# Finitude of physical measures for random maps

Pablo G. Barrientos  Instituto de Matemática e Estatística, UFF Rua Mário
Santos Braga s/n—Campus Valonguinhos, Niterói, Brazil<EMAIL_ADDRESS>,  Fumihiko Nakamura  Faculty of Engineering, Kitami Institute of Technology,
Hokkaido, 090-8507, JAPAN<EMAIL_ADDRESS>,  Yushi Nakano
Department of Mathematics, Tokai University, Kanagawa, 259-1292, JAPAN
<EMAIL_ADDRESS>and  Hisayoshi Toyokawa  Faculty of
Engineering, Kitami Institute of Technology, Hokkaido, 090-8507, JAPAN
<EMAIL_ADDRESS>
###### Abstract.

For random compositions of independent and identically distributed measurable
maps on a Polish space, we study the existence and finitude of absolutely
continuous ergodic stationary probability measures (which are, in particular,
physical measures) whose basins of attraction cover the whole space almost
everywhere. We characterize and hierarchize such random maps in terms of their
associated Markov operators, as well as show the difference between classes in
the hierarchy by plenty of examples, including additive noise, multiplicative
noise, and iterated function systems. We also provide sufficient practical
conditions for a random map to belong to these classes. For instance, we
establish that any continuous random map on a compact Riemannian manifold with
absolutely continuous transition probability has finitely many physical
measures whose basins of attraction cover Lebesgue almost all the manifold.

###### Key words and phrases:

Palis conjecture; physical measures; random maps; absolutely continuous
ergodic stationary probability measures; Markov operators; Markov processes

###### 2020 Mathematics Subject Classification:

Primary 37A30, 37C40, 37H05; Secondary 37A50, 37C30, 60J05

##  1\. Introduction

Uniform hyperbolicity was originally intended to encompass a residual, or at
least a dense subset of all smooth dynamical systems [5, 83], although it was
soon realized that this is not true [1, 72]. Uniformly hyperbolic systems are
structurally stable [5] and admit a very precise topological description of
their behavior: there are finitely many compact transitive invariant subsets
such that every forward orbit of the system accumulates on one of them [83].
The dynamics near such attractors may be quite chaotic and thus essentially
unpredictable after a long period of time. However, Sinai, Ruelle and Bowen
demonstrated that these attractors are the supports of physical measures and
thus behave well from a statistical point of view [28, 80, 82]. Kifer further
showed that such systems are stochastically stable (i.e. the physical measures
continuously vary) under small random perturbations [50]. Building on this
background, in a conference in honor of Douady in 1995 (refer to [75, 76]),
Palis developed a global picture recovering, in a more probabilistic
formulation, much of the paradigm of uniform hyperbolic systems. Namely, Palis
conjectured that every smooth dynamical system can be approximated by systems
having only finitely many attractors, which support physical measures that
describe the time averages for Lebesgue almost all points and are
stochastically stable under small random perturbations.

An important contribution to the stochastic part of the global Palis
conjecture was provided by Brin–Kifer [30] and Araújo [6]. Under some natural
nondegenerate assumptions on noise, they found finitely many absolutely
continuous ergodic stationary measures (in particular physical measures) with
pairwise disjoint supports and whose statistical basins of attraction cover
the whole ambient space almost everywhere. From now on, we refer to this
property as _finitude of physical measures_ or (FPM) for short111All the
measures considered in this paper will be probabilities. Also, the context of
this paper is the study of absolutely continuous ergodic stationary
probabilities (which are, in particular, physical). For that reason, the
property is simply called “finitude of physical measures”.. See Definition 1.1
for a more precise description. To be more specific on the nondegenerate
assumptions, Brin and Kifer assumed that the transition probability has a
continuous density, while Araújo assumed that the transition probability has a
density whose support includes a ball of which the diameter is uniformly
bounded from below. In the last two decades, their nondegenerate conditions
appeared as the main assumption of many works on stochastic stability around
the Palis conjecture (especially of systems without uniform hyperbolicity);
see e.g. [7, 9, 10, 16, 22].

In this paper, we will refine the Brin–Kifer and Araújo conditions by showing
that (FPM) follows merely if one assumes that the transition probability is
absolutely continuous. This is a significant improvement for applications. For
instance, random dynamical systems generated by additive noise (the most
common noise in applications; see Remark 1.4 for details) are some examples
that satisfy these conditions. Our proof is quite different from [6, 30]. It
is based on Markov operators theory, which enables us to obtain much stronger
properties than (FPM), such as the exponential decay of the annealed
correlation functions of each physical measure for some iterate of the random
map. Moreover, we give a _necessary and sufficient_ condition for (FPM) in
terms of Markov operators, extending a previous work by Inoue and Ishitani
[45] for Perron–Frobenius operators. That is, we introduce the notion of _mean
constrictivity_ for Markov operators and show its equivalence with both (FPM)
and the property of _asymptotic periodicity in mean_ introduced by Inoue and
Ishitani. This is also a generalization of the result by Lasota, Li and Yorke
in [62] for the equivalence between constrictivity and asymptotic periodicity
of Markov operators, which served as a stepping-stone for several later papers
studying the existence of absolutely continuous invariant density for
stochastic operators (see e.g. [63]). This functional approach allows us to
give a hierarchy of classes of Markov operators, which implies, together with
plenty of examples indicating the difference between the classes, that (FPM)
are much weaker than the Brin–Kifer and Araújo conditions, refer to Figures 1
and 2 (and Remark 6.3). For instance, we include some random dynamical systems
generated by finitely many continuous maps (so-called iterated function
systems) and by multiplicative noise with a common fixed point (another
important class of noise in applications). These systems never have absolutely
continuous transition probabilities.

###  1.1. Finitude of physical measures (FPM)

Let $X$ be a Polish space equipped with a probability measure $m$ on the Borel
$\sigma$-field $\mathscr{B}$ of $X$. Let $(T,\mathscr{A},p)$ be a probability
space and consider the product space
$(\Omega,\mathscr{F},\mathbb{P})=(T^{\mathbb{N}},\mathscr{A}^{\mathbb{N}},p^{\mathbb{N}})$.
In this paper, we will deal with a measurable map $f:T\times X\to X$ where we
denote $f_{t}=f(t,\cdot)$ for $t\in T$ and consider the following
nonautonomous iterations

$f^{0}_{\omega}=\mathrm{id}\quad\text{and}\quad
f^{n}_{\omega}=f_{\omega_{n}}\circ\dots\circ f_{\omega_{1}}\ \ \text{for}\
n\in\mathbb{N}\ \text{and $\omega=(\omega_{1},\omega_{2},\ldots)\in\Omega$}.$

Since we consider the Bernoulli probability $\mathbb{P}=p^{\mathbb{N}}$ on
$\Omega$, the sequence
$\\{\omega=(\omega_{1},\omega_{2},\ldots)\mapsto\omega_{n}\\}_{n\geq 1}$ of
noises at each step is an independent and identically distributed random
process. Thus, the sequence $\\{f_{\omega}^{n}(x_{0})\\}_{n\geq 0}$ can be
view as a (time-homogeneous) discrete Markov chain $\\{X_{n}\\}_{n\geq 0}$ on
$(\Omega,\mathscr{F},\mathbb{P})$ with initial distribution
$X_{0}(\omega)=x_{0}$ and transition probability given by

$P(x,A)=p(\\{t\in T:\,f_{t}(x)\in
A\\})=\mathbb{P}(\\{\omega\in\Omega:f_{\omega}(x)\in A\\})\quad\text{for $x\in
X$, \ $A\in\mathscr{B}$}.$ (1.1)

Recall that a nonnegative function $Q(x,A)$ defined for $x\in X$ and
$A\in\mathscr{B}$ is called a _(Markov) transition probability_ if

  1. (i)

$Q(x,\cdot)$ is a probability measure for every fixed $x\in X$,

  2. (ii)

$Q(\cdot,A)$ is a $\mathscr{B}$-measurable function for every fixed
$A\in\mathscr{B}$.

We shall also use the $n$-th transition probability $P^{n}(x,A)$ of the
process $\\{X_{n}\\}_{n\geq 0}$ given by

$\displaystyle P^{n}(x,A)$
$\displaystyle=\mathbb{P}(\\{\omega\in\Omega:f^{n}_{\omega}(x)\in A\\})$

which is also a Markov transition probability. We refer to [11] and [69] for
general theories of random dynamical systems and Markov processes,
respectively. A probability measure $\mu$ on $X$ is said to be a _stationary
measure_ of $f$ if

$\mu(A)=\int(f_{t})_{*}\mu(A)\,dp(t)\quad\text{for all $A\in\mathscr{B}$}.$
(1.2)

Here, $g_{*}\nu$ denotes the pushforward of a measure $\nu$ by a measurable
map $g$, that is, $g_{*}\nu(A)=\nu(g^{-1}A)$ for $A\in\mathscr{B}$. It is well
known that $\mu$ is a stationary measure of $f$ if and only if
$\mathbb{P}\times\mu$ is an invariant measure for the skew-product

$F:\Omega\times X\to\Omega\times X,\quad
F(\omega,x)=(\sigma\omega,f_{\omega}(x))$ (1.3)

where $\sigma$ is the shift operator defined on $\Omega$ (see [74]). Moreover,
we say that $\mu$ is ergodic if $\mathbb{P}\times\mu$ is an ergodic
probability measure of $F$. This is equivalent to asking that any
$A\in\mathscr{B}$ such that $P(x,A)\geq 1_{A}(x)$ where $1_{A}$ is the
indicator function has $\mu$-measure 0 or 1 (see [51, Appendix A.1] and
Appendix C). Finally, recall that a $\nu$-null set is a measurable set that
has $\nu$-measure zero, and a measure $\nu$ is said to be absolutely
continuous with respect to a measure $\lambda$ if any $\lambda$-null set is
$\nu$-null.

Our goal is to find a condition on $f$ under which (FPM) holds.

######  Definition 1.1.

We say that $f$ satisfies (FPM) if there exist finitely many ergodic
stationary probability measures $\mu_{1},\dots,\mu_{r}$ of $f$ such that

  1. 1)

they are absolutely continuous with respect to $m$;

  2. 2)

they have pairwise disjoint supports (up to an $m$-null set);

  3. 3)

the union of the fiberwise statistical basins of attraction of these measures
$\mathbb{P}$-almost surely covers $X$ up to a set of null $m$-measure. That
is,

$m\left(X\setminus(B_{\omega}(\mu_{1})\cup\dots\cup
B_{\omega}(\mu_{r}))\right)=0\quad\text{for $\mathbb{P}$-almost every
$\omega\in\Omega$}$

where

$B_{\omega}(\mu_{i})=\left\\{x\in
X:\,\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f_{\omega}^{j}(x)}=\mu_{i}\right\\}\quad\text{for
$i=1,\dots,r$}.$

Here, the limits of measures are taken in the weak*-topology.

It is not difficult to show (see Lemma 5.1) that, for the measure $\mu_{i}$
above, $B_{\omega}(\mu_{i})$ has full $\mu_{i}$-measure for
$\mathbb{P}$-almost all $\omega\in\Omega$ and $i=1,\dots,r$. Moreover, since
$\mu_{i}$ is absolutely continuous with respect to $m$, it holds that
$m(B_{\omega}(\mu_{i}))>0$ for $\mathbb{P}$-almost every $\omega\in\Omega$.
Therefore, $\mu_{i}$ is a _physical measure_ with respect to the reference
measure $m$ in the sense that $\mathbb{P}$-almost surely the fiberwise
statistical basin of attraction $B_{\omega}(\mu_{i})$ has a positive
$m$-measure for each $i=1,\ldots,r$. See more details on the notion of
physical measures for random maps in Remark 5.3.

In the deterministic case, that is, when $f_{t}=g$ for all $t\in T$ with some
$g:X\to X$, (FPM) means that the dynamics of $g$ can be statistically
understood by finitely many ergodic invariant physical probability measures
that describe the time average of almost all points in $X$. When and where
this property holds attracted great interest in dynamical systems theory, and
we refer e.g. to [27, 75]. Notice that there are some obstacles to the
finitude of physical ergodic invariant probability measures, such as the so-
called Newhouse domains [32, 39, 64, 73, 77] in which generically infinitely
many attractors coexist. Moreover, recently Berger has realized that the
coexistence of infinitely many attractors is locally Kolmogorov typical in
parametric families of endomorphisms of surfaces and diffeomorphisms of higher
dimensional manifolds, see [18, 19, 23, 24]. In contrast to the Palis
conjecture, the so-called Takens’ last problem asked if it is possible to
construct a persistent class of dynamics of a compact manifold where time
averages do not exist (named as historic behavior; cf. [81, 87]) on a Lebesgue
positive measure set. Some important advances in this question have been
obtained in [53] where the authors constructed a locally dense class of
$C^{r}$-surface diffeomorphisms ($r\geq 2$) with historic behavior on a
positive Lebesgue measure set. This result has been extended to $C^{\infty}$
and real analytic surface diffeomorphisms in [25] and to
$C^{r}$-diffeomorphisms with $r\geq 1$ in dimension three and higher in [17]
(see also [52] for a specific three-dimensional example).

Araújo [6] gave a quite useful sufficient condition (see (1) below) for (FPM)
for the following class of random maps. First of all, here $X$ is a compact
Riemannian manifold and $T$ is the unit ball of a Euclidean space with $m$ and
$p$ being the normalized Lebesgue measures on $X$ and $T$, respectively. Now,

  1. (1)

$f:T\times X\to X$ is a continuous map and $f_{t}$ is a $C^{1}$-diffeomorphism
for every $t\in T$; there are $n_{0}\in\mathbb{N}$ and a positive number
$\xi_{0}$ such that for all $n\geq n_{0}$ and $x\in X$

    1. (A)

$\left\\{f^{n}_{\omega}(x):\omega\in\Omega\right\\}$ contains the ball of
radius $\xi_{0}$ and centered at $f^{n}_{0}(x)$;

    2. (B)

$P^{n}(x,\cdot)$ is absolutely continuous with respect to $m$.

Here, we wrote $f^{n}_{0}$ for the usual $n$-th iteration of the single map
$f_{0}:X\to X$. Condition (A) above is a topological requirement. On the other
hand, since the $n$-th and $(n-1)$-th transition probabilities are related by
the recurrence

$P^{n}(x,A)=\int P^{n-1}(y,A)P(x,dy)$ (1.4)

we have condition (B) by only requiring that $P^{n_{0}}(x,\cdot)$ be
absolutely continuous with respect to $m$ for all $x\in X$. A greater
requirement is the continuity of $f$ in item (1). For instance, such a
requirement is not necessary for the approach by Brin and Kifer [30, Section
2] to get (FPM). These authors dealt with abstract Markov chains
$\\{X_{n}\\}_{n\geq 0}$ on a compact Riemannian manifold $X$ with transition
probability $P(x,A)$ having a continuous density in the following sense:

  1. (2)

there are an integer $n_{0}\geq 1$ and a nonnegative function $p(x,y)$ that is
continuous in both variables such that for any $x\in X$ and $A\in\mathscr{B}$,

$P^{n_{0}}(x,A)=\int_{A}p(x,y)\,dm(y).$

Thus, $P^{n_{0}}(x,\cdot)$ is absolutely continuous with respect to the
normalized Lebesgue measure $m$ _for all_ $x\in X$ as in the Araújo’s
condition (B). Although this abstract approach seems more general, this is not
the case because any Markov chain in a Polish space can be represented by a
random map as in (1.1) (cf. [49, 51]).

We will prove that condition (1) and condition (2), respectively, imply that
there are $n_{0}\in\mathbb{N}$ and a nonnegative function $p(x,y)$ such that
$P^{n_{0}}(x,dy)=p(x,y)\,dm(y)$ and $x\mapsto p(x,\cdot)$ is a continuous map
from $X$ to $L^{1}(m)$. See Remarks 1.2 and 1.3. Here
$L^{1}(m)=L^{1}(X,\mathscr{B},m)$ denotes, as usual, the Banach space of all
real-valued $\mathscr{B}$-measurable functions $\varphi$ on $X$ whose
$L^{1}$-norm $\|\varphi\|\coloneqq\int_{X}|\varphi|dm$ is bounded, where two
functions that coincide with each other $m$-almost everywhere are identified.
In view of this, the following theorem is a notable improvement of
Brin–Kifer’s and Araújo’s sufficient conditions to get (FPM):

######  Theorem A.

Let $(X,\mathscr{B},m)$ and $(T,\mathscr{A},p)$ be a compact Polish
probability space and a probabilistic space, respectively. Consider a
measurable map $f:T\times X\to X$ and let $P^{n}(x,A)$ be the $n$-th
transition probabilities for the associated Markov chain induced by $f$.
Assume that for some $n_{0}\in\mathbb{N}$,

$P^{n_{0}}(x,dy)=p(x,y)\,dm(y)\quad\text{such that}\quad x\in X\mapsto
p(x,\cdot)\in L^{1}(m)\ \ \text{is continuous}.$

Then, $f$ satisfies (FPM).

In the following series of remarks, we will show some easily checkable
assumptions from which Theorem A is applicable, allowing us to compare with
the previous sufficient conditions from the literature.

We say that $f:T\times X\to X$ is a _continuous random map_ if $f$ is
measurable and $f_{t}=f(t,\cdot):X\to X$ is continuous for $p$-almost every
$t\in T$.

######  Remark 1.2.

Firstly, (FPM) follows from the following assumption.

  1. (i)

Let $(X,\mathscr{B},m)$ and $(T,\mathscr{A},p)$ be a compact Polish
probability space and a probabilistic space, respectively, such that for some
$n_{0}\in\mathbb{N}$, it holds that

    1. (a)

$f:T\times X\to X$ is a continuous random map,

    2. (b)

$P^{n_{0}}(x,\cdot)$ is absolutely continuous with respect to $m$ for all
$x\in X$.

This generalizes Araújo’s result in [6]. The fact that (i) implies (FPM)
follows immediately from Corollary 2.11 and Theorem A. We will prove in
Proposition 1.13 that in the context of (i), neither condition (a) nor
condition (b) is sufficient to get (FPM). Moreover, in Proposition 1.13 we
also show that these conditions are not necessary to obtain (FPM). Notice also
that (i) generalizes Araújo’s result by removing condition (A) in his result,
and we will give in Theorem 6.4 (1) another practically sufficient condition
for (FPM) which utilizes condition (A).

######  Remark 1.3.

Secondly, (FPM) holds under the following assumption:

  1. (ii)

_Let $(X,\mathscr{B},m)$ be a compact probability Polish space and
$P^{n_{0}}(x,dy)=p(x,y)\,dm(y)$ for some $n_{0}\in\mathbb{N}$, where
$p(\cdot,y)$ is a continuous function on $X$ for $m$-almost every $y\in X$._

This clearly generalizes Brin–Kifer’s result in [30]. The proof of this
observation follows from the Schaffé–Riez theorem, cf. [61]. Indeed, if
$\\{x_{n}\\}_{n\in\mathbb{N}}$ is a converging sequence to $x$, then by the
continuity of $p(\cdot,y)$ we get $p(x_{n},y)\to p(x,y)$ and thus the
Schaffé–Riez theorem implies that $\|p(x_{n},\cdot)-p(x,\cdot)\|\to 0$ as
$n\to\infty$. This means that $p(x,\cdot)$ varies continuously on $L^{1}(m)$
with respect to $x$. Now, Theorem A implies (FPM). See also Theorem 6.4 (2)
for another weakening of Brin–Kifer’s condition.

######  Remark 1.4.

One of the most important noises in real applications that satisfy the
assumption in Theorem A is the random dynamical system generated by _additive
noise with absolutely continuous distribution_. Here, $X=T$ is a compact Lie
group where the algebraic multiplication is denoted by "$+$", $m$ is the Haar
measure, and $p$ is an absolutely continuous probability with respect to $m$.
The torus $\mathbb{R}^{d}/\mathbb{Z}^{d}$ is often considered in the
literature as an example. The random map $f$ is defined by
$f_{t}(x)=f_{0}(x)+t$ for some continuous map $f_{0}:X\to X$. As a slight
abuse of notation, we denote the density function of $p$ by $p(x)$. Then,
$P(x,A)$ can be written as

$P(x,A)=\int_{A}p(y-f_{0}(x))\,dm(y)\quad\text{for $x\in X$ and
$A\in\mathscr{B}$}$

(cf. [63, Equation (10.5.5)]). Hence, $P(x,\cdot)$ is absolutely continuous
with respect to $m$ for all $x\in X$. Notice that if the support of $p(x)$
does not include any open ball centered at $0$ nor $p(x)$ is not continuous,
then both Araújo’s condition (A) and Brin–Kifer’s condition (2) are violated
in general. Furthermore, it seems difficult to have the condition in Theorem A
with $n_{0}=1$ because $x\in X\mapsto p(y-f_{0}(x))\in\mathbb{R}$ is not
continuous for some $y\in X$ in general. However, surprisingly, by virtue of
Remark 1.2 we know that the condition in Theorem A is always satisfied (and
thus (FPM) holds). In fact, by Corollary 2.11, the condition in Theorem A
holds with $n_{0}=2$.

######  Remark 1.5.

Another important class of noise that appears in real applications is
_multiplicative noise with absolutely continuous distribution_ (cf. [12, 86]).
For instance, consider the case when $X=T=[0,1]$, $m$ is the Lebesgue measure
and $p$ is an absolutely continuous probability with respect to $m$ with
density function $p(x)$. Define $f_{t}(x)=tg(x)$, where $g$ is some continuous
map on $X$. It is straightforward to see that $P(x,A)$ is of the form

$P(x,A)=\int_{A}p\left(\frac{y}{g(x)}\right)dm(y)\quad\text{for $x\in X$ and
$A\in\mathscr{B}$}$

(cf. [63, Equation (10.7.5)]). Therefore, if $g$ is bounded away from zero,
then $P(x,\cdot)$ is absolutely continuous with respect to $m$, and by Remark
1.2, (FPM) holds. Actually, the condition in Theorem A holds (with $n_{0}=2$).
In fact, this (together with Theorem B) generalizes [63, Theorem 10.7.1]. On
the other hand, in contrast to additive noise, if $g(x)=0$ for some $x\in X$,
then the absolute continuity of $P(x,\cdot)$, the condition in Theorem A and
(FPM) can fail to hold. See Section 1.4.2 for details.

######  Remark 1.6.

Finally, we remark that the important class of random dynamical systems
generated by iterated function systems with probabilities (see Section 1.4 for
its formal definition) does not meet the assumption in Theorem A. Indeed, in
this case, we have that $T$ is a finite set $\\{1,\dots,k\\}$ and thus

$P(x,\cdot)=\sum_{i=1}^{k}p_{i}\delta_{f_{i}(x)}\quad\text{where
$p_{i}=p(\\{i\\})>0$}$

which cannot be absolutely continuous with respect to $m$ in general. However,
some of them will be in the range of Section 1.2 to get (FPM), in which
several weaker versions of the condition in Theorem A are given.

The proof of Theorem A is based on the analysis of the annealed
Perron–Frobenius operator associated with the random dynamical system
generated by $f$. We will show that this operator belongs to the class of
constrictive Markov operators, which has been extensively studied in the
literature. This observation allows us to generalize Theorem A in terms of
Markov operators. In particular, we can obtain new practical sufficient
conditions implying (FPM), as indicated in Theorem 6.4. As we will explain in
Remark 6.5, such conditions generalize the sufficient condition studied in the
work by Araújo and Aytaç [8] to get uniform ergodicity (a stronger property
than (FPM)).

###  1.2. Markov operators

In order to provide a general definition of Markov operator, assume that
$(X,\mathscr{B},m)$ is any abstract probability space (not necessarily a
Polish space as in the previous subsection). Let $D(m)=D(X,\mathscr{B},m)$ be
the space of density functions, that is,

$D(m)=\left\\{h\in L^{1}(m):h\geq 0\ \text{$m$-almost everywhere and
$\|h\|=1$}\right\\}.$

An operator $P:L^{1}(m)\to L^{1}(m)$ is called a _Markov operator_ if $P$ is
linear, positive (i.e. $P\varphi\geq 0$ $m$-almost everywhere if $\varphi\geq
0$ $m$-almost everywhere) and

$\int P\varphi\,dm=\int\varphi\,dm\quad\text{ for all $\varphi\in L^{1}(m)$}.$
(1.5)

Note that a positive linear operator $P$ on $L^{1}(m)$ is a Markov
operator222Any Markov operator $P$ is a bounded operator: Given $\varphi\in
L^{1}(m)$, consider
$\varphi_{+}\coloneqq\max\\{\varphi,0\\},\varphi_{-}\coloneqq\max\\{-\varphi,0\\}$.
Then, $P\varphi_{+},P\varphi_{-}\geq 0$, and thus
$\|P\varphi\|\leq\int(P\varphi_{+}+P\varphi_{-})\,dm=\int\varphi_{+}dm+\int\varphi_{-}dm=\|\varphi\|$.
if and only if $P(D(m))\subset D(m)$. It is not difficult to see that this is
equivalent to $P^{*}1_{X}=1_{X}$, where $P^{*}$ is the adjoint operator of
$P$, that is, $P^{*}$ is a bounded linear operator on
$L^{\infty}(m)\cong(L^{1}(m))^{*}$ given by $\int
P^{*}\psi\cdot\varphi\,dm=\int\psi\cdot P\varphi\,dm$ for $\psi\in
L^{\infty}(m)$, $\varphi\in L^{1}(m)$. Recall that
$L^{\infty}(m)=L^{\infty}(X,\mathscr{B},m)$ is the Banach space of bounded
real-valued $\mathscr{B}$-measurable $m$-essentially bounded functions defined
on $X$. As usual, two functions that coincide with each other $m$-almost
everywhere, are identified.

A key property of Markov operators for the purpose of this paper is
_constrictivity_.

######  Definition 1.7.

A sequence $(Q_{n})_{n\geq 1}$ of Markov operators on $L^{1}(m)$ is called

  1. (1)

constrictive if there exists a compact set $F$ of $L^{1}(m)$ such that for any
$h\in D(m)$,

$\lim_{n\to\infty}d(Q_{n}h,F)=0;$

  2. (2)

uniformly constrictive if there exists a compact set $F$ of $L^{1}(m)$ such
that

$\lim_{n\to\infty}\sup_{h\in D(m)}d(Q_{n}h,F)=0.$

Here $d(\varphi,F)=\inf_{\psi\in F}\|\varphi-\psi\|$. In particular, a Markov
operator $P$ on $L^{1}(m)$ is called

  1. (3)

(uniformly) constrictive if the sequence $(P^{n})_{n\geq 1}$ is (uniformly)
constrictive.

  2. (4)

mean constrictive if the sequence $(A_{n})_{n\geq 1}$ is constrictive, where
$A_{n}$ is given by

$A_{n}\varphi=\frac{1}{n}\sum_{i=0}^{n-1}P^{i}\varphi\quad\text{for
$\varphi\in L^{1}(m)$}.$

The compact set $F$ above is called a _constrictor_. These conditions appeared
in the context of mean ergodic theorems, see [36, 63]. Notice that, by
definition, we have

uniformly constrictive $\Rightarrow$ constrictive $\Rightarrow$ mean
constrictive .

See also Figure 1 for a global picture. Furthermore, the uniform
constrictivity of $P$ is known to be equivalent to the quasi-compactness of
$P$ in $L^{1}(m)$, cf. [20, Theorem 2]. Recall that an operator $Q$ is called
_quasi-compact_ if there is a compact linear operator $R$ such that
$\|Q^{n}-R\|_{\rm op}<1$ for some $n\in\mathbb{N}$. Hence, the class of quasi-
compact operators contains the subclass of _eventually compact operators_ ,
that is, the linear operators $Q$ such that $Q^{n}$ is compact for some
$n\in\mathbb{N}$.

####  1.2.1. Perron–Frobenius operators

One of the most important examples of Markov operators is the
_Perron–Frobenius operator_ induced by a nonsingular transformation $g:X\to
X$. Recall that $g$ is nonsingular (with respect to the reference measure $m$
on $X$) if the preimage of any $m$-null set by $g$ is $m$-null. The
Perron–Frobenius operator $\mathcal{L}_{g}:L^{1}(m)\to L^{1}(m)$ of $g$ is
defined by the formula

$g_{*}m_{\varphi}(A)=\int_{A}\mathcal{L}_{g}\varphi\,dm\quad\text{ for all
$\varphi\in L^{1}(m)$ and $A\in\mathscr{B}$}$

where $m_{\varphi}$ is the finite signed measure given by

$m_{\varphi}(A)=\int_{A}\varphi\,dm\quad\text{for $A\in\mathscr{B}$.}$ (1.6)

It is easy to see that $\mathcal{L}_{g}$ is a Markov operator and
$m_{\mathcal{L}_{g}\varphi}=g_{*}m_{\varphi}$. As in this example, Markov
operators $P$ naturally appear in the study of (random) dynamical systems, and
$(P^{n}\varphi)_{n\geq 0}$ is interpreted as the evolution of density
functions driven by the system. We refer to [13, 14, 36, 37, 63].

Let $f:T\times X\to X$ be as in Section 1.1. We simply write $\mathcal{L}_{t}$
for the Perron–Frobenius operator of $f_{t}=f(t,\cdot)$ for $t\in T$ and
define the _annealed Perron–Frobenius operator_ ${\mathcal{L}}_{f}:L^{1}(m)\to
L^{1}(m)$ by

${\mathcal{L}}_{f}\varphi(x)=\int\mathcal{L}_{t}\varphi(x)\,dp(t).$

Then, it is straightforward to see that ${\mathcal{L}}_{f}$ is also a Markov
operator. Now we are in a position to state two of our main results.

######  Theorem B.

Under the assumption of Theorem A, $\mathcal{L}_{f}$ is eventually compact, in
particular, uniformly constrictive.

We will obtain the above result by proving an equivalent but apparently more
general theorem stated in terms of Markov operators and Markov processes. See
Theorem 2.3 together with Proposition 2.2 and Remark 2.5.

In the setting of Theorem B, since $\mathcal{L}_{f}$ is quasi-compact, we can
obtain exponential decay of annealed correlation functions, in the sense that
there are finitely many absolutely continuous ergodic stationary measures
$\mu_{1},\ldots,\mu_{r}$ with pairwise disjoint supports and constants
$k\in\mathbb{N}$, $C>0$, $\rho\in(0,1)$ such that for any $\varphi,\psi\in
L^{\infty}(m)$, $n\in\mathbb{N}$ and $j=1,\ldots,r$, we have

$\left|\int\psi\circ
f^{kn}_{\omega}\cdot\varphi\,d(\mathbb{P}\times\mu_{j})-\int\psi\,d\mu_{j}\int\varphi\,d\mu_{j}\right|\leq
C\rho^{kn}\|\varphi\psi\|_{\infty}.$ (1.7)

Refer to, for example, [31, 67] for background and, [15] for proof333Although
the paper [15] only dealt with deterministic maps, one can show the claim by
literally repeating the argument in [15, Appendix B] with $\mathcal{L}_{f}$
instead of a Perron–Frobenius operator $\mathcal{L}_{g}$ of a nonsingular map
$g:X\to X$, after realizing the duality $\int\psi\circ
f^{n}_{\omega}\cdot\varphi\,d(\mathbb{P}\times
m)=\int\psi\cdot\mathcal{L}_{f}^{n}\varphi\,dm$ corresponding to
$\int\psi\circ
g^{n}\cdot\varphi\,dm=\int\psi\cdot\mathcal{L}_{g}^{n}\varphi\,dm$.. Actually,
this claim with $r=k=1$ was shown by Araújo and Aytaç in [8] under a bit
stronger assumption than Araújo’s conditions (A) and (B) in a different manner
(using a purely probability-theoretic technique); see also Remark 6.5.
Furthermore, the quasi-compactness of $\mathcal{L}_{f}$ may lead to other
several limit theorems (such as central limit theorem, large deviation
principle, local limit theorem and almost sure invariance principle) via the
so-called Nagaev–Guivarc’h perturbative spectral method, refer to [2, 40, 44].

######  Theorem C.

Let $(X,\mathscr{B},m)$ and
$(\Omega,\mathscr{F},\mathbb{P})=(T^{\mathbb{N}},\mathscr{A}^{\mathbb{N}},p^{\mathbb{N}})$
be a locally compact Polish probability space and the infinite product space
of a probability space $(T,\mathscr{A},p)$, respectively. Consider a
measurable map $f:T\times X\to X$. Then, the followings are equivalent:

  1. (i)

${\mathcal{L}}_{f}$ is mean constrictive;

  2. (ii)

$f$ satisfies (FPM).

Moreover, if $f$ satisfies any of the above equivalent conditions and
$\mu_{1},\dots,\mu_{r}$ denote the measures that appear in the property (FPM),
then

  1. (1)

$(\mathbb{P}\times m)\left(B(\mu_{1})\cup\dots\cup B(\mu_{r})\right)=1$;

  2. (2)

$\mathbb{P}\left(B_{x}(\mu_{1})\cup\dots\cup B_{x}(\mu_{r})\right)=1$ for
$m$-almost every $x\in X$;

where for each $i=1,\dots,r$,

$B(\mu_{i})=\left\\{(\omega,x)\in\Omega\times
X:\,\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f_{\omega}^{j}(x)}=\mu_{i}\right\\}$

and

$\displaystyle
B_{x}(\mu_{i})=\left\\{\omega\in\Omega:\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{j}_{\omega}(x)}=\mu_{i}\right\\}.$

We note that since uniform constrictivity implies mean constrictivity, Theorem
A is just a consequence of Theorems B and C. In fact, conclusions (1) and (2)
in Theorem C also hold under the assumptions of Theorem A.

Notice that $B(\mu_{i})$ is basically the statistical basin of attraction of
the measure $\mathbb{P}\times\mu_{i}$ for the skew-product map $F$ in (1.3).
Thus, the above theorem is actually a characterization of an absolutely
continuous version of the Palis conjecture for the class of deterministic
systems of the form (1.3), in terms of the annealed Perron–Frobenius operator
of the random map $f$. In particular, when $T$ (or $\Omega$) is a singleton,
Theorem C provides a characterization of the existence of finitely many
$m$-absolutely continuous invariant probability measures for deterministic
dynamics such that the union of their basins has full $m$-measure, in terms of
the Perron–Frobenius operator.

####  1.2.2. General Markov operators

We next generalize the equivalence in Theorem C to general Markov operators.
However, to do this, we need some preliminaries.

Let $P$ be a Markov operator on $L^{1}(m)$ and consider the adjoint operator
$P^{*}$ on $L^{\infty}(m)$. We define the support of a real-valued function
$h$ (up to an $m$-null set) by $\operatorname*{supp}h=\\{x\in
X:\,h(x)\not=0\\}$. We say that $h$ is an _invariant density_ of $P$ (or a
_$P$ -invariant density_) if $h\in D(m)$ and $Ph=h$. As usual, $P^{n}$ and
$(P^{*})^{n}$ denote the $n$-th iterated of $P$ and $P^{*}$ respectively. Let
$(P^{n})^{*}$ be the adjoint operator of $P^{n}$. Using recursively the
duality relation, it is not difficult to see that $(P^{n})^{*}=(P^{*})^{n}$.
For simplicity of notation, we will simply write $P^{n*}$ when no confusion
can arise. Having this notation in mind, we say that a $P$-invariant density
$h$ has the _maximal support_ if

$\displaystyle\lim_{n\to\infty}P^{n*}1_{\operatorname*{supp}h}(x)=1\quad\text{for
$m$-almost all $x\in X$.}$ (1.8)

A probability measure $\mu$ is called _ergodic_ if any $A\in\mathscr{B}$ such
that $P^{*}1_{A}\geq 1_{A}$ satisfies $\mu(A)\in\\{0,1\\}$. We will say that a
$P$-invariant density $h\in D(m)$ is _ergodic_ if the probability measure
$m_{h}$ given in (1.6) is ergodic. See Appendix C for more details on
equivalent definitions.

######  Theorem D.

Let $(X,\mathscr{B},m)$ be an abstract probability space and consider a Markov
operator $P:L^{1}(m)\to L^{1}(m)$. Then, the following conditions are
equivalent:

  1. (MC)

$P$ is mean constrictive;

  2. (FED)

$P$ admits finitely many ergodic $P$-invariant densities $h_{1},\dots,h_{r}$
with mutually disjoint supports (up to an $m$-null set) and the invariant
density function $h=\frac{1}{r}(h_{1}+\dots+h_{r})$ has the maximal support;

  3. (APM)

There exist finitely many ergodic $P$-invariant densities $h_{1},\dots,h_{r}$
with mutually disjoint supports (up to an $m$-null set) and positive bounded
linear functionals $\lambda_{1},\dots,\lambda_{r}$ on $L^{1}(m)$ such that

$\displaystyle\lim_{n\to\infty}\big{\|}A_{n}\varphi-\sum_{i=1}^{r}\lambda_{i}(\varphi)h_{i}\big{\|}=0\quad\text{for
any $\varphi\in L^{1}(m)$}.$

The condition (APM), named as _asymptotic periodicity in mean_ , was
introduced by Inoue and Ishitani for Perron–Frobenius operators in [45] as a
weaker version of the classic property _asymptotic periodicity_. In turn,
asymptotic periodicity was introduced and shown to be equivalent to
constrictivity in [62]. The equivalence between the conditions (FED) and (APM)
is the generalization of Inoue–Ishitani [45] to the case of general Markov
operators. Similarly, the equivalence between (APM) and (MC) is a
generalization of the spectral decomposition theorem (on $L^{1}(m)$) that has
been developed by Lasota, Li, Yorke, Komorník, Bartoszek during the eighties
and nineties, and by Storozhuk, Toyokawa more recently. Refer to [21, 88] and
references therein. Actually, we will prove in Theorem 4.1 a more complete
version of Theorem D where we will provide a sequence of equivalences between
(MC) and, a priori, weaker conditions.

######  Remark 1.8.

If $1_{X}$ is an ergodic $P$-invariant density, then $P$ is (MC). Indeed,
since any two ergodic $P$-invariant densities are equal or have disjoint
support up to an $m$-null set, see Proposition C.10 in Appendix C, $1_{X}$ is
actually the unique invariant ergodic density of $P$. Then $P$ satisfies (FEM)
and consequently, by Theorem D, $P$ is (MC).

###  1.3. Hierarchy of classes of Markov operators

We have considered the classes of _uniform constrictive_ (UC), _constrictive_
(C) and _mean constrictive_ (MC) Markov operators introduced in Definition
1.7. In this subsection, we introduce in Definition 1.10 a new class between
them that we call _asymptotic constrictivity_ (AC). In the next subsection, we
will provide examples of random maps that show the difference between these
classes. First, to show the global picture, we characterize the class of
Markov operators in $L^{1}(m)$ for which there is an invariant density, which
we denote by (S).

####  1.3.1. Straube class

In [85] Straube studied the existence of invariant densities for the
Perron–Frobenius operator associated with a nonsingular transformation $g:X\to
X$. Namely, Straube showed that there exists a $g$-invariant probability
measure which is absolutely continuous with respect to $m$ if and only if
there exist $\delta>0$ and $0<\alpha<1$ such that $m(A)<\delta$ implies
$m(g^{-k}(A))<\alpha$ for all $k\geq 0$. Recall that a $g$-invariant
absolutely continuous probability measure corresponds to an invariant density
of the Perron–Frobenius operator $\mathcal{L}_{g}$. Moreover,

$m\left(g^{-k}(A)\right)=\int 1_{A}\circ
g^{k}(x)\,dm=\int_{A}\mathcal{L}^{k}_{g}1_{X}\,dm.$

More recently, Islam, Góra and Boyarsky in [46] proved also similar necessary
and sufficient conditions in terms of Markov operator for the existence of
absolutely continuous stationary measures for a certain class of iterated
function systems with probabilities. The following theorem finally provides a
characterization of the class (S) in the spirit of (MC), generalizing the
previous results in [46, 85]. A more complete version will be given in Theorem
3.4.

######  Theorem E.

For a Markov operator $P:L^{1}(m)\to L^{1}(m)$, the following assertions are
equivalent:

  1. (1)

There exists an invariant density for $P$;

  2. (2)

There exist $\alpha\in(0,1)$ and $\delta>0$ such that

$\sup_{n\geq 0}\int_{A}P^{n}1_{X}\,dm<\alpha\quad\text{for any
$A\in\mathscr{B}$ with $m(A)<\delta$};$

  3. (3)

There exist $\alpha\in(0,1)$ and $\delta>0$ such that

$\sup_{n\geq 0}\int_{A}A_{n}1_{X}\,dm<\alpha\quad\text{for any
$A\in\mathscr{B}$ with $m(A)<\delta$};$

Moreover, the equivalence also holds by taking $\alpha=1$ in all the above
items.

####  1.3.2. Weakly almost periodic class

Consider the class of Markov operators on $L^{1}(m)$ having an invariant
density with the maximal support as in (1.8). This class was characterized in
[88] as the well-known class of Markov operators in $L^{1}(m)$ called weakly
almost periodic.

######  Definition 1.9.

A Markov operator $P$ on $L^{1}(m)$ is weakly almost periodic if
$(P^{n}\varphi)_{n\geq 1}$ is weakly precompact for any $\varphi\in L^{1}(m)$,
that is, if any sequence $(P^{n_{k}}\varphi)_{k\geq 1}$ contains a further
subsequence $(P^{n_{k_{j}}}\varphi)_{j\geq 1}$ that weakly converges to a
function in $L^{1}(m)$.

By the Dunford–Pettis theorem444 The theorem states that $F\subset L^{1}(m)$
is weakly precompact (i.e. any sequence in $F$ contains a weakly converging
subsequence in $L^{1}(m)$) if and only if $F$ is bounded (i.e. there is $M>0$
such that $\|\varphi\|<M$ for all $\varphi\in F$) and uniformly integrable
(i.e. for any $\varepsilon>0$, there is $\delta>0$ such that $\int_{A}\varphi
dm<\epsilon$ for any $\varphi\in F$ and $A\in\mathscr{B}$ with $m(A)<\delta$).
Refer to e.g. [26, Theorem 4.7.18]., a Markov operator $P$ is weakly almost
periodic if and only if

  1. (WAP)

for any $\varepsilon>0$ and $\varphi\in L^{1}(m)$, there is $\delta>0$ such
that

$\int_{A}P^{n}\varphi\,dm<\varepsilon\quad\text{for any $n\in\mathbb{N}$ and
$A\in\mathscr{B}$ with $m(A)<\delta$.}$

As mentioned, due to [88, Theorem 3.1], we have that (WAP) is equivalent to
the existence of a $P$-invariant density with the maximal support. Therefore,
(WAP) implies (S), and moreover, (MC) implies (WAP) from Theorem D.

####  1.3.3. Asymptotically constrictive class

An equivalent formulation for the class of constrictive Markov operators
$P:L^{1}(m)\to L^{1}(m)$ is

  1. (C)

for any $\varepsilon>0$ there is $\delta>0$ such that for any $\varphi\in
D(m)$, there is $n_{0}\in\mathbb{N}$ satisfying

$\int_{A}P^{n}\varphi\,dm<\varepsilon\quad\text{for any $n\geq n_{0}$ and
$A\in\mathscr{B}$ with $m(A)<\delta$.}$

One of the implications follows easily from Dunford–Pettis theorem. Indeed, if
$P$ is constrictive with a constrictor $F$, since $F$ is compact (in
particular, weakly precompact) in $L^{1}(m)$ and $\int_{A}P^{n}\varphi\,dm\leq
d(P^{n}\varphi,F)+\sup_{\phi\in F}\int_{A}\phi\,dm$ for each $\varphi\in
D(m)$, $n\in\mathbb{N}$ and $A\in\mathscr{B}$, we immediately obtain (C). One
can find the proof of the converse e.g. in [56]. Having in mind these
equivalent formulations of (C) and (WAP), we introduce the following middle
property between constrictivity and weak almost periodicity, which we could
come up with while looking for non-constrictive examples.

######  Definition 1.10.

A Markov operator $P$ on $L^{1}(m)$ is said to be asymptotically constrictive
(AC) if for any $\varepsilon>0$, there is $\delta>0$ such that

$\limsup_{n\rightarrow\infty}\int_{A}P^{n}\varphi\,dm<\varepsilon\quad\text{for
any $\varphi\in D(m)$ and $A\in\mathscr{B}$ with\leavevmode\nobreak\
$m(A)<\delta$.}$ (1.9)

It is straightforward to see that (C) implies (AC). We prove in Theorem 4.9
that (AC) implies (MC). We summarize the hierarchy between classes in the
following figure.

(UC)
$\overset{\text{Obvious}}{\longrightarrow}$$\underset{\text{Prop.\leavevmode\nobreak\
\ref{prop:figa}}}{\longleftarrow}$$/$ (C)
$\overset{\text{Obvious}}{\longrightarrow}$$\underset{\text{Prop.\leavevmode\nobreak\
\ref{prop:contracting}}}{\longleftarrow}$$/$ (AC)
$\overset{\text{Thm.\leavevmode\nobreak\
\ref{propC2}}}{\longrightarrow}$$\underset{\text{Prop.\leavevmode\nobreak\
\ref{prop:rotations}}}{\longleftarrow}$$/$ (MC)
$\overset{\text{Thm.\leavevmode\nobreak\
\ref{thm:D}}}{\longrightarrow}$$\underset{\text{Prop.\leavevmode\nobreak\
\ref{prop:rotations}}}{\longleftarrow}$$/$ (WAP)
$\overset{\text{Obvious}}{\longrightarrow}$$\underset{\text{Prop.\leavevmode\nobreak\
\ref{prop:figb}}}{\longleftarrow}$$/$ (S).

Figure 1. The hierarchy of the classes between (UC) and (S).

####  1.3.4. Sub-hierarchy in (UC)

A transition probability $P(x,A)$ on $X\times\mathscr{B}$ is said to be _$m$
-nonsingular_ if $P(x,A)=0$ for $m$-almost every $x\in X$ whenever $m(A)=0$.
It is well-known that the theories of Markov operators and Markov processes
(transition probabilities) are intimately related. Indeed, given an
$m$-nonsingular transition probability $P(x,A)$ for $x\in X$ and
$A\in\mathscr{B}$, we can induce a Markov operator $P$ on $L^{1}(m)$ such that
$P\varphi$ is the Radon–Nikodým derivative with respect to $m$ of the finite
signed measure

$\mu_{\varphi}(A)=\int\varphi(x)P(x,A)\,dm,\quad\text{for $A\in\mathscr{B}$}.$

See [47] or [70, Proposition V.4.2] for more details on the construction.
Conversely, given a Markov operator $P$ on $L^{1}(m)$ one can define

$P^{n}(\cdot,A)=P^{n*}1_{A}\quad\text{for all $A\in\mathscr{B}$ and $n\geq
1$}.$

Notice that $P^{n}(\cdot,A)$ is an equivalent class and thus as a real-valued
function is only defined up to $m$-null sets. It is not hard to see that one
may choose in each equivalent class $P^{n}(\cdot,A)$ a
$\mathscr{B}$-measurable function $P^{n}(x,A)$ on $X$ for every fixed
$A\in\mathscr{B}$ such that it differs from an $m$-nonsingular transition
probability only on a negligible set of points. Nevertheless, if $X$ is a
Polish space, Neveu proved in [70, Proposition V.4.4] that these
representations can be chosen appropriately to get that $P^{n}(x,A)$ is a
transition probability which induces the Markov operator $P^{n}$ on $L^{1}(m)$
as explained above. Taking into account this relation between Markov operators
and Markov processes, in what follows we will introduce two subclasses in
(UC).

Let $(X,\mathscr{B},m)$ be a Polish probability space and consider a Markov
operator $P$ on $L^{1}(m)$. Let $P^{n}(x,A)$ be an $n$-transition probability
that induces $P^{n}$ as explained above. We first introduce the following
classical condition from the theory of Markov chain adapted to Markov
operators in $L^{1}(m)$:555 Recall that $P(x,A)$ is said to satisfy the
_Doeblin_ condition if there exist $n_{0}\geq 1$, $\varepsilon>0$, $\delta<1$
and a probability $\mu$ such that $P^{n_{0}}(x,A)>\varepsilon$ for all $x\in
X$ and $A\in\mathscr{B}$ with $\mu(A)>\delta$ (cf. [69]). If $\mu$ is in
addition required to be absolutely continuous with respect to $m$, then this
condition is equivalent to (D).

  1. (D)

there exist $n_{0}\geq 1$, $0<\varepsilon<1$, $\delta>0$ and a probability
$\mu$ absolutely continuous with respect to $m$ such that
$P^{n_{0}}(x,A)<\varepsilon$ for all $x\in X$ and $A\in\mathscr{B}$ with
$\mu(A)<\delta$.

The second class that we introduce is also (an equivalent condition to) a
classical property widely discussed and studied in the literature of Markov
processes under the name of _uniform ergodicity_ that we adapt to Markov
operators on $L^{1}(m)$:

  1. (D*)

there exist $n_{0}\geq 1$, $0<\varepsilon<1$, $\delta>\frac{1}{2}$ and a
probability $\mu$ absolutely continuous with respect to $m$ such that
$P^{n_{0}}(x,A)<\varepsilon$ for all $x\in X$ and $A\in\mathscr{B}$ with
$\mu(A)<\delta$.

Dorea and Pereira [33] introduced (D*) without the absolute continuity of
$\mu$ as an equivalent condition to uniform ergodicity. In Proposition 6.6 we
will show equivalent conditions to (D*) including uniform ergodicity adapted
to Markov operators in $L^{1}(m)$. Furthermore, we will show that the
condition (D*) implies that $P$ only admits a unique invariant density, see
Remark 6.7.

In Proposition 6.1 we will prove that (UC) is the class of Markov operators
$P$ on $L^{1}(m)$ that satisfy the following condition:

  1. (UC)

there are $n_{0}\geq 1$, $0<\varepsilon<1$, $\delta>0$ and a probability $\mu$
absolutely continuous with respect to $m$ such that
$P^{n_{0}}(x,A)<\varepsilon$ for all $A\in\mathscr{B}$ with $\mu(A)<\delta$
and $m$-almost every $x\in X$ (depending on $A$).

In view of these characterizations, one immediately obtains the following
relations:

(D*) $\overset{\text{Obvious}}{\longrightarrow}$$\underset{\text{Prop
\ref{prop:dd}}}{\longleftarrow}$$/$ (D)
$\overset{\text{Prop\leavevmode\nobreak\
\ref{prop:UC}}}{\longrightarrow}$$\underset{\text{Prop
\ref{prop:ucd}}}{\longleftarrow}$$/$ (UC).

Figure 2. The subhierarchy in (UC).

In the next subsection of examples, we will show that neither of the converse
implications above holds. However, if we restrict ourselves to the class of
strong Feller operators, then (UC) implies (D). See Proposition 6.2. Due to
this result, the annealed Perron–Frobenius operator $\mathcal{L}_{f}$
satisfies (D) when $f$ satisfies the Araújo or Brin–Kifer conditions (see
Remark 6.3). Furthermore, $\mathcal{L}_{f}$ satisfies (D*) for the conditions
in Araújo–Aytaç [8]. In fact, we will generalize the argument in Araújo–Aytaç
to show that a version of Araújo’s condition (other than the condition in
Remark 1.2) is sufficient to obtain (UC). See Remark 6.5 for details.

###  1.4. Examples and counterexamples

In this subsection, to complete Remark 1.2 and the list that reverse
implications in Figures 1 and 2 fail, we will consider several examples coming
from the three aforementioned important classes of random dynamical systems,
i.e. additive noise, multiplicative noise and iterated function systems. We
will also explain that some examples can be easily modified to be
deterministic systems.

####  1.4.1. Additive type noise

First, we consider some perturbed systems with additive type noise, which will
indicate the reverse implications in Figures 2. Let $X$ and $T$ be the closed
interval $[0,1]$ equipped with the Lebesgue measure, denoted by $m$ and $p$,
respectively. Consider a measurable map $f_{0}:X\to X$ and consider the random
map $f:T\times X\to X$ given by $f(t,x)=f_{t}(x)$,

$f_{t}(x)=\begin{cases}0\quad&\text{for}\ x=0,\\\
f_{0}(x)+t\pmod{1}\quad&\text{for}\ x\neq 0.\end{cases}$

Then the following proposition holds for this $f$.

######  Proposition 1.11.

$\mathcal{L}_{f}$ satisfies (UC), but does not satisfy (D).

Next, let us consider $X=X_{-}\cup X_{+}$ with $X_{-}=(-1,0]$, $X_{+}=(0,1]$
and $T=X_{+}$. We equip $X$ and $T$ with the Lebesgue measures $m$ and $p$,
respectively. Let $\iota:X\to X$ be the involution given by $\iota(x)=x+1$ for
$x\in X_{-}$ and $\iota(x)=x-1$ for $x\in X_{+}$, so that $\iota(X_{-})=X_{+}$
and $\iota(X_{+})=X_{-}$. Let $f_{0}:X\to X$ be a measurable map satisfying
that $f_{0}(X_{-})\subset X_{+}$, $f_{0}(X_{+})\subset X_{-}$ and
$f_{0}\circ\iota=\iota\circ f_{0}$ on $X_{+}$. Define the random map
$\widetilde{f}:T\times X_{+}\to X_{+}$ by

$\widetilde{f}_{t}(x)=\iota\circ f_{0}(x)+t\pmod{1}$

and let $f:T\times X\to X$ be the random map given by

$f_{t}(x)=\begin{cases}\iota\circ\widetilde{f}_{t}(x)&\text{for $x\in
X_{+}$},\\\ \widetilde{f}_{t}\circ\iota(x)&\text{for $x\in
X_{-}$}.\end{cases}$ (1.10)

See Figure 5. Then, the following proposition holds for this $f$.

######  Proposition 1.12.

$\mathcal{L}_{f}$ satisfies (D), but does not satisfy (D*).

####  1.4.2. Multiplicative noise

We secondly focus on a family of perturbed systems with multiplicative noise.
As announced, these examples prove the assertions in Remark 1.2. Let us
consider a random map $f:T\times X\to X$ given by the following multiplicative
noise,

$f_{t}(x)=(1-\varepsilon t)f_{0}(x),\quad x\in[0,1],\ \ t\in[0,1]\ \
\text{and}\ \ 0<\varepsilon<1.$ (1.11)

Here $f_{0}:[0,1]\to[0,1]$ is a measurable map and $X=T=[0,1]$ is equipped
with the Lebesgue measure which, as before, we denote by $m$ and $p$
respectively. The next three examples in Proposition 1.13 show that both
conditions (a) and (b) in Remark 1.2 are neither sufficient nor necessary to
get (FPM). Realize that the first example gives an important warning: the
natural relaxation of the condition (b) from “for all $x\in X$” to “for
$m$-almost every $x\in X$” (denoted by _almost-(b)_) is not useful to get
(FPM).

######  Proposition 1.13.

Consider the random map given in (1.11).

  1. (1)

Let $f_{0}(x)=\frac{x}{2}$. Then, $f$ does not satisfy (b), but satisfies (a)
and almost-(b). Moreover, (S) does not hold. In particular, (FPM) does not
hold. However, we have that

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{j}_{\omega}(x)}=\delta_{0}\quad\text{for
all $x\in X$ and $\omega\in\Omega$.}$

  2. (2)

Let $f_{0}(x)=\frac{x}{2}$ if $x\not=0$ and $f_{0}(0)=\frac{1}{2}$. Then, $f$
does not satisfy (a), but satisfies (b). Moreover, (S) does not hold. In
particular, (FPM) does not hold.

  3. (3)

Let $f_{0}(x)=2x\text{ (mod 1)}$. Then, $f$ does not satisfy (a) and (b), but
$\mathcal{L}_{f}$ satisfies (C). In particular, (FPM) holds.

Proposition 1.13 (1) provides a class of examples that does not satisfy (FPM)
but almost-(b) holds. However, this example still has a unique non-absolutely
continuous physical measure. The next example shows that there is a drastic
gap between (b) and almost-(b): finitude versus infinitude.

Consider a random map $f$ under a multiplicative type noise, given by

$f_{t}(x)=tx+(1-t)f_{0}(x)\quad x\in X=[0,1],\ \ t\in T=[0,1].$ (1.12)

Now, we consider a concrete $C^{1}$ map $f_{0}:X\to X$ with infinitely many
sinks, which was essentially given by Araújo in [7, Example 1]. Let
$\phi:X\to\mathbb{R}$ be a $C^{1}$ function given by

$\phi(x)=X^{4}\sin\frac{1}{X}\qquad\text{where $\displaystyle
X=\frac{2}{\pi}\left(x-\frac{1}{2}\right)$}.$ (1.13)

Note that $\phi$ can be seen as a $C^{1}$ function on the Lie group
$S^{1}=\mathbb{R}/\mathbb{Z}$ under the identification of $S^{1}$ with
$X=[0,1]$. Notice also that $\phi$ has (countably) infinitely many local
maxima and minima, which accumulates to $\frac{1}{2}$. Therefore, the one-time
map $f_{0}$ of the gradient flow given by $\dot{x}=\nabla\phi(x)$ has
infinitely many sinks that accumulate to $\frac{1}{2}$ and whose basin covers
$X$ except the sources of $f_{0}$.

######  Proposition 1.14.

Consider the random map given in (1.12) with the time-one map $f_{0}$ of the
gradient flow induced by the potential function (1.13). Then, $f$ does not
satisfy (b), but satisfies (a) and almost-(b). Moreover, there are infinitely
many points $(s_{k})_{k\geq 1}\subset X$ such that

$G(\delta_{s_{k}})=\left\\{x\in
X\,:\,\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\delta_{f^{n}_{\omega}(x)}=\delta_{s_{k}}\quad\text{for
all $\omega\in\Omega$}\right\\}$

is a non-empty open set (in particular, has an $m$-positive measure) for all
$k\geq 1$, and $\bigcup_{k=1}^{\infty}G(\delta_{s_{k}})=X$ up to an $m$-zero
measure set. In particular, (FPM) does not hold.

####  1.4.3. Iterated function systems

Finally, we consider some random maps generated by iterated function systems
(IFS). These examples will disprove the reverse implications in Figures 1. As
explained in Remark 1.6, an IFS with probabilities is a random map $f:T\times
X\to X$ on a finite set $T=\\{1,2,\ldots,k\\}$ with a probability measure $p$
where $p(\\{i\\})=p_{i}>0$ for $i=1,\dots,k$. Notice that this setting allows
the deterministic case, that is, the case $k=1$. As before, set
$\Omega=T^{\mathbb{N}}$ and $\mathbb{P}=p^{\mathbb{N}}$. Then, $\mathbb{P}$ is
the Bernoulli probability on $\Omega$, and the corresponding annealed
Perron–Frobenius operator is of the form

${\mathcal{L}}_{f}\varphi=\int\mathcal{L}_{t}\varphi\,dp(t)=\sum_{i=1}^{k}p_{i}\mathcal{L}_{i}\varphi\quad\text{for
$\varphi\in L^{1}(m)$}$

where $\mathcal{L}_{i}$ is the Perron–Frobenius operator of $f_{i}=f(i,\cdot)$
for $i=1,\dots,k$. Throughout this subsection, we keep the setting and
notations.

(a) Random expanding maps. Let $X$ be the unit interval $[0,1]$ equipped with
the Borel $\sigma$-field $\mathscr{B}$ and the normalized Lebesgue measure
$m$. Let $f_{i}$ be a piecewise $C^{2}$ nonsingular transformation with a
finite partition for each $i=1,\dots,k$. Assume the following expanding (on
average) condition:

$\displaystyle\sum_{i=1}^{k}\frac{p_{i}}{\left\lvert
f^{\prime}_{i}(x)\right\rvert}<1\quad\text{for all $x\in X$.}$ (1.14)

When $x$ is a discontinuity point of some $f_{i}$, the left and right limits
of $f^{\prime}_{i}(x)$ are considered and (1.14) with these limits instead of
$f_{i}^{\prime}(x)$ are required. Then, the following proposition holds for
this $f$.

######  Proposition 1.15.

${\mathcal{L}}_{f}$ satisfies (C), but does not satisfy (UC).

(b) Random contracting maps. Let $(X,\mathscr{B},m)$ be as in the previous
example. Consider the case $k=2$ and

$f_{1}(x)=\frac{x}{2},\quad f_{2}(x)=\frac{x}{2}+\frac{1}{2}\quad\text{($x\in
X$)}\quad\text{with}\quad p_{1}=p_{2}=\frac{1}{2}$

known as a special case of Bernoulli convolution [78]. Then the following
proposition holds for this $f$.

######  Proposition 1.16.

${\mathcal{L}}_{f}$ satisfies (AC), but does not satisfy (C).

######  Remark 1.17.

Note that each $f_{i}$ does not satisfy even (S): the Dirac measure at $0$
(resp. $1$) is the only invariant probability measure of $f_{1}$ (resp.
$f_{2}$), but it is not absolutely continuous with respect to Lebesgue
measure. Namely, the random map $f$ has a property that each deterministic map
$f_{\omega}$ does not have (such behaviors are called _noise-induced
phenomena_ , and have been attracting attention among physicists, cf. [38]).
This is contrastive to other examples in Section 1.4.3, which causes us to
work a bit harder in Section 1.4.4.

######  Remark 1.18.

A slightly weaker condition than (AC) (i.e. “some $\varepsilon>0$” instead of
“any $\varepsilon>0$”) appeared in [55, 58] with the name of smoothing
property, and was later called almost constrictivity in [56]. Note that the
definition of smoothing in [57] is stronger than ones in [55, 58], whence the
one in [57] is indeed equivalent to constrictivity. The author of [55]
asserted that a Markov operator $P$ is asymptotically periodic (or
equivalently constrictive) when $P$ is smoothing. However, this claim cannot
be true because the example in Proposition 1.16 satisfies the smoothing
property, but does not satisfy (C). We also remark that it was proven in [56]
that the almost constrictivity implies (WAP). Figure 1 shows that the results
in this paper largely improved it.

(c) Random rotations. Now let $X=\mathbb{T}^{1}=\mathbb{R}/\mathbb{Z}$
equipped with the Borel $\sigma$-field $\mathscr{B}$ and the normalized
Lebesgue measure $m$. Let $k=2$ and consider two irrational rotations $f_{1}$
and $f_{2}$ with angles $\alpha$ and $\beta$, that is,

$f_{1}(x)=x+\alpha\pmod{1}\quad{\rm and}\quad f_{2}(x)=x+\beta\pmod{1}$

where $\alpha,\beta\in[0,1]$. Let $p_{1}=p_{2}=\frac{1}{2}$.

######  Proposition 1.19.

The following hold.

  1. (1)

If $\alpha-\beta$ is irrational, then ${\mathcal{L}}_{f}$ satisfies (C).

  2. (2)

If $\alpha$ and $\beta$ are irrational numbers such that $\alpha-\beta$ is
rational, then ${\mathcal{L}}_{f}$ satisfies (MC) but does not satisfy (AC).

  3. (3)

If $\alpha$ and $\beta$ are rational numbers, then ${\mathcal{L}}_{f}$
satisfies (WAP) but does not satisfy (MC).

(d) Direct sums of random contraction and expanding map. Let $k=2$ and
consider the direct sum of random transformations, where one satisfies (S) and
the other does not. That is, let $X=X_{-}\sqcup X_{+}$ and equip $X$ with a
probability measure $m$ for which both $X_{-}$ and $X_{+}$ have a positive
measure. With $p_{1}=p_{2}=\frac{1}{2}$, define

$f_{1}(x)=\begin{cases}\tau_{1}^{-}(x)&\text{for $x\in X_{-}$}\\\
\tau_{1}^{+}(x)&\text{for $x\in X_{+}$}\\\ \end{cases}\quad{\rm and}\quad
f_{2}(x)=\begin{cases}\tau_{2}^{-}(x)&\text{for $x\in X_{-}$}\\\
\tau_{2}^{+}(x)&\text{for $x\in X_{+}$}\\\ \end{cases}$

where the random dynamics generated by $\tau_{1}^{+},\tau_{2}^{+}:X_{+}\to
X_{+}$ (with equal probabilities) satisfies (S), and the random dynamics
generated by $\tau_{1}^{-},\tau_{2}^{-}:X_{-}\to X_{-}$ does not satisfy (S).
A typical example is the case when $\tau_{1}^{+},\tau_{2}^{+}$ are one-
dimensional piecewise $C^{2}$ nonsingular transformations with finite
partitions satisfying the expanding condition (1.14) (with $\tau_{i}^{+}$
instead of $f_{i}$), and $\tau_{1}^{-}(x)=\tau_{2}^{-}(x)=\frac{x}{2}$ on
$X_{-}=[0,1]$. Then, the following proposition holds for this $f$.

######  Proposition 1.20.

${\mathcal{L}}_{f}$ satisfies (S), but does not satisfy (WAP).

####  1.4.4. Deterministic systems

By slightly modifying examples in Section 1.4.3, one can easily make examples
of deterministic dynamical systems that disprove the reverse implications in
Figures 1, as follows. Recall that the baker’s transformation is a map $g:X\to
X$ on $X=[0,1]^{2}$ (equipped with the Lebesgue measure $m$) defined by
$g(x,y)=(2x,\frac{y}{2})$ when $0\leq x<\frac{1}{2}$ and
$g(x,y)=(2x-2,\frac{y}{2}+\frac{1}{2})$ when $\frac{1}{2}\leq x\leq 1$.

######  Proposition 1.21.

The following hold.

  1. (1)

Any one-dimensional piecewise $C^{2}$ nonsingular transformation with a finite
partition satisfying (1.14) is (C) but not (UC).

  2. (2)

The baker’s transformation is (AC) but not (C).

  3. (3)

Any irrational rotations are (MC) but not (AC).

  4. (4)

Any rational rotations are (WAP) but not (MC).

  5. (5)

Define $g:X\to X$ as a measurable map preserving a splitting of $X$ by two
positive measure sets $X_{-}$, $X_{+}$ such that the restriction of $g$ on
$X_{+}$ satisfies (S) and the restriction of $g$ on $X_{-}$ does not. Then,
$g$ is (S) but not (WAP).

###  1.5. Questions

We would like to leave questions on the notion of physical noise, a possible
generalization of Theorem C for more general probabilities, and statistical
properties of Markov operators satisfying (AC).

####  1.5.1. On the physical noise

We call noise satisfying condition (b) in Remark 1.2 as _physical noise_. This
kind of noise is frequently used in the literature as indicated in Remarks 1.4
and 1.5. As an important consequence of Remark 1.2, any continuous random
perturbations of a dynamics on a compact space $X$ by a physical noise
satisfies (FPM). Recall that (FPM) asks the existence of finitely many ergodic
measures with disjoint supports which are absolutely continuous with respect
to $m$ and the union of its fiberwise statistical basin of attraction covers
almost surly $X$. Proposition 1.13 provides a class of examples of continuous
random perturbation of a dynamics by an _almost physical noise_ (that is,
condition almost-(b)) which does not satisfy (FPM). However, this example
still has a unique non-absolutely continuous physical measure where any
fiberwise statistical basin of attraction covers the full space up to a null
set. Then, a natural question is whether (FPM) could be weakened by asking
finitely many physical ergodic stationary measures instead of finitely many
absolutely continuous ergodic stationary measures. Call this property (fpm)
for short. On the other hand, Proposition 1.14 shows that considering random
perturbations by almost physical noise is still insufficient to get (fpm).

######  Question 1.

Find a reasonable sufficient condition to get (fpm) generalizing the physical
noise.

####  1.5.2. Generalization of Theorem C

Let $f:T\times X\to X$ be a $\mathbb{P}$-random map as defined in Appendix A
where $\mathbb{P}$ is a shift-invariant probability measure on
$(\Omega,\mathscr{F})=(T^{\mathbb{N}},\mathscr{A}^{\mathbb{N}})$ but not
necessarily a Bernoulli probability as before. Consider the annealed
Perron–Frobenius operator ${\mathcal{L}}_{f}$ given by

${\mathcal{L}}_{f}\varphi=\int\mathcal{L}_{\omega}\varphi\,d\mathbb{P}(\omega)\quad\text{for
$\varphi\in L^{1}(m)$}$

where $\mathcal{L}_{\omega}$ is the Perron–Frobenius of $f_{\omega}=f_{t}$ for
$(\omega_{i})_{i\geq 0}\in\Omega$ with $\omega_{0}=t$. Then Theorem D applies
to ${\mathcal{L}}_{f}$. However, we cannot conclude Theorem C from this
because an invariant density of ${\mathcal{L}}_{f}$ does not correspond in
general with a stationary measure with respect to $\mathbb{P}$. To be more
clear, an invariant density $h$ for ${\mathcal{L}}_{f}$ defines a probability
measure $\mu$ by $d\mu=h\,dm$ which satisfies that

$\mu(A)=\int(f_{\omega})_{*}\mu(A)\,d\mathbb{P}(\omega)\quad\text{for all
$A\in\mathscr{B}$}.$ (1.15)

However, a stationary measure $\mu$ with respect to $\mathbb{P}$ is not
necessarily a measure that satisfies (1.15). See, for instance, [68] where
stationary measures for general Markov probabilities are characterized. This
stationary is necessary to obtain invariant measures for the skew-product $F$
in (1.3) associated with $f$. One can see that if $\mu$ satisfies (1.15) and
$\mathbb{P}$ is not a Bernoulli measure, $\mathbb{P}\times\mu$ is not in
general an invariant measure of $F$ (cf. [11, Example 1.4.7]). Since the
$F$-invariance of $\mathbb{P}\times\mu$ is essential for us to prove Theorem
C, we do not know how to obtain this result in this case.

######  Question 2.

Is it possible to obtain a similar result to Theorem C for a random iteration
driving by a general probability $\mathbb{P}$ on $\Omega$ of a measurable map
$f$ as above?

####  1.5.3. Equivalent conditions to (AC) and (UC)

We next consider the counterpart for (AC) in the position of (APM) for (MC).
First, we introduce some definitions.

An invariant density $h$ of a Markov operator $P:L^{1}(m)\to L^{1}(m)$ is
called _mixing_ (resp.  _exact_) if

$\lim_{n\to\infty}P^{n}\varphi=h\int\varphi\,dm\quad\text{weakly
(resp.\leavevmode\nobreak\ strongly) in $L^{1}(m)$}$

for any $\varphi\in L^{1}(m)$ whose support is included in
$\operatorname*{supp}h$. If $h$ is exact and the above converge occurs
exponentially fast (i.e. there are constant $C>0$, $0<\rho<1$ independently of
$\varphi$ such that $\|P^{n}\varphi-h\int\varphi\,dm\|\leq
C\rho^{n}\|\varphi\|$ for each $n$), we say that $P$ is exponentially exact.
In particular, given a non-singular measurable map $g:X\to X$ and an
$\mathcal{L}_{g}$-invariant density $h$, a probability measure $m_{h}$ given
in (1.6) is mixing (resp. exact) for $g$ if and only if $h$ is mixing (resp.
exact) for $\mathcal{L}_{g}$. See Appendix D for more details.

We also recall the well-known fact that a Markov operator $P$ is (C) if and
only if

  * (AP)

$P$ is _asymptotically periodic_ , that is, there exist finitely many
densities $g_{1},\dots,g_{r}$ with mutually disjoint supports (up to an
$m$-null set), positive bounded linear functionals
$\lambda_{1},\dots,\lambda_{r}$ on $L^{1}(m)$ and a permutation $\rho$ of
$\\{1,\ldots,r\\}$ such that $Pg_{i}=g_{\rho(i)}$ for each $i=1,\dots,r$ and
for $\varphi\in L^{1}(m)$,

$\lim_{n\to\infty}P^{n}\big{(}\varphi-\sum_{i=1}^{r}\lambda_{i}(\varphi)g_{i}\big{)}\to
0\quad\text{strongly in $L^{1}(m)$.}$ (1.16)

It is also known that if $P$ satisfies (AP) with $r=1$, then $g_{1}$ is exact.
Refer to e.g. [63].

Mimicking the above asymptotically periodic condition, we introduce the
following class of Markov operators:

  * (APW)

$P$ is _asymptotically periodic weakly_ , that is, (AP) holds with “weakly”
instead of “strongly” in (1.16).

######  Question 3.

Is (AC) equivalent to (APW)? Moreover, is $g_{1}$ mixing if $r=1$?

The different classes considered in this paper can be classified into three
categories according to the type of conditions involved in the definition of
the class: conditions on _constrictor_ , conditions à la _Dunford–Pettis_ or
conditions on _periodicity_ in the limit. Since many of these conditions are
ultimately equivalent, we have avoided as far as possible introducing many
names to indicate the different equivalent definitions. In Table 1 we organize
this classification of classes, but first we introduce two more classes:

  1. (EAP)

$P$ is _exponentially asymptotically periodic_ , that is, (AP) holds with

$\bigg{\|}P^{n}\big{(}\varphi-\sum_{j=1}^{r}\lambda_{j}(\varphi)g_{j}\big{)}\bigg{\|}\leq
C\rho^{n}\|\varphi\|\quad\text{for any $\varphi\in L^{1}(m)$}$

for some $C>0$, $0<\rho<1$ taken independently of $n$, $\varphi$, instead of
(1.16).

  2. (CW)

$P$ is _constrictive weakly_ 666 There exists a quite similar name in
literature, called _weak constrictivity_. $P$ is said to be weakly
constrictive if there is a weakly compact set $F$ of
$\lim_{n\to\infty}d(P^{n}h,F)=0$ for any $h\in D(m)$. It is known that weak
constrictivity is an equivalent condition to (C) (cf. [56]). , that is, there
exists a weakly compact set $F$ of $D(m)$ such that for any $h\in D(m)$ there
exists $(\psi_{n})_{n\in\mathbb{N}}\subset F$ satisfying that

$\lim_{n\to\infty}(P^{n}h-\psi_{n})=0\quad\text{weakly in $L^{1}(m)$}$

In Proposition 4.15 we will show that (APW) $\Rightarrow$ (CW) $\Rightarrow$
(AC). As we see below, these implications have some consequences. The first
obvious consequence is that Question 3 is actually reduced to prove (AC)
$\Rightarrow$ (APW). Moreover,

(C) $\Rightarrow$ (CW) $\Rightarrow$ (MC) and (AP) $\Rightarrow$ (APW)
$\Rightarrow$ (APM). (1.17)

Indeed, notice first that “$\lim_{n\to\infty}d(P^{n}h,F)=0$” in the notion of
constrictive Markov operator given in Definition 1.7 can be rephrased as there
is $(\psi_{n})_{n\in\mathbb{N}}\in F$ such that $P^{n}h-\psi_{n}\to 0$
strongly in $L^{1}(m)$. Thus, clearly (C) implies (CW). Also, since (AC)
implies (MC), from Proposition 4.15 and Theorem D, we have (CW) $\Rightarrow$
(MC). Similarly, we have (AP) $\Rightarrow$ (APW) $\Rightarrow$ (APM).
Finally, we mention that neither of the converses of the implications in
(1.17) holds. To see this, we will prove in Proposition 7.3 that in example
(b) of Section 1.4.3, $1_{X}$ is a $P$-invariant density (called the trivial
density) which is mixing, but not exact. In particular, $P$ satisfies (APW)
with $r=1$ and consequently (CW) from Proposition 4.15. However, since any
Markov operator with the mixing and non-exact trivial density is not
constrictive (cf. [63]), $P$ in the example does not satisfy (C). Also, (MC)
(resp. (APM)) does not imply (CW) (resp. (APW)) since (AC) is not equivalent
to (MC) (and thus (APM)) from Propositions 1.19 (2) and 1.21 (3).

constrictor |  | (UC)
---
Def. 1.7
| (C)
---
Def. 1.7
| (CW)?
---
p. (CW)
| (MC)
---
Def. 1.7
Dunford–Pettis |  | (UC)
---
Prop. 6.1 (3)
| (C)
---
p. (C)
| (AC)
---
Def. 1.10
| (MC)
---
Thm. 4.1
periodicity |  |  (EAP)?
---
p. (EAP)
| (AP)
---
p. (AP)
|  (APW)?
---
p. (APW)
| (APM)
---
Thm. D
$r=1$ | exp. exact ? | exact | mixing ? | ergodic
Table 1. Classification of different notions of constrictive type classes.

In Theorem 4.1 we prove the equivalence between the different definitions for
the class of constrictive in mean Markov operators. In Proposition 6.1, we
prove the equivalence between the definitions of uniformly constrictive à la
Danford–Pettis and the version on the constrictor. Finally, to complete Table
1 it remains to solve (together with (AC) $\Rightarrow$ (APW)) the following
question:

######  Question 4.

Is (UC) equivalent with (EAP)? Moreover, if (EAP) holds with $r=1$, then is
$g_{1}$ exponentially exact?

###  1.6. Organization of the paper

In Section 2 we will prove Theorem B and show the generalization of Araújo’s
result mentioned after Theorem A. The proof of the most general versions of
Theorems D and E is carried on in Section 4 and 3 respectively. Theorem C is
proved in Section 5. In Section 6 we study the sub-hierarchy in (UC). Finally,
in Section 7 we provide the proof of the propositions of the examples
discussed in the introduction. We also include four appendices that may be of
independent interest. In Appendix A we briefly mention in a general framework
some properties of the annealed Perron–Frobenius operator that we will use.
Appendix B studies and generalizes the restriction of a Markov operator to the
support of a density. In Appendix C, we study the ergodicity of invariant
densities. Finally, in Appendix D we relate the definition of mixing and
exactness in this section with the classical definition of mixing and
exactness for deterministic maps.

##  2\. Feller continuity and quasi-compactness: proof of Theorem B

A Markov transition probability $P(x,A)$ is said to be _Feller continuous_ ,
_strong Feller continuous_ and _ultra Feller continuous_ if the family of
probabilities $P(x,\cdot)$ varies continuously with respect to the weak*
topology, setwise convergence and total variation distance on the space of
probabilities respectively. Namely, for any $x\in X$ and every sequence
$\\{x_{n}\\}_{n\geq 1}$ with $x_{n}\to x$,

  1. (1)

$P(x,A)$ is Feller if $P(x_{n},\cdot)\to P(x,\cdot)$ in the weak* topology,
i.e., if

$\int\varphi(y)\,P(x_{n},dy)\to\int\varphi(y)\,P(x,dy)$

for all bounded continuous real-valued function $\varphi$ on $X$.

  2. (2)

$P(x,A)$ is strong Feller if $P(x_{n},\cdot)\to P(x,\cdot)$ in setwise
convergence, i.e., if

$P(x_{n},A)\to P(x,A)\quad\text{for all $A\in\mathscr{B}$}.$

  3. (3)

$P(x,A)$ is ultra Feller if $P(x_{n},\cdot)\to P(x,\cdot)$ in total variation
distance. i.e., if

$\|P(x_{n},A)-P(x,A)\|_{TV}\to 0$

where the total variation distance of two Borel probability measures $\mu$ and
$\nu$ on $X$ is given by

$\|\mu-\nu\|_{TV}=2\cdot\sup_{A\in{\mathscr{B}}}|\mu(A)-\nu(A)|.$

It is clear by definition that ultra Feller continuity implies strong Feller
continuity. Moreover, since $X$ is a Polish space, an equivalent way of
describing the setwise convergence of a sequence of measures $(\mu_{n})_{n\geq
1}$ to $\mu$ is the following:

$\lim_{n\to\infty}\int\varphi\,d\mu_{n}=\int\varphi\,d\mu,$

for all bounded Borel measurable real-valued function $\varphi$ on $X$. This
is because the simple functions are dense among the bounded Borel measurable
real-valued functions on $X$ under the supremum norm. Thus, as a consequence,
strong Feller continuity implies Feller continuity. The converse of these
implications is not true in general. However, although ultra Feller continuity
seems, at first sight, to be stronger than the strong Feller continuity, it
turns out that the two are almost equivalent. More precisely, according to
[42, Theorem. 3.37], if two Markov transition probabilities $Q(x,A)$ and
$R(x,A)$ are strong Feller, then the convolution $QR$ given by

$QR(x,A)=\int R(y,A)\,Q(x,dy)$

is an ultra Feller continuous Markov transition probability. In view of
Chapman–Kolmogorov relation,

$P^{n+k}(x,A)=\int P^{n}(y,A)P^{k}(x,dy)\quad\text{for all
$n,k\in\mathbb{N}$},$

we have that $P^{n+k}(x,A)$ is the convolution of $P^{n}(x,A)$ and
$P^{k}(x,A)$. In particular, we get the following remark:

######  Remark 2.1.

If $P^{n_{0}}(x,A)$ is strong Feller continuous for some $n_{0}\geq 1$, then
$P^{2n_{0}}(x,A)$ is ultra Feller continuous.

To prove Theorem B we need the following proposition that shows some
equivalent formulation of the assumption of Theorem A. First, recall that a
Markov transition probability $P(x,A)$ is said to be $m$-nonsigular if
$m(A)=0$ implies that $P(x,A)=0$ for $m$-almost every $x\in X$. We also
address the reader to recall how a transition probability can be associated
with a Markov operator explained at the beginning of Section 1.3.4, see also
[70, Chapter V.4].

######  Proposition 2.2.

Let $(X,\mathscr{B},m)$ be a Polish probability space. Consider a Markov
transition probability $P(x,A)$ with $x\in X$ and $A\in\mathscr{B}$. Then, the
following conditions are equivalent:

  1. (a)

there exists $n_{0}\geq 1$ such that

    1. (1)

$P^{n_{0}}(x,A)$ is strong Feller continuous, and

    2. (2)

$P^{n_{0}}(x,\cdot)$ is $m$-nonsigular;

  2. (b)

there exists $n_{0}\geq 1$ such that

    1. (i)

$P^{n_{0}}(x,A)$ is ultra Feller continuous, and

    2. (ii)

$P^{n_{0}}(x,\cdot)$ is absolutely continuous with respect to $m$ for all
$x\in X$;

  3. (c)

there exists $n_{0}\geq 1$ such that

$P^{n_{0}}(x,dy)=p(x,y)\,dm(y)\ \ \text{with}\ \ x\in X\mapsto p(x,\cdot)\in
L^{1}(m)\ \ \text{continuous.}$

Moreover, if $P^{n}(x,A)$ is an $n$-th Markov transition probability
associated to a Markov operator $P:L^{1}(m)\to L^{1}(m)$, then $P^{n}(x,A)$ is
$m$-nonsingular for each $n\geq 1$.

###### Proof.

Assume the condition (a), and show (b). According to Remark 2.1, the condition
(1) implies that $P^{2n_{0}}(x,A)$ is ultra Feller continuous. Moreover, since
$P^{n}(x,A)$ is the convolution of $P$ and $P^{n-1}$ (see (1.4)) we get from
(2) that $P^{n}(x,\cdot)$ is $m$-nonsingular for all $n\geq n_{0}$ and
$m$-almost every $x\in X$. In particular, $P^{2n_{0}}(x,\cdot)$ is
$m$-nonsingular. In fact, the continuity with respect to the total variation
distance implies that $P^{2n_{0}}(x,\cdot)$ is actually absolutely continuous
with respect to $m$ for all $x\in X$. Indeed, take $A\in\mathscr{B}$ with
$m(A)=0$, $x\in X$ and consider $x_{n}\to x$ with $x_{n}\in X$ such that
$P^{2n_{0}}(x_{n},A)=0$ for all $n\geq 1$. By the ultra Feller continuity of
$P^{2n_{0}}(\cdot,A)$ we have that $P^{2n_{0}}(x_{n},A)\to P^{2n_{0}}(x,A)$ as
$n\to\infty$. Then $P^{2n_{0}}(x,A)=0$ as required. Consequently, we get (i)
and (ii) for the positive integer $2n_{0}$.

Conversely, (i) and (ii) clearly imply (1) and (2). We now prove that (i) and
(ii) are equivalent to $L^{1}(m)$-continuity of the Radon–Nikodým derivative
$p(x,\cdot)$ of $P^{n_{0}}(x,\cdot)$ with respect to $m$. But this follows
immediately from the well-known fact that

$\left\lVert
P^{n_{0}}(x,\cdot)-P^{n_{0}}(x^{\prime},\cdot)\right\rVert_{TV}=\left\lVert
p(x,\cdot)-p(x^{\prime},\cdot)\right\rVert\quad\text{for all $x,x^{\prime}\in
X$}.$

See the equation before Lemma 2 of [61]. This completes the proof of the
equivalences.

The last assertion follows immediately from the fact that
$P^{n}(x,A)=P^{n*}1_{A}(x)$ for $m$-almost every $x\in X$. Indeed, by duality
$\int P^{n}(x,A)\,dm=\int_{A}P^{n}1_{X}\,dm=0$ whenever $m(A)=0$.
Consequently, $P^{n}(x,A)=0$ for $m$-almost every $x\in X$ concluding that
$P^{n}(x,A)$ is $m$-singular. ∎

The following result is Theorem B stated in terms of Markov processes and
Markov operators.

######  Theorem 2.3.

Let $(X,\mathscr{B},m)$ be a compact Polish probability space. Consider a
Markov operator $P:L^{1}(m)\to L^{1}(m)$ and let $P^{n}(x,A)$ be an associated
$n$-th transition probability. Assume that there exists $n_{0}\in\mathbb{N}$
such that $P^{n_{0}}(x,A)$ is strong Feller continuous. Then $P$ is eventually
compact.

###### Proof.

According to Proposition 2.2, we can assume that $P^{n_{0}}(x,A)$ satisfies
conditions (i) and (ii) in that proposition. Now, from [43, Lemma 1], we have
that the absolute continuity of $P^{n_{0}}(x,\cdot)$ with respect to $m$ is
equivalent to the uniformly absolutely continuity of such measure with respect
to $m$ for all $x\in X$. That is, for any $x\in X$ and $\varepsilon>0$, there
is $\delta_{x}=\delta_{x}(\varepsilon)>0$ such that

$P^{n_{0}}(x,A)<\varepsilon\quad\text{for all $A\in\mathscr{B}$ with
$m(A)<\delta_{x}$.}$

######  Claim 2.4.

For each $\varepsilon>0$, there is $\delta=\delta(\varepsilon)>0$ such that

$P^{n_{0}}(x,A)<\varepsilon$ for all $A\in\mathscr{B}$ with $m(A)<\delta$ and
$x\in X$.

###### Proof.

By the continuity of the family of probability measures $P^{n_{0}}(x,\cdot)$
with respect to the total variation, for each $x\in X$, there exists a
neighborhood $V(x,\varepsilon)$ of $x$ such that
$P^{n_{0}}(x^{\prime},A)<\varepsilon$ for all $x^{\prime}\in V(x,\varepsilon)$
and $A\in\mathscr{B}$ with $m(A)<\delta_{x}$. Since the union of the open sets
$V(x,\varepsilon)$ for $x\in X$ covers the compact set $X$, we can extract a
finite subcover $V(x_{1},\varepsilon),\dots,V(x_{k},\varepsilon)$. Thus, the
claim follows by taking
$\delta=\min\\{\delta_{x_{1}},\dots,\delta_{x_{k}}\\}$. ∎

Finally, let us conclude the proof. First, recall that a _bounded_ family
$F\subset L^{1}(m)$ is said to be _uniform integrable_ (in $L^{1}(m)$) if for
every $\varepsilon>0$ there is $\delta>0$ such that

$\int_{A}|g|\,dm<\varepsilon\quad\text{for all $g\in F$ and $A\in\mathscr{B}$
with $m(A)<\delta$}.$

Writing $P^{n_{0}}(x,dy)=p(x,y)\,dm(y)$ and take $F=\\{p(x,\cdot):x\in
X\\}\subset D(m)$. Claim 2.4 implies that $F$ is uniform integrable (in
$L^{1}(m)$). Then, according to [91, Corollary 2.5 (b)], the operator
$\pi^{2}$ is compact where $\pi:L^{\infty}(m)\to L^{\infty}(m)$ is given by

$\pi\psi(x)=\int\psi(y)p(x,y)\,dm(y)=\int\psi(y)P^{n_{0}}(x,dy)=P^{n_{0}*}\psi(x),\quad\psi\in
L^{\infty}(m).$

That is, $\pi$ is the adjoint operator $P^{n_{0}*}$ of $P^{n_{0}}$. Thus,
$P^{2n_{0}*}=(P^{n_{0}*})^{2}=\pi^{2}$ is a compact operator. Hence, by
Schauder’s theorem (cf. [79, Theorem 4.19]), $P^{2n_{0}}$ is also compact,
concluding the proof. ∎

We just indicate that Theorem B follows immediately from Proposition 2.2 and
Theorem 2.3. Actually, Theorem B and Theorem 2.3 are equivalent from the
following observation.

######  Remark 2.5.

As indicated in the introduction, it is well known that the theory of Markov
operators and Markov processes are intimately related. Less known perhaps is
that the general theory of Markov operators is actually equivalent to the
particular theory of annealed Perron–Frobenius operator associated with random
maps on Polish probability space $(X,\mathscr{B},m)$. Indeed, clearly given a
random map $f$ we define a Markov operator by means of the annealed
Perron–Frobenius operator $\mathcal{L}_{f}$. Conversely, given a Markov
operator $P:L^{1}(m)\to L^{1}(m)$, we consider a transition probability
$P(x,A)$ which induces $P$. See [70, Proposition V.4.4] and Section 1.3.4 for
more details. Notice that

$P(x,A)=P^{*}1_{A}(x)\quad\text{for $m$-almost every $x\in X$}$

where $P^{*}$ is the adjoint operator of $P$. Now, Kifer proved in [51,
Theorem 1.1] that any transition probability in a Polish777Actually the only
requirement is that $(X,\mathscr{B})$ will be countably generated measurable
space, i.e., that the $\sigma$-algebra $\mathscr{B}=\sigma(\mathcal{A})$ for
some countable subset $\mathcal{A}$ of $\mathscr{B}$. space $X$ can be
represented by a random product of independent and identically distributed
measurable maps. That is, there exists a probability space $(T,\mathscr{A},p)$
and a measurable map $f:T\times X\to X$ such that

$P(x,A)=p(\\{t\in T:f_{t}(x)\in A\\})$

where $f_{t}=f(t,\cdot)$. This implies that
$P(x,A)=\mathcal{L}^{*}_{f}1_{A}(x)$ for $m$-almost every $x\in X$ where
$\mathcal{L}^{*}_{f}$ is the adjoint operator of the annealed Perron–Frobenius
operator $\mathcal{L}_{f}$ associated with $f$. Hence,
$P^{*}1_{A}=\mathcal{L}^{*}_{f}1_{A}$ and therefore $P=\mathcal{L}_{f}$. To
see this final assertion, observe first that any $g\in L^{\infty}(m)$ can be
approximated uniformly by simple functions
$g_{n}=\sum_{i=1}^{N}a_{i}1_{A_{i}}$ where $a_{i}=a_{i}(n)\in\mathbb{R}$,
$A_{i}=A_{i}(n)\in\mathscr{B}$ and $N=N(n)\in\mathbb{N}$. Then, $g_{n}$
converges to $g$ in $L^{\infty}(m)$-norm and hence

$P^{*}g=\lim_{n\to\infty}P^{*}g_{n}=\lim_{n\to\infty}\sum_{i=1}^{N}a_{i}\,P^{*}1_{A_{i}}=\lim_{n\to\infty}\sum_{i=1}^{N}a_{i}\,\mathcal{L}_{f}^{*}1_{A_{i}}=\lim_{n\to\infty}\mathcal{L}^{*}_{f}g_{n}=\mathcal{L}^{*}_{f}g$

This implies that $P^{*}=\mathcal{L}_{f}^{*}$ as well as $P=\mathcal{L}_{f}$
as required.

###  2.1. Generalization of Araújo’s result

We will prove that under the condition (i) mentioned in Remark 1.2, the
assumptions of Theorems A and B hold. To do this, we need some preliminaries.

The following proposition provides a well-known sufficient condition to obtain
that a given Markov transition probability is Feller continuous. The proof is
straightforward and could be founded in [41, Theorem 4.22]. Recall that a
continuous random map is a measurable map $f:T\times X\to X$ such that
$f_{t}=f(t,\cdot)$ is continuous for $p$-almost every $t\in T$.

######  Proposition 2.6.

If $P(x,A)$ is the transition probability associated to a continuous random
map, then $P(x,A)$ is Feller continuous.

The following result extends [60, Theorem 1.2 in Section 4] to the case of
compact Polish probability spaces.

######  Theorem 2.7.

Let $(X,\mathscr{B},m)$ be a compact Polish probability space. Consider a
Markov transition probability $P(x,A)$ with $x\in X$ and $A\in\mathscr{B}$.
Assume that

  1. (1)

$P(x,A)$ is Feller continuous, and

  2. (2)

$P(x,\cdot)$ is absolutely continuous with respect to $m$ for all $x\in X$.

Then $P(x,A)$ is a strong Feller continuous transition probability.

We will prove the above result by modifying the argument in [60] where Theorem
2.7 was shown with $\mathbb{R}^{d}$ instead of $X$. For this modification, we
need the following.

######  Lemma 2.8.

Let $(X,\mathscr{B},m)$ be a Polish probability space. Then, for any
$A\in\mathscr{B}$ and $\delta>0$, there exists $B\in\mathscr{B}$ such that

$m(\partial B)=0\quad\text{and}\quad m(A\Delta B)<\delta$

where $\partial B$ is the boundary of $B$ and $A\Delta B$ is the symmetric
difference of $A$ and $B$.

###### Proof.

Consider the subalgebra $\mathcal{A}$ in $\mathscr{B}$ of $m$-continuity sets,
i.e., $B\in\mathcal{A}$ if and only if $B\in\mathscr{B}$ and $m(\partial
B)=0$. Let us prove that the $\sigma$-algebra generated by the $\mathcal{A}$
is $\mathscr{B}$. To see this, fix any metric $d$ compatible with the topology
of $X$. Note that for $x\in X$, $\partial{B}_{r}(x)\subset\\{y\in
X:d(x,y)=r\\}$ where $B_{r}(x)$ denotes the ball centered at $x$ and of radius
$r>0$. In particular, $\partial{B}_{r}(x)\cap\partial B_{s}(s)=\emptyset$ for
$r\not=s$. But in a space of finite measure, there can only be countably many
pairwise disjoint sets of positive measure. Hence, $B_{r}(x)\in\mathcal{A}$
for all but at most countably many $r$. In particular, $\mathcal{A}$ contains
a neighborhood base of $x$. This shows that $\sigma(\mathcal{A})=\mathscr{B}$
as desired. Now, since the subalgebra $\mathcal{A}\subset\mathscr{B}$
generates $\mathscr{B}$, according to the well-known result on approximation
generating subalgebras (cf. [66, Theorem 1.1]), for every $A\in\mathscr{B}$
and $\delta>0$, there is $B\in\mathcal{A}$ such that $m(A\Delta B)<\delta$.
This concludes the proof of the lemma. ∎

Now, we are in a position to prove Theorem 2.7.

###### Proof of Theorem 2.7.

Let us fix $A\in\mathscr{B}$ and $x\in X$, and consider a sequence
$\\{x_{n}\\}_{n\geq 1}$ converging to $x$. We will prove that $P(x_{n},A)\to
P(x,A)$ as $n\to\infty$ concluding the strong Feller continuity of $P(x,A)$.
It follows from Portemanteau’s theorem (cf. [54, Theorem 13.16]) that
$P(x_{n},\cdot)$ converges to $P(x,\cdot)$ in the weak∗ topology if and only
if $P(x_{n},B)$ converges to $P(x,B)$ for any continuity set $B$ of
$P(x,\cdot)$, i.e. any $B\in\mathscr{B}$ satisfying $P(x,\partial B)=0$. The
former condition holds because we assumed that $P(x,A)$ is Feller continuous,
and thus, recalling that $P(x,\cdot)$ is absolutely continuous with respect to
$m$ by assumption,

$P(x_{n},B)\to P(x,B)\quad\text{for any $B\in\mathscr{B}$ satisfying
$m(\partial B)=0$.}$ (2.1)

Now, according to [43, Lemma 1], the absolute continuity of a probability
measure $\nu$ with respect to $m$ is equivalent to the uniform absolute
continuity of $\nu$ with respect to $m$. That is, $\nu(B)\to 0$ as $m(B)\to
0$. Thus, for each $\varepsilon>0$ and $n\geq 1$, by Lemma 2.8 and the
absolute continuity of $P(x_{n},\cdot)$ with respect to $m$, one can find
$B_{\varepsilon,n}\in\mathscr{B}$ such that

$m(\partial B_{\varepsilon,n})=0\quad\text{and}\quad P(x_{n},A\Delta
B_{\varepsilon,n})<\frac{\varepsilon}{2^{n}}.$

Set

$C_{\varepsilon}=\bigcup_{n\geq 1}B_{\varepsilon,n}\quad\text{and}\quad
D_{\varepsilon}=\bigcap_{n\geq 1}B_{\varepsilon,n}.$

######  Claim 2.9.

It holds that

$P(x,C_{\varepsilon})-2\varepsilon\leq\liminf_{n\to\infty}P(x_{n},A)\leq\limsup_{n\to\infty}P(x_{n},A)\leq
P(x,C_{\varepsilon}).$

###### Proof.

Since

$m(\partial(C_{\varepsilon}\setminus D_{\varepsilon}))\leq m(\partial
C_{\varepsilon}\cup\partial D_{\varepsilon})\leq
2\sum_{n=1}^{\infty}m(\partial B_{\varepsilon,n})=0$

it follows from (2.1) that

$\displaystyle\limsup_{n\to\infty}P(x_{n},C_{\varepsilon}\setminus
B_{\varepsilon,n})$
$\displaystyle\leq\limsup_{n\to\infty}P(x_{n},C_{\varepsilon}\setminus
D_{\varepsilon})=P(x,C_{\varepsilon}\setminus D_{\varepsilon})$
$\displaystyle\leq P(x,(A\Delta C_{\varepsilon})\cup(A\Delta
D_{\varepsilon}))\leq
2\sum_{n=1}^{\infty}\frac{\varepsilon}{2^{n}}=2\varepsilon.$

Therefore, since

$\displaystyle P(x_{n},C_{\varepsilon})$ $\displaystyle\leq
P(x_{n},C_{\varepsilon}\setminus B_{\varepsilon,n})+P(x_{n},A\cup
B_{\varepsilon,n})$ $\displaystyle\leq P(x_{n},C_{\varepsilon}\setminus
B_{\varepsilon,n})+P(x_{n},A\Delta B_{\varepsilon,n})+P(x_{n},A)$

by applying (2.1) again (note that $m(\partial
C_{\varepsilon})\leq\sum_{n=1}^{\infty}m(\partial B_{\varepsilon,n})=0$) we
have

$\liminf_{n\to\infty}P(x_{n},A)\geq\liminf_{n\to\infty}P(x_{n},C_{\varepsilon})-2\varepsilon-\lim_{n\to\infty}\frac{\varepsilon}{2^{n}}=P(x,C_{\varepsilon})-2\varepsilon.$

On the other hand, since

$P(x_{n},A)\leq P(x_{n},B_{\varepsilon,n})+P(x_{n},A\Delta
B_{\varepsilon,n})\leq P(x_{n},C_{\varepsilon})+\frac{\varepsilon}{2^{n}}$

we have

$\displaystyle\limsup_{n\to\infty}P(x_{n},A)\leq P(x,C_{\varepsilon})$

concluding the proof of the claim. ∎

Finally, observe that

$|P(x,C_{\varepsilon})-P(x,A)|\leq P(x,A\Delta
C_{\varepsilon})\leq\sum_{n=1}^{\infty}\frac{\varepsilon}{2^{n}}=\varepsilon.$

Thus, $P(x,C_{\varepsilon})\to P(x,A)$ as $\varepsilon\to 0$. Hence, by Claim
2.9, we get that

$\lim_{n\to\infty}P(x_{n},A)=\lim_{\varepsilon\to\infty}P(x,C_{\varepsilon})=P(x,A).$

This concludes the proof of the theorem. ∎

######  Remark 2.10.

A converse of Theorem 2.7 follows from [34, Proposition 12.1.7]. Also, from
[47, Remark 2], by arguing as in the implication of (b) from (a) in
Proposition 2.2, we immediately get the following more specific converse: if
$P(x,A)$ is a strong Feller continuous Markov transition probability on a
Polish probability space $(X,\mathscr{B},m)$, then there exists a probability
measure $m^{*}$ on $(X,\mathscr{B})$ such that $m$ is absolutely continuous
with respect to $m^{*}$ and $P(x,A)$ satisfies (1) and (2) with $m^{*}$
instead $m$.

######  Corollary 2.11.

Let $(X,\mathscr{B},m)$ be a compact Polish probability space. Consider a
continuous random map $f:T\times X\to X$ where $(T,\mathscr{A},p)$ is a
probability space and let $P^{n}(x,A)$ be the associated $n$-th transition
probability. Assume that there is $n_{0}\geq 1$ such that

$P^{n_{0}}(x,\cdot)\ \ \text{is absolutely continuous with respect to $m$ for
all $x\in X$}.$

Then, for each $x\in X$, there exists $p(x,\cdot)\in D(m)$ such that

$P^{2n_{0}}(x,A)=p(x,y)\,dm(y)\quad\text{and}\quad x\in X\mapsto p(x,\cdot)\in
L^{1}(m)\ \ \text{is continuous}.$

Moreover, the annealed Perron–Frobenius operator $\mathcal{L}_{f}$ associated
with $f$ is eventually compact.

###### Proof.

According to Proposition 2.6, $P^{n_{0}}(x,A)$ is a Feller continuous Markov
transition probability. Hence, by Theorem 2.7, we have that $P^{n_{0}}(x,A)$
is actually strong Feller continuous. Therefore, Remark 2.1, Proposition 2.2
and Theorem 2.3 immediately imply the conclusion of the corollary. ∎

##  3\. Existence of invariant measures: proof of Theorem E

In what follows, $(X,\mathscr{B},m)$ denotes any abstract probability space.
We will prove in this section a generalization of Theorem E which will be used
to prove Theorem D in the next section. But first, to prove these results we
will need some preliminaries that can be interesting by themselves.

###  3.1. Fundamental lemma

If $\phi$ belongs to $L^{1}(m)$, then the support of $\phi$, denoted by
$\operatorname*{supp}\phi$, is not defined in a unique way, since $\phi$ can
be represented by two functions whose values are different in a zero
$m$-measuring set. However, since it is common to simplify terminology, we
consider $\operatorname*{supp}\phi$ as a set defined by the relation
$\phi\not=0$. Moreover, if we want to emphasize that a relationship between
sets holds except a set of zero $m$-measure, we say that it holds _up to an
$m$-null set_. For instance, $A\subseteq B$ up to an $m$-null set if
$m(A\setminus B)=0$.

######  Lemma 3.1.

Let $E$ be either $L^{1}(m)$ or $L^{\infty}(m)$. Let $P:E\to E$ be a positive
linear bounded operator and consider $\phi,\psi\in E$ with $\psi\geq 0$. It
holds that

  1. (1)

$\operatorname*{supp}P\phi\subset\operatorname*{supp}P1_{\operatorname*{supp}\phi}$
and
$\operatorname*{supp}P\psi=\operatorname*{supp}P1_{\operatorname*{supp}\psi}$,

  2. (2)

if $\operatorname*{supp}\phi\subset\operatorname*{supp}\psi$, then
$\operatorname*{supp}P\phi\subset\operatorname*{supp}P\psi$.

###### Proof.

It is easy to see that (1) implies (2). Indeed, if
$\operatorname*{supp}\phi\subset\operatorname*{supp}\psi$, we have that
$1_{\operatorname*{supp}\phi}\leq 1_{\operatorname*{supp}\psi}$ and since $P$
is positive linear operator, $P1_{\operatorname*{supp}\phi}\leq
P1_{\operatorname*{supp}\psi}$. This implies that
$\operatorname*{supp}P1_{\operatorname*{supp}\phi}\subset\operatorname*{supp}P1_{\operatorname*{supp}\psi}$.
Now, (1) immediately implies the conclusion of (2). But, actually, if (2)
holds for any pair $\phi$ and $\psi$ in the assumptions of the lemma, we also
get (1) as follows. Since
$\operatorname*{supp}\phi=\operatorname*{supp}1_{\operatorname*{supp}\phi}$,
by (2) we get that
$\operatorname*{supp}P\phi\subset\operatorname*{supp}P1_{\operatorname*{supp}\phi}$.
Same argument and inclusion hold for $\psi$. But, in this case, taking into
account that $\psi\geq 0$, we can also apply (2) to
$\operatorname*{supp}1_{\operatorname*{supp}\psi}=\operatorname*{supp}\psi$
getting the other inclusion and proving (1).

In view of the above observation, the proof of the lemma is reduced to show
(2). To do this, let $\phi$ and $\psi$ be as in the statement. From the
approximation by simple functions, $\phi$ and $\psi$ can be written as the
pointwise limit of a functions

$\phi_{n}=\sum_{k=1}^{N}a_{k}1_{F_{k}}\quad\text{and}\quad\psi_{n}=\sum_{k=1}^{M}b_{k}1_{G_{k}}$

respectively, where $N=N(n),M=M(n)\in\mathbb{N}$, $a_{k}=a_{k}(n)\not=0$,
$b_{k}=b_{k}(n)>0$ and $F_{k}=F_{k}(n),G_{k}=G_{k}(n)\in\mathscr{B}$ such that
$F_{\ell}\cap F_{j}=\emptyset$ and $G_{\ell}\cap G_{j}=\emptyset$ if
$\ell\not=j$. Moreover, $\phi_{n}$ and $\psi_{n}$ can be chosen888For a non-
negative measurable function $g$, we consider the mesh with width $2^{-n}$
from the level $2^{-n}$ up to $2^{n}$, i.e.,
$g_{n}=\sum_{k=0}^{2^{2n}-1}(k+1)2^{-n}1_{g^{-1}(k2^{-n},\,(k+1)2^{-n}]}+2^{n}1_{g^{-1}(2^{n},\,\infty)}.$
Then we can see that $g_{n}$ pointwise converges to $g$ since $0\leq
g_{n}(x)-g(x)<2^{-n}$ on the set where $g\leq 2^{n}$. Moreover,
$\operatorname*{supp}g=\operatorname*{supp}g_{n}$. For a general $g$, we
consider $g=g^{+}-g^{-}$ where $g^{+}$ and $g^{-}$ are the positive and
negative parts of $g$ which we approximate by simple functions $g^{+}_{n}$ and
$g_{n}^{-}$ as before. Finally, it is simple to verify that
$g_{n}=g^{+}_{n}-g^{-}_{n}$ converges in the norm $L^{1}(m)$ or
$L^{\infty}(m)$ where appropriate, to $g$ and
$\operatorname*{supp}g_{n}=\operatorname*{supp}g$. so that

  1. (i)

$\operatorname*{supp}\phi_{n}=\operatorname*{supp}\phi\subset\operatorname*{supp}\psi=\operatorname*{supp}\psi_{n}$,
and

  2. (ii)

$\phi_{n}$ and $\psi_{n}$ converges in norm of $E$ to $\phi$ and $\psi$,
respectively.

Since $P$ is a bounded operator, (ii) implies that $P\phi_{n}\to P\phi$ and
$P\psi_{n}\to P\psi$. Then

  1. (iii)

$m((\operatorname*{supp}P\phi_{n})\Delta(\operatorname*{supp}P\phi))\to 0$ and
$m((\operatorname*{supp}P\psi_{n})\Delta(\operatorname*{supp}P\psi))\to 0$ as
$n\to\infty$.

Note that by (i), $\phi_{n}$ and $\psi_{n}$ satisfy the assumption in (2). It
is easy to see that if (2) holds for $\phi_{n}$ and $\psi_{n}$, then (iii)
implies that $\operatorname*{supp}P\phi\subset\operatorname*{supp}P\psi$ up to
an $m$-null set. Hence, we reduce the proof to check the conclusion of (2) for
$\phi_{n}$ and $\psi_{n}$.

Taking into account $a_{k}\not=0$, $P1_{F_{k}}\geq 0$ and the linearity of
$P$, we see that

$\displaystyle\operatorname*{supp}P\phi_{n}=\operatorname*{supp}\sum_{k=1}^{N}a_{k}P1_{F_{k}}$
$\displaystyle\subset\bigcup_{k=1}^{N}\operatorname*{supp}P1_{F_{k}}$ (3.1)
$\displaystyle=\operatorname*{supp}\sum_{k=1}^{N}P1_{F_{k}}=\operatorname*{supp}P1_{\cup_{k=1}^{N}F_{k}}=\operatorname*{supp}P1_{\operatorname*{supp}\phi_{n}}.$

Now, using that $b_{k}>0$, $P1_{G_{k}}\geq 0$ and the linearity of $P$,

$\operatorname*{supp}P1_{\operatorname*{supp}\psi_{n}}=\operatorname*{supp}P1_{\cup_{k=1}^{M}G_{k}}=\operatorname*{supp}\sum_{k=1}^{M}P1_{G_{k}}=\operatorname*{supp}\sum_{k=1}^{M}b_{k}P1_{G_{k}}=\operatorname*{supp}P\psi_{n}.$
(3.2)

Observe that (3.1) and (3.2) implies (1) for $\phi_{n}$ and $\psi_{n}$. As we
see at the beginning of the proof, (1) implies (2) and we conclude the lemma.
∎

###  3.2. Restrictions of Markov operatos

Fix $S\in\mathscr{B}$ with $m(S)>0$. Let us consider the probability space
$(S,\mathscr{B}_{S},m_{S})$ where

$\mathscr{B}_{S}=\\{A\in\mathscr{B}:A\subset S\\}\quad\text{and}\quad
m_{S}(A)=\frac{m(A)}{m(S)},\ \ A\in\mathscr{B}_{S}\ \ \ \left(\text{i.e.,}\
dm_{S}=\frac{1_{S}}{m(S)}\,dm\right).$

Denote $L^{1}(S,\mathscr{B}_{S},m_{S})$ by $L^{1}(m_{S})$. Observe that
$L^{1}(m_{S})\hookrightarrow L^{1}(m)$ by means of the inclusion

$1_{S}:\phi\in L^{1}(m_{S})\mapsto 1_{S}\phi\in L^{1}(m)\quad\text{given
by}\quad 1_{S}\phi(x)=\begin{cases}\phi(x)&x\in S,\\\ 0&x\in X\setminus
S.\end{cases}$ (3.3)

Although we are using the same notation $1_{S}$ to designate the
characteristic function of $S$ and the inclusion by the characteristic
function, it should cause no confusion. Now, given a Markov operator
$P:L^{1}(m)\to L^{1}(m)$, we define an operator

$P_{S}:L^{1}(m_{S})\to L^{1}(m_{S}),\qquad P_{S}\phi=1_{S}\cdot
P(1_{S}\phi)\quad\text{for}\ \phi\in L^{1}(m_{S}).$ (3.4)

It is not difficult to see that $P_{S}$ is a positive contraction of
$L^{1}(m_{S})$, that is, $P_{S}\phi\geq 0$ if $\phi\geq 0$ and $\|P_{S}\|\leq
1$. Taking advantage of the abuse of notation, we can extend $P_{S}$ to
$L^{1}(m)$ as follows:

$P_{S}\phi=1_{S}\cdot P(1_{S}\cdot\phi)\quad\text{for}\ \phi\in L^{1}(m).$

When no confusion can arise, we will omit the dot in the above expression and
in (3.4). We also identify $L^{1}(m_{S})$ with

$1_{S}(L^{1}(m_{S}))=\\{\phi\in L^{1}(m):\operatorname*{supp}\phi\subset
S\\}\subset L^{1}(m).$

######  Lemma 3.2.

Let $P:L^{1}(m)\to L^{1}(m)$ be a Markov operator and consider
$S\in\mathscr{B}$ with $m(S)>0$. If $\operatorname*{supp}P1_{S}\subset S$ up
to an $m$-null set, then $P_{S}:L^{1}(m_{S})\to L^{1}(m_{S})$ defined in (3.4)
is a Markov operator and

$P_{S}\phi=P(1_{S}\phi)\quad\text{for all $\phi\in L^{1}(m)$}.$

###### Proof.

Let $\phi\in L^{1}(m)$. Note that $\operatorname*{supp}1_{S}\phi\subset
S=\operatorname*{supp}1_{S}$. Thus, by Lemma 3.1,
$\operatorname*{supp}P(1_{S}\phi)\subset\operatorname*{supp}P1_{S}\subset S$
up to an $m$-null set. In particular, $1_{S}P(1_{S}\phi)=P(1_{S}\phi)$. Then,
for any $\phi\in L^{1}(m)$,

$\int
P_{S}\phi\,dm_{S}=\frac{1}{m(S)}\int_{S}P(1_{S}\phi)\,dm=\frac{1}{m(S)}\int
P(1_{S}\phi)\,dm=\frac{1}{m(S)}\int_{S}\phi\,dm=\int\phi\,dm_{S}.$

Therefore, $P_{S}$ is a Markov operator. ∎

The implication in the previous lemma is actually an equivalence. To see this
observation and other interesting equivalent conditions see Proposition B.1 in
the Appendix B where the theory of the restriction of a Markov operator is
extended and clarified. For the following result, recall the notion of weak
almost periodicity (WAP) in Definition 1.9. As mentioned, according to [88,
Theorem 3.1], (WAP) is equivalent to the existence of an invariant density
with the maximal support.

######  Proposition 3.3.

Let $P:L^{1}(m)\to L^{1}(m)$ be a Markov operator and consider $h\in D(m)$
such that $Ph=h$ and set $S=\operatorname*{supp}h$. Then,
$P_{S}:L^{1}(m_{S})\to L^{1}(m_{S})$ is a weakly almost periodic Markov
operator. Moreover, $P_{S}h=h$ and $P_{S}\phi=P(1_{S}\phi)$ for all $\phi\in
L^{1}(m)$.

###### Proof.

Note that $\operatorname*{supp}h=S=\operatorname*{supp}1_{S}$ and by Lemma 3.1
and the $P$-invariance of $h$, it follows that
$S=\operatorname*{supp}h=\operatorname*{supp}Ph=\operatorname*{supp}P1_{S}$ up
to an $m$-null set as well as $m(S)>0$. Hence, Lemma 3.2 implies that $P_{S}$
is a Markov operator and $P_{S}\phi=P(1_{S}\phi)$ for all $\phi\in L^{1}(m)$.
To prove that $P_{S}$ is also weakly almost periodic, it suffices to see that
$P_{S}h=h$ since $P_{S}^{*}1_{S}=1_{S}$ by the Markov property and thus $h$
has trivially the maximal support (as a fixed point of $P_{S}$). But this is
proved as follows: $P_{S}h=P(1_{S}h)=Ph=h.$ ∎

###  3.3. Proof of Theorem E

In this subsection, we prove a more general version of Theorem E. In
particular, we remark that the new item (5) below was shown in [84] to be a
sufficient condition for the existence of a $P$-invariant density, and below
we show it is indeed also a necessary condition.

######  Theorem 3.4.

Let $(X,\mathscr{B},m)$ be an abstract probability space and consider a Markov
operator $P:L^{1}(m)\to L^{1}(m)$. Then, the following conditions are
equivalent:

  1. (1)

There exists an invariant density for $P$;

  2. (2)

There exist $\alpha\in(0,1)$ and $\delta>0$ such that

$\sup_{n\geq 0}\int_{A}P^{n}1_{X}\,dm<\alpha\quad\text{for any
$A\in\mathscr{B}$ with $m(A)<\delta$};$

  3. (3)

There exist $\alpha\in(0,1)$ and $\delta>0$ such that

$\sup_{n\geq 0}\int_{A}A_{n}1_{X}\,dm<\alpha\quad\text{for any
$A\in\mathscr{B}$ with $m(A)<\delta$};$

  4. (4)

There exist $\alpha\in(0,1)$ and $\delta\geq 0$ such that

$\inf_{n\geq 1}\int_{A}A_{n}1_{X}\,dm>1-\alpha\quad\text{for any
$A\in\mathscr{B}$ with $m(A)>\delta$}.$

  5. (5)

There exist $\alpha\in(0,1)$ and $\delta>0$ such that

$\limsup_{n\to\infty}\int_{A}P^{n}1_{X}\,dm<\alpha\quad\text{for any
$A\in\mathscr{B}$ with $m(A)<\delta$};$

  6. (6)

There exist $\alpha\in(0,1)$ and $\delta>0$ such that

$\limsup_{n\to\infty}\int_{A}A_{n}1_{X}\,dm<\alpha\quad\text{for any
$A\in\mathscr{B}$ with $m(A)<\delta$};$

The function $1_{X}$ in (5) and (6) could be substituted by any arbitrary
density $h\in D(m)$. Moreover, the equivalence also holds by taking $\alpha=1$
in all the above items.

###### Proof.

We first prove (1) implies (2). Let $g\in D(m)$ be an invariant density of
$P$, that is, $Pg=g$. Hence, Proposition 3.3 implies that
$P_{S}:L^{1}(m_{S})\to L^{1}(m_{S})$ is a weak almost periodic Markov operator
where $S=\operatorname*{supp}g$. According to [88, Theorem 3.1], this is
equivalent to weak compactness of $\\{P_{S}^{n}1_{S}\\}_{n\geq 0}$ in
$L^{1}(m_{S})$. Hence, by Dunford–Pettis characterization, for each
$\varepsilon>0$, there exists $\delta>0$ such that

$\sup_{n\geq 0}\int_{A}P_{S}^{n}1_{S}\,dm_{S}<\varepsilon\quad\text{for any
$A\in\mathscr{B}_{S}$ with $m_{S}(A)<\delta$}.$ (3.5)

Note that clearly $m(S)>0$ since $g$ is a density function. Thus,
$m(X\setminus S)<1$ and hence, we can take $\varepsilon>0$ small enough such
that $\alpha\coloneqq\varepsilon\cdot m(S)+m(X\setminus S)<1$. Then, by
Dunford–Pettis characterization, there is $\delta>0$ such that (3.5) holds.
Let $\delta^{\prime}=\delta\cdot m(S)>0$. Observe that, according to
Proposition 3.3, $P_{S}1_{S}=P1_{S}$ and then, $P^{n}_{S}1_{S}=P^{n}1_{S}$ and
$\operatorname*{supp}P^{n}1_{S}\subset S$ up to an $m$-null set for all $n\geq
0$. Finally, for any $A\in\mathscr{B}$ with $m(A)<\delta^{\prime}$ it holds
that $m_{S}(A\cap S)<\delta$ and then,

$\displaystyle\sup_{n\geq 0}\int_{A}P^{n}1_{X}\,dm$ $\displaystyle=\sup_{n\geq
0}\left(\int_{A}P^{n}1_{S}\,dm+\int_{A}P^{n}1_{X\setminus S}\,dm\right)$
$\displaystyle=\sup_{n\geq 0}\left(m(S)\cdot\int_{A\cap
S}P^{n}_{S}1_{S}\,dm_{S}+\int_{A}P^{n}1_{X\setminus S}\,dm\right)$
$\displaystyle<m(S)\cdot\varepsilon+\sup_{n\geq 0}\int_{X\setminus
S}P^{n*}1_{X}\,dm$ $\displaystyle<\varepsilon\cdot m(S)+m(X\setminus
S)=\alpha.$

The implication from the condition (2) to (3) follows immediately taking into
account that

$\int_{A}A_{n}1_{X}\,dm=\frac{1}{n}\sum_{i=0}^{n-1}\int_{A}P^{i}1_{X}\,dm<\frac{1}{n}\sum_{i=0}^{n-1}\alpha=\alpha$

for all $n\geq 1$ and $A\in\mathscr{B}$ with $m(A)<\delta$.

We show the implication from (3) to (4). Observe that since $P$ is a Markov
operator, then $P^{*}1_{X}=1_{X}$ and thus,

$\int A_{n}1_{X}\,dm=\int A^{*}_{n}1_{X}\,dm=\frac{1}{n}\sum_{k=0}^{n-1}\int
P^{k*}1_{X}\,dm=1.$

From this and assumption (3), we have

$\displaystyle\inf_{n\geq 1}\int_{X\setminus E}A_{n}1_{X}\,dm=\inf_{n\geq
1}\int A_{n}1_{X}\,dm-\sup_{n\geq 1}\int_{E}A_{n}1_{X}\,dm>1-\alpha$

for all $E\in\mathscr{B}$ with $m(E)<\delta$. Setting $\delta$ in (4) as
$1-\delta$ in (3), we get the desired implication.

Next we show the implication from (4) to (1). Suppose contrarily that there is
no invariant density for $P$. Then, according to [59, Theorem 4.6 and Lemma
4.5], there exists $\psi\in L^{\infty}(m)$ with $\psi\geq 0$, fully supported
on $X$ (i.e., $\operatorname*{supp}\psi=X$) and $\|A^{*}_{n}\psi\|_{\infty}\to
0$ as $n\to 0$. Now, since $h$ is fully supported on $X$, for $\delta\geq 0$
as in the condition (4), we can find $\eta>0$ such that the set
$E=\\{h\geq\eta\\}$ satisfies $m(E)>\delta$. Since $\psi\geq\eta$ on $E$,
$\eta^{-1}\psi\geq 1_{E}$. Hence,

$\displaystyle\int_{E}A_{n}1_{X}\,dm$ $\displaystyle=\int
1_{E}A_{n}1_{X}\,dm\leq\int\frac{\psi}{\eta}\,A_{n}1_{X}\,dm=\frac{1}{\eta}\int
A_{n}^{*}\psi\,dm\leq\frac{1}{\eta}\left\lVert
A_{n}^{*}\psi\right\rVert_{\infty}\to 0$

as $n\to\infty$. This contradicts (4) and the proof is done. Finally, observe
that the previous arguments work assuming $\alpha=1$ in all of the items.

To complete the proof we will see now the equivalence with (5), (6). Clearly
(2) implies (5). Also, (5) immediately implies (6). On the other hand, from
[84, Theorem 1], we have that (6) implies (1). Moreover, a slight modification
of the argument of [84, Theorem 1] also shows that (6) implies (1). Indeed,
assume (6) instead of (5) (which is called (C) in [84] and where $1_{X}$ could
be substituted by any density in $D(m)$). Define

$\lambda(\varphi)=\mathrm{Lim}\,\int A_{n}1_{X}\,\varphi\,dm\quad\text{for
$\varphi\in L^{\infty}(m)$}$

where $\mathrm{Lim}$ is a Banach limit (i.e. replace $P^{n}$ in the definition
of $\lambda(h)$ of [84] with $A_{n}$). The rest of the proof in [84, Theorem
1] literally works to conclude (1) in our case, except proving
$\lambda(P^{*}\varphi)=\lambda(\varphi)$ for each $\varphi\in L^{\infty}(m)$,
that is the unique calculation that one needs to do. To see it,

$\displaystyle\lambda(P^{*}\varphi)$ $\displaystyle=\mathrm{Lim}\,\int
A_{n}1_{X}\,P^{*}\varphi\,dm=\mathrm{Lim}\,\int\frac{1}{n}\sum_{i=0}^{n-1}P^{i+1}1_{X}\,\varphi\,dm$
$\displaystyle=\mathrm{Lim}\,\int\left(A_{n}1_{X}+\frac{1}{n}P^{n}1_{X}-\frac{1}{n}1_{X}\right)\,\varphi\,dm=\lambda(\varphi)+\mathrm{Lim}\,\frac{1}{n}\int(P^{n}1_{X}-1_{X})\,\varphi\,dm.$

In the last equality we used the linearity of Banach limits. On the other
hand,

$\left|\int(P^{n}1_{X}-1_{X})\,\varphi\,dm\right|\leq\|\varphi\|_{\infty}\
\int(P^{n}1_{X}+1_{X})\,dm=2\|\varphi\|_{\infty}$

since $P$ is a Markov operator. Hence we get
$\frac{1}{n}\int(P^{n}1_{X}-1_{X})\,\varphi\,dm\to 0$ as $n\to\infty$, and
obtain the desired equality $\lambda(P^{*}\varphi)=\lambda(\varphi)$. The
version for $\alpha=1$ follows since, actually, the assumption in [84, Theorem
1] is (5) for $\alpha=1$ and where $1_{X}$ could be substituted by any density
in $D(m)$. This completes the proof. ∎

The implications between (1)–(4) in Theorem 3.4 require strongly that $P$ is a
Markov operator. However, the equivalence between (1) and (4) for $\alpha=1$
holds even under the weaker assumption that the linear positive operator $P$
is just a contraction (i.e., $\|P\|_{\rm op}\leq 1$). As a final result of
this section, we will prove it as follows. Compare with [59, Theorem 4.2 and
Corollary 4.7 in Chapter 3] and [71, Theorem 2].

######  Theorem 3.5.

If $P:L^{1}(m)\to L^{1}(m)$ is a linear positive contraction, then the
following are equivalent:

  1. (i)

there exists a $P$-invariant density;

  2. (ii)

there exists $\delta\geq 0$ such that

$\inf_{n\geq 0}\int_{E}P^{n}1_{X}\,dm>0\quad\text{for all $E\in\mathscr{B}$
with $m(E)>\delta$;}$

  3. (iii)

there exists $\delta\geq 0$ such that

$\inf_{n\geq 0}\int_{E}A^{n}1_{X}\,dm>0\quad\text{for all $E\in\mathscr{B}$
with $m(E)>\delta$.}$

###### Proof.

Observe that the proof of the implication of (1) from (4) in Theorem 3.4 does
not require that $\int P\varphi\,dm=\int\varphi\,dm$ for all $\varphi\in
L^{1}(m)$. This shows that (iii) implies (i). The equivalence between (i) and
(ii) follows from [59, Theorem 4.2] as follows. This result says that if $h$
is a $P$-invariant density, then

$\inf_{n\geq 0}\int_{\tilde{E}}P^{n}1_{X}\,dm>0\quad\text{for all
$\tilde{E}\subset\operatorname*{supp}h$ with $m(\tilde{E})>0$}.$

Taking $\delta=1-m(\operatorname*{supp}h)\geq 0$, we have that if $E\subset X$
with $m(E)>\delta$, then $m(\tilde{E})>0$ where
$\tilde{E}=E\cap\operatorname*{supp}h$. Hence, from the above inequality it
holds that

$\inf_{n\geq 0}\int_{E}P^{n}1_{X}\,dm\geq\inf_{n\geq
0}\int_{\tilde{E}}P^{n}1_{X}\,dm>0.$

Finally, since (ii) implies (iii) immediately, we conclude the cycle of
implications and thus the proof of the theorem. ∎

##  4\. Mean constrictivity: proof of Theorem D

In the sequel, we will prove the following result which is a more general
version of Theorem D.

######  Theorem 4.1.

Let $(X,\mathscr{B},m)$ be an abstract probability space and consider a Markov
operator $P:L^{1}(m)\to L^{1}(m)$. Then the following conditions are
equivalent:

  1. (MC)

$P$ is _mean constrictive_ ;

  2. (MC2)

There is a compact set $F\subset L^{1}(m)$ and $\kappa<1$ such that

$\displaystyle\limsup_{n\to\infty}d(A_{n}\varphi,F)\leq\kappa\quad\text{for
any $\varphi\in D(m)$};$

  3. (WMC)

$P$ is _weakly mean constrictive_ , i.e., there is a weakly compact set
$F\subset L^{1}(m)$ such that

$\displaystyle\lim_{n\to\infty}d(A_{n}\varphi,F)=0\quad\text{for any
$\varphi\in D(m)$};$

  4. (WMC2)

There is a weakly compact set $F\subset L^{1}(m)$ and $\kappa<1$ such that

$\displaystyle\limsup_{n\to\infty}d(A_{n}\varphi,F)\leq\kappa\quad\text{for
any $\varphi\in D(m)$};$

  5. (MCDP)

$P$ is _mean constrictive à la Dunford–Pettis_ , i.e., for every
$\varepsilon>0$, there exists $\delta>0$ such that for any $\varphi\in D(m)$,
there is $n_{0}=n_{0}(\varepsilon,\varphi)\geq 1$ satisfying that

$\displaystyle\int_{E}A_{n}\varphi\,dm<\varepsilon\quad\text{for any $n\geq
n_{0}$ and $E\in\mathscr{B}$ with $m(E)<\delta$};$

  6. (MCDP2)

There exist $\kappa<1$ and $\delta>0$ such that for any $\varphi\in D(m)$,
there is $n_{0}=n_{0}(\varphi)\geq 1$ satisfying that

$\displaystyle\int_{E}A_{n}\varphi\,dm\leq\kappa\quad\text{for any $n\geq
n_{0}$ and $E\in\mathscr{B}$ with $m(E)<\delta$};$

  7. (AMC)

$P$ is _asymptotically mean constrictive_ , i.e., for every $\varepsilon>0$,
there is $\delta>0$ such that

$\displaystyle\limsup_{n\to\infty}\int_{E}A_{n}\varphi\,dm<\varepsilon\quad\text{for
any $\varphi\in D(m)$ and $A\in\mathscr{B}$ with\leavevmode\nobreak\
$m(A)<\delta$};$

  8. (AMC2)

There exist $\kappa<1$ and $\delta>0$ such that

$\displaystyle\limsup_{n\to\infty}\int_{E}A_{n}\varphi\,dm<\kappa\quad\text{for
any $\varphi\in D(m)$ and $A\in\mathscr{B}$ with\leavevmode\nobreak\
$m(A)<\delta$};$

  9. (FED)

$P$ admits finitely many ergodic invariant densities $h_{1},\dots,h_{r}$ with
mutually disjoint supports (up to an $m$-null set) and the invariant density
function $h=\frac{1}{r}(h_{1}+\dots+h_{r})$ has the maximal support;

  10. (APM)

$P$ is _asymptotically periodic in mean_ , i.e., there exist finitely many
ergodic invariant densities $h_{1},\dots,h_{r}$ with mutually disjoint
supports (up to an $m$-null set) and positive bounded linear functionals
$\lambda_{1},\dots,\lambda_{r}$ on $L^{1}(m)$ such that

$\displaystyle\lim_{n\to\infty}\left\lVert
A_{n}\varphi-\sum_{i=1}^{r}\lambda_{i}(\varphi)h_{i}\right\rVert=0\quad\text{for
any $\varphi\in L^{1}(m)$}.$

$\textstyle{\tt(MCDP)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{trivial}$$\scriptstyle{trivial}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\tt(WMC)}$$\scriptstyle{Lemma\leavevmode\nobreak\
\tiny\ref{lemC1}}$$\textstyle{\tt(ACM)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{trivial}$$\textstyle{\tt(MCDP2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{trivial}$$\textstyle{\tt(AMC2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Theorem\leavevmode\nobreak\
\tiny\ref{propC2}}$$\textstyle{\tt(FED)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Lemma\leavevmode\nobreak\
\tiny\ref{lemC3}}$$\textstyle{\tt(APM)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Lemma\leavevmode\nobreak\
\tiny\ref{lemC4}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\tt(MC)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{trivial}$$\scriptstyle{trivial}$$\textstyle{\tt(WMC2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Lemma\leavevmode\nobreak\
\tiny\ref{lemC1}}$$\textstyle{\tt(MC2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{trivial}$
Figure 3. Diagram of the relations between the conditions of Theorem 4.1.

We will prove Theorem 4.1 following the implications described in Figure 3. To
organize the proof, we divide the proof into two subsections.

###  4.1. Proof of the lemmas

######  Lemma 4.2.

Condition (WMC) (resp. (WMC2)) implies condition (MCDP) (resp. (MCDP2)).

###### Proof.

Let $F$ be as in the condition (WMC). Fix $\varepsilon>0$. The condition (WMC)
guarantees that for any $\varphi\in D(m)$ there exists
$n_{0}=n_{0}(\varepsilon,\varphi)\geq 1$ such that

$\displaystyle\inf_{\phi\in F}\left\lVert
A_{n}\varphi-\phi\right\rVert<\frac{\varepsilon}{2}\quad\text{for any $n\geq
n_{0}$}.$

Since $F$ is weakly compact, there exists $\delta=\delta(\varepsilon)>0$ such
that for any $\phi\in F$ and $E\in\mathscr{B}$ with $m(E)<\delta$, we have
$\int_{E}|\phi|\,dm<\frac{\varepsilon}{2}$. Thus, for any set $E$ with
$m(E)<\delta$

$\displaystyle\int_{E}A_{n}\varphi\,dm$ $\displaystyle\leq\inf_{\phi\in
F}\left\lVert
A_{n}\varphi-\phi\right\rVert+\int_{E}|\phi|\,dm<\varepsilon\quad\text{for any
$n\geq n_{0}$.}$

This proves (MCDP).

The implication (WMC2) implies (MCDP2) follows analogously. Namely, we put
$\kappa$ in (MCDP2) as the sum of $\kappa$ in (WMC2) and
$\varepsilon=\frac{1-\kappa}{2}$ from the weak compactness of $F$. ∎

######  Lemma 4.3.

The condition (FED) implies the condition (APM).

###### Proof.

Suppose the condition (FED). In particular, the existence of an invariant
density with the maximal support implies that $P$ is (WAP), cf. [88, Theorem
3.1]. Yosida and Kakutani proved in [92] that (WAP) implies mean
ergodicity.999The converse of this implication was recently proved in [88,
Proposition 3.9]. Recall that a Markov operator $P:L^{1}(m)\to L^{1}(m)$ is
called mean ergodic if $A_{n}\varphi$ converges in $L^{1}(m)$-norm for any
$\varphi\in L^{1}(m)$. For each density $h\in D(m)$, denote by $h^{*}$ the
limit in $L^{1}(m)$-norm of $A_{n}h$. Observe that $h^{*}$ is an invariant
density of $P$. Hence, from the finitude of ergodic invariant densities
$h_{1},\dots,h_{r}$, or equivalently extremal points in the space of invariant
densities (see Proposition C.10 in Appendix C), we can find
$\lambda_{1}(h),\dots\lambda_{r}(h)$ with $\lambda_{i}(h)\geq 0$,
$\lambda_{1}(h)+\dots+\lambda_{r}(h)=1$ such
$h^{*}=\lambda_{1}(h)h_{1}+\dots+\lambda_{r}(h)h_{r}$. We have the equation in
(APM) for any $h\in D(m)$ and also for $\varphi\in L^{1}(m)$ with $\varphi\geq
0$. Then, considering the positive part and negative part of any function in
$L^{1}(m)$, we can extend the above argument to $L^{1}(m)$. Obviously
$\lambda_{i}:L^{1}(m)\to\mathbb{R}$ is bounded and linear so that the proof is
done. ∎

######  Lemma 4.4.

The condition (APM) implies the condition (MC).

###### Proof.

The proof is straightforward, taking the compact constrictor for $A_{n}$ by

$\displaystyle F=\left\\{\varphi\in L^{1}(m):\ \varphi\text{ is a convex
combination of }h_{1},\dots,h_{r}\right\\}.$

Therefore, the proof is done. ∎

###  4.2. Proof of the theorem

In this subsection, we will show that (AMC2) implies (FED).

######  Remark 4.5.

Observe that from Proposition 3.3, the restriction $P_{S}$ of a Markov
operator $P$ to the support $S$ of a $P$-invariant density is a Markov
operator. Hence, Lemma B.1 in the Appendix B implies that $P^{*}1_{S}\geq
1_{S}$. In particular the sequence $(P^{*}1_{S})_{n\geq 1}$ is increasing.

######  Proposition 4.6.

(AMC2) implies that the set of ergodic invariant densities of $P$ is nonempty
and finite. Moreover, these ergodic invariant densities have mutually disjoint
support.

###### Proof.

Observe first that by virtue of Theorem 3.4 (6), it immediately follows from
(AMC2) that there exists a $P$-invariant density. Thus, by Corollary C.10,

$\mathcal{E}=\\{g\in L^{1}(m):\ \text{$g$ is an ergodic $P$-invariant
density}\\}\not=\emptyset.$

The following claim is the key to obtaining the finitude of ergodic
$P$-invariant densities.

######  Claim 4.7.

Consider $E\in\mathscr{B}$ such that $m(E)>0$ and $P^{*}1_{E}\geq 1_{E}$. Let
$\delta>0$ be the constant given in (AMC2). Then, $m(E)\geq\delta$.

###### Proof.

Assume that $m(E)<\delta$. Then, applying (AMC2) to $h=\frac{1_{E}}{m(E)}\in
D(m)$, we have

$\limsup_{n\to\infty}\int_{E}A_{n}\frac{1_{E}}{m(E)}\,dm\leq\kappa<1.$

On the other hand, for any $n\geq 1$,

$\displaystyle\int_{E}A_{n}\frac{1_{E}}{m(E)}\,dm$
$\displaystyle=\frac{1}{m(E)}\int_{E}A^{*}_{n}1_{E}\,dm=\frac{1}{m(E)}\int_{E}\frac{1}{n}\sum_{i=0}^{n-1}P^{i*}1_{E}\,dm\geq\frac{1}{m(E)}\int_{E}1_{E}\,dm=1.$

This is a contradiction, and thus, we have $m(E)\geq\delta$. ∎

Now, by Remark 4.5, the support $S$ of any invariant density satisfies
$P^{*}1_{S}\geq 1_{S}$ and by Corollary C.10 and Remark C.11 two distinct
ergodic $P$-invariant densities have disjoint support. Hence, it follows from
Claim 4.7 that the cardinality of $\mathcal{E}$ is less than $\delta^{-1}$.
Set $r$ be this cardinality (notice that $r>0$ since
$\mathcal{E}\not=\emptyset$). Thus, we get that $P$ admits finitely many
ergodic invariant densities $h_{1},\dots,h_{r}$ in $D(m)$ with mutually
disjoint supports. ∎

######  Proposition 4.8.

(WAP) holds if (AMC2) holds.

We postpone the proof of the above proposition and show how to use it to
conclude the following main result of this subsection:

######  Theorem 4.9.

(AMC2) implies (FED). In particular, (AC) implies (MC).

###### Proof.

Let $h_{1},\dots,h_{r}$ be the ergodic invariant densities of $P$ obtained in
Proposition 4.6. To conclude (FED) we need to prove that the $P$-invariant
density $h=\frac{1}{r}(h_{1}+\dots+h_{r})$ has the maximal support. According
to Proposition 4.8, $P$ is (WAP). Then there is a $P$-invariant density $g$
with the maximal support, i.e., such that
$\lim_{n\to\infty}P^{n*}1_{\operatorname*{supp}g}=1_{X}$. Since the set of
$P$-invariant densities is convex and the ergodic densities are its extremal
points, it follows that $g=\lambda_{1}h_{1}+\dots+\lambda_{r}h_{r}$ with
$\lambda_{i}\geq 0$ and $\lambda_{1}+\dots+\lambda_{r}=1$. Then,
$\operatorname*{supp}g\subset\operatorname*{supp}h$ and thus we also have
$\lim_{n\to\infty}P^{n*}1_{\operatorname*{supp}h}=1_{X}$. This shows that $h$
has the maximal support concluding the proof.

The final implication follows immediately from the first part of the theorem
since, trivially, (AC) $\Rightarrow$ (AMC2) and (FED) $\Rightarrow$ (MC) by
Lemmas 4.3 and 4.4. ∎

####  4.2.1. Proof of Proposition 4.8

Let $h$ be a $P$-invariant density. Set $S=\operatorname*{supp}h$. Define

$\varphi\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{n\to\infty}P^{n*}1_{S}.$

This limit exists since the sequence $(P^{n*}1_{S})_{n\geq 1}$ is increasing.
See Remark 4.5. The first observation is that $0\leq\varphi\leq 1$. Also, by
the monotone continuity property of $P^{*}$ (see [70, Proposition V.4.1]),
$P^{*}\varphi=\varphi$. Moreover, since $P^{*}1_{S}\geq 1_{S}$, one has that
$\varphi=1$ on $S$. Let us split

$X=E_{0}\cup E\cup E_{1},\quad E=\\{x\in X:0<\varphi<1\\}\ \ \text{and}\ \
E_{i}=\\{x\in X:\varphi=i\\}\ \text{for $i=0,1$}.$

######  Lemma 4.10.

We have $P^{*}1_{E}\leq 1_{E}$ and $P^{*}1_{E_{0}}\geq 1_{E_{0}}$.

###### Proof.

Note that $\operatorname*{supp}1_{E}\subset\operatorname*{supp}\varphi$ and
$\operatorname*{supp}1_{E}\subset\operatorname*{supp}(1_{X}-\varphi)$. Then,
by Lemma 3.1, it follows that

$\operatorname*{supp}P^{*}1_{E}\subset\operatorname*{supp}P^{*}\varphi\cap\operatorname*{supp}P^{*}(1_{X}-\varphi)=\operatorname*{supp}\varphi\cap\operatorname*{supp}(1_{X}-\varphi)=E.$

Then, Proposition B.1 in Appendix B implies that $P^{*}1_{E}\leq 1_{E}$. On
the other hand, since
$\lim_{n\to\infty}\min\\{1_{X},n\varphi\\}=1_{\operatorname*{supp}\varphi}$,
it holds that

$P^{*}1_{\operatorname*{supp}\varphi}=\lim_{n\to\infty}P^{*}\min\\{1_{X},n\varphi\\}\leq\lim_{n\to\infty}\min\\{1_{X},n\varphi\\}=1_{\operatorname*{supp}\varphi}.$

Equivalently, $P^{*}1_{E_{0}}\geq 1_{E_{0}}$ since
$E_{0}=X\setminus\operatorname*{supp}\varphi$. ∎

######  Lemma 4.11.

For every $A\subset\operatorname*{supp}\varphi\setminus S$,
$\lim_{n\to\infty}P^{n*}(\varphi 1_{A})=0$.

###### Proof.

By assumption, $1_{A}\leq 1_{\operatorname*{supp}\varphi\setminus S}$. Hence,
$0\leq\varphi 1_{A}\leq\varphi 1_{\operatorname*{supp}\varphi\setminus S}$.
Then, since

$\lim_{n\to\infty}P^{n*}(\varphi 1_{\operatorname*{supp}\varphi\setminus
S})=\lim_{n\to\infty}P^{n*}(\varphi 1_{\operatorname*{supp}\varphi}-\varphi
1_{S})=\lim_{n\to\infty}P^{n*}\varphi-\lim_{n\to\infty}P^{n*}1_{S}=\varphi-\varphi=0,$

it also follows that $\lim_{n\to\infty}P^{n*}(\varphi 1_{A})=0$. ∎

Let us define

$\psi\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\lim_{n\to\infty}P^{n*}1_{E}.$

As above, notice that this limit exists since, from Lemma 4.10, the sequence
$(P^{n*}1_{E})_{n\geq 1}$ is decreasing. Moreover, again, by the monotone
continuity property of $P^{*}$, we have that $P^{*}\psi=\psi$ and
$\operatorname*{supp}\psi\subset E$.

######  Lemma 4.12.

If $m(E_{0})=0$, then $\varphi=1_{X}-\psi$.

###### Proof.

Since $m(E_{0})=0$,

$1_{X}=\lim_{n\to\infty}P^{n*}1_{X}=\lim_{n\to\infty}P^{n*}(1_{S}+1_{E_{1}\setminus
S}+1_{E})=\varphi+\lim_{n\to\infty}P^{n*}1_{E_{1}\setminus S}+\psi.$

Having it in mind that $1_{E_{1}\setminus S}=\varphi 1_{E_{1}\setminus S}$ and
$E_{1}\setminus S\subset\operatorname*{supp}\varphi\setminus S$, Lemma 4.11
implies that $\lim_{n\to\infty}P^{n*}1_{E_{1}\setminus S}=0$. Substituting
above, we get $1_{X}=\varphi+\psi$ concluding the proof. ∎

For each $\varepsilon>0$, let us denote $B_{\varepsilon}=\\{x\in
X:1-\varepsilon\leq\psi(x)\\}.$

######  Lemma 4.13.

If $\varphi=1_{X}-\psi$, then $\lim_{n\to\infty}P^{n*}(\psi
1_{B_{\varepsilon}})=\psi$ for every $\varepsilon>0$. Moreover, if
$m(B_{\varepsilon})=0$ for some $\varepsilon>0$, then $\psi=0$ (and hence,
$\varphi=1$) $m$-almost everywhere.

###### Proof.

For each $k\geq 1$, define $C_{k}=\\{x\in
X:\,\frac{1}{k+1}\leq\varphi(x)<\frac{1}{k}\\}$. Since $\varphi=1_{X}-\psi$,
we have

$C_{k}=\left\\{x\in X:\,1-\frac{1}{k}<\psi(x)\leq 1-\frac{1}{k+1}\right\\}.$

Hence,

$1_{C_{k}}\psi\leq
1_{C_{k}}\left(1-\frac{1}{k+1}\right)=1_{C_{k}}\cdot\frac{k}{k+1}\leq
k\,1_{C_{k}}\varphi.$

Moreover, from Lemma 4.11, since $C_{k}\subset
E\subset\operatorname*{supp}\varphi\setminus S$, we have that

$0\leq\lim_{n\to\infty}P^{n*}\left(1_{C_{k}}\psi\right)\leq
k\lim_{n\to\infty}P^{n*}\left(1_{C_{k}}\varphi\right)=0$

and thus, for each $k\geq 1$,

$\lim_{n\to\infty}P^{n*}\left(1_{C_{k}}\psi\right)=0.$ (4.1)

On the other hand, for each $\varepsilon>0$, there exists $k_{0}\geq 1$ such
that

$E\setminus B_{\varepsilon}=\\{x\in
X:0<\psi(x)<1-\varepsilon\\}\subset\bigcup_{k=1}^{k_{0}}C_{k}.$

Then, from (4.1) we get

$\lim_{n\to\infty}P^{n*}\left(1_{E\setminus
B_{\varepsilon}}\psi\right)=0\quad\text{for all $\varepsilon>0$.}$ (4.2)

Now, observe that $\psi=\psi 1_{E}=\psi 1_{B_{\varepsilon}}+\psi 1_{E\setminus
B_{\varepsilon}}$. Hence, by (4.2), we conclude that

$\displaystyle\lim_{n\to\infty}P^{n*}\left(\psi
1_{B_{\varepsilon}}\right)=\lim_{n\to\infty}P^{n*}\psi=\psi.$

Finally, note that if $m(B_{\varepsilon})=0$ for some $\varepsilon>0$ then
$\psi<1-\varepsilon$ $m$-almost everywhere. From this, it follows that

$\displaystyle\psi=P^{n*}\psi=P^{n*}(\psi
1_{E})<(1-\varepsilon)P^{n*}1_{E}\to(1-\varepsilon)\psi\quad\text{as
$n\to\infty$.}$

Then $\psi=0$ $m$-almost everywhere. ∎

Now we set

$\mathcal{F}=\\{S\subset X:S\ \ \text{is the support of a}\ P\text{-invariant
density}\\}.$

Then, according to Proposition 4.6 there is at least one $P$-invariant density
and thus $\mathcal{F}\not=\emptyset$. The inclusion up to an $m$-null set
induces a partial order in $\mathcal{F}$. Given a chain, by the Zorn lemma,
there exists a maximal element $S\in\mathcal{F}$ of the chain in the sense
that if $S\subset S^{\prime}$ and $S^{\prime}\in\mathcal{F}$ then
$S^{\prime}=S$ up to an $m$-null set. Let $h\in D(m)$ be a $P$-invariant
density such that $S=\operatorname*{supp}h$ and consider for this density the
sets $E_{0}$, $E$ and $E_{1}$ defined above.

######  Lemma 4.14.

(AMC2) implies that $m(E_{0})=0$ and $m(B_{\varepsilon})=0$ for some
$\varepsilon>0$.

###### Proof.

By Lemma 4.10, $P^{*}1_{E_{0}}\geq 1_{E_{0}}$. If $m(E_{0})>0$, then
Proposition B.1 implies that $P_{E_{0}}:L^{1}(m_{E_{0}})\to L^{1}(m_{E_{0}})$
is a Markov operator. Therefore, by virtue of Theorem 3.4 (6), it immediately
follows from (AMC2) that there exists a $P_{E_{0}}$-invariant density $g\in
D(m_{E_{0}})$. In particular, it satisfies that $P(1_{E_{0}}g)=1_{E_{0}}g$ and
$\int 1_{E_{0}}g\,dm=m(E_{0})$. Hence,
$\phi=\frac{1}{2}(h+\frac{1_{E_{0}}g}{m(E_{0})})$ is a $P$-invariant density
and thus $S^{\prime}=\operatorname*{supp}\phi\in\mathcal{F}$ and
$S=\operatorname*{supp}h\subsetneq S^{\prime}$. This contradicts the
maximality of $S$, concluding that $m(E_{0})=0$.

Next we will see that there exists $\varepsilon>0$ such that
$m(B_{\varepsilon})=0$. From the assumption (AMC2), there are $0<\kappa<1$ and
$\delta>0$ so that

$\displaystyle\limsup_{n\to\infty}\int_{B}A_{n}\phi\,dm\leq\kappa\quad\text{for
each $\phi\in D(m)$ and $B\in\mathscr{B}$ with $m(B)<\delta$.}$ (4.3)

On the other hand, one can find $\varepsilon_{0}>0$ such that for any
$0<\varepsilon\leq\varepsilon_{0}$ it holds $m(B_{\varepsilon})<\delta$ since
$m(B_{\varepsilon})\to 0$ as $\varepsilon\to 0$. We fix
$0<\varepsilon<\min\\{\varepsilon_{0},1-\kappa\\}$. We will see that
$m(B_{\varepsilon})=0$. Arguing by contradiction, we assume that
$m(B_{\varepsilon})>0$. Then for a function
$\phi=\frac{1_{B_{\varepsilon}}}{m(B_{\varepsilon})}$, it follows from (4.3)
that

$\displaystyle\kappa$
$\displaystyle\geq\limsup_{n\to\infty}\int_{B_{\varepsilon}}A_{n}\phi\,dm$
$\displaystyle=\limsup_{n\to\infty}\int_{X}\phi\cdot
A_{n}^{*}1_{B_{\varepsilon}}\,dm\geq\limsup_{n\to\infty}\int_{X}\phi\cdot
A_{n}^{*}(\psi 1_{B_{\varepsilon}})\,dm=\int_{X}\phi\psi\,dm$

by the Lebesgue dominated convergence theorem and Lemma 4.13 (see
$\varphi=1_{X}-\psi$ due to $m(E_{0})=0$ and Lemma 4.12). But we also have

$\displaystyle\int_{X}\phi\psi\,dm=\frac{1}{m\left(B_{\varepsilon}\right)}\int_{B_{\varepsilon}}\psi\,dm\geq\frac{1}{m\left(B_{\varepsilon}\right)}\int_{B_{\varepsilon}}(1-\varepsilon)\,dm>\kappa.$

Therefore, we conclude $m(B_{\varepsilon})=0$. ∎

###### Proof of Proposition 4.8.

The result follows from Lemmas 4.12, 4.13 and 4.14. ∎

###  4.3. On the classes (AC), (CW) and (APW)

In Theorem 4.9 we have proved that (AC) $\Rightarrow$ (MC) and, in particular
from Theorem D, (AC) implies (APM). The following proposition concludes that
conditions (APW) and (CW) introduced in Section 1.5.3 are sufficient for (AC).
Thus, as a consequence, (APW) $\Rightarrow$ (APM) and (CW) $\Rightarrow$ (MC).

######  Proposition 4.15.

It holds that (CW) implies (AC). Furthermore, (APW) implies (CW).

###### Proof.

Assume that (CW) holds. Hence, given $h\in D(m)$, there is $(\psi_{n})_{n\geq
1}\subset F$ such that $P^{n}h-\psi_{n}\to 0$ weakly in $L^{1}(m)$ as
$n\to\infty$. Therefore,

$\displaystyle\limsup_{n\to\infty}\int_{A}P^{n}h\,dm$
$\displaystyle=\limsup_{n\to\infty}\left(\int_{A}\psi_{n}\,dm+\int_{A}(P^{n}h-\psi_{n})\,dm\right)$
$\displaystyle=\limsup_{n\to\infty}\int_{A}\psi_{n}\,dm\leq\sup_{\psi\in
F}\int_{A}\psi\,dm.$

Since $F$ is a weakly compact set, it follows from Dunford–Pettis theorem that
for any $\varepsilon>0$, there is $\delta>0$ such that $\sup_{\psi\in
F}\int_{A}\psi\,dm<\varepsilon$ for any $A\in\mathscr{B}$ with $m(A)<\delta$.
This concludes that $P$ satisfies (AC).

Next we assume that (APW) holds. Then, for $h\in D(m)$, since $P$ is a Markov
operator, due to the weak convergence in (APW),

$\displaystyle 0$ $\displaystyle=\lim_{n\to\infty}\int
P^{n}\big{(}h-\sum_{i=1}^{r}\lambda_{i}(h)g_{i}\big{)}\,dm=\int\big{(}\varphi-\sum_{i=1}^{r}\lambda_{i}(h)g_{i}\big{)}\,dm=1-\sum_{i=1}^{r}\lambda_{i}(h).$
(4.4)

Set

$F\coloneqq\left\\{\sum_{i=1}^{r}a_{i}\,g_{i}:\
a_{i}\in\mathbb{R},\;\sum_{i=1}^{r}a_{i}=1\right\\}\subset D(m).$

Notice that $F$ is a weakly compact set and also $P$-invariant since
$Pg_{i}=g_{\rho(i)}$ where $\rho$ is the permutation of $\\{1,\dots,r\\}$ in
the definition of (APW). Moreover, in view of (4.4), the weak convergence in
(APW) means that for any $h\in D(m)$, there exists

$\psi_{n}\coloneqq\sum_{i=1}^{r}\lambda_{i}(h)P^{n}g_{i}=\sum_{i=1}^{r}\lambda_{i}(h)g_{\rho^{n}(i)}\in
F$

such that $P^{n}h-\psi_{n}\to 0$ weakly as $n\to\infty$. This concludes that
$P$ satisfies (CW). ∎

##  5\. Characterization of finitude of physical measures: proof of Theorem C

Let $(X,\mathscr{B},m)$ and
$(\Omega,\mathscr{F},\mathbb{P})=(T^{\mathbb{N}},\mathscr{A}^{\mathbb{N}},p^{\mathbb{N}})$
be a locally compact Polish probability space and the infinite product space
of a probability space $(T,\mathscr{A},p)$, respectively. Consider a
measurable map $f:T\times X\to X$ and

$f^{0}_{\omega}=\mathrm{id}\quad\text{and}\quad
f^{n}_{\omega}=f_{\omega_{n-1}}\circ\dots\circ f_{\omega_{0}}\ \ \text{for}\
n>0\ \text{and $\omega=(\omega_{0},\omega_{1},\ldots)\in\Omega$},$