# Invariant measures for random piecewise convex maps

Tomoki INOUE  Graduate School of Science and Engineering
Ehime University
3, Bunkyo-cho, Matsuyama, Ehime
790-8577, JAPAN<EMAIL_ADDRESS>and  Hisayoshi TOYOKAWA
Faculty of Engineering
Kitami Institute of Technology
165 Koen-cho, Kitami, Hokkaido
090-8507, JAPAN<EMAIL_ADDRESS>
###### Abstract.

We show the existence of Lebesgue-equivalent conservative and ergodic
$\sigma$-finite invariant measures for a wide class of one-dimensional random
maps consisting of piecewise convex maps. We also estimate the size of
invariant measures around a small neighborhood of a fixed point where the
invariant density functions may diverge. Application covers random
intermittent maps with critical points or flat points. We also illustrate that
the size of invariant measures tends to infinite for random maps whose right
branches exhibit a strongly contracting property on average, so that they have
a strong recurrence to a fixed point.

###### Key words and phrases:

Invariant Measures; Infinite Invariant Measures; Random Dynamical Systems;
Piecewise Convex Maps; Random Piecewise Convex Maps

###### 2020 Mathematics Subject Classification:

Primary 37A40; Secondary 37H12, 37A05

##  1\. Introduction

For a given non-singular map on a probability space, the question whether an
invariant measure, which is absolutely continuous with respect to the
reference measure, exists or not is one of the fundamental problems in ergodic
theory. The same question for a random map (in terms of both annealed and
quenched sense) makes sense and is also important passing through ergodic
theory for Markov operators, skew-product transformations or Markov operator
cocycles. See [4, 13, 22, 21] and references therein. On the one hand, if a
system, whether deterministic or random, admits an absolutely continuous
finite invariant measure, some classical ergodic theorems such as the Birkhoff
ergodic theorem are applicable. Moreover some limit theorems such as the
central limit theorem may be expected [3, 10, 11]. On the other hand, if a
system possesses only an absolutely continuous $\sigma$-finite and infinite
invariant measure, such a system is within the scope of infinite ergodic
theory and has been paid attention for recent decades [2, 25, 28]. Typical
examples of such systems have indifferent or neutral, but weakly repelling,
fixed points and are known as an intermittent model, as well as a model of
non-uniformly hyperbolic systems. The existence of absolutely continuous
$\sigma$-finite invariant measures and several statistical properties for
random versions of intermittent maps thus have been recently enthusiastically
studied [5, 6, 7, 18, 29, 30].

The subject of the present paper is a certain class of one-dimensional random
dynamical systems called _random piecewise convex maps_ in annealed (or
i.i.d.) sense (see the conditions ( ‣ 2)–(2) and (A) or (B) precisely). The
existence of invariant measures and their ergodic properties of the
deterministic piecewise convex maps were firstly studied by Lasota and Yorke
in [23] for the case when maps are uniformly expanding on the first branch and
other branches have positive derivative, and then studied by Inoue in [15, 16]
for more general cases. The aim of this paper is to generalize them,
demonstrating that random piecewise convex maps admit Lebesgue-equivalent
ergodic $\sigma$-finite invariant measures. (For some interesting studies of
random generalization of piecewise convex maps with ``position dependent''
probability measures, we refer to [8] (cf., [20]), while we do not deal with
position dependent random maps but we handle random maps consisting of
potentially uncountably many maps with infinite invariant measures.) We also
estimate the size of invariant measures, from which it is revealed whether the
$\sigma$-finite invariant measures for random piecewise convex maps are finite
or infinite. The phenomenon that an invariant measure varies from finite to
infinite as a parameter of a system varies is well-known for (deterministic)
intermittent maps employed by Thaler in [27] or Liverani–Saussol–Vaienti (see
(1.1) below) in [24]. Although our random piecewise convex maps also admit
both finite and $\sigma$-finite infinite invariant measures depending on
parameters and probabilities of choice of maps, some examples of them (e.g.,
Example 4.1) have neither a common indifferent fixed point nor a common
critical point, which is very different from deterministic cases. The
mechanism is, roughly speaking, derived from strong contracting property on
average, which never occurs for deterministic systems and is unique to random
dynamical systems.

We then briefly review the LSV map, named after Liverani–Saussol–Vaienti from
[24], which has been analyzed as a simple model of intermittency, and we will
compare them (and known random versions of them, see also [5, 6, 7]) with our
random piecewise convex maps. For $\alpha>0$, the LSV map
$T_{\alpha}:[0,1]\to[0,1]$ is defined by

$\displaystyle
T_{\alpha}x=\begin{cases}x\left(1+2^{\alpha}x^{\alpha}\right)&x\in[0,\frac{1}{2}],\\\
2x-1&x\in(\frac{1}{2},1]\end{cases}$ (1.1)

which has an indifferent fixed point 0 and Lebesgue-typical orbits would be
trapped around a small neighborhood of 0 for a long time. For this map
$T_{\alpha}$, it is well-known that a Lebesgue-equivalent ergodic invariant
measure exists and that the invariant density function is of order
$x^{-\alpha}$ as $x\to 0$ [24, 27, 32]. Thus $T_{\alpha}$ possesses an
equivalent finite invariant measure for $0<\alpha<1$ and an equivalent
$\sigma$-finite and infinite invariant measure for $\alpha\geq 1$. The order
of invariant measures radically affects their statistical properties, such as
the central limit theorem or the mixing rate in the finite measure-preserving
case (cf., [7, 14, 26]) and the wandering rate in the Darling–Kac theorem or
the arcsine law in the infinite measure-preserving case (cf., [2, 25, 28]).
Therefore it is certainly worth establishing invariant measures and analyzing
the asymptotics of the invariant measures for given systems. The asymptotics
of the invariant density for $T_{\alpha}$ is also tightly related to the decay
of the inverse images of the disconnected point $\frac{1}{2}$ by the left
branch. If we set $x_{1}(\alpha)=\frac{1}{2}$ and
$x_{n}(\alpha)\in[0,\frac{1}{2})$ for $n\geq 2$ such that
$T_{\alpha}(x_{n+1}(\alpha))=x_{n}(\alpha)$ for $n\geq 1$, then it follows
from the results in [32, 27] that

$\displaystyle x_{n}(\alpha)\sim Cn^{-\frac{1}{\alpha}},$ (1.2)

for some $C>0$, where $a_{n}\sim b_{n}$ for positive sequences
$\\{a_{n}\\}_{n=1}^{\infty},\\{b_{n}\\}_{n=1}^{\infty}\subset\mathbb{R}$
stands for $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}=1$. In this paper, for random
piecewise convex maps, we establish the existence of Lebesgue-equivalent,
conservative and ergodic $\sigma$-finite invariant measures and evaluate the
asymptotics of the invariant measures, which is a generalization of results
for random LSV maps as in [5, 6]. The advantage of our results is that we do
not restrict ourselves to constituent maps of random dynamical systems to be
at most countable nor expanding on average outside of a small neighborhood of
the common indifferent fixed point. As an application, we can modify a random
LSV map to admit uniformly contracting branches and moreover to admit a
critical or flat point around the inverse image of an indifferent fixed point
(see Example 4.3–4.6). The key point in the estimate of the invariant measures
for random piecewise convex maps is, on the contrary to the LSV maps, the
decay of random inverse images of the disconnected point by the _right_
branches (see Definition 2.1, Theorem 3.2 and Theorem 3.3 precisely). That is,
one needs to take the contracting effect by right branches into account.
Indeed, the induced (random) map/the first return (random) map for (random)
LSV maps satisfy uniformly expanding property (on average), whereas those for
our random piecewise convex maps do not in general. Hence we cannot expect so-
called spectral decomposition method from the Lasota–Yorke type inequality any
longer. We refer to [30] for similar arguments and some background.

###  1.1. Notation

Throughout the paper, all sets and functions mentioned are measurable and any
difference on null sets with respect to a measure under consideration is
ignored. As usual, $L^{1}(X,\lambda)$, for a set $X$ with measurable structure
and a measure $\lambda$ over $X$, stands for the set of all
$\lambda$-integrable functions over $X$ where functions differing only on
$\lambda$-null sets are identified. For a measurable set $A$, $1_{A}$ always
denotes the indicator function on $A$.

Let $\\{a_{n}\\}_{n=1}^{\infty},\\{b_{n}\\}_{n=1}^{\infty}\subset\mathbb{R}$
be positive sequences. The notation $a_{n}\sim b_{n}$ is explained below
(1.2). For the notational convention, we further use $a_{n}\gtrapprox b_{n}$,
equivalently $b_{n}\lessapprox a_{n}$, by meaning that there exists a constant
$M>0$ independent of $n\in\mathbb{N}$ such that $a_{n}\geq Mb_{n}$ holds.
$a_{n}\approx b_{n}$ is used when $a_{n}\gtrapprox b_{n}$ and
$a_{n}\lessapprox b_{n}$ hold.

###  1.2. Organization

The present paper is organized as follows. In §2, we give necessary
preliminaries and define random piecewise convex maps. §3 is devoted to our
main result. We establish in Theorem 3.1 the existence of Lebesgue-equivalent,
conservative and ergodic $\sigma$-finite invariant measures for random
piecewise convex maps. Theorem 3.2 and Theorem 3.3 show the asymptotics of the
invariant measures given in Theorem 3.1. We illustrate in §4 several examples
of random piecewise convex maps. §4 also provides some counterexample which
possesses an infinite derivative.

##  2\. The model of random piecewise convex maps

In this section, we define random picewise convex maps. Let $X=[0,1]$ be the
unit interval equipped with the Lebsgue measure $\lambda$ over the Borel
$\sigma$-algebra $\mathcal{B}$ and let $\mathbb{A}$ and $\mathbb{B}$ be some
parameter regions with some measurable structure (usually they are subspaces
of $\mathbb{N}$ or $\mathbb{R}$). For each $\alpha\in\mathbb{A}$ and
$\beta\in\mathbb{B}$, we define a non-singular maps $T_{\alpha,\beta}$, i.e.,
$\lambda(T_{\alpha,\beta}^{-1}N)=0$ whenever $\lambda(N)$=0, on $X$ by

$T_{\alpha,\beta}x=\begin{cases}\tau_{\alpha}(x)&x\in[0,\frac{1}{2}],\\\
S_{\beta}(x)&x\in(\frac{1}{2},1]\end{cases}$

where $\tau_{\alpha}:[0,\frac{1}{2}]\to X$ and $S_{\beta}:(\frac{1}{2},1]\to
X$ for $\alpha\in\mathbb{A}$ and $\beta\in\mathbb{B}$ are injective and
continuous maps with some conditions (see the conditions ( ‣ 2)–(2), (A) and
(B) precisely). The standing assumption on
$\\{\tau_{\alpha}:\alpha\in\mathbb{A}\\}$ and
$\\{S_{\beta}:\beta\in\mathbb{B}\\}$ is the following:

  1. (0)

The map $T:\mathbb{A}\times\mathbb{B}\times X\to X$; $(\alpha,\beta,x)\mapsto
T_{\alpha,\beta}(x)$ is measurable with respect to each variable.

Note that the above condition ( ‣ 2) is fulfilled if $\mathbb{A}$ and
$\mathbb{B}$ are topological spaces endowed with their Borel structures and
the maps $\mathbb{A}\ni\alpha\mapsto\tau_{\alpha}$ and
$\mathbb{B}\ni\beta\mapsto S_{\beta}$ are continuous.

Our random dynamical systems are defined as random compositions of maps
$\\{T_{\alpha,\beta}:\alpha\in\mathbb{A},\ \beta\in\mathbb{B}\\}$ with the
condition ( ‣ 2) in the annealed sense. In order to define our random
dynamical systems, we set probability measures $\nu_{\mathbb{A}}$ and
$\nu_{\mathbb{B}}$ on $\mathbb{A}$ and $\mathbb{B}$, respectively.
$\nu_{\mathbb{A}}^{\infty}$ denotes the infinite product of the probability
measure of $\nu_{\mathbb{A}}$ over $\mathbb{A}^{\mathbb{N}}$. Then, for the
family of maps $\\{T_{\alpha,\beta}:\alpha\in\mathbb{A},\
\beta\in\mathbb{B}\\}$ and probability measures $\nu_{\mathbb{A}}$ and
$\nu_{\mathbb{B}}$ over the parameter spaces $\mathbb{A}$ and $\mathbb{B}$, we
consider the following transition function

$\displaystyle\mathbb{P}\left(x,A\right)=\int_{\mathbb{A}\times\mathbb{B}}1_{A}\left(T_{\alpha,\beta}x\right)d\nu_{\mathbb{A}}(\alpha)d\nu_{\mathbb{B}}(\beta)$
(2.1)

for each $A\in\mathcal{B}$ and $\lambda$-almost every $x\in X$. By the
condition ( ‣ 2) and non-singularity of each $T_{\alpha,\beta}$ with respect
to $\lambda$, it is straightforward to see that this transition function is
null-preserving, i.e., $\lambda(N)=0$ implies $\mathbb{P}(x,N)=0$ for
$\lambda$-almost every $x\in X$. Thus, we can define the corresponding Markov
operator $P:L^{1}(X,\lambda)\to L^{1}(X,\lambda)$ (i.e., $Pf\geq 0$ and
$\lVert Pf\rVert_{L^{1}}=\lVert f\rVert_{L^{1}}$ for each $f\in
L^{1}(X,\lambda)$ non-negative) by

$\int_{A}Pfd\lambda=\int_{X}f\cdot\mathbb{P}(\,\cdot\,,A)d\lambda$

for each $f\in L^{1}(X,\lambda)$ and $A\in\mathcal{B}$. The adjoint operator
of $P$ acting on $L^{\infty}(X,\lambda)$ is denoted by $P^{*}$ which is
characterized by

$\int_{X}Pf\cdot gd\lambda=\int_{X}f\cdot P^{*}gd\lambda$

for each $f\in L^{1}(X,\lambda)$ and $g\in L^{\infty}(X,\lambda)$.

In order to make more precise assumptions on random piecewise convex maps, we
introduce some notations. As in the previous section (note that
$\tau_{\alpha}$ is not necessarily the same as the LSV map
$T_{\alpha}|_{[0,\frac{1}{2}]}$ from §1), for
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\dots)\in\mathbb{A}^{\mathbb{N}}$, let
$x_{1}^{\bm{\alpha}}=x_{1}\coloneqq\frac{1}{2}$ and
$x_{n+1}^{\bm{\alpha}}\coloneqq\tau_{\alpha_{n-1}}^{-1}\circ\cdots\circ\tau_{\alpha_{1}}^{-1}(x_{1}^{\bm{\alpha}})$
for $n\geq 1$. For simplicity, let $x_{0}^{\bm{\alpha}}=x_{0}\coloneqq 1$ and
set $X_{n}^{\bm{\alpha}}\coloneqq(x_{n+1}^{\bm{\alpha}},x_{n}^{\bm{\alpha}}]$
for $\bm{\alpha}\in\mathbb{A}^{\mathbb{N}}$ and $n\geq 0$.

For considering the inverses by the right branch of $x_{n}^{\bm{\alpha}}$'s as
well, we need the following definition:

######  Definition 2.1.

$\eta$ is defined by a map from $\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$ to
$\mathbb{N}\cup\\{0\\}$ satisfying $S_{\beta}(1)\in
X_{\eta(\bm{\alpha},\beta)}^{\bm{\alpha}}$.

We always assume that $\eta$ is measurable as a standing hypothesis. Then, for
$\bm{\alpha}\in\mathbb{A}$ and $\beta\in\mathbb{B}$, let
$y_{1}^{\bm{\alpha},\beta}=y_{1}\coloneqq 1$ and $y_{n+1}^{\bm{\alpha},\beta}$
be the inverse of $x_{\eta(\bm{\alpha},\beta)+n}^{\bm{\alpha}}$ by the right
branch of $T_{\alpha,\beta}$, namely,

$y_{n+1}^{\bm{\alpha},\beta}\coloneqq
S_{\beta}^{-1}\left(x_{\eta(\bm{\alpha},\beta)+n}^{\bm{\alpha}}\right)\quad\text{for
$n\geq 1$.}$

We set
$Y_{n}^{\bm{\alpha},\beta}\coloneqq(y_{n+1}^{\bm{\alpha},\beta},y_{n}^{\bm{\alpha},\beta}]$
for $n\geq 1$ and $Y\coloneqq[\frac{1}{2},1]$.

Throughout the paper we assume, together with the condition ( ‣ 2), that a
family of maps $\\{T_{\alpha,\beta}:\alpha\in\mathbb{A},\
\beta\in\mathbb{B}\\}$ satisfies the following conditions (piecewise convex
property, see also Figure 1): for $\nu_{\mathbb{A}}$-almost every
$\alpha\in\mathbb{A}$ and $\nu_{\mathbb{B}}$-almost every
$\beta\in\mathbb{B}$,

  1. (1)

$\tau_{\alpha}$ and $S_{\beta}$ are $C^{1}$-functions and $S_{\beta}$ can be
extended to a continuous function on $Y$ (the extension is also denoted by the
same symbol $S_{\beta}$) with $\tau_{\alpha}(0)=0$,
$\tau_{\alpha}(\frac{1}{2})=1$ and $S_{\beta}(\frac{1}{2})=0$;

  2. (2)

$\tau_{\alpha}^{\prime}$ and $S_{\beta}^{\prime}$ are non-decreasing on
$(0,\frac{1}{2})$ and $(\frac{1}{2},1)$, respectively, with
$\tau_{\alpha}^{\prime}(0)\geq 1$, $\tau_{\alpha}^{\prime}(x)>1$ for
$x\in(0,\frac{1}{2})$, $S_{\beta}^{\prime}(\frac{1}{2})\geq 0$ and
$S_{\beta}^{\prime}(x)>0$ for $x\in(\frac{1}{2},1)$, where
$\tau_{\alpha}^{\prime}(0)$ and $S_{\beta}^{\prime}(\frac{1}{2})$ are taken as
the right derivatives.

By our assumptions (1) and (2), for $\nu_{\mathbb{A}}^{\infty}$-almost every
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\dots)\in\mathbb{A}^{\mathbb{N}}$ and
$\nu_{\mathbb{B}}$-almost every $\beta\in\mathbb{B}$, we have
$T_{\alpha_{n},\beta}X_{n}^{\bm{\alpha}}=X_{n-1}^{\bm{\alpha}}$ and
$T_{\alpha,\beta}Y_{n+1}^{\bm{\alpha},\beta}=X_{\eta(\bm{\alpha},\beta)+n}^{\bm{\alpha}}$
for any $n\geq 1$ and
$T_{\alpha_{0},\beta}Y_{1}^{\bm{\alpha},\beta}=(x_{\eta(\bm{\alpha},\beta)+1}^{\bm{\alpha}},S_{\beta}(1)]\subset
X_{\eta(\bm{\alpha},\beta)}^{\bm{\alpha}}$, where $\alpha_{0}\in\mathbb{A}$ is
arbitrary.

######  Remark 2.1.

(I) The phase space $X$ is of course not necessarily $[0,1]$ but just needs to
be a bounded interval in $\mathbb{R}$. Similarly, the choice of the
discontinuous point $\frac{1}{2}$ is just for simplicity, that is, we can take
an arbitrary $c\in(0,1)$ instead of $\frac{1}{2}$ so that $\tau_{\alpha}$ and
$S_{\beta}$ are defined on $[0,c]$ and $(c,1]$ respectively. Other similar
generalizations, such as increasing the number of partitions to more than two
or the case when $\tau_{\alpha}$'s are not surjective too, may be considered
without big difficulty. For instance, if we decompose $X$ into
$\\{X_{i}\\}_{i=0}^{n}$ with $X_{i}=[a_{i},a_{i+1})$ with
$0=a_{0}<a_{1}<a_{2}<\cdots<a_{n}<a_{n+1}=1$ for some $n\geq 2$, where maps on
$X_{0}$ satisfy the conditions on $\tau_{\alpha}$ and maps on $X_{i}$ for
$i=1,\dots,n$ satisfy the conditions on $S_{\beta}$ from (1) and (2), then the
strongest contracting property in $\\{X_{i}\\}_{i=1}^{n}$ would dominate the
statistical laws of the random system.

(II) In the condition (1), the assumption that $\tau_{\alpha}$ and $S_{\beta}$
are $C^{1}$ can be relaxed to the following condition: there are families of
countable open subintervals $\\{I_{n}^{L}\\}_{n}$ and $\\{I_{n}^{R}\\}_{n}$,
with the closure of their union being $X$, such that, for
$\nu_{\mathbb{A}}$-almost every $\alpha\in\mathbb{A}$ and
$\nu_{\mathbb{B}}$-almost every $\beta\in\mathbb{B}$, $\tau_{\alpha}$ and
$S_{\beta}$ are $C^{1}$ on $I_{n}^{L}$ and $I_{n}^{R}$, respectively for each
$n$. Hence some (but not all) examples from [30] are also in sight of the
present paper.

(III) In the above conditions (1) and (2), we do not exclude
$\tau_{\alpha}^{\prime}(0)=1$ nor $S_{\beta}^{\prime}(\frac{1}{2})=0$ for
$\alpha\in\mathbb{A}$ and $\beta\in\mathbb{B}$. Furthermore, we will consider
a random map with the common indifferent fixed point and the common flat point
i.e., $\tau_{\alpha}^{\prime}(0)=1$ and $S_{\beta}^{(n)}(\frac{1}{2})=0$ for
any $\alpha\in\mathbb{A}$, $\beta\in\mathbb{B}$ and $n\geq 1$ (see Example 4.4
and 4.5).

Figure 1. A graph of a possible $T_{\alpha,\beta}$: in this case
$\eta(\bm{\alpha},\beta)=1$ where $\bm{\alpha}=(\alpha,\alpha,\dots)$.

Recall that for a Markov operator $P$, a measure $\mu$ over $(X,\mathcal{B})$
is called an _absolutely continuous_ (resp.  _equivalent_) _$\sigma$ -finite
invariant measure_ if $\mu$ is a $\sigma$-finite measure which is absolutely
continuous (resp. equivalent) with respect to $\lambda$ and its Radon–Nikodým
derivative $\frac{d\mu}{d\lambda}$ is a (non-trivial) fixed point of $P$. Note
that by positivity of Markov operators the domain of any Markov operator can
be naturally extended to the set of non-negative and locally integrable
functions and hence the definition of absolutely continuous $\sigma$-finite
_infinite_ invariant measures makes sense.

We then consider the following technical conditions on our random dynamical
systems, which are important in establishing the existence of equivalent
$\sigma$-finite invariant measures.

  1. (A)

$\displaystyle\operatorname*{ess\,sup}_{\alpha\in\mathbb{A}}\tau^{\prime}_{\alpha}\left(\tfrac{1}{2}\right)<\infty$;

  2. (B)

$\displaystyle\int_{\mathbb{A}}\frac{1}{x_{1}-x_{2}^{\bm{\alpha}}}d\nu_{\mathbb{A}}(\alpha)<\infty$.

Obviously the condition (A) implies the condition (B).

######  Lemma 2.1.

Under the assumption ( ‣ 2)–(2), the condition (B) implies the following: for
any $\delta>0$, there exists $N_{0}\geq 1$ such that

$\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\frac{y_{N_{0}+1}^{\bm{\alpha},\beta}-\frac{1}{2}}{x_{1}^{\bm{\alpha}}-x_{2}^{\bm{\alpha}}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})\nu_{\mathbb{B}}(\beta)<\delta.$

###### Proof.

It follows from (1) and (2) that $y_{N+1}^{\bm{\alpha},\beta}-\frac{1}{2}\to
0$ as $N\to\infty$ for $\nu_{\mathbb{A}}^{\infty}$-almost every
$\bm{\alpha}\in\mathbb{A}^{\mathbb{N}}$ and $\nu_{\mathbb{B}}$-almost every
$\beta\in\mathbb{B}$. Then we have

$\displaystyle\lim_{N\to\infty}\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\frac{y_{N+1}^{\bm{\alpha},\beta}-\frac{1}{2}}{x_{1}^{\bm{\alpha}}-x_{2}^{\bm{\alpha}}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})\nu_{\mathbb{B}}(\beta)$
$\displaystyle=\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\lim_{N\to\infty}\frac{y_{N+1}^{\bm{\alpha},\beta}-\frac{1}{2}}{x_{1}^{\bm{\alpha}}-x_{2}^{\bm{\alpha}}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})\nu_{\mathbb{B}}(\beta)$
$\displaystyle=0$

from the Lebesgue dominated convergence theorem, which proves the lemma. ∎

In what follows,
$\\{T_{\alpha,\beta};\nu_{\mathbb{A}},\nu_{\mathbb{B}}:\alpha\in\mathbb{A},\
\beta\in\mathbb{B}\\}$ denotes a random dynamical system given by the
transition function (2.1) with conditions ( ‣ 2)–(2) and is referred as a
_random piecewise convex map_. In the next section, we prove the existence and
uniqueness of equivalent $\sigma$-finite invariant measures for random
piecewise convex maps. Furthermore, we show the asymptotics of the invariant
measures.

##  3\. Equivalent $\sigma$-finite invariant measures

Before stating the main results in the paper, some basic definitions are
listed. Let $\mu$ be an absolutely continuous measure with respect to
$\lambda$. Recall that an _invariant set_ for a Markov operator $P$ is a
measurable set $E\in\mathcal{B}$ with the property $P^{*}1_{E}=1_{E}$
$\lambda$-almost everywhere and $P$ is called _ergodic_ with respect to $\mu$
if each invariant set $E$ satisfies either $\mu(E)=0$ or $\mu(X\setminus
E)=0$. $P$ is called _conservative_ with respect to $\mu$ if any function $h$
supported on $\operatorname*{supp}\mu$ with $h\geq P^{*}h$ satisfies
$h=P^{*}h$. Other equivalent characterization of conservativity are found in
[22]*§3.1.

The following main theorem establishes the existence of Lebesgue-equivalent,
conservative and ergodic $\sigma$-finite invariant measures for random
piecewise convex maps defined in the previous section.

######  Theorem 3.1.

Let
$\\{T_{\alpha,\beta};\nu_{\mathbb{A}},\nu_{\mathbb{B}}:\alpha\in\mathbb{A},\
\beta\in\mathbb{B}\\}$ be a random piecewise convex map satisfying the
conditions ( ‣ 2)–(2) and (B) in §2. Then, for the random piecewise convex
map, there exists a conservative and ergodic $\sigma$-finite invariant measure
$\mu$ which is equivalent to the Lebesgue measure $\lambda$. Moreover, the
invariant density function of $\mu$, $\frac{d\mu}{d\lambda}$, satisfies

  1. (D)

$\frac{d\mu}{d\lambda}$ restricted on $(0,\frac{1}{2})$ is non-increasing
$\lambda$-almost everywhere, and

  2. (U)

for any $0<\varepsilon<\frac{1}{2}$, there is a constant $C=C(\varepsilon)>0$
such that $\frac{d\mu}{d\lambda}\leq C$, $\lambda$-almost everywhere on
$X\setminus[0,\varepsilon)$.

If we suppose (A) (and hence (B) is automatically fulfilled), then it also
holds

  1. (L)

there is a constant $c>0$ such that $\frac{d\mu}{d\lambda}\geq c$,
$\lambda$-almost everywhere on $X$.

######  Remark 3.1.

(I) Since an equivalent $\sigma$-finite measure in Theorem 3.1 is conservative
and ergodic, it is unique (up to a multiplicative constant) by [12]*Theorem A
in Chapter VI.

(II) When we do not suppose the condition (A), there does no longer exist
lower bound for $\frac{d\mu}{d\lambda}$, namely the condition (L), in general.
See Example 4.7 for a counterexample.

We further state two theorems of Theorem 3.1 which tell us when the invariant
measure becomes an infinite measure. The first one deals with a specific case
when $\mathbb{A}$ is a point set, from which we can show the general case as
in Theorem 3.3.

######  Theorem 3.2.

Let
$\\{T_{\alpha,\beta};\nu_{\mathbb{A}},\nu_{\mathbb{B}}:\alpha\in\mathbb{A},\
\beta\in\mathbb{B}\\}$ be as in Theorem 3.1 and assume (A). Suppose
$\mathbb{A}=\\{\alpha\\}$ is a singleton, and set $X_{n}=X_{n}^{\bm{\alpha}}$
and $\eta(\beta)=\eta(\bm{\alpha},\beta)$ where
$\bm{\alpha}=(\alpha,\alpha,\dots)$. Then the asymptotics of the invariant
measure $\mu$ given in Theorem 3.1 is of order

$\mu\left(X_{n}\right)\approx\int_{\\{\beta\in\mathbb{B}:\eta(\beta)<n\\}}\left(y_{n-\eta(\beta)}^{\beta}-\tfrac{1}{2}\right)d\nu_{\mathbb{B}}(\beta)+\nu_{\mathbb{B}}\left\\{\beta\in\mathbb{B}:\eta(\beta)\geq
n\right\\}$

for $n$ large enough.

######  Remark 3.2.

In Theorem 3.2, if $S_{\beta}$ is surjective for $\nu_{\mathbb{B}}$-almost
every $\beta\in\mathbb{B}$ then by the definition of $\eta$ we have
$\eta(\beta)=0$ and the invariant measure $\mu$ is of order

$\mu\left(X_{n}\right)\approx\int_{\mathbb{B}}\left(y_{n+1}^{\beta}-\tfrac{1}{2}\right)d\nu_{\mathbb{B}}(\beta).$

Simultaneously, the second term
$\nu_{\mathbb{B}}\\{\beta\in\mathbb{B}:\eta(\beta)\geq n\\}$ is negligible
when $\operatorname*{ess\,inf}_{\beta\in\mathbb{B}}S_{\beta}(1)>0$ and hence
$\operatorname*{ess\,sup}_{\beta\in\mathbb{B}}\eta(\beta)<\infty$ (e.g., when
$\\#\mathbb{B}<\infty$).

When $\mathbb{A}$ is an uncountable set, the form of the invariant density is
complicated in general. However, combining Theorem 3.2 and the comparison
theorem from [19], we can estimate the size of the $\sigma$-finite invariant
measure $\mu$ in Theorem 3.1 even when $\mathbb{A}$ is not singleton, by
reducing to the case of singleton. In order to clarify our statement, we need
to introduce the following condition. A random piecewise convex map
$\\{T_{\alpha,\beta};\nu_{\mathbb{A}},\nu_{\mathbb{B}}:\alpha\in\mathbb{A},\
\beta\in\mathbb{B}\\}$ is said to satisfy the condition ($\dagger$) if there
are some $c\in(0,\frac{1}{2})$ and $\alpha_{1},\alpha_{2}\in\mathbb{A}$ such
that

$\displaystyle\nu_{\mathbb{A}}\left\\{\alpha\in\mathbb{A}:\tau_{\alpha}(0,\varepsilon)\subset\tau_{\alpha_{1}}(0,\varepsilon)\text{
for any }\varepsilon\in\left(0,c\right)\right\\}=1\text{ and}$
$\displaystyle\nu_{\mathbb{A}}\left\\{\alpha\in\mathbb{A}:\tau_{\alpha}(0,\varepsilon)\supset\tau_{\alpha_{2}}(0,\varepsilon)\text{
for any }\varepsilon\in\left(0,c\right)\right\\}>0.$

These conditions are of course equivalent to that

$\displaystyle\nu_{\mathbb{A}}\left\\{\alpha\in\mathbb{A}:\tau_{\alpha}\leq\tau_{\alpha_{1}}\text{
on }\left(0,c\right)\right\\}=1\text{ and}$
$\displaystyle\nu_{\mathbb{A}}\left\\{\alpha\in\mathbb{A}:\tau_{\alpha}\geq\tau_{\alpha_{2}}\text{
on }\left(0,c\right)\right\\}>0.$

With some abuse of notation, for a fixed $\bar{\alpha}\in\mathbb{A}$,
$\\{T_{\bar{\alpha},\beta};\nu_{\mathbb{B}}:\beta\in\mathbb{B}\\}$ denotes a
random piecewise convex map
$\\{T_{\alpha,\beta};\nu_{\bar{\alpha}},\nu_{\mathbb{B}}:\alpha\in\\{\bar{\alpha}\\},\beta\in\mathbb{B}\\}$
where $\nu_{\bar{\alpha}}$ is the Dirac measure on $\bar{\alpha}$.

######  Theorem 3.3.

Let
$\\{T_{\alpha,\beta};\nu_{\mathbb{A}},\nu_{\mathbb{B}}:\alpha\in\mathbb{A},\
\beta\in\mathbb{B}\\}$ and $\mu$ be as in Theorem 3.1 with the assumption (A)
and satisfy the condition $(\dagger)$ with some
$\alpha_{1},\alpha_{2}\in\mathbb{A}$. Let $\mu_{i}$'s be $\sigma$-finite
invariant measures for random piecewise convex maps
$\\{T_{\alpha_{i},\beta};\nu_{\mathbb{B}}:\beta\in\mathbb{B}\\}$ $(i=1,2)$
given in Theorem 3.2. Then there is a constant $M>0$ such that for any $a$ and
$b$ with $0\leq a<b\leq\frac{1}{2}$

$M^{-1}\mu_{1}\left([a,b]\right)\leq\mu\left([a,b]\right)\leq
M\mu_{2}\left([a,b]\right).$

Consequently, if $\mu_{1}(X)=\infty$ then $\mu(X)=\infty$, and if
$\mu_{2}(X)<\infty$ then $\mu(X)<\infty$.

######  Remark 3.3.

(I) In Theorem 3.3, $\alpha_{1}$ is chosen to be a parameter for which
$\tau_{\alpha_{1}}$ dominates any other $\tau_{\alpha}$ for
$\alpha\in\mathbb{A}$ from above and $\alpha_{2}$ should be chosen in the way
that $\tau_{\alpha_{2}}$ is close to $\tau_{\alpha_{1}}$ as much as possible
so that the inequality becomes sharper. For instance, see Example 4.2 for the
choice of parameters.

(II) If $\\#\mathbb{A}<\infty$ and $\nu_{\mathbb{A}}(\alpha)>0$ for all
$\alpha\in\mathbb{A}$, then one can have $\alpha_{1}=\alpha_{2}$ in Theorem
3.3. Similarly, if there is a parameter $\alpha^{\prime}\in\mathbb{A}$ such
that $\tau_{\alpha}\leq\tau_{\alpha^{\prime}}$ on $(0,\frac{1}{2})$ for
$\nu_{\mathbb{A}}$-almost every $\alpha\in\mathbb{A}$ and
$\nu_{\mathbb{A}}(\\{\alpha^{\prime}\\})>0$, then both $\alpha_{1}$ and
$\alpha_{2}$ in Theorem 3.3 can be taken as $\alpha^{\prime}$. That is, the
invariant measure $\mu$ in Theorem 3.3 is of same order of
$\mu_{\alpha^{\prime}}$, where $\mu_{\alpha^{\prime}}$ is the invariant
measure for
$\\{T_{\alpha^{\prime},\beta};\nu_{\mathbb{B}}:\beta\in\mathbb{B}\\}$.

Before proving Theorem 3.1, we recall the key tool, called the induced
operator (or the first return map in the sense of [18]), to construct an
absolutely continuous $\sigma$-finite invariant measure and we also prepare
lemmas.

As in the previous section, we let $Y=[\frac{1}{2},1]$ and recall (see also
[13, 29]) that the induced operator (on $Y$) $P_{Y}$ is defined by

$\displaystyle P_{Y}=I_{Y}P\sum_{n=0}^{\infty}\left(I_{Y^{c}}P\right)^{n}$
(3.1)

where $I_{Y}$ and $I_{Y^{c}}$ are the restriction operators on $Y$ (i.e.,
$I_{Y}f=1_{Y}f$ for each measurable function $f$) and $Y^{c}$, respectively.
The operator $P_{Y}$ is a well-defined Markov operator over $L^{1}(X,\lambda)$
since $Y$ is a $P$-sweep-out set with respect to $\lambda$ (see Lemma 4.7 in
[29] precisely). The induced operator for a Markov operator is a
generalization of the induced map for a non-singular map.

For $(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$,
$\mathcal{L}_{T_{Y}^{(\bm{\alpha},\beta)}}$ denotes the Perron–Frobenius
operator associated with the induced (random) map
$T_{Y}^{(\bm{\alpha},\beta)}x\coloneqq\tau_{\alpha_{1}}\circ\cdots\circ\tau_{\alpha_{n(x)}}\circ
S_{\beta}x$ where $n(x)\geq 1$ is the minimum number satisfying
$\tau_{\alpha_{1}}\circ\cdots\circ\tau_{\alpha_{n(x)}}\circ S_{\beta}x\in Y$
(such $n(x)$ exists for $x\in Y\setminus\\{\frac{1}{2}\\}$).

######  Lemma 3.1.

The induced operator $P_{Y}$ defined by the equation (3.1) satisfies

$P_{Y}f=\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\mathcal{L}_{T_{Y}^{(\bm{\alpha},\beta)}}fd\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})\nu_{\mathbb{B}}(\beta)$

for each $f\in L^{1}(Y,\lambda)$ with $f=0$ $\lambda$-almost everywhere on
$Y^{c}$.

###### Proof.

As in the equality (3.1), the induced operator on $Y$ and its adjoint operator
are defined by

$P_{Y}=I_{Y}P\sum_{n=0}^{\infty}(I_{Y^{C}}P)^{n}\quad\text{and}\quad
P^{*}_{Y}=\sum_{n=0}^{\infty}\left({P^{*}}I_{Y^{c}}\right)^{n}\left(P^{*}I_{Y}\right).$

On the other hand, by Proposition 4.1 (iv) of [18], $P^{*}_{Y}1_{A}(x)$ equals
to the transition function from $x$ into $A$ which defines the induced map on
$Y$ of the original random map. ∎

We then prove the following key lemma.

######  Lemma 3.2.

Suppose the condition (B). If $f$ is non-negative, bounded and non-increasing
on $Y$ and satisfies $f=0$ $\lambda$-almost everywhere on $Y^{c}$, then so is
$P_{Y}f$. Moreover, if $\lVert f\rVert_{L^{1}}\leq 1$ then there is some
positive constant $K>0$, independent of $f$, such that for any
$\delta\in(0,1)$ and $\lambda$-almost every $x\in Y$,

$\displaystyle P_{Y}f(x)<\delta f\left(\tfrac{1}{2}\right)+K.$ (3.2)

###### Proof.

We follow the proof of Proposition 5.1 in [16]. Let
$X_{n}^{\bm{\alpha}}\coloneqq(x_{n+1}^{\bm{\alpha}},x_{n}^{\bm{\alpha}}]$ and
$Y_{n}^{\bm{\alpha},\beta}\coloneqq(y_{n+1}^{\bm{\alpha},\beta},y_{n}^{\bm{\alpha},\beta}]$
as before. Then for each
$(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$, the induced
map $T_{Y}^{(\bm{\alpha},\beta)}$ is piecewise convex such that
$T_{Y}^{(\bm{\alpha},\beta)}|_{Y_{1}^{\bm{\alpha},\beta}}=\tau_{\alpha_{1}}\circ\cdots\circ\tau_{\alpha_{\eta(\bm{\alpha},\beta)}}\circ
S_{\beta}|_{Y_{1}^{\bm{\alpha},\beta}}$ maps from $Y_{1}^{\bm{\alpha},\beta}$
onto $[\frac{1}{2},T_{Y}^{(\bm{\alpha},\beta)}(1)]\subset Y$ and
$T_{Y}^{(\bm{\alpha},\beta)}|_{Y_{n+1}^{\bm{\alpha},\beta}}=\tau_{\alpha_{1}}\circ\cdots\circ\tau_{\eta(\bm{\alpha},\beta)+n}\circ
S_{\beta}|_{Y_{n+1}^{\bm{\alpha},\beta}}$ maps from
$Y_{n+1}^{\bm{\alpha},\beta}$ onto $Y$ for $n\geq 1$ by construction.

If we set

$\varphi_{1}^{(\bm{\alpha},\beta)}(x)=\begin{cases}\dfrac{1}{\left(T_{Y}^{(\bm{\alpha},\beta)}\right)^{\prime}\circ\left(T_{Y}^{(\bm{\alpha},\beta)}\big{|}_{Y_{1}^{\bm{\alpha},\beta}}\right)^{-1}(x)}&\left(x\in
T_{Y}^{(\bm{\alpha},\beta)}\left(Y_{1}^{\bm{\alpha},\beta}\right)\right),\\\
0&\left(\text{otherwise}\right)\end{cases}$

and

$\varphi_{n+1}^{(\bm{\alpha},\beta)}(x)=\frac{1}{\left(T_{Y}^{(\bm{\alpha},\beta)}\right)^{\prime}\circ\left(T_{Y}^{(\bm{\alpha},\beta)}\big{|}_{Y_{n+1}^{\bm{\alpha},\beta}}\right)^{-1}(x)}\quad(x\in
X)$

for $(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$ and $n\geq
1$, then $\varphi_{n}^{(\bm{\alpha},\beta)}$ is non-increasing on $Y$ for each
$(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$ and $n\geq 1$.
Since for any non-negative and non-increasing function $f$ on $Y$ we have

$\displaystyle P_{Y}f$
$\displaystyle=\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\Bigg{(}\sum_{n\geq
2}\varphi_{n}^{(\bm{\alpha},\beta)}f\circ\left(T_{Y}^{(\bm{\alpha},\beta)}\big{|}_{Y_{n}^{\bm{\alpha},\beta}}\right)^{-1}$
$\displaystyle\qquad\qquad\qquad+\varphi_{1}^{(\bm{\alpha},\beta)}f\circ\left(T_{Y}^{(\bm{\alpha},\beta)}\big{|}_{Y_{1}^{\bm{\alpha},\beta}}\right)^{-1}1_{T_{Y}^{(\bm{\alpha},\beta)}\left(Y_{1}^{\bm{\alpha},\beta}\right)}\Bigg{)}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})\nu_{\mathbb{B}}(\beta)$

from Lemma 3.1, $P_{Y}f$ is also non-negative and non-increasing and the
former part of the lemma is proven.

Now from the convexity of $T_{\alpha,\beta}$ we can easily see that

$\displaystyle
T_{\alpha_{\eta(\bm{\alpha},\beta)+n},\beta}^{\prime}\big{|}_{Y_{n}^{\bm{\alpha},\beta}}\geq\frac{\lambda\left(X_{\eta(\bm{\alpha},\beta)+n}^{\bm{\alpha}}\right)}{\lambda\left(Y_{n+1}^{\bm{\alpha},\beta}\right)}\quad\text{and}\quad
T_{\alpha_{n},\beta}^{\prime}\big{|}_{X_{n}^{\bm{\alpha}}}\geq\frac{\lambda\left(X_{n}^{\bm{\alpha}}\right)}{\lambda\left(X_{n+1}^{\bm{\alpha}}\right)}$
(3.3)

for any $(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$ and
$n\geq 1$. Thus it follows from (3.3) that for any
$(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$ and $n\geq 1$

$\displaystyle\left(T_{Y}^{(\bm{\alpha},\beta)}\right)^{\prime}\bigg{|}_{Y_{n}^{\bm{\alpha},\beta}}$
$\displaystyle=T^{\prime}_{\alpha_{\eta(\bm{\alpha},\beta)+n},\beta}\big{|}_{Y_{n}^{\bm{\alpha},\beta}}\prod_{k=1}^{\eta(\bm{\alpha},\beta)+n-1}T^{\prime}_{\alpha_{\eta(\bm{\alpha},\beta)+n-k},\beta}\big{|}_{X_{\eta(\bm{\alpha},\beta)+n-k}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\circ
T_{\alpha_{\eta(\bm{\alpha},\beta)+n-k+1},\beta}\circ\cdots\circ
T_{\alpha_{\eta(\bm{\alpha},\beta)+n-1},\beta}$
$\displaystyle\geq\frac{\lambda\left(X_{1}^{\bm{\alpha}}\right)}{\lambda\left(Y_{n+1}^{\bm{\alpha},\beta}\right)}.$

Then it holds for each $N\geq 1$ that

$\displaystyle\sum_{n=N}^{\infty}\varphi_{n}^{(\bm{\alpha},\beta)}\left(\tfrac{1}{2}\right)$
$\displaystyle=\sum_{n=N}^{\infty}\frac{1}{\left(T_{Y}^{(\bm{\alpha},\beta)}\right)^{\prime}\left(y_{n+1}^{\bm{\alpha},\beta}\right)}$
$\displaystyle\leq\sum_{n=N}^{\infty}\frac{\lambda\left(Y_{n+1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{1}^{\bm{\alpha}}\right)}$
$\displaystyle=\frac{y_{N+1}^{\bm{\alpha},\beta}-\frac{1}{2}}{x_{1}^{\bm{\alpha}}-x_{2}^{\bm{\alpha}}}.$

By Lemma 2.1, for any fixed $0<\delta<1$ there exists $N_{0}\geq 1$ such that
we have

$\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\sum_{n=N_{0}}^{\infty}\varphi_{n}^{(\bm{\alpha},\beta)}\left(\tfrac{1}{2}\right)d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})\nu_{\mathbb{B}}(\beta)<\delta.$

Since any non-increasing density function on $Y$ cannot exceed
$(x-\frac{1}{2})^{-1}$ (see also [23, Step III in Proof of Theorem 4]), it
holds that, for any non-negative, bounded and non-increasing function $f$ on
$Y$ with $\lVert f\rVert_{L^{1}}\leq 1$,

$\displaystyle P_{Y}f(x)$ $\displaystyle\leq P_{Y}f\left(\tfrac{1}{2}\right)$
$\displaystyle=\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\left(\sum_{n=N_{0}}^{\infty}\varphi_{n}^{(\bm{\alpha},\beta)}\left(\tfrac{1}{2}\right)f\left(y_{n+1}^{\bm{\alpha},\beta}\right)+\sum_{n=1}^{N_{0}-1}\varphi_{n}^{(\bm{\alpha},\beta)}\left(\tfrac{1}{2}\right)f\left(y_{n+1}^{\bm{\alpha},\beta}\right)\right)d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)$
$\displaystyle<\delta
f\left(\tfrac{1}{2}\right)+\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\sum_{n=1}^{N_{0}-1}\frac{\varphi_{n}^{(\bm{\alpha},\beta)}\left(\frac{1}{2}\right)}{y_{n+1}^{\bm{\alpha},\beta}-\tfrac{1}{2}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)$

for $\lambda$-almost every $x\in Y$. Therefore, putting
$K=\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\sum_{n=1}^{N_{0}-1}\frac{\varphi_{n}^{(\bm{\alpha},\beta)}(\frac{1}{2})}{y_{n+1}^{\bm{\alpha},\beta}-\frac{1}{2}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)<\infty$,
we have obtained the inequality (3.2). ∎

We now emphasize that the left branches $\tau_{\alpha}$'s map points
surjectively onto $[0,1]$. This together with the condition (B) guarantees
that an invariant density for the induced operator $P_{Y}$ is fully supported
on $Y$ as well as bounded above. Furthermore, (A) ensures the invariant
density to be bounded away from zero on $Y$. Henceforth $\lambda|_{Y}$ denotes
the measure $\lambda$ restricted on $Y$.

######  Lemma 3.3.

Under the assumption (B), the induced operator $P_{Y}$ is ergodic with respect
to the Lebesgue measure $\lambda$ and admits a unique
$\lambda|_{Y}$-equivalent invariant probability measure whose density is non-
increasing and bounded above on $Y$. Moreover, if we assume (A) then the
density function is bounded away from zero on $Y$.

###### Proof.

First of all, ergodicity follows from the following argument. For each
$(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}$, the map
$T_{Y}^{(\bm{\alpha},\beta)}$ satisfies the conditions in [16, Proposition
5.1] and is ergodic (or moreover exact) with respect to $\lambda|_{Y}$. If $D$
is an invariant set for $P_{Y}$, then

$P^{*}_{Y}1_{D}=\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}1_{D}\circ
T_{Y}^{(\bm{\alpha},\beta)}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)=1_{D}.$

Thus $D$ is a $T_{Y}^{(\bm{\alpha},\beta)}$-invariant set for
$\nu_{\mathbb{A}}^{\infty}\times\nu_{\mathbb{B}}$-almost everywhere. Now it is
straightforward to see that $D=\emptyset$ or $X\pmod{\lambda}$ since each
$T_{Y}^{(\bm{\alpha},\beta)}$ is ergodic.

From Lemma 3.2, $P_{Y}^{n}1_{Y}$ is non-increasing for $n\geq 0$ and we apply
(3.2) repeatedly to get for any fixed $\delta\in(0,1)$ and for $x\in Y$

$P_{Y}^{n}1_{Y}(x)<\delta
P_{Y}^{n-1}1_{Y}\left(\tfrac{1}{2}\right)+K<\delta^{2}P_{Y}^{n-2}1_{Y}\left(\tfrac{1}{2}\right)+\delta
K+K$

and so on. Eventually, we have for $n\geq 1$

$\displaystyle P_{Y}^{n}1_{Y}<\delta^{n}+\frac{K}{1-\delta},$ (3.4)

that is, $P_{Y}^{n}1_{Y}$ is bounded above by $C_{0}\coloneqq
1+K(1-\delta)^{-1}<\infty$ for any $n\geq 1$. Therefore, by [29, Theorem 3.1
and Proposition 3.9], the limiting point

$h_{0}\coloneqq\frac{1}{\lambda(Y)}\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}P_{Y}^{i}1_{Y}=2\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}P_{Y}^{i}1_{Y}$

exists and is an invariant density of $P_{Y}$, which is conservative and
ergodic implying uniqueness of the invariant density.

We then show this $h_{0}$ satisfies the conditions in the statement of the
lemma. From Lemma 3.2 and (3.4), $h_{0}$ is non-increasing and bounded above
by $C_{1}\coloneqq 2C_{0}$ on $Y$. For the lower bound, notice that from the
fact that $P_{Y}^{n}1_{Y}$ is non-increasing and the above inequality (3.4)

$P_{Y}^{n}1_{Y}\geq 1$

on $[\frac{1}{2},\frac{1}{2}+\frac{1}{C_{1}}]$ for $n\geq 1$ so that

$\displaystyle h_{0}\geq 2\quad\text{on
}\left[\tfrac{1}{2},\tfrac{1}{2}+\tfrac{1}{C_{1}}\right].$ (3.5)

On the other hand, it follows from the Lebsgue dominated convergence theorem
(see also the proof to Lemma 2.1), we get $N_{0}\geq 2$ such that

$\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\left(y_{N_{0}}^{\bm{\alpha},\beta}-\tfrac{1}{2}\right)d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)<\frac{1}{C_{1}}.$

We define

$E\coloneqq\left\\{(\bm{\alpha},\beta)\in\mathbb{A}^{\mathbb{N}}\times\mathbb{B}:\bigcup_{n=N_{0}}^{\infty}Y_{n}^{\bm{\alpha},\beta}\subset\left[\frac{1}{2},\frac{1}{2}+\frac{1}{C_{1}}\right]\right\\}.$

Then since it holds that

$y_{N_{0}}^{\bm{\alpha},\beta}-\frac{1}{2}=\lambda\left(\bigcup_{n=N_{0}}^{\infty}Y_{n}^{\bm{\alpha},\beta}\right),$

we have $\nu_{\mathbb{A}}^{\infty}\times\nu_{\mathbb{B}}(E)>0$. Combining
(3.5) with the above argument, we have

$h_{0}\geq 2\sum_{n=N_{0}}^{\infty}1_{Y_{n}^{\bm{\alpha},\beta}}\quad\text{for
any }(\bm{\alpha},\beta)\in E.$

Thus, taking $\varphi_{n}$ defined in the proof to Lemma 3.2 into account, we
have by Lemma 3.1

$\displaystyle h_{0}$
$\displaystyle=P_{Y}h_{0}=\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\mathcal{L}_{T_{Y}^{(\bm{\alpha},\beta)}}h_{0}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)$
$\displaystyle\geq
2\int_{E}\sum_{n=N_{0}}^{\infty}\mathcal{L}_{T_{Y}^{(\bm{\alpha},\beta)}}1_{Y_{n}^{(\bm{\alpha},\beta)}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)$
$\displaystyle=2\int_{E}\sum_{n=N_{0}}^{\infty}\varphi_{n}^{(\bm{\alpha},\beta)}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)$

on $Y$ since $T_{Y}^{(\bm{\alpha},\beta)}|_{Y_{n}^{(\bm{\alpha},\beta)}}$ is
surjective for each $n\geq 2$. The conditions (1) and (2) imply that

$C_{2}(x)\coloneqq
2\int_{E}\sum_{n=N_{0}}^{\infty}\varphi_{n}^{(\bm{\alpha},\beta)}(x)d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)>0$

$\lambda|_{Y}$-almost every $x\in Y$. Therefore, we conclude $C_{1}\geq
h_{0}(x)\geq C_{2}(x)$ on $Y$. Moreover, if we assume (A) then
$\operatorname*{ess\,inf}_{x\in Y}C_{2}(x)>0$. The proof is completed. ∎

###### Proof of Theorem 3.1.

The well-known formula of invariant measures via the induced operators (see
Proposition 4.14 in [29] for example) shows that

$h=\sum_{n=0}^{\infty}(I_{Y^{c}}P)^{n}h_{0}=h_{0}+I_{Y^{c}}\sum_{n=0}^{\infty}(PI_{Y^{c}})^{n}Ph_{0}.$

gives an invariant density function of an absolutely continuous
$\sigma$-finite invariant measure $\mu$ for $P$ where $h_{0}$ is the invariant
density of $P_{Y}$ obtained in Lemma 3.3. Then it follows from the fact that
$h_{0}$ is supported on $Y$ that

$\displaystyle
h=h_{0}+\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\sum_{n=0}^{\infty}I_{Y^{c}}\mathcal{L}_{\tau_{\alpha_{0}}}\cdots\mathcal{L}_{\tau_{\alpha_{n-1}}}\mathcal{L}_{S_{\beta}}h_{0}d\nu_{\mathbb{A}}^{\infty}(\alpha_{0},\alpha_{1},\dots)d\nu_{\mathbb{B}}(\beta).$
(3.6)

Since $\tau_{\alpha}$ for each $\alpha\in\mathbb{A}$ is surjective and the
support of $h_{0}$ is $Y$ up to $\lambda$-measure zero sets, $h$ is evidently
fully supported on $X$ and thus the invariant measure is equivalent to
$\lambda$. Now that $h_{0}$ is non-increasing and so is
$(PI_{Y^{c}})^{n}h_{0}$ for $n\geq 0$ from the similar argument of the proof
to Lemma 3.2 together with the assumption (2), we have (D). Then (U) follows
from (D) and the fact that $\mu$ is $\sigma$-finite.

If we assume (A), then we have $C^{-1}1_{Y}\leq h_{0}\leq C1_{Y}$ for some
$C\geq 1$ by Lemma 3.3. Or equivalently, for each
$\bm{\alpha}\in\mathbb{A}^{\mathbb{N}}$ and $\beta\in\mathbb{B}$

$C^{-1}\sum_{n=1}^{\infty}1_{Y_{n}^{\bm{\alpha},\beta}}\leq h_{0}\leq
C\sum_{n=1}^{\infty}1_{Y_{n}^{\bm{\alpha},\beta}}.$

Note that for the bound above (or the desired consequence (U)), we only need
the condition (B).

We first observe that for $n\geq 2$ and $\beta\in\mathbb{B}$,

$\displaystyle\mathcal{L}_{T_{\alpha,\beta}}1_{Y_{n}^{\bm{\alpha},\beta}}$
$\displaystyle=\frac{1_{Y_{n}^{\bm{\alpha},\beta}}\circ
S_{\beta}|_{Y_{n}^{\bm{\alpha},\beta}}^{-1}}{S_{\beta}^{\prime}\circ
S_{\beta}|_{Y_{n}^{\bm{\alpha},\beta}}^{-1}}\leq\dfrac{\lambda\left(Y_{n+1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{\eta(\bm{\alpha},\beta)+n}\right)}1_{X_{\eta(\bm{\alpha},\beta)+n-1}},\text{
and}$
$\displaystyle\mathcal{L}_{T_{\alpha,\beta}}1_{Y_{n}^{\bm{\alpha},\beta}}$
$\displaystyle\geq\dfrac{\lambda\left(Y_{n-1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{\eta(\bm{\alpha},\beta)+n-2}\right)}1_{X_{\eta(\bm{\alpha},\beta)+n-1}}$

by the convexity of $S_{\beta}$. Hence, taking it into account that

$\bigcup_{n=\eta(\bm{\alpha},\beta)+1}^{\infty}X_{n}^{\bm{\alpha}}\subset
S_{\beta}Y\subset\bigcup_{n=\eta(\bm{\alpha},\beta)}^{\infty}X_{n}^{\bm{\alpha}},$

for each $\bm{\alpha}\in\mathbb{A}^{\mathbb{N}}$ we have

$\displaystyle Ph_{0}$
$\displaystyle=\int_{\mathbb{B}}\mathcal{L}_{T_{\alpha,\beta}}h_{0}d\nu_{\mathbb{B}}(\beta)\leq
C\int_{\mathbb{B}}\sum_{n=1}^{\infty}\frac{\lambda\left(Y_{n+1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{\eta(\bm{\alpha},\beta)+n}^{\bm{\alpha}}\right)}1_{X_{\eta(\bm{\alpha},\beta)+n-1}^{\bm{\alpha}}}d\nu_{\mathbb{B}}(\beta),\text{
and}$ (3.7) $\displaystyle Ph_{0}$ $\displaystyle\geq
C^{-1}\int_{\mathbb{B}}\sum_{n=2}^{\infty}\frac{\lambda\left(Y_{n-1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{\eta(\bm{\alpha},\beta)+n-2}^{\bm{\alpha}}\right)}1_{X_{\eta(\bm{\alpha},\beta)+n-1}^{\bm{\alpha}}}d\nu_{\mathbb{B}}(\beta)$
(3.8)

where we define $X_{0}^{\bm{\alpha}}\coloneqq Y$. Note that for each $n\geq
2$,
$\tau_{\bm{\alpha}}^{k}\coloneqq\tau_{\alpha_{n-k+1}}\circ\tau_{\alpha_{n-k+2}}\circ\cdots\circ\tau_{\alpha_{n}}|_{X_{n}^{\bm{\alpha}}}:X_{n}^{\bm{\alpha}}\to
X_{n-k}^{\bm{\alpha}}$ is convex for $1\leq k\leq n-1$ where
$\bm{\alpha}=(\alpha_{1},\alpha_{2},\dots)$. Thus we have

$\displaystyle\frac{\lambda\left(X_{n-1}^{\bm{\alpha}}\right)}{\lambda\left(X_{n-1-k}^{\bm{\alpha}}\right)}1_{X_{n-k}^{\bm{\alpha}}}\leq\mathcal{L}_{\tau_{\bm{\alpha}}^{k}}1_{X_{n}^{\bm{\alpha}}}\leq\frac{\lambda\left(X_{n+1}^{\bm{\alpha}}\right)}{\lambda\left(X_{n+1-k}^{\bm{\alpha}}\right)}1_{X_{n-k}^{\bm{\alpha}}}$
(3.9)

for $1\leq k\leq n-2$ and

$\displaystyle\frac{1}{\tau_{\alpha_{1}}^{\prime}\left(\frac{1}{2}\right)}\cdot\frac{\lambda\left(X_{n-1}^{\bm{\alpha}}\right)}{\lambda\left(X_{1}^{\bm{\alpha}}\right)}1_{X_{1}^{\bm{\alpha}}}\leq\mathcal{L}_{\tau_{\bm{\alpha}}^{n-1}}1_{X_{n}^{\bm{\alpha}}}\leq\frac{\lambda\left(X_{n+1}^{\bm{\alpha}}\right)}{\lambda\left(X_{1}^{\bm{\alpha}}\right)}1_{X_{1}^{\bm{\alpha}}}.$
(3.10)

Note that $h-h_{0}$ is supported on
$Y^{c}=\bigcup_{n=1}^{\infty}X_{n}^{\bm{\alpha}}\pmod{\lambda}$ for any
$\bm{\alpha}\in\mathbb{A}^{\mathbb{N}}$, where $h$ is a $P$-invariant locally
integrable function given in (3.6). Then by combining the inequality (3.7) and
(3.8) with (3.9) and (3.10), it also follows from
$I_{Y^{c}}\mathcal{L}_{\tau_{\alpha}1_{X_{1}^{\bm{\alpha}}}}=0$ for any
$\alpha\in\mathbb{A}$ that for each $N\geq 1$

$\displaystyle h-h_{0}$ $\displaystyle\leq
C\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}{\sum_{n=1+\delta_{0,\eta(\bm{\alpha},\beta)}}^{\infty}}\sum_{k=0}^{\eta(\bm{\alpha},\beta)+n-2}\frac{\lambda\left(Y_{n+1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{\eta(\bm{\alpha},\beta)+n-k}^{\bm{\alpha}}\right)}1_{X_{\eta(\bm{\alpha},\beta)+n-k-1}^{\bm{\alpha}}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta),\text{
and}$ $\displaystyle h-h_{0}$ $\displaystyle\geq
C^{-1}\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}{\sum_{n=2}^{\infty}}\sum_{k=0}^{\eta(\bm{\alpha},\beta)+n-2}\frac{1}{\tau_{\alpha_{1}}^{\prime}\left(\frac{1}{2}\right)}\cdot\frac{\lambda\left(Y_{n-1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{\eta(\bm{\alpha},\beta)+n-k-1}^{\bm{\alpha}}\right)}1_{X_{\eta(\bm{\alpha},\beta)+n-k-1}^{\bm{\alpha}}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta)$

where $\delta_{0,\eta(\bm{\alpha},\beta)}$ is the Dirac delta function. Here
$n$ for the summand of the upper bound for $h-h_{0}$ runs from
$1+\delta_{0,\eta(\bm{\alpha},\beta)}$ in order that the union of
$X_{\eta(\bm{\alpha},\beta)+n-k-1}^{\bm{\alpha}}$'s coincides with $Y^{c}$.
Comparing the coefficients of $1_{X_{m}}$ above, we have

$\displaystyle h-h_{0}$ $\displaystyle\leq
C\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\sum_{m=1}^{\infty}\underset{\eta(\bm{\alpha},\beta)+n\geq
m+1}{\sum_{n\geq
1+\delta_{0,\eta(\bm{\alpha},\beta)}}}\frac{\lambda\left(Y_{n+1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{m+1}^{\bm{\alpha}}\right)}1_{X_{m}^{\bm{\alpha}}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta),\text{
and}$ $\displaystyle h-h_{0}$ $\displaystyle\geq
C^{-1}\int_{\mathbb{A}^{\mathbb{N}}\times\mathbb{B}}\sum_{m=1}^{\infty}\underset{\eta(\bm{\alpha},\beta)+n\geq
m}{\sum_{n\geq
2}}\frac{1}{\tau_{\alpha_{1}}^{\prime}\left(\frac{1}{2}\right)}\cdot\frac{\lambda\left(Y_{n-1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{m}^{\bm{\alpha}}\right)}1_{X_{m}^{\bm{\alpha}}}d\nu_{\mathbb{A}}^{\infty}(\bm{\alpha})d\nu_{\mathbb{B}}(\beta).$

For fixed $m\geq 1$, we have that

$\underset{\eta(\bm{\alpha},\beta)+n\geq m}{\sum_{n\geq
2}}\frac{\lambda\left(Y_{n-1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{m}^{\bm{\alpha}}\right)}=\sum_{n=\max\\{2,m-\eta(\bm{\alpha},\beta)\\}}^{\infty}\frac{\lambda\left(Y_{n-1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{m}^{\bm{\alpha}}\right)}=\frac{y_{\max\\{1,m-\eta(\bm{\alpha},\beta)-1\\}}^{\bm{\alpha},\beta}-\frac{1}{2}}{\lambda\left(X_{m}^{\bm{\alpha}}\right)}.$

Note that

$y_{\max\\{1,m-\eta(\bm{\alpha},\beta)-1\\}}^{\bm{\alpha},\beta}=\begin{cases}1&\text{if
}\eta(\bm{\alpha},\beta)+2>m,\\\
S_{\beta}^{-1}\left(x_{m-1}^{\bm{\alpha}}\right)&\text{if }m\geq
2+\eta(\bm{\alpha},\beta).\end{cases}$

If $\eta(\bm{\alpha},\beta)+2>m$ then

$\underset{\eta(\bm{\alpha},\beta)+n\geq m}{\sum_{n\geq
2}}\frac{\lambda\left(Y_{n-1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{m}^{\bm{\alpha}}\right)}=\frac{1}{2\lambda\left(X_{m}^{\bm{\alpha}}\right)}\geq\frac{1}{2}$

and if $m\geq\eta(\bm{\alpha},\beta)+2$ then

$\displaystyle\underset{\eta(\bm{\alpha},\beta)+n\geq m}{\sum_{n\geq
2}}\frac{\lambda\left(Y_{n-1}^{\bm{\alpha},\beta}\right)}{\lambda\left(X_{m}^{\bm{\alpha}}\right)}$
$\displaystyle=\frac{S_{\beta}^{-1}\left(x_{m-1}^{\bm{\alpha}}\right)-S_{\beta}^{-1}(0)}{\lambda\left(X_{m}^{\bm{\alpha}}\right)}$
$\displaystyle\geq\dfrac{x_{m-1}^{\bm{\alpha}}-0}{2\left(x_{m}^{\bm{\alpha}}-x_{m+1}^{\bm{\alpha}}\right)}\geq\dfrac{x_{m-1}^{\bm{\alpha}}}{2x_{m}^{\bm{\alpha}}}\geq\dfrac{1}{2}$

again from the convexity of $S_{\beta}$. Under the assumption (A), we also
have a lower bound $\frac{1}{2C}$ for $h$. Therefore, we conclude $h$ is
bounded above on the complement of each small neighborhood of $0$ and, under
the assumption (A), bounded away from zero on $X$ as well.

The conservativity of $\mu$ follows from [29, Remark 12] and ergodicity
follows from Lemma 3.3 and [30, Proposition 2.1]. ∎

###### Proof of Theorem 3.2.

In this proof, since $\mathbb{A}$ is a singleton, we write $X_{n}$ and
$Y_{n}^{\beta}$ instead of $X_{n}^{\bm{\alpha}}$ and
$Y_{n}^{\bm{\alpha},\beta}$. As shown in the proof of Theorem 3.1, the
invariant density function of $\mu$ satisfies

$\displaystyle h-h_{0}$ $\displaystyle\leq
C\int_{\mathbb{B}}\sum_{m=1}^{\infty}\underset{\eta(\beta)+n\geq
m+1}{\sum_{n\geq
1+\delta_{0,\eta(\beta)}}}\frac{\lambda\left(Y_{n+1}^{\beta}\right)}{\lambda\left(X_{m+1}\right)}1_{X_{m}}d\nu_{\mathbb{B}}(\beta),\text{
and}$ $\displaystyle h-h_{0}$ $\displaystyle\geq
C^{-1}\int_{\mathbb{B}}\sum_{m=1}^{\infty}\underset{\eta(\beta)+n\geq
m}{\sum_{n\geq
2}}\frac{\lambda\left(Y_{n-1}^{\beta}\right)}{\lambda\left(X_{m}\right)}1_{X_{m}}d\nu_{\mathbb{B}}(\beta)$

for some $C>0$. Therefore, integrating the above inequalities over $X_{m}$,
for $m\geq 1$ large enough, we have

$\displaystyle\mu(X_{m})$
$\displaystyle\gtrapprox\int_{\mathbb{B}}\underset{\eta(\beta)+n\geq
m}{\sum_{n\geq 2}}\lambda\left(Y_{n-1}^{\beta}\right)d\nu_{\mathbb{B}}(\beta)$
$\displaystyle\gtrapprox\int_{\left\\{\beta\in\mathbb{B}:\eta(\beta)<m\right\\}}\left(y_{m-\eta(\beta)}^{\beta}-\tfrac{1}{2}\right)d\nu_{\mathbb{B}}(\beta)+\nu_{\mathbb{B}}\left\\{\beta\in\mathbb{B}:\eta(\beta)\geq
m\right\\}.$

The upper estimate of asymptotics of $\mu(X_{m})$ is almost same and omitted.
The proof is completed. ∎

###### Proof of Theorem 3.3.

Since the density functions of $\mu$, $\mu_{1}$ and $\mu_{2}$ restricted on
$Y$ are all bounded above and away from zero, the assumptions of comparison
theorems (Theorem 6.2 and Theorem 6.5) in [19] are fulfilled. ∎

##  4\. Examples

In this section, we apply our result to several random piecewsise convex maps.

###  4.1. Random piecewise linear maps with low slopes

Let $\mathbb{B}\subset\mathbb{N}$ and
$p_{\beta}\coloneqq\nu_{\mathbb{B}}(\\{\beta\\})$ a point mass on
$\mathbb{B}$. We define for $\beta\in\mathbb{B}$

$\displaystyle T_{\beta}x=\begin{cases}2x&x\in\left[0,\frac{1}{2}\right],\\\
2^{-\beta}(2x-1)&x\in\left(\frac{1}{2},1\right].\end{cases}$ (4.1)

This obviously satisfies ( ‣ 2)–(2) and (A). Note that the left branch of
$T_{\beta}x=2x$ does not vary at all, and $\mathbb{A}$ is interpreted as a
singleton. By the definition of $T_{\beta}$, $x_{n}=\frac{1}{2^{n}}$ and
$X_{n}=(\tfrac{1}{2^{n+1}},\tfrac{1}{2^{n}}]$ for $n\geq 1$. Thus we have
$\eta(\beta)=\beta$ and

$y_{n+1}^{\beta}=\frac{1}{2}+\frac{1}{2^{n+1}}.$

Then we can apply Theorem 3.1 and Theorem 3.2 to get

######  Proposition 4.1.

The random piecewise convex map given by (4.1) admits a $\lambda$-equivalent,
conservative and ergodic $\sigma$-finite invariant measure $\mu$ such that

$\mu\left(\left(\tfrac{1}{2^{n+1}},\tfrac{1}{2^{n}}\right]\right)\approx\frac{\sum_{\beta\in\mathbb{B}:\beta<n}2^{\beta}p_{\beta}}{2^{n}}+\sum_{\beta\in\mathbb{B}:\beta\geq
n}p_{\beta}$

for $n$ large enough.

######  Remark 4.1.

By the above proposition, for example, when $\mathbb{B}=\\{[k^{n}]:n\geq 1\\}$
and $p_{[k^{n}]}=2^{-n}$ for some $k\geq 2$, $\mu$ is infinite if $k\geq 2$,
where $[\,\cdot\,]$ denotes its integer part. Indeed, in this setting, we can
write

$\displaystyle\mu\left(\left(\tfrac{1}{2^{n+1}},\tfrac{1}{2^{n}}\right]\right)$
$\displaystyle\approx 2^{-n}\sum_{1\leq
s<\log_{k}n}2^{[k^{s}]}2^{-s}+\sum_{s\geq\log_{k}n}2^{-s}$
$\displaystyle\gtrapprox n^{-1}$

and this shows the desired conclusion.

###  4.2. Random weakly expanding map with positive derivative

We first note that this example contains random LSV maps. Let
$\mathbb{A}=[\alpha_{0},\alpha_{1}]$ for some $0<\alpha_{0}<\alpha_{1}<\infty$
and $\mathbb{B}$ be some parameter space. We set probability measures
$\nu_{\mathbb{A}}$ and $\nu_{\mathbb{B}}$ on the parameter spaces $\mathbb{A}$
and $\mathbb{B}$ respectively. For $\alpha\in\mathbb{A}$ and
$\beta\in\mathbb{B}$, we define

$\displaystyle
T_{\alpha,\beta}x=\begin{cases}x\left(1+2^{\alpha}x^{\alpha}\right)&x\in\left[0,\frac{1}{2}\right],\\\
S_{\beta}x&x\in\left(\frac{1}{2},1\right]\end{cases}$ (4.2)

where we assume the conditions ( ‣ 2)–(2). The condition (A) holds since
$\alpha_{1}<\infty$. Suppose further that there exists $\gamma>0$ such that
for $\nu_{\mathbb{B}}$-almost every $\beta\in\mathbb{B}$ it holds
$S_{\beta}^{\prime}x>\gamma$. This implies
$\operatorname*{ess\,sup}_{\mathbb{B}}\eta(\beta)<\infty$. Moreover, since
$\gamma(x-\frac{1}{2})\leq T|_{(\frac{1}{2},1]}\leq 2(x-\frac{1}{2})$ by the
convexity of $S_{\beta}$, for each
$\bm{\alpha}=(\alpha,\alpha,\dots)\in\mathbb{A}^{\mathbb{N}}$ we have

$\frac{x_{n}^{\bm{\alpha}}}{2}\leq
y_{n+1}^{\bm{\alpha},\beta}-\frac{1}{2}\leq\frac{x_{n}^{\bm{\alpha}}}{\gamma}$

for large $n\geq 1$. According to the asymptotic approximation (1.2), we have

$y_{n+1}^{\bm{\alpha},\beta}-\frac{1}{2}\approx n^{-\frac{1}{\alpha}}$

for each $\bm{\alpha}=(\alpha,\alpha,\dots)\in\mathbb{A}^{\mathbb{N}}$ and
$\beta\in\mathbb{B}$. Note that if $\bar{\alpha}\geq\alpha$ then
$T_{\alpha,\beta}(x)\geq T_{\bar{\alpha},\beta}(x)$ for any
$x\in[0,\frac{1}{2}]$ and $\beta\in\mathbb{B}$. Then applying Theorem 3.3 to
this model, we have the following.

######  Proposition 4.2.

The random piecewise convex map derived from (4.2) admits a
$\lambda$-equivalent, conservative and ergodic $\sigma$-finite invariant
measure $\mu$ such that for any $\alpha^{\prime}\in\mathbb{A}$ with
$\nu_{\mathbb{A}}\\{\alpha\in\mathbb{A}:\alpha^{\prime}\geq\alpha\\}>0$

$n^{-\frac{1}{\alpha_{0}}}\lessapprox\mu\left(X_{n}^{\bm{\alpha_{0}}}\right)\text{
and }\mu\left(X_{n}^{\bm{\alpha^{\prime}}}\right)\lessapprox
n^{-\frac{1}{\alpha^{\prime}}}$

for $n$ large enough, where $\bm{\alpha}_{0}=(\alpha_{0},\alpha_{0},\dots)$
and $\bm{\alpha^{\prime}}=(\alpha^{\prime},\alpha^{\prime},\dots)$.

As consequences of Proposition 4.2, $\mu([0,1])=\infty$ if $\alpha_{0}\geq 1$.
Also if there is some $\alpha^{\prime}\in[\alpha_{0},1)$ such that
$\nu_{\mathbb{A}}\\{\alpha\in\mathbb{A}:\alpha^{\prime}\geq\alpha\\}>0$ then
$\mu([0,1])<\infty$.

###  4.3. Random weakly expanding maps with uniformly contracting branches

Let $\mathbb{A}=[\alpha_{0},\alpha_{1}]$ for some
$0<\alpha_{0}<\alpha_{1}<\infty$ and $\mathbb{B}=[0,1]$. We set probability
measures $\nu_{\mathbb{A}}$ and $\nu_{\mathbb{B}}$ on the parameter spaces
$\mathbb{A}$ and $\mathbb{B}$ respectively. For $\alpha\in\mathbb{A}$ and
$\beta\in\mathbb{B}$, we define

$\displaystyle
T_{\alpha,\beta}x=\begin{cases}x\left(1+2^{\alpha}x^{\alpha}\right)&x\in\left[0,\frac{1}{2}\right],\\\
\beta\left(x-\frac{1}{2}\right)&x\in\left(\frac{1}{2},1\right].\end{cases}$
(4.3)

Then ( ‣ 2)–(2) and (A) are satisfied. If we set
$\mathbb{B}_{k}(\bm{\alpha})\coloneqq\\{\beta\in\mathbb{B}:\eta(\bm{\alpha},\beta)=k\\}$
for $\bm{\alpha}=(\alpha,\alpha,\dots)\in\mathbb{A}^{\mathbb{N}}$, then
$\mathbb{B}=\bigcup_{k=1}^{\infty}\mathbb{B}_{k}(\bm{\alpha})$ (disjoint) for
each $\bm{\alpha}\in\mathbb{A}^{\mathbb{N}}$. For
$\beta\in\mathbb{B}_{k}(\bm{\alpha})$, by (1.1), we have

$y_{n+1}^{\bm{\alpha},\beta}-\frac{1}{2}\approx\frac{(n+k)^{-\frac{1}{\alpha}}}{\beta}\quad\text{and}\quad
y_{n-\eta(\bm{\alpha},\beta)+1}^{\bm{\alpha},\beta}-\frac{1}{2}\approx\frac{n^{-\frac{1}{\alpha}}}{\beta}.$

Hence, Theorem 3.2 ensures a random piecewise convex map
$\\{T_{\alpha,\beta};\nu_{\mathbb{B}}:\beta\in\mathbb{B}\\}$, where
$\alpha\in\mathbb{A}$ is fixed with
$\bm{\alpha}=(\alpha,\alpha,\dots)\in\mathbb{A}^{\mathbb{N}}$, to have an
invariant measure $\mu_{\bm{\alpha}}$ such that

$\displaystyle\mu_{\bm{\alpha}}\left(X_{n}^{\bm{\alpha}}\right)$
$\displaystyle\approx\sum_{k=1}^{n-1}\int_{\\{\beta:\eta(\beta)<n\\}\cap\mathbb{B}_{k}(\bm{\alpha})}\frac{n^{-\frac{1}{\alpha}}}{\beta}d\nu_{\mathbb{B}}(\beta)+\nu_{\mathbb{B}}\left\\{\beta\in\mathbb{B}:\eta(\beta)\geq
n\right\\}$ $\displaystyle\approx
n^{-\frac{1}{\alpha}}\sum_{k=1}^{n-1}\int_{\mathbb{B}_{k}(\bm{\alpha})}\frac{1}{\beta}d\nu_{\mathbb{B}}(\beta)+\sum_{k=n}^{\infty}\nu_{\mathbb{B}}\left(\mathbb{B}_{k}(\bm{\alpha})\right)$

for $n$ large enough. Thus we have

######  Proposition 4.3.

The random piecewise convex map derived from (4.3) admits a
$\lambda$-equivalent, conservative and ergodic $\sigma$-finite invariant
measure $\mu$ such that for any $\alpha^{\prime}\in\mathbb{A}$ with
$\nu_{\mathbb{A}}\\{\alpha\in\mathbb{A}:\alpha^{\prime}\geq\alpha\\}>0$

$\displaystyle
n^{-\frac{1}{\alpha_{0}}}\sum_{k=1}^{n-1}\int_{\mathbb{B}_{k}(\bm{\alpha}_{0})}\frac{1}{\beta}d\nu_{\mathbb{B}}(\beta)+\sum_{k=n}^{\infty}\nu_{\mathbb{B}}\left(\mathbb{B}_{k}(\bm{\alpha}_{0})\right)\lessapprox\mu\left(X_{n}^{\bm{\alpha_{0}}}\right)\quad\text{and}$
$\displaystyle\qquad\qquad\qquad\mu\left(X_{n}^{\bm{\alpha^{\prime}}}\right)\lessapprox
n^{-\frac{1}{\alpha^{\prime}}}\sum_{k=1}^{n-1}\int_{\mathbb{B}_{k}(\bm{\alpha^{\prime}})}\frac{1}{\beta}d\nu_{\mathbb{B}}(\beta)+\sum_{k=n}^{\infty}\nu_{\mathbb{B}}\left(\mathbb{B}_{k}(\bm{\alpha^{\prime}})\right)$

for $n$ large enough, where $\bm{\alpha}_{0}=(\alpha_{0},\alpha_{0},\dots)$
and $\bm{\alpha^{\prime}}=(\alpha^{\prime},\alpha^{\prime},\dots)$.

######  Remark 4.2.

As an example, let $\nu_{\mathbb{A}}$ be the normalized Lebesgue measure on
$\mathbb{A}=[\frac{1}{2},2]$ (that is, $\alpha_{0}=\frac{1}{2}$,
$\alpha_{1}=2$ and $\alpha^{\prime}$ can be taken an arbitrary number in
$(\alpha_{0},\alpha_{1}]$) and
$d\nu_{\mathbb{B}}(\beta)=(1-\ell)\beta^{-\ell}$ on $\mathbb{B}=[0,1]$ for
some $\ell\in(0,1)$. Then Proposition 4.3 tells us that $\mu([0,1])=\infty$ if
$\ell>\frac{1}{2}$. Indeed, for each $\bm{\alpha}=(\alpha,\alpha,\dots)$ where
$\alpha\in(\frac{1}{2},2]$, since
$\sup_{\mathbb{B}_{k+1}(\bm{\alpha})}\beta=\inf_{\mathbb{B}_{k}(\bm{\alpha})}\beta$
for $k\geq 1$ and $\sup_{\mathbb{B}_{1}(\bm{\alpha})}\beta=1$, we have

$\displaystyle
n^{-\frac{1}{\alpha}}\sum_{k=1}^{n-1}\int_{\mathbb{B}_{k}(\bm{\alpha})}\frac{1}{\beta}d\nu_{\mathbb{B}}(\beta)$
$\displaystyle=n^{-\frac{1}{\alpha}}\int_{\inf_{\mathbb{B}_{n-1}(\bm{\alpha})}\beta}^{1}\frac{1-\ell}{\beta^{1+\ell}}d\beta$
$\displaystyle\approx
n^{-\frac{1}{\alpha}}\left(\sup_{\beta\in\mathbb{B}_{n}(\bm{\alpha})}\beta^{-\ell}-1\right)$
$\displaystyle\approx n^{-\frac{1-\ell}{\alpha}}$

and

$\displaystyle\sum_{k=n}^{\infty}\nu_{\mathbb{B}}\left(\mathbb{B}_{k}(\bm{\alpha})\right)$
$\displaystyle=\sum_{k=n}^{\infty}\int_{\mathbb{B}_{k}(\bm{\alpha})}\frac{1-\ell}{\beta^{\ell}}d\beta=\sum_{k=n}^{\infty}\left(\sup_{\beta\in\mathbb{B}_{k}(\bm{\alpha})}\beta^{1-\ell}-\inf_{\beta\in\mathbb{B}_{k}(\bm{\alpha})}\beta^{1-\ell}\right)$
$\displaystyle=\sup_{\beta\in\mathbb{B}_{n}(\bm{\alpha})}\beta^{1-\ell}$
$\displaystyle\approx n^{-\frac{1-\ell}{\alpha}}$

for large $n$. Here we used that any point $\beta$ in
$\mathbb{B}_{k}(\bm{\alpha})$ can be approximated by

$(k+1)^{-\frac{1}{\alpha}}\lessapprox\beta\lessapprox k^{-\frac{1}{\alpha}}$

asymptotically. Then, since we can choose $\alpha$ arbitrary close to
$\frac{1}{2}$, the claim follows.

###  4.4. Random weakly expanding maps with a critical point

Let $\mathbb{A}\subset(0,+\infty)$ and $\mathbb{B}\subset(1,+\infty)$ be
compact sets and $\nu_{\mathbb{A}}$ and $\nu_{\mathbb{B}}$ be probability
measures on $\mathbb{A}$ and $\mathbb{B}$, respectively. We let
$\alpha_{0}\coloneqq\min_{\mathbb{A}}\alpha$. For $\alpha\in\mathbb{A}$ and
$\beta\in\mathbb{B}$, define

$\displaystyle
T_{\alpha,\beta}x=\begin{cases}x\left(1+2^{\alpha}x^{\alpha}\right)&x\in\left[0,\frac{1}{2}\right],\\\
2^{\beta}\left(x-\frac{1}{2}\right)^{\beta}&x\in\left(\frac{1}{2},1\right].\end{cases}$
(4.4)

Figure 2. The graph of $T_{\alpha,\beta}$ from (4.4)

Note that $T_{\alpha,\beta}$ has an indifferent fixed point at $0$ for each
$\alpha\in\mathbb{A}$ and $\beta\in\mathbb{B}$. $T_{\alpha,\beta}$ also has
derivative $0$ at $\frac{1}{2}$, around an inverse image of the indifferent
fixed point, for any $\alpha>0$ and $\beta>1$ (see also Figure 2). According
to the asymptotic equation (1.2) we have

$y_{n+1}^{\bm{\alpha},\beta}-\frac{1}{2}\approx n^{-\frac{1}{\alpha\beta}}.$

Then applying Theorem 3.3 to this model, since $\eta(\bm{\alpha},\beta)=0$ for
each $\bm{\alpha}\in\mathbb{A}^{\mathbb{N}}$ and $\beta\in\mathbb{B}$, we have
the following.

######  Proposition 4.4.

The random piecewise convex map derived from (4.4) admits a
$\lambda$-equivalent, conservative and ergodic $\sigma$-finite invariant
measure $\mu$ such that for any $\alpha^{\prime}\in\mathbb{A}$ with
$\nu_{\mathbb{A}}\\{\alpha\in\mathbb{A}:\alpha^{\prime}\leq\alpha\\}>0$

$\int_{\mathbb{B}}n^{-\frac{1}{\alpha_{0}\beta}}d\nu_{\mathbb{B}}(\beta)\lessapprox\mu\left(X_{n}^{\bm{\alpha_{0}}}\right)\quad\text{and}\quad\mu\left(X_{n}^{\bm{\alpha^{\prime}}}\right)\lessapprox\int_{\mathbb{B}}n^{-\frac{1}{\alpha^{\prime}\beta}}d\nu_{\mathbb{B}}(\beta)$

for $n$ large enough.

######  Remark 4.3.

(I) From Proposition 4.4, if
$\nu_{\mathbb{B}}\left\\{\beta\in\mathbb{B}:\alpha_{0}\beta\geq 1\right\\}>0$
then $\mu([0,1])=\infty$ and if
$\alpha^{\prime}<(\max_{\mathbb{B}}\beta)^{-1}=1$ for some
$\alpha^{\prime}\in\mathbb{A}$ then $\mu([0,1])<\infty$. We remark that the
invariant measure of (4.4) tends to become infinite rather than that of (4.2).

(II) [9, Theorem 1.1] showed that an upper bound for the invariant density

$\frac{d\mu}{d\lambda}(x)\lessapprox x^{-(1+\alpha-\frac{1}{\beta})}$

holds for the deterministic map (4.4) with a fixed parameter such that
$1<\beta<\frac{1}{\alpha}$ (and hence only finite invariant measures are dealt
with in [9]). This also implies that $\mu(X_{n}^{\bm{\alpha}})\lessapprox
n^{-\frac{1}{\alpha\beta}}$. Thus Proposition 4.4 is a random generalization
of [9] (note that our result can admit parameter $\beta\geq\frac{1}{\alpha}$)
as well as showing lower bound of $\mu$ for large $n\geq 1$.

###  4.5. Random weakly expanding maps with a flat point

Let $\mathbb{A}\subset(0,+\infty)$ and $\mathbb{B}\subset[1,+\infty)$111If
$\beta<1$, then the convex property of $S_{\beta}$ may be violated. compact
sets and $\nu_{\mathbb{A}}$ and $\nu_{\mathbb{B}}$ be probability measures on
$\mathbb{A}$ and $\mathbb{B}$, respectively. For $\alpha\in\mathbb{A}$ and
$\beta\in\mathbb{B}$, define

$\displaystyle
T_{\alpha,\beta}x=\begin{cases}x\left(1+2^{\alpha}x^{\alpha}\right)&x\in\left[0,\frac{1}{2}\right],\\\
\exp\left(2^{\beta}-\left(x-\frac{1}{2}\right)^{-\beta}\right)&x\in\left(\frac{1}{2},1\right].\end{cases}$
(4.5)

Then we can see that $\frac{1}{2}$, the inverse image of $0$, is a flat point
in the sense that $T_{\alpha,\beta}^{(n)}(\frac{1}{2})=0$ for any $\alpha>0$
and $\beta\geq 1$. Using the same notation, we have

$y_{n+1}^{\bm{\alpha},\beta}-\frac{1}{2}\approx\left(\log
n\right)^{-\frac{1}{\beta}}$

for large $n\geq 1$. We can again apply Theorem 3.3 to this model.

######  Proposition 4.5.

The random piecewise convex map derived from (4.5) admits a
$\lambda$-equivalent, conservative and ergodic $\sigma$-finite invariant
measure $\mu$ such that for any
$\bm{\alpha}=(\alpha,\alpha,\dots)\in\mathbb{A}$

$\mu\left(X_{n}^{\bm{\alpha}}\right)\approx\int_{\mathbb{B}}\left(\log
n\right)^{-\frac{1}{\beta}}d\nu_{\mathbb{B}}(\beta)$

for $n$ large enough. Consequently, we always have $\mu([0,1])=\infty$ for any
$\mathbb{A}$, $\mathbb{B}$, $\nu_{\mathbb{A}}$ and $\nu_{\mathbb{B}}$.

######  Remark 4.4.

We remark that even a modification of $\alpha\to 0$ leaves no space for $\mu$
to be finite. That is, if $\mathbb{A}=\\{0\\}$ (so we drop a symbol $\alpha$
henceforth in this remark) and $\nu_{\mathbb{A}}$ is a point mass on
$\mathbb{A}$ then $T_{\beta}x=2x$ for $x\in[0,\frac{1}{2}]$ and
$y_{n+1}^{\beta}-\frac{1}{2}\approx n^{-\frac{1}{\beta}}$. This means that
$\mu(X_{n})\approx\int_{\mathbb{B}}n^{-\frac{1}{\beta}}d\nu_{\mathbb{B}(\beta)}$
and we still always have $\mu([0,1])=\infty$ because
$\min_{\mathbb{B}}\beta\geq 1$.

###  4.6. Random weakly expanding maps with a wide entrance

Our example is defined as follows, which is similar to the examples (4.1) and
(4.4). Let $\mathbb{A}=[0,\alpha_{1}]$ for some $0<\alpha_{1}<\infty$ and
$\mathbb{B}=[1,+\infty)$ and $\nu_{\mathbb{A}}$ and $\nu_{\mathbb{B}}$ be
probability measures on $\mathbb{A}$ and $\mathbb{B}$, respectively. For
$\alpha\in\mathbb{A}$ and $\beta\in\mathbb{B}$, define

$\displaystyle
T_{\beta}x=\begin{cases}x\left(1+2^{\alpha}x^{\alpha}\right)&x\in\left[0,\frac{1}{2}\right],\\\
\left(x-\frac{1}{2}\right)^{\beta}&x\in\left(\frac{1}{2},1\right].\end{cases}$
(4.6)

From the definition $T_{\beta}(1)=2^{-\beta}$ and hence the image of the right
half part $(\frac{1}{2},1]$ will vanish as $\beta$ tends to infinity. Again,
let
$\mathbb{B}_{k}(\bm{\alpha})=\\{\beta\in\mathbb{B}:\eta(\bm{\alpha},\beta)=k\\}$
and $\mathbb{B}=\bigcup_{k=1}^{\infty}\mathbb{B}_{k}(\bm{\alpha})$ (disjoint)
for each $\bm{\alpha}=(\alpha,\alpha,\dots)\in\mathbb{A}^{\mathbb{N}}$.

We consider two cases of $\alpha=0$ and $\alpha>0$, and we first observe when
$\alpha=0$ which gives a lower bound for the invariant measure for the random
piecewise convex map given by (4.6). In this case, it is straightforward to
see that for each $\beta\in\mathbb{B}_{k}(\bm{0})$

$x_{n}^{\bm{0}}=2^{-n}\quad\text{and}\quad
y_{n+1}^{\bm{0},\beta}-\tfrac{1}{2}=2^{-\frac{n+k}{\beta}}$

with notation $\bm{0}=(0,0,\dots)\in\mathbb{A}^{\mathbb{N}}$. We also have
$\mathbb{B}_{k}(\bm{0})=[k,k+1)$ for $k\geq 1$. Thus the invariant measure
$\mu_{\bm{0}}$ for a random piecewise convex map
$\\{T_{0,\beta};\nu_{\mathbb{B}}:\beta\in\mathbb{B}\\}$ satisfies that

$\displaystyle\mu_{\bm{0}}\left(X_{n}^{\bm{0}}\right)$
$\displaystyle\approx\sum_{k=1}^{n-1}\int_{\mathbb{B}_{k}(\bm{0})}\left(y_{n-k+1}^{\bm{0},\beta}-\tfrac{1}{2}\right)d\nu_{\mathbb{B}}(\beta)+\sum_{k=n}^{\infty}\nu_{\mathbb{B}}\left(\mathbb{B}_{k}(\bm{0})\right)$
$\displaystyle=\sum_{k=1}^{n-1}\int_{[k,k+1)}2^{-\frac{n}{\beta}}d\nu_{\mathbb{B}}(\beta)+\nu_{\mathbb{B}}\left([n,\infty)\right)$
$\displaystyle=\int_{[1,n)}2^{-\frac{n}{\beta}}d\nu_{\mathbb{B}}(\beta)+\nu_{\mathbb{B}}\left([n,\infty)\right)$

for large $n$ by Theorem 3.2.

We secondly consider the case when $\alpha>0$ which is in need for an upper
bound for the invariant measure. Then for each
$\bm{\alpha}=(\alpha,\alpha,\dots)\in\mathbb{A}^{\mathbb{N}}$ and
$\beta\in\mathbb{B}_{k}(\bm{\alpha})$ we have

$y_{n+1}^{\bm{\alpha},\beta}-\frac{1}{2}\approx(n+k)^{-\frac{1}{\alpha\beta}}$

as $n\to\infty$. Then the invariant measure $\mu_{\bm{\alpha}}$ for the random
piecewise convex map
$\\{T_{\alpha,\beta};\nu_{\mathbb{B}}:\beta\in\mathbb{B}\\}$ satisfies that

$\displaystyle\mu_{\bm{\alpha}}\left(X_{n}^{\bm{\alpha}}\right)$
$\displaystyle\approx\sum_{k=1}^{n-1}\int_{\mathbb{B}_{k}(\bm{\alpha})}n^{-\frac{1}{\alpha\beta}}d\nu_{\mathbb{B}}(\beta)+\sum_{k=n}^{\infty}\nu_{\mathbb{B}}\left(\mathbb{B}_{k}(\bm{\alpha})\right)$
$\displaystyle\approx\int_{\left[1,\sup_{\mathbb{B}_{n-1}(\bm{\alpha})}\beta\right)}n^{-\frac{1}{\alpha\beta}}d\nu_{\mathbb{B}}(\beta)+\sum_{k=n}^{\infty}\nu_{\mathbb{B}}\left(\mathbb{B}_{k}(\bm{\alpha})\right)$

From these observations, we have the following.

######  Proposition 4.6.

The random piecewise convex map derived from (4.6) admits a
$\lambda$-equivalent, conservative and ergodic $\sigma$-finite invariant
measure $\mu$ such that for any $\alpha^{\prime}\in\mathbb{A}$ with
$\nu_{\mathbb{A}}\\{\alpha\in\mathbb{A}:\alpha^{\prime}\leq\alpha\\}>0$

$\displaystyle\int_{[1,n)}2^{-\frac{n}{\beta}}d\nu_{\mathbb{B}}(\beta)+\nu_{\mathbb{B}}\left([n,\infty)\right)\lessapprox\mu\left(X_{n}^{\bm{0}}\right)\quad\text{and}$
$\displaystyle\qquad\qquad\qquad\mu\left(X_{n}^{\bm{\alpha^{\prime}}}\right)\lessapprox\int_{\left[1,\inf_{\mathbb{B}_{n}(\bm{\alpha^{\prime}})}\beta\right)}n^{-\frac{1}{\alpha^{\prime}\beta}}d\nu_{\mathbb{B}}(\beta)+\nu_{\mathbb{B}}\left(\bigg{[}\inf_{\mathbb{B}_{n}(\bm{\alpha^{\prime}})}\beta,\infty\bigg{)}\right)$

for $n$ large enough, where $\bm{0}=(0,0,\dots)\in\mathbb{A}^{\mathbb{N}}$ and
$\bm{\alpha^{\prime}}=(\alpha^{\prime},\alpha^{\prime},\dots)\in\mathbb{A}^{\mathbb{N}}$.

######  Remark 4.5.

As an example of this proposition, if
$d\nu_{\mathbb{B}}(\beta)=(\ell-1)\beta^{-\ell}d\beta$ for some $\ell>1$ on
$\mathbb{B}=[1,+\infty)$ then the invariant measure $\mu$ is infinite when
$\ell\leq 2$, independent of the choice of $\nu_{\mathbb{A}}$. The calculation
is similar as in Remark 4.2: To see this, we need to show the lower bound in
Proposition 4.6 is proportional to or greater than $n^{-1}$.

$\displaystyle\int_{[1,n)}2^{-\frac{n}{\beta}}d\nu_{\mathbb{B}}(\beta)+\nu_{\mathbb{B}}\left([n,\infty)\right)$
$\displaystyle\geq
2^{-n}\int_{1}^{n}(\ell-1)\beta^{-\ell}d\beta+\int_{n}^{\infty}(\ell-1)\beta^{-\ell}d\beta$
$\displaystyle=2^{-n}\left(1-n^{-(\ell-1)}\right)+n^{-(\ell-1)}$
$\displaystyle>n^{-1}$

and our conclusion is valid.

###  4.7. Counterexample with infinite derivative

We finally illustrate an example which does not satisfy (A) in Theorem 3.1.
The example below still admits an equivalent $\sigma$-finite invariant measure
but the density function of the invariant measure is no longer bounded away
from zero.

Let $\mathbb{B}$ be a parameter space and $\nu_{\mathbb{B}}$ be a probability
measure on $\mathbb{B}$. For $\beta\in\mathbb{B}$, define

$\displaystyle
T_{\beta}x=\begin{cases}1-\sqrt{1-2x}&\left[0,\frac{1}{2}\right],\\\
S_{\beta}&\left(\frac{1}{2},1\right]\end{cases}$ (4.7)

where $S_{\beta}$'s satisfy ( ‣ 2)–(2). Note that (A) does not hold, while (B)
holds. We then assume that there is some $\kappa\in(0,1)$ such that for
$\nu_{\mathbb{B}}$-almost every $\beta\in\mathbb{B}$, $S_{\beta}(1)\leq
1-\kappa$ holds.

Since (4.7) satisfies (B), there is an equivalent $\sigma$-finite invariant
measure $\mu$ for this random piecewise convex map. Then we have for any
$0<\varepsilon<\kappa$

$\mu\left([1-\varepsilon,1]\right)=\int_{\mathbb{B}}\mu\left(T_{\beta}^{-1}[1-\varepsilon,1]\right)d\nu_{\mathbb{B}}(\beta)=\mu\left(\left[\tfrac{1-\varepsilon^{2}}{2},\tfrac{1}{2}\right]\right).$

Since $\frac{d\mu}{d\lambda}\leq C_{1}$ on $[\frac{1-\kappa^{2}}{2},1]$ for
some $C_{1}>0$,

$\mu\left(\left[\tfrac{1-\varepsilon^{2}}{2},\tfrac{1}{2}\right]\right)\leq\frac{C_{1}\varepsilon^{2}}{2}.$

If $\frac{d\mu}{d\lambda}$ were bounded away from zero on $Y$, then we also
have

$\mu\left([1-\varepsilon,1]\right)\geq C_{2}\varepsilon$

for some $C_{2}>0$. However this implies
$\varepsilon\geq\frac{2C_{2}}{C_{1}}$, which is contradiction since
$\varepsilon>0$ can be taken arbitrary small. Therefore, we conclude that
$\frac{d\mu}{d\lambda}(x)\to 0$ as $x\to 1$.

### Acknowledgements

This work was supported by the Research Institute for Mathematical Sciences,
an International Joint Usage/Research Center located in Kyoto University.
Hisayoshi Toyokawa was supported by JSPS KAKENHI Grant Number 21K20330.

## References

  * [1]
  * [2] J. Aaronson,  _An introduction to infinite ergodic theory_ ,  Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, (1997), xii+284 pp.
  * [3] R. Aimino, M. Nicol and S. Vaienti,  Annealed and quenched limit theorems for random expanding dynamical systems,  _Probab. Theory Related Fields_ , 162 (2015), 233–274.
  * [4] L. Arnold,  _Random dynamical systems_ ,  Springer (1995).
  * [5] W. Bahsoun and C. Bose,  Mixing rates and limit theorems for random intermittent maps,  _Nonlinearity_ 29 (2016), 1417–1433.
  * [6] W. Bahsoun, C. Bose and Y. Duan,  Decay of correlation for random intermittent maps, _Nonlinearity_ 27 (2014), 1543–1554.
  * [7] W. Bahsoun, C. Bose and M. Ruziboev,  Quenched decay of correlations for slowly mixing systems,  _Trans. Amer. Math. Soc._ , 372 (2019), 6547–6587.
  * [8] W. Bahsoun and P. Góra,  Weakly convex and concave random maps with position dependent probabilities  _Stochastic Anal. Appl._ , 21 (2003), 983–994.
  * [9] H. Cui,  Invariant densities for intermittent maps with critical points,  _J. Difference Equ. Appl._ , 27 (2021), 404–421.
  * [10] D. Dragičević, G. Froyland, C. González-Tokman and S. Vaienti,  A spectral approach for quenched limit theorems for random expanding dynamical systems,  _Comm. in Math. Phy._ , 360 (2018), 1121–1187.
  * [11] D. Dragičević, G. Froyland, C. González-Tokman and S. Vaienti,  Almost sure invariance principle for random piecewise expanding maps,  _Nonlinearity_ , 31 (2018), 2252–2280.
  * [12] S. R. Foguel,  _The ergodic theory of Markov processes_ ,  Van Nostrand Mathematical Studies, No. 21. Van Nostrand Reinhold Co., New York-Toronto, Ont.-London 1969 v+102 pp.
  * [13] S. R. Foguel,  _Selected topics in the study of Markov operators_ ,  Carolina Lecture Series, 9. University of North Carolina, Department of Mathematics, Chapel Hill, N.C., 1980.
  * [14] S. Gouëzel,  Sharp polynomial estimates for the decay of correlations,  _Israel J. Math._ , 139 (2004), 29–65.
  * [15] T. Inoue,  Asymptotic stability of densities for piecewise convex maps,  _Ann. Polon. Math._ , 57 (1992), 83–90.
  * [16] T. Inoue,  Weakly attracting repellors for piecewise convex maps,  _Japan J. Indust. Appl. Math._ , 9 (1992), 413–430.
  * [17] T. Inoue,  Invariant measures for position dependent random maps with continuous random parameters,  _Studia Math._ , 208 (2012), 11–29.
  * [18] T. Inoue,  First return maps of random maps and invariant measures,  _Nonlinearity_ , 33 (2020), 249–275.
  * [19] T. Inoue,  Comparison theorems for invariant measures of random dynamical systems,  _arXiv reprint_ , arXiv:2303.09784.
  * [20] M. S. Islam,  Piecewise convex deterministic dynamical systems and weakly convex random dynamical systems and their invariant measures,  _Int. J. Appl. Comput. Math._ , 7 (2021), 25 pp.
  * [21] Y. Kifer,  _Ergodic theory of random transformations_ ,  Progress in Probability and Statistics, 10. Birkhäuser Boston, Inc., Boston, MA, 1986. x+210 pp.
  * [22] U. Krengel,  _Ergodic theorems_ ,  De Gruyter Studies in Mathematics, 6. Walter de Gruyter & Co., Berlin, 1985. viii+357 pp.
  * [23] A. Lasota and J. A. Yorke,  Exact dynamical systems and the Frobenius-Perron operator,  _Trans. Amer. Math. Soc._ , 273 (1982), 375–384.
  * [24] C. Liverani, B. Saussol and S. Vaienti,  A probabilistic approach to intermittency,  _Ergodic Theory Dynam. Systems_ , 19 (1999), 671–685.
  * [25] F. Nakamura, Y. Nakano, H. Toyokawa and K. Yano,  Arcsine law for random dynamics with a core,  _Nonlinearity_ , 36 (2023), 1491–1509.
  * [26] O. Sarig,  Subexponential decay of correlations,  _Invent. Math._ , 150 (2002), 629–653.
  * [27] M. Thaler,  Estimates of the invariant densities of endomorphisms with indifferent fixed points,  _Israel J. Math._ , 37 (1980), 303–314.
  * [28] M. Thaler and R. Zweimüller,  Distributional limit theorems in infinite ergodic theory,  _Probab. Theory Related Fields_ 135 (2006), 15–52.
  * [29] H. Toyokawa,  $\sigma$-finite invariant densities for eventually conservative Markov operators,  _Discrete Continuous Dynamical Systems- A_ , 40 (2020): 2641–2669.
  * [30] H. Toyokawa,  On the existence of a $\sigma$-finite acim for a random iteration of intermittent Markov maps with uniformly contractive part,  _Stochastics and Dynamics_ , 21, (2021), 14 pages.
  * [31] K. Yosida,  _Functional analysis_ ,  Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, (1995).
  * [32] L.-S. Young,  Recurrence times and rates of mixing  _Israel J. Math._ , 110 (1999), 153–188.