# IFS measures on generalized Bratteli diagrams

Sergey Bezuglyi and Palle E.T. Jorgensen

###### Abstract

The purpose of the paper is a general analysis of path space measures. Our
focus is a certain path space analysis on generalized Bratteli diagrams. We
use this in a systematic study of systems of self-similar measures (the term
“IFS measures” is used in the paper) for both types of such diagrams, discrete
and continuous. In special cases, such measures arise in the study of iterated
function systems (IFS). In the literature, similarity may be defined by, e.g.,
systems of affine maps (Sierpinski), or systems of conformal maps (Julia). We
study new classes of semi-branching function systems related to stationary
Bratteli diagrams. The latter plays a big role in our understanding of new
forms of harmonic analysis on fractals. The measures considered here arise in
classes of discrete-time, multi-level dynamical systems where similarity is
specified between levels. These structures are made precise by prescribed
systems of functions which in turn serve to define self-similarity, i.e., the
similarity of large scales, and small scales. For path space systems, in our
main result, we give a necessary and sufficient condition for the existence of
such generalized IFS measures. For the corresponding semi-branching function
systems, we further identify the measures which are also shift-invariant.

Dedicated to the memory of Robert Strichartz

Keywords— Bratteli diagrams, invariant path space measures, self-similar
measures, iterated and semibranching function systems, IFS measures, Perron-
Frobenius.

AMS Mathematics Subject Classification: 28A33, 28A80, 37B10, 46G12, 60J22

###### Contents

  1. 1 Introduction
  2. 2 Basics on generalized Bratteli diagrams
    1. 2.1 Main definitions
    2. 2.2 Measures on the path space of a Bratteli diagram
    3. 2.3 Semibranching function systems on Bratteli diagrams
    4. 2.4 Shift invariant measures on stationary Bratteli diagrams
  3. 3 IFS measures on discrete generalized Bratteli diagrams
    1. 3.1 Iterated function systems and measures
    2. 3.2 IFS measures on generalized stationary Bratteli diagrams
  4. 4 Measurable Bratteli diagrams and IFS measures
    1. 4.1 Measurable Bratteli diagrams and path space measures
    2. 4.2 IFS measures on measurable Bratteli diagrams

##  1 Introduction

We present new results in the study of self-similar measures on path space for
infinite graph models,here called generalized Bratteli diagrams. By this, we
mean particular graph systems $B$ with the property that the corresponding
sets $V$ of vertices, and $E$ of edges, admit discrete level-structures,
defined below. This means that $V$ is a disjoint union of the sets $V_{n}$ and
$E$ is a disjoint union of the sets $E_{n}$. We emphasize that the class of
generalized Bratteli diagrams include two cases: (1) all the sets
$V_{n},E_{n}$ are countable (if all $V_{n}$’s are finite, then we have a
standard Bratteli diagram); (2) all the set $V_{n},E_{n}$ are standard Borel
spaces. In case (2), we will say that $B$ is a measurable (or continuous)
Bratteli diagram.

Our present framework is motivated by, but more versatile than, the more
familiar setting in earlier studies of Bratteli diagrams. We use the discrete
levels in order to identify a class of self-similar path space measures,
called iterated function system (IFS) measures. The latter in turn are
inspired by earlier studies of IFS systems arising the analysis of fractals,
such as various Sierpinski constructs, or conformal attractors. However, by
contrast our analysis centers around a new path space analysis for discrete-
time random walk models in generalized Bratteli diagrams. We also discuss IFS
measures for the case when the system of levels making up $B$ are instead
standard measure spaces, so non-discrete.

We recall that a measure $\mu$ on a standard Borel space $X$ is called self-
similar with respect to an iterated function system $f_{i}:X\to X,i\in I,$ if

$\mu=\sum_{i\in I}p_{i}\mu\circ f_{i}^{-1},$

where $p=(p_{i})_{i\in I}$ is a probability vector and $I$ is finite or
countable.

A main aim of our paper is an identification of, and an analysis of, iterated
function system (IFS) measures on the path space $X_{B}$ of generalized
Bratteli diagrams. This entails two tasks, (a) an analysis of IFS structure of
the path spaces, and (b) a study of the particular IFS measures on them.

Our main results consist of finding an explicit construction of IFS measures
for both discrete and measurable generalized Bratteli diagrams.

In order to motivate, and to place this in context, we add the following three
comments here: (i) Previously, IFS measures have been considered in special
geometries of self-similar fractals (see e.g., [Jor18] and the papers cited
there). These standard self-similar fractals may be realized in finite-
dimensional Euclidean space. By contrast, there are no previous studies of IFS
measures in path space, i.e., IFS measures realized in the class of path space
structures considered here, standard and generalized Bratteli diagrams. (ii)
While there are earlier results on other, different but related, classes of
measures on path space of generalized Bratteli diagrams, e.g., tail invariant
measures, and Markov measures, our present identification of IFS measures on
path space is new. (iii) In order to prepare the reader for the IFS path space
measures, it will be necessary for us to begin with an account of tail
invariant measures, and Markov measures. The tools involved there for
generalized Bratteli diagrams are also needed in our introduction of the new
IFS measures. But this means our main results for IFS measures will be
postponed to sections 3 and 4 below, after the necessary preliminaries have
been presented.

Generalized Bratteli diagrams considered here arise in various areas and have
many applications. We mention Cantor and Borel dynamics where they are used to
construct models of transformations, see [HPS92], [GPS95], [Dur10], [BK16],
[BDK06]. Measurable Bratteli diagrams can be met in the theory of Markov
chains, we refer to [Num84] as an example of such literature. Several other
applications come to mind, (1) models from financial mathematics, and (2)
neural networks. In each case, the role of the IFS measures must be specified.
In some applications, there are important limit theorems, for example for
financial derivative models both discrete and continuous pricing formulas are
important. And it may be stated in the setting of generalized Bratteli
diagrams, measurable setting. We mean first of all financial derivative
models, e.g., binomial models; vs continuous/measurable (cm), e.g., pricing of
options via Ito calculus, see e.g., [BS13], [KS98]. In these applications, we
also have theorems to the effect that the continuous models are limits of
discrete counterparts. Typically, the limit arguments involve the Central
Limit Theorem from probability. Another case of discrete models take the form
of deep neural network models, deep means a “large number of levels”, so many
steps through the levels in our diagrams.

The organization of the paper is the following. Section 2 contains the basic
definitions related to the concept of generalized Bratteli diagram (discrete
case), and the description of various classes of measures on the path space of
a generalized Bratteli diagrams. We consider tail invariant, shift-invariant,
and Markov measures and their relations to a semibranching function system
generated by a stationary generalized Bratteli diagram. In section 3, we
consider a semibranching function system $\\{\tau_{e}\\}$ defined on the path
space of a stationary generalized Bratteli diagram and indexed by the edge set
$E$. We prove there one of the main results by giving necessary and sufficient
conditions on the existence of an IFS measure for $\\{\tau_{e}\\}$, see
Theorem 3.4. Section 4 focuses on measurable Bratteli diagrams. Since this
notion is relatively new, we give detailed definitions and discuss the
properties of such diagrams. Then we prove a measurable version of the main
theorem about the existence of an IFS measure, Theorem 4.7.

At the end of this introduction, we mention the literature that may be
interesting for the reader. The literature on standard Bratteli diagrams their
applications in dynamics is very extensive. We mention here the following
pioneering papers [HPS92], [GPS95], [DHS99], a recent book [Put18], and
surveys [Dur10], [BK16], [BK20]. Generalized Bratteli diagrams are less
studied. The stationary case uses the Perron-Frobenius theory for infinite
matrices. We refer to the book [Kit98] and the literature there. These
diagrams are discussed in [BJ22] and [BJS22]. More references and numerous
connections with other areas can be found therein. In particular, the
following papers on IFS measures and fractals are related to the current paper
[BJ99], [BJ02], [DJ09], [DJ10], [DJ14b], [Jor06], [Jor18], [RS16], [ARCG+20],
[CHlQ+21].

##  2 Basics on generalized Bratteli diagrams

We consider the fundamentals of path space for generalized Bratteli diagrams
in this section. This notion was first introduced in [BDK06] under the name of
Borel-Bratteli diagram. More detailed exposition of this concept can be found
in [BJ22]. For the reader’s convenience we give a concise version here.

###  2.1 Main definitions

In the introduction, we described the notion of a Bratteli diagram as an
infinite graded graph. A natural extension of this concept consists of
consideration of diagrams with countably infinite levels.

######  Definition 2.1.

(Generalized Bratteli diagrams, vertices, edges, incidence matrices) Let
$V_{0}$ be a countable set (which can be identified with either $\mathbb{N}$
or $\mathbb{Z}$ if necessary). Set $V_{i}=V_{0}$ for all $i\geq 1$, and
$V=\bigsqcup_{i=0}^{\infty}V_{i}$. A countable graded graph
$B=(\mathcal{V,E})$ is called a generalized Bratteli diagram if it has the
following properties.

(i) The set of edges $\mathcal{E}$ of $B$ is represented as
$\bigsqcup_{i=0}^{\infty}E_{i}$ where $E_{i}$ is the set of edges between the
vertices of levels $V_{i}$ and $V_{i+1},i\geq 0$.

(ii) The set $E(w,v)$ of edges $e$ between the vertices $w\in V_{i}$ and $v\in
V_{i+1}$ is finite (or empty). Let $f^{(i)}_{v,w}=|E(w,v)|$ where $|\cdot|$
denotes the cardinality of a set. It defines a sequence of infinite
(countable-by-countable) incidence matrices $(F_{i}:i\in\mathbb{N}_{0})$ with
entries $F_{i}=(f^{(i)}_{v,w}:v\in V_{i+1},w\in V_{i}),\ \
f^{(i)}_{v,w}\in\mathbb{N}_{0}.$

(iii) It is required that the matrices $F_{i}$ have at most finitely many non-
zero entries in each row. In general, we do not impose any restrictions on the
columns of $F_{i}$.

(iv) The maps $r,s:E\to V$ are defined on the diagram $B$: for every
$e\in\mathcal{E}$, there are $w,v$ such that $e\in E(w,v)$; then $s(e)=w$ and
$r(e)=v$. They are called the range ($r$) and source ($s$) maps.

(v) For every $w\in V_{i},\;i\geq 0$, there exist an edge $e\in E_{i}$ such
that $s(e)=w$; for every $v\in V_{i},\;i>1$, there exists an edge
$e^{\prime}\in E_{i-1}$ such that $r(e^{\prime})=w$. In other words, every row
and every column of the incidence matrix $F_{i}$ has non-zero entries.

(vi) If all entries of incidence matrices $F_{n}$ are zero or ones, the
corresponding generalized Bratteli diagram is called a 0-1 diagram.

######  Remark 2.2.

(1) It follows from Definition 2.1 that the structure of every generalized
Bratteli diagram is completely determined by a sequence of matrices $(F_{n})$
such that every matrix $F_{n}$ satisfies (iii) and (iv). Indeed, the entry
$f_{v,w}^{(n)}$ indicates the number of edges between the vertex $w\in V_{n}$
and vertex $v\in V_{n+1}$. It defines the set $E(w,v)$; then one takes

$E_{n}=\bigcup_{w\in V_{n},v\in V_{n+1}}E(w,v).$

In this case, we write $B=B(F_{n})$. If all $F_{n}=F$, the corresponding
generalized Bratteli diagram $B(F)$ is called stationary.

(2) If $V_{0}$ is a singleton, and each $V_{n}$ is a finite set, then we
obtain the standard definition of a Bratteli diagram originated in [Bra72].
Later it was used in the theory of $C^{*}$-algebras and dynamical systems for
solving important classification problems of Cantor dynamics and construction
of various models of homeomorphisms of a Cantor set (for references, see
Introduction).

######  Definition 2.3.

(Path space and cylinder sets) A finite or infinite path in a generalized
Bratteli diagram $B=(V,E)$ is a sequence of edges $(e_{i}:i\geq 0)$ such that
$r(e_{i})=s(e_{i+1})$ for all $i\geq 0$. Denote by $X_{B}$ the set of all
infinite paths. Every finite path $\overline{e}=(e_{0},\ldots,e_{n})$
determines a cylinder subset $[\overline{e}]$ of $X_{B}$:

$[\overline{e}]:=\\{x=(x_{i})\in X_{B}:x_{0}=e_{0},\ldots,x_{n}=e_{n}\\}.$

The collection of all cylinder subsets forms a base of neighborhoods for a
topology on $X_{B}$. In this topology, $X_{B}$ is a Polish zero-dimensional
space, and every cylinder set is clopen. In general, $X_{B}$ is not locally
compact. But if the set $s^{-1}(v)$ is finite for every vertex
$v\in\mathcal{V}$, then the path space $X_{B}$ is locally compact.

In the following remark, we formulate several statements about properties of
generalized Bratteli diagrams and their path spaces.

######  Remark 2.4.

(1) If $x=(x_{i})$ is a point in $X_{B}$, then it is obviously represented as
the intersection of clopen sets:

$\\{x\\}=\bigcap_{n\geq 0}[\overline{e}]_{n}$ (2.1)

where $[\overline{e}]_{n}=[x_{0},\ldots,x_{n}]$.

(2) Define a metric on $X_{B}$ compatible with the clopen topology: for
$x=(x_{i})$ and $y=(y_{i})$ from $X_{B}$, we set

$\mathrm{dist}(x,y)=\frac{1}{2^{N}},\ \ \
N=\min\\{i\in\mathbb{N}_{0}:x_{i}\neq y_{i}\\}.$

(3) We will assume that the diagram $B$ is chosen so that the space $X_{B}$
has no isolated points. This means that every column of the incidence matrix
$F_{n},n\in\mathbb{N}_{0},$ has more than one non-zero entry.

(4) Let $\mathbbm{1}=(...,1,1,....)$ be the vector indexed by $v\in V_{0}$
such that every entry equals 1. Define $H^{(n)}:=F_{n-1}\cdots
F_{0}\mathbbm{1}$. Let $E(V_{0},v),v\in V_{n},$ denote the set of all finite
paths between $V_{0}$ and a fixed vertex $v\in V_{n}$. Then
$H^{(n)}_{v}=|E(V_{0},v)|$.

(5) Let $X_{v}^{(n)}$ be a subset of $X_{B}$ such that

$X_{v}^{(n)}=\bigcup_{v_{0}\in V_{0}}\bigcup_{\overline{e}\in
E(v_{0},v)}[\overline{e}].$ (2.2)

For every level $V_{n}$, the collection $\\{X_{v}^{(n)}:v\in V_{n}\\}$ forms a
partition $\zeta_{n}$ of $X_{B}$ into disjoint clopen sets. Every set
$X_{v}^{(n)}$ is a finite union of cylinder sets. The number of the cylinder
sets here is exactly $H^{(n)}_{v}$. The sequence of partitions $(\zeta_{n})$
is refining. According to (2.2), the cylinder sets from all $X_{v}^{(n)}$
generate the topology (and Borel $\sigma$-algebra) on $X_{B}$.

For a generalized Bratteli diagram $B$, define the tail equivalence relation
$\mathcal{R}$ on the path space $X_{B}$.

######  Definition 2.5.

(Tail equivalence relation) It is said that two infinite paths $x=(x_{i})$ and
$y=(y_{i})$ are tail equivalent if there exists $m\in\mathbb{N}$ such that
$x_{i}=y_{i}$ for all $i\geq m$. Let $[x]_{\mathcal{R}}:=\\{y\in
X_{B}:(x,y)\in\mathcal{R}\\}$ be the set of points tail equivalent to $x$. We
say that a point $x$ is periodic if $|[x]_{\mathcal{R}}|<\infty$. If there is
no periodic points, then the tail equivalence relation is called aperiodic.

Without loss of generality, we will consider generalized Bratteli diagrams
with aperiodic $\mathcal{R}$. Clearly, $\mathcal{R}$ is a hyperfinite
countable Borel equivalence relation, see [DJK94] for definitions.

######  Definition 2.6.

It is said that a generalized Bratteli diagram $B$ is irreducible if, for any
two vertices $v$ and $w$ and any $n\in\mathbb{N}_{0}$, there exists a level
$V_{m}(m>n)$ such that $w\in V_{n}$ and $v\in V_{m}$ are connected by a finite
path. This is equivalent to the property that, for any fixed $v,w$, there
exists $m\in\mathbb{N}$ such that the product of matrices $F_{m-1}\cdots
F_{n}$ has a non-zero $(v,w)$-entry.

###  2.2 Measures on the path space of a Bratteli diagram

In this subsection, we will consider two classes of Borel measures on the path
space $X_{B}$ of a generalized Bratteli diagram. They are tail invariant
measures and Markov measures.

######  Definition 2.7.

(Tail equivalent measures) Let $B=(\mathcal{\mathcal{V,E}})$ be a generalized
Bratteli diagram, and $X_{B}$ the path space of $B$. A Borel measure $\mu$ on
$X_{B}$ (finite or $\sigma$-finite) is called tail invariant if, for any two
finite paths $\overline{e}$ and $\overline{e}^{\prime}$ such that
$r(\overline{e})=r(\overline{e}^{\prime})$, one has

$\mu([\overline{e}])=\mu([\overline{e}^{\prime}]),$ (2.3)

where $[e]$ and $[e^{\prime}]$ denote the corresponding cylinder sets.

If $\mu(X_{B})=1$, then the property of tail invariance means that the
probability to arrive at a vertex $v\in V_{n}$ does not depend on a starting
point $w\in V_{0}$ and does not depend on the path connecting $w$ and $v$.

Let $\mu$ be a Borel tail invariant measure on $X_{B}$. Relation (2.3) defines
a sequence of non-negative vectors $(\mu^{(n)})$ where
$\mu^{(n)}=(\mu^{(n)}_{v}:v\in V_{n})$:

$\mu^{(n)}_{v}=\mu([\overline{e}]),\ \ \ \overline{e}\in E(V_{0},v),\ v\in
V_{n}.$ (2.4)

Because $\mu$ is tail invariant the value $\mu^{(n)}_{v}$ does not depend on
the choice of $\overline{e}\in E(V_{0},v)$.

The following theorem is a key tool in the study of tail invariant measures,
see [BKMS10], [Dur10], [BK16], [BJ22].

######  Theorem 2.8.

Let $B=(\mathcal{V},\mathcal{E})$ be a generalized Bratteli diagram defined by
a sequence $(F_{n})$ of incidence matrices. Let $\mu$ be a Borel probability
measure on the path space $X_{B}$ of $B$ which is tail invariant. Then the
corresponding sequence of vectors $\mu^{(n)}$ (defined as in (2.4)) satisfies
the property

$A_{n}\mu^{(n+1)}=\mu^{(n)},$ (2.5)

where $A_{n}=F_{n}^{T}$ is the transpose of $F_{n}$.

Conversely, if a sequence of vectors $\mu^{(n)}$ satisfies (2.5), then it
defines a unique tail invariant measure $\mu$.

The theorem remains true for $\sigma$-finite measures $\nu$ satisfying the
property $\nu([\overline{e}])<\infty$ for every cylinder set $[\overline{e}]$.

The other interesting class of measures on Bratteli diagrams is Markov
measures. In the context of Bratteli diagrams, these measures were considered
in [DH03], [Ren18], [BJ22] and some other papers.

######  Definition 2.9.

(Markov measures) Let $B=(\mathcal{V,E})$ be a generalized Bratteli diagram
constructed by a sequence of incidence matrices $(F_{n})$. Let $q=(q_{v})$ be
a strictly positive vector, $q_{v}>0,v\in V_{0}$, and let $(P_{n})$ be a
sequence of non-negative infinite matrices with entries $(p^{(n)}_{v,e})$
where $v\in V_{n},e\in E_{n},n=0,1,2,\ldots$. To define a Markov measure $m$,
we require that the sequence $(P_{n})$ satisfies the following properties:

$(a)\ \ p^{(n)}_{v,e}>0\ \Longleftrightarrow\ (s(e)=v);\ \ \ \ (b)\ \
\sum_{e:s(e)=v}p^{(n)}_{v,e}=1.$ (2.6)

Condition (2.6)(a) shows that $p^{(n)}_{v,e}$ is positive only on the edges
outgoing from the vertex $v$, and therefore the matrices $P_{n}$ and
$A_{n}=F_{n}^{T}$ share the same set of zero entries. For any cylinder set
$[\overline{e}]=[(e_{0},e_{1},\ldots,e_{n})]$ generated by the path
$\overline{e}$ with $v=s(e_{0})\in V_{0}$, we set

$m([\overline{e}])=q_{s(e_{0})}p^{(0)}_{s(e_{0}),e_{0}}\cdots
p^{(n)}_{s(e_{n}),e_{n}}.$ (2.7)

Relation (2.7) defines the value of the measure $m$ of the set
$[\overline{e}]$. By (2.6)(b), this measure satisfies the Kolmogorov
consistency condition and can be extended to the $\sigma$-algebra of Borel
sets. To emphasize that $m$ is generated by a sequence of stochastic matrices,
we will also write $m=m(P_{n})$.

If all stochastic matrices $P_{n}$ are equal to a matrix $P$, then the
corresponding measure $m(P)$ is called stationary Markov measure.

We refer to [BJ22] for a detailed study of Markov measures. We mention here
only the following result.

######  Theorem 2.10.

Let $\nu$ be a tail invariant probability measure on the path space $X_{B}$ of
a generalized Bratteli diagram $B=(\mathcal{V,E})$. Then there exists a
sequence of Markov matrices $(P_{n})$ such that $\nu=m(P_{n})$.

For stationary generalized Bratteli diagrams, we can find explicit formulas
for tail invariant measures. In the following statement, we use the
terminology from the Perron-Frobenius theory, see [Kit98] for details.

######  Theorem 2.11.

[BJ22] (1) Let $B=B(F)$ be a stationary Bratteli diagram such that the
incidence matrix $F$ (and therefore $A=F^{T}$) is irreducible, aperiodic, and
recurrent. Let $t=(t_{v}:v\in V_{0})$ be a right eigenvector corresponding to
the Perron eigenvalue $\lambda$ for $A$, $At=\lambda t$. Then there exists a
tail invariant measure $\mu$ on the path space $X_{B}$ whose values on
cylinder sets are determined by the following rule: for every finite path
$\overline{e}(w,v)$ that begins at $w\in V_{0}$ and terminates at $v\in
V_{n}$, $n\in\mathbb{N}_{0}$, we set

$\mu^{(n)}_{v}=\mu([\overline{e}(w,v)])=\frac{t_{v}}{\lambda^{n}}.$ (2.8)

(2) The measure $\mu$ is finite if and only if the right eigenvector
$t=(t_{v})$ has the property $\sum_{v}t_{v}<\infty$.

In particular, $\mu(X_{w}^{(0)})=t_{w},w\in V_{0}$, and

$\mu(X_{v}^{(n)})=H_{v}^{(n)}\frac{t_{v}}{\lambda^{n}}.$ (2.9)

###  2.3 Semibranching function systems on Bratteli diagrams

Here we give the definition of a semibranching function system following
[MP11] and [BJ15]. This notion was used in the literature, in particular, for
the construction of representations of Cuntz-Krieger algebras, see [BJ15],
[FGJ+18a], [FGJ+18b].

######  Definition 2.12.

(Semibranching function systems and coding maps)

(1) Let $(X,\mu)$ be a probability measure space with non-atomic measure
$\mu$. We consider a family $\\{\sigma_{i}:i\in\Lambda\\}$ of one-to-one
$\mu$-measurable maps indexed by a finite (or countable) set $\Lambda$. The
family $\\{\sigma_{i}\\}$ is called a semibranching function system (s.f.s.)
if the following conditions hold:

(i) $\sigma_{i}$ is defined on a subset $D_{i}$ of $X$ and takes values in
$R_{i}=\sigma_{i}(D_{i})$ such that $\mu(R_{i}\cap R_{j})=0$ for $i\neq j$ and
$\mu(X\setminus\bigcup_{i\in\Lambda}R_{i})=0$;

(ii) the measure $\mu\circ\sigma_{i}$ is equivalent to $\mu$ and, i.e.,

$\rho{{}_{\mu}}(x,\sigma_{i}):=\frac{d\mu\circ\sigma_{i}}{d\mu}(x)>0\ \
\mbox{for $\mu$-a.e. $x\in D_{i}$};$

(iii) there exists an endomorphism $\sigma:X\to X$ (called a coding map) such
that $\sigma\circ\sigma_{i}(x)=x$ for $\mu$-a.e. $x\in D_{i},\ i\in\Lambda$.

If, additionally to properties (i) - (iii), we have
$\bigcup_{i\in\Lambda}D_{i}=X$ ($\mu$-a.e.), then the s.f.s.
$\\{\sigma_{i}:i\in\Lambda\\}$ is called saturated.

(2) It is said that a saturated s.f.s. satisfies condition C-K 111C-K stands
for Cuntz-Krieger. if for any $i\in\Lambda$ there exists a subset
$\Lambda_{i}\subset\Lambda$ such that up to a set of measure zero

$D_{i}=\bigcup_{j\in\Lambda_{i}}R_{j}.$

In this case, condition C-K defines a 0-1 matrix $\widetilde{A}$ by the rule:

$\widetilde{a}_{i,j}=1\ \ \Longleftrightarrow\ \ j\in\Lambda_{i},\ \
i\in\Lambda.$ (2.10)

Then the matrix $\widetilde{A}$ is of the size $|\Lambda|\times|\Lambda|$.

Semibranching function system associated with a generalized stationary
Bratteli diagram.

Let $B$ be a generalized stationary 0-1 Bratteli diagram. We construct an
s.f.s. $\Sigma$ which is defined on the path space $X_{B}$ endowed with a
Markov measure $m$. As we will see below, this measure must have some
additional properties to satisfy Definition 2.12. The role of the index set
$\Lambda$ for this s.f.s. is played by the edge set $E$ which is the set of
edges between any two consecutive levels of $B$. For any $e\in E$, we denote

$D_{e}=\\{y=(y_{i})\in X_{B}:s(y)=s(y_{0})=r(e)\\},$ (2.11)
$R_{e}=\\{y=(y_{i})\in X_{B}:y_{0}=e\\}.$ (2.12)

We see that $D_{e}$ depends on $r(e)$ only so that $D_{e}=D_{e^{\prime}}$ if
and only if $r(e)=r(e^{\prime})$.

The collection of maps $\\{\tau_{e}:e\in E\\},\tau_{e}:D_{e}\to R_{e}$, is
defined by the formula

$\tau_{e}(y):=(e,y_{0},y_{1},....),\quad y=(y_{i}).$ (2.13)

Since $s(y_{0})=r(e)$, the map $\tau_{e}$ is well defined on $D_{e}$.

######  Remark 2.13.

We defined the metric dist in Remark 2.4. It follows from the definition of
$\tau_{e}$ that

$\mbox{dist}(\tau_{e}(x),\tau_{e}(y))=\frac{1}{2}\mbox{dist}(x,y),\ e\in E,$

that is $\tau_{e}$ is a contractive map for every $e$.

######  Proposition 2.14.

The system $\\{D_{e},R_{e},\tau_{e}:e\in E\\}$ is a saturated s.f.s. on the
path space $X_{B}$ of a generalized stationary Bratteli diagram satisfying
conditions (i), (iii) of Definition 2.12 and the C-K condition.

###### Proof.

Let $\sigma:X_{B}\to X_{B}$ be defined as follows: for any $x=(x_{i})_{i\geq
0}\in X_{B}$,

$\sigma(x):=(x_{1},x_{2},...)$ (2.14)

It follows from (2.13) and (2.14) that the map $\sigma$ is onto and

$\sigma\circ\tau_{e}(x)=x,\ \ \ x\in D_{e};$

Hence $\sigma$ is a coding map.

We deduce from (2.12) that $\\{R_{e}:e\in E\\}$ constitutes a partition of
$X_{B}$ into clopen sets. Relation (2.11) implies that $\\{\tau_{e}:e\in E\\}$
is a saturated s.f.s. Moreover, we claim that it satisfies condition C-K.
Indeed,

$D_{e}=\bigcup_{f:s(f)=r(e)}R_{f},\ \ \ e\in E,$ (2.15)

because $y=(y_{i})\in D_{e}\ \Longleftrightarrow\ s(y_{0})=r(e)\
\Longleftrightarrow\ \exists f=y_{0}\ \mbox{such\ that}\ y=(f,y_{1},...)\
\Longleftrightarrow\ y\in\bigcup_{f:s(f)=r(e)}R_{f}$. Thus,
$\Lambda_{e}=\\{f:s(f)=r(e)\\}$, see (2.10).

Relation (2.15) shows that the non-zero entries of the 0-1 matrix
$\widetilde{A}$ from Definition 2.12 are defined by the rule:

$(\widetilde{a}_{e,f}=1)\ \Longleftrightarrow\ (s(f)=r(e)).$ (2.16)

We observe that the matrix $\widetilde{A}$ has the following property: there
are finitely many nonzero entries in every column of $\widetilde{A}$, but the
rows of $\widetilde{A}$ may contain infinitely many nonzero entries.

Next, we observe that $\sigma:X_{B}\to X_{B}$ is a finite-to-one continuous
map. Indeed,

$|\sigma^{-1}(x)|=|r^{-1}(r(x_{1}))|=\sum_{u\in V_{0}}f_{v,u}.$

The latter is finite by Definition 2.1. ∎

Thus, it remains to find out under what conditions property (ii) of Definition
2.12 holds. We consider here two classes: tail invariant measures and Markov
measures.

Let $m$ be a Borel probability measure on $X_{B}$. Since $X_{B}$ is naturally
partitioned into a refining sequence of clopen partitions $\mathcal{Q}_{n}$
formed by cylinder sets of length $n$, we can apply de Possel’s theorem (see,
for instance, [SG77]). We have that for $m$-a.a $x$,

$\rho_{m}(x,\tau_{e})=\lim_{n\to\infty}\frac{m(\tau_{e}([\overline{e}(n)])}{m[\overline{e}(n)]}$
(2.17)

where $\\{x\\}=\bigcap_{n}[\overline{e}(n)]$.

Tail invariant measure. We first consider the case when $m$ is the tail
invariant measure $\mu$ determined in Theorem 2.11. Let $At=\lambda t$ where
$t=(t_{v})$. If $\overline{f}=(f_{0},f_{1},...,f_{n})\in D_{e}$, then
$r(\overline{f})=r(f_{n})$ and $\tau_{e}(\overline{f})=(e,f_{0},...,f_{n})$.
By (2.8), we have

$\mu([\overline{f}])=\frac{t_{r(\overline{f})}}{\lambda^{n}},\ \ \
\mu(\tau_{e}([\overline{f}]))=\frac{t_{r(\overline{f})}}{\lambda^{n+1}},$

and therefore

$\rho_{\mu}(x,\tau_{e})=\lambda^{-1}.$ (2.18)

Stationary Markov measure. Let $m$ be a stationary Markov measure determined
by a stochastic matrix $P$, $m=m(P)$ as in (2.7). If
$\\{x\\}=\bigcap_{n}[f(n)]\in D_{e}$, then

$m([\overline{f}(n)])=q_{s(f_{0})}p_{s(f_{0}),f_{0}}\cdots
p_{s(f_{n}),f_{n}},$
$m(\tau_{e}([\overline{f}(n)]))=q_{s(e)}p_{s(e),e}p_{s(f_{0}),f_{0}}\cdots
p_{s(f_{n}),f_{n}},$

and the Radon-Nikodym derivative can be found by

$\rho_{m}(x,\tau_{e})=\frac{q_{s(e)}p_{s(e),e}}{q_{s(f_{0})}}.$ (2.19)

It follows from (2.18) and (2.19) that $\rho_{m}$ is positive on $D_{e}$ if
and only if all entries of the vector $q=(q_{v})$ are positive. The latter
means that the support of $m$ is the entire space $X_{B}$.

Non-stationary Markov measure. In this case, we need some additional
conditions to guarantee that the Radon-Nikodym derivative
$\rho_{m}(x,\tau_{e})$ is positive on $D_{e}$. As above, we represent
$x=(x_{i})\in D_{e}$ by means of the sequence
$[\overline{f}(n)]=[(f_{0},f_{1},...,f_{n})]$ such that
$x_{i}=f_{i},i=0,1,...,n$, and find

$m([f(n)])=q_{s(f_{0})}p^{(0)}_{s(f_{0}),f_{0}}\cdots
p^{(n)}_{s(f_{n}),f_{n}}$

and

$m(\tau_{e}([f(n)]))=q_{s(e)}p^{(0)}_{s(e),e}p^{(1)}_{s(f_{0}),f_{0}}\cdots
p^{(n+1)}_{s(f_{n}),f_{n}}.$

A direct computation gives the following result.

######  Lemma 2.15.

Let $m$ be a Markov measure on the path space of a generalized stationary 0-1
Bratteli diagram. Then $\rho_{m}(x,\tau_{e})>0$ on $D_{e}$ if and only if

$0<\prod_{i=1}^{\infty}\frac{p^{(i+1)}_{s(f_{i}),f_{i}}}{p^{(i)}_{s(f_{i}),f_{i}}}<\infty$
(2.20)

for any $x=\bigcap_{n}[\overline{f}(n)]\in D_{e}$

A Markov measure satisfying (2.20) is called a quasi-stationary measure.
Condition (2.20) appeared first in [DJ14a], [DJ15] in a different context.

We summarize the above results in the following theorem.

######  Theorem 2.16.

Given a generalized stationary 0-1 Bratteli diagram $B$ with the edge set $E$,
the collection of maps $\\{\tau_{e}:D_{e}\to R_{e}\\},\ e\in E$, defined in
(2.13) on the space $(X_{B},m)$, forms a saturated s.f.s. $\Sigma$ satisfying
C-K condition where the Markov measure $m$ is either the tail invariant
measure $\mu$, or a stationary Markov measure $m(P)$ of full support, or a
quasi-stationary measure Markov measure of full support.

######  Remark 2.17.

Let $B$ be a generalized stationary Bratteli diagram of bounded size, see
[BJKS] for the definition. In particular, if the incidence matrix $A$ is
banded, then $B$ is of bounded size. In this case, the definition of the
Cuntz-Krieger algebra $\mathcal{O}_{A}$ can be given similar to the case of
finite matrices. Then we can use the methods developed in [BJ15, Theorem 4.12]
to construct a representation of the Cuntz-Krieger algebra $\mathcal{O}_{A}$
generated by the s.f.s. defined in Theorem 2.16. We omit the details.

###  2.4 Shift invariant measures on stationary Bratteli diagrams

Let $B=(\mathcal{V,E})$ be a generalized stationary Bratteli diagram defined
by the incidence infinite matrix $F$ and $A=F^{T}$. In this subsection, we
discuss $\sigma$-invariant measures on $X_{B}$. We will consider two cases,
tail invariant measures and Markov measures.

Recall that, for every $v\in V_{1}$, the row sum $H^{(1)}_{v}=\sum_{w\in
V_{0}}f_{v,w}$ is finite, see Remark 2.4 for notation. In (2.14), we defined a
finite-to-one endomorphism $\sigma$ acting on the path space $X_{B}$. For
$x=(e_{0}.e_{1},...)$, we have

$\sigma^{-1}(x)=\\{y=(y_{i})\in X_{B}:r(y_{0})=s(e_{0}),y_{i}=e_{i-1},i\geq
1\\}$ (2.21)

and $|\sigma^{-1}(x)|=H^{(1)}_{s(e_{0})}$.

Tail invariant measures. There is a special case of a generalized stationary
Bratteli diagram such that the tail invariant measure is also shift-invariant.

######  Proposition 2.18.

Let the matrix $A$ has a Perron-Frobenius eigenpair $(\xi,\lambda)$. Let $\mu$
be the tail invariant measure on $X_{B}$ defined as in Theorem 2.11. Then
$\mu$ is $\sigma$-invariant if and only if $H^{(1)}_{v}=\lambda$ for all $v\in
V_{0}$.

###### Proof.

For a cylinder set $[\overline{e}]=[e_{0},e_{1},...,e_{n}]$, calculate
$m([\overline{e}])$ and $m(\sigma^{-1}([\overline{e}]))$. it follows from
(2.21) that

$\sigma^{-1}([\overline{e}])=\bigcup_{f:r(f)=s(e_{0})}[f,e_{0},...,e_{n}]=\bigcup_{f:r(f)=s(e_{0})}[f,\overline{e}]$
(2.22)

and this union is disjoint. Let $\lambda$ and $\xi$ be a Peron-Frobenius
eigenpair, $A\xi=\lambda\xi$. By Theorem 2.11, we have

$\mu([\overline{e}])=\frac{\xi_{v}}{\lambda^{n}},\qquad\mu([f,\overline{e}])=\frac{\xi_{v}}{\lambda^{n+1}},\
v=r(\overline{e})$

Then

$\mu(\sigma^{-1}[\overline{e}])=\sum_{f:r(f)=s(e_{0})}\mu([f,\overline{e}])=\sum_{f:r(f)=s(e_{0})}\frac{\xi_{v}}{\lambda^{n+1}}=|\sigma^{-1}([\overline{e}])|\frac{\xi_{v}}{\lambda^{n+1}}=H^{(1)}_{w}\frac{\xi_{v}}{\lambda^{n+1}}$

where $w=s(e_{0})\in V_{1}$. By de Possel’s theorem (2.17),

$\frac{d\mu\circ\sigma^{-1}}{d\mu}(x)=\lim_{n\to\infty}\frac{\mu\circ\sigma^{-1}(\overline{e}]_{n})}{\mu([\overline{e}]_{n})}=H^{(1)}_{w}\lambda^{-1}$

where $x=\bigcap_{n}[\overline{e}]_{n}$. Therefore,

$\mu\circ\sigma^{-1}=\mu\ \Longleftrightarrow\ H^{(1)}_{w}=\sum_{u\in
V_{0}}f_{w,u}=\lambda,\ \forall w\in V_{0}.$

∎

######  Remark 2.19.

If $\max\\{H^{(1)}_{v}:v\in V_{1}\\}=M<\infty$, then

$\lambda^{-1}\leq\frac{d\mu\circ\sigma^{-1}}{d\mu}(x)\leq M\lambda^{-1}.$

This means that $\mu$ is equivalent to a $\sigma$-invariant measure
$\mu^{\prime}$. In particular, this is true for bounded size diagrams.

Markov measures. Consider first a stationary Markov measure on the path space
of a generalized stationary Bratteli diagram, see Definition 2.9. For
simplicity we will work with a $0-1$ diagram; the general case is considered
similarly.

Let $q=(q_{v}:v\in V_{0})$ be a positive probability vector (called initial
distribution). Let $P$ be a Markov matrix. For a cylinder set
$[\overline{e}]=[e_{0},e_{1},...,e_{n}]$, we use (2.7) to determine the value
of the corresponding Markov measure $m(P)$.

######  Theorem 2.20.

For $B,q,P,m(P)$ as above, the measure $m=m(P)$ is $\sigma$-invariant if and
only if $qP=P$.

###### Proof.

Suppose that $qP=q$. We know that
$m([\overline{e}])=q_{s(e_{0})}p_{s(e_{0}),r(e_{0})}\cdots
p_{s(e_{n}),r(e_{n})}$ and

$\displaystyle m(\sigma^{-1}[\overline{e}])=$
$\displaystyle\sum_{f:r(f)=s(e_{0})}m([f,\overline{e}])$ (2.23)
$\displaystyle=$
$\displaystyle\sum_{f:r(f)=s(e_{0})}q_{s(f)}p_{s(f),r(f)}p_{s(e_{0}),r(e_{0})}\cdots
p_{s(e_{n}),r(e_{n})}$ $\displaystyle=$ $\displaystyle
q_{r(f)}p_{s(e_{0}),r(e_{0})}\cdots p_{s(e_{n}),r(e_{n})}$ $\displaystyle=$
$\displaystyle m([\overline{e}])$

because $q_{r(f)}=q_{s(e_{0})}$ and $q$ is $P$-invariant. Since $\overline{e}$
is arbitrary, we see that $m$ is $\sigma$-invariant.

The converse statement follows from (2.23): the equality
$m([\overline{e}])=m(\sigma^{-1}[\overline{e}])$ implies $qP=q$. ∎

It remains to consider a non-stationary Markov measure $m$ defined on the path
space of a generalized stationary Bratteli diagram $B$. In this case, the
measure $m$ is defined by a sequence of Markov matrices $(P_{n})$ and an
initial distribution $q=(q_{v})$:

$m([e_{0},...,e_{n}])=q_{s(e_{0})}p^{(0)}_{s(e_{0}),r(e_{0})}\cdots
p^{(n)}_{s(e_{n}),r(e_{n})}.$

######  Proposition 2.21.

Suppose that $qP_{0}=q$. Then the measures $m=m(P_{n})$ is $\sigma$-invariant
if and only if

$\prod_{i=1}^{\infty}\frac{p^{(i+1)}_{s(e_{i}),r(e_{i})}}{p^{(i)}_{s(i_{i}),r(e_{i})}}=1.$

###### Proof.

Using the same technique, we can show that, under the condition $qP=q$,

$\frac{d\mu\circ\sigma^{-1}}{d\mu}(x)=\prod_{i=1}^{\infty}\frac{p^{(i+1)}_{s(e_{i}),r(e_{i})}}{p^{(i)}_{s(i_{i}),r(e_{i})}}$

where $x=\bigcap_{n}[e_{0},...,e_{n}$. We omit the details. ∎

##  3 IFS measures on discrete generalized Bratteli diagrams

In this section, we consider the notion of iterated function system (IFS).
This concept have been discussed in many books and papers. We mention here
only several related references such as [Hut96], [HR00], [Jor06], [Bar06],
[Jor18], [DJ07], [DJ09], [JKS11], [MS21]. Our goal is to describe IFS measures
defined on the path space of a generalized Bratteli diagram.

###  3.1 Iterated function systems and measures

By an endomorphism $\sigma$ we mean a finite-to-one (or countable-to-one)
Borel map of a standard Borel space $(X,\mathcal{B})$ onto itself. In this
case, there exists a family of one-to-one maps $\\{\tau_{i}\\}_{i\in\Lambda}$
such that $\tau_{i}:X\to X$ and $\sigma\circ\tau_{i}=\mbox{id}_{X}$ where
$\Lambda$ is at most countable. The maps $\tau_{i}$ are called the _inverse
branches_ for $\sigma$. The collection of maps $(\tau_{i}:1\in\Lambda)$ gives
an example of iterated function system (IFS). In general, an IFS is defined by
a collection of maps $\\{\tau_{i}:i\in\Lambda\\}$ of a complete metric space
(or a compact space) such that $\tau_{i}$’s are continuous (or even
contractions). In this work, we focus on the following features of the family
$\\{\tau_{i}:i\in\Lambda\\}$: (i) each $\tau_{i}$ is a one-to-one map defined
on a subset $D_{i}$ of $X$, and (ii) there exists an endomorphism $\sigma$ of
$X$ such that $\sigma\circ\tau_{i}=\mathrm{id}_{X}$ for each $i$.

The general theory of infinite iterated function systems is more detailed and
requires additional assumptions (see, for example, the expository article
[Mau95]). As an example, one can consider the Gauss map $x\mapsto\\{1/x\\}$ of
the unit interval with the piecewise monotone inverse branches
$\tau_{i}:1/(i+x)$ defined on $(1/(i+1),1/i)$.

######  Definition 3.1.

(IFS measures) Suppose that $(\tau_{i}:i\in\Lambda)$ is a given IFS on a Borel
space $(X,\mathcal{B})$. Let $p=(p_{i}:i\in\Lambda)$ be a strictly positive
vector indexed by a countable set $\Lambda$. A measure $\mu_{p}$ on
$(X,\mathcal{B})$ is called an _IFS measure_ for $(\tau_{i})$ if

$\mu_{p}=\sum_{i\in\Lambda}p_{i}\;\mu_{p}\circ\tau_{i}^{-1},$ (3.1)

or, equivalently,

$\int_{X}f(x)\;d\mu_{p}(x)=\sum_{i\in\Lambda}p_{i}\int_{X}f(\tau_{i}(x))\;d\mu_{p}(x),\
\qquad f\in L^{1}(\mu_{p}).$

The main tool in the study of an IFS $(\tau_{i}:i\in\Lambda)$ on a space $X$
is a realization of the IFS as the full one-sided shift $S$ on a symbolic
product space $\Omega$. Then any $S$-invariant and ergodic product-measure
$\mathbb{P}$ on $\Omega$ can be pulled back to $X$. This construction gives
ergodic invariant measures for IFSs.

In the next subsection, we will find an explicit method for construction of an
IFS measure on the path space of a generalized stationary Bratteli diagram.

We discuss here a method that leads to construction of IFS measures. This
approach works perfectly for many specific applications under some additional
conditions on $X$ and maps $\tau_{i}$.

Suppose that $(X;\tau_{i},i\in\lambda)$ is a given IFS. Let $\Omega$ be the
infinite direct product

$\Omega=\prod_{i\in\mathbb{N}_{0}}\Lambda_{i},\qquad\Lambda_{i}=\Lambda.$

For $\omega=(\omega_{1},\omega_{2},...)\in\Omega$, let $\omega|_{n}$ denote
the finite word $(\omega_{0},...,\omega_{n})$. Then, $\omega|_{n}$ defines a
map $\tau_{\omega|_{n}}$ acting on $X$ by the formula:

$\tau_{\omega|_{n}}(x):=\tau_{\omega_{0}}\cdots\tau_{\omega_{n}}(x),\qquad
x\in X,\ \ \omega\in\Omega,\ n\in\mathbb{N}_{0}.$

It is said that $\Omega$ is an _encoding space_ if, for every
$\omega\in\Omega$,

$F(\omega)=\bigcap_{n\geq 1}\tau_{\omega|_{n}}(X)$ (3.2)

is a singleton. Relation (3.2) defines a Borel map $F:\Omega\to F(\Omega)$. If
each $\tau_{i}$ is a contraction and $X$ is a complete metric, then a coding
map $F:\Omega\to X$ always exists.

Let $S$ be the left shift on $\Omega$:

$S(\omega_{0},\omega_{1},...)=(\omega_{1},\omega_{2},...).$

The inverse branches $s_{i},i\in\Lambda,$ of $S$ are

$s_{i}(\omega_{0},\omega_{1},...)=(i,\omega_{0},\omega_{1},...).$

Clearly,

$s_{i}(\Omega)=C(i)=\\{\omega\in\Omega:\omega_{0}=i\\},$

and the space $\Omega$ is partitioned into the sets $C(i),\in\Lambda$.

Let $p=(p_{i}:i\in\Lambda)$ be a positive probability vector. It defines the
product measure $\mathbb{P}=p\times p\times\cdots$ on $\Omega$. We observe
that the maps $(s_{i}:i\in\Lambda)$ constitute an IFS on $\Omega$ such that
$\mathbb{P}$ is an IFS measure:

$\mathbb{P}=\sum_{i\in\Lambda}p_{i}\;\mathbb{P}\circ s_{i}^{-1}.$ (3.3)

The following result shows how IFS measures arise.

######  Proposition 3.2.

Suppose that $(X;\tau_{i},i\in\Lambda)$ is an IFS that admits a coding map
$F:\Omega\to X$. Let $p=(p_{i})$ be a probability vector generating the
product measure $\mathbb{P}=p\times p\times\cdots$. Then the measure
$\mu:=\mathbb{P}\circ F^{-1}$ is an IFS measure satisfying

$\mu=\sum_{i=1}^{N}p_{i}\mu\circ\tau_{i}^{-1}.$

Moreover, if $F$ is continuous, then $\mu$ has full support.

The proof can be found in [BJ18].

In Section 4, we will consider a analogue of the above construction for
measurable Bratteli diagrams.

###  3.2 IFS measures on generalized stationary Bratteli diagrams

In this subsection we discuss an explicit formula for an IFS measure on the
path space of a generalized stationary 0-1 Bratteli diagram
$B=(\mathcal{V},\mathcal{E})$.

In the following remark, we show that the requirement to be a 0-1 Bratteli
diagram is not restrictive.

######  Remark 3.3.

Every generalized stationary Bratteli diagram $B=(\mathcal{V},\mathcal{E})$
can be represented as a 0-1 Bratteli diagram. Indeed, we can use the matrix
$\widetilde{A}$ defined in (2.16) to built a new generalized Bratteli diagram
$\widetilde{B}=(\widetilde{V},\widetilde{E})$. Note that $\widetilde{A}$ is a
0-1 matrix such that every row of $\widetilde{F}=\widetilde{A}^{T}$ has
finitely many non-zero entries so that $\widetilde{F}$ can be viewed as an
incidence matrix of $\widetilde{B}$. In this case, the set of vertices
$\widetilde{V}$ for each level is coincides with $E$ and two vertices
$\widetilde{v}=f\in E$ and $\widetilde{w}=f\in E$ are connected by a singe
edge if and only if $r(e)=s(f)$. Moreover, the path spaces $X_{\widetilde{B}}$
coincides with $X_{B}$.

The above argument can be applied to any non-stationary generalized Bratteli
diagram.

We can now apply the results of Subsection 2.3 and construct an s.f.s.
$\Psi=(X_{B};\tau_{e},e\in E)$ where $\tau_{e}:D_{e}\to R_{e}$ is defined in
(2.13).

######  Theorem 3.4.

Let $B=(\mathcal{V},\mathcal{E})$ be a generalized stationary 0-1 Bratteli
diagram, $p=(p_{e}:e\in E)$ a probability vector, and $\Psi$ an s.f.s. defined
by $\tau_{e}:D_{e}\to R_{e}$, $e\in E$. We identify every edge $e\in E$ with
the pair of vertices $(s(e),r(e))$. Let $\nu$ be a Borel probability full
measure on $X_{B}$; we define a probability positive vector $q=(q_{v}:v\in
V_{0})$ by setting $q_{v}=\nu([w])$ where $[w]=\\{xinX_{B}:s(x)=w\\}$. Then

$\nu=\sum_{e\in E}p_{e}\;\nu\circ\tau_{e}^{-1}$ (3.4)

if and only if

$q_{w}=\sum_{w\in V_{0}}p_{w,v}q_{v}=\sum_{e:s(e)=w}p_{s(e),r(e)}q_{r(e)},$
(3.5)

that is $Pq=q$ where the matrix $P$ has the entries $(p_{s(e),r(e)}:e\in E)$.

###### Proof.

We use in the proof notation from (2.11) \- (2.15). We note that relation
(3.4) will be proved if we show that it holds for any cylinder set $C\subset
X_{B}$. Recall that it follows from the definition of $\tau_{e}$ that
$\tau_{e}^{-1}$ is uniquely determined on $R_{e}$ and
$\tau_{e}^{-1}(e,y_{1},y_{2},...)=(y_{1},y_{2},...)$.

Let $q=(q_{w})>0$ be a solution to $q=Pq$. Define $\nu([w])=q_{w}$, $w\in
V_{0}$. For any edge $f\in E$ and the corresponding cylinder set $[f]$, we set

$\nu([f])=q_{r(f)}p_{f}.$ (3.6)

Check that this definition of $\nu$ satisfies (3.4) for the cylinder sets
$[f]$. Indeed, since $\tau_{e}^{-1}$ is defined only on $R_{e}=[e]$ and
$\tau_{f}^{-1}([f])=[r(f)]$, we have

$\sum_{e\in
E}p_{e}\nu\circ\tau_{e}^{-1}([f])=p_{f}\nu\circ\tau_{f}^{-1}([f])=p_{f}\nu([r(f)])=q_{r(f)}p_{f}=\nu([f]).$

Then, by induction, we define the values of $\nu$ on all cylinder sets of
length $n$:

$\nu([f_{0},...,f_{n-1}]):=p_{f_{0}}\cdots p_{f_{n-1}}q_{r(f_{n-1})}.$

Verify that this definition satisfies the Kolmogorov extension theorem.
Because

$[f_{0},...,f_{n-1}]=\bigcup_{e:s(e)=r(f_{n-1})}[f_{0},...,f_{n-1},e],$

we find that

$\displaystyle\sum_{e:s(e)=r(f_{n-1})}\nu([f_{0},...,f_{n-1},e])=$
$\displaystyle\sum_{e:s(e)=r(f_{n-1})}p_{f_{0}}\cdots
p_{f_{n-1}}p_{e}q_{r(e)}$ $\displaystyle=$ $\displaystyle p_{f_{0}}\cdots
p_{f_{n-1}}\sum_{e:s(e)=r(f_{n-1})}p_{e}q_{r(e)}$ $\displaystyle=$
$\displaystyle p_{f_{0}}\cdots p_{f_{n-1}}q_{r(f_{n-1})}$ $\displaystyle=$
$\displaystyle\nu([f_{0},...,f_{n-1}])$

(we used here (3.5)).

It remains to show that (3.4) holds for any cylinder $[f_{0},...,f_{n}]$.
Indeed,

$\displaystyle\sum_{e\in E}p_{e}\nu\circ\tau_{e}^{-1}([f_{0},...,f_{n}])=$
$\displaystyle p_{f_{0}}\nu([f_{1},...,f_{n}])$ $\displaystyle=$
$\displaystyle p_{f_{0}}p_{f_{1}}\cdots p_{f_{n}}q_{r(f_{n})}$
$\displaystyle=$ $\displaystyle\nu([f_{0},...,f_{n}]).$

Remark that the condition $Pq=q$ is used to check that the measure $\nu$
defined inductively on cylinder sets can be extended to all Borel sets. ∎

######  Theorem 3.5.

Let $B$ be an irreducible generalized stationary Bratteli diagram, and let
$\nu$ be the IFS measure defined in Theorem 3.4. Then:

(i) $\nu$ is not tail invariant;

(ii) $\nu$ is shift-invariant.

###### Proof.

(i) Suppose that $\nu$ is tail invariant. Let $f,e\in E$ be two edges with
$r(f)=r(e)$, i.e., $f,e$ are tail-equivalent. Then it follows from (3.6) that
$\nu([f])=\nu([e])$ or

$q_{r(f)}p_{f}=q_{r(e)}p_{e}.$

This means that $p_{f}=p_{e}$. In other words, the matrix

$P=(p_{w,v}):e=(w,v)\in E)$

has constant columns.

Consider two cylinder sets $[f_{0},f_{1}]$ and $[e_{0},e_{1}]$ such that
$r(f_{1})=r(e_{1})$ and there exist edges $g,h$ satisfying the properties:
$s(g)=s(h)$, $r(g)=r_{f_{0}}$, and $r(h)=r_{e_{0}}$. From the tail invariance
of $\nu$ and the case considered above, we obtain that $p_{f_{1}}=p_{e_{1}}$
and

$q_{r(f_{1})}p_{f_{0}}p_{f_{1}}=q_{r(e_{1})}p_{e_{0}}p_{e_{1}}$

Hence, $p_{f_{0}}=p_{e_{0}}$ and therefore $p_{g}=p_{h}$. This proves that the
rows of the matrix $P$ are constant. But the vector $(p_{e}:e\in E)$ is
probability, contradiction.

(ii) Let $\varphi(x)$ be a bounded positive Borel function on the path space
$X_{B}$. For the left shift $\sigma$ on $X_{B}$, compute

$\displaystyle\int_{X_{B}}\varphi(x)\;d\nu\circ\sigma^{-1}(x)=$
$\displaystyle\int_{X_{B}}\varphi(\sigma x)\;d\nu(x)$ (3.7) $\displaystyle=$
$\displaystyle\sum_{e\in E}\int_{X_{B}}p_{e}\varphi(\sigma
x)\;d\nu\circ\tau_{e}^{-1}(x)$ $\displaystyle=$ $\displaystyle\sum_{e\in
E}\int_{X_{B}}p_{e}\varphi(\sigma(\tau_{e}x))\;d\nu(x)$ $\displaystyle=$
$\displaystyle\sum_{e\in E}\int_{X_{B}}p_{e}\varphi(x)\;d\nu(x)$
$\displaystyle=$ $\displaystyle\int_{X_{B}}\varphi(x)\;d\nu(x).$

This proves that $\nu\circ\sigma^{-1}=\nu$. ∎

##  4 Measurable Bratteli diagrams and IFS measures

In this section, we discuss a measurable analogue of generalized Bratteli
diagrams. The principal differences between discrete and measurable
generalized Bratteli diagrams are: (a) the levels $V_{n}$ of a measurable
Bratteli diagram are formed by standard Borel spaces
$(X_{n},\mathcal{A}_{n})$, and (b) the sets of edges $E_{n}$ are Borel subsets
of $X_{n}\times X_{n+1}$. Because these objects are non-discrete, we need to
use new methods and techniques.

###  4.1 Measurable Bratteli diagrams and path space measures

We give the definitions of main objects in this subsection.

######  Definition 4.1.

(Measurable Bratteli diagrams) Let
$\\{(X_{n},\mathcal{A}_{n}):n\in\mathbb{N}_{0}\\}$ be a sequence of standard
Borel spaces. Let $\\{E_{n}:n\in N_{0}\\}$ be a sequence of Borel subsets such
that $E_{n}\subset X_{n}\times X_{n+1}$. Denote by
$\mathcal{E}=\bigsqcup_{i\geq 0}E_{i}$ and $\mathcal{X}=\bigsqcup_{i\geq
0}X_{i}$ the sets of “edges” and “vertices”, respectively. For $e=(x,y)\in
E_{i}$, the maps $s_{i}(x,y)=x$ and $r_{i}(x,y)=y$ are onto projections of
$E_{i}$ to $X_{i}$ and $X_{i+1}$. Define $s,r$ on $\mathcal{E}$ by setting
$s=s_{i},r=r_{i}$ on $E_{i}$. Then we call
$\mathcal{B}=(\mathcal{X},\mathcal{E})$ a measurable Bratteli diagram. The
pair $(X_{n},E_{n})$ is called the $n$-th level of the measurable Bratteli
diagram $\mathcal{B}$.

If all $E_{n}=E$, then the measurable Bratteli diagram $\mathcal{B}$ is called
stationary.

######  Remark 4.2.

(1) We will identify the standard Borel spaces
$\\{(X_{n},\mathcal{A}_{n}):n\in\mathbb{N}_{0}\\}$ with an uncountable
standard Borel space $(X,\mathcal{A})$. This means that a paint $x$ can be
seen in all levels $(X_{n},\mathcal{A}_{n})$. This fact explains why we do not
require that all levels $X_{n}=X_{0}$ in the definition of a stationary
Bratteli diagram. Nevertheless, we will keep using subindeces to indicate the
level of a measurable Bratteli diagram.

(2) The set of edges $E_{i},i\geq 0,$ can be represented as follows:

$E_{i}=\bigcup_{x\in X_{i}}s^{-1}(x)=\bigcup_{x\in X_{i+1}}r^{-1}(x).$

This means that $E_{i}$ can be seen as the union of “vertical” and
“horizontal” sections. The fact that $r$ is an onto map says that $\forall
y\in X_{i+1}\ \exists x\in X_{i}\ \mbox{such\ that}\ (x,y)\in E_{i}$. A
similar property holds for the map $s$.

(3) To define the path space $\mathcal{X}_{\mathcal{B}}$ of a measurable
Bratteli diagram $\mathcal{B}=(\mathcal{X},\mathcal{E})$, we take a sequence
$\overline{x}=\\{e_{i}=(x_{i},x_{i+1})\\},e_{i}\in E_{i}$, such that
$r(e_{i})=s(e_{i+1})$ for all $i$. Equivalently,
$\overline{x}=(x_{0},x_{1},x_{2},...)$ where every pair $(x_{i},x_{i+1})$ is
in $E_{i}$, $i\in\mathbb{N}_{0}$. Then $s(\overline{x})=x_{0}$.

(4) We denote

$\mathcal{X}_{\mathcal{B}}(w)=\\{\overline{x}\in\mathcal{X}_{\mathcal{B}}:s(\overline{x})=w\\}.$
(4.1)

Clearly, the sets $\mathcal{X}_{\mathcal{B}}(w)$ form a partition $\eta$ of
$\mathcal{X}_{\mathcal{B}}$, and $X_{0}$ is the quotient space with respect to
this partition.

We will consider below  measurable stationary Bratteli diagrams, i.e.,
$E_{i}=E$. For every $e=(w,v)\in E$, we set

$D_{e}=\\{\overline{x}\in\mathcal{X}_{\mathcal{B}}:s(\overline{x})=r(e)=v\\},$
$R_{e}=\\{\overline{x}\in\mathcal{X}_{\mathcal{B}}:(x_{0},x_{1})=e\\}.$

For $\overline{x}=(x_{0},x_{1},...)\in D_{e}$, define the map
$\tau_{e}:D_{e}\to R_{e}$ by setting

$\tau_{e}(\overline{x})=(w,v,x_{1},x_{2},...).$

We note that $\overline{x}\in D_{e}$ means that $x_{0}=v$ so that
$\tau_{e}(\overline{x})$ is well defined and belongs to $R_{e}$. Moreover, the
map $\tau_{e}$ is one-to-one on its domain and the map $\tau_{e}^{-1}:R_{e}\to
D_{e}$ is defined by

$\tau_{e}^{-1}(e,\overline{y})=\overline{y}.$

We can consider the metric dist on the path space $\mathcal{X}_{\mathcal{B}}$
similarly to the case of generalized Bratteli diagrams. As shown in Remark
2.13, the maps $\tau_{e}$ are contractive for all $e\in E$.

Let $\mathcal{B}$ be a measurable Bratteli diagram with the path space
$\mathcal{X}_{\mathcal{B}}$. By a measure $\mu$ on $X_{\mathcal{B}}$, we mean
a Borel positive (finite or sigma-finite) measure.

Consider a sequence of Borel sets $C_{i}\subset X_{i}:i\in\mathbb{N}_{0}$.
Then $(C_{0}\times C_{1}\times\cdots\times
C_{N})\cap\mathcal{X}_{\mathcal{B}}$ is called a cylinder subset of
$X_{\mathcal{B}}$ of length $N$ and denoted by $[C_{0},...,C_{N}]$. In
particular, we can consider the set $R_{e}$ as a cylinder set $[e]$ defined by
the edge $e$, $e\in E$. It defines the partition

$\xi=\\{[e]:e\in E\\}$ (4.2)

of $\mathcal{X}_{\mathcal{B}}$. We remark that the quotient
$\mathcal{X}_{\mathcal{B}}/\xi$ is isomorphic to the set $E$. Since
$D_{e}=\mathcal{X}_{\mathcal{B}}(r(e))$ for every $e\in E$, the partition
$\eta$ (see Remark 4.2 (4)) is an enlargement of $\xi$ because

$D_{e}=\bigcup_{f:s(f)=r(e)}[f].$

Observation. The collection of cylinder sets $[C_{0},...,C_{N}]$ of any finite
length generates the Borel sigma-algebra of $X_{\mathcal{B}}$. A measure $\mu$
on $X_{\mathcal{B}}$ is completely determined by its values on cylinder sets.
These facts are obvious.

###  4.2 IFS measures on measurable Bratteli diagrams

Let $\mathcal{B}=(\mathcal{X},\mathcal{E})$ be a stationary measurable
Bratteli diagram. Recall that $E$ denotes a Borel subset of $X_{0}\times
X_{1}$ (in fact, $E$ can be viewed as a subset of $X_{0}\times X_{0}$).
Suppose that $p$ is a Borel measure on $E$, then $(E,p)$ becomes a standard
measure space. We consider both cases, finite and sigma-finite measures $p$.

The key tool of this subsection is the disintegration theorem for a finite or
sigma-finite measure with respect to a measurable partition of a measure
space. The literature on this subject is very extensive. We refer to the
original paper by Rokhlin [Roh49] and more recent paper [Sim12] (see also
[BJ18]).

######  Definition 4.3.

Suppose that $(Z,\mathcal{C},\mu)$ and $(Y,\mathcal{D},\nu)$ are standard
$\sigma$-finite measure spaces. Let $\pi:Z\to Y$ be a measurable function. A
system of conditional measures for $\mu$ with respect to $\pi$ is a collection
of measures $\\{\mu_{y}:y\in Y\\}$ such that
(i) $\mu_{y}$ is a Borel measure on $\pi^{-1}(y)$,
(ii) for every $B\in\mathcal{C}$, the function $y\mapsto\mu_{y}(B)$ is
measurable and $\mu(B)=\int_{Y}\mu_{y}(B)\;d\nu(y)$.

######  Theorem 4.4 ([Sim12]).

Let $(Z,\mathcal{C},\mu)$ and $(Y,\mathcal{D},\nu)$ be as above. For any
measurable function $\pi:Z\to Y$ such that $\mu\circ\pi^{-1}\ll\nu$ there
exists a uniquely determined system of conditional measures $(\mu_{y})_{y\in
Y}$ which disintegrates the measure $\mu$, i.e.

$\mu=\int_{Y}\mu_{y}d\nu(y).$

######  Remark 4.5.

For the edge set $E$ of a stationary measurable Bratteli diagram $\mathcal{B}$
and a Borel measure $p$ on $E$, consider the partition of $E$ into the sets
$r(s^{-1}(x)),x\in X_{0}$. This partition is measurable in the sense of
[Roh49]. We can apply the disintegration theorem cited above. More precisely,
the sets $r^{-1}(x)=\\{(x,y)\in E:y\in X_{1}\\}$, where $x\in X_{0}$, are
vertical sections of the set $E$. Setting
$\widehat{p}(\cdot)=p(s^{-1}(\cdot))$, we have the projection of the measure
$p$ onto $X_{0}$. Denoting by $x\mapsto p(x,\cdot)$ the corresponding system
of conditional measures, we have

$p=\int_{X_{0}}p(x,dy)\;d\widehat{p}(x)$

where the measure $p(x,dy)$ is supported by the set $r(s^{-1}(x))$.

We will consider several versions of disintegration theorem related to
measures on the path space $\mathcal{X}_{\mathcal{B}}$ of a measurable
Bratteli diagram $\mathcal{B}$. We assume that the measures will satisfy
conditions (i) and (ii) formulated below.

(i) For a given measure $m$ on $\mathcal{X}_{\mathcal{B}}$ and the set
$\mathcal{X}_{\mathcal{B}}(y)$ defined in (4.1), the measurable function

$q_{m}(y)=m(\mathcal{X}_{\mathcal{B}}(y)),\ y\in X_{0},$

takes finite values.

(ii) The partition $\xi$ defined in (4.2) is obviously measurable, and the
measure $m$ on $\mathcal{X}_{\mathcal{B}}$ can be disintegrated with respect
to $\xi$. Denote by $\widehat{m}$ the projection of $m$ onto $E=X/\xi$. Let
$p$ be a measure on $E$. Assuming that $\widehat{m}\ll p$ and applying Theorem
4.4, we have

$m=\int_{E}m_{e}\;dp(e),$ (4.3)

where $e\mapsto m_{e}$ is the system of conditional measures of $m$ with
respect to $(E,p)$. We note that the conditional measure $m_{e}$ is supported
by the set $[e]$, $e\in E$.

The class of IFS measures is determined by a special form of measures $m_{e}$
defined in (4.3). We will assume that the measures considered on the path
space $\mathcal{X}_{\mathcal{B}}$ satisfy conditions (i) and (ii).

######  Definition 4.6.

(IFS measures and disintegration) Let $\mathcal{B}$ be a stationary measurable
Bratteli diagram, and let $\\{\tau_{e},e\in E\\}$ be the system of contractive
maps defined in subsection 4.1. Suppose $p$ is a fixed probability measure on
the set $E$. A Borel measure $\mu$ is called an IFS measure with respect to
$p$ if

$\mu=\int_{E}\mu\circ\tau_{e}^{-1}\;dp(e).$ (4.4)

Our goal is to find conditions under which a measure $m$ defined on the path
space of a stationary measurable Bratteli diagram is an IFS measure.

######  Theorem 4.7.

Let $p$ be a Borel probability measure on $E$ where $E$ is the edge set of a
stationary measurable Bratteli diagram
$\mathcal{B}=(\mathcal{X},\mathcal{E})$. Let $\mu$ be a measure on
$\mathcal{X}_{\mathcal{B}}$ and $q(x)=\mu(\mathcal{X}_{\mathcal{B}}(x))$. Then
$\mu$ is and IFS measure if and only if the following condition holds:

$\int_{X_{1}}p(x,dy)q(y)=q(x).$ (4.5)

###### Proof.

In the proof of the theorem, we apply the idea used in Theorem 3.4.

We will construct an IFS measure $\mu$ by defining its values on cylinder sets
of consequently increasing length. This construction will be based on the
application of (4.4) so that the measure $\mu$ will be an IFS measure
automatically. Relation (4.5) is used to satisfy the Kolmogorov extension
theorem.

As mentioned in Remark 4.5, the projection of $p$ onto $X_{0}$ defines the
measure $\widehat{p}$. This allows us to define $\mu$ on cylinder subsets of
$\mathcal{X}_{\mathcal{B}}$ generated by Borel sets on $X_{0}$:

$\mu([C_{0}])=\int_{C_{0}}q(x_{0})\;d\widehat{p}(x_{0}),$

where $C_{0}$ is a Borel subset of $X_{0}$.

For a cylinder set
$[C_{0},C_{1}]=\\{x=(x_{i})\in\mathcal{X}_{\mathcal{B}}:x_{0}\in
C_{0},x_{1}\in C_{1}\\}$ of length two, we define

$\mu([C_{0},C_{1}])=\int_{C_{0}}\left(\int_{C_{1}\cap
r(s^{-1}(x_{0}))}p(x_{0},dy)q(y)\right)\;d\widehat{p}(x_{0}).$ (4.6)

Show that this definition of the measure $\mu$ satisfies (4.4):

$\displaystyle\int_{E}\mu\circ\tau_{e}^{-1}([C_{0},C_{1}])\;dp(e)=$
$\displaystyle\int_{E}\mu\circ\tau_{e}^{-1}([C_{0},C_{1}]\cap[e])\;dp(e)$
$\displaystyle=$ $\displaystyle\int_{C_{0}}\int_{C_{1}\cap
r(s^{-1}(x_{0}))}\mu(\mathcal{X}_{\mathcal{B}}(y)\;p(x_{0},dy)\;d\widehat{p}(x_{0})$
$\displaystyle=$ $\displaystyle\int_{C_{0}}\int_{C_{1}\cap
r(s^{-1}(x_{0}))}q(y)\;p(x_{0},dy)\;d\widehat{p}(x_{0})$ $\displaystyle=$
$\displaystyle\mu([C_{0},C_{1}])$

In general, if we defined the measure $\mu$ on cylinder sets of the form
$[C_{0},...,C_{k}],k=1,...,n-1,$, then we set

$\mu([C_{0},...,C_{n}])=\int_{E}\mu\circ\tau_{e}^{-1}([C_{0},...,C_{n-1}])\;dp(e).$

This means that this definition of $\mu$ shows that relation (4.4) holds
automatically.

It remains to show that the measure $\mu$ is well defined, that is it
satisfies the Kolmogorov extension theorem. We check this property for the
cylinder sets of length two. Indeed, in this case we take $C_{1}=X_{1}$ and
compute

$\displaystyle\mu([C_{0},X_{1}])=$
$\displaystyle\int_{C_{0}}\left(\int_{X_{1}\cap
r(s^{-1}(x_{0}))}q(y)p(x_{0},dy)\right)d\widehat{p}(y)$ $\displaystyle=$
$\displaystyle\int_{C_{0}}\left(\int_{r(s^{-1}(x_{0}))}q(y)p(x_{0},dy)\right)d\widehat{p}(y)$
$\displaystyle=$ $\displaystyle\int_{C_{0}}q(x_{0})\;d\widehat{p}(x_{0})$
$\displaystyle=$ $\displaystyle\mu([C_{0}]).$

The general case is proved analogously. ∎

######  Remark 4.8.

We note that the existence of an IFS measure on $\mathcal{X}_{\mathcal{B}}$
can be proved by using a fixed point theorem following [Hut81]. We consider
the metric $\mathrm{dist}$ on $\mathcal{X}_{\mathcal{B}}$ defined in Remark
2.4. Let

$\mbox{Lip}_{1}=\\{f:\mathcal{X}_{\mathcal{B}}\to\mathbb{R}:|f(x)-f(y)\leq\
\mbox{dist}(x,y)\\}.$

For a measure $\nu$ on $\mathcal{X}_{\mathcal{B}}$, set

$L(\nu)=\int_{E}\nu\circ\tau_{e}^{-1}\;dp(e).$

Since $\tau_{e}$ is a contractive map for every $e$, one can show that
$\rho(L(\nu),L(\mu))\leq\rho(\nu,\mu)$ where the metric $\rho$ is defined by

$\rho(\nu,\mu)=\sup\Bigl{\\{}\big{|}\int f\;d\nu-\int
f\;d\mu\;\big{|}:f\in\mathrm{Lip}_{1}\Bigr{\\}}.$

Then we conclude that there exists a measure $\mu_{0}$ such that
$L(\mu_{0})=\mu_{0}$.

######  Remark 4.9.

The IFS measure $\mu$ defined in Theorem 4.7 is shift invariant. The proof of
this fact is similar to that giving in Theorem 3.5.

Acknowledgements. The authors are pleased to thank our colleagues and
collaborators, especially, R. Curto, H. Karpel, P. Muhly, W. Polyzou, S.
Sanadhya. We are thankful to the members of the seminars in Mathematical
Physics and Operator Theory at the University of Iowa for many helpful
conversations.

Declaration. The authors declare that they have no conflict of interest.

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