# Regularity and singularity for conjugate equations driven by linear
fractional transformations

Kazuki Okamura  Department of Mathematics, Faculty of Science, Shizuoka
University<EMAIL_ADDRESS>
###### Abstract.

We consider the conjugate equation driven by two families of finite maps on
the unit interval satisfying a compatibility condition. This framework
contains de Rham’s functional equations. We consider some real analytic
properties of the solution in the case that the equation is driven by non-
affine maps, in particular, linear fractional transformations. We give
sufficient conditions for the regularity in the sense of Ullman-Stahl-Totik
and for the singularity of the solution.

##  1\. Introduction

Let $N\geq 2$ and let $\mathcal{I}_{N}:=\\{0,1,\cdots,N-1\\}$. Let
$\\{f_{i}\\}_{i\in\mathcal{I}_{N}}$ and $\\{g_{i}\\}_{i\in\mathcal{I}_{N}}$ be
two families of $N$ maps on $[0,1]$ satisfying a compatibility condition. We
consider the following conjugate equation for $\varphi:[0,1]\to[0,1]$:

$g_{i}\circ\varphi=\varphi\circ f_{i},\ i\in\mathcal{I}_{N}.$ (1.1)

See Section 2 below for the definition of the compatibility condition. An
alternative expression is given by

$\varphi(x)=\begin{cases}g_{0}(\varphi(f_{0}^{-1}(x)))&0\leq x\leq f_{0}(1)\\\
\vdots&\\\ g_{N-1}(\varphi(f_{N-1}^{-1}(x)))&f_{N-1}(0)\leq x\leq
1\end{cases}.$ (1.2)

This is known as de Rham’s functional equation [7, 8]. This functional
equation is generally fractal in nature, and solutions of (1.1) can be
singular functions.

Assume that $N=2$, $0<p<1$, $f_{0}(x)=x/2$, $f_{1}(x)=(x+1)/2$, $g_{0}(x)=px$
and $g_{1}(x)=(1-p)x+p$. Then, (1.2) is expressed as

$\varphi(x)=\begin{cases}p\varphi(2x)&0\leq x\leq 1/2\\\
(1-p)\varphi(2x-1)+p&1/2\leq x\leq 1\end{cases}.$

There exists a unique continuous strictly increasing solution $\varphi_{p}$
for this equation, and $\varphi_{p}$ is singular if $p\neq 1/2$ and
$\varphi_{p}(x)=x$ if $p=1/2$. $\varphi_{p}$ is called the Lebesgue singular
function. This function is also a solution of the functional equation by
Matkowski and Wesołowski [16, 24, 25].

Minkowski’s question-mark function [23] is also obtained by a solution of
(1.1) for the case that
$\displaystyle(f_{0}(x),f_{1}(x),g_{0}(x),g_{1}(x))=\left(x/(x+1),1/(2-x),x/2,(x+1)/2\right)$.
It can also be expressed in the form

$\varphi(x)=\begin{cases}\frac{1}{2}\varphi(\frac{x}{1-x})&0\leq x\leq 1/2\\\
\frac{1}{2}\varphi\left(2-\frac{1}{x}\right)+\frac{1}{2}&1/2\leq x\leq
1\end{cases}.$

This function has appeared in various areas of mathematics such as number
theory and dynamical systems. The real analytic properties of this function
have been considered ([15, 19, 21]). The conjugate equation of the de Rham
type has been investigated by many authors ([2, 4, 6, 11, 12, 13, 17, 28, 29,
30, 34]). We refer the reader to [27, Subsection 3.4] for a more detailed
review of de Rham’s functional equations.

The main purpose of this paper is to investigate some properties of the
solution of (1.1). Specifically, we give sufficient conditions for the
solution to be regular or singular. Since $\varphi$ is an increasing function
on $[0,1]$, we deal with the probability measure $\mu_{\varphi}$, which is the
Borel probability measure on $[0,1]$ whose distribution function is $\varphi$.
We consider the case that both of $\\{f_{i}\\}_{i}$ and $\\{g_{i}\\}_{i}$ are
non-affine. This case is more difficult to analyze than the case that both
$\\{f_{i}\\}_{i}$ and $\\{g_{i}\\}_{i}$ are affine.

This paper deals with the regularity of $\mu_{\varphi}$ in the sense of
Ullman-Stahl-Totik, which is related to the asymptotic properties of the
orthonormal polynomials of $\mu_{\varphi}$ ([32]). The regularity holds under
a certain measure-theoretic criterion. (See [22, Criterion 1.1] or Definition
3.1 below.) By using the criterion, [22, Theorems 1.2 and 1.3] establish the
regularity of the Minkowski question-mark function and its inverse function
respectively. Our result, Theorem 3.2 below, gives a general regularity result
which simultaneously shows [22, Theorems 1.2 and 1.3]. Theorem 3.2 holds for
the case that both of $\\{f_{i}\\}_{i}$ and $\\{g_{i}\\}_{i}$ are classes of
linear fractional transformations. This is outside the scope of [22, Theorem
8.1]. The proof of [22, Theorem 1.2] is more complicated than the proof of
[22, Theorem 1.3]. Our proof is simpler than the proof of [22, Theorem 1.2].
Theorem 3.2 is stable for the exchange of $\\{f_{i}\\}_{i}$ with
$\\{g_{i}\\}_{i}$. The viewpoint from the conjugate equation works well.

It is interesting to know a necessary and sufficient condition of
$\\{f_{i}\\}_{i}$ and $\\{g_{i}\\}_{i}$ for the singularity. This topic has
been considered in [2, 3, 10, 13, 18, 25]. In [27], the author dealt with the
case that $\\{g_{i}\\}_{i}$ are linear fractional transformations and
$\\{f_{i}\\}_{i}$ are affine. Our results, Theorems 4.2 and 4.3 below, deal
with the case that $\\{f_{i}\\}_{i}$ are not necessarily affine. We do not
assume the differentiability of $\\{f_{i}\\}_{i}$. For the proof, we use a
martingale approach as in [27], but here we need more delicate quantitative
estimates.

The structure of this paper is as follows. In Section 2, we give some
terminology and basic results of the conjugate equation on the unit interval.
In Sections 3 and 4, we deal with the regularity and singularity,
respectively. Finally, we give an example in Section 5.

##  2\. Preliminaries

###  2.1. Existence and uniqueness

We first discuss the existence and uniqueness of the solution of the conjugate
equation (1.1).

We say that a family of maps $\\{h_{i}\\}_{i\in\mathcal{I}_{N}}$ on $[0,1]$ is
a compatible system or satisfies a compatibility condition if they are
strictly increasing continuous functions such that $h_{0}(0)=0$,
$h_{i-1}(1)=h_{i}(0)$, $1\leq i\leq N-1$, $h_{N-1}(1)=1$, and, a compatible
system $\\{h_{i}\\}_{i\in\mathcal{I}_{N}}$ is a D-system on $[0,1]$ if

$\mathcal{D}_{h}:=\\{h_{i_{1}}\circ\cdots\circ h_{i_{n}}(j):i_{1},\cdots
i_{n}\in\mathcal{I}_{N},j\in\\{0,1\\}\\}$

is dense in $[0,1]$.

######  Lemma 2.1.

Let $\\{h_{i}\\}_{i\in\mathcal{I}_{N}}$ be a D-system on $[0,1]$. Then, for
every $x\in[0,1]$, there exists an infinite sequence $(i_{n})_{n}$ of
$\mathcal{I}_{N}$ such that

$x=\lim_{n\to\infty}h_{i_{1}}\circ\cdots\circ
h_{i_{n}}(0)=\lim_{n\to\infty}h_{i_{1}}\circ\cdots\circ h_{i_{n}}(1).$ (2.1)

If $x\in\mathcal{D}_{h}\setminus\\{0,1\\}$, then, there exist two such
infinite sequences. Otherwise, there exists exactly one such infinite
sequence.

###### Proof.

We first show that for every infinite sequence $(i_{n})_{n}$,

$\lim_{n\to\infty}|h_{i_{1}}\circ\cdots\circ
h_{i_{n}}(0)-h_{i_{1}}\circ\cdots\circ h_{i_{n}}(1)|=0.$ (2.2)

Assume this fails. By the definition of $\\{h_{i}\\}_{i}$,
$\left(h_{i_{1}}\circ\cdots\circ h_{i_{n}}(0)\right)_{n}$ is increasing and
$\left(h_{i_{1}}\circ\cdots\circ h_{i_{n}}(1)\right)_{n}$ is decreasing. Since
$\mathcal{D}_{h}$ is dense in $[0,1]$, there exists $k_{1},\cdots,k_{N}$ and
$j\in\\{0,1\\}$ such that

$h_{i_{1}}\circ\cdots\circ
h_{i_{N}}(0)\leq\lim_{n\to\infty}h_{i_{1}}\circ\cdots\circ
h_{i_{n}}(0)<h_{k_{1}}\circ\cdots\circ h_{k_{N}}(j)$
$<\lim_{n\to\infty}h_{i_{1}}\circ\cdots\circ h_{i_{n}}(1)\leq
h_{i_{1}}\circ\cdots\circ h_{i_{N}}(1).$

This cannot occur. Hence (2.2) holds.

Case 1. Assume $x\in\mathcal{D}_{h}\setminus\\{0,1\\}$. Assume that
$x=h_{i_{1}}\circ\cdots\circ h_{i_{n}}(0)$. Since $h_{0}(0)=0$ and $x>0$, we
can assume that $i_{n}\geq 1$. Let $i_{k}:=0$ for every $k\geq n+1$. Then,
(2.1) holds. By replacing $i_{n}$ with $i_{n}-1$ and let $i_{k}:=N-1$ for
every $k\geq n+1$, then, (2.1) holds. Assume that $x=h_{i_{1}}\circ\cdots\circ
h_{i_{n}}(1)$. Since $h_{N-1}(1)=1$ and $x<1$, we can assume that $i_{n}\leq
N-2$. Let $i_{k}:=N-1$ for every $k\geq n+1$. Then, (2.1) holds. By replacing
$i_{n}$ with $i_{n}+1$ and let $i_{k}:=0$ for every $k\geq n+1$, then, (2.1)
holds.

Case 2. Assume $x\in\\{0,1\\}$. Assume $x=0$. Since each $h_{i}$ is strictly
increasing, there is no other choice than $i_{k}=0$ for every $k\geq 1$ for
(2.1). Assume $x=1$. Then, there is no other choice than $i_{k}=N-1$ for every
$k\geq 1$ for (2.1).

Case 3. Assume $x\notin\mathcal{D}_{h}$. Then, for each $n\geq 1$, there
exists a unique finite sequence $(i_{n,1},\cdots,i_{n,n})$ such that

$h_{i_{n,1}}\circ\cdots\circ h_{i_{n,n}}(0)<x<h_{i_{n,1}}\circ\cdots\circ
h_{i_{n,n}}(1).$

Then, $i_{n-1,k}=i_{n,k}$ for every $k\leq n-1$. Let $i_{k}:=i_{n,k}$. By
(2.2), (2.1) holds. If (2.1) holds, then, for every $N\geq 1$,

$h_{i_{1}}\circ\cdots\circ
h_{i_{N}}(0)\leq\lim_{n\to\infty}h_{i_{1}}\circ\cdots\circ
h_{i_{n}}(0)=x=\lim_{n\to\infty}h_{i_{1}}\circ\cdots\circ h_{i_{n}}(1)\leq
h_{i_{1}}\circ\cdots\circ h_{i_{N}}(1).$

Hence,

$h_{i_{1}}\circ\cdots\circ h_{i_{N}}(0)<x<h_{i_{1}}\circ\cdots\circ
h_{i_{N}}(1).$

Hence the uniqueness holds. ∎

######  Remark 2.2.

The existence and uniqueness of the bounded solutions of conjugate equations
on more general spaces than $[0,1]$ have been discussed in [6, 26, 28, 30].

######  Lemma 2.3.

Assume that $\\{f_{i}\\}_{i\in\mathcal{I}_{N}}$ and
$\\{g_{i}\\}_{i\in\mathcal{I}_{N}}$ are D-systems on $[0,1]$. Then, there
exists a unique continuous strictly increasing solution of (1.1).

Hereafter, we let $\mu_{\varphi}$ be the continuous probability measure on
$[0,1]$ whose distribution function is $\varphi$. This is the central object
considered in this paper.

###### Proof.

We first see that

$\varphi\left(f_{i_{1}}\circ\cdots\circ
f_{i_{n}}(j)\right)=g_{i_{1}}\circ\cdots\circ g_{i_{n}}(j),\
i_{1},\cdots,i_{n}\in\mathcal{I}_{N},\ j\in\\{0,1\\},$ (2.3)

by induction in $n$. Hence, $\varphi(\mathcal{D}_{f})=\mathcal{D}_{g}$.

Step 1. We define $\varphi(x)$ for $x\notin\mathcal{D}_{f}$. By the
assumption, there exists an infinite sequence $(i_{n})_{n}$ of
$\mathcal{I}_{N}$ such that $x\in(f_{i_{1}}\circ\cdots\circ
f_{i_{n}}(0),f_{i_{1}}\circ\cdots\circ f_{i_{n}}(1))$. Since
$\left[g_{i_{1}}\circ\cdots\circ g_{i_{n}}(0),g_{i_{1}}\circ\cdots\circ
g_{i_{n}}(1)\right]$ is decreasing with respect to $n$,

$\bigcap_{n\geq 1}\left[g_{i_{1}}\circ\cdots\circ
g_{i_{n}}(0),g_{i_{1}}\circ\cdots\circ
g_{i_{n}}(1)\right]=\left[\lim_{n\to\infty}g_{i_{1}}\circ\cdots\circ
g_{i_{n}}(0),\lim_{n\to\infty}g_{i_{1}}\circ\cdots\circ g_{i_{n}}(1)\right].$

By the density of $\mathcal{D}_{g}$ in $[0,1]$, we can apply (2.2), and obtain
that

$\lim_{n\to\infty}g_{i_{1}}\circ\cdots\circ
g_{i_{n}}(0)=\lim_{n\to\infty}g_{i_{1}}\circ\cdots\circ g_{i_{n}}(1).$ (2.4)

We let $\varphi(x):=\lim_{n\to\infty}g_{i_{1}}\circ\cdots\circ g_{i_{n}}(0)$.

Step 2. We show $\varphi$ is strictly increasing on $[0,1]$. By (2.3),
$\varphi$ is strictly increasing on $\mathcal{D}_{f}$. Let $x<y$. By (2.4),
for some $n$,

$f_{i_{1}}\circ\cdots\circ f_{i_{n}}(0)\leq x\leq f_{i_{1}}\circ\cdots\circ
f_{i_{n}}(1)<f_{j_{1}}\circ\cdots\circ f_{j_{n}}(0)\leq y\leq
f_{j_{1}}\circ\cdots\circ f_{j_{n}}(1).$

By this and Step 1,

$\varphi(x)\leq g_{i_{1}}\circ\cdots\circ
g_{i_{n}}(1)<g_{j_{1}}\circ\cdots\circ g_{j_{n}}(0)\leq\varphi(y).$

Step 3. We show that $\varphi$ is continuous on $[0,1]$. Assume that there is
a discontinuous point $x_{0}\in(0,1)$ of $\varphi$. Then,

$b_{0}:=\lim_{x\to x_{0}+0}\varphi(x)>\lim_{x\to x_{0}-0}\varphi(x)=:a_{0}.$

By Step 2, $\varphi([0,1])\cap(a_{0},b_{0})=\emptyset$, in particular,
$\mathcal{D}_{g}\cap(a_{0},b_{0})=\emptyset$. This contradicts the density of
$\mathcal{D}_{g}$ in $[0,1]$. The same argument is applied to the case that
$x_{0}\in\\{0,1\\}$. ∎

It is natural to consider the inverse function of the solution $\varphi$. We
easily see that

######  Lemma 2.4.

Assume that $\\{f_{i}\\}_{i\in\mathcal{I}_{N}}$ and
$\\{g_{i}\\}_{i\in\mathcal{I}_{N}}$ are D-systems on $[0,1]$. Let $\varphi$ be
a unique solution of (1.1). Then, the inverse function $\varphi^{-1}$ of
$\varphi$ is well-defined and satisfies the following conjugate equation

$\varphi^{-1}\circ g_{i}=f_{i}\circ\varphi^{-1},\ i\in\mathcal{I}_{N},$

which is equivalent with (1.1) where $\\{f_{i}\\}_{i}$ and $\\{g_{i}\\}_{i}$
are exchanged.

This fact will be important in Section 3. By [4, Proposition 3.1], if
$\varphi$ is singular, then, $\varphi^{-1}$ is also singular.

###  2.2. Weak contraction

Hereafter, we assume more additional conditions for the D-systems
$\\{f_{i}\\}_{i}$ and $\\{g_{i}\\}_{i}$. We say that a function
$h:[0,1]\to[0,1]$ is a contraction on $[0,1]$ if

$\textup{Lip}(h):=\sup_{x\neq y}\left|\frac{h(x)-h(y)}{x-y}\right|<1.$

$\textup{Lip}(h)$ is called the Lipschitz constant of $h$. We say that a
function $h:[0,1]\to[0,1]$ is a weak contraction on $[0,1]$ if

$\phi_{h}(t):=\lim_{s\to t+0}\sup\left\\{|h(x)-h(y)|:|x-y|\leq s\right\\}<t,\
\ t>0.$

The following is well-known. See [5, 14].

######  Lemma 2.5.

Let $h$ be a weak contraction. Then, $\lim_{n\to\infty}\phi_{h}^{n}(t)=0$ for
every $t>0$ and $|h(x)-h(y)|\leq\phi_{h}(|x-y|)$ for $x,y\in[0,1]$.

###### Proof.

Let

$\omega_{h}(s):=\sup\left\\{|h(x)-h(y)|:|x-y|\leq s\right\\}.$

Since this function is increasing, $\phi_{h}$ is right-continuous and
increasing on $[0,1]$. Hence,
$|h(x)-h(y)|\leq\lim_{s\to|x-y|+0}\phi_{h}(s)=\phi_{h}(|x-y|)$.

By the assumption, for every $t>0$, $(\phi_{h}^{n}(t))_{n}$ is strictly
decreasing. There exists the limit
$\phi^{\infty}_{h}(t):=\lim_{n\to\infty}\phi_{h}^{n}(t)$. By the right-
continuity of $\phi_{h}$,
$\phi_{h}\left(\phi^{\infty}_{h}(t)\right)=\phi^{\infty}_{h}(t)$. Hence,
$\phi^{\infty}_{h}(t)=0$. ∎

######  Lemma 2.6.

Let $\\{h_{i}\\}_{i}$ be a compatible system on $[0,1]$ such that each $h_{i}$
is a weak contraction on $[0,1]$. Then, $\\{h_{i}\\}_{i\in\mathcal{I}_{N}}$ is
a D-system.

We say that a compatible system $\\{h_{i}\\}_{i}$ on $[0,1]$ is an (weakly)
contractive system on $[0,1]$ if each $h_{i}$ is an (weak) contraction on
$[0,1]$. Every weakly contractive system is a D-system.

###### Proof.

Let $\phi:=\max_{i\in\mathcal{I}_{N}}\phi_{h_{i}}$. Then, $\phi$ is right-
continuous111The assumption that $N$ is finite would be important. on $[0,1]$,
$\phi(t)<t$ for every $t>0$, and
$\max_{i\in\mathcal{I}_{N}}|h_{i}(x)-h_{i}(y)|\leq\phi(|x-y|)$, $x,y\in[0,1]$.
Then, we see that $\lim_{n\to\infty}\phi^{n}(t)=0$ for every $t>0$. Hence,

$\left|h_{i_{1}}\circ\cdots\circ h_{i_{n}}(1)-h_{i_{1}}\circ\cdots\circ
h_{i_{n}}(0)\right|\leq\phi^{n}(1).$
$\lim_{n\to\infty}\max_{i_{1},\cdots,i_{n}\in\mathcal{I}_{N}}|h_{i_{1}}\circ\cdots\circ
h_{i_{n}}(1)-h_{i_{1}}\circ\cdots\circ h_{i_{n}}(0)|=0.$

By the definition, for each $n\geq 1$,

$[0,1]=\bigcup_{i_{1},\cdots,i_{n}\in\mathcal{I}_{N}}\left[h_{i_{1}}\circ\cdots\circ
h_{i_{n}}(0),h_{i_{1}}\circ\cdots\circ h_{i_{n}}(1)\right],$

and their interiors are disjoint. ∎

###  2.3. Linear fractional transforms

######  Definition 2.7.

We say that a family of functions $\\{h_{i}\\}_{i}$ is an LF system on $[0,1]$
if $h_{i}(x):=\dfrac{a_{i}x+b_{i}}{c_{i}x+d_{i}},x\in[0,1]$, for each
$i\in\mathcal{I}_{N}$, where $a_{i},b_{i},c_{i},d_{i},i\in\mathcal{I}_{N}$,
are real numbers such that
(1) (well-defined) $c_{i}+d_{i}>0$, $d_{i}>0$, $i\in\mathcal{I}_{N}$.
(2) (strictly increasing, weak contraction)

$0<a_{i}d_{i}-b_{i}c_{i}\leq\min\\{d_{i}^{2},(c_{i}+d_{i})^{2}\\},\
i\in\mathcal{I}_{N}.$

(3) (compatibility) $b_{0}=0$,
$\dfrac{a_{i}+b_{i}}{c_{i}+d_{i}}=\dfrac{b_{i+1}}{d_{i+1}}$,
$i\in\mathcal{I}_{N}$, where we let $\dfrac{b_{N}}{d_{N}}:=1$.

For condition (2), we may consider a stronger condition that
(2’) $0<a_{i}d_{i}-b_{i}c_{i}<\min\\{d_{i}^{2},(d_{i}+c_{i})^{2}\\},\
i\in\mathcal{I}_{N}$.

Since $h_{i}$ is invariant if we multiply a positive constant by each of
$a_{i},b_{i},c_{i}$ and $d_{i}$, we may assume that $d_{i}=1$, or,
$a_{i}d_{i}-b_{i}c_{i}=1$.

######  Lemma 2.8.

Every LF system is a weakly contractive system. If (2’) holds instead of (2),
then, it is a contractive system.

###### Proof.

Let $\\{h_{i}\\}_{i\in\mathcal{I}_{N}}$ be an LF-system. We assume that
$d_{i}=1$ for every $i$. By Lemma 2.6, it suffices to show that each $h_{i}$
is a weak contraction. We first remark that

$|h_{i}(x)-h_{i}(y)|=\frac{a_{i}-b_{i}c_{i}}{(c_{i}x+1)(c_{i}y+1)}|x-y|.$

We consider three cases: (i) $c_{i}>0$, (ii) $c_{i}<0$, and (iii) $c_{i}=0$.

(i) Assume $c_{i}>0$. Then, $a_{i}-b_{i}c_{i}\leq 1$ and

$|h_{i}(x)-h_{i}(y)|\leq\frac{a_{i}-b_{i}c_{i}}{c_{i}|x-y|+1}|x-y|\leq\frac{|x-y|}{c_{i}|x-y|+1}.$

Hence,

$\omega_{h_{i}}(s)\leq\frac{s}{c_{i}s+1},\ \ s\geq 0.$

By this and $c_{i}>0$,

$\phi_{h_{i}}(t)\leq\frac{t}{c_{i}t+1}<t,\ t>0.$

(ii) Assume $c_{i}<0$. Then, $a_{i}-b_{i}c_{i}\leq(c_{i}+1)^{2}$ and

$|h_{i}(x)-h_{i}(y)|\leq\frac{a_{i}-b_{i}c_{i}}{(c_{i}+1)(c_{i}(1-|x-y|)+1)}|x-y|\leq\frac{(1+c_{i})|x-y|}{-c_{i}|x-y|+1+c_{i}}.$

Hence,

$\omega_{h_{i}}(s)\leq\frac{(1+c_{i})s}{-c_{i}s+1+c_{i}},\ \ s\geq 0.$

By this and $c_{i}<0$,

$\phi_{h_{i}}(t)\leq\frac{(1+c_{i})t}{-c_{i}t+1+c_{i}}<t,\ t>0.$

(iii) Assume $c_{i}=0$. Then, $0<a_{i}\leq 1$ and
$|h_{i}(x)-h_{i}(y)|=a_{i}|x-y|$. By the assumption for $a_{i},b_{i},c_{i}$,
$h_{j}(1)>h_{j}(0)$ for every $j$. Hence, by $N\geq 2$,
$a_{i}=h_{i}(1)-h_{i}(0)<1$. ∎

######  Remark 2.9.

The technique used above is related to a characterization of Rakotch
contractions. See [20, Lemma 2.10 (iii)].

##  3\. Regularity

In this section, we deal with the regularity of $\mu_{\varphi}$ in the sense
of Ullman-Stahl-Totik. The following is the same as [22, Criterion 1.1].

######  Definition 3.1.

We say that criterion $\lambda^{*}$ holds for a continuous Borel probability
measure $\mu$ on $[0,1]$ if for every $\eta>0$,

$\ell\left(\left\\{x\in(0,1):\mu\left([x-s^{-1},x+s^{-1}]\right)\geq\exp(-\eta
s)\right\\}\right)\to 1,s\to+\infty,$

where we let $\ell$ be the Lebesgue measure on $[0,1]$.

Informally speaking, the criterion $\lambda^{*}$ assures that $\mu$ is not too
thin on $[0,1]$. Not only absolutely continuous measures but also some
singular measures satisfy this criterion. It is known that this criterion
implies the regularity of $\mu_{\varphi}$ in the sense of Ullman-Stahl-Totik
([31, Section 4.3]).

The main result in this section is

######  Theorem 3.2.

Assume that $\\{f_{i}\\}_{i\in\mathcal{I}_{N}}$ and
$\\{g_{i}\\}_{i\in\mathcal{I}_{N}}$ are both LF systems and let $\varphi$ be
the solution of (1.1). Then, $\mu_{\varphi}$ satisfies the criterion
$\lambda^{*}$.

This assertion includes [22, Theorems 1.2 and 1.3] as a corollary. Let $N=2$.
If
$\displaystyle(f_{0}(x),f_{1}(x),g_{0}(x),g_{1}(x))=\left(\frac{x}{x+1},\frac{1}{2-x},\frac{x}{2},\frac{x+1}{2}\right)$,
then, $\mu_{\varphi}$ is the probability measure of the Minkowski question-
mark function. If
$\displaystyle(f_{0}(x),f_{1}(x),g_{0}(x),g_{1}(x))=\left(\frac{x}{2},\frac{x+1}{2},\frac{x}{x+1},\frac{1}{2-x}\right)$,
then, $\mu_{\varphi}$ is the probability measure of the inverse of the
Minkowski question-mark function. These cases correspond to each other by
Lemma 2.4. The viewpoint from functional equations works well here.

If

$(f_{0}(x),f_{1}(x),f_{2}(x),g_{0}(x),g_{1}(x),g_{2}(x))=\left(\frac{x}{4-x},\frac{1}{3-x},\frac{3x+1}{2x+2},\frac{x}{3},\frac{x+1}{3},\frac{x+2}{3}\right),$

then, $\\{f_{i}\\}_{i}$ and $\\{g_{i}\\}_{i}$ are both LF systems, and,
$f_{0}$ and $f_{1}$ are convex and $f_{2}$ is concave. Hence this case is
outside of the scope of [22, Theorem 8.1].

Our proof is much simpler than the proof of [22, Theorem 1.2]. We can show
Theorem 3.2 by following the proof of [22, Theorem 1.3], thanks to Lemma 3.4
below.

###  3.1. Proof

Let $(\mathcal{I}_{N})^{*}:=\cup_{n\in\mathbb{N}}(\mathcal{I}_{N})^{n}$. We
denote the length of $\sigma\in(\mathcal{I}_{N})^{*}$ by $|\sigma|$. For an LF
system $\\{h_{i}\\}_{i}$ and
$\sigma=(\sigma_{1},\cdots,\sigma_{n})\in(\mathcal{I}_{N})^{*}$, we let

$h_{\sigma}(x)=h_{\sigma_{1}}\circ\cdots\circ h_{\sigma_{n}}(x),\ x\in[0,1].$

######  Lemma 3.3.

Let $\\{h_{i}\\}_{i}$ be an LF system. Then, there exists $c\in(0,1)$ such
that for every $\sigma\in(\mathcal{I}_{N})^{*}$,

$h_{\sigma}(1)-h_{\sigma}(0)\geq c^{|\sigma|}.$

###### Proof.

Let $A_{i}:=\begin{pmatrix}a_{i}&b_{i}\\\ c_{i}&d_{i}\end{pmatrix}$,
$i\in\mathcal{I}_{N}$. We assume that $a_{i}d_{i}-b_{i}c_{i}=1$ for each $i$.
Let $(i_{n})_{n}\in(\mathcal{I}_{N})^{\mathbb{N}}$. Let

$\begin{pmatrix}p_{n}&q_{n}\\\ r_{n}&s_{n}\end{pmatrix}=A_{i_{1}}\cdots
A_{i_{n}},\ n\geq 1.$

Let $C_{1}:=\max\\{|a_{i}|,|b_{i}|,|c_{i}|,|d_{i}|:i\in\mathcal{I}_{N}\\}$.
Then, for every $n$, $|r_{n+1}|+|s_{n+1}|\leq C_{1}(|r_{n}|+|s_{n}|)$. Hence,
$|r_{n}|+|s_{n}|\leq 2C_{1}^{n}$ for every $n$.

Assume that $\sigma=(i_{1},\cdots,i_{n})$. Then, $|\sigma|=n$ and

$h_{\sigma}(1)-h_{\sigma}(0)=\frac{1}{s_{n}(r_{n}+s_{n})}\geq\frac{1}{(|r_{n}|+|s_{n}|)^{2}}\geq\frac{1}{4|C_{1}|^{2n}}.$

∎

We give an upper bound for $h_{\sigma}(1)-h_{\sigma}(0)$.

######  Lemma 3.4.

Let $\\{h_{i}\\}_{i}$ be an LF system. Then, there exists $C>0$ such that for
every $\sigma\in(\mathcal{I}_{N})^{*}$,

$h_{\sigma}(1)-h_{\sigma}(0)\leq\frac{C}{|\sigma|}.$

###### Proof.

We first remark that

$(c_{i}x+d_{i})(c_{i}y+d_{i})\geq 1+|c_{i}||x-y|,\ x,y\in[0,1].$ (3.1)

We assume that $x\leq y$. Assume $c_{i}>0$. Then,

$c_{i}y+d_{i}\geq
c_{i}x+d_{i}\geq\min_{x\in[0,1]}c_{i}x+d_{i}=\min\\{d_{i},c_{i}+d_{i}\\}=d_{i}\geq
1.$

Hence,

$(c_{i}x+d_{i})(c_{i}y+d_{i})=(c_{i}x+d_{i})^{2}+c_{i}(y-x)(c_{i}x+d_{i})\geq
1+c_{i}(y-x).$

Assume $c_{i}<0$. Then,

$(c_{i}x+d_{i})(c_{i}y+d_{i})=(c_{i}y+d_{i})^{2}-c_{i}(y-x)(c_{i}x+d_{i})\geq
1-c_{i}(y-x).$

If $c_{i}=0$, then, $(c_{i}x+d_{i})(c_{i}y+d_{i})=d_{i}^{2}\geq
1=1+|c_{i}||x-y|$. Thus we see (3.1).

By (3.1),

$|h_{i}(x)-h_{i}(y)|\leq\frac{|x-y|}{1+|c_{i}||x-y|}.$

If $c_{i}=0$, then,

$|h_{i}(x)-h_{i}(y)|=\frac{a_{i}}{d_{i}}|x-y|.$

Since $h_{i}(1)-h_{i}(0)<1$, $a_{i}<d_{i}$. Let
$C_{1}:=\min\\{|c_{i}|:c_{i}\neq 0\\}$ and
$C_{2}:=\max\left\\{\dfrac{a_{i}}{d_{i}}:c_{i}=0\right\\}$. Then, $C_{1}>0$,
$C_{2}<1$ and

$\max_{i}|h_{i}(x)-h_{i}(y)|\leq\max\left\\{\frac{|x-y|}{1+C_{1}|x-y|},C_{2}|x-y|\right\\}.$

We can take $C_{3}\in(0,C_{1})$ such that

$\max_{i}|h_{i}(x)-h_{i}(y)|\leq\frac{|x-y|}{1+C_{3}|x-y|},\ x,y\in[0,1].$

Let
$a_{n}:=\max\left\\{h_{\sigma}(1)-h_{\sigma}(0):\sigma\in(\mathcal{I}_{N})^{*},|\sigma|=n\right\\}$.
Then, $a_{1}\leq\dfrac{1}{1+C_{3}}$ and
$a_{n+1}\leq\dfrac{a_{n}}{1+C_{3}a_{n}},n\geq 1$. Hence,

$a_{n}\leq\frac{1}{nC_{3}+1}\leq\frac{1}{nC_{3}}.$

∎

###### Proof of Theorem 3.2.

We follow the strategy of the proof of [22, Theorem 1.3]. We apply Lemma 3.3
to $\\{g_{i}\\}_{i}$ and Lemma 3.4 to $\\{f_{i}\\}_{i}$. Let $C$ be the
constant appearing in Lemma 3.4. Take $y\in(0,1)$ and $r>0$ such that $0\leq
y-r<y+r\leq 1$ arbitralily. Let $n$ be a natural number such that
$\frac{C}{n}<r\leq\frac{C}{n-1}$. Then, by the compatibility condition and
Lemma 3.4, there exist $i_{1},\cdots,i_{n}\in\mathcal{I}_{N}$ such that

$[f_{i_{1}}\circ\cdots\circ f_{i_{n}}(0),f_{i_{1}}\circ\cdots\circ
f_{i_{n}}(1)]\subset[y-r,y+r].$

Hence, by (2.3),

$\mu_{\varphi}([y-r,y+r])\geq g_{i_{1}}\circ\cdots\circ
g_{i_{n}}(1)-g_{i_{1}}\circ\cdots\circ g_{i_{n}}(0).$

Let $c$ be the constant appearing in Lemma 3.3. By Lemma 3.3,

$\mu_{\varphi}([y-r,y+r])\geq c^{n}\geq c^{1+C/r}.$

Hence, criterion $\lambda^{*}$ holds for $\mu_{\varphi}$. ∎

##  4\. Singularity

In this section, we consider (1.1) under the setting that $\\{g_{i}\\}_{i}$ is
an LF-system on $[0,1]$, and, $\\{f_{i}\\}_{i}$ is a weakly contractive
D-system on $[0,1]$. By Lemma 2.8, both of $\\{f_{i}\\}_{i}$ and
$\\{g_{i}\\}_{i}$ are D-systems. Hence, by Lemma 2.3, there exists a unique
continuous strictly increasing solution $\varphi$ of (1.1).

We consider singularity for the solution $\varphi$. We will give sufficient
conditions for $\dim_{H}\mu_{\varphi}<1$. It is known that if
$\dim_{H}\mu_{\varphi}<1$, then, $\varphi$ is a singular function on $[0,1]$,
specifically, $\varphi$ is a strictly increasing and continuous function on
$[0,1]$ such that its derivative is equal to zero almost everywhere.

The main purpose of this section is to extend [27, Theorem 43] to the case
that $f_{i}$ can be non-linear. In [27], the author dealt with many objects
other than the conjugate functional equation, so the arguments were not
specialized for the functional equation. Some parts dealing with the
functional equation in [27] are somewhat sketchy, so we give more details
here. We introduce some new notations and the formulations of our main results
and some of the arguments in the proofs are different from [27, Theorem 43].
These changes will make arguments more transparent.

We assume that an LF-system $\\{g_{i}\\}_{i}$ is defined by
$g_{i}(x):=\dfrac{a_{i}x+b_{i}}{c_{i}x+d_{i}},x\in[0,1]$,
$i\in\mathcal{I}_{N}$, where $a_{i},b_{i},c_{i},d_{i},i\in\mathcal{I}_{N}$,
are real numbers satisfying the conditions (1), (2) and (3) in Definition 2.7.
In this section, we assume $d_{i}=1$ for each $i$.

Let

$\alpha:=\min\left\\{0,\frac{c_{0}}{1-a_{0}},\frac{a_{i}-1+\sqrt{(1-a_{i})^{2}+4b_{i}c_{i}}}{2b_{i}}\middle|1\leq
i\leq N-1\right\\},$

where we delete222This is different from [27]. $\frac{c_{0}}{1-a_{0}}$ if
$a_{0}=1$. Let

$\beta:=\max\left\\{0,\frac{c_{0}}{1-a_{0}},\frac{a_{i}-1+\sqrt{(1-a_{i})^{2}+4b_{i}c_{i}}}{2b_{i}}\middle|1\leq
i\leq N-1\right\\},$

where $\dfrac{c_{0}}{1-a_{0}}:=+\infty$ if $a_{0}=1$.

Let $Y:=[\alpha,\beta]$. Let

$G_{i}(y):=\frac{(a_{i}-b_{i}c_{i})(y+1)}{(b_{i}y+1)((a_{i}+b_{i})y+(c_{i}+1))}=\frac{(b_{i+1}-b_{i})(y+1)}{(b_{i+1}y+1)(b_{i}y+1)},\
y\in[\alpha,\beta].$

Let

$H_{i}(y):=\frac{a_{i}y+c_{i}}{b_{i}y+1},\ y\in[\alpha,\beta].$

Let $G_{i}(+\infty)$ and $H_{i}(+\infty)$ be the limits. Specifically,

$G_{0}(+\infty):=1,G_{i}(+\infty):=0,H_{0}(+\infty):=+\infty,H_{i}(+\infty):=\frac{a_{i}}{b_{i}},1\leq
i\leq N-1.$

The following is the same as [27, Lemma 27], so we will not repeat the proof.

######  Lemma 4.1.

(i) $0\in Y\subset[-1,+\infty]$.
(ii) Each $H_{i}$ is well-defined on $Y$.
(iii) $\alpha\leq H_{i}(\alpha)\leq H_{i}(\beta)\leq\beta$, that is,
$H_{i}(Y)\subset Y$.
(iv) $\sum_{i\in\Sigma_{N}}G_{i}(y)=1,\ y\in Y.$

By (iii), we can compose $H_{i}$ freely on $Y$. We consider the
compactification when $\beta=+\infty$. Then, $Y=[\alpha,\beta]$ is compact,
and $G_{i}:Y\to[0,1]$ and $H_{i}:Y\to Y$ are continuous. We remark that
$Y,G_{i},H_{i}$ are defined by $(g_{i})_{i}$, not by $(f_{i})_{i}$.

$\textup{Fix}(H_{i})$ denotes the set of fixed points of $H_{i}$ on
$[\alpha,\beta]$. This is uniquely determined and

$\textup{Fix}(H_{0})=\left\\{\dfrac{c_{0}}{1-a_{0}}\right\\},\
\textup{Fix}(H_{i})=\left\\{\dfrac{a_{i}-1+\sqrt{(1-a_{i})^{2}+4b_{i}c_{i}}}{2b_{i}}\right\\},\
i\geq 1.$

Let

$\mathcal{P}_{N}:=\left\\{(p_{i})_{i\in\mathcal{I}_{N}}:p_{i}>0,\sum_{i}p_{i}=1\right\\}$

and

$\overline{\mathcal{P}_{N}}:=\left\\{(p_{i})_{i\in\mathcal{I}_{N}}:p_{i}\geq
0,\sum_{i}p_{i}=1\right\\}.$

Let

$\mathfrak{p}_{i}:=\left(G_{j}(\textup{Fix}(H_{i}))\right)_{j}\in\overline{\mathcal{P}_{N}},\
i\in\mathcal{I}_{N}.$

Let

$\dim_{H}\mu_{\varphi}:=\inf\left\\{\dim_{H}A:A\in\mathcal{B}([0,1]),\mu_{\varphi}(A)=1\right\\}.$

The following results extend [27, Theorem 43 (i)] to the case that
$\\{f_{i}\\}_{i}$ may be non-affine.

######  Theorem 4.2.

If $\left|\\{\mathfrak{p}_{i}\\}_{i\in\mathcal{I}_{N}}\right|\geq 2$, then,
for every $\epsilon\in(0,1/2)$, there exists $\delta>0$ such that it holds
that $\dim_{H}\mu_{\varphi}<1$ for every contractive system $(f_{i})_{i}$ on
$[0,1]$ such that $\sum_{i}r_{i}<1+\delta$ and $\epsilon\leq r_{i}\leq
1-\epsilon$ for each $i$, where we let $r_{i}:=\textup{Lip}(f_{i})$.

The probability vectors $\mathfrak{p}_{i},i\in\mathcal{I}_{N}$, are determined
only by $(\alpha,\beta,\\{G_{i}\\}_{i},\\{H_{i}\\}i)$ and date back to
$\\{g_{i}\\}_{i}$. $\\{f_{i}\\}_{i}$ is not related to this. This contains the
affine case that $f_{i}(x)=a_{i}x+(a_{0}+\cdots+a_{i-1})$ for each $i$. The
reason why we have introduced $Y,G_{i}$ and $H_{i}$ will be clarified in the
proof. Contrary to the regularity, we cannot exchange $\\{f_{i}\\}_{i}$ with
$\\{g_{i}\\}_{i}$.

We also deal with the case that
$\left|\\{\mathfrak{p}_{i}\\}_{i\in\mathcal{I}_{N}}\right|=1$. Let the total
distance between probability vectors be

$\|{\bf p}-{\bf q}\|:=\sum_{i\in\mathcal{I}_{N}}|p_{i}-q_{i}|,\ \ {\bf
p}=(p_{i})_{i},{\bf q}=(q_{i})_{i}\in\overline{\mathcal{P}_{N}}.$

######  Theorem 4.3.

If $\left|\\{\mathfrak{p}_{i}\\}_{i\in\mathcal{I}_{N}}\right|=1$, then, for
every $\epsilon\in(0,1/2)$, there exists $\delta>0$ such that it holds that
$\dim_{H}\mu_{\varphi}<1$ for every contractive system $(f_{i})_{i}$ on
$[0,1]$ such that $\sum_{i}r_{i}<1+\delta$, $\epsilon\leq r_{i}\leq
1-\epsilon$ for each $i$, and $\|{\bf r}-\mathfrak{p}_{0}\|\geq\epsilon$,
where we let $r_{i}:=\textup{Lip}(f_{i})$ and ${\bf
r}:=(r_{0},\cdots,r_{N-1})$.

If $\left|\\{\mathfrak{p}_{i}\\}_{i\in\mathcal{I}_{N}}\right|=1$, then,
$\mathfrak{p}_{0}=\cdots=\mathfrak{p}_{N-1}$. If
$f_{i}(x)=a_{i}x+(a_{0}+\cdots+a_{i-1})$ for each $i$, and
$\mathfrak{p}_{0}={\bf q}$, then, the solution $\varphi$ is smooth on $[0,1]$
and we also have an explicit expression for $\varphi$. See [27, Theorem 43
(ii)].

###  4.1. Proofs

We first give notations for symbolic dynamics. Let
$\mathbb{N}:=\\{1,2,\cdots\\}$. We define a map
$\pi:(\mathcal{I}_{N})^{\mathbb{N}}\to[0,1]$ by

$\pi(i_{1}i_{2}\cdots):=\lim_{k\to\infty}f_{i_{1}}\circ
f_{i_{2}}\circ\cdots\circ f_{i_{k}}(0).$

This is well-defined because each $f_{i}$ is a contraction and
$\left(f_{i_{1}}\circ f_{i_{2}}\circ\cdots\circ f_{i_{k}}(0)\right)_{k}$ is a
Cauchy sequence.

Let $\xi_{j}:(\mathcal{I}_{N})^{\mathbb{N}}\to\mathcal{I}_{N}$ be the
projection to the $j$-th coordinate. Let

$I(i_{1},\cdots,i_{n}):=\left\\{w\in(\mathcal{I}_{N})^{\mathbb{N}}:\xi_{j}(w)=i_{j},\
1\leq j\leq n\right\\}.$

Then, by Lemma 2.1,

$\pi\left(I(i_{1},\cdots,i_{k})\right)=\left[f_{i_{1}}\circ
f_{i_{2}}\circ\cdots\circ f_{i_{k}}(0),f_{i_{1}}\circ
f_{i_{2}}\circ\cdots\circ f_{i_{k}}(1)\right].$

Hence, by (2.3),

$\displaystyle\mu_{\varphi}\left(\pi(I(i_{1},\cdots,i_{k}))\right)$
$\displaystyle=\varphi\left(f_{i_{1}}\circ\cdots\circ
f_{i_{k}}(1)\right)-\varphi\left(f_{i_{1}}\circ\cdots\circ
f_{i_{k}}(0)\right)$ $\displaystyle=g_{i_{1}}\circ\cdots\circ
g_{i_{k}}(1)-g_{i_{1}}\circ\cdots\circ g_{i_{k}}(0).$

For $y\in Y$, let $\nu_{y}$ be a probability measure on
$(\mathcal{I}_{N})^{\mathbb{N}}$ such that $\nu_{y}(I(i))=G_{i}(y)$,
$i\in\mathcal{I}_{N}$, and,

$\frac{\nu_{y}(I(i_{1},\cdots,i_{n-1},i_{n}))}{\nu_{y}(I(i_{1},\cdots,i_{n-1}))}=G_{i_{n}}\circ
H_{i_{n-1}}\circ\cdots\circ H_{i_{1}}(y),\ n\geq 2.$ (4.1)

Since $\sum_{i\in\mathcal{I}_{N}}G_{i}(y)=1$ for each $y\in Y$, the
consistency condition holds. Therefore, a probability measure on
$(\mathcal{I}_{N})^{\mathbb{N}}$ satisfying (4.1) uniquely exists by the
Kolmogorov extension theorem.

For $i\in\mathcal{I}_{N}$, let
$F_{i}:(\mathcal{I}_{N})^{\mathbb{N}}\to(\mathcal{I}_{N})^{\mathbb{N}}$ be a
map defined by $F_{i}(z)=iz$. Now we see that

$\nu_{y}=\sum_{i\in\mathcal{I}_{N}}G_{i}(y)\nu_{H_{i}(y)}\circ F_{i}^{-1}.$
(4.2)

Let $\mu_{y}:=\nu_{y}\circ\pi^{-1}$. Then, $\mu_{y}$ is a Borel probability
measure on $[0,1]$. By $\pi\circ F_{i}=f_{i}\circ\pi$ and (4.2),

$\mu_{y}=\sum_{i\in\mathcal{I}_{N}}G_{i}(y)\mu_{H_{i}(y)}\circ f_{i}^{-1}.$
(4.3)

We recall that $0\in Y=[\alpha,\beta]$. Then, by (4.3),

$\mu_{0}=\sum_{i\in\mathcal{I}_{N}}G_{i}(0)\mu_{H_{i}(0)}\circ f_{i}^{-1}.$

Since $H_{i}(0)\neq 0$ can occur, this does not define an iterated function
system. (4.3) is in the framework of a place-dependent iterated function
system [9].

By [27, Lemma 30],

$\nu_{0}\circ\pi^{-1}=\mu_{0}=\mu_{\varphi}.$ (4.4)

Let $\epsilon\in(0,1/2)$. We assume that
$\epsilon\leq\min_{i}r_{i}\leq\max_{i}r_{i}\leq 1-\epsilon$. Since
$\\{f_{i}\\}_{i}$ is a compatible system, $\sum_{i}r_{i}\geq 1$ and $r_{i}>0$
for each $i$. Let $s\geq 1$ is a constant such that
$\sum_{i\in\mathcal{I}_{N}}r_{i}^{s}=1$.

As the following assertion shows, the constant $s$ is close to $1$ if
$\delta>0$ is sufficiently small.

######  Lemma 4.4.

Let $\epsilon_{2}>0$. Then, for $\delta>0$ such that

$\frac{\log N}{\log(N/(1+\delta))}<1+\frac{\epsilon_{2}}{\log 1/\epsilon},$
(4.5)

it holds that $s<1+\dfrac{\epsilon_{2}}{\log 1/\epsilon}$ for every choice of
$(r_{i})_{i}\in\mathcal{P}_{N}$ such that $\epsilon\leq r_{i}\leq 1-\epsilon$
for each $i$.

###### Proof.

Let $u_{i}:=r_{i}/\sum_{i}r_{i}$. Then, $\sum_{i}u_{i}=1$. We see that
$\sum_{i}r_{i}^{s}=1$ if and only if $\sum_{i}u_{i}^{s}=(\sum_{i}r_{i})^{-s}$.
Hence, if $\sum_{i}r_{i}\leq 1+\delta$, then,

$\frac{\log(\sum_{i}u_{i}^{s})}{s}\leq-\log(1+\delta).$

$\sum_{i}u_{i}^{s}$ takes its minimum at $u_{0}=\cdots=u_{N-1}=\dfrac{1}{N}$.
We see this by the method of Lagrange multipliers. The minimum is $N^{1-s}\leq
1$. Hence, $-\dfrac{s-1}{s}\log N\geq-\log(\sum_{i}r_{i})$, which is
equivalent to $s<\dfrac{\log N}{\log(N/(1+\delta))}$. Therefore, if (4.5)
holds, then, $s<1+\dfrac{\epsilon_{2}}{\log 1/\epsilon}$. ∎

Let

$R_{n}({\bf q};z):=\frac{\nu_{0}(I_{n}(z))}{q_{\xi_{1}(z)}\cdots
q_{\xi_{n}(z)}},\ \ {\bf
q}=(q_{i})_{i}\in\mathcal{P}_{N},z\in(\mathcal{I}_{N})^{\mathbb{N}}.$

By a measure-theoretic argument, we can show that

######  Lemma 4.5.

Let $\widetilde{q}_{i}:=r_{i}^{s}>0$ and
${\bf\widetilde{q}}:=(\widetilde{q}_{i})_{i}\in\mathcal{P}_{N}$. If there
exists $\epsilon_{2}>0$ such that

$\limsup_{n\to\infty}\frac{-\log
R_{n}({\bf\widetilde{q}};z)}{n}\leq-\epsilon_{2},\ \textup{
$\nu_{0}$-a.s.$z\in(\mathcal{I}_{N})^{\mathbb{N}}$.}$ (4.6)

then,

$\dim_{H}\mu_{\varphi}\leq s-\frac{\epsilon_{2}}{\log 1/\epsilon}.$

We omit the proof of this assertion. See the proof of [27, Lemma 7]. (4.4)
plays an important role.

By Lemmas 4.4 and 4.5, we obtain that

######  Proposition 4.6.

Assume that there exists $\epsilon_{2}>0$ such that for every ${\bf
q}\in\mathcal{P}_{N}$,

$\limsup_{n\to\infty}\frac{-\log R_{n}({\bf q};z)}{n}\leq-\epsilon_{2},\
\textup{ $\nu_{0}$-a.s.$z\in(\mathcal{I}_{N})^{\mathbb{N}}$.}$ (4.7)

Then, for $\delta=\delta(N,\epsilon,\epsilon_{2})>0$ satisfying (4.5),
$\dim_{H}\mu_{\varphi}<1$.

Now it suffices to establish (4.7). We give some definitions for the relative
entropy. Let

$s_{N}\left({\bf p}|{\bf
q}\right):=\sum_{i\in\mathcal{I}_{N}}p_{i}\log\frac{q_{i}}{p_{i}},\ \ {\bf
p}=(p_{i})_{i},{\bf q}=(q_{i})_{i}\in\overline{\mathcal{P}_{N}}.$

If $p_{i}=0$, then, $p_{i}\log\frac{q_{i}}{p_{i}}:=0$. $s_{N}$ is continuous
on $\overline{\mathcal{P}_{N}}$. Then, by Jensen’s inequality, $s_{N}({\bf
p}|{\bf q})\leq 0$. It holds that $s_{N}\left({\bf p}|{\bf q}\right)=0$ if and
only if ${\bf p}={\bf q}$.

For $y\in Y,z\in(\mathcal{I}_{N})^{\mathbb{N}}$, let

$p_{j}(y;z):=\left(G_{i}\circ H_{\xi_{j}(z)}\circ\cdots\circ
H_{\xi_{1}(z)}(y)\right)_{i\in\mathcal{I}_{N}}\in\overline{\mathcal{P}_{N}},\
j\geq 1.$

and

$p_{0}(y;z):=(G_{i}(y))_{i\in\mathcal{I}_{N}}\in\overline{\mathcal{P}_{N}}.$

By a martingale argument, we can show that

######  Proposition 4.7.

It holds that for every ${\bf q}\in\mathcal{P}_{N}$,

$\limsup_{n\to\infty}\frac{-\log R_{n}({\bf
q};z)}{n}=\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}s_{N}(p_{j}(0;z)|{\bf
q}),\ \textup{ $\nu_{0}$-a.s.$z\in(\mathcal{I}_{N})^{\mathbb{N}}$.}$

We omit the proof of this assertion. See the proof of [27, Proposition 6]. It
is important to assume that ${\bf q}\in\mathcal{P}_{N}$, not ${\bf
q}\in\overline{\mathcal{P}_{N}}$.

For $\epsilon\in(0,1)$, let
$\overline{\mathcal{P}_{N}(\epsilon)}:=\left\\{(p_{i})_{i\in\mathcal{I}_{N}}:p_{i}\geq\epsilon,\sum_{i}p_{i}=1\right\\}$.
Then,
$\overline{\mathcal{P}_{N}(\epsilon)}\subset\mathcal{P}_{N}\subset\overline{\mathcal{P}_{N}}$.

Recall Lemma 4.4. If

$\frac{\log N}{\log(N/(1+\delta))}\leq 1+\frac{\log 2}{\log(1/\epsilon)},$

then, ${\bf\widetilde{q}}\in\overline{\mathcal{P}_{N}(\epsilon/2)}$. In other
words, ${\bf\widetilde{q}}$ is apart from the boundary
$\overline{\mathcal{P}_{N}}\setminus\mathcal{P}_{N}$

Assume that

######  Proposition 4.8.

If $|\\{\mathfrak{p}_{i}\\}_{i}|\geq 2$, then, there exists $\epsilon_{2}>0$
such that for every ${\bf q}\in\overline{\mathcal{P}_{N}(\epsilon/2)}$,

$\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}s_{N}\left(p_{j}(0;z)|{\bf
q}\right)\leq-\epsilon_{2},\ \textup{
$\nu_{0}$-a.s.$z\in(\mathcal{I}_{N})^{\mathbb{N}}$.}$

Theorem 4.2 follows from Propositions 4.6, 4.7 and 4.8. We now show
Proposition 4.8. We give a strategy of the proof. By the definition,
$s_{N}\left(p_{j}(0;z)|{\bf q}\right)\leq 0$. Hence, it suffices to show the
number of $j$ satisfying that $s_{N}\left(p_{j}(0;z)|{\bf q}\right)$ is
strictly negative has positive density in $\mathbb{N}$.

We first give some further definitions for quantities of the relative entropy.

######  Lemma 4.9.

Let

$V\left({\bf q};\epsilon\right):=\max\left\\{s_{N}\left({\bf p}|{\bf
q}\right)\middle|{\bf p}\in\overline{\mathcal{P}_{N}},\|{\bf p}-{\bf
q}\|\geq\epsilon\right\\},\ \epsilon>0.$

Then,
(i) $V\left({\bf q};\epsilon\right)<0$.
(ii) $V\left({\bf q};\epsilon\right)$ is decreasing with respect to
$\epsilon$.
(iii) $V\left({\bf q};\epsilon\right)$ is right-continuous with respect to
$\epsilon$.
(iv) $V\left({\bf q};\epsilon\right)$ is continuous with respect to ${\bf q}$.

###### Proof.

(i) and (ii) are easy to see. (iii) Assume that for some
$\epsilon_{0}\in(0,1/3)$, $V\left({\bf q};\epsilon\right)$ is not right-
continuous. Then, $V\left({\bf
q};\epsilon_{0}\right)>\lim_{\epsilon\to\epsilon_{0}+0}V\left({\bf
q};\epsilon\right)$. Assume that $V\left({\bf
q};\epsilon_{0}\right)=s_{N}\left({\bf p}|{\bf q}\right)$. If $\|{\bf p}-{\bf
q}\|>\epsilon_{0}$, then, this cannot occur. Assume $\|{\bf p}-{\bf
q}\|=\epsilon_{0}$. Then, there exists a sequence $\\{{\bf p}^{(n)}\\}_{n}$ in
$\overline{\mathcal{P}_{N}}$ such that $\|{\bf p}^{(n)}-{\bf
q}\|\geq\epsilon_{0}+N^{-n}$ for each $n$ and $\lim_{n\to\infty}\|{\bf
p}^{(n)}-{\bf q}\|=\epsilon_{0}$. Hence,

$\liminf_{n\to\infty}V\left({\bf
q};\epsilon_{0}+N^{-n}\right)\geq\lim_{n\to\infty}s_{N}\left({\bf
p}^{(n)}|{\bf q}\right)=s_{N}\left({\bf p}|{\bf q}\right)=V\left({\bf
q};\epsilon_{0}\right).$

This is a contradiction.

(iv) follows from (iii) and

$V\left({\bf q};\epsilon+\|{\bf p}-{\bf q}\|\right)\leq V\left({\bf
p};\epsilon\right)$

∎

By Lemma 4.9, we can let

$V(\epsilon):=\max_{{\bf q}\in\overline{\mathcal{P}_{N}}}V\left({\bf
q};\epsilon\right)<0.$

We now give a quantity defined by $Y,\\{G_{i}\\}_{i}$ and $\\{H_{i}\\}_{i}$.

######  Lemma 4.10.

Let

$W_{i}\left({\bf
q},y\right):=\max\left\\{\max_{k\in\mathcal{I}_{N}}|G_{k}(y)-q_{k}|,\max_{k\in\mathcal{I}_{N}}|G_{k}(H_{i}(y))-q_{k}|\right\\},\
{\bf q}\in\overline{\mathcal{P}_{N}},y\in Y,i\in\mathcal{I}_{N}.$

Then,
(i) $W_{i}({\bf q},y)$ is continuous on $\overline{\mathcal{P}_{N}}\times Y$.
(ii) $W_{i}({\bf q},y)=0$ if and only if ${\bf q}=\mathfrak{p}_{i}$ and
$y=\textup{Fix}(H_{i})$.

This assertion is easily shown. If
$\max_{k\in\mathcal{I}_{N}}|G_{k}(y)-q_{k}|>0$, then,
$s_{N}\left((G_{k}(y))_{k}|{\bf q}\right)<0$. However, in our setting, we
cannot exclude the possibility that $|G_{k}(y)-q_{k}|=0$ for every
$k\in\mathcal{I}_{N}$. Assume that
$\left|\\{\mathfrak{p}_{i}\\}_{i}\right|\geq 2$. If
$\max_{k\in\mathcal{I}_{N}}|G_{k}(y)-q_{k}|$ is close to $0$, then,
$\max_{k\in\mathcal{I}_{N}}|G_{k}(H_{i}(y))-q_{k}|$ is strictly positive.
Hence, $W_{i}\left({\bf q},y\right)>0$.

The following corresponds to [27, Lemma 8]. Informally speaking, if
$W_{i}({\bf q},y)$ is positive uniformly to $y$, then, a sum of two successive
terms of $s_{N}$ is negative.

######  Lemma 4.11.

Let ${\bf q}\in\mathcal{P}_{N}$. Let $i_{1}\in\mathcal{I}_{N}$. Assume that
for some $\epsilon_{1}>0$,

$\inf_{y\in Y}W_{i_{1}}\left({\bf q},y\right)\geq\epsilon_{1}.$ (4.8)

Assume that $X_{j+1}(z)=i_{1}$. Then,

$s_{N}\left(p_{j}(0;z)|{\bf q}\right)+s_{N}\left(p_{j+1}(0;z)|{\bf
q}\right)\leq V\left({\bf q};\epsilon_{1}\right)\leq V(\epsilon_{1})<0.$

###### Proof.

The proof is essentially the same as the proof of [27, Lemma 8]333It contains
a typo.. Since $X_{j+1}(z)=i_{1}$,

$p_{j}(0;z)=\left(G_{k}(H_{X_{j}(z)}\circ\cdots\circ
H_{X_{1}(z)}(0))\right)_{k\in\mathcal{I}_{N}},$

and

$p_{j+1}(0;z)=\left(G_{k}(H_{i_{1}}\circ H_{X_{j}(z)}\circ\cdots\circ
H_{X_{1}(z)}(0))\right)_{k\in\mathcal{I}_{N}}.$

Let ${\bf q}=(q_{k})_{k}$. Then, by (4.8), either

$\left|G_{k_{1}}(H_{X_{j}(z)}\circ\cdots\circ
H_{X_{1}(z)}(0))-q_{k_{1}}\right|\geq\epsilon_{1},\textup{ for some
}k_{1}\in\mathcal{I}_{N},$

or,

$\left|G_{k_{2}}(H_{i_{1}}\circ H_{X_{j}(z)}\circ\cdots\circ
H_{X_{1}(z)}(0))-q_{k_{2}}\right|\geq\epsilon_{1},\textup{ for some
}k_{2}\in\mathcal{I}_{N},$

holds. If the former holds, then, $s_{N}\left(p_{j}(0;z)|{\bf q}\right)\leq
V\left({\bf q};\epsilon_{1}\right)$. If the latter holds, then,
$s_{N}\left(p_{j+1}(0;z)|{\bf q}\right)\leq V\left({\bf
q};\epsilon_{1}\right)$. Now it suffices to recall that $s_{N}$ takes non-
positive values. ∎

Let $\ell:=|\\{\mathfrak{p}_{i}\\}_{i}|\geq 2$. We assume that
$\\{\mathfrak{p}_{i}\\}_{i}=\\{\mathfrak{p}_{j_{k}}:1\leq k\leq\ell\\}$ where
$0\leq j_{1}<\cdots<j_{\ell}\leq N-1$. Let
$\delta_{0}:=\min\\{\|\mathfrak{p}_{j_{k_{1}}}-\mathfrak{p}_{j_{k_{2}}}\|:k_{1}\neq
k_{2}\\}>0$. Let $B_{j}:=\left\\{{\bf p}\in\overline{\mathcal{P}_{N}}:\|{\bf
p}-\mathfrak{p}_{j}\|\leq\delta_{0}/4\right\\}$. $B_{j}=B_{k}$ can occur even
if $j\neq k$. Define a coloring map
$\mathscr{A}:\overline{\mathcal{P}_{N}}\to\mathcal{I}_{N}$ by

$\mathscr{A}\left({\bf p}\right):=\begin{cases}j_{2}&{\bf p}\in B_{j_{1}}\\\
\vdots&\\\ j_{\ell}&{\bf p}\in B_{j_{\ell-1}}\\\ j_{1}&{\bf p}\in
B_{j_{\ell}}\\\ 0&\textup{ otherwise}\end{cases}.$

Under these definitions, we see that

######  Lemma 4.12.

$\widetilde{\epsilon_{1}}:=\inf\left\\{W_{\mathscr{A}\left({\bf
q}\right)}({\bf q},y)\middle|{\bf q}\in\overline{\mathcal{P}_{N}},y\in
Y\right\\}>0.$

###### Proof.

Fix ${\bf q}\in\overline{\mathcal{P}_{N}}$. By Lemma 4.10,

$W_{\mathscr{A}\left({\bf q}\right)}({\bf
q},y)\geq\frac{1}{2}\max_{j}\left|G_{j}(H_{\mathscr{A}\left({\bf
q}\right)}(y))-G_{j}(y)\right|.$

If $(G_{j}(y))_{j}\notin B_{\mathscr{A}({\bf q})}$, then,
$y\neq\mathfrak{p}_{\mathscr{A}({\bf
q})}=\left(G_{j}(\textup{Fix}(H_{\mathscr{A}\left({\bf
q}\right)}))\right)_{j}$ and hence, $y$ is not contained in an open interval
$J_{\mathscr{A}({\bf q})}$ in $Y$ containing
$\textup{Fix}(H_{\mathscr{A}\left({\bf q}\right)})$. Since $G_{j}$ and
$H_{\mathscr{A}\left({\bf q}\right)}$ are continuous,
$\max_{j}\left|G_{j}(H_{\mathscr{A}\left({\bf q}\right)}(y))-G_{j}(y)\right|$
is continuous with respect to $y$ on $Y$. Hence,

$\inf_{(G_{j}(y))_{j}\notin B_{\mathscr{A}({\bf
q})}}\max_{j}\left|G_{j}(H_{\mathscr{A}\left({\bf
q}\right)}(y))-G_{j}(y)\right|$ $\geq\inf_{y\notin J_{\mathscr{A}({\bf
q})}}\max_{j}\left|G_{j}(H_{\mathscr{A}\left({\bf
q}\right)}(y))-G_{j}(y)\right|=:\epsilon_{\mathscr{A}\left({\bf q}\right)}>0.$

Assume that $(G_{j}(y))_{j}\in B_{\mathscr{A}({\bf q})}$. Since ${\bf q}\notin
B_{\mathscr{A}({\bf q})}$,

$W_{\mathscr{A}\left({\bf q}\right)}({\bf
q},y)\geq\frac{1}{N}\sum_{j}|G_{j}(y)-q_{j}|\geq\frac{\delta_{0}}{4N}.$

Hence,

$\inf_{y\in Y}W_{\mathscr{A}\left({\bf q}\right)}({\bf
q},y)\geq\min\left\\{\frac{\delta_{0}}{4N},\epsilon_{j_{k}}\middle|1\leq
k\leq\ell\right\\}=:\tilde{\epsilon_{1}}.$

∎

We discuss the frequency of occurrence of some elements in $\mathcal{I}_{N}$
in $(\xi_{n})_{n}$.

######  Lemma 4.13.

Let $I\subset\mathcal{I}_{N}$ be a non-empty set. Let $c_{I}:=\inf_{y\in
Y}\sum_{i\in I}G_{i}(y)$. Then,

$\liminf_{n\to\infty}\frac{|\\{1\leq j\leq n:\xi_{j}(z)\in I\\}|}{n}\geq
c_{I},\ \textup{ $\nu_{0}$-a.s.$z\in(\mathcal{I}_{N})^{\mathbb{N}}$.}$

This corresponds to [27, Lemma 9], but the formulation differs.

###### Proof.

If $c_{I}=0$, then, the assertion is obvious. Assume $c_{I}>0$. We denote the
expectation with respect to $\nu_{0}$ by $E^{\nu_{0}}$. Let
$\mathcal{F}_{m}:=\sigma(\xi_{1},\cdots,\xi_{m})$ for $m\geq 1$ and
$\mathcal{F}_{0}$ be the trivial $\sigma$-algebra on
$(\mathcal{I}_{N})^{\mathbb{N}}$. We denote the indicator function of
$A\subset(\mathcal{I}_{N})^{\mathbb{N}}$ by ${\bf 1}_{A}$.

Let $M_{0}:=0$ and

$M_{n}-M_{n-1}:={\bf 1}_{\\{\xi_{n}\in I\\}}-E^{\nu_{0}}\left[{\bf
1}_{\\{\xi_{n}\in I\\}}|\mathcal{F}_{n-1}\right],n\geq 1.$

Then, $\\{M_{n},\mathcal{F}_{n}\\}_{n}$ is a martingale.

By (4.1), $E^{\nu_{0}}\left[{\bf 1}_{\\{\xi_{n}\in
I\\}}|\mathcal{F}_{n-1}\right]\geq c_{I}$, $\nu_{0}$-a.s. Therefore, for every
$\eta\in(0,1)$,

$\nu_{0}\left(\frac{|\\{1\leq j\leq n:\xi_{j}\in
I\\}|}{n}\leq(1-\eta)c_{I}\right)\leq\nu_{0}\left(M_{n}\leq-n\eta
c_{I}\right).$

We remark that $|M_{n}-M_{n-1}|\leq 1$ for each $n\geq 1$. By Azuma’s
inequality ([1], [33, E14.2]),

$\nu_{0}\left(M_{n}\leq-n\eta
c_{I}\right)\leq\exp\left(-\frac{n\eta^{2}c_{I}^{2}}{2}\right).$

Using $c_{I}>0$ and Borel-Cantelli’s lemma,

$\liminf_{n\to\infty}\frac{|\\{1\leq j\leq n:\xi_{j}(z)\in
I\\}|}{n}\geq(1-\eta)c_{I},\ \textup{
$\nu_{0}$-a.s.$z\in(\mathcal{I}_{N})^{\mathbb{N}}$.}$

The assertion follows if $\eta\to+0$. ∎

We proceed to the proof of Proposition 4.8. The proof is divided into two
cases. The following is easily shown.

######  Lemma 4.14.

(i) $a_{0}\leq 1$ and $b_{N-1}+c_{N-1}\geq 0$.
(ii) If $a_{0}=1$ and $b_{N-1}+c_{N-1}=0$, then, $\alpha=-1,\beta=+\infty$.
(iii) If $a_{0}=1$ and $b_{N-1}+c_{N-1}>0$, then, $\alpha>-1,\beta=+\infty$.
(iv) If $a_{0}<1$ and $b_{N-1}+c_{N-1}=0$, then, $\alpha=-1,\beta<+\infty$.
(v) If $a_{0}<1$ and $b_{N-1}+c_{N-1}>0$, then, $\alpha>-1,\beta<+\infty$.
(vi) If $a_{0}=1$, then, $\textup{Fix}(H_{0})=\\{+\infty\\}$ and
$\mathfrak{p}_{0}=(1,0,\cdots,0)\in\overline{\mathcal{P}_{N}}\setminus\mathcal{P}_{N}$.
(vii) If $b_{N-1}+c_{N-1}=0$, then, $\textup{Fix}(H_{N-1})=\\{-1\\}$ and
$\mathfrak{p}_{N-1}=(0,\cdots,0,1)\in\overline{\mathcal{P}_{N}}\setminus\mathcal{P}_{N}$.

$\alpha=-1$ or $\beta=+\infty$ occurs according to the values of $a_{0}$ and
$b_{N-1}+c_{N-1}$.

Case 1. We first deal with the case that $a_{0}<1$ and $b_{N-1}+c_{N-1}>0$.

In this case, we do not need to assume ${\bf
q}\in\overline{\mathcal{P}_{N}(\epsilon/2)}$.

###### Proof of Proposition 4.8, Case 1.

By the assumption that $a_{0}<1$ and $b_{N-1}+c_{N-1}>0$ and Lemma 4.14 (v),
$-1<\alpha\leq\beta<+\infty$. Hence,

$0<\inf_{y\in Y,i\in\mathcal{I}_{N}}G_{i}(y)\leq\sup_{y\in
Y,i\in\mathcal{I}_{N}}G_{i}(y)<1.$

Fix ${\bf q}\in\overline{\mathcal{P}_{N}}$. Let $\widetilde{c}:=\inf_{y\in
Y,i\in\mathcal{I}_{N}}G_{i}(y)$. Then, $c_{\\{\mathscr{A}({\bf
q})\\}}\geq\widetilde{c}>0$ and by Lemma 4.13 for $I=\\{\mathscr{A}({\bf
q})\\}$,

$\liminf_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}{\bf
1}_{\\{\xi_{k}=\mathscr{A}({\bf q})\\}}(z)\geq\widetilde{c},\ \textup{
$\nu_{0}$-a.s.$z\in(\mathcal{I}_{N})^{\mathbb{N}}$.}$

The number $i_{1}\in\mathcal{I}_{N}$ in Lemma 4.11 will be chosen according to
the position of ${\bf q}$. By applying Lemma 4.11 to $i_{1}=\mathscr{A}({\bf
q})$ and $\epsilon_{1}=\widetilde{\epsilon_{1}}$ where
$\widetilde{\epsilon_{1}}$ is the constant in Lemma 4.12, we see that if
$X_{i+1}(z)=\mathscr{A}({\bf q})$, then,

$s_{N}\left(p_{i}(0;z)|{\bf q}\right)+s_{N}\left(p_{i+1}(0;z)|{\bf
q}\right)\leq V(\widetilde{\epsilon_{1}})<0.$

Now the assertion follows if we let $\epsilon_{2}:=\dfrac{\widetilde{c}\
V(\widetilde{\epsilon_{1}})}{2}$. ∎

Case 2. We now deal with the case that $a_{0}=1$ or $b_{N-1}+c_{N-1}=0$.

If $a_{0}=1$ and $b_{N-1}+c_{N-1}=0$, then, $c_{\\{0\\}}=c_{\\{N-1\\}}=0$, so
we consider $c_{\\{0,N-1\\}}$. In the following, we do not use Lemma 4.12 or
the assumption that $|\\{\mathfrak{p}_{i}\\}_{i}|\geq 2$. On the other hand,
we assume ${\bf q}\in\overline{\mathcal{P}_{N}(\epsilon/2)}$.

###### Proof of Proposition 4.8, Case 2.

Case 2.1. We first consider the case that $a_{0}=1$ and $b_{N-1}+c_{N-1}>0$.

By Lemma 4.14, $\alpha>-1$ and $\beta=+\infty$, $H_{0}(y)=y+c_{0}$, and
$\textup{Fix}(H_{0})=\\{+\infty\\}$.
$\mathfrak{p}_{0}=(1,0,\cdots,0)\notin\mathcal{P}_{N}$. On the other hand,
$\mathfrak{p}_{i}\in\mathcal{P}_{N}$ for $i\geq 1$.

Since $G_{0}(+\infty)=G_{0}(H_{0}(+\infty))=1$ and $q_{0}<1-\epsilon/2$,

$\inf_{y\in Y}W_{0}({\bf q},y)>c_{0}(\epsilon/2)>0,$

where the lower bound $c_{0}(\epsilon/2)$ is uniform with respect to ${\bf
q}\in\overline{\mathcal{P}_{N}(\epsilon/2)}$.

Since $\alpha>-1$, $\inf_{y\in Y}G_{0}(y)\geq c>0$. By using Lemma 4.11 and
Lemma 4.13 for $I=\\{0\\}$, we have the assertion.

Case 2.2. We secondly consider the case that $a_{0}<1$ and
$b_{N-1}+c_{N-1}=0$.

We recall that $\alpha=-1$ and $\beta<+\infty$, By Lemma 4.14,
$H_{N-1}(y)=\dfrac{(1-2b_{N-1})y+b_{N-1}}{1-b_{N-1}}$, and
$\textup{Fix}(H_{N-1})=\\{-1\\}$.
$\mathfrak{p}_{N-1}=(0,\cdots,0,1)\notin\mathcal{P}_{N}$. On the other hand,
$\mathfrak{p}_{i}\in\mathcal{P}_{N}$ for $i\leq N-2$.

Since $G_{N-1}(-1)=G_{N-1}(H_{N-1}(-1))=1$ and $q_{N-1}<1-\epsilon/2$,

$\inf_{y\in Y}W_{N-1}({\bf q},y)>c_{N-1}(\epsilon/2)>0,$

where the lower bound $c_{N-1}(\epsilon)$ is uniform with respect to ${\bf
q}\in\overline{\mathcal{P}_{N}(\epsilon/2)}$. Since $\beta<+\infty$,
$\inf_{y\in Y}G_{N-1}(y)\geq c>0$. The rest of the argument is the same as in
the above case.

Case 2.3. We finally consider the case that $a_{0}=1$ and $b_{N-1}+c_{N-1}=0$.

Since $\textup{Fix}(H_{0})=\\{+\infty\\}$, $\textup{Fix}(H_{N-1})=\\{-1\\}$,
$q_{0}<1-\epsilon/2$ and $q_{N-1}<1-\epsilon/2$, it holds that

$\inf_{y\in Y}W_{0}({\bf q},y)>c_{0}(\epsilon/2)>0$

and

$\inf_{y\in Y}W_{N-1}({\bf q},y)>c_{N-1}(\epsilon/2)>0.$

Since

$G_{0}(y)+G_{N-1}(y)=\frac{b_{1}b_{N-1}y^{2}+2b_{1}y1+b_{1}-b_{N-1}}{(b_{1}y+1)(b_{N-1}y+1)},$

it holds that

$\inf_{y\in Y}G_{0}(y)+G_{N-1}(y)>0.$

By using Lemma 4.11 and Lemma 4.13 for $I=\\{0,N-1\\}$, we have the assertion.
∎

Now the proof of Theorem 4.2 completes.

######  Remark 4.15.

(i) In the proof of [27, Theorem 43], the details of Case 2 were not given.
(ii) Assume that $\\{g_{i}\\}_{i}$ are affine functions such that
$g_{i}(x)=b_{i}x+(b_{0}+\cdots+b_{i-1})$. Since $Y=\\{0\\}$, there is no need
to introduce $H_{i}$, and $W_{i}({\bf
q},y)=\max_{k}|G_{k}(y)-q_{k}|=\max_{k}|b_{k}-q_{k}|$. However,
$\mathscr{p}_{0}=\cdots=\mathscr{p}_{N-1}=(b_{i})_{i}$, and hence this is not
in the scope of Theorem 4.2. This case is considered in Theorem 4.3.

We now proceed to the proof of Theorem 4.3.

By (4.6), if we take $\delta>0$ such that

$\frac{\log N}{\log(N/(1+\delta))}\leq
1+\frac{\log\left(1+\epsilon/2\right)}{\log(1/\epsilon)},$ (4.9)

then, $s\leq 1+\dfrac{\log\left(1+\epsilon/2\right)}{\log(1/\epsilon)}$.

Assume that $s\leq 1+\dfrac{\log\left(1+\epsilon/2\right)}{\log(1/\epsilon)}$.
Since $\min_{i}r_{i}\geq\epsilon$,
$\sum_{i}|r_{i}-r_{i}^{s}|\leq\epsilon^{1-s}-1\leq\epsilon/2$. Hence,
$\left\|{\bf\widetilde{q}}-\mathfrak{p}_{0}\right\|\geq\epsilon/2$, where we
let $\widetilde{q}_{i}:=r_{i}^{s}$ for each $i$ and
${\bf\widetilde{q}}:=(\widetilde{q}_{i})_{i}$. Since $\epsilon\leq 1/2$,
$\widetilde{q}_{i}\geq\epsilon^{s}\geq\epsilon/2$ for each $i$. Hence,
${\bf\widetilde{q}}\in\overline{\mathcal{P}_{N}(\epsilon/2)}$.

By Lemma 4.10 (i) and (ii), it holds that

$\epsilon_{3}:=\inf\left\\{W_{i}({\bf q},y)\middle|i\in\mathcal{I}_{N},y\in
Y,{\bf q}\in\overline{\mathcal{P}_{N}(\epsilon/2)},\left\|{\bf
q}-\mathfrak{p}_{0}\right\|\geq\epsilon/2\right\\}>0.$

By Lemma 4.11, we see that for every $j\geq 0$, every
$z\in(\mathcal{I}_{N})^{\mathbb{N}}$, and every ${\bf
q}\in\overline{\mathcal{P}_{N}(\epsilon/2)}$ with $\left\|{\bf
q}-\mathfrak{p}_{0}\right\|\geq\epsilon/2$,

$s_{N}\left(p_{j}(0;z)|{\bf q}\right)+s_{N}\left(p_{j+1}(0;z)|{\bf
q}\right)\leq V\left({\bf q};\epsilon_{3}\right).$

By Lemma 4.9, we can let

$V_{\epsilon}(\epsilon_{3}):=\max\left\\{V\left({\bf
q};\epsilon_{3}\right)\middle|{\bf
q}\in\overline{\mathcal{P}_{N}(\epsilon/2)},\left\|{\bf
q}-\mathfrak{p}_{0}\right\|\geq\epsilon/2\right\\}<0.$

Then,

$\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}s_{N}\left(p_{j}(0;z)|{\bf
q}\right)\leq-\frac{V_{\epsilon}(\epsilon_{3})}{2},\ \textup{
$\nu_{0}$-a.s.$z\in(\mathcal{I}_{N})^{\mathbb{N}}$.}$

Assume that $\delta>0$ satisfies not only (4.9) but also (4.5) for
$\epsilon_{2}:=V_{\epsilon}(\epsilon_{3})/2$. Then, by Propositions 4.6 and
4.7, $\dim_{H}\mu_{\varphi}<1$.

######  Remark 4.16.

(i) We do not need to use Lemma 4.13. For every small $\epsilon>0$ and every
$i\in\mathcal{I}_{N}$,

$\min\left\\{W_{i}({\bf q},y)\middle|y\in Y,{\bf
q}\in\overline{\mathcal{P}_{N}(\epsilon/2)}\right\\}=0,$

by Lemma 4.10 (ii).
(ii) The case that both of $\\{f_{i}\\}_{i}$ and $\\{g_{i}\\}_{i}$ are affine
is easily handled by the dimension formula. However, if some of the functions
$\\{f_{i},g_{i}\\}_{i}$ are non-affine, then, it is difficult to judge the
singularity by a generalized dimension formula in [9].

##  5\. Example

We give an example that $\\{g_{i}\\}_{i}$ is an LF system and
$\\{f_{i}\\}_{i}$ is non-affine and the solution $\varphi$ is singular. The
result depends on the quantitative estimates in Section 4.

Let $N=2$. Then,

$G_{0}(y)=\frac{b_{1}(y+1)}{b_{1}y+1},\ H_{0}(y)=a_{0}y+c_{0},\
H_{1}(y)=\frac{(1-b_{1}+c_{1})y+c_{1}}{b_{1}y+1},$

Let

$g_{0}(x):=\frac{x}{-x+6},\ g_{1}(x):=\frac{3x+1}{5-x}.$

Then, $g_{0}$ is contractive, and $g_{1}$ is weakly contractive but not
contractive. It holds that $a_{0}=1/6$, $c_{0}=-1/6$, $a_{1}=3/5$,
$b_{1}=1/5$, and $c_{1}=-1/5$. Hence,

$\alpha=\min\\{0,-1/5,-1\\}=-1,\beta=\max\\{0,-1/5,-1\\}=0.$

Hence,

$\sup_{y\in[-1,0]}G_{0}(y)=1-\inf_{y\in[-1,0]}G_{1}(y)=\frac{1}{5}.$

Let $\mathcal{P}[4/5]:=\left\\{(p_{0},p_{1})\in\mathcal{P}_{2}:p_{0}\geq
4/5\right\\}$. Let ${\bf q}=(q_{0},q_{1})\in\mathcal{P}[4/5]$. Then,

$|G_{0}(y)-q_{0}|=|G_{1}(y)-q_{1}|\geq\frac{3}{5},\ y\in[-1,0].$

Hence,

$s_{2}\left(p_{j}(0,z)|{\bf q}\right)\leq V\left({\bf q};\frac{6}{5}\right),\
j\geq 0,z\in(\mathcal{I}_{N})^{\mathbb{N}}.$ (5.1)

Since $s_{2}({\bf p}|{\bf
q})=s_{2}\left((p_{0},1-p_{0})|(q_{0},1-q_{0})\right)$ is increasing with
respect to $p_{0}$ on $[0,q_{0}]$,

$V\left({\bf q};\frac{6}{5}\right)\leq
s_{2}\left(\left(\frac{1}{5},\frac{4}{5}\right)\middle|(q_{0},1-q_{0})\right)\leq
s_{2}\left(\left(\frac{1}{5},\frac{4}{5}\right)\middle|\left(\frac{4}{5},\frac{1}{5}\right)\right)\leq-\frac{6}{5}\log
2\leq-0.83.$

By Proposition 4.6, it holds that

$\limsup_{n\to\infty}\frac{\log R_{n}({\bf q};z)}{n}\leq-0.83,\ \textup{
$\nu_{0}$-a.s.$z$, \ ${\bf q}\in\mathcal{P}[4/5]$.}$ (5.2)

Let $\epsilon\in(0,1/2)$. Let $\\{f_{0},f_{1}\\}$ be a contractive system such
that $\epsilon\leq r_{i}\leq 1-\epsilon$, where $r_{i}:=\textup{Lip}(f_{i})$,
$i=0,1$. Let $s\geq 1$ be a constant such that $r_{0}^{s}+r_{1}^{s}=1$. Assume
that $(r_{0}^{s},r_{1}^{s})\in\mathcal{P}[4/5]$. Then, by (5.2) and Lemma 4.5,

$\dim_{H}\mu_{\varphi}\leq s-\frac{0.83}{\log(1/\epsilon)}.$

We see that $\dfrac{0.83}{\log 10}\geq 0.36$ and $\dfrac{\log 2}{\log 2-\log
1.2}\leq 1.357$. Hence, (4.5) holds for $\epsilon=0.1$, $\epsilon_{2}=0.83$
and $\delta=0.2$. By Lemma 4.4, if $\\{f_{0},f_{1}\\}$ is a contractive system
such that $0.1\leq r_{i}\leq 0.9$, $i=0,1$, $r_{0}+r_{1}\leq 1.2$, and
$(r_{0}^{s},r_{1}^{s})\in\mathcal{P}[4/5]$, then,

$\dim_{H}\mu_{\varphi}\leq s-\frac{0.83}{\log(1/\epsilon)}\leq\dfrac{\log
2}{\log 2-\log 1.2}-\frac{0.83}{\log 10}<1.$

Now we can let $\\{f_{0},f_{1}\\}$ be a compatible system such that $0.85\leq
r_{0}\leq 0.9$ and $0.1\leq r_{1}\leq 0.3$. Then, $r_{0}+r_{1}\leq 1.2$. By
the compatibility condition, $r_{0}+r_{1}\geq 1$. The maps $f_{0}$ and $f_{1}$
may not be affine. Since $s\leq\frac{\log 2}{\log 2-\log 1.2}\leq 1.357$,
$r_{0}^{s}\geq(0.85)^{1.357}\geq 0.8$. Hence,
$(r_{0}^{s},r_{1}^{s})\in\mathcal{P}[4/5]$.

######  Remark 5.1.

We see that $a_{0}<1$ and $b_{1}+c_{1}=0$. Hence, this case corresponds to
Case 2.2. However, we do not need to apply Lemma 4.11 since we have the upper
bound for the relative entropy in (5.1).

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