# Measurable multiresolution systems, endomorphisms, and representations of
Cuntz relations

Sergey Bezuglyi and Palle E.T. Jorgensen  University of Iowa, Iowa City, Iowa,
USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to present new classes of function systems as
part of multiresolution analyses. Our approach is representation theoretic,
and it makes use of generalized multiresolution function systems (MRSs). It
further entails new ideas from measurable endomorphisms-dynamics. Our results
yield applications that are not amenable to more traditional techniques used
on metric spaces. As the main tool in our approach, we make precise new
classes of generalized MRSs which arise directly from a dynamical theory
approach to the study of surjective endomorphisms on measure spaces. In
particular, we give the necessary and sufficient conditions for a family of
functions to define generators of Cuntz relations. We find an explicit
description of the set of generalized wavelet filters. Our results are
motivated in part by analyses of sub-band filters in signal/image processing.
But our paper goes further, and it applies to such wider contexts as
measurable dynamical systems, and complex dynamics.

A unifying theme in our results is a new analysis of endomorphisms in general
measure space, and its connection to multi-resolutions, to representation
theory, and generalized wavelet systems.

###### Key words and phrases:

Measure spaces, endomorphisms, multiresolution, wavelet filters, Cuntz
relations, Radon-Nikodym derivative, Markovian functions

###### 2020 Mathematics Subject Classification:

46G12, 37D45, 20K30, 22E66

###### Contents

  1. 1 Introduction
  2. 2 Basics on endomorphisms
    1. 2.1 Endomorphisms of a measure space
    2. 2.2 Measurable partition
    3. 2.3 Radon-Nikodym derivatives and Markovian functions
    4. 2.4 Properties of endomorphisms
    5. 2.5 Markovian functions
  3. 3 Operators generated by endomorphisms
    1. 3.1 Transfer operators and endomorphisms
    2. 3.2 Composition operators and Markovian functions
    3. 3.3 Radon-Nikodym derivatives and conditional expectations
  4. 4 Cuntz relations for invariant and quasi-invariant measures
    1. 4.1 Quasi-invariant measure
    2. 4.2 Invariant measure
  5. 5 The set of wavelet filters
    1. 5.1 Actions of loop groups on wavelet filters
    2. 5.2 Endomorphisms, wavelet filters and cyclic representations

##  1\. Introduction

The present paper is focused on relationships between the following objects:
surjective endomorphisms of a measure space $(X,\mathcal{B},\mu)$, transfer
operators in $L^{2}(X,\mathcal{B},\mu)$, generalized wavelet filters,
Markovian functions, and representations of the Cuntz relations.

Our analysis of transformations of a measure space, associated transfer
operators, and representations of the Cuntz algebras derive from a new
analysis of endomorphisms in general measure space, presented here. While
various special cases of endomorphisms have been studied in earlier work, we
call attention here to three new elements: (i) our context is that of the most
general transformations of a measure spaces; (ii) we introduce a new harmonic
analysis into the problem via representations of Cuntz algebras; and (iii) our
representations of Cuntz algebras for the purpose arise directly from the
endomorphism at hand, $\sigma$, and a choice of a quasi-invariant measure
$\mu$. From this, we then identify a new construction of an infinite-
dimensional manifold $\mathfrak{M}$ of generalized wavelet filters, and a
transitive action of a canonical group $\mathcal{G}$, acting on
$\mathfrak{M}$, and depending only on the given pair $\sigma$ (endomorphisms),
and $\mu$ (measure). The representations of the particular Cuntz algebra
(depending on $\sigma$) in turn define endomorphisms of $B(L^{2}(\mu))$, the
$C^{*}$-algebra of all bounded operators in $L^{2}(\mu))$, and with each non-
commutative endomorphism extending the initial endomorphism $\sigma$.

We present a new class of function systems $\mathfrak{M}$ as part of
multiresolution analyses (MRAs), motivated in part by [ACSS22], [PR22],
[CD22]. Our approach is based on methods of the representation theory, and it
makes use of generalized iterated function systems related to a surjective
endomorphism of a measure space. For background on the representation theory,
we refer the readers to [Jor18], [DJS12], [JS18]. Our results for these
generalized MRSs further entail new ideas from measurable dynamics, see e.g.,
[AJL17], [AJL18], [And22], [BD22]. While our general focus is on branching
systems in the measurable category, our applications are not amenable to more
traditional metric techniques such as e.g., [FS22], [RR22], [MV23]. We turn to
new classes of generalized iterated function systems which arise directly from
a more general dynamical theory approach via a systematic study of
endomorphisms in measure spaces. We are motivated in part by analyses of sub-
band filters in signal/image processing, see e.g., [DJ07], [AJL18], [BLP+10],
[BMPR12]. But our paper goes further, and it encompasses such wider contexts
as measurable dynamical systems and complex dynamics. The corresponding
literature on wavelet filters, representations of Cuntz algebras, iterated
function systems, transfer operators, and other adjacent areas are very
extensive; we mention here the following sources where the reader can find
more details and alternative approaches: [BJ97a], [BJ99] [BJ97b], [JT19],
[BJ97c], [Jor96], [Jor01], [DJ07], [DJ14b], [DJ14a], [DJ06], [DJ14b], [DJ15],
[JP11], [JKS07], [JKS11], [JT17].

As noted in the papers cited above, the traditional approach to iterated
function systems, or more generally to semi-branching function systems, the
starting point is typically a fixed system of maps that can be shown to admit
limits in the form of attractors and measures invariant with respect to
iterated function systems. These constructions are typically based on metric
considerations, and they play a big role in such diverse applications as
(fractal) harmonic analysis, graph Laplacian, boundaries, and analysis of
geometries which are given by classes of self-similarity. Cantor and
Sierpinski constructions are cases in point. The corresponding IFS can be
shown in turn admit realizations in shift dynamical systems.

Our present approach is the opposite: we begin with a consideration of
endomorphisms in measure spaces (see the definition in Subsection 2.1), and of
associated measurable partitions (Subsection 2.2). With this as starting
point, we then introduce several bounded operators which will define
representations of a non-abelian $C^{*}$-algebras given by generators and
relations, called Cuntz algebras, Sections 3 and 4. and 4. There are two
advantages to this approach, (i) it allows for a much wider family of
(generalized) MRSs, and (ii) it also offers new and direct tools for attacking
the corresponding harmonic analysis questions. Finally, we give an explicit
description of the set of generalized wavelet filters, see Section 5.

Measurable transformations of a standard measure space $(X,\mathcal{B},\mu)$
is the central concept of the ergodic theory. Invertible transformations
(automorphisms) and their properties have been extensively studied from
various points of view. The study of non-invertible transformations
(endomorphisms) has been less popular than that of automorphisms although
their role in dynamics and adjoint areas is extremely important. In
particular, they are used in the construction of iterated function systems
(IFSs), transfer operators, and wavelet filters. We refer here to several
recent books dealing with endomorphisms and their applications in the operator
theory [Jor06], [PU10], [BJ18], [Haw21], [EFHN15], [Bru22], [URM22].

Using an analysis of endomorphisms of a standard measure space, we associate
with every endomorphism several bounded operators acting in
$L^{2}(X,\mathcal{B},\mu)$. Non-singular endomorphisms $\sigma$ of a measure
space $(X,\mathcal{B},\mu)$ are naturally divided into two classes: if
$\mu=\mu\circ\sigma^{-1}$, then $\sigma$ is called measure-preserving; if
$\mu\sim\mu\circ\sigma^{-1}$, then the measure $\mu$ is called quasi-invariant
with respect to $\sigma$. If additionally, $\mu\circ\sigma\sim\mu$ on
${\sigma^{-1}(\mathcal{B})}$, then $\sigma$ is called forward quasi-invariant.
In this case, there are Borel functions $\varphi$, called Markovian functions,
such that

(1.1) $\int_{X}(f\circ\sigma)\varphi\;d\mu=\int_{X}f\;d\mu.$

This relation is the basis for defining isometric operators: the composition
operator $S_{\sigma}:f\mapsto f\circ\sigma$ in the case of an invariant
measure $\mu$ and the weighted composition operator
$S_{\varphi}:f\mapsto\sqrt{\varphi}(f\circ\sigma)$ for a quasi-invariant
measure $\mu$ (here $\varphi$ is Markovian). These operators, together with
transfer operators, will play key roles in our constructions.

We formulate now our main results and outline the paper’s organization. In
Section 2, we define the main objects of this paper. They are: a standard
measure space, measurable partitions, canonical systems of measures,
subjective endomorphisms, invariant and quasi-invariant measures, and
Markovian functions. All these notions are used in the next sections. Section
2 should be viewed as a brief survey on endomorphisms and related notions.
Section 3 is focusing on the study of linear operators generated by an
endomorphism of a measure space. We define abstract transfer operators acting
on bounded Borel functions and consider their properties. If the operator
$S_{\sigma}$ is considered in $L^{2}(X,\mathcal{B},\mu)$ where $\mu$ is
$\sigma$-invariant, then $S^{*}_{\sigma}$ is a transfer operator which has
interesting properties, see Theorems 3.9, 3.14, 3.15, and 3.16. In particular,
the operator $S^{*}_{\sigma}$ coincides with a transfer operator $R_{\sigma}$
defined by the measurable partition into pre-images of $\sigma$. We prove also
similar results for weighted composition operators. Section 4 contains the
principal theorems connecting wavelet filters with representations of the
Cuntz relations. We recall that a family of isometries
$\\{T_{i}:i\in\Lambda\\}$ defines the Cuntz relations if
$\sum_{i\in\Lambda}T_{i}T^{*}_{i}=\mathbb{I}$ and the projections
$T_{i}T^{*}_{i}$ are mutually orthogonal for different indexes where $\Lambda$
is finite or countable. Let $\varphi$ be a Markovian function and $m$ a
complex-valued function such that
$S^{*}_{\varphi}(\sqrt{\varphi}|m|^{2})=\mathbbm{1}$. Define
$T_{m}(f)=mS_{\varphi}(f)$. Then $T_{m}$ an isometry in $L^{2}(\mu)$. We prove
the following results (see Theorem 4.11).

######  Theorem 1.1.

Let $\sigma$ be an onto endomorphism of $(X,\mathcal{B},\mu)$ where $\mu$ is
quasi-invariant with respect to $\sigma$. Let $\\{m_{i}:i\in\Lambda\\}$ be a
family of complex-valued functions. The operators $\\{T_{m_{i}},i\in\Lambda$}
generate a representation of the Cuntz algebra $\mathcal{O}_{|\Lambda|}$ if
and only if

$(i)\ \
S^{*}_{\varphi}(\sqrt{\varphi}\;\overline{m}_{j}m_{i})=\delta_{ij}\mathbbm{1},$
$(ii)\ \ \sum_{i\in\Lambda}m_{i}\mathbb{E}_{\varphi}(\overline{m}_{i}f)=f,\ \
\ f\in L^{2}(\mu).$

Here $\mathbb{E}_{\varphi}=S_{\varphi}S^{*}_{\varphi}$ is the orthogonal
projection from $L^{2}(\mu)$ onto a subspace $\mathcal{H}_{\varphi}$.

As a corollary, we have the following decomposition of $L^{2}(\mu)$ in the
case of $\sigma$-invariant measure $\mu$:

$L^{2}(\mu)=\bigoplus_{i\in\Lambda}\
m_{i}L^{2}(X,{\sigma^{-1}(\mathcal{B})},\mu).$

In Section 5, we focus on finding a description of the families of functions
$\underline{m}=(m_{i})\in\mathfrak{M}_{\varphi}$ satisfying the above theorem.
Let $\mathcal{G}$ be the group of Borel functions with values in the unitary
operators on $\ell^{2}(\Lambda)$. Then we prove the following:

######  Theorem 1.2 (Theorem 5.5).

(1) The set $\mathfrak{M}_{\varphi}$ is isomorphic (as a set) to the loop
group $\mathcal{G}$.

(2) For every element $G=(g_{ij})$ of the loop group, there exists a wavelet
filter $\underline{m}$ such that
$g_{ij}=S^{*}_{\varphi}(\sqrt{\varphi}\;m_{i}\overline{m}_{j})$.

Let functions $(m_{i}:i\in\Lambda)$ satisfy the property.

(1.2) $S_{\sigma}^{*}(|m_{i}|^{2})=\mathbbm{1}$

and $S_{i}(f)=m_{i}(f\circ\sigma)$ is an isometry on $L^{2}(\mu)$.

######  Theorem 1.3 (Theorem 5.8).

Let $(m_{i}:i\in\Lambda)$ be a set of cyclic vectors for a representation of
$L^{\infty}(X,{\sigma^{-1}(\mathcal{B})},\mu)$ on $L^{2}(\mu)$ that satisfies
(1.2). Then $\underline{m}=(m_{i})$ is a wavelet filter if and only if
$\sum_{i\in\Lambda}S^{*}_{i}S_{i}=\mathbb{I}$. In other words,
$\underline{m}\in\mathfrak{M}$ if and only if the operators $S_{i}$ are the
generators of a representation of the Cuntz algebra $\mathcal{O}_{|\Lambda|}$.

##  2\. Basics on endomorphisms

In this section, we give basic definitions and facts from the theory of
endomorphisms of a standard measure space $(X,\mathcal{B},\mu)$. The notion of
an endomorphism is one of the central concepts of ergodic theory; the
foundations and more advanced results on endomorphisms can be found in some
pioneering papers in ergodic theory and in more recent papers and books, see
e.g., [Roh61], [Haw94], [CFS82], [HS91], [Bén96], [BH09], [PU10], [Haw21] and
the papers cited therein. The study of endomorphisms is mostly based on the
notion of a measurable partition of a measure space and associated subalgebras
of Borel sets. A systematic study of measurable dynamical systems based on
applications of measurable partitions was initiated by Rokhlin in [Roh49a,
Roh49b, Roh61].

###  2.1. Endomorphisms of a measure space

We begin with the definitions of the main objects considered in the paper.

Let $(X,\mathcal{B})$ be a standard Borel space, i.e., $(X,\mathcal{B})$ is
Borel isomorphic to a Polish space with the sigma-algebra of Borel sets. If
$\mu$ is a non-atomic Borel positive measure on $(X,\mathcal{B})$, then
$(X,\mathcal{B},\mu)$ is called a standard Borel space.

By an endomorphism $\sigma$ of $(X,\mathcal{B},\mu)$ (or $(X,\mathcal{B})$) we
mean a measurable (or Borel) map of $X$ onto itself ($\sigma$ is subjective).
We will discuss various properties of endomorphisms below. In particular,
$\sigma$ defines a partition of $X$ into subsets $\\{\sigma^{-1}(x):x\in
X\\}$. Depending on the cardinality of the sets $\sigma^{-1}(x)$, we call
$\sigma$ either finite-to-one, or countable-to-one, or continuum-to-one.
Without loss of generality, we can assume that the cardinality
$|\sigma^{-1}(x)|$ is constant. The collection of sets
$\sigma^{-1}(A),A\in\mathcal{B}$, forms a $\sigma$-subalgebra of Borel sets
which is denoted ${\sigma^{-1}(\mathcal{B})}$ (below it will be also denoted
by $\mathcal{A}$ to shorten formulas). In general, the set $\sigma(A)$ is not
Borel for every $A\in\mathcal{B}$, but if $\sigma$ is at most countable-to-one
then $\sigma(A)$ is automatically Borel.

Let $End(X,\mathcal{B})$ denote the set (semigroup) of all surjective
endomorphisms of a standard Borel space $(X,\mathcal{B})$. By $M_{1}(X)$, we
denote the set of all Borel probability non-atomic measures. An element of
$M_{1}(X)$ will be simply called a measure in the paper. For $\sigma\in
End(X,\mathcal{B})$ and $\mu\in M_{1}(X)$, we define the measure
$\mu\circ\sigma^{-1}$ where $\mu\circ\sigma^{-1}(A):=\mu(\sigma^{-1}(A))$.
Then, the map $\mu\mapsto\mu\circ\sigma^{-1}$ defines an action of
$End(X,\mathcal{B})$ on $M_{1}(X)$. We will be interested in the following
cases: (i) the measure $\mu\circ\sigma^{-1}$ is equivalent to $\mu$, and (ii)
$\mu$ is $\sigma$-invariant. In case (i), we say that an endomorphism $\sigma$
is non-singular, i.e.,

(2.1) $\mu(A)=0\ \Longleftrightarrow\ \mu(\sigma^{-1}(A))=0,\ \ \ \forall
A\in\mathcal{B}.$

In other words, the measure $\mu$ is called (backward) $\sigma$-quasi-
invariant, in symbols, $\mu\circ\sigma^{-1}\sim\mu$. We use the notation
$End(X,\mathcal{B},\mu)$ to denote the semigroup of all surjective
endomorphisms $\sigma$ such that $\mu$ is quasi-invariant with respect to
$\sigma$. For a fixed $\mu$, the set $End(X,\mathcal{B},\mu)$ contains the
sub-semigroup $End_{\mu}$ which consists of the endomorphisms preserving
$\mu$, i.e., $\mu(A)=\mu\circ\sigma^{-1}(A)$. The set $End_{\mu}$ can be
viewed as the stabilizer of the action of $End(X,\mathcal{B})$ on $M_{1}(X)$
at $\mu$.

We will need also the notion of a (forward) quasi-invariant measure $\mu$.
This means that for every $\mu$-measurable set $A$, the set $\sigma(A)$ is
measurable and $\mu(A)=0\ \Longleftrightarrow\ \mu(\sigma(A))=0$.

######  Lemma 2.1.

Let $\sigma$ be a surjective endomorphism of a standard Borel space
$(X,\mathcal{B})$. Then $M_{1}(X)$ always contains a $\sigma$-quasi-invariant
measure $\mu$.

###### Proof.

Every endomorphism $\sigma$ generates a countable Borel equivalence relation
$E(\sigma)$ whose classes are the orbits of $\sigma$. By definition, $(x,y)\in
E(\sigma)$ if there exist $m,n\in\mathbb{N}_{0}$ such that
$\sigma^{n}(x)=\sigma^{m}(y)$. Quasi-invariant measures for $\sigma$ coincide
with quasi-invariant measures for $E(\sigma)$. Then we can use [DJK94,
Proposition 3.1] where the existence of $E(\sigma)$-quasi-invariant measures
was proved. ∎

######  Example 2.2.

Let
$(X,\mathcal{B},\mu)=\prod_{i\in\mathbb{N}}(X_{i},\mathcal{B}_{i},\mu_{i})$.
Define the left shift $\sigma$ on $X$:
$\sigma(x_{1},x_{2},x_{3},\cdots)=(x_{2},x_{3},\cdots)$. Then $\sigma$ is an
endomorphism. If $|X_{i}|=N$ for all $i$, then $\sigma$ is $N$-to-one.
Clearly, this construction can give other types of endomorphisms classified by
the cardinality of $\sigma^{-1}(x)$. If all
$(X_{i},\mathcal{B}_{i},\mu_{i})=(Y,\mathcal{C},\nu)$ are the same and $\mu$
is the product-measure $\otimes_{i}\nu$, then $\mu$ is $\sigma$-invariant. The
measure $\mu$ can be quasi-invariant with respect to the left shift $\sigma$
if we use different measures $\mu_{i}$. More details are in [HS91], [DH94],
and other papers of these authors.

###  2.2. Measurable partition

We refer to [Roh49b] (or [CFS82]) for the definition of a measurable
partition.

Let $\xi=\\{C_{\alpha}:\alpha\in I\\}$ be a partition of a standard
probability measure space $(X,\mathcal{B},\mu)$ such that
$C_{\alpha}\in\mathcal{B}$ (the index set $I$ can be either countable or
uncountable; we focus on the case of an uncountable set). A Borel set $A$ of
the form $\bigcup_{\alpha\in I^{\prime}}C_{\alpha}$, $I^{\prime}\subset I$, is
called a _$\xi$ -set_. Let $\mathcal{B}(\xi)$ be the sigma-algebra generated
by all $\xi$-sets.

A partition $\xi$ is called _measurable_ if $\mathcal{B}(\xi)$ contains a
countable subset $\\{D_{j}\\}$ of $\xi$-sets that separates any two elements
$C,C^{\prime}$ of $\xi$. This means that there exists $D_{i}$ such that either
$C\subset D_{i}$ and $C^{\prime}\subset X\setminus D_{i}$ or
$C^{\prime}\subset D_{i}$ and $C\subset X\setminus D_{i}$. Let $\pi$ be the
natural projection from $X$ to $X/\xi$, i.e., $\pi(x)=C_{x}$ where $C_{x}$ is
the element of $\xi$ containing $x$. Using the projection $\pi:X\to X/\xi$,
one can define a measure space $(X/\xi,\mathcal{B}/\xi,\mu_{\xi})$ where
$E\in\mathcal{B}/\xi$ if and only if the $\xi$-set $\pi^{-1}(E)$ is in
$\mathcal{B}$ and $\mu_{\xi}=\mu\circ\pi^{-1}$.

The following result was proved in [Roh49b].

######  Lemma 2.3.

A partition $\xi$ of a standard measure space $(X,\mathcal{B},\mu)$ is
measurable if and only if $(X/\xi,\mathcal{B}/\xi,\mu_{\xi})$ is a standard
measure space.

It is said that a partition $\zeta$ _refines_ $\xi$ (in symbols,
$\xi\prec\zeta$) if every element $C$ of $\xi$ is a $\zeta$-set. It turns out
that every partition $\zeta$ has a _measurable hull_ , that is a measurable
partition $\xi$ such that $\xi\prec\zeta$ and $\xi$ is a maximal measurable
partition with this property. If $\xi_{\alpha}$ is a family of measurable
partitions, then their product $\bigvee_{\alpha}\xi_{\alpha}$ is a measurable
partition $\xi$ which is uniquely determined by the conditions: (i)
$\xi_{\alpha}\prec\xi$ for all $\alpha$, and (ii) if $\eta$ is a measurable
partition such that $\xi_{\alpha}\prec\eta$, then $\xi\prec\eta$. Similarly,
one defines the intersection $\bigwedge_{\alpha}\xi_{\alpha}$ of measurable
partitions. There is a one-to-one correspondence between the set of measurable
partitions of a standard measure space $(X,\mathcal{B},\mu)$ and the set of
complete sigma-subalgebras $\mathcal{B}^{\prime}$ of $\mathcal{B}$.

The role of measurable partitions becomes clear from Theorem 2.5 given below.
This famous result uses the notion of measure disintegration.

######  Definition 2.4.

For a standard probability measure space $(X,\mathcal{B},\mu)$ and a
measurable partition $\xi$ of $X$, it is said that a collection of measures
$(\mu_{C})_{C\in X/\xi}$ is a _system of conditional measures_ with respect to
$(X,\mathcal{B},\mu)$ and $\xi$ if

(i) for each $C\in X/\xi$, $\mu_{C}$ is a measure on the sigma-algebra
$\mathcal{B}_{C}:=\mathcal{B}\cap C$ such that $(C,\mathcal{B}_{C},\mu_{C})$
is a standard probability measure space;

(ii) for any $B\in\mathcal{B}$, the function $C\mapsto\mu_{C}(B\cap C)$ is
$\mu_{\xi}$-measurable;

(iii) for any $B\in\mathcal{B}$,

(2.2) $\mu(B)=\int_{X/\xi}\mu_{C}(B\cap C)\;d\mu_{\xi}(C).$

Condition (2.2) can be rewritten in the equivalent form:

(2.3)
$\int_{X}f(x)\;d\mu(x)=\int_{X/\xi}\left(\int_{C}f(y)\;d\mu_{C}(y)\right)\;d\mu_{\xi}(C).$

Measurable partitions are characterized by the following result.

######  Theorem 2.5 ([Roh49b]).

For any measurable partition $\xi$ of a standard probability measure space
$(X,\mathcal{B},\mu)$, there exists a unique system of conditional measures
$(\mu_{C})$. Conversely, if $(\mu_{C})_{C\in X/\xi}$ is a system of
conditional measures with respect to $((X,\mathcal{B},\mu),\xi)$, then $\xi$
is a measurable partition.

We apply Theorem 2.5 to the case of an endomorphism $\sigma\in
End(X,\mathcal{B},\mu)$. Let $\xi_{\sigma}$ be the measurable partition of
$(X,\mathcal{B},\mu)$ into preimages $\sigma^{-1}(x)=C_{x}$ of points $x\in
X$. Let $(\mu_{C})$ be the system of conditional measures defined by
$\xi_{\sigma}$. In the case when $(X/\xi,\mu_{\xi})$ is isomorphic to
$(X,\mu)$ (for example, when $\sigma$ is the left shift or $\sigma:z\mapsto
z^{N}$, $z\in\mathbb{T}^{1}$), we see that relations (2.2) and (2.3) have the
form

(2.4)
$\int_{X}f(x)\;d\mu(x)=\int_{X}\left(\int_{C}f(y)\;d\mu_{x}(y)\right)\,d\mu(x).$

This decomposition is the key fact in our representation of the transfer
operator generated by $\sigma$.

In most important cases, the disintegration of a measure is applied to
probability (finite) measures. The problem of measure disintegration is
discussed in many books and articles. We refer here to [Bog07], [CFS82],
[Fab87, Fab00], [Kec95]. The case of an infinite sigma-finite measure was
considered by several authors, see e.g. [Sim12].

######  Theorem 2.6 ([Sim12]).

Let $(X,\mathcal{B},\mu)$ and $(Y,\mathcal{A},\nu)$ be standard measure spaces
with sigma-finite measures, and suppose that $\pi:X\to Y$ is a measurable map.
Let $(X,\mathcal{B},\mu)$ and $(Y,\mathcal{A},\nu)$ be as above. Suppose that
$\widehat{\mu}=\mu\circ\pi^{-1}\ll\nu$. Then there exists a unique system of
conditional measures $(\nu_{y})_{y\in Y}$ for $\mu$. For $\nu$-a.e., $\nu_{y}$
is a sigma-finite measure.

###  2.3. Radon-Nikodym derivatives and Markovian functions

Suppose $\sigma\in End(X,\mathcal{B},\mu)$. Recall that
$\mu\circ\sigma^{-1}\sim\mu$ in this case. Then we can define the Radon-
Nikodym derivative $\rho_{\mu}(x):=\frac{d\mu\circ\sigma^{-1}}{d\mu}(x)$,
which is a Borel function such that

(2.5) $\int_{X}f(\sigma x)\;d\mu=\int_{X}f(x)\rho_{\mu}(x)\;d\mu,\quad f\in
L^{1}(\mu).$

Setting $\rho_{n}(x)=\frac{d\mu\circ\sigma^{-n}}{d\mu}(x)$, we obtain the
Radon-Nikodym cocycle satisfying the equation
$\rho_{n+m}(x)=\rho_{m}(\sigma^{n}(x))\rho_{n}(x)$.

######  Remark 2.7.

Suppose that $\sigma\in End(X,\mathcal{B},\mu)$ and $\nu$ is a measure
equivalent to $\mu$, i.e., there exists a Borel function $\xi$ such that
$d\nu(x)=h(x)d\mu(x)$. Then $\sigma$ is also non-singular with respect to
$\nu$, and $\rho_{\nu}(x)=h(\sigma x)\rho_{\mu}(x)h^{-1}(x)$.

Every $\sigma\in End(X,\mathcal{B})$ defines a linear operator $S_{\sigma}$
called a composition operator on the space of bounded Borel functions:

$S_{\sigma}(f)=f\circ\sigma,\qquad f\in\mathcal{F}(X,\mathcal{B}).$

This operator is also known by the name of a Koopman operator when it is
considered in a $L^{2}$ space.

The map $A\mapsto\mu(\sigma(A)),A\in\mathcal{B},$ defines a measure on the
subalgebra $\sigma^{-1}(\mathcal{B})$. If $\mu$ is a forward quasi-invariant
measure, then there exists a unique $\sigma^{-1}(\mathcal{B})$-measurable
function $\omega_{\mu}(x)=\frac{d\mu\circ\sigma}{d\mu}(x)$ such that

(2.6) $\int_{X}f(\sigma x)\omega_{\mu}(x)\;d\mu=\int_{X}f(x)\;d\mu,\quad f\in
L^{1}(\mu).$

It can be deduced from the uniqueness of the Radon-Nikodym derivative that

$\omega_{\mu}(x)=\rho_{\mu}(\sigma(x))^{-1}$

when these functions are considered as functions measurable with respect to
$\sigma^{-1}(\mathcal{B})$.

The following statement is well-known and we omit its proof.

######  Lemma 2.8.

(1) The composition operator $S_{\sigma}:L^{2}(X,\mathcal{B},\mu)\to
L^{2}(X,{\sigma^{-1}(\mathcal{B})},\mu)$ is an isometry if and only if
$\mu\circ\sigma^{-1}=\mu$.

(2) The operator $S_{\sigma}$ on $L^{2}(\mu)$ is bounded if and only if there
exists a constant $k>0$ such that

$\frac{\mu(\sigma^{-1}(A))}{\mu(A)}\leq k,\quad A\in\mathcal{B}.$

(3) If $\mu$ is a forward quasi-invariant measure, then

$T_{\sigma}:f\longmapsto\sqrt{\omega_{\mu}}(f\circ\sigma)$

is an isometry from $L^{2}(X,\mathcal{B},\mu)$ onto
$L^{2}(X,{\sigma^{-1}(\mathcal{B})},\mu)$.

###  2.4. Properties of endomorphisms

In this subsection, we collected the properties of endomorphisms of a measure
space for the reader’s convenience. Here and below we implicitly use the
$\mathrm{mod}\ 0$-convention which means that a property (formula, relation,
etc) holds almost everywhere with respect to a fixed measure.

######  Definition 2.9.

Let $\sigma$ be a surjective endomorphism of $(X,\mathcal{B},\mu)$ with quasi-
invariant measure $\mu$.

(i) The endomorphism $\sigma$ is called conservative if for any set $A$ of
positive measure there exists $n>0$ such that $\mu(\sigma^{n}(A)\cap A)>0$.

(ii) The endomorphism $\sigma$ is called ergodic if whenever $A$ is
$\sigma$-invariant, i.e., $\sigma^{-1}(A)=A$, then either $A$ or $X\setminus
A$ is of measure zero. Equivalently, $\sigma$ is ergodic if, for a bounded
Borel function $f$, the condition $f\circ\sigma=f$ implies that $f$ is a
constant $\textrm{mod}\ 0$.

(iii) Any endomorphism $\sigma\in End(X,\mathcal{B},\mu)$ generates the
sequence of sub-algebras:

$\mathcal{B}\supset\sigma^{-1}(\mathcal{B})\ \cdots\
\supset\sigma^{-i}(\mathcal{B})\supset\ \cdots$

Then $\sigma\in End(X,\mathcal{B},\mu)$ is called exact if

$\mathcal{B}_{\infty}:=\bigcap_{k\in\mathbb{N}}\sigma^{-k}(\mathcal{B})=\\{\emptyset,X\\}\
\mod 0.$

(iv) A non-singular endomorphism $\sigma$ of $(X,\mathcal{B},\mu)$ is said to
be $\mu$-recurrent if for every non- negative Borel function $f$, the function

$\Sigma(f)=\sum_{n\geq 0}f(\sigma^{n}(x))\omega_{n}(x)$

takes only the values 0 and $\infty$ $\mu$-a.e. where
$\omega_{n}(x)=\dfrac{d\mu\circ\sigma^{n}}{d\mu}(x)$.

In the next remark, we include several results illustrating the properties of
endomorphisms given in Definition (2.9). We use some results from [Sil88],
[HS91], [Haw94].

######  Remark 2.10.

(1) Every exact endomorphism is ergodic. There are examples of ergodic
endomorphisms which are not exact. As it is customary in ergodic theory, we
can always assume, without loss of generality, that an endomorphism is
ergodic.

(2) There are examples of one-sided shifts ($n$-to-one endomorphisms) which
are not exact.

(3) We note that there are ergodic endomorphisms that are not conservative.

(4) An endomorphism $\sigma$ is recurrent with respect to a finite measure
$\mu$ if and only if $\sum_{n\geq 0}\omega_{n}(x)=\infty$.

(5) A $\mu$-recurrent endomorphism is conservative.

Since every surjective endomorphism defines an isometry $S_{\sigma}$ (or
$T_{\sigma}$), see Lemma 2.8, then we can apply Wold’s theorem to these
objects.

######  Theorem 2.11 (Wold’s theorem).

Let $S$ be an isometric operator in a Hilbert space $\mathcal{H}$. Define

$\mathcal{H}_{\infty}=\bigcap_{n}S^{n}\mathcal{H},$

and

$\mathcal{H}_{shift}=N_{S^{*}}\oplus SN_{S^{*}}\oplus\cdots\oplus
S^{k}N_{S^{*}}\oplus\cdots.$

Then the following statements hold.

(1) The space $\mathcal{H}$ is decomposed into the orthogonal direct sum

$\mathcal{H}=\mathcal{H}_{\infty}\oplus\mathcal{H}_{shift}.$

(2)

$\mathcal{H}_{\infty}=\\{x\in\mathcal{H}:\|(S^{*})^{n}x\|=\|x\|,\ \forall
n\in\mathbb{N}\\}.$

(3) The operator $S$ restricted on $\mathcal{H}_{\infty}$ is a unitary
operator, and $S$ is a unilateral shift in the space $\mathcal{H}_{shift}$.

For $\sigma\in End(X,\mathcal{B},\mu)$, define

$\mathcal{B}_{\infty}=\bigcap_{n=0}^{\infty}\sigma^{-n}(\mathcal{B}),$

and let $\mathcal{A}_{\sigma}=\\{A\in\mathcal{B}:\sigma^{-1}(A)=A\\}$ be the
subalgebra of $\sigma$-invariant subsets of $X$.

Let $\zeta$ be a partition of $(X,\mathcal{B},\mu)$ into orbits of $\sigma$
(recall that $x\stackrel{{\scriptstyle\zeta}}{{\sim}}y$ if there are $n,m$
such that $\sigma^{n}(x)=\sigma^{m}(y)$). Let $\eta$ be the partition of
$(X,\mathcal{B},\mu)$ such that $x\stackrel{{\scriptstyle\eta}}{{\sim}}y$ if
there is $n$ such that $\sigma^{n}(x)=\sigma^{n}(y)$. By $\zeta^{\prime}$ and
$\eta^{\prime}$ we denote the measurable halls of $\zeta$ and $\eta$,
respectively.

If $\epsilon$ denotes the partition of $X$ into points, then we have the
sequence of decreasing measurable partitions
$\\{\sigma^{-i}(\epsilon)\\}_{i=0}^{\infty}$:

$\epsilon\succeq\sigma^{-1}(\epsilon)\succeq\sigma^{-2}(\epsilon)\cdots.$

As shown in [Roh61], the following results hold

$\zeta^{\prime}\preceq\eta^{\prime},\qquad\ \ \
\eta^{\prime}=\bigwedge_{n}\sigma^{-n}(\epsilon)$

and

$\mathcal{A}(\zeta^{\prime})=\mathcal{A}_{\sigma},\qquad\ \ \
\mathcal{A}(\eta^{\prime})=\mathcal{B}_{\infty}.$

In particular, $\sigma$ is ergodic if the partition $\zeta^{\prime}$ is
trivial, and $\sigma$ is exact if the partition $\eta^{\prime}$ is trivial.

Since $\eta^{\prime}$ is a measurable partition, we can define the quotient
measure space
$(Y,\nu)=(X/\eta^{\prime},\mathcal{B}/\eta^{\prime},\mu_{\eta^{\prime}})$
where $\mathcal{B}/\eta^{\prime}=\mathcal{B}_{\infty}$.

The following result is deduced from Wold’s theorem, see details in [BJ18].

######  Corollary 2.12.

(1) Let $\pi:X\to Y$ be the natural projection. Then there exists a measure-
preserving automorphism $\widetilde{\sigma}:(Y,\nu)\to(Y,\nu)$ such that
$\widetilde{\sigma}$ is an automorphic factor of $\sigma$, i.e.,

$\widetilde{\sigma}\circ\pi=\pi\circ\sigma.$

(2) Let $S_{\sigma}:f\to f\circ\sigma$ be the isometry on
$\mathcal{H}=L^{2}(\mu)$. Then, in the Wold decomposition
$\mathcal{H}=\mathcal{H}_{\infty}\oplus\mathcal{H}_{\infty}^{\bot}$ for
$S_{\sigma}$, we have

$\mathcal{H}_{\infty}=L^{2}(Y,\nu),$

and the restriction of $S_{\sigma}$ to $\mathcal{H}_{\infty}$ corresponds to
the unitary operator $U$ defined by $\widetilde{\sigma}$,
$U(f)=f\circ\widetilde{\sigma}$.

It turns out that every non-singular endomorphism is a factor of an invertible
dynamical system. It is said that an automorphism $T\in
Aut(Y,\mathcal{C},\nu)$ is a natural extension of an endomorphism $\sigma\in
End(X,\mathcal{B},\mu)$ if there exists a map
$\tau:(Y,\mathcal{C},\nu)\to(X,\mathcal{B},\mu)$ such that

(i) $\mathcal{C}=\bigvee_{n=0}^{\infty}T^{n}(\tau^{-1}\mathcal{B}),\ \ \mod
0$,

(ii) there exists a measure $\nu^{\prime}\sim\nu$ such that
$\omega_{\nu^{\prime}}=\omega_{\mu}\circ\tau$.

We recall an important fact saying that every non-singular endomorphism
$\sigma$ of $(X,\mathcal{B},\mu)$ admits a a natural extension, see [Roh49a],
[Sil88], [Bén96], and Example 2.13.

######  Example 2.13.

Let $\sigma$ be an onto endomorphism of a standard Borel space
$(X,\mathcal{B})$. Define the set $\widehat{X}$ as follows: $\widehat{X}$ is a
subset of $X\times X\times X\times\cdots$ such that

$\widehat{x}=(x_{i})_{i\geq 0}\in\widehat{X}\ \Longleftrightarrow\
\sigma(x_{i+1})=x_{i}\ \ \forall i\geq 0.$

The set $\widehat{X}$ is often called the solenoid constructed by $(X,\sigma)$
and denoted $Sol_{(X,\sigma)}$. Note that $\widehat{X}$ is a closed subset in
the product space $X\times X\times\cdots$. Since $\widehat{X}$ is Borel, it
inherits the Borel structure $\widehat{\mathcal{B}}$ from the product space.

Define the map $\widehat{\sigma}:\widehat{X}\to\widehat{X}$ by setting

$\widehat{\sigma}(x_{0},x_{1},x_{2},\dots)=(\sigma(x_{0}),x_{0},x_{1},\dots).$

Then one can easily verify that $\widehat{\sigma}$ is a one-to-one Borel map
of $\widehat{X}$ onto itself, and the shift

$\tau:(x_{0},x_{1},x_{2},\dots)\mapsto(x_{1},x_{2},x_{3},\dots)$

is inverse to $\widehat{\sigma}$, $\tau=\widehat{\sigma}^{-1}$.

Let $\pi_{n}:\widehat{X}\to X$ be the projection from $\widehat{X}$ onto the
$n$-th coordinate: for $\widehat{x}=(x_{n})$, set
$\pi_{n}(\widehat{x})=x_{n}$, $n\geq 0$. Then $\pi_{n}$ can be extended to a
map $f\mapsto f\circ\pi_{n}$ from $\mathcal{F}(X,\mathcal{B})$ to
$\mathcal{F}(\widehat{X},\widehat{\mathcal{B}})$. It follows from the above
definitions that
$\pi_{n+1}\widehat{\sigma}(\widehat{x})=\pi_{n}(\widehat{x})$, and $\pi_{0}$
is a factor map from $(\widehat{X},\widehat{\sigma})$ to $(X,\sigma)$, i.e.,

$\pi_{0}\widehat{\sigma}=\sigma\pi_{0}.$

######  Proposition 2.14.

Let $\mu$ be a Borel continuous measure on a standard Borel space
$(X,\mathcal{B})$ which is quasi-invariant with respect to an endomorphism
$\sigma$. Let the measure $\mathbb{P}$ on $(\widehat{X},\widehat{B})$ be
defined by the relation $\mathbb{P}\circ\pi_{0}^{-1}=\mu$. Then the map

$L^{2}(\mu)\ni f\stackrel{{\scriptstyle
V_{0}}}{{\longrightarrow}}f\circ\pi_{0}\in L^{2}(\mathbb{P})$

is an isometry. Moreover, the operator $U:L^{2}(\mathbb{P})\to
L^{2}(\mathbb{P})$ defined by the formula

$U=V_{0}S_{\sigma}V_{0}^{*}$

is an isometry.

###  2.5. Markovian functions

In this subsection, we consider a class of functions defined by an
endomorphism $\sigma\in End(X,\mathcal{B},\mu)$. The following definition is
motivated by relation (2.6).

######  Definition 2.15.

Let $\sigma$ be an onto endomorphism of $(X,\mathcal{B},\mu)$. A function
$\varphi\in{\mathcal{F}(X,\mathcal{B})}$ satisfying

(2.7) $\int_{X}(f\circ\sigma)\varphi\;d\mu=\int_{X}f\;d\mu$

for all $f$ in $L^{1}(\mu)$ is called a Markovian function. The set of all
Markovian functions is denoted by $M(\sigma,\mu)$. We denote
$M_{2}(\sigma,\mu)=M(\sigma,\mu)\cap L^{2}(\mu)$.

The Markovian functions were considered in a series of papers [Haw94], [HS91],
[DH94], and others.

######  Remark 2.16.

(1) The set $M(\sigma,\mu)$ is convex.

(2) Suppose that $\mu$ is a forward quasi-invariant measure for $\sigma\in
End(X,\mathcal{B},\mu)$. Then the set of Markovian functions $M(\sigma,\mu)$
is not empty because $\omega_{\mu}\in M(\sigma,\mu)$ due to relation (2.6).

(3) For two equivalent measures $\mu$ and $\nu$, we discuss relation between
the functions $\omega_{\mu}$ and $\omega_{\nu}$ in Theorem 3.15. As above, the
function $\omega_{\mu}$ generates a cocycle by setting
$\omega_{n}(x)=\frac{d\mu\circ\sigma^{n}}{d\mu}(x)$ (where
$\omega_{1}=\omega_{\mu}$).

(4) Let $g$ be a function from $L^{1}(\mu)$. It was shown in [Haw94] that, for
the measure $d\nu=gd\mu$, the following holds:

$\int_{X}f(\sigma
x)\dfrac{g\omega_{\nu}}{g\circ\sigma}(x)\;d\mu=\int_{X}f(x)\;d\mu,\quad f\in
L^{1}(\mu).$

This means that the function $\dfrac{g\omega_{\nu}}{g\circ\sigma}$ is
Markovian with respect to $(\sigma,\mu)$.

Based on the facts from Remark 2.16, we prove the following result.

######  Proposition 2.17.

Let $\sigma\in End(X,\mathcal{B},\mu)$. Suppose a measure $\nu$ is equivalent
to $\mu$ and $g=\frac{d\nu}{d\mu}$. Then

$M(\sigma,\nu)=g^{-1}M(\sigma,\mu)(g\circ\sigma).$

###### Proof.

For a function $\varphi\in M(\sigma,\mu)$, show that
$h=g^{-1}\varphi(g\circ\sigma)\in M(\sigma,\nu)$. For any $f\in L^{1}(\nu)$,
we have

$\displaystyle\int_{X}(f\circ\sigma)g^{-1}\varphi(g\circ\sigma)\;d\nu=$
$\displaystyle\int_{X}(fg)\circ\sigma\varphi\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}fg\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}f\;d\nu.$

This shows that the function $h$ is in $M(\sigma,\nu)$.

Conversely, let $\varphi$ be a Markovian function from $M(\sigma,\nu)$. Since
$d\nu=gd\mu$,

$\displaystyle\int_{X}fg\;d\mu=$ $\displaystyle\int_{X}(f\circ\sigma)\varphi
g\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}(f\circ\sigma)(g\circ\sigma)[(g\circ\sigma)^{-1}\varphi
g]\;d\mu.$

The latter means that $(g\circ\sigma)^{-1}\varphi g\in M(\sigma,\mu)$. ∎

######  Lemma 2.18.

Let $\varphi_{1},...,\varphi_{k}$ be Markovian functions from $M(\sigma,\mu)$.
Then the function
$\psi=(\varphi_{k}\circ\sigma^{k-1})\cdots(\varphi_{2}\circ\sigma)\varphi_{1}$
belongs to $M(\sigma^{k},\mu)$.

###### Proof.

We compute

$\displaystyle\int_{X}(f\circ\sigma^{k})\psi\;d\mu=$
$\displaystyle\int_{X}[(f\circ\sigma^{k-1})\circ\sigma(\varphi_{k}\circ\sigma^{k-2})\circ\sigma\cdots(\varphi_{2}\circ\sigma)]\varphi_{1}\;d\mu$
$\displaystyle=$
$\displaystyle\int_{X}(f\circ\sigma^{k-1})(\varphi_{k}\circ\sigma^{k-2})\cdots(\varphi_{3}\circ\sigma)\varphi_{2}\;d\mu$
$\displaystyle\ \ \cdots\cdots$ $\displaystyle=$
$\displaystyle\int_{X}f\;d\mu.$

∎

##  3\. Operators generated by endomorphisms

This section considers several linear operators naturally defined by
surjective endomorphisms of $(X,\mathcal{B},\mu)$. These operators act in
$L^{2}(\mu)$ and other functional spaces.

###  3.1. Transfer operators and endomorphisms

We define a transfer operator in general settings using only the Borel
structure of the space $(X,\mathcal{B})$. Transfer operators are extensively
studied for various dynamical systems applying the properties of phase spaces.

######  Definition 3.1.

Let ${\mathcal{F}(X,\mathcal{B})}$ be the set of all bounded Borel functions
111In this section, we will consider real-valued functions for definiteness;
the case of complex-valued functions can be done similarly. and let
$R:{\mathcal{F}(X,\mathcal{B})}\to{\mathcal{F}(X,\mathcal{B})}$ be a linear
operator. Then $R$ is called a transfer operator if it satisfies the following
properties:

(i) $f\geq 0\ \Longrightarrow\ R(f)\geq 0$ (i.e., $R$ is a positive operator);

(ii) for any Borel functions $f,g\in\mathcal{F}(X,\mathcal{B})$, the pull-out
property holds

(3.1) $R((f\circ\sigma)g)=fR(g).$

To emphasize that a transfer operator $R$ is defined by an onto endomorphism
$\sigma$, we will also write $R$ as $(R,\sigma)$.

Let $\mathbbm{1}$ be a function on $(X,\mathcal{B})$ that takes the only value
1. If $R(\mathbbm{1})(x)>0$ for all $x\in X$, then we say that $R$ is a
_strict transfer operator_. If $R(\mathbbm{1})=\mathbbm{1}$, then the transfer
operator $R$ is called normalized.

Every transfer operator $R$ defines an action on the set of probability
measures $M_{1}(X)$: given $\mu\in M_{1}(X)$, set

$(\mu R)(f)=\int_{X}R(f)\;d\mu,\quad f\in{\mathcal{F}(X,\mathcal{B})}.$

If $\mu=\mu R$, then $\mu$ is called $R$-invariant. A measure $\mu\in
M_{1}(X)$ is called strongly invariant with respect to a transfer operator
$(R,\sigma)$ if $\mu$ $\sigma$-invariant and $\mu R=\mu$.

Restrictions of transfer operators on Banach or Hilbert spaces give more
possibilities to study their properties.

######  Lemma 3.2.

Let $(X,\mathcal{B},\mu)$ be a probability standard measure space. Suppose
that $(R,\sigma)$ is a normalized transfer operator acting in
$L^{2}(X,\mathcal{B},\mu)$ where $\mu\in M_{1}(X)$. Then $\mu$ is strongly
invariant with respect to $(R,\sigma)$ if and only if
$R^{*}(\mathbbm{1})=\mathbbm{1}$.

###### Proof.

Here and below we denote by $\langle\cdot,\cdot\rangle_{\mu}$ the inner
product in $L^{2}(\mu)$. Since $R$ is normalized, we can write

$\langle f,\mathbbm{1}\rangle_{\mu\circ\sigma^{-1}}=\langle
f\circ\sigma,R^{*}(\mathbbm{1})\rangle_{\mu}=\langle
R(f\circ\sigma),\mathbbm{1}\rangle_{\mu}=\langle
fR(\mathbbm{1}),\mathbbm{1}\rangle_{\mu}=\langle f,\mathbbm{1}\rangle_{\mu},$

that is $\mu$ is $\sigma$-invariant if $R^{*}(\mathbbm{1})=\mathbbm{1}$.

Similarly, we see that the condition $R^{*}(\mathbbm{1})=1$ is equivalent to
$\mu=\mu R$:

$\langle f,\mathbbm{1}\rangle_{\mu R}=\langle
R(f),\mathbbm{1}\rangle_{\mu}=\langle
f,R^{*}(\mathbbm{1})\rangle_{\mu}=\langle f,\mathbbm{1}\rangle_{\mu}.$

∎

######  Example 3.3.

Let $\sigma\in End(X,\mathcal{B},\mu)$, $\mu(X)=1$, and let the partition
$\xi_{\sigma}$ of $X$ be defined by preimages $\\{\sigma^{-1}(x):x\in X\\}$ of
$\sigma$. We give an example of a transfer operator $R_{\sigma}$ which is
determined by the system of conditional measures $\\{\mu_{C}\\}$ (see
Subsection 2.2) over the partition $\xi_{\sigma}$ of $X$. Define a linear
operator $R_{\sigma}$ acting on Borel bounded functions over the standard
probability measure space $(X,\mathcal{B},\mu)$ by setting

(3.2) $R_{\sigma}(f)(x):=\int_{C_{x}}f(y)\;d\mu_{C_{x}}(y)$

where $C_{x}=\sigma^{-1}(x)$.

######  Lemma 3.4.

The operator
$R_{\sigma}:{\mathcal{F}(X,\mathcal{B})}\to{\mathcal{F}(X,\mathcal{B})}$
defined by (3.2) is a transfer operator.

###### Proof.

Clearly, $R_{\sigma}$ is a positive normalized operator. To see that (3.1)
holds, we calculate

$\displaystyle R_{\sigma}((f\circ\sigma)g)(x)$ $\displaystyle=$
$\displaystyle\int_{C_{x}}(f\circ\sigma)(y)g(y)\;d\mu_{C_{x}}$
$\displaystyle=$ $\displaystyle f(x)\int_{C_{x}}g(y)\;d\mu_{C_{x}}(y)$
$\displaystyle=$ $\displaystyle f(x)(R_{\sigma}g)(x).$

Here we used the fact that $f(\sigma(y))=f(x)$ for $y\in
C_{x}=\sigma^{-1}(x)$. ∎

For an onto endomorphism $\sigma$ acting on the space $(X,\mathcal{B},\mu)$,
we consider the subalgebra $\mathcal{A}=\\{\sigma^{-1}(B):B\in\mathcal{B}\\}$
of $\mathcal{B}$. It is a well-known fact that there exists the conditional
expectation $\mathbb{E}_{\sigma}:L^{2}(X,\mathcal{B},\mu)\to
L^{2}(X,{\sigma^{-1}(\mathcal{B})},\mu)$. To simplify the formulas, we will
use also the following notations: $\mathcal{A}={\sigma^{-1}(\mathcal{B})}$,
$L^{2}(\mu)=L^{2}(X,\mathcal{B},\mu)$, and
$L^{2}(\mu_{\mathcal{A}})=L^{2}(X,{\sigma^{-1}(\mathcal{B})},\mu)$. Below, we
will describe the operator $\mathbb{E}_{\sigma}$ explicitly in terms of the
composition operator $S_{\sigma}$.

It turns out that, for every transfer operator $R$, one can define another
operator which is, in some sense, analogous to the conditional expectation.
For this, let $(R,\sigma)$ be a normalized transfer operator. We define

(3.3)
$E:{\mathcal{F}(X,\mathcal{B})}\to\mathcal{F}(X,{\sigma^{-1}(\mathcal{B})}):f\mapsto
R(f)\circ\sigma.$

We discuss the properties of the operator $E$ in Proposition 3.5. Some of them
are proved in [BJ18].

######  Proposition 3.5.

Let $R$ be a normalized transfer operator and $E(f)=R(f)\circ\sigma$. Then the
following properties hold:

(1) $E$ is positive and $E^{2}=E$,

$E({\mathcal{F}(X,\mathcal{B})})=\mathcal{F}(X,{\sigma^{-1}(\mathcal{B})})$,

$E|_{\mathcal{F}(X,{\sigma^{-1}(\mathcal{B})})}=id$,

$R\circ E\circ R=R^{2}$ and $R\circ E=R$.

(2) For $S_{\sigma}(f)=f\circ\sigma$, we have

$(RS_{\sigma})(f)=f,\ \ \ \ \ (S_{\sigma}R)(f)=E(f).$

(3) A bounded Borel function $f$ belongs to
$\mathcal{F}(X,{\sigma^{-1}(\mathcal{B})})$ if and only if there exists a
function $g\in{\mathcal{F}(X,\mathcal{B})}$ such that $f=g\circ\sigma$.

###### Proof.

These properties are proved directly. We only check that $R\circ E\circ
R=R^{2}$. Indeed,

$R[E(R(f))]=R[R(R(f)\circ\sigma]=R^{2}(f)R(\mathbbm{1})=R^{2}(f).$

∎

######  Corollary 3.6.

Let $R$ be the transfer operator defined in (3.2). Then the conditional
expectation
$E:{\mathcal{F}(X,\mathcal{B})}\to\mathcal{F}(X,{\sigma^{-1}(\mathcal{B})}):f\mapsto
R(f)\circ\sigma$ acts by the formula

$E(f)=R(f)\circ\sigma=\int_{\sigma^{-1}(\sigma(x))}f(y)\;d\mu_{C_{\sigma(x)}}(y).$

###  3.2. Composition operators and Markovian functions

We recall our notation: $\sigma$ is a surjective endomorphism of a probability
measure space $(X,\mathcal{B},\mu)$, $S_{\sigma}:f\mapsto f\circ\sigma$ is the
composition operator, and $M(\sigma,\mu)$ is the set of Markovian functions.

Let $t=t(x)$ be a bounded Borel function, and $\mu$ a $\sigma$-invariant
probability measure on $(X,\mathcal{B})$. We define the operator $P_{t}$ on
$L^{2}(\mu)$ by setting

$P_{t}(f)=t(f\circ\sigma),\quad f\in L^{2}(\mu).$

We call the operator $P_{t}$ a weighted composition operator. Clearly, $P_{t}$
is a bounded operator in $L^{2}(\mu)$.

######  Theorem 3.7.

(1) For $\sigma\in End(X,\mathcal{B},\mu)$ as above, consider the composition
operator $S_{\sigma}$ in the Hilbert space $L^{2}(\mu)$. Then the adjoint
operator $S_{\sigma}^{*}$ acts by the formula:

(3.4) $S_{\sigma}^{*}(g)=\frac{(gd\mu)\circ\sigma^{-1}}{d\mu},\quad g\in
L^{2}(\mu_{\mathcal{A}}).$

(2) The adjoint operator $S_{\sigma}^{*}$ is a transfer operator

$S_{\sigma}^{*}(g(f\circ\sigma))=fS_{\sigma}^{*}(g).$

The transfer operator $S_{\sigma}^{*}$ is normalized if and only if $\mu$ is
$\sigma$-invariant.

(3) For a function $t\in\mathcal{F}(X,\mathcal{B})$, the adjoint operator
$P_{t}^{*}$ is a non-normalized transfer operator.

###### Proof.

(1) For functions $f,g\in L^{2}(\mu)$, we have

$\displaystyle\langle S_{\sigma}f,g\rangle_{\mu}=$
$\displaystyle\int_{X}(f\circ\sigma)g\;d\mu$ $\displaystyle=$
$\displaystyle\int f\;(gd\mu)\circ\sigma^{-1}$ $\displaystyle=$
$\displaystyle\int f\frac{(gd\mu)\circ\sigma^{-1}}{d\mu}\;d\mu$
$\displaystyle=$ $\displaystyle\langle f,S_{\sigma}^{*}g\rangle_{\mu}$

which proves (3.4).

(2) To prove the pull-out property, we compute, for arbitrary functions
$f,g,h\in L^{2}(\mu)$,

$\displaystyle\int_{X}hS_{\sigma}^{*}((f\circ\sigma)g)\;d\mu=$
$\displaystyle\int_{X}S_{\sigma}(h)(f\circ\sigma)g\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}((hf)\circ\sigma)\;g\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}hfS_{\sigma}^{*}(g)\;d\mu.$

Finally, we see that

$\int_{X}f\;d\mu\circ\sigma^{-1}=\int_{X}(f\circ\sigma)\mathbbm{1}\;d\mu=\int_{X}fS_{\sigma}^{*}(\mathbbm{1})\;d\mu,$

and $S_{\sigma}^{*}(\mathbbm{1})=\mathbbm{1}\ \Longleftrightarrow\
\mu\circ\sigma^{-1}=\mu$.

(3) The same proof as in (1) gives the formula

$P_{t}^{*}(g)=\frac{(tgd\mu)\circ\sigma^{-1}}{d\mu},\quad g\in
L^{2}(\mu_{\mathcal{A}}).$

This shows that $P^{*}_{t}$ is not normalized.

To prove that $P_{t}^{*}$ satisfies the pull-out property, we write

$\displaystyle\langle h,P_{t}^{*}(g(f\circ\sigma))\rangle_{\mu}=$
$\displaystyle\langle P_{t}(h),g(f\circ\sigma)\rangle_{\mu}$ $\displaystyle=$
$\displaystyle\langle t(h\circ\sigma),g(f\circ\sigma)\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle P_{t}(hf),g\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle h,fP_{t}^{*}(g)\rangle_{\mu}.$

∎

######  Corollary 3.8.

(1) Let $\sigma\in End(X,\mathcal{B},\mu)$ be such that
$\mu=\mu\circ\sigma^{-1}$. Then the conditional expectation
$\mathbb{E}_{\sigma}:L^{2}(X,\mathcal{B},\mu)\to
L^{2}(X,\mathcal{A},\mu_{\mathcal{A}})$ can be represented as
$S_{\sigma}S^{*}_{\sigma}$.

(2) If $\mu$ is forward quasi-invariant, then the conditional expectation
$\mathbb{E}_{\sigma}$ coincides with $T_{\sigma}T^{*}_{\sigma}$ where the
isometry $T_{\sigma}(f)=\sqrt{\omega_{\mu}}(f\circ\sigma)$ is defined in Lemma
2.8.

###### Proof.

The fact that $(S_{\sigma}S^{*}_{\sigma})^{2}=S_{\sigma}S_{\sigma}^{*}$
follows from the identity $S_{\sigma}^{*}S_{\sigma}=\mathbb{I}$.

Next, we verify that $\langle
f,S_{\sigma}S_{\sigma}^{*}(h)\rangle_{\mu}=\langle f,h\rangle_{\mu}$ for $h\in
L^{2}(\mu_{\mathcal{A}})$. Recall that if $f\in L^{2}(\mu_{\mathcal{A}})$,
then there exists $g\in L^{2}(\mu)$ such that $f=g\circ\sigma$. Then, using
the fact that $S_{\sigma}^{*}$ is a normalized transfer operator (Theorem
3.7), we have

$\displaystyle\langle g\circ\sigma,S_{\sigma}S_{\sigma}^{*}(h)\rangle_{\mu}=$
$\displaystyle\langle
S_{\sigma}^{*}(g\circ\sigma),S_{\sigma}^{*}(h)\rangle_{\mu}$ $\displaystyle=$
$\displaystyle\langle
gS_{\sigma}^{*}(\mathbbm{1}),S_{\sigma}^{*}(h)\rangle_{\mu}$ $\displaystyle=$
$\displaystyle\langle g,S_{\sigma}^{*}(h)\rangle_{\mu}$ $\displaystyle=$
$\displaystyle\langle g\circ\sigma,h\rangle_{\mu}$ $\displaystyle=$
$\displaystyle\langle f,h\rangle_{\mu}$

The orthogonality of the projection $S_{\sigma}S_{\sigma}^{*}$ is obtained
from the relation

$\langle S_{\sigma}S_{\sigma}^{*}f,g\circ\sigma\rangle_{\mu}=\langle
S^{*}_{\sigma}f,g\rangle_{\mu}=\langle f,g\circ\sigma\rangle_{\mu}.$

It proves that $\mathbb{E}_{\sigma}=S_{\sigma}S_{\sigma}^{*}$.

(2) The case of a quasi-invariant measure $\mu$ is considered similarly. We
note that $\omega_{\mu}$ is a Borel function measurable with respect to
${\sigma^{-1}(\mathcal{B})}$. Hence, every function $f\in
L^{2}(\mu_{\mathcal{A}})$ there exists a function $g\in L^{2}(\mu)$ such that
$f=\sqrt{\omega_{\mu}}(g\circ\sigma)$. Then we repeat the above calculations.
We leave the details to the reader. ∎

We associated with every endomorphism $\sigma\in End(X,\mathcal{B},\mu)$ two
transfer operators $R$ and $S^{*}_{\sigma}$. It turns out that they coincide
in $L^{2}(\mu)$.

######  Theorem 3.9.

Let $\sigma\in End(X,\mathcal{B},\mu)$. Then the transfer operators
$R_{\sigma}$ and $S^{*}_{\sigma}$ coincide in $L^{2}(\mu)$, where $R_{\sigma}$
is defined in (3.2) and $S_{\sigma}^{*}$ satisfies (3.4).

###### Proof.

We compute $S^{*}_{\sigma}(f)$ using the disintegration of $\mu$ with respect
to the conditional measures $\mu_{x}$ on $C_{x}=\sigma^{-1}(x)$:

$\displaystyle\langle S(g),f\rangle_{\mu}=$
$\displaystyle\int_{X}(g\circ\sigma)f\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}\left(\int_{C_{x}}g(\sigma(y))f(y)\;d\mu_{x}(y)\right)\;d\mu(x)$
$\displaystyle=$
$\displaystyle\int_{X}g(x)\left(\int_{C_{x}}f(y)\;d\mu_{x}(y)\right)\;d\mu(x)$
$\displaystyle=$ $\displaystyle\langle g,S^{*}_{\sigma}(f)\rangle_{\mu}$

From the latter, we see that

$S^{*}_{\sigma}(f)=\int_{C_{x}}f(y)\;d\mu_{x}(y)=R_{\sigma}(f).$

∎

######  Remark 3.10.

(1) It follows from Lemma 3.2 and Theorem 3.9 that, for the transfer operator
$S^{*}_{\sigma}$ in $L^{2}(\mu)$, the measure $\mu$ is strong invariant if and
only if it is $\sigma$-invariant.

(2) Theorem 3.9 implies that $E(f)=R(f)\circ\sigma$ coincides with
$\mathbb{E}_{\sigma}$ for $R=S^{*}_{\sigma}$.

We now consider weighted composition operators where the weight function is
Markovian; for consistency, we will write $P_{\varphi}$ for such a weighted
composition operator. We will continue discussing the properties of weighted
composition operators in the next section.

Let $\sigma\in End(X,\mathcal{B},\mu)$ be a surjective endomorphism, and $\mu$
is a forward quasi-invariant measure. Consider the operator

(3.5) $P_{\varphi}:f\to\varphi(f\circ\sigma)$

which is formally defined in ${\mathcal{F}(X,\mathcal{B})}$. It can be also
written as

$P_{\varphi}(f)=M_{\varphi}S_{\sigma}(f),\quad
f\in{\mathcal{F}(X,\mathcal{B})}$

where $M_{\varphi}$ is the multiplication operator.

######  Lemma 3.11.

Let $\sigma,\varphi,$ and $P_{\varphi}$ be as above. Then $\varphi$ is a
Markovian function ($\varphi\in M(\sigma,\mu)$) if and only if $\mu$ is
$P_{\varphi}$-invariant, i.e., $\mu P_{\varphi}=\mu$ where

$\mu P_{\varphi}(f)=\int_{X}P_{\varphi}(f)\;d\mu.$

In particular, $\mu=\mu P_{\omega_{\mu}}$.

###### Proof.

Indeed, if $\varphi$ is Markovian, then

$\int_{X}P_{\varphi}(f)\;d\mu=\int_{X}(f\circ\sigma)\varphi\;d\mu=\int_{X}f\;d\mu$

which means that $\mu$ is $P_{\varphi}$-invariant. The converse statement also
follows from the relation above. ∎

######  Lemma 3.12.

Let $\varphi\in M(\sigma,\mu)$ and $P_{\varphi}$ a weighted composition
operator. Them $P_{\varphi}:L^{1}(\mu)\to L^{1}(\mu_{\mathcal{A}})$ and
$P_{\sqrt{\varphi}}:L^{2}(\mu)\to L^{2}(\mu_{\mathcal{A}})$ are isometric
operators.

###### Proof.

Straightforward. ∎

######  Proposition 3.13.

Let $\sigma\in End(X,\mathcal{B},\mu)$ and $\varphi$ is a function from
$L^{2}(\mu)$. Then $\varphi$ is a Markovian function if and only if
$S^{*}_{\sigma}(\varphi)=\mathbbm{1}$.

###### Proof.

Let $f\in L^{2}(\mu)$. Then the result follows from the following relations:

$`\begin{aligned} \int_{X}f\;d\mu=&\int_{X}(f\circ\sigma)\varphi\;d\mu\\\
=&\int S_{\sigma}(f)\varphi\;d\mu\\\
=&\int_{X}fS_{\sigma}^{*}(\varphi)\;d\mu.\end{aligned}$

∎

######  Theorem 3.14.

Let $P_{\varphi}$ be defined in $L^{2}(\mu)$ according to (3.5) where
$\varphi>0$. (1) The following statements are equivalent:

(i) $P_{\varphi}$ is an isometry in $L^{2}(\mu)$;

(ii) the composition operator $S_{\sigma}$ is an isometry in
$L^{2}(\varphi\mu)$;

(iii) $S^{*}_{\sigma}(\varphi)=\mathbbm{1}$;

(iv)

$\frac{(\varphi d\mu)\circ\sigma^{-1}}{d\mu}=1\ \ \mathrm{a.e.}$

(2) If $\varphi$ is a positive Borel function from $L^{2}(\mu)$, then the
adjoint operator $P^{*}_{\varphi}$ is a transfer operator and
$P^{*}_{\varphi}$ is normalized if and only if $\varphi$ is Markovian.

###### Proof.

(1) The proof of the first statement uses the arguments given in the proofs of
Theorems 3.13 and 3.7. We leave the details for the reader.

(2) It is clear that $P^{*}_{\varphi}$ is positive because we have the
following formula for $P^{*}_{\varphi}$:

$P^{*}_{\varphi}(g)=\frac{(\varphi gd\mu)\circ\sigma^{-1}}{d\mu}.$

Show that it satisfies the pull-out property. Since
$P_{\varphi}=M_{\varphi}S_{\sigma}$ and $S_{\sigma}^{*}$ is a transfer
operator, we obtain

$\displaystyle P^{*}_{\varphi}((f\circ\sigma)g)=$ $\displaystyle\
S^{*}_{\sigma}M^{*}_{\varphi}[(f\circ\sigma)g]$ $\displaystyle=$
$\displaystyle\ S^{*}_{\sigma}[\overline{\varphi}g(f\circ\sigma)]$
$\displaystyle=$ $\displaystyle\ fS^{*}_{\sigma}(\overline{\varphi}g)$
$\displaystyle=$ $\displaystyle\ fS^{*}_{\sigma}M^{*}_{\varphi}(g)$
$\displaystyle=$ $\displaystyle\ fP^{*}_{\varphi}(g).$

To finish the proof, we note that

$\displaystyle\int_{X}(f\circ\sigma)\varphi\mathbbm{1}\;d\mu=$
$\displaystyle\int_{X}P_{\varphi}(f)\mathbbm{1}\;d\mu$ $\displaystyle=$
$\displaystyle\ \int_{X}fP^{*}_{\varphi}(\mathbbm{1})\;d\mu.$

Hence

$\int_{X}(f\circ\sigma)\varphi\;d\mu=\int_{X}f\;d\mu$

if and only if $P^{*}_{\varphi}(\mathbbm{1})=\mathbbm{1}$. ∎

###  3.3. Radon-Nikodym derivatives and conditional expectations

As above, $\sigma\in End(X,\mathcal{B},\mu)$ is an onto endomorphism with
quasi-invariant measure $\mu$. Equation (2.6) defines a uniquely determined
Radon-Nikodym derivative $\omega_{\mu}$ which is
$\sigma^{-1}(\mathcal{B})$-measurable function.

Let $\mathbb{E}_{\sigma}$ denote the conditional expectation from $L^{2}(\mu)$
onto $L^{2}(\mu_{\mathcal{A}})$, where
$\mathcal{A}={\sigma^{-1}(\mathcal{B})}$ and $\mu_{\mathcal{A}}$ is the
projection of $\mu$ onto the sigma-algebra $\mathcal{A}$. We recall that
$\mathbb{E}_{\sigma}=S_{\sigma}S^{*}_{\sigma}$ for $\sigma$-invariant measure
$\mu$ and $\mathbb{E}_{\sigma}=T_{\sigma}T^{*}_{\sigma}$ for $\sigma$-quasi-
invariant measure $\mu$, see Corollary 3.8.

Suppose that $\nu$ is another measure on $(X,\mathcal{B})$ which is equivalent
to the measure $\mu$. In the next theorem, we show how Markovian functions
with respect to the measures $\mu$ and $\nu$ are related (see also Remark
2.7).

######  Theorem 3.15.

Let $\sigma\in End(X,\mathcal{B},\mu)$ and $d\nu(x)=h(x)d\mu(x)$ where
$h(x)>0$ $\mu$-a.e. Let $\psi\in M(\sigma,\nu)$ and $\varphi\in
M(\sigma,\mu)$. Then

(3.6) $\mathbb{E}_{\sigma}(h)\psi=\varphi(h\circ\sigma).$

Hence, relation (3.6) establishes a one-to-one correspondence between the sets
of Markovian functions $M(\sigma,\mu)$ and $M(\sigma,\nu)$.

###### Proof.

Since the both sides of (3.6) are measurable with respect to
${\sigma^{-1}(\mathcal{B})}$, it suffices to prove that, for every function
$g\in{\mathcal{F}(X,\mathcal{B})}$,

(3.7)
$\int_{X}(g\circ\sigma)\mathbb{E}_{\sigma}(h)\;\psi\;d\mu=\int_{X}(g\circ\sigma)(h\circ\sigma)\;\varphi\;d\mu.$

We compute the left-hand side and the right-hand side in (3.7) separately
using the definition of Markovian functions and the properties of conditional
expectations. For the RHS:

$\displaystyle\int_{X}(g\circ\sigma)(h\circ\sigma)\;\varphi\;d\mu=$
$\displaystyle\int_{X}(gh)\circ\sigma\;\varphi\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}gh\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}g\;d\nu.$

For the LHS, we use the fact that $\mathbb{E}_{\sigma}$ is the conditional
expectation and the function $(g\circ\sigma)\psi$ is
${\sigma^{-1}(\mathcal{B})}$-measurable:

$\displaystyle\int_{X}(g\circ\sigma)\mathbb{E}_{\sigma}(h)\;\psi\;d\mu=$
$\displaystyle\int_{X}(g\circ\sigma)h\;\psi\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}(g\circ\sigma)\;\psi\;d\nu$ $\displaystyle=$
$\displaystyle\int_{X}g\;d\nu.$

∎

Let $R=(R,\sigma)$ be a transfer operator where $\sigma$ is an onto
endomorphism of $(X,\mathcal{B},\mu)$. Recall that we have defined in (3.3)
the operator
$E=R(f)\circ\sigma:{\mathcal{F}(X,\mathcal{B})}\to\mathcal{F}(X,{\sigma^{-1}(\mathcal{B})})$
which is an analog of the conditional expectation $\mathbb{E}_{\sigma}$. In
the following statement, we find out under what conditions on the measure
$\mu$ and the transfer operator $R$ the operator $E$ coincides with the
genuine conditional expectation $\mathbb{E}_{\sigma}$.

######  Theorem 3.16.

In the setting formulated above, the operator $E=R(f)\circ\sigma$ coincides
with $\mathbb{E}_{\sigma}=S_{\sigma}S^{*}_{\sigma}$ in $L^{2}(\mu)$ if and
only if

(3.8)
$R(f)\frac{d\mu\circ\sigma^{-1}}{d\mu}=\frac{(fd\mu)\circ\sigma^{-1}}{d\mu},\quad
f\in L^{2}(\mu).$

Relation (3.8) can be also written as $\rho_{\mu}R(f)=S^{*}_{\sigma}(f)$.

###### Proof.

It was shown in Proposition 3.5 that $E^{2}=E$. It remains to find out under
what conditions the relation $E=E^{*}$ hold. Clearly, it is equivalent to the
property

(3.9) $\int_{X}gE(f)\;d\mu=\int_{X}gf\;d\mu,\quad\forall
g\in\mathcal{F}(X,{\sigma^{-1}(\mathcal{B})}).$

Representing $g$ as $h\circ\sigma$ ($h\in{\mathcal{F}(X,\mathcal{B})}$), we
obtain that (3.9) is equivalent to

$\int_{X}(h\circ\sigma)R(f)\circ\sigma\;d\mu=\int_{X}(h\circ\sigma)f\;d\mu$

or

$\int_{X}hR(f)\;d\mu\circ\sigma^{-1}=\int_{X}h\;(fd\mu)\circ\sigma^{-1}.$

This means that

$R(f)=\frac{(fd\mu)\circ\sigma^{-1}}{d\mu\circ\sigma^{-1}}$

which is equivalent to (3.8). This proves that $E=\mathbb{E}_{\sigma}$ if and
only if $S^{*}_{\sigma}(f)=\rho_{\mu}R(f)$.

We note that if $\mu$ is a $\sigma$-invariant measure, then

$R(f)=\frac{(fd\mu)\circ\sigma^{-1}}{d\mu}$

which coincides with $S^{*}_{\sigma}(f)$ by (3.4). It follows then that
$\mathbb{E}_{\sigma}=E=R(f)\circ\sigma$ in the case of $\sigma$-invariant
measure $\mu$. Moreover, it is obvious that the condition
$R(f)=S^{*}_{\sigma}(f)$ implies the invariance of $\mu$ with respect to
$\sigma$. ∎

######  Corollary 3.17.

In notation given above, the operator
$E_{\rho_{\mu}}=(\rho_{\mu}R)\circ\sigma$ coincides with
$\mathbb{E}_{\sigma}$.

###### Proof.

We noted that $\rho_{\mu}R$ is a transfer operator coinciding with
$S^{*}_{\sigma}$, and therefore we can define
$E_{\rho_{\mu}}=S_{\sigma}R_{\rho_{\mu}}$. By Corollary 3.8, we obtain that
$S_{\sigma}R_{\rho_{\mu}}=S_{\sigma}S^{*}_{\sigma}=\mathbb{E}_{\sigma}$ which
proves the statement. ∎

######  Remark 3.18.

The results of Proposition 2.17 and Theorem 3.15 can be interpreted as
follows.

Let $G$ denote the group $\mathcal{F}_{+}(X,\mathcal{B})$ of Borel bounded
strictly positive functions. Then $G$ acts on the set
$\\{M(\sigma,\nu):\nu\sim\mu\\}$. This action $\alpha=\\{\alpha_{f}:f\in G\\}$
is defined by the rule:

$\alpha_{f}(\varphi)=\frac{f\circ\sigma}{f}\varphi,\ \quad\varphi\in
M(\sigma,\mu).$

Clearly,

$\alpha_{f}(M(\sigma,\mu))=M(\sigma,\nu),$

where $d\nu=fd\mu$, and $\alpha_{f}\alpha_{g}=\alpha_{fg}$. The action
$\alpha$ is free in the sense that it satisfies the property
$M(\sigma,\mu)\cap M(\sigma,\nu)=\emptyset$ if $\nu\sim\mu$. Moreover,
$\alpha$ is transitive.

##  4\. Cuntz relations for invariant and quasi-invariant measures

Starting with the measurable category, and disintegration of the appropriate
measures, we showed above that careful choice of Hilbert spaces allows for a
powerful tool in the analysis of endomorphisms and branching systems (in the
measurable setting). In more detail, the steps from transformations in measure
space to $L^{2}$ spaces and operators are often called “passing to the Koopman
operators”. In our context, the non-commutativity for the operators under
consideration is captured well with the Cuntz relations, or rather their
representations; see [Cun77], [JT20], [BJ15]. Recall that, following J. Cuntz,
for every $n$, one introduces a $C^{*}$-algebra $\mathcal{O}_{|\Lambda|}$
defined by a system of $|\Lambda|$ generators $T_{i}$. These generators may be
realized as operators in Hilbert space, say $H$ as follows: The relations
(Cuntz-relations) state that the $T_{i}$ system is represented by isometries
with orthogonal ranges in $H$ such that the sum of these ranges is $H$. (Think
of the subspaces as sub-bands.) In other words, via the isometries, $H$ arises
as an orthogonal sum of copies of itself. As a $C^{*}$-algebra,
$\mathcal{O}_{|\Lambda|}$ is simple. Its representations are important, and
they play a crucial role in the study of self-similar dynamics and self-
similar geometries.

###  4.1. Quasi-invariant measure

Let $\sigma\in End(X,\mathcal{B},\mu)$ where $\mu$ is a forward and backward
quasi-invariant measure. We recall that, in this case, the corresponding
Radon-Nikodym derivatives $\omega_{\mu}$ and $\rho_{\mu}$ are well-defined
functions satisfying (2.6) and (2.5). In the remaining sections of this paper,
we will consider $L^{2}$-spaces of complex-valued functions.

Let $\varphi$ be a positive Markovian function, $\varphi\in M(\sigma,\mu)$.
Then we define a weighted composition operator $S_{\varphi}$ acting on
$L^{2}(\mu)$ by

(4.1) $S_{\varphi}(f)=\sqrt{\varphi}(f\circ\sigma)$

Equivalently, $S_{\varphi}=M_{\sqrt{\varphi}}S_{\sigma}$ where
$M_{\sqrt{\varphi}}$ denotes the operator of multiplication, and $S_{\sigma}$
is the composition operator.

We mention two important particular cases of (4.1) when (a)
$\varphi=\omega_{\mu}$ and (b) $\varphi=\mathbbm{1}$. Case (b) occurs if and
only if $\mu\circ\sigma^{-1}=d\mu$.

######  Lemma 4.1.

(1) Let $\varphi$ be a Markovian function from
$M_{2}(\sigma,\mu)=M(\sigma,\mu)\cap L^{2}(\mu)$, and $S_{\sigma}$ the
composition operator. Then $S_{\sigma}^{*}(\varphi)=\mathbbm{1}$.

(2) The function $\varphi$ is Markovian with respect to $\sigma$ and $\mu$ if
and only if $(\varphi d\mu)\circ\sigma^{-1}=\mu$.

###### Proof.

The first statement follows from the definition of a Markovian function:

$\int_{V}f\;d\mu=\int_{X}(f\circ\sigma)\varphi\;d\mu=\int_{X}S_{\sigma}(f)\varphi\;d\mu=\int_{V}fS_{\sigma}^{*}(\varphi)\;d\mu.$

The second statement is a reformulation of relation (2.7). ∎

######  Lemma 4.2.

The operator $S_{\varphi}$ is an isometry in $L^{2}(\mu)$.

###### Proof.

It follows from (2.7) that the function $\sqrt{\varphi}(f\circ\sigma)\in
L^{2}(\mu)$ if $f\in L^{2}(\mu)$. Since $\varphi$ is Markovian, we have

$\displaystyle\langle S_{\varphi}(f),S_{\varphi}(g)\rangle_{\mu}=$
$\displaystyle\int_{X}\sqrt{\varphi}(f\circ\sigma)\sqrt{\varphi}(\overline{g}\circ\sigma)\;d\mu$
$\displaystyle=$
$\displaystyle\int_{X}(f\overline{g}\circ\sigma)\varphi\;d\mu$
$\displaystyle=$ $\displaystyle\int_{X}f\overline{g}\;d\mu,\qquad f,g\in
L^{2}(\mu).$

∎

######  Theorem 4.3.

(1) For $\sigma\in End(X,\mathcal{B},\mu)$ as above and the operator
$S_{\varphi}$, the adjoint operator $S_{\sigma}^{*}$ acts by the formula:

(4.2)
$S_{\varphi}^{*}(g)=\frac{(g\sqrt{\varphi}d\mu)\circ\sigma^{-1}}{d\mu},\quad
g\in L^{2}(\mu).$

(2) The adjoint operator $S_{\varphi}^{*}$ is a transfer operator satisfying
the pull-out property:

(4.3) $S_{\varphi}^{*}(g(f\circ\sigma))=fS_{\varphi}^{*}(g).$

The operator $S_{\varphi}^{*}$ is normalized if and only if $\mu$ is
$\sigma$-invariant.

###### Proof.

(1) For functions $f,g\in L^{2}(\mu)$, we have

$\int_{X}\sqrt{\varphi}(f\circ\sigma)g\;d\mu=\int
f\;(\sqrt{\varphi}gd\mu)\circ\sigma^{-1}=\int
f\frac{(gd\mu)\circ\sigma^{-1}}{d\mu}\;d\mu$

which proves (4.2).

(2) The operator $S^{*}_{\varphi}$ is obviously positive. To prove the pull-
out property, we compute, for arbitrary functions $f,g,h\in L^{2}(\mu)$,

$\displaystyle\int_{X}hS_{\varphi}^{*}((f\circ\sigma)g)\;d\mu=$
$\displaystyle\int_{X}S_{\varphi}(h)(f\circ\sigma)g\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}\sqrt{\varphi}(hf)\circ\sigma\;g\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}S_{\varphi}(hf)g\;d\mu$ $\displaystyle=$
$\displaystyle\int_{X}hfS^{*}_{\varphi}(g)\;d\mu.$

This proves that (4.3) holds.

We see that

$S_{\varphi}^{*}(\mathbbm{1})=\frac{(\sqrt{\varphi}d\mu)\circ\sigma^{-1}}{d\mu}.$

Hence, $S_{\varphi}^{*}$ is normalized if and only if
$\mu=\mu\circ\sigma^{-1}$ and $\varphi=\mathbbm{1}$. This can be proved as
follows

$\int_{X}f\;d\mu\circ\sigma^{-1}=\int_{X}(f\circ\sigma)\mathbbm{1}\;d\mu=\int_{X}fS_{\sigma}^{*}(\mathbbm{1})\;d\mu,$

and $S_{\sigma}^{*}(\mathbbm{1})=\mathbbm{1}\ \Longleftrightarrow\
\mu\circ\sigma^{-1}=\mu$. ∎

######  Remark 4.4.

One can easily check that, for $\varphi=\omega_{\mu}$,

$S^{*}_{\omega_{\mu}}(\frac{1}{\sqrt{\omega_{\mu}}})=\rho_{\mu}.$

We recall that $\mathcal{A}$ denotes the subalgebra $\sigma^{-1}(\mathcal{B})$
and $\mu_{\mathcal{A}}$ denotes the restriction of $\mu$ onto $\mathcal{A}$.
It is an important observation that a function $f$ is $\mathcal{A}$-measurable
if and only if there exists a $\mathcal{B}$-measurable function $g$ such that
$f=g\circ\sigma$.

For a fixed Markovian function $\varphi$, consider the subspace
$\mathcal{H}_{\varphi}$ of function spanned by $\sqrt{\varphi}$ and
$\mathcal{A}$-measurable functions:

$\mathcal{H}_{\varphi}=\\{\sqrt{\varphi}(f\circ\sigma):f\in L^{2}(\mu)\\}$

######  Proposition 4.5.

Let $\sigma,\varphi,$ and $S_{\varphi}$ be as above. Then

(1) $\mathbb{E}_{\varphi}:=S_{\varphi}S^{*}_{\varphi}$ is an orthogonal
projection from $L^{2}(\mu)$ onto $\mathcal{H}_{\varphi}$;

(2)
$\mathbb{E}_{\varphi}((f\circ\sigma)g)=(f\circ\sigma)\mathbb{E}_{\varphi}(g)$

(3)
$S^{*}_{\varphi}(f\mathbb{E}_{\varphi}(g))=S^{*}_{\varphi}(g)S^{*}_{\varphi}(f\sqrt{\varphi})$.

###### Proof.

(1) Let $f,g\in L^{2}(\mu)$. Then we write

$\displaystyle\langle
S_{\varphi}S^{*}_{\varphi}f,\sqrt{\varphi}(g\circ\sigma)\rangle_{\mu}=$
$\displaystyle\langle S_{\varphi}S^{*}_{\varphi}f,S_{\varphi}g\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle S^{*}_{\varphi}f,g\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle f,S_{\varphi}g\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle
f,\sqrt{\varphi}(g\circ\sigma)\rangle_{\mu}.$

Hence, $(f-S_{\varphi}S^{*}_{\varphi}f)\perp\mathcal{H}_{\varphi}$.

For (2), we first note that
$S_{\varphi}(fg)=\sqrt{\varphi}(g\circ\sigma)(f\circ\sigma)=S_{\varphi}(g)S_{\sigma}(f)$,
and then

$\displaystyle\mathbb{E}_{\varphi}((f\circ\sigma)g)=$ $\displaystyle
S_{\varphi}S^{*}_{\varphi}((f\circ\sigma)g)$ $\displaystyle=$ $\displaystyle
S_{\varphi}(S^{*}_{\varphi}(g)f)$ $\displaystyle=$
$\displaystyle(S_{\varphi}S^{*}_{\varphi})(g)S_{\sigma}(f)$ $\displaystyle=$
$\displaystyle(f\circ\sigma)\mathbb{E}_{\varphi}(g).$

For (3), we use the pull-out property of $S^{*}_{\varphi}$:

$\displaystyle S^{*}_{\varphi}(f\mathbb{E}_{\varphi}(g))=$ $\displaystyle
S^{*}_{\varphi}(fS_{\varphi}S^{*}_{\varphi}(g))$ $\displaystyle=$
$\displaystyle
S^{*}_{\varphi}(f\sqrt{\varphi}(S^{*}_{\varphi}(g)\circ\sigma))$
$\displaystyle=$ $\displaystyle
S^{*}_{\varphi}(g)S^{*}_{\varphi}(f\sqrt{\varphi}).$

∎

Let $\\{m_{i}:i\in\Lambda\\}$ be a collection of complex-valued functions from
$L^{2}(\mu)$. We fix a Markovian function $\varphi$. For every $i\in\Lambda$,
we define

(4.4) $T_{m_{i}}(f)=m_{i}\sqrt{\varphi}(f\circ\sigma)=M_{m_{i}}S_{\varphi}(f)$

where $M_{m_{i}}$ is the multiplication operator. Then $T_{m_{i}}$ is an
operator acting from $L^{2}(\mu)$ onto $m_{i}\mathcal{H}_{\varphi}$.

######  Lemma 4.6.

The operator $T_{m}(f)=m\sqrt{\varphi}(f\circ\sigma)$ is bounded on
$L^{2}(\mu)$ if and only if $|m|^{2}\in L^{\infty}(\mu)$.

###### Proof.

For $f\in L^{2}(\mu)$, we have

$\displaystyle||T_{m}(f)||^{2}=$
$\displaystyle\int_{X}(m\sqrt{\varphi}(f\circ\sigma))(\overline{m}\sqrt{\varphi}(\overline{f}\circ\sigma)\;d\mu$
$\displaystyle=$
$\displaystyle\int_{X}|m|^{2}(|f|^{2}\circ\sigma)\varphi\;d\mu$
$\displaystyle\leq$
$\displaystyle\sup|m|^{2}\int_{X}(|f|^{2}\circ\sigma)\varphi\;d\mu$
$\displaystyle=$ $\displaystyle\sup|m|^{2}||f||^{2}.$

∎

######  Lemma 4.7.

The operator $T_{m}$ is an isometry in $L^{2}(\mu)$ if and only if

$S^{*}_{\varphi}(\sqrt{\varphi}|m|^{2})=\mathbbm{1}.$

###### Proof.

We note that $T^{*}_{m}(f)=S^{*}_{\varphi}M_{\overline{m}}(f)$. Using the
pull-out property for $S^{*}_{\varphi}$, we can write

$T^{*}_{m}T_{m}(f)=S^{*}_{\varphi}M_{\overline{m}}(m_{i}\sqrt{\varphi}(f\circ\sigma))=fS^{*}_{\varphi}(|m|^{2}\sqrt{\varphi})$

where $f\in L^{2}(\mu)$. This proves the lemma. ∎

######  Remark 4.8.

It follows from Lemma 4.7 that $T_{m}$ is an isometry if and only if

$\mathbb{E}_{\varphi}(\sqrt{\varphi}|m|^{2})=\sqrt{\varphi}.$

######  Lemma 4.9.

The operators $T_{m},S_{\varphi},$ and $\mathbb{E}_{\varphi}$ satisfy the
properties:

(4.5)
$T^{*}_{m_{1}}T_{m_{2}}(f)=fS^{*}_{\varphi}(\sqrt{\varphi}\;\overline{m}_{1}m_{2}),$
(4.6)
$T_{m_{1}}T^{*}_{m_{2}}(f)=m_{1}\mathbb{E}_{\varphi}(\overline{m}_{2}f).$

###### Proof.

Indeed, we have

$\displaystyle T^{*}_{m_{1}}T_{m_{2}}(f)=$ $\displaystyle
T^{*}_{m_{1}}(\sqrt{\varphi}\;m_{2}(f\circ\sigma))$ $\displaystyle=$
$\displaystyle
S^{*}_{m_{2}}M_{\overline{m}_{1}}(\sqrt{\varphi}\;m_{2}(f\circ\sigma))$
$\displaystyle=$ $\displaystyle
fS^{*}_{m_{2}}(\sqrt{\varphi}\;\overline{m}_{1}m_{2}),$

and

$T_{m_{1}}T^{*}_{m_{2}}(f)=M_{m_{1}}S_{\varphi}S^{*}_{\varphi}M_{\overline{m}_{2}}(f)=m_{1}\mathbb{E}_{\varphi}(\overline{m}_{2}f).$

∎

######  Remark 4.10.

Let $g$ be a bounded positive Borel function. Take a Markovian function
$\varphi\in M(\sigma,\mu)$ and consider $\psi=(g\circ\sigma)\varphi g^{-1}$.
By Proposition 2.17 we see that $\psi\in M(\sigma,\nu)$ where $d\nu=gd\mu$.
Then

$S_{\psi}=M^{-1}_{\sqrt{g}}S_{\varphi}M_{\sqrt{g}}.$

Then a direct computation shows that $S_{\psi}$ is an isometry in $L^{2}(\nu)$
where $d\nu=g\,d\mu$.

Denote by $\widetilde{T}_{m}$ the operator acting on $L^{2}(\nu)$: $f\mapsto
mS_{\psi}(f)$. By Lemma 4.7, $\widetilde{T}_{m}$ is an isometry in
$L^{2}(\nu)$ if and only if $S^{*}_{\psi}(\sqrt{\psi}|m|^{2})=\mathbbm{1}$. It
follows from the definition of the operators $T_{m}$ and $\widetilde{T}_{m}$
that

$\widetilde{T}_{m}=M^{-1}_{\sqrt{g}}T_{m}M_{\sqrt{g}}.$

######  Theorem 4.11.

Let $\sigma$ be an onto endomorphism of $(X,\mathcal{B},\mu)$ where $\mu$ is
quasi-invariant with respect to $\sigma$. Let $\\{m_{i}:i\in\Lambda\\}$ be a
family of complex-valued functions. The operators $\\{T_{m_{i}},i\in\Lambda$}
generate a representation of the Cuntz algebra $\mathcal{O}_{|\Lambda|}$ if
and only if

$(i)\ \
S^{*}_{\varphi}(\sqrt{\varphi}\;\overline{m}_{j}m_{i})=\delta_{ij}\mathbbm{1},$
$(ii)\ \ \sum_{i\in\Lambda}m_{i}\mathbb{E}_{\varphi}(\overline{m}_{i}f)=f,\ \
\ f\in L^{2}(\mu).$

###### Proof.

We first note that condition (i) of the theorem implies that the operators
$T_{m_{i}}$ are isometries because of Lemma 4.7. The Cuntz relations for
isometries $\\{T_{m_{i}}\\}_{i\in\Lambda}$ mean that

$\sum_{i\in\Lambda}T_{m_{i}}T_{m_{i}}^{*}=\mathbb{I}$

and the projections $T_{m_{i}}T_{m_{i}}^{*}$ are mutually orthogonal.

We will show that, for $f,g\in L^{2}(\mu)$ and $i\neq j$, the vectors
$T_{m_{i}}T_{m_{i}}^{*}(f)$ and $T_{m_{j}}T_{m_{j}}^{*}(g)$ are orthogonal if
and only if condition (i) of the theorem holds. In the following computation,
we use (4.5).

$\displaystyle\langle
T_{m_{i}}T_{m_{i}}^{*}(f),T_{m_{j}}T_{m_{j}}^{*}(g)\rangle_{\mu}=$
$\displaystyle\langle(T_{m_{j}}^{*}T_{m_{i}})T_{m_{i}}^{*}(f)),T_{m_{j}}^{*}(g)\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle
S^{*}_{\varphi}(\sqrt{\varphi}\;\overline{m}_{j}m_{i})(\xi),\eta\rangle_{\mu}$

where $\xi=T_{m_{i}}^{*}(f)$ and $\eta=T_{m_{j}}^{*}(g)$. Since $f,g$ are
arbitrary functions, the left-hand side is zero if and only if
$S^{*}_{\varphi}(\sqrt{\varphi}\;\overline{m}_{j}m_{i})=\delta_{ij}\mathbb{I}$.

Next, we use (4.6) to see when the identity operator is decomposed in the sum
of orthogonal projections $T_{m_{i}}T_{m_{i}}^{*}$. It follows from Lemma 4.9
that, for any $f\in L^{2}(\mu)$,

$\sum_{i\in\Lambda}T_{m_{i}}T_{m_{i}}^{*}(f)=\sum_{i\in\Lambda}m_{i}\mathbb{E}_{\varphi}(\overline{m}_{i}f)$

Hence, the property $\sum_{i\in\Lambda}T_{m_{i}}T_{m_{i}}^{*}=\mathbb{I}$ is
equivalent to condition (ii) of the theorem. ∎

######  Corollary 4.12.

It follows from Theorem 4.11 that

$L^{2}(\mu)=\bigoplus_{i\in\Lambda}\ m_{i}\mathcal{H}_{\varphi}.$

In other words, $\\{m_{i}:i\in\Lambda\\}$ is an orthogonal module basis for
$L^{2}(\mu)$ over
$\mathcal{H}_{\varphi}=\sqrt{\varphi}\;L^{2}(\mu_{\mathcal{A}})$ where
$\mathcal{A}=\sigma^{-1}(\mathcal{B})$.

###  4.2. Invariant measure

In what follows we will consider the case of a $\sigma$-invariant measure
$\mu$.

######  Lemma 4.13.

Let $\sigma\in End(X,\mathcal{B},\mu)$ be an onto endomorphism. Then if $\mu$
is $\sigma$-invariant probability measure, then the only
${\sigma^{-1}(\mathcal{B})}$-measurable Markovian function is the constant
function $\mathbbm{1}$.

###### Proof.

Indeed, let $\varphi$ be a Markovian function. Then we have

$\int_{X}(f\circ\sigma)\varphi\;d\mu=\int_{X}f\;d\mu=\int_{X}f\;d(\mu\circ\sigma)^{-1}=\int_{X}(f\circ\sigma)\;d\mu.$

Since $f$ is arbitrary, we get $\varphi=\mathbbm{1}$ on
${\sigma^{-1}(\mathcal{B})}$. ∎

We will apply the constructions of Subsection 4.1 to the case when
$\varphi=\mathbbm{1}$. Because the results of Subsection 4.1 are proved in
more general settings, we will just formulate the corresponding facts without
proof.

Let $\underline{m}=\\{m_{i}:i\in\Lambda\\}$ be a collection of complex-valued
bounded functions, and let

$S_{m_{i}}(f)=m_{i}(f\circ\sigma),\quad f\in L^{2}(\mu)$

be the corresponding weighted composition operators.

In this lemma, we collect the properties of $S_{m_{i}}$.

######  Lemma 4.14.

Let $f,g\in L^{2}(\mu)$.

(1) $S^{*}_{m}$ is an isometry on $L^{2}(\mu)$ if and only if
$S^{*}_{\sigma}(|m|^{2})=\mathbbm{1}$.

(2) $\mathbb{E}_{\sigma}:=S_{\sigma}S^{*}_{\sigma}$ is the conditional
expectation from $L^{2}(\mu)$ onto $L^{2}(\mu_{\mathcal{A}})$.

(3) $E_{\sigma}((f\circ\sigma)g)=(f\circ\sigma)\mathbb{E}_{\sigma}(g)$.

(4)
$S^{*}_{\sigma}(f\mathbb{E}_{\sigma}(g))=S^{*}_{\sigma}(f)S^{*}_{\sigma}(g)$.

(5) $S_{m_{1}}^{*}S_{m_{2}}(f)=S^{*}_{\sigma}(\overline{m}_{1}m_{2})f$.

(6) $S_{m_{1}}S_{m_{2}}^{*}(f)=m_{1}\mathbb{E}(\overline{m}_{2}f)$.

Here is the modified version of Theorem 4.11.

######  Theorem 4.15.

Let $\sigma$ be a surjective endomorphism of $(X,\mathcal{B},\mu)$ with
$\sigma$-invariant measure $\mu$. Let $\\{m_{i}:i\in\Lambda\\}$ be a family of
bounded complex-valued functions. The operators $\\{S_{m_{i}},i\in\Lambda$}
generate a representation of the Cuntz algebra $\mathcal{O}_{|\Lambda|}$ if
and only if

$(i)\ \ S^{*}_{\sigma}(\overline{m}_{j}m_{i})=\delta_{ij}\mathbbm{1},$ $(ii)\
\ \sum_{i\in\Lambda}m_{i}\mathbb{E}_{\sigma}(\overline{m}_{i}f)=f,\ \ \ f\in
L^{2}(\mu).$

It follows from Theorem 4.15 that the Hilbert space $L^{2}(\mu)$ admits the
decomposition

$L^{2}(\mu)=\bigoplus_{i\in\Lambda}\ m_{i}L^{2}(\mu_{\mathcal{A}})$

if and only if the operators $S_{m_{i}}$ are the generators of a
representation of the Cuntz algebra. In other words, $\\{m_{i}:i\in\Lambda\\}$
is an orthogonal module basis for $L^{2}(\mu)$ over $L^{2}(\mu_{\mathcal{A}})$
where $\mathcal{A}=\sigma^{-1}(\mathcal{B})$.

We finish this section with a discussion of the relations between the
operators $S_{\sigma}$, $S_{\varphi}$ and $\mathbb{E}_{\sigma}$,
$\mathbb{E}_{\varphi}$ where $\varphi$ is a Markovian function.

######  Proposition 4.16.

The following formulas hold: for $g\in L^{2}(\mu)$

$(1)\ \ S^{*}_{\varphi}(g)=S^{*}_{\sigma}(\sqrt{\varphi}\;g),\quad(2)\ \
\mathbb{E}_{\varphi}(g)=\sqrt{\varphi}\;\mathbb{E}_{\sigma}(\sqrt{\varphi}\;g).$

###### Proof.

For (1), take any functions $f,g\in L^{2}(\mu)$ and compute

$\displaystyle\langle f,S^{*}_{\varphi}(g)\rangle_{\mu}=$
$\displaystyle\langle S^{*}_{\varphi}(f),g\rangle_{\mu}$ $\displaystyle=$
$\displaystyle\langle\sqrt{\varphi}\;(f\circ\sigma),g\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle
S_{\sigma}(f),\sqrt{\varphi}\;g\rangle_{\mu}$ $\displaystyle=$
$\displaystyle\langle f,S^{*}_{\sigma}(\sqrt{\varphi}\;g)\rangle_{\mu}.$

To see that (2) is true, we recall that
$S_{\varphi}(f)=\sqrt{\varphi}\;S_{\sigma}(f)$, and then we can write

$\mathbb{E}_{\varphi}(f)=S_{\varphi}S^{*}_{\varphi}(f)=S_{\varphi}S^{*}_{\sigma}(\sqrt{\varphi}\;f)=\sqrt{\varphi}\;S_{\sigma}S^{*}_{\sigma}(\sqrt{\varphi}\;f)=\sqrt{\varphi}\;\mathbb{E}_{\sigma}(\sqrt{\varphi}\;f).$

∎

The results of Proposition 4.16 will be used in the next section.

##  5\. The set of wavelet filters

In this section, we answer the question about the structure of the set of
wavelet filters, i.e., we describe the set of functions satisfying the
conditions of Theorem 4.11. We consider two different approaches: (i) it will
be shown that there the set $\mathfrak{M}$ is isomorphic to the so called loop
group $\mathcal{G}$; (ii) using the decomposition into cyclic representations,
we describe elements $\underline{m}\in\mathfrak{M}$ as the collection of
cyclic vectors for the representation of $L^{\infty}(\mathcal{A})$ in
$L^{2}(X,\mathcal{B},\mu)$.

###  5.1. Actions of loop groups on wavelet filters

Consider bounded operators $B(l^{2}(\Lambda))$ in the Hilbert space
$l^{2}(\Lambda)$ where $\Lambda$ is a countably infinite set. If
$\\{e_{i}:i\in\Lambda\\}$ is the canonical orthonormal basis in
$l^{2}(\Lambda)$, then we define the infinite $|\Lambda|\times|\Lambda|$
matrix $\widehat{A}=(a_{ij})$ by setting $a_{ij}=\langle e_{i},Ae_{j}\rangle$.
This observation will allow us to work with matrix notation in the
computations below.

Let $\mathcal{U}$ be the group of all unitary operators in
$B(l^{2}(\Lambda))$. Denote by $\mathcal{G}$ the group of Borel functions on
$(X,\mathcal{B})$ with values in $\mathcal{U}$ (we recall that $\mathcal{U}$
is a Polish group). We use the notation $G=(g_{ij}(x))$ for elements of
$\mathcal{G}$. Then every entry $g_{ij}(x)$ is a Borel complex-valued matrix.
The group $\mathcal{G}$ is called the loop group. We remark that for
$G\in\mathcal{G}$, $G^{*}G=\mathbb{I}$ where the matrix
$G^{*}=\overline{G}^{T}$. We will use the relation

(5.1) $\sum_{l\in\Lambda}\overline{g}_{li}g_{lj}=\delta_{ij}$

where $G=(g_{ij})\in\mathcal{G}$.

Let $\varphi$ be a Markovian function. We recall that, in this case, the
operator $S_{\varphi}$ is isometric. We denote the set of generalized wavelet
filters by

(5.2)
$\mathfrak{M}_{\varphi}:=\\{\underline{m}=(m_{i})_{i\in\Lambda}:S_{\varphi}^{*}(\sqrt{\varphi}\;m_{i}\overline{m}_{j})=\delta_{ij}\mathbbm{1},\
\ \ \sum_{i\in\Lambda}m_{i}\mathbb{E}_{\varphi}(\overline{m}_{j}f)=f\\}.$

We consider simultaneously the case of $\sigma$-invariant measure $\mu$ and
the corresponding operators $S_{\sigma}$ and $\mathbb{E}_{\sigma}$, see
Section 4 for properties of these operators. In this case, we use the set

(5.3)
$\mathfrak{M}_{\sigma}:=\\{\underline{m}=(m_{i})_{i\in\Lambda}:S_{\sigma}^{*}(m_{i}\overline{m}_{j})=\delta_{ij}\mathbbm{1},\
\ \ \sum_{i\in\Lambda}m_{i}\mathbb{E}_{\sigma}(\overline{m}_{j}f)=f\\}.$

It turns out that the group $\mathcal{G}$ acts on the sets
$\mathfrak{M}_{\varphi}$ and $\mathfrak{M}_{\sigma}$. Indeed, for a fixed
$\underline{m}\in\mathfrak{M}_{\varphi}$ (or
$\underline{m}\in\mathfrak{M}_{\sigma}$) and $G\in\mathcal{G}$, we define
$\underline{m}^{G}:=(m^{G}_{i}(x):i\in\Lambda)$ by the formula

(5.4)
$m^{G}_{i}(x)=\sum_{j\in\Lambda}(\overline{g}_{ji}\circ\sigma)(x)m_{j}(x)$

or $\underline{m}^{G}=(G^{*}\circ\sigma)\underline{m}$ in a short form.

In the next statements, we will study the properties of this action of
$\mathcal{G}$ on the sets of wavelet filters. For definiteness, we formulate
these results for the set $\mathfrak{M}_{\varphi}$. The same proofs work for
the action of $\mathcal{G}$ on $\mathfrak{M}_{\sigma}$, we will omit them. We
will show in the next lemmas that: (i) the set $\mathfrak{M}_{\varphi}$ is
invariant with respect to the action of group $\mathcal{G}$; (ii) formula
(5.4) defines a group action on $\mathfrak{M}_{\varphi}$; (iii) this action of
$\mathcal{G}$ is free and transitive.

######  Lemma 5.1.

If $\underline{m}\in\mathfrak{M}_{\varphi}$, then
$\underline{m}^{G}\in\mathfrak{M}_{\varphi}$.

###### Proof.

We will verify that the family of functions
$\underline{m}^{G}=(m_{i}^{G}(x):i\in\Lambda)$ satisfies conditions (i) and
(ii) of Theorem 4.11. For (i), we use (5.1) and the fact that
$S^{*}_{\varphi}$ is a transfer operator satisfying the pull-out property:

$\displaystyle
S^{*}_{\varphi}(\sqrt{\varphi}\;m^{G}_{i}\overline{m}^{G}_{j})=$
$\displaystyle
S^{*}_{\varphi}\left(\sqrt{\varphi}\sum_{k\in\Lambda}(\overline{g}_{ki}\circ\sigma)m_{k}\cdot\sum_{l\in\Lambda}\overline{(\overline{g}_{jl}\circ\sigma)m_{l}}\right)$
$\displaystyle=$
$\displaystyle\sum_{k\in\Lambda}\sum_{l\in\Lambda}\overline{g}_{ki}g_{lj}S^{*}_{\varphi}(\sqrt{\varphi}\overline{m}_{l}m_{k})$
$\displaystyle=$
$\displaystyle\sum_{k\in\Lambda}\sum_{l\in\Lambda}\overline{g}_{ki}g_{lj}\delta_{kl}$
$\displaystyle=$ $\displaystyle\sum_{k\in\Lambda}\overline{g}_{ki}g_{kj}$
$\displaystyle=$ $\displaystyle\delta_{ij}.$

For (ii), let $f\in L^{2}(\mu)$, then

$\displaystyle\sum_{i\in\Lambda}m_{i}^{G}\mathbb{E}_{\varphi}(\overline{m}^{G}_{i}f)=$
$\displaystyle\sum_{i\in\Lambda}\sum_{k\in\Lambda}(\overline{g}_{ki}\circ\sigma)m_{k}\mathbb{E}_{\varphi}\left(\sum_{l\in\Lambda}(g_{li}\circ\sigma)\overline{m}_{l}f\right)$
$\displaystyle=$
$\displaystyle\sum_{i\in\Lambda}\sum_{k\in\Lambda}\sum_{l\in\Lambda}(\overline{g}_{ki}\circ\sigma)(g_{li}\circ\sigma)m_{k}\mathbb{E}_{\varphi}(\overline{m}_{l}f)$
$\displaystyle=$
$\displaystyle\sum_{i\in\Lambda}\sum_{k\in\Lambda}\sum_{l\in\Lambda}((\overline{g}_{ki}g_{li})\circ\sigma)m_{k}\mathbb{E}_{\varphi}(\overline{m}_{l}f)$
$\displaystyle=$
$\displaystyle\sum_{i\in\Lambda}\sum_{k\in\Lambda}\delta_{kl}m_{k}\mathbb{E}_{\varphi}(\overline{m}_{l}f)$
$\displaystyle=$
$\displaystyle\sum_{k\in\Lambda}m_{k}\mathbb{E}_{\varphi}(\overline{m}_{l}f)$
$\displaystyle=$ $\displaystyle f.$

We used here the equality $GG^{*}=\mathbb{I}$ or
$\sum_{i}\overline{g}_{ki}g_{li}=\delta_{kl}$. ∎

In the next lemma, we show that (5.4) defines an action of the group
$\mathcal{G}$ on $\mathfrak{M}$.

######  Lemma 5.2.

For every $\underline{m}\in\mathfrak{M}_{\varphi}$ and every
$G,H\in\mathcal{G}$,

(5.5)
$(\underline{m}^{G})^{H}=\underline{m}^{GH},\quad\underline{m}^{\mathbb{I}}=\underline{m}.$

###### Proof.

Indeed, if $\mathbb{I}$ is the identity matrix, then
$\underline{m}^{\mathbb{I}}=\underline{m}$ for every
$\underline{m}\in\mathfrak{M}_{\varphi}$.

Let $G,H\in\mathfrak{M}_{\varphi}$. Show that (5.5) holds:

$\displaystyle(m^{G})^{H}_{i}=$
$\displaystyle\sum_{j\in\Lambda}(\overline{h}_{ji}\circ\sigma)m_{j}^{G}$
$\displaystyle=$
$\displaystyle\sum_{j\in\Lambda}(\overline{h}_{ji}\circ\sigma)\sum_{k\in\Lambda}(\overline{g}_{kj}\circ\sigma)m_{k}$
$\displaystyle=$
$\displaystyle\sum_{k\in\Lambda}\sum_{j\in\Lambda}((\overline{g}_{ij}\overline{h}_{jk})\circ\sigma)m_{k}$
$\displaystyle=$
$\displaystyle\sum_{k\in\Lambda}(\overline{f}_{ik}\circ\sigma)m_{k}$
$\displaystyle=$ $\displaystyle m^{GH}_{i},$

where the entries of $GH$ are denoted by $(f_{ik})$. ∎

######  Lemma 5.3.

Let $\varphi$ be a positive Markovian function.

(1) The map $\Phi:\underline{m}\mapsto\sqrt{\varphi}\;\underline{m}$ defines
an isomorphism between the sets $\mathfrak{M}_{\varphi}$ and
$\mathfrak{M}_{\sigma}$.

(2) The map $\Phi$ implements the conjugation of actions of $\mathcal{G}$ on
the sets $\mathfrak{M}_{\sigma}$ and $\mathfrak{M}_{\varphi}$.

###### Proof.

(1) We will show that if $\underline{m}$ satisfies (5.2), then
$\sqrt{\varphi}\underline{m}$ belongs to $\mathfrak{M}_{\sigma}$, i.e., (5.3)
holds. For this, we check that

$S^{*}_{\sigma}(\sqrt{\varphi}m_{i}\;\sqrt{\varphi}\overline{m}_{j})=S^{*}_{\varphi}(\sqrt{\varphi}\;m_{i}\overline{m}_{j})=\delta_{ij},\
\ i,j\in\Lambda,$

and, using Proposition 4.16,

$\displaystyle\sum_{i\in\Lambda}\sqrt{\varphi}m_{i}\mathbb{E}_{\sigma}(\sqrt{\varphi}\overline{m}_{i}f)=$
$\displaystyle\sum_{i\in\Lambda}\sqrt{\varphi}m_{i}S_{\sigma}S^{*}_{\sigma}(\sqrt{\varphi}\overline{m}_{i}f)$
$\displaystyle=$
$\displaystyle\sum_{i\in\Lambda}m_{i}S_{\varphi}S^{*}_{\varphi}(\overline{m}_{i}f)$
$\displaystyle=$ $\displaystyle f.$

(2) It can be checked directly that, for every $G\in\mathcal{G}$ and
$\underline{m}\in\mathfrak{M}_{\varphi}$,

$\Phi\underline{m}^{G}=(\Phi\underline{m})^{G}.$

∎

######  Lemma 5.4.

Let $\varphi$ be a positive Markovian function. The action of $\mathcal{G}$ on
$\mathfrak{M}_{\varphi}$ defined in (5.4) is free and transitive.

###### Proof.

It follows from Lemma 5.3 that it suffices to show that the action of
$\mathcal{G}$ on $\mathfrak{M}_{\sigma}$ is free and transitive.

Let $\underline{m}=(m_{i})$ and $\underline{n}=(n_{j})$ be two elements of the
set $\mathfrak{M}_{\sigma}$. Define an infinite matrix $G$ by setting

$g_{ij}=S^{*}_{\varphi}(m_{i}\overline{n}_{j}),\quad i,j\in\Lambda.$

We show that $\underline{m}^{G}=\underline{n}$ to prove that the action is
transitive. Indeed,

$\displaystyle m_{i}^{G}=$
$\displaystyle\sum_{k\in\Lambda}(\overline{g}_{k,i}\circ\sigma)m_{k}$
$\displaystyle=$
$\displaystyle\sum_{k\in\Lambda}(S^{*}_{\sigma}(\overline{m}_{k}n_{i})\circ\sigma)m_{k}$
$\displaystyle=$
$\displaystyle\sum_{k\in\Lambda}S_{\sigma}S^{*}_{\sigma}(\overline{m}_{k}n_{i})m_{k}$
$\displaystyle=$
$\displaystyle\sum_{k\in\Lambda}\mathbb{E}_{\sigma}(\overline{m}_{k}n_{i})m_{k}$
$\displaystyle=$ $\displaystyle n_{i}.$

We used here relation (5.3).

To see that this action is free, we assume that
$\underline{m}^{G}=\underline{m}$ for some
$\underline{m}\in\mathfrak{M}_{\sigma}$ and $G\in\mathcal{G}$. Then

$\sum_{k\in\Lambda}(\overline{g}_{ki}\circ\sigma)m_{k}=m_{i},\quad
i\in\Lambda.$

Multiply both sides by $\overline{m}_{j}$ and apply the operator
$S^{*}_{\sigma}$:

$\sum_{k\in\Lambda}S^{*}_{\sigma}((\overline{g}_{ki}\circ\sigma)m_{k}\overline{m}_{j})=S^{*}_{\sigma}(m_{i}\overline{m}_{j})$

Use the pull-out property and (5.3):

$\sum_{k\in\Lambda}\overline{g}_{ki}S^{*}_{\sigma}(m_{k}\overline{m}_{j})=\delta_{ij}.$

Hence,

$\sum_{k\in\Lambda}\overline{g}_{ki}\delta_{kj}=\delta_{ij}$

and $\overline{g}_{li}=\delta_{ij}$. This proves that $G=\mathbb{I}$ and the
action is free.

∎

The following theorem gives a complete description of the set
$\mathfrak{M}_{\varphi}$. This result immediately follows from the proven
Lemmas 5.1 \- 5.4.

######  Theorem 5.5.

(1) The set $\mathfrak{M}_{\varphi}$ is isomorphic (as a set) to the loop
group $\mathcal{G}$.

(2) For every element $G=(g_{ij})$ of the loop group, there exists a wavelet
filter $\underline{m}$ such that
$g_{ij}=S^{*}_{\varphi}(\sqrt{\varphi}\;m_{i}\overline{m}_{j})$.

###  5.2. Endomorphisms, wavelet filters and cyclic representations

In this subsection, we will use another approach to describe the set
$\mathfrak{M}$ of wavelet filters associated with an endomorphism $\sigma$ of
$(X,\mathcal{B},\mu)$. We will assume here that $\sigma$ is measure-
preserving.

Let $\mathcal{A}={\sigma^{-1}(\mathcal{B})}$ and $\mu_{\mathcal{A}}$ is the
restriction of $\mu$ onto the sigma-subalgebra $\mathcal{A}$. Denote by
$\mathfrak{A}$ the set $L^{\infty}(\mu_{\mathcal{A}})$ of bounded
${\sigma^{-1}(\mathcal{B})}$-measurable functions. The $*$-algebra
$\mathfrak{A}$ acts on $L^{2}(\mu)$ by multiplication operators: for every
$f\in L^{2}(\mu)$ and $\gamma\in\mathfrak{A}$

$f\longmapsto\gamma f=M_{\gamma}(f).$

This formula defines a representation $\pi$ of $\mathfrak{A}$ in $L^{2}(\mu)$.
Using the theorem about the decomposition of non-degenerate representations
into orthogonal cyclic representations, we can write

(5.6)
$L^{2}(\mu)=\bigoplus_{i\in\Lambda}\mathcal{H}_{i},\quad\pi=\bigoplus_{i\in\Lambda}\pi_{i}$

where $\pi_{i}$ is a cyclic representation and $\mathcal{H}_{i}$ is the
closure of $\\{\pi_{i}(\gamma)h_{i}:\gamma\in\mathfrak{A}\\}$. The vectors
$h_{i},i\in\Lambda$, are called cyclic. The set of indexes $\Lambda$ is
countably infinite.

We will use the cyclic vectors $(h_{i})$ to construct a wavelet filter
$\underline{m}=(m_{i})$ satisfying the conditions of Theorem 4.15.

######  Lemma 5.6.

There are functions $(g_{i})$ such that the cyclic vectors
$(m_{i}=h_{i}(g_{i}\circ\sigma))$ for representations of $\mathfrak{A}$ have
the property

(5.7) $S_{\sigma}^{*}(|m_{i}|^{2})=\mathbbm{1}.$

###### Proof.

We note that for every function $\gamma\in\mathfrak{A}$ there exists a
function $g\in L^{\infty}(\mu)$ such that $\gamma=g\circ\sigma$. Hence, if
$h_{i}$ is a cyclic vector for a representation of $\mathfrak{A}$, then
$m_{i}$ is cyclic, too.

We will determine the function $g_{i}$ such that
$m_{i}=h_{i}(g_{i}\circ\sigma)$ satisfies the statement of the lemma. For
this, we compute

$S^{*}_{\sigma}(|m_{i}|^{2})=S^{*}_{\sigma}(|h_{i}(g_{i}\circ\sigma)|^{2})=|g_{i}|^{2}S^{*}_{\sigma}(|h_{i}|^{2}),$

and take $g_{i}$ such that the above expression equals 1 a.e. ∎

For functions $f\in L^{2}(\mu)$, we define the operators

(5.8) $S_{i}(f)=m_{i}(f\circ\sigma)=M_{m_{i}}S_{\sigma}(f).$

######  Lemma 5.7.

Let the functions $(m_{i})$ satisfy (5.7). Then the operators
$S_{i},i\in\Lambda$, are isometries satisfying the property

(5.9) $S_{\sigma}^{*}(m_{i}\overline{m}_{j})=\delta_{ij},\quad i,j\in\Lambda.$

###### Proof.

We first note that $S^{*}_{i}=S^{*}_{\sigma}M_{\overline{m}_{i}}$, and

$S^{*}_{i}S_{i}=\mathbb{I}\ \ \Longleftrightarrow\ \
S_{\sigma}^{*}(|m_{i}|^{2})=\mathbbm{1}.$

Next, we use the fact that cyclic representations, which are defined by the
distinct vectors $m_{i}$ and $m_{j}$, are mutually orthogonal. Applying the
fact that the transfer operator $S^{*}_{\sigma}$ has the pull-out property, we
compute, for any $\gamma_{1},\gamma_{2}\in\mathfrak{A}$,

$\displaystyle\langle\gamma_{1}m_{i},\gamma_{2}m_{j}\rangle_{\mu}=$
$\displaystyle\langle(g_{1}\circ\sigma)\;m_{i},(g_{2}\circ\sigma)\;m_{j}\rangle_{\mu}$
$\displaystyle=$
$\displaystyle\langle(g_{1}\circ\sigma)m_{i}\overline{m}_{j},S_{\sigma}(g_{2})\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle
S^{*}_{\sigma}((g_{1}\circ\sigma)m_{i}\overline{m}_{j}),g_{2}\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\langle
g_{1}S^{*}_{\sigma}(m_{i}\overline{m}_{j}),g_{2}\rangle_{\mu}.$

Hence, the inner product is zero if and only if (5.9) holds. ∎

######  Theorem 5.8.

Let $(m_{i}:i\in\Lambda)$ be a set of cyclic vectors satisfying (5.7). Then
$\underline{m}=(m_{i})$ is a wavelet filter if and only if
$\sum_{i\in\Lambda}S^{*}_{i}S_{i}=\mathbb{I}$. In other words,
$\underline{m}\in\mathfrak{M}$ if and only if the operators $S_{i}$ are the
generators of a representation of the Cuntz algebra $\mathcal{O}_{|\Lambda|}$.

###### Proof.

To prove the theorem, we use the result of Theorem (4.15). For this, we should
check that $\\{S_{i}:i\in\Lambda\\}$ are the operators satisfying conditions
(i) and (ii) of Theorem (4.15). Condition (i) is proved in Lemma (5.7). To
verify (ii), we observe that, for any $f\in L^{2}(\mu)$ and $i\in\Lambda$,

$\mathbb{E}_{\sigma}(f\overline{m}_{i})m_{i}=S_{i}S_{i}^{*}(f).$

Indeed,

$\displaystyle\mathbb{E}_{\sigma}(f\overline{m}_{i})m_{i}=S_{i}S_{i}^{*}(f)=$
$\displaystyle
S_{\sigma}S_{\sigma}^{*}(f\overline{m}_{i})m_{i}=S_{i}S_{i}^{*}(f)$
$\displaystyle=$ $\displaystyle S_{i}(S_{\sigma}^{*}M_{\overline{m}_{i}}(f))$
$\displaystyle=$ $\displaystyle S_{i}S_{i}^{*}(f).$

It follows that $\sum_{i\in\Lambda}S^{*}_{i}S_{i}=\mathbb{I}$ if and only if
condition (ii) holds. This proves the theorem. ∎

######  Corollary 5.9.

Let $\sigma\in End(X,\mathcal{B},\mu)$ where $\mu$ is a probability measure.
Let $(m_{i}:i\in\Lambda)$ be a set of cyclic vectors satisfying (5.6) and
(5.7). Then, for a.e. $x\in X$,

$\sum_{i\in\Lambda}|m_{i}(x)|^{2}<\infty.$

###### Proof.

The result follows from the fact that

$\sum_{i\in\Lambda}||m_{i}||_{\mu}^{2}=1.$

Indeed, this relation is obtained when the function $\mathbbm{1}(x)$ is
substituted into the cyclic decomposition (5.6). ∎

## References

  * [ACSS22] Daniel Alpay, Fabrizio Colombo, Irene Sabadini, and Baruch Schneider.  Beurling-Lax type theorems and Cuntz relations.  Linear Algebra Appl., 633:152–212, 2022.
  * [AJL17] Daniel Alpay, Palle Jorgensen, and Izchak Lewkowicz.  Characterizations of families of rectangular, finite impulse response, para-unitary systems.  J. Appl. Math. Comput., 54(1-2):395–423, 2017.
  * [AJL18] Daniel Alpay, Palle Jorgensen, and Izchak Lewkowicz.  $W$-Markov measures, transfer operators, wavelets and multiresolutions.  In Frames and harmonic analysis, volume 706 of Contemp. Math., pages 293–343. Amer. Math. Soc., [Providence], RI, [2018] ©2018.
  * [And22] P. A. Andrianov.  Multidimensional periodic discrete wavelets.  Int. J. Wavelets Multiresolut. Inf. Process., 20(2):Paper No. 2150053, 20, 2022.
  * [BD22] M. Younus Bhat and Aamir H. Dar.  Fractional vector-valued nonuniform MRA and associated wavelet packets on $L^{2}(\mathbb{R},\mathbb{C}^{M})$.  Fract. Calc. Appl. Anal., 25(2):687–719, 2022.
  * [Bén96] Catherine Bénéteau.  A natural extension of a nonsingular endomorphism of a measure space.  Rocky Mountain J. Math., 26(4):1261–1273, 1996.
  * [BH09] Henk Bruin and Jane Hawkins.  Rigidity of smooth one-sided Bernoulli endomorphisms.  New York J. Math., 15:451–483, 2009.
  * [BJ97a] O. Bratteli and P. E. T. Jorgensen.  Endomorphisms of ${\mathcal{B}}({\mathcal{H}})$. II. Finitely correlated states on ${\mathcal{O}}_{n}$.  J. Funct. Anal., 145(2):323–373, 1997.
  * [BJ97b] Ola Bratteli and Palle E. T. Jorgensen.  A connection between multiresolution wavelet theory of scale $N$ and representations of the Cuntz algebra $\mathcal{O}_{N}$.  In Operator algebras and quantum field theory (Rome, 1996), pages 151–163. Int. Press, Cambridge, MA, 1997.
  * [BJ97c] Ola Bratteli and Palle E. T. Jorgensen.  Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale $N$.  Integral Equations Operator Theory, 28(4):382–443, 1997.
  * [BJ99] Ola Bratteli and Palle E. T. Jorgensen.  Iterated function systems and permutation representations of the Cuntz algebra.  Mem. Amer. Math. Soc., 139(663):x+89, 1999.
  * [BJ15] Sergey Bezuglyi and Palle E. T. Jorgensen.  Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures.  In Trends in harmonic analysis and its applications, volume 650 of Contemp. Math., pages 57–88. Amer. Math. Soc., Providence, RI, 2015\.
  * [BJ18] Sergey Bezuglyi and Palle E. T. Jorgensen.  Transfer operators, endomorphisms, and measurable partitions, volume 2217 of Lecture Notes in Mathematics.  Springer, Cham, 2018.
  * [BLP+10] Lawrence W. Baggett, Nadia S. Larsen, Judith A. Packer, Iain Raeburn, and Arlan Ramsay.  Direct limits, multiresolution analyses, and wavelets.  J. Funct. Anal., 258(8):2714–2738, 2010.
  * [BMPR12] Lawrence W. Baggett, Kathy D. Merrill, Judith A. Packer, and Arlan B. Ramsay.  Probability measures on solenoids corresponding to fractal wavelets.  Trans. Amer. Math. Soc., 364(5):2723–2748, 2012.
  * [Bog07] Vladimir I. Bogachev.  Measure theory. Vol. I, II.  Springer-Verlag, Berlin, 2007.
  * [Bru22] Henk Bruin.  Topological and ergodic theory of symbolic dynamics, volume 228 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, [2022] ©2022.
  * [CD22] Nicholas J. Christoffersen and Dorin Ervin Dutkay.  Representations of Cuntz algebras associated to random walks on graphs.  J. Operator Theory, 88(1):139–170, 2022.
  * [CFS82] Isaac P. Cornfeld, Sergei V. Fomin, and Yakov G. Sinaĭ.  Ergodic theory, volume 245 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, New York, 1982.  Translated from the Russian by A. B. Sosinskiĭ.
  * [Cun77] Joachim Cuntz.  Simple $C^{*}$-algebras generated by isometries.  Comm. Math. Phys., 57(2):173–185, 1977.
  * [DH94] Karma G. Dajani and Jane M. Hawkins.  Examples of natural extensions of nonsingular endomorphisms.  Proc. Amer. Math. Soc., 120(4):1211–1217, 1994.
  * [DJ06] Dorin Ervin Dutkay and Palle E. T. Jorgensen.  Hilbert spaces built on a similarity and on dynamical renormalization.  J. Math. Phys., 47(5):053504, 20, 2006.
  * [DJ07] Dorin Ervin Dutkay and Palle E. T. Jorgensen.  Martingales, endomorphisms, and covariant systems of operators in Hilbert space.  J. Operator Theory, 58(2):269–310, 2007.
  * [DJ14a] Dorin Ervin Dutkay and Palle E. T. Jorgensen.  Monic representations of the Cuntz algebra and Markov measures.  J. Funct. Anal., 267(4):1011–1034, 2014.
  * [DJ14b] Dorin Ervin Dutkay and Palle E. T. Jorgensen.  The role of transfer operators and shifts in the study of fractals: encoding-models, analysis and geometry, commutative and non-commutative.  In Geometry and analysis of fractals, volume 88 of Springer Proc. Math. Stat., pages 65–95. Springer, Heidelberg, 2014.
  * [DJ15] Dorin Ervin Dutkay and Palle E. T. Jorgensen.  Representations of Cuntz algebras associated to quasi-stationary Markov measures.  Ergodic Theory Dynam. Systems, 35(7):2080–2093, 2015.
  * [DJK94] R. Dougherty, S. Jackson, and A. S. Kechris.  The structure of hyperfinite Borel equivalence relations.  Trans. Amer. Math. Soc., 341(1):193–225, 1994.
  * [DJS12] Dorin Ervin Dutkay, Palle E. T. Jorgensen, and Sergei Silvestrov.  Decomposition of wavelet representations and Martin boundaries.  J. Funct. Anal., 262(3):1043–1061, 2012.
  * [EFHN15] Tanja Eisner, Bálint Farkas, Markus Haase, and Rainer Nagel.  Operator theoretic aspects of ergodic theory, volume 272 of Graduate Texts in Mathematics.  Springer, Cham, 2015.
  * [Fab87] Raymond C. Fabec.  Induced group actions, representations and fibered skew product extensions.  Trans. Amer. Math. Soc., 301(2):489–513, 1987.
  * [Fab00] Raymond C. Fabec.  Fundamentals of infinite dimensional representation theory, volume 114 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics.  Chapman & Hall/CRC, Boca Raton, FL, 2000.
  * [FS22] De-Jun Feng and Károly Simon.  Dimension estimates for $C^{1}$ iterated function systems and repellers. Part II.  Ergodic Theory Dynam. Systems, 42(11):3357–3392, 2022.
  * [Haw94] Jane M. Hawkins.  Amenable relations for endomorphisms.  Trans. Amer. Math. Soc., 343(1):169–191, 1994.
  * [Haw21] Jane Hawkins.  Ergodic dynamics—from basic theory to applications, volume 289 of Graduate Texts in Mathematics.  Springer, Cham, [2021] ©2021.
  * [HS91] Jane M. Hawkins and Cesar E. Silva.  Noninvertible transformations admitting no absolutely continuous $\sigma$-finite invariant measure.  Proc. Amer. Math. Soc., 111(2):455–463, 1991.
  * [JKS07] Palle E. T. Jorgensen, Keri Kornelson, and Karen Shuman.  Harmonic analysis of iterated function systems with overlap.  J. Math. Phys., 48(8):083511, 35, 2007.
  * [JKS11] Palle E. T. Jorgensen, Keri A. Kornelson, and Karen L. Shuman.  Iterated function systems, moments, and transformations of infinite matrices.  Mem. Amer. Math. Soc., 213(1003):x+105, 2011.
  * [Jor96] Palle E. T. Jorgensen.  A duality for endomorphisms of von Neumann algebras.  J. Math. Phys., 37(3):1521–1538, 1996.
  * [Jor01] Palle E. T. Jorgensen.  Ruelle operators: functions which are harmonic with respect to a transfer operator.  Mem. Amer. Math. Soc., 152(720):viii+60, 2001.
  * [Jor06] Palle E. T. Jorgensen.  Analysis and probability: wavelets, signals, fractals, volume 234 of Graduate Texts in Mathematics.  Springer, New York, 2006.
  * [Jor18] Palle E. T. Jorgensen.  Harmonic analysis, volume 128 of CBMS Regional Conference Series in Mathematics.  American Mathematical Society, Providence, RI, 2018.  Smooth and non-smooth, Published for the Conference Board of the Mathematical Sciences.
  * [JP11] P. E. T. Jorgensen and A. M. Paolucci.  States on the Cuntz algebras and $p$-adic random walks.  J. Aust. Math. Soc., 90(2):197–211, 2011.
  * [JS18] Palle E. T. Jorgensen and Myung-Sin Song.  Markov chains and generalized wavelet multiresolutions.  J. Anal., 26(2):259–283, 2018.
  * [JT17] P. Jorgensen and F. Tian.  Transfer operators, induced probability spaces, and random walk models.  Markov Process. Related Fields, 23(2):187–210, 2017.
  * [JT19] Palle Jorgensen and Feng Tian.  Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations.  Discrete Contin. Dyn. Syst. Ser. S, 12(8):2307–2348, 2019.
  * [JT20] Palle Jorgensen and James Tian.  Noncommutative boundaries arising in dynamics and representations of the Cuntz relations.  Numer. Funct. Anal. Optim., 41(5):571–620, 2020.
  * [Kec95] Alexander S. Kechris.  Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1995.
  * [MV23] R. Medhi and P. Viswanathan.  On the code space and Hutchinson measure for countable iterated function system consisting of cyclic $\phi$-contractions.  Chaos Solitons Fractals, 167:Paper No. 113011, 2023.
  * [PR22] Matthew J. Picklo and Jennifer K. Ryan.  Enhanced multiresolution analysis for multidimensional data utilizing line filtering techniques.  SIAM J. Sci. Comput., 44(4):A2628–A2650, 2022.
  * [PU10] Feliks Przytycki and Mariusz Urbański.  Conformal fractals: ergodic theory methods, volume 371 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 2010.
  * [Roh49a] V. A. Rohlin.  Selected topics from the metric theory of dynamical systems.  Uspehi Matem. Nauk (N.S.), 4(2(30)):57–128, 1949.
  * [Roh49b] Vladimir A. Rohlin.  On the fundamental ideas of measure theory.  Mat. Sbornik N.S., 25(67):107–150, 1949.
  * [Roh61] Vladimir A. Rohlin.  Exact endomorphisms of a Lebesgue space.  Izv. Akad. Nauk SSSR Ser. Mat., 25:499–530, 1961.
  * [RR22] Lakshmi Roychowdhury and Mrinal Kanti Roychowdhury.  Quantization for a probability distribution generated by an infinite iterated function system.  Commun. Korean Math. Soc., 37(3):765–800, 2022.
  * [Sil88] Cesar E. Silva.  On $\mu$-recurrent nonsingular endomorphisms.  Israel J. Math., 61(1):1–13, 1988.
  * [Sim12] David Simmons.  Conditional measures and conditional expectation; Rohlin’s disintegration theorem.  Discrete Contin. Dyn. Syst., 32(7):2565–2582, 2012.
  * [URM22] Mariusz Urbański, Mario Roy, and Sara Munday.  Non-invertible dynamical systems. Vol. 1. Ergodic theory—finite and infinite, thermodynamic formalism, symbolic dynamics and distance expanding maps, volume 69 of De Gruyter Expositions in Mathematics.  De Gruyter, Berlin, [2022] ©2022.