset\mathbb{R}^{k}$ be a nonempty compact set such that
$X=cl{(intX)}.$ Then, $(X,d)$ is a compact metric space, where $d$ is an
Euclidean metric on $X$. Let $N\geq 2$ and $\\{f_{ij}:1\leq i,j\leq N\\}$ be a
system such that

$c_{ij}d(x,y)\leq d(f_{ij}(x),f_{ij}(y))\leq s_{ij}d(x,y)\text{ for all
}x,y\in X$

where $0<c_{ij}\leq s_{ij}<1$ for $1\leq i,j\leq N.$ Let $P=[p_{ij}]_{1\leq
i,j\leq N}$ be an $N\times N$ irreducible row stochastic matrix, that is for
all $i$, $\sum_{j=1}^{N}p_{ij}=1$ and $p_{ij}\geq 0$ for all $1\leq i,j\leq
N.$ Then, the collection $\mathcal{I}=\\{X;f_{ij},p_{ij}:1\leq i,j\leq N\\}$
is called a bi-Lipschitz recurrent iterated function system (RIFS). By using
the Banach’s fixed point theorem, there exist unique nonempty compact subsets
$E_{1},E_{2},\cdots,E_{N}$ of $X$ such that

(2) $E_{i}=\bigcup_{j,p_{ij}>0}f_{ij}(E_{j}).$

The set $E=\bigcup_{i=1}^{N}E_{i}$ is called the limit set of the RIFS
$\mathcal{I}.$

For $n\geq 2$, let
$\Omega_{n}=\\{(w_{1},w_{2},\cdots,w_{n}):p_{w_{i}w_{i+1}}>0\text{ for all
}1\leq i\leq n-1~{}~{}\text{and}~{}~{}1\leq w_{i}\leq N\\}$. Set
$\Omega^{*}=\bigcup_{n\geq 2}\Omega_{n}$. For
$w=(w_{1},w_{2},\cdots,w_{n})\in\Omega^{*}$, we define
$E_{w}=f_{w_{1}w_{2}}\circ f_{w_{2}w_{3}}\circ\cdots\circ
f_{w_{n-1}w_{n}}(E_{w_{n}})$. By using (2), the limit set $E$ satisfies the
following invariance equality

(3) $E=\bigcup_{w\in\Omega_{n}}E_{w}\text{ for all }n\geq 2.$

Define the code space $\Omega$ associated with the RIFS $\mathcal{I}$ as
follows

$\Omega=\\{(w_{1},w_{2},\cdots):p_{w_{i}w_{i+1}}>0\text{ for all
}i\in\mathbb{N}~{}~{}\text{and}~{}~{}1\leq w_{i}\leq N\\}.$

Let $\Pi:\Omega\times\Omega\to\mathbb{R}$ be defined as follows

$\Pi(w,\tau)=2^{-n}~{}~{}\text{if and only if
}~{}~{}n=\min\\{m:w_{m}\neq\tau_{m}\\}$

for $w=(w_{1},w_{2},\cdots),\tau=(\tau_{1},\tau_{2},\cdots)\in\Omega.$ Then,
$(\Omega,\Pi)$ is a compact metric space.

For $w=(w_{1},w_{2},\cdots)\in\Omega$, we denote
$w|_{n}=(w_{1},w_{2},\cdots,w_{n})$ and $w|_{0}=\emptyset$. Similarly, for
$w=(w_{1},w_{2},\cdots,w_{m})\in\Omega^{*}$, if $n\leq m$, then we write
$w|_{n}=(w_{1},w_{2},\cdots,w_{n}).$ For $w=(w_{1},w_{2},\cdots,w_{n})$ and
$\tau=(\tau_{1},\tau_{2},\cdots,\tau_{m})$ in $\Omega^{*}$, if
$p_{w_{n}\tau_{1}}>0$, we denote

$w\tau=(w_{1},w_{2},\cdots,w_{n},\tau_{1},\tau_{2},\cdots,\tau_{m})\in\Omega^{*}.$

We say $w$ is an extension of $\tau\in\Omega^{*}$ if $w|_{|\tau|}=\tau$, where
$|\tau|$ represents the length of $\tau.$ For any
$w=(w_{1},w_{2},\cdots,w_{n})\in\Omega^{*}$, we define
$w^{-}=(w_{1},w_{2},\cdots,w_{n-1}).$

We define the coding map $\pi:\Omega\to E$ as follows

$\pi(w)=\bigcap_{n\geq 2}f_{w_{1}w_{2}}\circ f_{w_{2}w_{3}}\circ\cdots\circ
f_{w_{n-1}w_{n}}(E_{w_{n}})$

where $w=(w_{1},w_{2},\cdots)\in\Omega.$ By the definition of $\pi$, it is
clear that $\pi$ is an onto and continuous mapping.

If $w=(w_{1},w_{2},\cdots,w_{n})\in\Omega^{*}$, then the set
$\\{\tau\in\Omega:\tau_{i}=w_{i}~{}~{}\text{for}~{}~{}i\in\\{1,2,\cdots,n\\}\\}$
is called the cylinder of length $n$ in $\Omega$ generated by
$w\in\Omega^{*}$, and is denoted by $[w]=[w_{1},w_{2},\cdots,w_{n}].$ A
cylinder of length $0$ is called the empty cylinder. The set of all elements
in $\Omega$ starting with the symbol $i$ is denoted by $C(i).$

Let $\mathcal{B}$ be the Borel sigma-algebra generated by the cylinders in
$\Omega.$ Since $P$ is an irreducible row stochastic matrix, there exists a
unique probability vector $p=(p_{1},p_{2},\cdots,p_{N})$ such that $pP=p$,
that is $\sum_{i=1}^{N}p_{i}p_{ij}=p_{j}$ for all $j\in\\{1,2,\cdots,N\\}.$ We
define

$\nu([w_{1},w_{2},\cdots,w_{n}])=p_{w_{1}}p_{w_{1}w_{2}}p_{w_{2}w_{3}}\cdots
p_{w_{n-1}w_{n}},$

where $[w_{1},w_{2},\cdots,w_{n}]$ is a cylinder in $\Omega.$ Using
Kolmogorov’s extension theorem, $\nu$ can be extended to a unique Borel
probability measure $\nu$ on $\mathcal{B}.$ The support of measure $\nu$ is
$\Omega.$ For $w=(w_{1},w_{2},\cdots,w_{n})\in\Omega^{*}$, $n\geq 2$, we
denote

$f_{w}=f_{w_{1}w_{2}}\circ f_{w_{2}w_{3}}\circ\cdots\circ
f_{w_{n-1}w_{n}},\quad c_{w}=c_{w_{1}w_{2}}c_{w_{2}w_{3}}\cdots
c_{w_{n-1}w_{n}},$ $s_{w}=s_{w_{1}w_{2}}s_{w_{2}w_{3}}\cdots
s_{w_{n-1}w_{n}},\quad p_{w}=p_{w_{1}}p_{w_{1}w_{2}}p_{w_{2}w_{3}}\cdots
p_{w_{n-1}w_{n}}.$

Also, let

$s_{\max}=\max\\{s_{ij}:p_{ij}>0,1\leq i,j\leq N\\},\quad
s_{\min}=\min\\{s_{ij}:p_{ij}>0,1\leq i,j\leq N\\},$
$c_{\max}=\max\\{c_{ij}:p_{ij}>0,1\leq i,j\leq N\\},\quad
c_{\min}=\min\\{c_{ij}:p_{ij}>0,1\leq i,j\leq N\\}$
$p_{\max}=\max\\{p_{1},p_{2},\cdots,p_{N}\\},\quad
p_{\min}=\min\\{p_{1},p_{2},\cdots,p_{N}\\},$ $P_{\max}=\max\\{p_{ij}:1\leq
i,j\leq N\\},\quad P_{\min}=\min\\{p_{ij}>0:1\leq i,j\leq N\\}$

Let $\mu$ be the push forward measure of $\nu$ under the coding map $\pi$ on
the limit set $E$ of the RIFS $\mathcal{I},$ that is $\mu=\nu\circ\pi^{-1}.$
Clearly the support of $\mu$ is $E.$ Our aim is to determine the quantization
dimension of this measure $\mu.$ Define

$\pi_{i}:=\pi|_{C(i)},\quad\mu_{i}=\mu|_{E_{i}}.$

Clearly $\mu_{i}=\nu\circ\pi_{i}^{-1}.$ For each $1\leq i\leq N$, the
probability measure $\mu_{i}$ satisfies the invariance
$\mu_{i}=\sum_{j=1}^{N}p_{ji}\mu_{j}\circ f_{ij}^{-1}.$ Thus, for each $n\geq
2$, $\mu$ satisfies the following invariance relation

$\mu=\sum_{w\in\Omega_{n}}p_{w}\mu\circ f_{w}^{-1}.$

Now, we define a matrix corresponding to the upper contraction ratios of the
mappings $f_{ij}$ for $1\leq i,j\leq N.$ Let us define an $N\times N$ matrix
$M_{r,t}=[(p_{ij}s_{ij}^{r})^{t}]_{1\leq i,j\leq N}$, where $r\in(0,\infty)$
and $t\in[0,\infty).$ Let $\Phi_{r}(t)$ be the spectral radius of the matrix
$M_{r,t}.$ Then, it is well-known that
$\Phi_{r}(t)=\lim_{n\to\infty}{\|M_{r,t}^{n}\|}_{1}^{\frac{1}{n}}$ and
$\Phi_{r}(t)$ is the largest eigenvalue of $M_{r,t}$. Let $m_{ij}^{(n)}(r,t)$
be the $(i,j)$th entry of the matrix $M_{r,t}^{n},$ then

$m_{ij}^{(n)}(r,t)=\sum_{j_{1},j_{2},\cdots,j_{n-1}=1}^{N}\Big{(}p_{ij_{1}}p_{j_{1}j_{2}}\cdots
p_{j_{n-1}j}\Big{(}s_{ij_{1}}s_{j_{1}j_{2}}\cdots
s_{j_{n-1},j}\Big{)}^{r}\Big{)}^{t}.$

Thus, we have

$\Phi_{r}(t)=\lim_{n\to\infty}\Big{(}\sum_{i,j=1}^{N}m_{ij}^{(n)}(r,t)\Big{)}^{\frac{1}{n}}.$

Similarly, corresponding to the lower contraction ratios, we define a matrix
$L_{r,t}=[(p_{ij}c_{ij}^{r})^{t}]_{1\leq i,j\leq N}$ and $\phi_{r}(t)$ is the
spectral radius of this matrix.

######  Definition 2.5.

The recurrent IFS $\mathcal{I}=\\{X;f_{ij},p_{ij}:1\leq i,j\leq N\\}$
satisfies the strong open set condition (SOSC), if there exist nonempty
bounded open sets $U_{1},U_{2},\cdots,U_{N}$ such that

$f_{ij}(U_{j})\cap
f_{kl}(U_{l})=\emptyset~{}~{}\text{for}~{}~{}(i,j)\neq(k,l),~{}~{}f_{ij}(U_{j})\subseteq
U_{i}~{}~{}\text{and}~{}~{}E_{i}\cap
U_{i}\neq\emptyset~{}~{}\text{where}~{}~{}1\leq i,j,k,l\leq N.$

######  Definition 2.6.

The recurrent IFS $\mathcal{I}=\\{X;f_{ij},p_{ij}:1\leq i,j\leq N\\}$
satisfies the strong separation condition (SSC), if

$f_{ij}(E_{j})\cap
f_{kl}(E_{l})=\emptyset~{}~{}\forall~{}~{}(i,j)\neq(k,l)~{}~{}\text{and}~{}~{}p_{ij}p_{kl}>0,~{}~{}\text{where}~{}~{}1\leq
i,j,k,l\leq N.$

A finite set $\Gamma\subset\Omega^{*}$ is said to be a finite maximal
antichain if each sequence in $\Omega$ is an extension of some element in
$\Gamma$, but no element of $\Gamma$ is an extension of another element of
$\Gamma.$

Let $\tau,w\in\Omega^{*}$ such that
$\tau=(\tau_{1},\tau_{2},\cdots,\tau_{m})\in\Omega_{m}$,
$w=(w_{1},w_{2},\cdots,w_{n})\in\Omega_{n}$ and $p_{w_{n}\tau_{1}}>0$, then we
have

$s_{w\tau}=s_{w}s_{\tau}s_{w_{n}\tau_{1}},\quad
c_{w\tau}=c_{w}c_{\tau}c_{w_{n}\tau_{1}},\quad
p_{w\tau}=p_{w}p_{\tau}\frac{p_{w_{n}\tau_{1}}}{p_{\tau_{1}}}.$

By the above, we obtain

$s_{w}s_{\tau}s_{\min}\leq s_{w\tau}\leq s_{w}s_{\tau}s_{\max},\quad
c_{w}c_{\tau}c_{\min}\leq c_{w\tau}\leq c_{w}c_{\tau}c_{\max},\quad
p_{w}p_{\tau}\frac{P_{\min}}{p_{\max}}\leq p_{w\tau}\leq
p_{w}p_{\tau}\frac{P_{\max}}{p_{\min}}.$

Therefore, we get

(4) $\displaystyle(p_{w}s_{w}^{r})(p_{\tau}s_{\tau}^{r})A_{r}\leq
p_{w\tau}s_{w\tau}^{r}\leq(p_{w}s_{w}^{r})(p_{\tau}s_{\tau}^{r})\tilde{A}_{r},$
(5) $\displaystyle(p_{w}c_{w}^{r})(p_{\tau}c_{\tau}^{r})B_{r}\leq
p_{w\tau}c_{w\tau}^{r}\leq(p_{w}c_{w}^{r})(p_{\tau}c_{\tau}^{r})\tilde{B}_{r},$

where
$A_{r}=\min\\{s_{\min}^{r}\frac{P_{\min}}{p_{\max}},1\\},\quad\tilde{A}_{r}=\max\\{s_{\max}^{r}\frac{P_{\max}}{p_{\min}},1\\},\quad
B_{r}=\min\\{c_{\min}^{r}\frac{P_{\min}}{p_{\max}},1\\}$ and
$\tilde{B}_{r}=\max\\{c_{\max}^{r}\frac{P_{\max}}{p_{\min}},1\\}.$

######  Remark 2.7.

In the above construction of the bi-Lipschitz RIFS, if we put
$f_{ij}=f_{i},c_{ij}=c_{i},s_{ij}=s_{i}$ and $p_{ij}=p_{j}$ for all $1\leq
i,j\leq N.$ Then, the bi-Lipschitz RIFS reduces to the bi-Lipschitz IFS
$\\{X;f_{1},f_{2},\cdots,f_{N}\\}$ with probability vector
$(p_{1},p_{2},\cdots,p_{N}).$ One can easily see that the limit set $E$
satisfies $E=\cup_{i=1}^{N}f_{i}(E)$ and $\mu$ becomes invariant measure
corresponding to this IFS such that $\mu=\sum_{i=1}^{N}p_{i}\mu\circ
f_{i}^{-1}.$ In this case, the matrix $M_{r,t}=[(p_{j}s_{i}^{r})^{t}]_{1\leq
i,j\leq N}.$ It is not difficult to see that the matrix $M_{r,t}$ is a rank-1
matrix. So, $0$ is an eigenvalue of $M_{r,t}$ with geometric multiplicity
$N-1.$ This implies that the spectral radius
$\Phi_{r}(t)=\sum_{i=1}^{N}(p_{i}s_{i}^{r})^{t}.$ Similarly,
$\phi_{r}(t)=\sum_{i=1}^{N}(p_{i}c_{i}^{r})^{t}.$

##  3\. Main result

In the following theorem, which is the main theorem of the paper, we determine
the upper and the lower bounds of the quantization dimensions of the Borel
probability measures supported on the limit sets of the bi-Lipschitz recurrent
IFS under the strong open set condition in terms of the spectral radius.

######  Theorem 3.1.

Let $\mathcal{I}=\\{X;f_{ij},p_{ij}:1\leq i,j\leq N\\}$ be a bi-Lipschitz
recurrent IFS satisfying the strong open set condition such that

$c_{ij}d(x,y)\leq d(f_{ij}(x),f_{ij}(y))\leq s_{ij}d(x,y)\quad\forall\quad
x,y\in X$

where $0<c_{ij}\leq s_{ij}<1$ for $1\leq i,j\leq N.$ Let $\mu$ be the Borel
probability measure supported on the limit set $E$ of the RIFS $\mathcal{I}$
defined as in the above section. Then,

$l_{r}\leq\underline{D}_{r}(\mu)\leq\overline{D}_{r}(\mu)\leq k_{r},$

where $l_{r}$ and $k_{r}$ are uniquely given by
$\phi_{r}\Big{(}\frac{l_{r}}{r+l_{r}}\Big{)}=1$ and
$\Phi_{r}\Big{(}\frac{k_{r}}{r+k_{r}}\Big{)}=1$, respectively.

######  Remark 3.2.

In [8], Graf and Luschgy gave a formula (1) for the quantization dimension of
the invariant measure of the similarity IFS under the OSC. However, in the
above theorem, if we consider $f_{ij}=f_{i},$ $p_{ij}=p_{j}$ and
$c_{ij}=s_{ij}=s_{i}$, then the bi-Lipschitz recurrent IFS reduces to the
similarity IFS with the OSC, and we get same formula (1) for the quantization
dimension. Thus, our result is a general version of the Graf-Luschgy’s result.

######  Remark 3.3.

In the construction of bi-Lipschitz RIFS, if we consider $s_{ij}=c_{ij}$, then
we get similarity RIFS. In this setting, we get same bounds of the
quantization dimension as in [23]. Thus, our result is more general than the
Roychowdhury’s result [23]. By Remark 3.2, it is clear that our result also
generalizes the result of Verma et al. [30].

######  Remark 3.4.

In [25], Roychowdhury determined the upper and the lower bound of the
quantization dimension of the invariant probability measure of the bi-
Lipschitz IFS under the SOSC and some restrictions on bi-Lipschitz constants.
Note that in the above theorem, we are not taking any restrictions on the bi-
Lipschitz constants as in [25]. By Remark 3.2 and the above theorem, we obtain
the same bound of the quantization dimension as in Roychowdhury’s paper [25].
Thus, our result generalizes the result of Roychowdhury [25] in more general
setting.

In order to prove Theorem 3.1, we need some lemmas and propositions.

######  Lemma 3.5.

Let $r\in(0,\infty)$ be fixed. Then, there exist unique
$k_{r},l_{r}\in(0,\infty)$ such that

$\Phi_{r}\Big{(}\frac{k_{r}}{r+k_{r}}\Big{)}=1,\quad\phi_{r}\Big{(}\frac{l_{r}}{r+l_{r}}\Big{)}=1.$

###### Proof.

The mapping $\Phi_{r}:[0,\infty)\to[0,\infty)$ is defined as

$\Phi_{r}(t)=\lim_{n\to\infty}\Big{(}\sum_{i,j=1}^{N}m_{ij}^{(n)}(r,t)\Big{)}^{\frac{1}{n}},$

where $m_{ij}^{(n)}(r,t)$ defined as in the above section. Clearly $\Phi_{r}$
is a continuous and strictly decreasing mapping, and

$\Phi_{r}(0)=\lim_{n\to\infty}\Big{(}\sum_{i,j=1}^{N}m_{ij}^{(n)}(r,0)\Big{)}^{\frac{1}{n}}=\lim_{n\to\infty}\Big{(}\sum_{i,j=1}^{N}N^{n-1}\Big{)}^{\frac{1}{n}}=N\geq
2.$

Moreover,

$\displaystyle\Phi_{r}(1)$
$\displaystyle=\lim_{n\to\infty}\Big{(}\sum_{i,j=1}^{N}m_{ij}^{(n)}(r,1)\Big{)}^{\frac{1}{n}}$
$\displaystyle=\lim_{n\to\infty}\Big{(}\sum_{i,j=1}^{N}\sum_{j_{1},j_{2},\cdots,j_{n-1}=1}^{N}(p_{ij_{1}}p_{j_{1}j_{2}}\cdots
p_{j_{n-1}j}(s_{ij_{1}}s_{j_{1}j_{2}}\cdots
s_{j_{n-1},j})^{r})\Big{)}^{\frac{1}{n}}$ $\displaystyle\leq s_{\max}^{r}<1.$

Since $\Phi_{r}$ is continuous, strictly decreasing, $\Phi_{r}(0)\geq 2$ and
$\Phi_{r}(1)<1$, there exists a unique $t_{r}\in(0,1)$ such that
$\Phi_{r}(t_{r})=1.$ By the uniqueness of $t_{r}$, there exists a unique
$k_{r}\in(0,\infty)$ such that $t_{r}=\frac{k_{r}}{r+k_{r}}$. Similarly, there
exits a unique $l_{r}\in(0,\infty)$ such that
$\phi_{r}\Big{(}\frac{l_{r}}{r+l_{r}}\Big{)}=1.$ This completes the proof. ∎

######  Lemma 3.6.

Let $r\in(0,\infty)$ be fixed and $k_{r},l_{r}$ be defined as in the above
lemma. Then,

$\lim_{n\to\infty}\frac{1}{n}\log\sum_{w\in\Omega_{n}}(p_{w}{c_{w}}^{r})^{\frac{l_{r}}{r+l_{r}}}=0,\quad\lim_{n\to\infty}\frac{1}{n}\log\sum_{w\in\Omega_{n}}(p_{w}{s_{w}}^{r})^{\frac{k_{r}}{r+k_{r}}}=0.$

###### Proof.

Consider the mapping $\theta_{r}:[0.\infty)\to\mathbb{R}$, defined by
$\theta_{r}(t)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{w\in\Omega_{n}}(p_{w}{s_{w}}^{r})^{t}.$
By (4), one can easily see that $\theta_{r}(t)$ exists. Thus,

$\displaystyle\theta_{r}(t)$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\log\Big{(}\sum_{i,j_{1},j_{2},\cdots,j_{n-1},j=1}^{N}(p_{i}p_{ij_{1}}p_{j_{1}j_{2}}\cdots
p_{j_{n-1}j}({s_{ij_{1}}s_{j_{1}j_{2}}\cdots s_{j_{n-1}j}})^{r})^{t}\Big{)}$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\log\Big{(}\sum_{i,j=1}^{N}p_{i}^{r}m_{ij}^{(n)}(r,t)\Big{)}.$

Therefore, we get

$\lim_{n\to\infty}\frac{1}{n}\log\Big{(}\sum_{i,j=1}^{N}p_{\min}^{r}m_{ij}^{(n)}(r,t)\Big{)}\leq\theta_{r}(t)\leq\lim_{n\to\infty}\frac{1}{n}\log\Big{(}\sum_{i,j=1}^{N}p_{\max}^{r}m_{ij}^{(n)}(r,t)\Big{)},$
$\lim_{n\to\infty}\frac{1}{n}\log\Big{(}\sum_{i,j=1}^{N}m_{ij}^{(n)}(r,t)\Big{)}\leq\theta_{r}(t)\leq\lim_{n\to\infty}\frac{1}{n}\log\Big{(}\sum_{i,j=1}^{N}m_{ij}^{(n)}(r,t)\Big{)}.$

Thus, we obtain $\log(\Phi_{r}(t))=\theta_{r}(t).$ By using the previous
lemma, we get $\theta_{r}(\frac{k_{r}}{r+k_{r}})=0.$ By using the similar
arguments as above, we obtain
$\lim\limits_{n\to\infty}\frac{1}{n}\log\sum\limits_{w\in\Omega_{n}}(p_{w}{c_{w}}^{r})^{\frac{l_{r}}{r+l_{r}}}=0.$
Thus, the proof is done. ∎

######  Lemma 3.7.

Let $r\in(0,\infty)$ be fixed and $k_{r},l_{r}$ be defined as in Lemma 3.5.
Then, for each $n\geq 2$

$F_{r}^{\frac{-k_{r}}{r+k_{r}}}\leq\sum_{w\in\Omega_{n}}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}\leq
F_{r}^{\frac{k_{r}}{r+k_{r}}},\text{ and
}G_{r}^{\frac{-l_{r}}{r+l_{r}}}\leq\sum_{w\in\Omega_{n}}(p_{w}s_{w}^{r})^{\frac{l_{r}}{r+l_{r}}}\leq
G_{r}^{\frac{l_{r}}{r+l_{r}}},$

where $F_{r}=\max\\{A_{r}^{-1},\tilde{A}_{r}\\}$ and
$G_{r}=\max\\{B_{r}^{-1},\tilde{B}_{r}\\}$

###### Proof.

We consider the same mapping $\theta_{r}$ as in the previous lemma. Then, we
have
$\theta_{r}(t)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{w\in\Omega_{n}}(p_{w}{s_{w}}^{r})^{t}.$
We can also write $\theta_{r}(t)$ as follows

$\theta_{r}(t)=\lim_{m\to\infty}\frac{1}{nm}\log\sum_{w\in\Omega_{nm}}(p_{w}{s_{w}}^{r})^{t}.$

Thus, by the above and inequality (4), we get

$\lim_{m\to\infty}\frac{1}{nm}\log(\sum_{w\in\Omega_{n}}(p_{w}{s_{w}}^{r}{A}_{r})^{t})^{m}\leq\theta_{r}(t)\leq\lim_{m\to\infty}\frac{1}{nm}\log(\sum_{w\in\Omega_{n}}(p_{w}{s_{w}}^{r}\tilde{A}_{r})^{t})^{m},$

this implies that

$\frac{1}{n}\log\sum_{w\in\Omega_{n}}(p_{w}{s_{w}}^{r}{A}_{r})^{t}\leq\theta_{r}(t)\leq\frac{1}{n}\log\sum_{w\in\Omega_{n}}(p_{w}{s_{w}}^{r}\tilde{A}_{r})^{t},$

therefore, we obtain

$e^{n\theta_{r}(t)}\tilde{A}_{r}^{-t}\leq\sum_{w\in\Omega_{n}}(p_{w}{s_{w}}^{r})^{t}\leq
e^{n\theta_{r}(t)}A_{r}^{-t}.$

Let $F_{r}=\max\\{A_{r}^{-1},\tilde{A}_{r}\\}$. By the previous Lemma,
$\theta_{r}(\frac{k_{r}}{r+k_{r}})=0.$ Thus, we get

$F_{r}^{\frac{-k_{r}}{r+k_{r}}}\leq\sum_{w\in\Omega_{n}}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}\leq
F_{r}^{\frac{k_{r}}{r+k_{r}}}.$

Similarly, we can get another inequality by taking $c_{w}$ instead of $s_{w}$
in the above. This completes the proof. ∎

######  Lemma 3.8.

Let $r\in(0,\infty)$ be fixed and $\Gamma$ be a finite maximal antichain.
Then, there exist constants $\eta_{r}\geq 1$ and $\tilde{\eta}_{r}\geq 1$ such
that

$\sum_{w\in\Gamma}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}\leq\eta_{r},\quad\sum_{w\in\Gamma}(p_{w}c_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}\leq\tilde{\eta}_{r}.$

###### Proof.

Since $\Gamma$ is a finite maximal antichain, we have
$\Gamma=\bigcup_{i=1}^{K}\Gamma_{n_{i}}$ for some $K>0$, where
$\Gamma_{n_{i}}=\\{w\in\Gamma:|w|=n_{i}\\}$ for all $i\in\\{1,2,\cdots,K\\}$.
Let $n>1+\max\\{n_{1},n_{2},\cdots,n_{K}\\}.$ Then, by the previous lemma, we
have

$\displaystyle F_{r}^{\frac{k_{r}}{r+k_{r}}}$
$\displaystyle\geq\sum_{\tau\in\Omega_{n}}(p_{\tau}s_{\tau}^{r})^{\frac{k_{r}}{r+k_{r}}}=\sum_{i=1}^{K}\sum_{w\in\Gamma_{n_{i}}}\sum_{\tau\in\Omega_{n},w\mathcal{<}\tau}(p_{\tau}s_{\tau}^{r})^{\frac{k_{r}}{r+k_{r}}}=\sum_{i=1}^{K}\sum_{w\in\Gamma_{n_{i}}}\sum_{\tau\in\Omega_{n-n_{i}}}(p_{w\tau}s_{w\tau}^{r})^{\frac{k_{r}}{r+k_{r}}}$
$\displaystyle=\sum_{i=1}^{K}\sum_{w\in\Gamma_{n_{i}}}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}\sum_{\tau\in\Omega_{n-n_{i}}}\Big{(}p_{w_{n_{i}}\tau_{1}}\frac{p_{\tau}}{p_{\tau_{1}}}s_{w_{n_{i}}\tau_{1}}^{r}s_{\tau}^{r}\Big{)}^{\frac{k_{r}}{r+k_{r}}}$
$\displaystyle\geq\sum_{i=1}^{K}\sum_{w\in\Gamma_{n_{i}}}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}(P_{\min}p_{\max}^{-1}s_{\min}^{r})^{\frac{k_{r}}{r+k_{r}}}\sum_{\tau\in\Omega_{n-n_{i}}}(p_{\tau}s_{\tau}^{r})^{\frac{k_{r}}{r+k_{r}}}.$

Again, using the previous lemma, we get

$F_{r}^{\frac{k_{r}}{r+k_{r}}}\geq\sum_{i=1}^{K}\sum_{w\in\Gamma_{n_{i}}}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}(P_{\min}p_{\max}^{-1}s_{\min}^{r})^{\frac{k_{r}}{r+k_{r}}}F_{r}^{\frac{-k_{r}}{r+k_{r}}}.$

This implies that

$\sum_{w\in\Gamma}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}\leq\eta_{r},$

where
$\eta_{r}=\max\Big{\\{}1,F_{r}^{\frac{2k_{r}}{r+k_{r}}}(P_{\min}p_{\max}^{-1}s_{\min}^{r})^{\frac{-k_{r}}{r+k_{r}}}\Big{\\}}.$
Similarly, one can get another inequality. Thus, the proof is done. ∎

######  Lemma 3.9.

Let $r\in(0,\infty)$ and $\mu$ be the Borel probability measure supported on
the limit set $E$ of the RIFS $\mathcal{I}$. Let $\Gamma\subset\Omega^{*}$ be
a finite maximal antichain. Then, for each $n\in\mathbb{N}$ with
$n\geq|\Gamma|$, we have

$V_{n,r}(\mu)\leq\inf\Big{\\{}\sum_{w\in\Gamma}p_{w}s_{w}^{r}V_{n_{w},r}(\mu):~{}~{}n_{w}\in\mathbb{N},\sum\limits_{w\in\Gamma}n_{w}\leq
n\Big{\\}}.$

###### Proof.

Since $\Gamma\in\Omega^{*}$ is a finite maximal antichain, we have
$\mu=\sum_{w\in\Gamma}p_{w}\mu\circ f_{w}^{-1}$. Let $A_{n_{w}}$ be an
$n_{w}$-optimal set of $V_{n_{w},r}(\mu)$ for each $w\in\Gamma.$ Since $f_{w}$
is a bi-Lipschitz map, we have $|f_{w}(A_{n_{w}})|=n_{w}$ and
$|\cup_{w\in\Gamma}f_{w}(A_{n_{w}})|\leq n.$ Then, we have

$\displaystyle V_{n,r}(\mu)$ $\displaystyle\leq\int
d\Big{(}x,\bigcup_{w\in\Gamma}f_{w}(A_{n_{w}})\Big{)}^{r}d{\mu}(x)$
$\displaystyle=\sum_{w\in\Gamma}p_{w}\int
d\Big{(}x,\bigcup_{w\in\Gamma}f_{w}(A_{n_{w}})\Big{)}^{r}d{(\mu\circ
f_{w}^{-1})}(x)$ $\displaystyle=\sum_{w\in\Gamma}p_{w}\int
d\Big{(}f_{w}(x),\bigcup_{w\in\Gamma}f_{w}(A_{n_{w}})\Big{)}^{r}d{\mu}(x)$
$\displaystyle\leq\sum_{w\in\Gamma}p_{w}\int
d\Big{(}f_{w}(x),f_{w}(A_{n_{w}})\Big{)}^{r}d{\mu}(x)$
$\displaystyle\leq\sum_{w\in\Gamma}p_{w}s_{w}^{r}\int
d(x,A_{n_{w}})^{r}d{\mu}(x)$
$\displaystyle=\sum_{w\in\Gamma}p_{w}s_{w}^{r}V_{n_{w},r}(\mu).$

This completes the proof. ∎

In the following proposition, we prove that the upper quantization coefficient
for the Borel probability measure supported on the limit set of the RIFS
$\mathcal{I}$ is finite without any separation condition.

######  Proposition 3.10.

Let $\mathcal{I}=\\{X;f_{ij},p_{ij}:1\leq i,j\leq N\\}$ be a bi-Lipschitz
recurrent IFS. Let $\mu$ be the Borel probability measure supported on the
limit set $E$ of the RIFS $\mathcal{I}$. Let $r\in(0,\infty)$ and
$k_{r}\in(0,\infty)$ be a unique number such that
$\Phi_{r}(\frac{k_{r}}{r+k_{r}})=1.$ Then,

$\limsup_{n\to\infty}ne_{n,r}^{k_{r}}(\mu)<\infty.$

###### Proof.

Let $\xi_{\min}=\\{(p_{ij}s_{ij}^{r})^{\frac{k_{r}}{r+k_{r}}}:p_{ij}>0,1\leq
i,j\leq N\\}$. Clearly $\xi_{\min}\in(0,1).$ Let $n_{0}\in\mathbb{N}$ be a
fixed number. Take any $n\in\mathbb{N}$ such that
$\frac{n_{0}\eta_{r}}{n}\leq\xi_{\min}^{2},$ this holds for all but finitely
many value of $n.$ Let $\epsilon=\frac{n_{0}\eta_{r}}{n}\xi_{\min}^{-1}.$
Clearly $\epsilon\in(0,1).$ Now, we define a subset $\Gamma_{\epsilon}$ of
$\Omega^{*}$ as follows

$\Gamma_{\epsilon}=\\{w\in\Omega^{*}:(p_{w^{-}}s_{w^{-}}^{r})^{\frac{k_{r}}{r+k_{r}}}\geq\epsilon>(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}\\}.$

Then, $\Gamma_{\epsilon}$ is a finite maximal antichain. Using Lemma 3.8, we
have

$\eta_{r}\geq\sum_{w\in\Gamma_{\epsilon}}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}\geq\sum_{w\in\Gamma_{\epsilon}}(p_{w^{-}}s_{w^{-}}^{r})^{\frac{k_{r}}{r+k_{r}}}\xi_{\min}\geq|\Gamma_{\epsilon}|\xi_{\min}\epsilon,$

which implies that $|\Gamma_{\epsilon}|\leq\frac{n}{n_{0}}.$ Thus, we have
$\sum_{w\in\Gamma_{\epsilon}}n_{0}\leq n.$ Therefore, by the above lemma, we
get

$\displaystyle V_{n,r}(\mu)$
$\displaystyle\leq\sum_{w\in\Gamma_{\epsilon}}p_{w}s_{w}^{r}V_{n_{0},r}(\mu)$
$\displaystyle=\sum_{w\in\Gamma_{\epsilon}}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}(p_{w}s_{w}^{r})^{\frac{r}{r+k_{r}}}V_{n_{0},r}(\mu)$
$\displaystyle\leq\sum_{w\in\Gamma_{\epsilon}}(p_{w}s_{w}^{r})^{\frac{k_{r}}{r+k_{r}}}{\epsilon}^{\frac{r}{k_{r}}}V_{n_{0},r}(\mu).$

Using Lemma 3.8 and $\epsilon=\frac{n_{0}\eta_{r}}{n}\xi_{\min}^{-1}$, we
obtain

$V_{n,r}(\mu)\leq\eta_{r}\Big{(}{\frac{n_{0}\eta_{r}}{n}\xi_{\min}^{-1}}\Big{)}^{\frac{r}{k_{r}}}V_{n_{0},r}(\mu),$

which implies that

$ne_{n,r}^{k_{r}}(\mu)\leq\eta_{r}^{(\frac{k_{r}}{r}+1)}n_{0}\xi_{\min}^{-1}e_{n_{0},r}^{k_{r}}(\mu).$

The above inequality holds for all but finitely many values of $n,$ and thus
we have

$\limsup_{n\to\infty}ne_{n,r}^{k_{r}}(\mu)<\infty,$

which completes the proof of the proposition. ∎

In the end, we will use the above proposition to determine the upper bound of
the quantization dimension. Now, we give some lemmas and propositions to
determine the lower bound of the quantization dimension.

######  Lemma 3.11.

Let $\mathcal{I}=\\{X;f_{ij},p_{ij}:1\leq i,j\leq N\\}$ be a bi-Lipschitz
recurrent IFS. Let $\mu$ be the Borel probability measure supported on the
limit set $E.$ Then, for each $\epsilon>0$

$\inf\\{\mu(B(x,\epsilon)):x\in E\\}>0.$

###### Proof.

Since $t\to\sum\limits_{i,j=1,p_{ij}>0}^{N}s_{ij}^{t}$ is a strictly
decreasing continuous function for $t\in[0,\infty)$, there exists a unique
$t_{o}\in(0,\infty)$ such that
$\sum\limits_{i,j=1,p_{ij}>0}^{N}s_{ij}^{t_{0}}=1.$ Let
$\xi_{0}=(\frac{\epsilon}{diam(E)})^{t_{0}}.$ We define a finite maximal
antichain $\Gamma_{\epsilon}$ as follows

$\Gamma_{\epsilon}=\\{w\in\Omega^{*}:s_{w^{-}}^{t_{0}}\geq\xi_{0}>s_{w}^{t_{0}}\\}.$

Since $E=\cup_{i=1}^{N}E_{i}$, we have $E=\cup_{w\in\Gamma_{\epsilon}}E_{w}.$
Let $x\in E$. Then, there exists a
$w_{0}=(w_{1},w_{2},\cdots,w_{n})\in\Gamma_{\epsilon}$ such that $x\in
E_{w_{0}}=f_{w_{0}}(E_{w_{n}}).$ Let $x_{1},x_{2}\in E_{w_{n}}.$ Then, we have

$\displaystyle d(f_{w_{0}}(x_{1}),f_{w_{0}}(x_{2}))\leq
s_{w_{0}}diam(E)<\epsilon.$

This implies that, $diam(E_{w_{0}})<\epsilon$. Since $x\in E_{w_{0}},$ we have
$E_{w_{0}}\subseteq B(x,\epsilon).$ Since
$\mu=\sum_{w\in\Gamma_{\epsilon}}p_{w}\mu\circ f_{w}^{-1},$ we have

$\displaystyle\mu{(E_{w_{0}})}=\sum_{w\in\Gamma_{\epsilon}}p_{w}\mu\circ
f_{w}^{-1}(E_{w_{0}})\geq
p_{w_{0}}\mu_{w_{n}}(E_{w_{n}})\geq\min_{w\in\Gamma_{\epsilon}}\\{P_{w}\\}\min_{1\leq
i\leq N}\\{\mu_{i}(E_{i})\\}.$

This implies that
$\mu(B(x,\epsilon))\geq\min_{w\in\Gamma_{\epsilon}}\\{P_{w}\\}\min_{1\leq
i\leq N}\\{\mu_{i}(E_{i})\\}$. This holds for all $x\in E$, thus
$\inf\\{\mu(B(x,\epsilon)):x\in E\\}>0.$ This completes the proof. ∎

######  Proposition 3.12.

Let $\mathcal{I}=\\{X;f_{ij},p_{ij}:1\leq i,j\leq N\\}$ be a bi-Lipschitz
recurrent IFS satisfying the strong open set condition. Then, for each
$m\in\mathbb{N}$, there is a RIFS $\mathcal{I}^{m}=\\{X;g_{w},p_{ij}:1\leq
i,j\leq N,w\in\Omega_{m}\\}$ satisfying the strong separation condition such
that $C_{w}d(x,y)\leq d(g_{w}(x),g_{w}(y))\leq S_{w}d(x,y)$ and $0<C_{w}\leq
S_{w}<1.$ Let $\mu_{m}$ be the Borel probability measure supported on the
limit set $F^{m}$ of the RIFS $\mathcal{I}^{m}.$ Then, there exists
$n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$ there exist
$n_{w}\in\mathbb{N}$ such that $\sum_{w\in\Omega_{m}}n_{w}\leq n$ and

$V_{n,r}(\mu_{m})\geq\sum_{w\in\Omega_{m}}p_{w}C_{w}^{r}V_{n_{w},r}(\mu_{m}).$

###### Proof.

Since the RIFS $\mathcal{I}$ satisfies the strong open set condition, there
exist nonempty bounded open sets $U_{1},U_{2},\cdots,U_{N}$ such that

$f_{ij}(U_{j})\cap
f_{kl}(U_{l})=\emptyset~{}~{}\text{for}~{}~{}(i,j)\neq(k,l),~{}~{}f_{ij}(U_{j})\subseteq
U_{i}~{}~{}\text{and}~{}~{}E_{i}\cap
U_{i}\neq\emptyset~{}~{}\text{where}~{}~{}1\leq i,j,k,l\leq N.$

This also implies that for any $w=(w_{1},w_{2},\cdots,w_{m})$ and
$\tau=(\tau_{1},\tau_{2},\cdots,\tau_{m})$ in $\Omega_{m}$ such that
$w\neq\tau$, we have $f_{w}(U_{w_{m}})\cap f_{\tau}(U_{\tau_{m}})=\emptyset.$

For each $i\in\\{1,2,\cdots,N\\}$, given that $E_{i}\cap U_{i}\neq\emptyset,$
there exists a sequence $\sigma=(i,\sigma_{2},\cdots,\sigma_{n})\in\Omega^{*}$
such that $E_{\sigma}=f_{\sigma}(E_{\sigma_{n}})\subseteq U_{i}.$ Since $P$ is
a irreducible matrix, there exists a finite sequence which starts from
$\sigma_{n}$ and ends at $i,$ say
$\sigma*=(\sigma_{n},\sigma_{n+1},\cdots,k,i)\in\Omega^{*}.$ Now, we define a
finite sequence
$\sigma^{i}=(i,\sigma_{2},\cdots,\sigma_{n},\sigma_{n+1},\cdots,i).$ Clearly
$\sigma^{i}\in\Omega^{*}$ and
$E_{\sigma^{i}}=f_{\sigma^{i}}(E_{i})=f_{i\sigma_{2}}\circ
f_{\sigma_{2}\sigma_{3}}\circ\cdots f_{\sigma_{n-1}\sigma_{n}}\circ
f_{\sigma_{n}\sigma_{n+1}}\circ\cdots\circ f_{ki}(E_{i})\subseteq U_{i}.$

For each $m\in\mathbb{N},$ we define a RIFS
$\mathcal{I}^{m}=\\{X;g_{w},p_{ij}:1\leq i,j\leq N,w\in\Omega_{m}\\}$, where
$g_{w}$ is defined as $g_{w}=f_{w}\circ f_{\sigma^{i}}$ for
$w=(w_{1},w_{2},\cdots,w_{m-1},i)\in\Omega_{m}.$ For $x,y\in X$, we have

$C_{w}d(x,y)\leq d(g_{w}(x),g_{w}(y))\leq S_{w}d(x,y),$

where $C_{w}=c_{w}c_{\sigma^{i}},S_{w}=s_{w}s_{\sigma^{i}}$ and
$w=(w_{1},w_{2},\cdots,w_{m-1},i)\in\Omega_{m}.$ Now, we claim that the RIFS
$\mathcal{I}^{m}$ satisfies the strong separation condition. Let $F^{m}$ be
the limit set of the RIFS $\mathcal{I}^{m}$. Then,

$F^{m}=\bigcup_{i=1}^{N}F_{i}^{m}\quad\text{and}\quad
F_{i}^{m}=\bigcup_{w\in\Omega_{m},w=(i,w_{2},\cdots,w_{m})}g_{iw_{2}\cdots
w_{m}}(F_{w_{m}}^{m}).$

By observing the code space of $\mathcal{I}^{m}$ and $\mathcal{I}$, it is
clear that the code space of $\mathcal{I}^{m}$ is contained in $\Omega.$ Then,
we have $F^{m}\subseteq E$ and $F_{i}^{m}\subseteq E_{i}$ for each
$i\in\\{1,2,\cdots,N\\}.$ Let
$w=(w_{1},w_{2},\cdots,w_{m})\neq\tau=(\tau_{1},\tau_{2},\cdots,\tau_{m})\in\Omega_{m}.$
Then, we have

$\displaystyle g_{w}(F_{w_{m}}^{m})=f_{w}\circ
f_{\sigma^{w_{m}}}(F_{w_{m}}^{m})\subseteq f_{w}\circ
f_{\sigma^{w_{m}}}(E_{w_{m}})\subseteq f_{w}(U_{w_{m}})$

and

$\displaystyle g_{\tau}(F_{\tau_{m}}^{m})=f_{\tau}\circ
f_{\sigma^{\tau_{m}}}(F_{\tau_{m}}^{m})\subseteq f_{\tau}\circ
f_{\sigma^{\tau_{m}}}(E_{\tau_{m}})\subseteq f_{\tau}(U_{\tau_{m}}).$

Since $w\neq\tau\in\Omega_{m}$, we obtain $g_{w}(F_{w_{m}}^{m})\cap
g_{\tau}(F_{\tau_{m}}^{m})=\emptyset.$ This implies that the RIFS
$\mathcal{I}^{m}$ satisfy the strong separation condition. Thus, the claim is
true. Let $\mu_{m}$ be the Borel probability measure supported on the limit
set $F^{m}$ of the RIFS $\mathcal{I}^{m}$ such that
$\mu_{m}=\sum_{w\in\Omega_{m}}p_{w}\mu_{m}\circ g_{w}^{-1}.$

Set
$\epsilon_{0}=\min_{w\neq\tau\in\Omega_{m}}\\{d(F_{w}^{m},F_{\tau}^{m})\\}$.
Since the RIFS $\mathcal{I}^{m}$ satisfies the strong open set condition, we
have $\epsilon_{0}>0.$ Let

$\delta=\inf\Big{\\{}\Big{(}\frac{\epsilon_{0}}{8}\Big{)}^{r}\mu_{m}\Big{(}B\Big{(}x,\frac{\epsilon_{0}}{8}\Big{)}\Big{)}:x\in
F^{m}\Big{\\}}.$

By using the previous lemma, we obtain that $\delta>0.$ Lemma 2.3 yields that
$\lim_{n\to\infty}V_{n,r}(\mu_{m})=0.$ Thus, there exists an
$n_{0}\in\mathbb{N}$ such that $V_{n,r}(\mu)<\delta\text{ for all }n\geq
n_{0}.$ Let $A_{n}$ be an $n$-optimal set for probability measure $\mu_{m}$ of
order $r$, then by Lemma 2.2 we get

$\Big{(}\frac{\|A_{n}\|_{\infty}}{2}\Big{)}^{r}\min\limits_{x\in
F^{m}}\mu_{m}\Big{(}B\Big{(}x,\frac{\|A_{n}\|_{\infty}}{2}\Big{)}\Big{)}<\delta.$

Since $t\to t^{r}\min_{x\in F^{m}}\mu_{m}(B(x,t))$ is an increasing function
for $t\in(0,\infty)$, we have

$\|A_{n}\|_{\infty}<\frac{\epsilon_{0}}{4}\text{ for all }n\geq n_{0}.$

We define a finite set $A_{n_{w}}=\\{a\in A_{n}:W(a|A_{n})\cap
F_{w}^{m}\neq\emptyset\\}$ for $w\in\Omega_{m}$ and let $n_{w}$ be the
cardinality of $A_{n_{w}}.$ Since
$\|A_{n}\|_{\infty}<\frac{\epsilon_{0}}{4}\text{ for all }n\geq n_{0}$, we
obtain that for all $n\geq n_{0},$ $A_{n_{w}}\cap A_{n_{\tau}}=\emptyset$ for
$w\neq\tau\in\Omega_{m}$ and $\sum_{w\in\Omega_{m}}n_{w}\leq n.$ Thus, for
$n\geq n_{0},$

$\displaystyle V_{n,r}(\mu_{m})$ $\displaystyle=\int
d\Big{(}x,A_{n}\Big{)}^{r}d{\mu_{m}}(x)$
$\displaystyle=\sum_{w\in\Omega_{m}}p_{w}\int
d\Big{(}g_{w}(x),A_{n}\Big{)}^{r}d{\mu_{m}}(x)$
$\displaystyle=\sum_{w\in\Omega_{m}}p_{w}\int
d\Big{(}g_{w}(x),A_{n_{w}}\Big{)}^{r}d{\mu_{m}}(x)$
$\displaystyle\geq\sum_{w\in\Omega_{m}}p_{w}C_{w}^{r}\int
d\Big{(}x,g_{w}^{-1}(A_{n_{w}})\Big{)}^{r}d{\mu_{m}}(x)$
$\displaystyle\geq\sum_{w\in\Omega_{m}}p_{w}C_{w}^{r}V_{n_{w},r}(\mu_{m}).$

Thus, the proof is done. ∎

######  Proposition 3.13.

Let $\mathcal{I}^{m}$ be a RIFS defined as in the previous proposition. Let
$r\in(0,\infty).$ Then,

$\liminf_{n\to\infty}ne_{n,r}^{l}(\mu_{m})>0\text{ for all }l\in(0,l_{m,r}),$

where $\l_{m,r}\in(0,\infty)$ is uniquely determined by
$\sum_{w\in\Omega_{m}}(p_{w}C_{w}^{r})^{\frac{l_{m,r}}{r+l_{m,r}}}=1$.
Moreover, $l_{m,r}\leq\underline{D}_{r}(\mu_{m}).$

###### Proof.

Since $p_{w}C_{w}^{r}<1$ for each $w\in\Omega_{m},$ we have
$t\to\sum_{w\in\Omega_{m}}(p_{w}C_{w}^{r})^{t}$ is a strictly decreasing
continuous function. It is not difficult to see that there exists a unique
$\l_{m,r}\in(0,\infty)$ such that
$\sum_{w\in\Omega_{m}}(p_{w}C_{w}^{r})^{\frac{l_{m,r}}{r+l_{m,r}}}=1.$ Let
$l\in(0,l_{m,r}),$ then
$\sum_{w\in\Omega_{m}}(p_{w}C_{w}^{r})^{\frac{l}{r+l}}>1.$ Let
$n_{0}\in\mathbb{N}$ be same as in the previous proposition. Let
$K_{0}=\inf\\{n^{\frac{r}{l}}V_{n,r}(\mu_{m}):n<n_{0}\\}$. Clearly $K_{0}>0.$
We claim that $K_{0}\leq n^{\frac{r}{l}}V_{n,r}(\mu_{m})$ for all
$n\in\mathbb{N}.$ We prove the claim by induction on $n\in\mathbb{N}.$ Let
$n\geq n_{0}$ and $K_{0}\leq k^{\frac{r}{l}}V_{k,r}(\mu_{m})$ for all $k<n.$
By using the previous proposition, we see that there exist
$n_{w}\in\mathbb{N}$ with $\sum_{w\in\Omega_{m}}n_{w}\leq n$. Then, we have

$\displaystyle n^{\frac{r}{l}}V_{n,r}(\mu_{m})$ $\displaystyle\geq
n^{\frac{r}{l}}\sum_{w\in\Omega_{m}}p_{w}C_{w}^{r}V_{n_{w},r}(\mu_{m})$
$\displaystyle\geq
n^{\frac{r}{l}}\sum_{w\in\Omega_{m}}p_{w}C_{w}^{r}K_{0}(n_{w})^{\frac{-r}{l}}$
$\displaystyle=K_{0}\sum_{w\in\Omega_{m}}p_{w}C_{w}^{r}\Big{(}\frac{n_{w}}{n}\Big{)}^{\frac{-r}{l}}.$

Now by using Hölder inequality for negative exponent, we obtain

$n^{\frac{r}{l}}V_{n,r}(\mu_{m})\geq
K_{0}\Big{(}\sum_{w\in\Omega_{m}}(p_{w}C_{w}^{r})^{\frac{l}{r+l}}\Big{)}^{\frac{l+r}{l}}\Big{(}\sum_{w\in\Omega_{m}}\Big{(}\frac{n_{w}}{n}\Big{)}^{\frac{-r}{l}\frac{-l}{r}}\Big{)}^{\frac{-l}{r}}.$

Using the fact that $\sum_{w\in\Omega_{m}}(p_{w}C_{w}^{r})^{\frac{l}{r+l}}>1$
and $\sum_{w\in\Omega_{m}}{n_{w}}\leq n,$ we get

$n^{\frac{r}{l}}V_{n,r}(\mu_{m})\geq K_{0}.$

Thus, by the principal of induction, we get
$n^{\frac{r}{l}}V_{n,r}(\mu_{m})\geq K_{0}$ holds for all $n\in\mathbb{N}$,
which is the claim. This implies that

$\liminf_{n\to\infty}ne_{n,r}^{l}(\mu_{m})\geq K_{0}^{\frac{l}{r}}>0\text{ for
all }l\in(0,l_{m,r}),$

Therefore, by Proposition 2.4, we have
$l_{m,r}\leq\underline{D}_{r}(\mu_{m}).$ This completes the proof. ∎

In the upcoming proposition, we determine the lower bound of the quantization
dimension of the Borel probability measure supported on the limit set of the
RIFS under the strong open set condition.

######  Proposition 3.14.

Let $\mathcal{I}=\\{X;f_{ij},p_{ij}:1\leq i,j\leq N\\}$ be a bi-Lipschitz
recurrent IFS satisfying the strong open set condition. Let $\mu$ be the Borel
probability measure supported on the limit set $E$ of the RIFS $\mathcal{I}$ .
Let $r\in(0,\infty)$ and $l_{r}\in(0,\infty)$ be a unique number such that
$\phi_{r}\Big{(}\frac{l_{r}}{r+l_{r}}\Big{)}=1.$ Then, we have
$l_{r}\leq\underline{D}_{r}(\mu).$

###### Proof.

Since the RIFS $\mathcal{I}$ satisfies the strong open set condition,
Proposition 3.12 yields that for each $m\in\mathbb{N},$ there exists a RIFS
$\mathcal{I}^{m}=\\{X;g_{w},p_{ij}:1\leq i,j\leq N,w\in\Omega_{m}\\}$ that
satisfy the strong separation condition, where $g_{w}=f_{w}\circ
f_{\sigma^{i}}$ for $w=(w_{1},w_{2},\cdots,w_{m-1},i)\in\Omega_{m}$ and
$\sigma^{i}$ is defined as, for each $i\in\\{1,2,\cdots,N\\}$ there exists a
finite sequence $\sigma=(i,\sigma_{2},\cdots,\sigma_{n})\in\Omega^{*}$ such
that $E_{\sigma}=f_{\sigma}(E_{\sigma_{n}})\subseteq U_{i}$ and
$\sigma^{*}=(\sigma_{n},\sigma_{n+1},\cdots,i)\in\Omega^{*},$ thus with the
help of $\sigma$ and $\sigma^{*}$ we define
$\sigma^{i}=(i,\sigma_{2},\cdots,\sigma_{n},\sigma_{n+1},\cdots,i).$ Let
$\mu_{m}$ be the probability measure supported on the limit set $F^{m}$ of the
RIFS $\mathcal{I}^{m}.$ By using Proposition 11.6 in [7], we have
$0<{\dim}_{\text{H}}(\mu)\leq\underline{D}_{r}(\mu).$ Thus,
$0<\underline{D}_{r}(\mu).$ If
$\underline{\dim}_{\text{B}}(F^{m})\leq\underline{D}_{r}(\mu)$ holds, then we
proceed. Otherwise, we may redefine $\sigma^{i}$ by choosing larger length of
$\sigma^{*}$ due to irreducibility of $P$ such that
$\underline{\dim}_{\text{B}}(F^{m})\leq\underline{D}_{r}(\mu)$. By the
previous proposition, $l_{m,r}\leq\underline{D}_{r}(\mu_{m}),$ where $l_{m,r}$
is uniquely determined by
$\sum_{w\in\Omega_{m}}(p_{w}C_{w}^{r})^{\frac{l_{m,r}}{r+l_{m,r}}}=1.$ Since
$\underline{D}_{r}(\mu_{m})\leq\underline{\dim}_{\text{B}}(F^{m})$ and
$\underline{\dim}_{\text{B}}(F^{m})\leq\underline{D}_{r}(\mu)$, we have
$\underline{D}_{r}(\mu_{m})\leq\underline{D}_{r}(\mu).$ Our aim is to show
that $l_{r}\leq\underline{D}_{r}(\mu).$ we prove it by contradiction. Let
$\underline{D}_{r}(\mu)<l_{r}.$ Let
$\eta=\min\\{c_{\sigma^{1}}^{r},c_{\sigma^{2}}^{r},\cdots,c_{\sigma^{N}}^{r}\\}$
and $\zeta=\max\\{p_{ij}c_{ij}^{r}:1\leq i,j\leq N\\}.$ Using the fact
$l_{m,r}\leq\underline{D}_{r}(\mu)$ and
$\sum_{w\in\Omega_{m}}(p_{w}C_{w}^{r})^{\frac{l_{m,r}}{r+l_{m,r}}}=1,$ we have

$\displaystyle{\eta}^{\frac{-l_{m,r}}{r+l_{m,r}}}$
$\displaystyle\geq\sum_{w\in\Omega_{m}}(p_{w}c_{w}^{r})^{\frac{l_{m,r}}{r+l_{m,r}}}$
$\displaystyle\geq\sum_{w\in\Omega_{m}}(p_{w}c_{w}^{r})^{\frac{\underline{D}_{r}(\mu)}{r+\underline{D}_{r}(\mu)}}$
$\displaystyle=\sum_{w\in\Omega_{m}}(p_{w}c_{w}^{r})^{\frac{\underline{D}_{r}(\mu)}{r+\underline{D}_{r}(\mu)}}(p_{w}c_{w}^{r})^{\frac{l_{r}}{r+l_{r}}}(p_{w}c_{w}^{r})^{\frac{-l_{r}}{r+l_{r}}}$
$\displaystyle\geq\sum_{w\in\Omega_{m}}(p_{w}c_{w}^{r})^{\frac{l_{r}}{r+l_{r}}}{\zeta}^{m\Big{(}{\frac{\underline{D}_{r}(\mu)}{r+\underline{D}_{r}(\mu)}}-{\frac{l_{r}}{r+l_{r}}}\Big{)}}.$

By Lemma 3.7 and $l_{m,r}\leq\underline{D}_{r}(\mu)<l_{r}$, we have

${\eta}^{\frac{-l_{r}}{r+l_{r}}}\geq{\zeta}^{m\Big{(}{\frac{\underline{D}_{r}(\mu)}{r+\underline{D}_{r}(\mu)}}-{\frac{l_{r}}{r+l_{r}}}\Big{)}}G_{r}^{\frac{-l_{r}}{r+l_{r}}},$

which is a contradiction for large $m\in\mathbb{N}.$ Thus,
$l_{r}\leq\underline{D}_{r}(\mu)$. This completes the proof. ∎

### Proof of Theorem 3.1

By Proposition 3.10 and $(1)$ of Proposition 2.4, we have
$\overline{D}_{r}(\mu)\leq k_{r}$. On the other hand, by Proposition 3.14, we
have $l_{r}\leq\underline{D}_{r}(\mu)$. Hence,
$l_{r}\leq\underline{D}_{r}(\mu)\leq\overline{D}_{r}(\mu)\leq k_{r}$, which
completes the proof of the theorem.

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